Rock Mechanics and Rock Engineering, 2-Volume Set 0367421658, 9780367421656

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Rock Mechanics and Rock Engineering

Rock Mechanics and Rock Engineering

Volume 1: Fundamentals of Rock Mechanics

Ömer Aydan Department of Civil Engineering, University of the Ryukyus, Nishihara, Okinawa, Japan

CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2020 Taylor & Francis Group, London, UK Typeset by Apex CoVantage, LLC All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Library of Congress Cataloging-in-Publication Data Applied for Published by: CRC Press/Balkema Schipholweg 107C, 2316 XC Leiden, The Netherlands e-mail: [email protected] www.crcpress.com – www.taylorandfrancis.com Volume 1 ISBN: 978-0-367-42162-5 (Hbk) ISBN: 978-0-367-82229-3 (eBook) DOI: https://doi.org/10.1201/9780367822293 Volume 2 ISBN: 978-0-367-42165-6 (Hbk) ISBN: 978-0-367-82230-9 (eBook) DOI: https://doi.org/10.1201/9780367822309 Two-volume set ISBN: 978-0-367-02935-7 (Hbk) ISBN: 978-0-429-00123-9 (eBook) DOI: https://doi.org/10.1201/9780429001239

Contents

Preface Author biography 1

Introduction and history of rock mechanics and rock engineering 1.1 Earlier traces of rock mechanics and rock engineering 1.2 Modern development of rock mechanics and rock engineering 1.3 Goals and content of this book

ix xi

1 1 3 3

2

Minerals, rocks, discontinuities and rock mass 2.1 Minerals 2.2 Rocks 2.3 Discontinuities 2.4 Rock mass

7 7 11 18 21

3

Fundamental definitions and measurement techniques 3.1 Physical parameters of rocks 3.2 Physical parameters of discontinuities 3.3 Rock mass

25 25 26 38

4

Fundamental governing equations 4.1 Fundamental governing equations for one-dimensional case 4.2 Multidimensional governing equations 4.3 Derivation of governing equations in integral form

43 43 51 54

5

Constitutive laws 5.1 One-dimensional constitutive laws 5.2 Multidimensional constitutive laws 5.3 Nonlinear behavior (elasto-plasticity and elasto-visco-plasticity) for solids

61 61 75 77

vi

Contents

6

Laboratory and in-situ tests 6.1 Laboratory tests on mechanical properties 6.2 In-situ mechanical tests 6.3 Thermal properties of rocks and their measurements 6.4 Tests for seepage parameters

7

In-situ stress estimation, measurement and inference methods 7.1 In-situ stress estimation methods 7.2 In-situ stress measurement methods 7.3 In-situ stress inference methods 7.4 Comparisons 7.5 Integration of various direct measurement and indirect techniques for in-situ stress estimation 7.6 Crustal stress changes

8

9

vi

99 99 121 125 132

147 147 157 162 169 179 180

Analytical methods 8.1 Basic approaches 8.2 Analytical solutions for solids 8.3 Analytical solutions for fluid flow through porous rocks 8.4 Analytical solutions for heat flow: temperature distribution in the vicinity of geological active faults 8.5 Analytical solutions for diffusion problems 8.6 Evaluation of creep-like deformation of semi-infinite soft rock layer

187 187 188 228

Numerical methods 9.1 Introduction 9.2 1-D hyperbolic problem: equation of motion 9.3 Parabolic problems: heat flow, seepage and diffusion 9.4 Finite element method for 1-D pseudo-coupled parabolic problems: heat flow and thermal stress; swelling and swelling pressure 9.5 Hydromechanical coupling: seepage and effective stress problem 9.6 Biot problem: coupled dynamic response of porous media 9.7 Introduction of boundary conditions in simultaneous equation system 9.8 Rayleigh damping and its implementation 9.9 Nonlinear problems 9.10 Special numerical procedures for rock mass having discontinuities

257 257 262 268

10 Ice mechanics and glacial flow 10.1 Physics of ice 10.2 Mechanical properties of ice 10.3 Glaciers and ice domes/sheets 10.4 Cliff and slope failures induced by glacial flow

241 245 251

272 279 286 295 297 298 298 303 303 305 305 307

vii

11

Contents

vii

10.5 Glacial cave failures 10.6 Moraine lakes and lake burst 10.7 Calving and iceberg formation

312 312 313

Extraterrestrial rock mechanics and rock engineering 11.1 Solar system 11.2 Moon 11.3 Mars 11.4 Venus 11.5 Issues of rock mechanics and rock engineering on the Moon, Mars and Venus 11.6 Conclusions and future studies

315 316 317 322 329

Appendices Appendix 1: Definitions of scalars, vectors and tensors and associated operations Appendix 2: Stress analysis Appendix 3: Deformation and strain Appendix 4: Gauss divergence theorem Appendix 5: Geometrical interpretation of Taylor expansion Appendix 6: Reynolds transport theorem Index

343

333 341

345 355 363 371 373 375 377

Preface

Rock is the main constituent of the crust of the Earth, and its behavior is the most complex one among all materials in the geosphere to be dealt with by humankind. Furthermore, it contains various discontinuities, which make the thermo-hydro-mechanical behavior of rocks more complex. These simply require a higher level of knowledge and intelligence within the Rock Mechanics and Rock Engineering (RMRE) community. Furthermore, the applications of the principles of rock mechanics to mining, civil, and petroleum engineering fields, as well as to earthquake science and engineering, are diverse, and it constitutes rock engineering. Recently, the International Society for Rock Mechanics (ISRM) added “Rock Engineering” in 2017 to its name while its acronym remains “ISRM.” Rock mechanics is concerned with the theoretical and applied science of the mechanical behavior of rock and rock masses, and it is one of branches of mechanics concerned with the response of rock and rock masses to their physical-chemical environment. Rock engineering is concerned with the application of the principles of mechanics to physical, chemical, and electromagnetic processes in the uppermost part of the Earth and the design of the rock structures associated with mining, civil, and petroleum engineering. This book is intended to be a fundamental text for younger generations and newcomers, as well as a reference source for experts specialized in rock mechanics and rock engineering. Due to the wide spectra of rock mechanics and rock engineering, the book is divided into two volumes: Rock Mechanics and Rock Engineering: Fundamentals of Rock Mechanics and Rock Mechanics and Rock Engineering: Applications of Rock Mechanics – Rock Engineering. In the first volume, the fundamental concepts, theories, analytical and numerical techniques and procedures of rock mechanics and rock engineering, together with some emphasis on new topics, are described as concisely as possible while keeping the mathematics simple. The second volume is concerned with the applications of rock mechanics and rock engineering in practice. It ranges from classic rock classifications, the response and stability of surface and underground structures, to model testing, monitoring, excavation techniques, and rock dynamics. Particularly, earthquake science and engineering, vibrations and nondestructive techniques are presented as a part of rock dynamics. Although the overall subject of Rock Mechanics and Rock Engineering is presented over two volumes, each volume is complete in its content and should serve the purposes of educators, students, experts, as well as practicing engineers. It is strongly hoped that these two volumes will fulfill the expectations and serve further advances in rock mechanics and rock engineering.

Author biography

Ömer Aydan, born in 1955, studied Mining Engineering at the Technical University of Istanbul, Turkey (BSc, 1979), studied Rock Mechanics and Excavation Engineering at the University of Newcastle upon Tyne, UK (MSc, 1982), and received his PhD in Geotechnical Engineering from Nagoya University, Japan, in 1989. Professor Aydan has worked at Nagoya University as a research associate (1987–1991) and then at the Department of Marine Civil Engineering at Tokai University, first as Assistant Professor (1991–1993), then as Associate Professor (1993–2001), and finally as Professor (2001–2010). He then became Professor of the Institute of Oceanic Research and Development at Tokai University and is currently Professor at the University of Ryukyus, Department of Civil Engineering & Architecture, Nishihara, Okinawa, Japan. He is also the director of the Disaster Prevention Research Center for Island Region of the University of the Ryukyus. Ömer Aydan has played an active role on numerous ISRM, JSCE, JGS, SRI, and Rock Mech. National Group of Japan committees and has organized several national and international symposia and conferences.

Chapter 1

Introduction and history of rock mechanics and rock engineering

1.1

Early traces of rock mechanics and rock engineering

The early traces of rock mechanics and rock engineering may be associated with archeological remains left by ancient peoples, such as Hattis, Sumerians, Egyptians, Hittites, Persians, Romans, and Native Americans. The quarries, open-pit mines, castles, underground quarries, semiunderground or underground cities in Anatolia (Anadolu) and underground tombs of Egyptians near Luxor, as well as pyramids, are all well preserved examples of rock engineering structures of the past, even though they did not have the excavation tools of modern times (Aydan, 2008, 2014). In Anadolu (Anatolia), there are traces of open-pit mining dated to 9000 years ago and of underground mines dated to 5000 years ago (Kaptan, 1992; Yener, 1997). Humankind has constructed underground or semiunderground openings in soft rocks in the past. However, one can also found such structures excavated in limestone in the form of irrigation tunnels. Hard stones (i.e. flints, diorite, obsidian) were used initially, and later, workers started to use metallic tools after gaining the knowledge of extracting metals from ores. Given the 9000-year-old archeological mining and metallurgic traces found in Anatolia, it is likely that the use of metallic tools could be as old as 9000 years (Hatti era). In view of recent findings in Göbekli Tepe in Şanlıurfa, the quarrying of limestone in the region extends at least 11000 BP (Fig. 1.1). It is estimated that Harran City was established at least 5000 years ago, during the period of the Sumerians, who came from Central Asia to Mesopotamia about 7000 years ago and governed the area until

Figure 1.1 Monuments in Central Asia and Anatolia

2 Introduction and history

2

BCE 2270. Sumerians were the pioneers in all aspects of the modern sciences, engineering, technology, culture and religion of humankind, including cuneiform script (Kramer, 1956). One can find also some earlier underground quarries in Anatolia and Thebes (Kulaksiz and Aydan, 2010; Kumsar et al., 2003; Aydan and Ulusay, 2003, 2013; Aydan et al., 2008a, 2008b; Aydan and Geniş, 2004; Hamada et al., 2014; Kumsar and Aydan, 2008; Tokashiki et al., 2008). At the Amenophis III Quarry at Qurna of the Thebes region of Egypt, marble mining started probably 3350–3500 years ago. The Bazda Quarry at Harran, Urfa region of Turkey, probably opened 4000 years ago by the Sumerians. The Bazda underground marble mine quarry is the oldest known underground quarry mine in Turkey (Fig. 1.2). Pyramids made of huge rock blocks to achieve structural stability for thousands of years under both static and dynamic loading conditions, particularly those in Egypt, are wellknown worldwide. However, some pyramids have been recently unearthed in Peru, Mexico, Bosnia and present China. Pyramids near Xianyang in present China were constructed by Proto-Turks (Proto-Uygurs) about 3000 BCE, which makes them the oldest pyramids in the world and confirms the hypothesis that pyramids in Egypt were built by people who migrated from Central Asia due to climate change and dried inland seas such as Taklamakan (in Uygur Turkish, is Döklemegen means “the point of no return”) and Gobi Desert. Besides the good mechanical interlocking of rock blocks, there are caverns within these pyramids. The roofs of these caverns consist of beams of hard rock (mainly granite) with blocks in the sidewalls put together to form inverted V-shaped or trapeze-shaped arches (like Sumerian arches). Of course, the beams were dimensioned in a way that they can resist tensile stresses induced by bending due to surcharge loads for thousands of years. Friction law, strength of rocks in tension and compressions, was undoubtedly known to ancestral civilizations (i.e. Sumerians, Turanians, Anatolians, Egyptians, Indians, Chinese, Peruvians, Maya, Aztecs, Persians and Roman) and measured by them precisely. The very advanced measurement systems developed by Sumerians have very likely direct connections to modern measurement systems. It is simply our disregard and ignorance of their knowledge and level of their advancement that lead us to quote, for example, Guillaume Amonton and Leonardo da Vinci as the pioneers of modern testing and measurements

Figure 1.2 Remnants of open-pit and underground quarries during the early stages of excavations: (a) pathway leading to the quarries, (b) initial underground quarries

3

Introduction and history

3

techniques. There is a need to initiate a working group from various countries on the history of testing and measurement techniques relevant to rock mechanics and rock engineering and to recognize the actual pioneers with due respect.

1.2

Modern development of rock mechanics and rock engineering

The principles of modern rock mechanics and rock engineering are associated with Talobre (1957), Terzaghi (1946) and Stini (1950), both from Austria (Müller, 1963). Their works were followed by Müller of Karlsruhe University in Germany, Talobre of Électricité de France in France, and Rocha of the Portuguese National Laboratory for Civil Engineering (LNEC), which is still home to the ISRM office. The book published by Jaeger and Cook (1979) was the first theoretical publication in rock mechanics. The books related to rock engineering by E. Hoek and his colleagues (Hoek and Bray, 1977; Hoek and Brown, 1980) at the Imperial College were other milestones in the advancement of rock mechanics and rock engineering. The International Society for Rock Mechanics (ISRM) has defined rock mechanics as the theoretical and applied science of the mechanical behavior of rock and rock masses; it is that branch of mechanics concerned with the response of rock and rock masses to the force fields of their physical environment. Rock mechanics itself forms part of the broader subject of geomechanics, which is concerned with the mechanical responses of all geological materials, including soils. Rock mechanics is concerned with the application of the principles of engineering mechanics to the design of the rock structures generated by mining, civil and petroleum activity, e.g. tunnels, mining shafts, underground excavations, open pit mines. While the acronym remains the same, the ISRM has changed its name to the International Society for Rock Mechanics and Rock Engineering. The International Society for Rock Mechanics (ISRM) was founded in Salzburg in 1962 as a result of the enlargement of the so-called Salzburger Kreis. Its foundation is mainly owed to Professor Leopold Müller who acted as president of the Society till September 1966. When one looks at the content of the proceedings of the first Congress, the spectrum of rock mechanics and rock engineering (RMRE) is very wide compared to that these days. In other words, the greater emphasis given to the applications in civil and mining engineering and the relation of rock mechanics with earth science or geoscience is almost nonexistent in the last three decades. The recent decrease of civil engineering constructions and mining activities due to economic reasons and environmental concerns in many countries resulted in the decrease of the interest of academia and the engineering community in RMRE. The overemphasis on nuclear waste disposal problems, which are only relevant to a limited number of countries worldwide, causes further decreases in interest in academia and the engineering community in RMRE.

1.3

Goals and content of this book

Rock is the main constituent of the crust of the Earth, and its behavior is the most complex one among all materials in the geosphere that humankind deals with. Furthermore, it contains various discontinuities, which make the thermo-hydro-mechanical mechanical

4 Introduction and history

4

behavior of rocks more complex. These simply require a higher level of knowledge and intelligence in the RMRE community. Rock mechanics is concerned with the theoretical and applied science of the mechanical behavior of rock and rock masses, and it is one of the branches of mechanics concerned with the response of rock and rock masses to their physical-chemical environment. Rock mechanics is concerned with the application of the principles of mechanics to physical, chemical and electromagnetic processes in the uppermost part of the Earth and the design of the rock structures associated with mining, civil and petroleum engineering. This book is intended to be a fundamental book for younger generations and newcomers, as well as a reference book for experts specialized in rock mechanics. The practitioners and experts of rock mechanics should have a profound knowledge of rock-constituting elements, the petrography of rocks, discontinuities and their causes to understand their behavior under various physical and chemical actions in nature. Several chapters are devoted to this issue. First, common rock-forming minerals, rocks, discontinuities and rock mass are explained, and fundamental definitions and their measurement techniques are presented. The governing equations, constitutive laws and experimental techniques are described. The fundamentals of techniques for solving the resulting partial differential equations of rock mechanics are explained, and some specific examples of applications are given. Second, the techniques for the characterization of rock masses, experimental techniques in situ, and the evaluation of the stress state in rock mass using direct and indirect techniques are described, and several specific examples of applications are given. Other chapters are devoted to ice mechanics and extraterrestrial rock mechanics as possible new directions of rock mechanics. This volume provides the fundamentals as well as many recent and relevant topics for younger generations, newcomers and experts specialized in rock mechanics, with some specific goals such as: 1

2

3

4

Understanding the basic components and features of rocks, discontinuities and rock masses and their physical characterization. This is a quite important aspect as some practitioners of rock mechanics lack this knowledge. The fundamental laws of mechanics for rock and rock masses, constitutive models and associated experimental techniques in laboratory and in situ, numerical techniques. Various physical modeling procedures used in the field of rock mechanics are described to help young generations as well as newcomers understand the fundamentals of rock mechanics. The evaluation of rock masses in nature. Many empirical, experimental and geophysical techniques are developed for this purpose of understanding this very complete subject. These techniques are described, and their applications in the practice are presented. Another important aspect in the design and construction of rock engineering structures is the evaluation of in-situ stress state before their construction. This aspect is presented from a broad perspective, and several direct and indirect techniques are explained. Rock excavations techniques are described, and some practical examples are given. The exploration and exploitation of natural resources under extreme climatic conditions on the Earth, Moon, planets, asteroids. Therefore, ice mechanics and extraterrestrial rock mechanics will become important fields of applications of rock mechanics. Current knowledge, findings and techniques are briefly described, and possible future aspects are discussed.

5

Introduction and history

5

References Aydan, Ö. (2008) New directions of rock mechanics and rock engineering: Geomechanics and Geoengineering. 5th Asian Rock Mechanics Symposium (ARMS5), Tehran. pp. 3–21. Aydan, Ö. (2014) Future advancement of rock mechanics and rock engineering (RMRE). ROCKMEC’2014-XIth Regional Rock Mechanics Symposium, Afyonkarahisar, Turkey. pp. 27–50. Aydan, Ö. & Geniş, M. (2004) Surrounding rock properties and openings stability of rock tomb of Amenhotep III (Egypt). ISRM Regional Rock Mechanics Symposium, Sivas. pp. 191–202. Aydan, Ö. & Ulusay, R. (2003) Geotechnical and geoenvironmental characteristics of man-made underground structures in Cappadocia, Turkey. Engineering Geology, 69, 245–272. Aydan, Ö. & Ulusay, R. (2013) Geomechanical evaluation of Derinkuyu Antique Underground City and its implications in geoengineering. In: Rock Mechanics and Rock Engineering. Springer Vienna. pp. 731–754. Aydan, Ö., Tano, H., Geniş, M., Sakamoto, I. & Hamada, M. (2008a) Environmental and rock mechanics investigations for the restoration of the tomb of Amenophis III. Japan-Egypt Joint Symposium New Horizons in Geotechnical and Geoenvironmental Engineering, Tanta, Egypt. pp. 151–162. Aydan, Ö., Tano, H., Ulusay, R. & Jeong, G.C. (2008b) Deterioration of historical structures in Cappadocia (Turkey) and in Thebes (Egypt) in soft rocks and possible remedial measures. 2008 International Symposium on Conservation Science for Cultural Heritage, Seoul. pp. 37–41. Aydan, Ö., Ohta, Y., Daido, M., Kumsar, H., Genis, M., Tokashiki, N., Ito, T. & Amini, M. (2011) Chapter 15: Earthquakes as a rock dynamic problem and their effects on rock engineering structures. In: Zhou, Y. and Zhao, J. (eds.) Advances in Rock Dynamics and Applications. CRC Press, London, Taylor and Francis Group, Boca Raton, FL. pp. 341–422. Geniş, M., Tokashiki, N. & Aydan, Ö. (2009) The stability assessment of karstic caves beneath Gushikawa Castle Remains (Japan). EUROCK, 2010, 449–454. Hamada, M., Aydan, Ö. & Tano, H. (2004) Rock Mechanical Investigation: Environmental and Rock Mechanical Investigations for the Conservation Project in the Royal Tomb of Amenophis III. Conservation of the Wall Paintings in the Royal Tomb of Amenophis III, First and Second Phases Report, UNESCO and Institute of Egyptology, Waseda University. pp. 83–138. Hoek, E. & Bray, J.W. (1977) Rock Slope Engineering, 2nd edition. Institution of Mining and Metallurgy, London. 402p. Hoek, E. & Brown, J.W. (1980) Underground Excavations in Rock. Institution of Mining and Metallurgy, London. 527p. International Society for Rock Mechanics. History of Society. www.isrm.net/. Jaeger, J.C. & Cook, N.G.W. (1979) Fundamentals of Rock Mechanics, 3rd edition. Chapman & Hall, London. pp. 79, 311. Kaptan, E. (1992) Tin and ancient underground tin mining in Anatolia (in Turkish with English abstract). Geological Engineering, 40, 15–19, Ankara. Kramer, S.N. (1956) History Begins at Sumer: Thirty-Nine Firsts in Man’s Recorded History. University of Pennsylvania Press, Philadelphia. Kulaksız, S. and Aydan, Ö. (2010) Characteristics of ancient underground quarries of Turkey and Egypt and their comparison. 22nd World Mining Congress, Istanbul. pp. 607–614. Kumsar, H. & Aydan, Ö. (2008) Preservation of some ancient cities in Aegean Region of Turkey with an emphasis on Hierapolis, Aphrodisias and Lagina. 2008 International Symposium on Conservation Science for Cultural Heritage, Seoul. pp. 47–50. Kumsar, H., Celik, S., Aydan, Ö. & Ulusay, R. (2003) Aphrodisias: Anatolian antique city of building and sculptural stones. International Symposium on Industrial Minerals and Building Stones, Istanbul. pp. 301–309. Müller, L. (1963) Der Felsbau. Ferdinand Enke Verlag, Stuttgart. Stini, I. (1950) Tunnelbaugeologie. Springer-Verlag, Vienna. 366p.

6 Introduction and history

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Talobre, J. (1957) La Mechanique des Rocheux. Dunod, Paris. Terzaghi, K. (1946) Rock defects and loads on tunnel supports. In: Proctor, R.V. & White, T. (eds.) Rock Tunneling with Steel Supports. Commercial Shearing and Stamping Co., Youngstown. pp. 15–99. Tokashiki, N., Aydan, Ö. & Jeong, G.C. (2008) Stone masonry historical structures in Ryukyu Islands and possible remedial measures. 2008 International Symposium on Conservation Science for Cultural Heritage, Seoul. pp. 51–55. Yener, K.A. (1997) Excavations at Kestel Mine, Turkey. The Final Season, 1996–1997 Annual Report. Orient Institute, Michigan University.

Chapter 2

Minerals, rocks, discontinuities and rock mass

2.1

Minerals

A mineral is an inorganic natural solid, which is found in nature. Its atoms are arranged in definite patterns (an ordered internal structure), and it has a specific chemical composition that may vary within certain limits. Minerals may be generally subdivided into two major groups (e.g. Goodman, 1989): 1 2

Silicates Non-silicates

2.1.1

Silicate minerals

Silicate minerals contain silica (SiO2) either contained or in free form within the mineral lattice structure. The major silicate minerals constituting rocks follow (see Figure 2.1): (a)

Quartz (SiO 2 )

Silica tetrahedra forms a neutral three-dimensional framework structure (trapezohedral) without other cations. This arrangement forms a very stable structure. It is a strong piezoelectric and pyroelectric mineral. (b)

Olivines ((Mg, Fe) 2 SiO 4 ) and garnets ((Mg, Fe, Mn) 3 (Fe, Al, Cr) 2 Si 3 O 12 )

Olivines and garnets consist of a series of isolated tetrahedra balanced by the cations magnesium (Mg), iron (Fe), and calcium (Ca). The olivines are orthorhombic. However, well formed crystals are rare. Common olivine is usually green or brownish-green in color. Olivine is an unsaturated mineral, and weathering or hydrothermal processes easily alter it. Garnets form crystals of trapezohedra or rhombidodecahedral habit. Metamorphic rocks are the commonest environment for garnets. Garnet is mineral that is resistant weathering. (c)

Pyroxenes

Pyroxenes have single chains of tetrahedra balanced by similar metal cations and sodium (Na). Most pyroxenes are monoclinic, and they are commonly found in basic or ultrabasic igneous rocks.

8 Minerals, rocks, discontinuities and rock mass

2

Figure 2.1 Views of major silica minerals

(d)

Amphiboles

Amphiboles are characterized by double chains of tetrahedra balanced by similar cations. They are monoclinic or orthorhombic and commonly found in igneous and metamorphic rocks. Asbestos is one of well-known amphibole mineral. Hornblende is a well-known mineral of the amphibole group. (e)

Micas

Sheets of tetrahedra are building blocks. Aluminum is also involved in these sheet structures, which are charge-balanced by the cations Mg, Na and K. They are divided into the muscovite and biotite groups. Muscovite is transparent and resistant to weathering. Biotite is characterized by shades of brown to black. They are commonly found in igneous and metamorphic rocks. (f)

Clay minerals

The atomic structure of clay minerals is basically similar to micas, and they generally occur as minute, platy crystals. An important characteristic is their ability to lose or take up water according to temperature and the amount of water present in a system. Some clay minerals contain loosely bonded cations, which can be easily exchanged for others. Clay minerals

3

Minerals, rocks, discontinuities and rock mass

9

Figure 2.2 Major feldspar minerals

are produced by the degradation of silicates or other silicate glasses. Kaolinite, illite, montmorillonite, vermiculite and palygorskite are well-known clay mineral groups. Kaolin is the main constituent of ceramics. Illite is a common clay mineral, and clay in soil mechanics is constituted by illite clay minerals. Montmorillonites (smectite group) are formed by the alteration of basic rocks or other silicates low in K under alkaline conditions. Na-type montmorillonite is especially notable for losing or taking up water and associated volumetric changes. (g)

Feldspars

Feldspars are the most important single group of rock-forming silicate minerals (Fig. 2.2). They consist of a second group of alumino-silicates and the tetrahedra to form three-dimensional frameworks with Ca, Na and K as the balancing cations. They are either monoclinic or triclinic. These very abundant feldspars are subdivided in the K-Na-bearing alkali feldspars and the Ca-Na solid-solution series, called the plagioclase feldspars. Alkali feldspars are found in alkali-rock rocks such as granites, syenites, while plagioclase feldspars are found in intermediate and basic rocks. (h)

Tourmaline

The tourmaline group belongs to cyclosilicates, in which SiO4 units are linked to form three-, four- or six-membered rings. Tourmaline is the best known six-membered ring silicate, and it is piezoelectric and pyroelectric. 2.1.2

Nonsilicate minerals

Nonsilicate minerals do not contain silica (SiO2) (Fig. 2.3).

10 Minerals, rocks, discontinuities and rock mass

4

Figure 2.3 Views of various nonsilicate minerals

(a)

Carbonates

The important carbonates are the minerals calcite (CaCO3) and dolomite (CaMg(CO3)2). They are significant rock-forming minerals in limestones and dolomites. (b)

Evaporates

The important groups of evaporate minerals are the halides, including the minerals halite (NaCl, rock salt), sylvite (KCl, potash), and fluorite (CaF2), as well as the sulfates including the minerals gypsum (CaSO4.2H2O) and anhydrite (CaSO4). Anhydrite alters to gypsum when it is attacked by watery solutions. (c)

Oxides

Many oxides (hematite and magnetite) and hydroxides (limonite and goethite) of iron are important minor constituents in rocks. The aluminum oxide bauxite can also occur as a rock-forming mineral. Oxides are common in geochemical environments poor in silica. Silicates form easily from magma, so if silica is used up in a magma chamber, then the oxides remain to be formed. Their structure is complex: octahedral and dodecahedral crystals. (d)

Sulfides

The mineral pyrite is the only sulfide that occurs commonly in rocks. Sulfides are most important as economic minerals providing the main sources of elements such as arsenic, copper, lead, nickel, mercury, molybdenum and zinc. (e)

Phosphates

Phosphates are relatively rare. The only important phosphate mineral is apatite.

5

2.2

Minerals, rocks, discontinuities and rock mass

11

Rocks

A rock is an aggregate of one or more minerals and is classified into three major groups: igneous, sedimentary and metamorphic.

2.2.1

Igneous rocks

Igneous rocks are created by melting and crystallization of magma. When the magma reaches the surface, the rocks are said to be extrusive (Fig. 2.4). Volcanic lava flows are examples of extrusive igneous rocks. If the magma cools within the Earth, it forms large bodies of crystalline rock called plutons or batholiths. These rocks are called intrusive igneous rocks. Igneous rocks are generally classified on the basis of three factors: (1) grain size and texture, (2) intrusive or extrusive, (3) silica content and mineral composition (Fig. 2.5). However, the classifications based on factors 2 and 3 are commonly used to describe igneous rocks as described next. 2.2.1.1

Intrusive igneous rocks

Igneous rocks slowly cooled inside the crust. (Plutonic rock means formed in the Earth.) They generally consist of large crystals of minerals.

Figure 2.4 Views of igneous rocks

12 Minerals, rocks, discontinuities and rock mass

6

Figure 2.5 Chemical composition of igneous rocks Source: From Pinet, 1992

(A) GRANITE

Granite constitutes the continental crust, and its density varies between 2.7 and 2.8 g cm
3. Granite is an acidic igneous rock and consists of quartz, mica (muscovite and biotite), potassic feldspar (orthoclase, microcline), plagioclase feldspars and hornblende and tourmaline. Crystals are intermingled, and the amount of quartz is about 30%. Granitic rock is much less common on the other terrestrial planets. (B) SYENITE

Syenite is coarse-grained intrusive igneous rock of the same general composition as granite. However, the quartz is either absent or present in relatively small amounts (less than 5%). The name of this rock is related to syenites in the Syene region of Egypt. (C) DIORITE

Diorite is a gray to dark gray intermediate intrusive igneous rock composed principally of plagioclase feldspar, hornblende and/or pyroxene. The amount of quartz is greater than 10%. (D) GRANODIORITE

Granodiorite is an intermediate form between granite and diorite. It usually contains abundant biotite and hornblende and orthoclase (potassium feldspar).

7 (E)

Minerals, rocks, discontinuities and rock mass

13

GABBRO

Gabbro is a dark, coarse-grained, intrusive igneous rock and contains varied percentages of pyroxene, plagioclase, amphibole and olivine. The amount of quartz is less than 5%. Gabbro is generally coarse grained, with crystal sizes in the range of 1 mm or greater. Gabbro is found in the ocean crust, underneath the basalt layer (0.5–2.5 km), from 2.5 to 6.3 km deep. The lunar highlands have many gabbros. (F)

PERIDOTITE

Peridotite is a dense, coarse-grained rock, consisting mostly of the minerals olivine and pyroxene. Peridotite is ultramafic and ultrabasic, as the rock contains less than 45% silica. Peridotite is the dominant rock of the upper mantle of the Earth. 2.2.1.2

Extrusive Igneous Rocks

These rocks are formed by rapid cooling of magma at the surface. As a result, crystals are either small or glassy. (A)

RHYOLITE

Rhyolite has medium silica content (intermediate), and it is a fine-grained volcanic rock of granitic composition. (B)

TRACHYTE

Trachyte is an igneous, extrusive volcanic rock with an aphanitic to porphyritic texture. The mineral assemblage is predominately potassium feldspar with relatively minor plagioclase. It is a fine-grained volcanic rock of syenitic composition. (C)

ANDESITE

Andesite is considered the extrusive equivalent to diorite. The name “andesite” is derived from the Andes mountain range. (D)

BASALT

Basalt is the extrusive equivalent to gabbro and is made up of feldspars and other minerals common in planetary crusts. It constitutes the ocean crust of the Earth. It is usually finegrained due to the rapid cooling of lava on the Earth’s surface. Unweathered basalt is black or gray. It has been identified as a major surface rock on the dark lunar planes and much of Mars, Venus and the asteroid Vesta. 2.2.1.3

Pyroclastic rocks

Pyroclastic rocks are formed by sedimentation and the welding of debris ejected by volcanoes.

14 Minerals, rocks, discontinuities and rock mass

8

(A) TUFF

Tuff is made of compacted debris from old volcanic ash showers. A tuff of recent origin is generally loose and incoherent. However, the older tuffs are cemented by pressure and the action of infiltrating water, resulting in strong enough yet not very hard material that can be extensively used for building purposes and creating cavities, as seen in Cappadocia region of Turkey. (B) VOLCANIC BRECCIA

Volcanic breccia (agglomerate) is composed of angular mineral fragments embedded in a matrix, the product of explosive eruptions. Agglomerates are accumulations of large blocks of volcanic material often found around vents. Agglomerates are coarser and less frequently well bedded. The blocks in agglomerates vary greatly in size. (C) IGNIMBRITES

Welded tuff or ignimbrite is a product of pyroclastic flows hot enough to fuse, or “weld,” still hot ash into a single uniform layer called a cooling unit. Ignimbrite is primarily composed of a matrix of volcanic ash, pumice fragments and crystals. 2.2.2

Sedimentary rocks

Sedimentary rocks are formed in layers deposited by wind, water or ice. They are the direct products of the weathering process. As sedimentary layers are buried, they are cemented and lithified. Sediments are subdivided into three types (Fig. 2.6): 1 2 3

Clastic sedimentary rocks Chemical sedimentary rocks Organic sedimentary rocks

2.2.2.1

Clastic sedimentary rocks

Clastic sedimentary rocks consist of rock and mineral grains derived from the chemical and mechanical breakdown (weathering) of preexisting rock. They contain rock fragments and, more commonly, particles of quartz and feldspar. The most common cementing materials are silica and calcium carbonate, Clastic rocks are further classified on the basis of grain size. Underneath each rock type, the Wentworth Scale of particle sizes is shown. (A) CONGLOMERATES AND BRECCIAS

Conglomerates are sedimentary rocks consisting of rounded fragments, whereas breccias consist of angular clasts. Conglomerates and breccias are characterized by clasts larger than 2 mm. (B) SANDSTONES

Sandstone is a sedimentary rock composed mainly of mineral or rock grains of sand size (0.062–2 mm). Most sandstone is composed of quartz and/or feldspar. Some sandstones are resistant to weathering, and they are porous. When sandstone contains roughly 60%

9

Minerals, rocks, discontinuities and rock mass

15

Figure 2.6 Views of various sedimentary rocks

quartz sand and 25% feldspar, it is called Arkose. If the percentage of quartz is greater than 95%, it is called either quartzite or quartzarenite. (C)

MUDSTONE

Mudstone consists of tiny particles less than 0.0062 mm in diameter. Individual grains are too small to be distinguished without a microscope. With increased pressure and time, the platy clay minerals may become aligned, with the appearance of fissility or parallel layering consolidated mud, rich in organic matter. (D)

ARGILLITE

Argillite is a sedimentary rock composed of clay particles, which have been hardened and cemented. 2.2.2.2

Chemical and organic sedimentary rocks

These rocks are formed either from minerals that precipitate directly from aqueous (water) solutions or from the accumulation of fossilized remains of organisms that become limestone.

16 Minerals, rocks, discontinuities and rock mass

10

(A) GYPSUM (CASO4.2H2O)

Gypsum is a very soft chemical rock, and it is composed of calcium sulfate dihydrate. (B) ANHYDRITE (CASO4)

Anhydrite, which is an evaporatic rock, is calcium sulfate. Interaction with water produces an increase in volume of the rock layer. This process causes a 30% volumetric expansion, termed as swelling. (C) ROCK SALT – HALITE (NACL)

Halite occurs in beds of sedimentary evaporite minerals that result from the drying up of enclosed lakes, playas and seas. It commonly occurs with other evaporite deposit minerals such as several of the sulfates, halides and borates. (D) LIMESTONE

Limestone is a sedimentary rock composed largely of the calcite mineral (CaCO3). Limestone may be crystalline, clastic, granular or massive, depending on the method of formation. The primary source of the calcite in limestone may be of organic origin or chemical solution. Limestone makes up about 10% of the total volume of all sedimentary rocks. Travertine is a variety of limestone formed along streams where there are waterfalls around hot or cold springs. Limestone is partially soluble, especially in acid, and therefore forms many erosional landforms, called karsts. Tufa is a thick, rock-like calcium carbonate deposit that forms by precipitation from bodies of water with a high dissolved calcium content. (E) CHERT

Chert is a fine-grained, silica-rich cryptocrystalline sedimentary rock. It varies greatly in color from white to black but most often manifests as gray, brown, grayish brown and light green to rusty red. (F) CHALK

Chalk is a soft, white, porous form of limestone composed of organic origin. Chalk is formed in shallow waters by the gradual accumulation of the calcite mineral remains of microorganisms such as planktonic green algae, associated with varying proportions of larger microscopic fragments of bivalves, foraminifera and ostracods. 2.2.3

Metamorphic rocks

Metamorphic rocks are rocks formed by the action of pressure (P), temperature (T), and fluids within the Earth. As sediments are deeply buried, they are deformed and new minerals recrystallize at the elevated temperatures and pressures to form metamorphic rocks. Metamorphic rocks are generated by recrystallization of either igneous or sedimentary rocks by the action of any or all of pressure, temperature and pore fluids. The lower limit of metamorphic temperatures is 150°Celsius. The upper limit is the melting temperature when magma forms. The type of metamorphic rock is determined by the parent rock and the P/T conditions. In general, metamorphism the growth of new

11

Minerals, rocks, discontinuities and rock mass

17

Table 2.1 Relation between parent rock and metamorphic rocks Rock Name

Type

Parent Rock

Characteristics

Phyllite Slate Schist Gneiss

Foliated Foliated Foliated Foliated

Silty sandstone Shales and mudstone Fine-grained rocks Coarse-grained rocks

Quartzite

Nonfoliated Sandstone

Marble

Nonfoliated Limestone

Splitting schistosity surfaces Prominent splitting surfaces Mica minerals, often crinkled or wavy Dark and light bands or layers of aligned minerals Interlocking almost fused quartz grains, little or no porosity Interlocking almost fused calcite grains, little or no porosity

Serpentine Nonfoliated Peridotite ultramafic rocks

Figure 2.7 Views of common metamorphic rocks

minerals, the deformation and rotation of mineral grains, the recrystallization of minerals, and the production of anisotropic rock (Table 2.1). Contact metamorphic rocks are recrystallized and rarely show foliation (Fig. 2.7). Shales baked by igneous contact form very hard fine-grained rocks called hornfels. Calcareous rocks, when subjected to contact metamorphism (an alteration by hot fluids), alter into rocks called skarns. Metamorphic rocks have been chemically altered by heat, pressure and deformation, while buried deep in the Earth’s crust. These rocks show changes in mineral composition or texture or both. This area of rock classification is highly specialized and complex. (a)

Phyllite

Phyllite was originally a fine-grained sedimentary rock such as sandy mudstone or shale composed mainly of clay minerals in a semirandom orientation. Phyllite is a common metamorphic rock, formed under a low-heat and high-pressure environment.

18 Minerals, rocks, discontinuities and rock mass

(b)

12

Shale

Shale is a fine-grained rock subjected to low-pressure metamorphism. Its original constituent is mudstone. It is characterized by thin laminae breaking with an irregular curving fracture. (c)

Slate

Slates are foliated rocks representing low-grade metamorphic alteration of shales (laminated clay). Slate is mainly composed of quartz and muscovite or illite, often along with other minerals. (d)

Schists

Schists are foliated medium-grade metamorphic rock with parallel layers, vertical to the direction of compaction. The schists contain lamellar minerals such as micas, chlorite, talc, hornblende, graphite and others. Quartz often occurs in drawn-out grains to such an extent that a particular form, called quartz schist, is produced. Schist contains more than 50% platy and elongated minerals, often finely interleaved with quartz and feldspar. Schist is characteristically foliated so that individual mineral grains split off easily into flakes or slabs. (e)

Gneiss

These are banded rocks consisting of alternating layers of quartz and feldspar of high metamorphic grade. The original rock formations may be igneous or sedimentary rocks. (f)

Quartzites

They represent metamorphosed quartzitic sandstone. Pure quartzite is usually white to gray. Quartzites often occur in various shades of pink and red due to varying amounts of iron oxide. Quartzite is very resistant to chemical weathering. (g)

Marble

Marble is metamorphosed limestone composed mostly of calcite. The metamorphism causes a complete recrystallization of the original rock into an interlocking mosaic of calcite, aragonite and/or dolomite crystals. (h)

Serpentinite

Serpentinite is comprised of serpentine minerals. Minerals in this group are formed by a hydration and metamorphic transformation of ultramafic rocks. Serpentinite is formed from olivine via several reactions. It can be easily weathered, resulting in swelling.

2.3

Discontinuities

Rocks, by nature, are geologically classified into three main groups: igneous rocks, sedimantery rocks and metamorphic rocks. Each of these rock classes may be further subdivided into several subclasses. For example, igneous rocks are subdivided into three classes: extrusive, intrusive and semi-intrusive rocks (such as dykes, sills etc.), although

13

Minerals, rocks, discontinuities and rock mass

19

the chemical composition of the three types of rocks may be same. The order of minerals and the internal structure of rocks are a result of chemical composition of rising magma, its velocity, and environmental conditions during the cooling process, which greatly affects the discontinuity formation in such rocks. The term “discontinuity” encompasses all types of interruptions of structural integrity of rock masses. They can be classified into three categories: 1 2 3

Intrinsic discontinuities: Bedding plane, schistosity, flow plane Volumetric-strain induced discontinuities: Sheeting, desiccation, cooling, erosion, freezing-thawing Plastic deformation-induced discontinuities: Faults, fracture zones, tension (T), Riedel (R-R0 ) shear cracks and Skempton (P) fractures.

Discontinuities in rocks are termed as cracks, fractures, joints, bedding planes, schistosity or foliation planes and faults. Discontinuities are products of some certain phenomena to which rocks were subjected in their geological past, and they are expected to be regularly distributed within rock mass. The discontinuities can be classified into the following four groups according to the mechanical or environmental process they were subjected to (Erguvanlı, 1973; Yüzer and Vardar, 1986; Miki, 1986; Ramsay and Huber, 1984; Aydan, 2018, etc.) (Fig. 2.8):

Figure 2.8 Views of discontinuities in situ

20 Minerals, rocks, discontinuities and rock mass

1

Tension discontinuities due to: • • • • • • •

2

(a)

Cooling Drying Freezing Bending Flexural slip Uplifting Faulting and stress relaxation due to erosion or glacier retreat or human-made excavation

Shear discontinuities due to: • •

3 4

14

Folding Faulting

Discontinuities due to periodic sedimentation Discontinuities due to metamorphism Joints

A joint is a discontinuity that is relatively planar and on which there has been no displacement. A series of joints in the same orientation are referred to as a joint set. Joints may be open, healed or filled. (b)

Bedding joints

Bedding joints that are parallel to the bedding are referred to as bedding joints. (c)

Foliation joints

Foliation joints are parallel to metamorphic foliation. (d)

Shear

Shears are structural breaks, where relative movement has occurred. The shear surfaces are characterized by the presence of slickensides, gouge, breccia, mylonite or a combination of these. Shears are in effect small faults and typically have displacements of less than 5 cm. (e)

Fault

Fault is a shear with significant continuity and evidence of large displacement. A fault can range from centimeters in width to a zone that is tens of meters thick. The fault may contain breccia, gouge and crushed rock. Fault zones are typically conduits for high groundwater flow. (f)

Contact

Contact is a geologic contact between two distinct lithologic units.

15

(g)

Minerals, rocks, discontinuities and rock mass

21

Vein

Vein is an infilling of a discontinuity caused by circulation of mineralized fluid and deposition of minerals. Veins can cause healing of the original discontinuity. Discontinuities, although they may be viewed as planes in large scale, have undulated surfaces varying in irregularities. As a result, they may be regarded as bands with a certain thickness in association with the amplitude of undulations. The discontinuities may be filled by some infilling materials such as calcite, quartzite or weathering products of host rock or transported materials, or they may exist from the beginning as thin films of clay deposits in sedimentary rocks along bedding planes.

2.4

Rock mass

Rock mass is considered to be the sum total of the rock as it exists in place, taking into account the intact rock material as well as joints, faults and other natural planes of weakness that can divide the rock into interlocking blocks of varying sizes and shapes. Rock mass is classified as intact rock mass, layered rock mass, blocky rock mass and sheared rock mass, which may be jointed. (a)

Intact rock mass

Intact rock mass contains neither joints nor hair cracks. Hence, if it breaks, it breaks across sound rock. (b)

Layered or foliated rock mass

Layered or foliated rock mass consists of individual layers or foliation with little or no resistance against separation along the boundaries between the layers. (c)

Blocky rock mass

Blocky rock mass consists of chemically intact or almost intact rock fragments, which are entirely separated from one another and are perfectly or imperfectly interlocked. (d)

Sheared rock mass (fracture zone)

Sheared (fracture zone) rock mass is rock mass that underwent shearing sufficient to create distinct fault gouge and various degree of fractures such as T, R-R0 , S fractures. A sheared rock mass generally consists of a crushed rock zone (fault gouge) and shears with slickensides and tension fractures, and the fragment size within the zone will vary with distance from the gouge zone. Because of the existence of discontinuities as a result of one or combined actions of these processes, the structure of rock mass in nature may look like an assemblage of blocks of some typical shapes (Figs. 2.9 and 2.10). Most common shapes of the blocks are rectangular, rhombohedral, hexagonal or pentagonal prisms. While hexagonal and/or pentagonal prismatic blocks are commonly observed in extrusive basic igneous rocks such as andesite or basalt, and some fine grained sedimentary rocks have undergone

Figure 2.9 views of rock mass in nature

Figure 2.10 Geometrical modeling of rock mass

17

Minerals, rocks, discontinuities and rock mass

23

cooling or drying processes, the most common block shapes are between rectangular prism and rhombohedric prism. The lower and upper bases of the blocks are usually limited by planes called flow planes, bedding planes and schistosity or foliation planes in igneous, sedimentary and metamorphic rocks, respectively. These discontinuities can be regarded as very continuous for most of the concerned rock structures. Other discontinuities are usually found in at least two or three sets, crossing these planes orthogonally or obliquely. These secondary sets, if present, may be very continuous or intermittent. As a result, the rock mass may be viewed as shown in Figure 2.10 (Goodman, 1989; Aydan et al., 1989, 2018): • • •

Continuous medium Tabular (layered) medium Blocky medium

Blocky medium can be further subdivided into two groups depending upon the continuity of secondary sets as follows (Aydan and Kawamoto, 1987; Aydan et al., 1989): • •

Cross-continuously arranged blocky medium Intermittently arranged blocky medium

References Aydan, Ö. (1989) The Stabilisation of Rock Engineering Structures by Rockbolts. Doctorate Thesis, Nagoya University, Faculty of Engineering, Nagoya, 204p. Aydan, Ö. (2018) Rock Reinforcement and Rock Support. CRC Press, London. 486p. Aydan, Ö. & Kawamoto, T. (1987) Toppling failure of discontinuous rock slopes and their stabilisation (in Japanese). Journal of Japan Mining Society, 103(1197), 763–770. Aydan, Ö., Shimizu, Y. & Ichikawa, Y. (1989) The effective failure modes and stability of slopes in rock mass with two discontinuity sets. Rock Mechanics and Rock Engineering, 22(3), 163–188. Erguvanlı, K. (1973) Engineering Geology (in Turkish). ITU Press, Istanbul. No. 966. 552p. Goodman, R.E. (1989) Introduction to Rock Mechanics. Wiley. New York. 576p. Miki, K. (1986) Introduction to Rock Mechanics (in Japanese). Kajima Press. Tokyo. 317p. Pinet, P.R. (1992) Oceanography: An Introduction to the Planet. West Publishing Company, Saint Paul. Ramsay, J.G. & Huber, M.I. (1984) The Techniques of Modern Structural Geology, Vol. 1: Strain Analysis. Academic Press, London. 306p. Yüzer, E. & Vardar, M. (1986) Rock Mechanics (in Turkish), ITU Foundation, Istanbul. No. 4. 154p.

Chapter 3

Fundamental definitions and measurement techniques

3.1 (a)

Physical parameters of rocks Bulk density

Bulk density is a property of materials and is defined as the ratio of the mass (m) of rock to the volume (V) it occupies. It is mathematically expressed as follows: r¼ (b)

m V

ð3:1Þ

Unit weight

Unit weight is defined as the ratio of the weight (W) of rock to the volume (V) it occupies. It is mathematically expressed as follows: g¼ (c)

W V

ð3:2Þ

Water content

Water content is a ratio to indicate the amount of water a rock contains. Water content can be either the volumetric (by volume) or the gravimetric (by weight) fraction of the total rock that is filled with liquid water. Volumetric water content is defined as volume of water per unit volume of soil: y¼

Vw V

ð3:3Þ

Gravimetric water content is defined as mass of water per unit mass of dry rock: w¼

mw ms

ð3:4Þ

26 Definitions and measurement techniques

(d)

2

Porosity

The porosity of a porous rock describes the fraction of void space in the material to its total volume: n¼

Vp V

ð3:5Þ

where Vp is the nonsolid volume, and V is the total volume of material, including the solid and nonsolid parts.

(e)

Longitudinal and traverse seismic wave velocity of rock (V s , V p )

Longitudinal and traverse seismic wave velocity of a rock sample with length are defined as follows: Vp ¼

L Dtp

and

Vp ¼

L Dtp

ð3:6Þ

where Dtp and Dts are the travel times of longitudinal and traverse waves.

3.2

Physical parameters of discontinuities

Rock discontinuities are characterized by their orientations, spacing and persistency, and surface topography. The methods used to evaluate the parameters are explained in this section.

3.2.1

Discontinuity orientation and its representation

Discontinuity orientation data consists of dip direction (or strike, which is perpendicular to the dip direction) and dip. It is measured either by directly clinometer or indirectly by photogrammetric techniques. If the photogrammetric technique is used, the images of discontinuities on three planes having different normal vectors are necessary. If clinometers are used, it is desirable to use a Clar-type clinometer (compass) as dip direction and dip of a discontinuity can be measured simultaneously. Also, new electronic clinometers store the measured data in digital form.

3.2.2

Discontinuity orientation representation

The stereographic projection method is used for the graphical presentation of orientation data of discontinuities. The method utilizes a sphere of a unit radius. The projections are done by using either equal area or equal angle approach. Equal angle projection is preferred for kinematic assessments of the stability of rock structures. The discontinuities are represented by great circles and/or poles, and they are projected onto equatorial plane using either upper or lower hemisphere projections. When the number of data is too large, the use of pole density projections is desirable.

3

Definitions and measurement techniques

3.2.3

27

Discontinuity spacing

Rock discontinuity spacing is also measured though direct measurement of scan lines on the outcrops of rock mass or photogrammetric methods. The number of scan lines should be sufficient to eliminate line bias on measurements, and it is desirable to have at least three outcrop surfaces with different unit vectors. Scan lines should be set up at near right angles to major discontinuity sets in order to avoid scan line–discontinuity set orientation bias. Otherwise, the measurements from scan lines would be apparent, and they have to be converted to true values. In recent years, new techniques of measuring spacing and storage and processing of the data will be added as alternative techniques. Particularly, laser-based techniques could be alternative procedures for dealing with the huge amount of data. 1 2

Manual techniques (Fig. 3.1) Digital techniques (manual or automated evaluation: Figure 3.2) a b

Photogrammetry (Fig. 3.2(a)) Laser scanning (Fig. 3.2(b))

3.2.4

Discontinuity persistency

Discontinuity persistence implies the areal extent or size of discontinuity within a plane. Scan lines must be set up on actual outcrops along the traces of each discontinuity set, and the lengths of rock bridges should be measured. The persistence (T) of the discontinuity is defined as follows: X lbi Þ  100 ð3:7Þ T ¼ ð1  L Where L is scan line length, and lbi is length of rock bridge i.

Figure 3.1 An example of line survey of chert layers in Ie island (Okinawa, Japan) as manual technique (scale is 1 m long)

28 Definitions and measurement techniques

4

Figure 3.2 Digital techniques for spacing measurement

Figure 3.3 (a) Thoroughgoing, (b) intermittent discontinuities

Although the original definition of persistence described is correct, it becomes meaningless for thoroughgoing discontinuities such as bedding planes, schistosity or sheeting joints (Fig. 3.3).

3.2.5

Discontinuity surface morphology and measurement techniques

Roughness is a geometrical parameter, and it has directional characteristics (Fig. 3.4). Many methods have been used to characterize the surface topography of rock discontinuities such

5

Definitions and measurement techniques

29

Figure 3.4 Some examples of surface roughness: (a) schistosity surface, (b) sheeting joint, (c) bedding plane in limestone, (d) and (e) striated shear discontinuity

Figure 3.5 Needle-type and roller-stylus for roughness measurements Source: Aydan et al., 1995

as asperity height, asperity inclination, profile length ratio, autocorrelation function, fractal dimension and so on. There are different techniques to measure surface morphology of discontinuities. While contact-type profilers are used for this purpose, some new noncontact-type techniques based on laser profiling and photogrammetric techniques will be added as alternative and more accurate systems. Furthermore, it should be noted that Joint Roughness Coefficient (JRC) proposed by Barton and Choubey (1977) is related to an additional friction component associated with the inclination of the asperity wall, and it is one of the parameters. 3.2.5.1

Profiler (needle or roller-pen-recorder)

The earliest profiler used for roughness measurement is the needle profiler. Another profiler utilizes a roller-type stylus. The needle- or roller-type profiler can be manual or automatic (Fig. 3.5). Nevertheless, automatic profilers need a power source, which makes them difficult to use on site.

30 Definitions and measurement techniques

3.2.5.2

6

Photogrammetry

The photos of the surface are used to quantify the surface morphology of rock discontinuities. This technique requires the identification of surface profiles from digital images, as shown in Figure 3.6. A special application of this technique is the shadow profilometry technique, and it is one of the techniques and scale-independent method available for 3-D characterization (Maerz, 1990; Maerz et al., 1990). Shadow profilometry technique is a technique in which an edge of light/shadow is used to trace a surface profile (Fig. 3.7). Multiple profiles produced by moving laterally across a fragment can be used to create a 3-D surface.

Figure 3.6 Profile image of bedding plane in limestone

Figure 3.7 Shadow profilometry technique Source: From Maerz and Hilgers, 2010

7

3.2.5.3

Definitions and measurement techniques

31

Laser technology

Laser technology for surface morphology can be used as an automatic modern procedure (Figs. 3.8 and 3.9). The simple technique utilizes 1-D scanning (Fig. 3.8); more sophisticated laser devices can scan for a given bandwidth (Fig. 3.9). Its limitation to in-situ applications will be also pointed out.

Figure 3.8 1-D scanning devices in laboratory Source: Aydan’s Laboratory

Figure 3.9 Laser scanning device for a given bandwidth in laboratory Source: From Maerz and Hilgers, 2010

32 Definitions and measurement techniques

3.2.5.4

8

Surface morphology characterization

Surface morphology is the geometry of the surface of discontinuities. Appropriate parameters are described here. The parameters associated with linear profiles are height of asperities, inclination of asperity walls, length of asperity wall relative to base length and periodicity of asperities (Aydan and Shimizu, 1995; Aydan et al., 1999; Myers, 1962; Sayles and Thomas, 1977; Tse and Cruden, 1979; Thomas, 1982; Türk et al., 1987).

(A) HEIGHT PARAMETERS

Center-line average height (CLAH) is defined as: Z 1 x¼L CLAH ¼ jφjdx L x¼0

ð3:8Þ

where L is measurement length; x is distance from origin; φ is height of the profile from the reference base line. Mean standard variation of height (MSVH) is defined as: Z 1 x¼L 2 φ dx ð3:9Þ MSVH ¼ L x¼0 Root mean-square of height (RMSH) is defined as:  Z x¼L 1=2 1 2 RMSH ¼ φ dx L x¼0

ð3:10Þ

(B) PROFILE LENGTH PARAMETERS

Ratio of profile length (RPL) is defined as:  1=2 Z Z 1 x¼L 1 x¼L dφ RPL ¼ ds ¼ 1þ dx L x¼0 L x¼0 dx

ð3:11Þ

(C) ASPERITY INCLINATION PARAMETERS

Weighted asperity inclination (WAI ) is defined as:   Z 1 x¼L  dφ   dx 5mmWAI ¼ tan1 ðWAI  Þ WAI ¼ L x¼0  dx  Weighted asperity inclination difference (WAID ) is defined as:   Z Z 1 x¼Lp dφ 1 x¼Ln  dφ  j jp dx  dx; WAID ¼ tan1 ðWAID Þ WAID ¼ Lp x¼0 dx Ln x¼0  dx n

ð3:12Þ

ð3:13Þ

where p and n stand for positive and negative, respectively. Furthermore, L ¼ Lp þ Ln .

9

Definitions and measurement techniques

Mean standard variation of inclination (MSVI) is defined as:  2 Z 1 x¼L dφ dx MSVI ¼ L x¼0 dx Root mean-square of inclination (RMSI) is defined as: " Z  2 #1=2 1 x¼L dφ RMSI ¼ dx L x¼0 dx (D)

33

ð3:14Þ

ð3:15Þ

PERIODICITY PARAMETERS

Autocorrelation function (ACF) is defined as: Z 1 x¼L φðxÞφðx þ tÞdx ACF ¼ L x¼0 Structure function (SF) is defined as: Z 1 x¼L 2 ðφðxÞ  φðx þ tÞÞ dx SF ¼ L x¼0

ð3:16Þ

ð3:17Þ

where t is a measure of the periodicity of asperities.

(E)

FRACTAL DIMENSION

Fractal dimension is defined as: N ¼ ClD

ð3:18Þ

where N is the number of steps, C is a constant, l is step length, and D is fractal dimension. Since the following relations holds between the total length of the profile and the step length: L ¼ Nl

ð3:19Þ

then the above equation is rewritten as: L ¼ Cl1D (F)

ð3:20Þ

ANISOTROPY OF SURFACE MORPHOLOGY PARAMETERS

To characterize the surface morphology of discontinuities, one may introduce an elliptical coordinate system so that the principal axes of the coordinate system coincide with those eigen directions (Fig. 3.10a). Such a coordinate system would be appropriate for many discontinuity types found in rock masses. However, Aydan et al. (1996) utilized a Cartesian coordinate system. Let us assume that axis X coincides with the axis of ridges and troughs and axis Y is perpendicular to that of the ridges and troughs. Let us further

Figure 3.10 (a) Notations for measuring profiles of discontinuities, (b) profiles of several discontinuity types measured by varying measuring direction Source: Aydan et al., 1996

11

Definitions and measurement techniques

35

assume that direction y is measured from axis X anticlockwise (Fig. 3.10a). A surface morphology parameter F as a function of measuring direction y may be assumed to be of the following form: FðyÞ ¼

n n X X ai cosi y þ bi sini y þ i¼1 n X

i¼1

ci cosi 2y þ

n X di sini 2y þ    þ

i¼1 n X

ð3:21Þ

i¼1

yi cosi Ny þ

i¼1

n X zi sini Ny i¼1

As a particular form of the preceding equation, we select the following: FðyÞ ¼ a1 cos y þ a2 cos2 y þ b1 sin y þ b2 sin2 y

ð3:22Þ

Furthermore, we assume that the spectra of the parameter are obtained experimentally along eigen directions as: Fðy¼ 0 Þ ¼ F0 ;

Fðy¼ 90 Þ ¼ F90 ;

Fðy¼ 180 Þ ¼ F180 ;

Fðy¼ 270 Þ ¼ F270 ð3:23Þ

With the preceding conditions, constants a1 ; a2 ; b1 ; b2 are obtained as follows: a1 ¼

F0  F180 ; 2

a2 ¼

F0 þ F180 ; 2

b1 ¼

F90  F270 ; 2

b2 ¼

F90 þ F270 2

ð3:24Þ

Nevertheless, it should be noted that constants a1 and b1 are expected to be zero for the parameters if the surface morphology is isotropic except for WAID and WAID unless some errors are caused by the measuring system and digitization. The length of profiles also influences computed results. For this purpose, the computations were carried out by varying the length of the digitized profile for a sampling interval of 2 mm. As seen from Figure 3.10(b), if the profile length is greater than the wave length of the main asperity of a given discontinuity, the influence of the profile length becomes less pronounced. The procedure to evaluate the anisotropy of some of fundamental surface morphology parameters; namely, CLAH, WAI and RPL are applied to profiles shown in Figure 3.10b. Figure 3.11 shows the computed parameters as a function of measuring direction y together with measured results for the sheeting joint in Nakatsukawa granite, whose original profiles were shown in Figure 3.10. Measured spectra of the discontinuity were used in plotting computed results, using Equation (3.22). As seen from the figure, the computed results closely fit the measured ones, although some slight differences between them exist. The differences may be attributable to errors caused while digitizing the measured profiles. Nevertheless, the proposed procedure to evaluate the anisotropy of surface morphology parameters has been concluded to be appropriate. If necessary, better fits to measured parameters can be obtained by using higher-order functions. It is also interesting to note

36 Definitions and measurement techniques

12

Figure 3.11 Variation of surface morphology parameters CLAH, WAI and RPL as a function of measuring direction (sheeting joint in Nakatsukawa granite)

that asperity inclinations have a minimum along the ridge axis and a maximum perpendicular to the ridge axis. 3.2.5.6

Aperture

Aperture is the separation of between discontinuity surfaces. Aperture can be also measured by borehole cameras (Figs. 3.12 and 3.13). 3.2.5.7

Filling

When discontinuity walls are separated, they may be filled (Fig. 3.14). The filling of discontinuities is directly related to sedimentation of solutions in groundwater. The clayey materials as filling in discontinuities are related either to shearing of discontinuities in the geological past or to weathering or hydrothermal alteration of rock material adjacent to the discontinuity walls. Therefore, when the effect of filling has to be taken into account, the character of the filling and its cause must also be taken into account.

Figure 3.12 Views of apertures of discontinuities in various rocks

Figure 3.13 Measurements of apertures and orientation using borehole camera Source: Courtesy of RAAX

38 Definitions and measurement techniques

14

Figure 3.14 Views of different infillings of rock discontinuities

3.3 3.3.1

Rock mass Physical properties of rock mass

The physical properties of rock mass are defined in the same manner as those for intact rocks. However, it is very rare to determine the physical properties of rock masses in practice except the seismic wave properties. As the wave velocity of rock mass is used to infer the properties of rock mass (Ikeda, 1970; Aydan et al., 1993, 1997; Aydan and Kawamoto, 2000), it is often measured in practice during the site exploration as well as during excavation. 3.3.2

Number of discontinuity sets

The number of discontinuity sets plays a major role on the overall behavior of rock mass as well as its stability and thermo-hydro-mechanical properties. In practice, the stereo technique used to plot contouring is carried out to evaluate the number of discontinuity sets, as shown in Figure 3.15(a). However, more descriptive procedures, such as geological interpretation, photogrammetric techniques, geological knowledge–based approach (unit block method), automatic identification of discontinuity sets from stereo projections for evaluating discontinuity sets, are carried out in practice (Aydan et al., 1991) (Fig. 3.15 and 3.16). Recently, borehole cameras have been improved to determine many fundamental parameters such as the orientation, aperture, infilling, spacing and roughness associated with discontinuities (Fig. 3.17).

15

Definitions and measurement techniques

39

Figure 3.15 Comparison of different techniques for determining number of sets

Figure 3.16 Photogrammetric technique

3.3.3

Block size index (I b ) and block volume (V b )

Block size index and block volume are defined as follows (ISRM, 1978; Peaker, 1990): Ib ¼

3 X

Si ; Vb ¼ S1  S2  S3

ð3:25Þ

i¼1

where Si is the average discontinuity spacing. 3.3.4

Volumetric joint count (J v )

Volumetric joint count is based on discontinuity set spacing (ISRM, 1978), and it is mathematically expressed as: Jv ¼

n X 1 i¼1

Si

ð3:26Þ

40 Definitions and measurement techniques

16

Figure 3.17 Reconstructed core logs from borehole-wall images Source: Courtesy of RAAX

References Aydan, Ö. & Kawamoto, T. (2000) The assessment of mechanical properties of rock masses through RMR rock classification system. GeoEng2000, Melbourne. p. UW0926. Aydan, Ö. & Shimizu, Y. (1995) Surface morphology characteristics of rock discontinuities with particular reference to their genesis. Proc., Int. Meeting on Fractography, Special Publication on Fractography, Geol. Soc. of UK, Geol. pp. 11–26. Aydan, Ö., Ichikawa, Y., Shimizu, Y. & Murata, K. (1991) An integrated system for the stability of rock slopes. The 5th Int. Conf. on Computer Methods and Advances in Geomechanics, Cairns, 1. pp. 469–465. Aydan, T., Akagi, T. & Kawamoto, T. (1993) Squeezing potential of rocks around tunnels: Theory and prediction. Rock Mechanics and Rock Engineering, 26(2), 137–163. Aydan, Ö., Shimizu, Y. & Kawamoto, T. (1995) A portable system for in-situ characterization of surface morphology and frictional properties of rock discontinuities. The 4th Int. Symp. on Field Measurements in Geomechanics, Bergamo. pp. 463–470. Aydan, Ö., Shimizu, Y. & Kawamoto, T. (1996) The anisotropy of surface morphology characteristics of rock discontinuities. Rock Mechanics and Rock Engineering, 29(1), 47–59. Aydan, Ö., Ulusay, R. & Kawamoto, T. (1997) Assessment of rock mass strength for underground excavations. The 36th US Rock Mechanics Symposium. New York. pp. 777–786.

17

Definitions and measurement techniques

41

Aydan, Ö., Tokashiki, N., Shimizu, Y. & Kawamoto, T. (1999) A simple system for measuring the surface morphology characteristics of rock discontinuities (in Japanese). The 29th Rock Mechanics Symposium of Japan. Tokyo. 136–140. Barton, N. & Choubey, V. (1977) The shear strength of rock joints in theory and practice. Rock Mechanics. Vienna, 10, 1–54. Ikeda, K. (1970) A classification of rock conditions for tunnelling. 1st Int. Congr. Eng. Geology, IAEG, Paris. pp. 1258–1265. ISRM (1978) Suggested methods for the quantitative description of discontinuities in rock masses. International Journal of Rock Mechanics and Mining Science Geomechanics Abstracts, 15, 319–368. Maerz, N.H. (1990) Photoanalysis of Rock Fabric. Ph.D. dissertation, Department of Earth Sciences, University of Waterloo, Waterloo, Ontario, Canada. Maerz, N.H. & Hilgers, M.C. (2010) A method for matching fractured surfaces using shadow profilometry. Third International Conference on Tribology and Design 2010, May 11–13 2010, Algarve, Portugal. pp. 237–248. Maerz, N.H., Franklin, J.A. & Bennett, C.A. (1990) Joint roughness measurement using shadow profilometry. International Journal of Rock Mechanics and Mining Science Geomechanics Abstracts, 27, 329–343. Myers, M.O. (1962) Characterization of surface roughness. Wear, 5, 182–189. Peaker, S.M. (1990) Development of a Simple Block Size Distribution Model for the Classification of Rock Masses. M.Sc. Thesis, Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada. Sayles, R.S. & Thomas, T.R. (1977) The spatial representation of surface roughness by means of the structure function: A practical alternative to correlation. Wear, 42, 263–276. Thomas, T.R. (1982) Rough Surfaces. Longman. London. Tse, R. & Cruden, D.M. (1979) Estimating joint roughness coefficients. International Journal of Rock Mechanics and Mining Science, 16, 303–307. Türk, N., Gerig, M.J., Dearman, W.R. & Amin, F.F. (1987) Characterization of rock joint surfaces by fractal dimension. Proc., 28th US Symposium on Rock Mechanics. Tucson, Arizona. pp. 1223– 1236.

Chapter 4

Fundamental governing equations

Various actual applications of rock mechanics involve mass transportation phenomena such as seepage, diffusion, static and dynamic stability assessment of structures, and heat flow. Principles of fundamental laws presented herein follow basically the laws of continuum mechanics (e.g. Eringen, 1980; Mase, 1970). In this section, the fundamental governing equation of each phenomenon is presented. The governing equations are developed for the one-dimensional case, and they are extended to multidimensional situations.

4.1

Fundamental governing equations for one-dimensional case

4.1.1

Mass conservation law

Mass conservation law is stated as: gained mass ¼ input flux  output flux þ mass generated Let us consider an infinitely small cubic element as shown in Figure 4.1. The preceding statement can be written in the following form for x-direction as: Dm ¼ qx DyDzDt  qxþDx DyDzDt þ gDxDyDzDt

ð4:1Þ

where m is mass, q is flux, and g is mass generated per unit volume per unit time. Terms Dm and q are explicitly written as: Dm ¼ DrDxDyDz

ð4:2Þ

q ¼ rv

ð4:3Þ

Figure 4.1 Illustration of mass-conservation law

44 Fundamental governing equations

2

where r is density and v is velocity. The quantity qxþDx can be written in the following form using the Taylor expansion: qxþDx ¼ qx þ

@q Dx þ 02 @x

ð4:4Þ

Inserting this relation in Equation (4.1) and dividing both sides by DxDyDzDt yields the following: Dr DðrvÞ ¼ þg Dt Dx

ð4:5Þ

Taking the limit results in mass conservation law for 1-D case as: @r @ðrvÞ ¼ þg @t @x 4.1.2

ð4:6Þ

Momentum conservation law

Momentum balance for a 1-D case can be written for a typical infinitely small control volume (simple momentum concept: p = m  v) (Fig. 4.2): sx DyDzDt þ sxþDx DyDzDt þ bDxDyDzDt  DðrvÞDxDyDz  Dðrv2 ÞDyDzDt ¼ 0

ð4:7Þ

where s is stress, b is body force, r is density, and v is velocity. Stress sxþDx can be expressed using the Taylor’s expansion as: sxþDx ¼ sx þ

@s Dx þ 02 @x

ð4:8Þ

Dividing by DxDyDzDt and taking the limits yield the following: @s @ðrvÞ @ðrv2 Þ þb¼ þ @x @t @x The preceding equation may be rewritten as:

 @s @ðrÞ @ðrvÞ @v @v þb¼v þ þr þv @x @t @x @t @x

Figure 4.2 Illustration of momentum-conservation law

ð4:9Þ

ð4:10Þ

3

Fundamental governing equations

45

Introducing the following operator, which is called material derivatization operator: d @ @ ¼ þv dt @t @x

ð4:11Þ

and using the mass conservation law for no mass generation, Equation (4.10) becomes: @s dv þb¼r @x dt

ð4:12Þ

If we define acceleration a as: a¼

dv dt

ð4:13Þ

Equation (4.12) can be rewritten as: @s þ b ¼ ra @x

ð4:14Þ

Acceleration a may also be expressed in terms of displacement u as: a¼

d2 u ¼ u€ dt2

ð4:15Þ

Accordingly, Equation (4.14) can be reexpressed in terms of u as: @s þ b ¼ r€ u @x

4.1.3

ð4:16Þ

Energy conservation laws

Energy balance law for a 1-D case can be written for a typical infinitely small control volume: DðU þ KÞDxDyDz ¼ ððU þ KÞvÞx DyDzDt  ððU þ KÞvÞxþDx DyDzDtþ qx DyDzDt  qxþDx DyDzDt

ð4:17Þ

ðsvÞx DyDzDt þ ðsvÞxþDx DyDzDtþ ðbvÞDxDyDzDt þ QDxDyDzDt where U is internal energy, K is kinetic energy, v is velocity, q is flux, s is stress, b is body force, and Q is energy generated per unit volume per unit time. Energy ðU þ KÞxþDx and momentum ðsvÞxþDx can be expressed using the Taylor’s expansion as: ððU þ KÞvÞxþDx ¼ ððU þ KÞvÞx þ ðsvÞxþDx ¼ ðsvÞx þ

@ðsvÞ Dx þ 02 @x

@ððU þ KÞvÞ Dx þ 02 @x

ð4:18Þ ð4:19Þ

46 Fundamental governing equations

4

Dividing by DxDyDzDt and taking the limits yields the following: @ðU þ KÞ @ððU þ KÞvÞ @q @ðsvÞ ¼ þ þ þ ðbvÞ þ Q @t @x @x @x

ð4:20Þ

Expressing internal energy U and kinetic energy K as: U ¼ re;

1 K ¼ rv2 2

ð4:21Þ

and rearranging Equation (4.20) yields:    @ðe þ 12 v2 Þ @ðe þ 12 v2 Þ @r @ðrvÞ 1 2 @q @ðsvÞ þ rv ¼ þ þ eþ v þr þ ðbvÞ þ Q @t @x @t @x 2 @x @x ð4:22Þ where e is specific internal energy per unit mass. Using the mass conservation law and momentum balance law, Equation (4.22) becomes:
 @e @e @q @v r þv ¼ þs þQ ð4:23Þ @t @x @x @x Introducing the following operator: d @ @ ¼ þv dt @t @x

ð4:24Þ

Equation (4.23) can be rewritten as r

de @q @v ¼ þs þQ dt @x @x

ð4:25Þ

Denoting the gradient of velocity by strain rate as: ε_ ¼

@v @u_ ¼ ; @x @x

u_ ¼

dv dt

ð4:26Þ

Equation (4.25) is rewritten in the following form: r

de @q ¼  þ sε_ þ Q dt @x

ð4:27Þ

If we express free energy e by cT where c is specific heat capacity and T is temperature, we have the following form for energy balance law: r

dðcTÞ @q @v ¼ þs þQ dt @x @x

ð4:28Þ

If we further introduce Fourier’s law as a constitutive law between heat flux q and temperature T as: q ¼ k

@T @x

ð4:29Þ

5

Fundamental governing equations

47

Figure 4.3 Illustration of energy conservation law

Equation (4.28) takes the following well-known 1-D energy balance law in thermodynamics by assuming that heat conductivity coefficient and specific heat capacity are constant: rc

dT @2T ¼ k 2 þ sε_ þ Q dt @x

ð4:30Þ

See Figure 4.3. 4.1.5 4.1.5.1

Fundamental governing equations for coupled hydromechanical behavior 1-D mass conservation law for a mixture of solid and fluid

The mass conservation law for a mixture without mass generation is stated as it was previously: gained mass ¼ input flux  output flux Let us consider an infinitely small cubic element as shown in Figure 4.4. The final equation for mass conservation law of mixture for 1-D case is fundamentally same: @p @q ¼ @t @x

ð4:31Þ

However, the average density of mixture is defined as: r ¼ ð1  nÞrs þ nrf

ð4:32Þ

where n is the volume fraction of fluid, rf is density of fluid, and rs is density of solid. q ¼ rv ¼ ð1  nÞrs vs þ nrf vf

ð4:33Þ

Let us also define a relative velocity between the solid phase and fluid phase as: vr ¼ nðvf  vs Þ

or

nv ¼ vr  nvs

ð4:34Þ

The mass conservation laws for each constituent can be written as: Solid phase @ @ ðð1  nÞrs Þ ¼  ðð1  nÞrs vs Þ @t @x

ð4:35Þ

48 Fundamental governing equations

6

Defining the following operator: ds @ @ ¼ þ vs @x dt @t

ð4:36Þ

The preceding equation becomes: ds n ð1  nÞ ds rs @v þ ð1  nÞ s ¼ dt @x dt rs

ð4:38Þ

Fluid phase @ @ ðnrf Þ ¼¼  ðrf vr þ nrf vs Þ @t @x

ð4:39Þ

The preceding equation becomes: ds n n ds rf 1 @ðrf vr Þ @ ¼   n ðrf vf Þ dt rf dt rf @x @x

ð4:40Þ

Equating Equations (4.38) and (4.40), we have: ð1  nÞ ds rs @vs n ds rf 1 @ðrf vr Þ þ ¼  rs rf dt rf @x dt @x

ð4:41Þ

Let us write constitutive laws for the volumetric response of each constituent in the following forms: 1 ds rs 1 ds p ¼ rs dt Ks dt

ð4:42Þ

1 ds rf 1 ds p ¼ rf dt Kf dt

ð4:43Þ

Inserting these constitutive laws in Equation (4.41), we obtain: ! @εv ð1  nÞ n ds p 1 @  ðr v Þ ¼ þ Ks Kf dt rf @x f r @t

ð4:44Þ

where

 @vs @ @us @ @us @ε ¼ ¼ ¼ v @x @x @t @t @t @x

ð4:45Þ

Let us introduce Darcy’s law given by: vr ¼ 

k @p @p ¼ K m @x @x

ð4:46Þ

7

Fundamental governing equations

Thus Equation (4.44) becomes: !
 @εv ð1  nÞ n ds p 1 @ @p þ rK ¼ þ ks Kf dt rf @x f @x @t Some particular cases of the preceding equation are: Undrained condition and slow deformation: vr ¼ 0 and vs ¼ 0 ! @εv ð1  nÞ n @p @p ¼ Cu ¼ þ Ks Kf @t @t @t Drained condition and slow deformation: vs ¼ 0
 @εv ð1  nÞ @p @p ¼ Cd ¼ Ks @t @t @t

49

ð4:47Þ

ð4:48Þ

ð4:49Þ

If the solid and fluid are incompressible materials, Equation (4.42) or (4.43) becomes:
 @εv @v @ @p @2p K ¼K 2 ¼ r¼ ð4:50Þ @x @x @t @x @x Equation 4.50 becomes Terzaghi’s or Biot’s equation used in coupled problems in which densities are assumed to be constant. 4.1.5.2

1-D force conservation law for a mixture of solid and fluid

Total force equilibrium in terms of total stress without inertia form yields the following relation: @s þb¼0 @x

ð4:51Þ

Let us introduce the concept of effective stress as: s ¼ s0  ap

ð4:52Þ

where a is a physical nondimensional quantity. When a is equal to 1, it corresponds to Terzaghi’s effective stress law. On the other hand, if it is not equal to 1, it corresponds to Biot’s effective stress law. Accordingly, the equilibrium equation becomes: @s0 @p a þb¼0 @x @x

ð4:53Þ

Taking the time derivative of the preceding expression, we have: @s_0 @p_  a þ b_ ¼ 0 @x @x

ð4:54Þ

Equation 4.54 is used together with Equation 4.47 for the hydromechanical response of rock masses.

50 Fundamental governing equations

8

Under dynamic conditions, which require the consideration of inertia components, Equation (4.51) is replaced by the following equation: df vf @s dv þ rg ¼ ð1  nÞrs s s þ nrf @x dt dt

ð4:55Þ

@sf 1 þ rf g ¼ rf ⃛uf þ tsf n @x

ð5:56Þ

where s ¼ ð1  nÞss þ nsf n tsf ¼  vr K

ð4:57Þ ð4:58Þ

The coefficient K is called the hydraulic conductivity coefficient (wrongly called permeability in many publications), which appears in Darcy’s law. Equations (4.55) and (4.56) can be rewritten as: Total system @s @2u @2w þ rg ¼ r 2s þ rf 2 @x @t @t

ð4:59Þ

Fluid phase @sf @ 2 u rf @ 2 w 1 @w þ rf g ¼ rf 2s þ þ @x @t n @t2 K @t

ð4:60Þ

Where w ¼ nðvf  vs Þ. 4.1.5.3

1-D seepage

If fluid density does not change with position and no mass is gained or lost, the preceding equations can be rewritten as: @r @ ¼ r ð vÞ @t @x

ð4:61Þ

Although it is known that the relationship between average velocity ( v) and pressure gradient (@p=@x) becomes nonlinear with increasing values of Reynolds numbers, Darcy’s law is widely used for analyzing test results on soil and rock. Darcy’s law can be given as: v ¼ 

k dp Z dx

ð4:62Þ

where k is permeability, Z is viscosity, and p is pressure. Inserting Equation (4.62) into Equation (4.61) yields the governing equations for fluid flow as: @r rk @ @p ¼ ð Þ @t Z @x @x

ð4:63Þ

9

Fundamental governing equations

51

If the following relation exists between fluid density and pressure as: r ¼ cp

ð4:64Þ

Equation (4.63) takes the following form, which is the governing equation used in seepage: @p k @2p ¼ @t cZ @x2

or

@p @2p ¼b 2 @t @x

ð4:65Þ

k

where b ¼ cZ.

4.2

Multidimensional governing equations

4.2.1

Mass conservation laws for seepage and diffusion phenomena

Water is always present in rock mass, and it strongly affects the stability of rock engineering structures. Furthermore, rock excavations, which are generally of large scale, may disturb groundwater regime and may have some environmental impacts. In any case, the governing equation for seepage flow in rock mass is derived from the mass conservation law if the rock mass is considered a porous medium. The final form of the governing equation for seepage flow may be written as: S

@h ¼ r  q  Q @t

ð4:66Þ

where S; h; q and Q are the storativity coefficient, water head, flux vector and source or sink, respectively. r is the directional derivative operator. As for the diffusion phenomenon of a certain substance, one may obtain the governing equation as: d ¼ r  f  P ð4:67Þ dt where , f and P are concentration of substance, diffusion flux vector and source or sink, respectively. 4.2.2

Momentum conservation law

Momentum conservation law for rock mass can be derived as done in continuum mechanics, and the final form can be written as: r

@v ¼ r  σ þ b @t

ð4:68Þ

where r; v; σ and b are density, velocity, stress tensor and body force, respectively. 4.2.3

Angular momentum conservation law

The final form of the angular momentum equation indicates that the stress tensor is a symmetric tensor, that is: sij ¼ sji

ð4:69Þ

52 Fundamental governing equations

4.2.4

10

Energy conservation law

The energy conservation law of rock mass may be written in the following form: rc

@T ¼ r  qh þ σ  ε_ þ Qh @t

ð4:70Þ

where c; T; qh ; ε_ and Qh are specific heat, temperature, strain rate tensor and heat source or sink, respectively. 4.2.5

Fundamental equation of fluid flow in porous media

The mass conservation law for fluid flowing through the pores within rock may be given in the following form with the use of the mixture theory and assuming that a coordinate system is fixed to the solid phase (i.e. Aydan, 1998, 2001a, 2001b): @ðrf Þ ¼ r  ðqf Þ @t

ð4:71Þ

where r ¼ @x@ i ei , i ¼ 1; 3; ρf is fluid density,  is porosity, and qf is fluid flux. One may write the following relation for fluid flux in terms of relative velocity vr of the fluid and the velocity vs of solid phase as: qf ¼ rf ðvr þ vs Þ

ð4:72Þ

Let us assume that the flow of fluid obeys Darcy’s law. Thus we have: k vr ¼  rp Z

ð4:73Þ

where k is permeability, and η is viscosity of fluid. Inserting Equations (4.73) and (4.72) into Equation (4.71) yields the following: @ðrf Þ k ¼ r  ðrf ð rp  vs ÞÞ Z @t

ð4:74Þ

The material derivative operator according to Eulerian description may be written as (Eringen, 1980): ds @ ¼ þ vs  rðÞ dt @t

ð4:75Þ

Introducing this operator into Equation (4.74), we have the following relation: ds   ds rf k 1 þ ¼ r  ð rpÞ  r  ðvs Þ dt rf dt Z rf

ð4:76Þ

The following constitutive relations are assumed to hold among porosity, fluid and solid densities and pressure (i.e. Zimmerman et al., 1986): ds  d p 1 ds rf dp ¼ ðCb  ð1 þ ÞCs Þ s ; ¼ Cf s dt dt rf dt dt

ð4:77Þ

11

Fundamental governing equations

53

If the velocity of solid phase is assumed to be small so that it can be neglected, Equation (4.76) takes the following form with the use of Equation (4.77): b

@p ¼ r  ðrpÞ @t

ð4:78Þ

where b ¼ ½ðCb  Cs Þ þ ðCf  Cs Þ (a)

Z k

ð4:79Þ

Special forms of Equation 4.78 for longitudinal and axisymmetric cases

Equation (4.78) can be rewritten for one-dimensional longitudinal flow as: b

@p @ 2 p ¼ @t @x2

ð4:80Þ

Similarly Equation (5.8) can be also written for axisymmetric radial flow as:
 @p 1 @ @p ¼ r b @t r @r @r (b)

ð4:81Þ

Governing equations of fluid in reservoirs attached to sample

Using the mass conservation law and the constitutive relation between pressure and fluid density, the velocities v1, v2 of fluid contained in reservoirs numbered (1) and (2) and attached to the ends of a sample can be written as: v1 ¼ Cf V1

@p1 @p ; v2 ¼ Cf V2 2 @t @t

ð4:82Þ

where V1 and V2 are volumes of reservoirs, and p1 and p2 are pressures acting on the fluid reservoirs. 4.2.6

Modeling of water absorption/desorption processes and associated volumetric changes in rocks

Some rocks such fine-grain sandstone, mudstone and siltstone start to fracture when losing its water content, as observed in many laboratory tests and in-situ. The situation is similar to the reverse problem of swelling. It is considered that rock shrinks as it loses its water content. This consequently results in shrinkage strain, leading to the fracturing of rock in tension. Therefore, a coupled formulation of the problems is necessary. The water content variation in rock can be modeled as a diffusion problem. Thus the governing equation is written as: dy ¼ r  q þ Q dt

ð4:83Þ

where y; q, Q and t are water content, water content flux, water content source and time, respectively. If the water content migration obeys Fick’s law, the relation between flux q

54 Fundamental governing equations

12

and water content is written in the following form: q ¼ kry

ð4:84Þ

where k is the water diffusion coefficient. If some of water content is transported by the groundwater seepage or airflow in open space, this may be taken into account through the material derivative operator in Equation (4.83). However, it would be necessary to describe or evaluate the seepage velocity or airflow. If the stress variations occur at slow rates, the equation of motion without inertial term may be used in incremental form as: r  σ_ ¼ 0

ð4:85Þ

The simplest constitutive law for rock between stress and strain fields would be a linear law, in which the properties of rocks may be related to the water content in the following form (i.e. Aydan et al., 2006): σ_ ¼ DðyÞε_ e

ð4:86Þ

The volumetric strain variations associated with shrinkage (inversely swelling) may be related to the strain field in the following form: ε_ e ¼ ε_  ε_ s

4.2.7

ð4:87Þ

Thermo-mechanical modeling heat transport in rocks

The well-known governing equation of energy conservation (4.70), rewritten for porous media, takes the following form: rc

@T ¼ r  ðkrTÞ þ σ  ε_ þ Qh @t

ð4:88Þ

The incremental form of the equation of motion without inertia from Equation (4.68) is given by: r  σ_ ¼ 0

ð4:89Þ

The well-known equation of strain component induced by temperature variation is given by the following equation: ε_ T ¼ lDTI

ð4:90Þ

where I is the Kronecker delta tensor. The constitutive law in terms of net-strain is generally written in the following incremental form: σ_ ¼ Dðε_  ε_ T Þ

4.3

ð4:91Þ

Derivation of governing equations in integral form

In this section, fundamental conservation laws are derived using the approach of integral form.

13

Fundamental governing equations

4.3.1

55

Mass conservation law

Mass is defined as: Z m ¼ rðx; tÞdO

ð4:92Þ

O

Mass conservation law requires: Z Z dm d dðdOÞ ¼ rdO þ r dt dt dt O

ð4:93Þ

O

as dO ¼ JdOo

and

dðdOÞ ¼¼ ðr  xÞdO dt

ð4:94Þ

With the use of the Reynolds transport theorem (Appendices 4 and 6) together with Equation (4.94), Equation (4.93) becomes:  Z
dm dr ¼ þ rr  v dO ð4:95Þ dt dt O

To satisfy this condition, the integrand should be zero, so that we have the following: dr þ rr  v ¼ 0 dt

ð4:96Þ

The time derivative in Lagrangian and Eulerian descriptions are given in the following forms: Lagrangian description dðÞ @ðÞ ¼ dt @t

ð4:97aÞ

Eulerian description dðÞ @ðÞ ¼ þ v  rðÞ dt @t

ð4:97bÞ

With the use of Equation (4.97a) in Equation (4.96), we obtain the following relation: @r þ v  rr þ rr  v ¼ 0 @t

or

@r þ r  ðrvÞ ¼ 0 @t

ð4:98Þ

Using index notation, we have: v ¼ vk ek

and

r¼

@ e @xk k

ð4:99Þ

56 Fundamental governing equations

14

Equation 4.98 can be rewritten as: @r @ðrvk Þ @r þ þ ðrvk Þ;k ¼ 0 ¼ @t @xk @t if

dr dt

ð4:100Þ

¼ 0 media is incompressible so that rðr  vÞ ¼ 0 or r  v ¼ 0.

4.3.2

Momentum conservation law

The definition of momentum is: Z p ¼ rvdO

ð4:101Þ

O

Preliminary relations Z Z dðdOÞ ¼¼ ðr  xÞdO; t ¼ σ  n r  σdO ¼ σ  ndG; dt O

ð4:102Þ

G

Conservation of momentum is written in the following form in view of Equation (4.102), which is also known Reynolds transport theorem: Z Z Z d rvdO ¼ tdG þ bdO ð4:103Þ dt O

G

O

Equation (4.103) may be rewritten as:  Z
Z Z dðrvÞ þ ðrvÞr  v dO ¼ r  σdO þ þ bdO dt O

O

O

Carrying out the derivation in Equation (4), we have the following:   Z Z Z

dr dv þ rðr  vÞ v dO þ r dO ¼ ðr  σ þ bÞdO dt dt O

O

ð4:104Þ

ð4:105Þ

O

The first term on left-hand side disappears by virtue of mass conservation law, and the equation takes the following form: Z Z dv r dO ¼ ðr  σ þ bÞdO ð4:106Þ dt O

O

Equation (4.106) may be rewritten as:  Z  dv r  ðr  σ þ bÞ dO ¼ 0 dt

ð4:107Þ

O

To satisfy Equation (4.107), the integrand should be zero, so that we have the following relation: dv r ¼rσþb ð4:108Þ dt

15

Fundamental governing equations

57

Furthermore, the derivation on the left-hand side may be related to acceleration or displacement vectors as follows: dv ¼a dt

4.3.3

dv d 2 u ¼ 2 dt dt

or

ð4:109Þ

Angular momentum conservation law

Some preliminary relations are: Z Z Z dr v  v ¼ 0; r  ndG ¼ r  rdO ¼ IdO; r  σ  n ¼ ðr  σÞ  n; ¼ v dt G O Z Z Z O r  tdG ¼ r  ðσ  nÞdG ¼ r  ðr  σÞdO G

G

Z

Z

O

ðr  rÞ  σdOþ

¼

r  ðr  σÞdO

O

O

O

G

dr  rvdO þ dt Z

Z O

dðrvÞ dO þ r dt Z

O

Z

Z v  rvdO þ

r O

Z

r  ðrvÞ O

dðdOÞ dt

ð4:113Þ

r  ðr  σ þ bÞdO


 Z dr dv vþr dO þ ðr  rvÞr  vdO dt dt

O

ð4:114Þ

O

Z

ðr  rÞ  σdO þ

¼

Z

O

or

O

ð4:112Þ

O

ðr  rÞ  σdO þ

¼

ð4:111Þ

O

Angular momentum law requires the following condition to be valid: Z Z Z d r  rvdO ¼ r  tdG þ r  bdO dt Z

ð4:110Þ

r  ðr  σ þ bÞdO O

Thus

 Z Z Z
dr dv v  rvdO þ r  r  r  r  σ  b dO ð4:115Þ þ rr  v vdO þ dt dt O

Z

O

ðr  rÞ  σdO ¼ 0

 O

O

58 Fundamental governing equations

16

By virtue of the mass conservation law, momentum conservation law, and preliminary relations, Equation (4.115) reduces to the following form: Z Z ðr  rÞ  σdO ¼ ε  σdO ¼ 0 ð4:116Þ O

O

where ε is known as the permutation symbol (see Appendix 1), and it is a rank 3 tensor. It is given in index notation as: εijk sjk ¼ 0

ð4:117Þ

The rank 3 permutation tensor has the following properties: 8 i 6¼ j 6¼ k; ; 1; 2; 3; 1; 2ðεijk ¼ 1Þ > < i 6¼ j 6¼ k; ; 3; 2; 1; 3; 2ðεijk ¼ 1Þ εijk ¼ > : i ¼ j 6¼ k; i ¼ k 6¼ j; i 6¼ j ¼ k; εijk ¼ 0

ð4:118Þ

With this property, it is shown that the stress tensor is symmetric: sjk ¼ skj

4.4.4

ð4:119Þ

Energy conservation law

The time variation of internal and kinetic energies of a body with a given volume and surface area should be equal to the energy input from surface traction and body force and heat flux and volumetric heat production: d ðU þ KÞÞ ¼ W þ H dt

ð4:120Þ

where Z U¼

redO; K ¼ O

1 2

Z

Z

Z

O

q  ndG 

H¼ G

Z rv  vdO; W ¼

Z t  vdG þ

G

b  vdO;

ð4:121Þ

O

QdO O

With the use of Equation (4.121), one may write the following relations:  Z
dU dðreÞ ¼ þ rer  v dO dt dt

ð4:122Þ

O

dK 1 ¼ dt 2

 Z
dðrvÞ dv  v þ rv  þ ðrv  vÞr  v dO dt dt O

ð4:123Þ

17

Fundamental governing equations

Z

59

Z

W¼

ðr  σ þ bÞ  vdO þ O

σ : rvdO

ð4:124Þ

O

Z ðr  q  QÞdO

H¼

ð4:125Þ

O

Inserting Equation results in: Z d 1 ðU þ KÞÞ ¼ dt 2

O

(4.122) to Equation (4.123) into the left-hand side of Equation (4.120) dr ð þ rr  vÞv  vdO þ dt

Z O

dr ð þ rr  vÞedO þ dt

Z O

de r dO þ dt

Z rv  O

dv dt

ð4:126Þ Inserting Equations (4.124) and (4.125) into the right-hand side of Equation (4.120) and requiring that the mass conservation and momentum conservation laws are satisfied, the final for Equation (4.120) takes the following form:  Z
de ð4:127Þ r  σ : rv þ r  q  Q dO ¼ 0 dt O

To satisfy the Equation (4.127), the integrand should be zero so that the following relation is obtained: r

de ¼ r  q þ σ : rv  Q dt

ð4:128Þ

As ε_ ¼ rv, Equation (4.128) is rewritten as: r

de ¼ r  q þ σ : ε_  Q dt

ð4:129Þ

If internal energy e is related to temperature (T) together with the specific heat coefficient c as cT, Equation (4.129) becomes: rc

dT ¼ r  q þ σ : ε_  Q dt

ð4:130Þ

Equation (4.130) is known as the first law of thermodynamics.

References Aydan, Ö. (1998) Finite element analysis of transient pulse method tests for permeability measurements. The 4th European Conf. on Numerical Methods in Geotechnical EngineeringNUMGE98, Udine. pp. 719–727. Aydan, Ö. (2001a) Modelling and analysis of fully coupled hydro-thermo-diffusion phenomena. Int. Symp. On Clay Science for Engineering, Balkema, IS-SHIZUOKA. pp. 353–360. Aydan, Ö. (2001b) A finite element method for fully coupled hydro-thermo-diffusion problems and its applications to geoscience and geoengineering. 10th IACMAG Conference, Austin.

60 Fundamental governing equations

18

Aydan, Ö., Daido, M., Tano, H., Nakama, S. & Matsui, H. (2006) The failure mechanism of around horizontal boreholes excavated in sedimentary rock. 50th US Rock mechanics Symposium, Paper No. 06-130 (on CD). Eringen, A.C. (1980) Mechanics of Continua. Robert E. Krieger Publishing Co., Huntington, NY, 606p. Mase, G. (1970) Theory and Problems of Continuum Mechanics, Schaum Outline Series. McGraw Hill Co., New York, 230p. Zimmerman, R.W., Somerton, W.H. & King, M.S. (1986) Compressibility of porous rocks. Journal of Geophysical Research Atmospheres, 91, 12765–12777.

Chapter 5

Constitutive laws

The fundamental governing equations cannot be solved in their original forms as the number of equations is lower than the number of variables to be determined. If constitutive laws, which are fundamentally determined from experiments, are introduced, their solution becomes possible. In this chapter, the well-known constitutive laws are introduced

5.1

One-dimensional constitutive laws

5.1.1

1-D Linear constitutive laws

In this subsection, linear constitutive laws heat, seepage, diffusion and mechanical behavior of rocks and rock masses are given. 5.1.1.1

Fourier’s law

Fourier’s law states that heat flux q is linearly proportional to the gradient of temperature T, that is (Fig. 5.1): q ¼ k

@T @x

ð5:1Þ

where k is called thermal heat conductivity. 5.1.1.2

Fick’s law

Fick’s law is essentially the same as that of Fourier’s, and it is used in diffusion problems. Fick’s law states that mass flux q is linearly proportional to the gradient of mass concentration C, that is (Fig. 5.2): q ¼ k

@C @x

ð5:2Þ

where k is called the diffusion coefficient. 5.1.1.3

Darcy’s law

Darcy’s law is also essentially the same as that of Fourier’s, and it is used in ground-water seepage problems. Darcy’s law states that seepage flux q is linearly proportional to the

62 Constitutive laws

2

Figure 5.1 One-dimensional illustration of Fourier’s law

Figure 5.2 One-dimensional illustration of Fick’s law

gradient of groundwater pressure p, that is (Fig. 5.3): q ¼ k

@p @x

ð5:3Þ

where k is called the hydraulic conductivity, and it is sometimes wrongly used as the permeability coefficient.

3

Constitutive laws

63

Figure 5.3 One-dimensional illustration of Darcy’s law

Figure 5.4 One-dimensional illustration of Hooke’s law

5.1.1.4

Hooke’s law

Hooke’s law is used in the theory of elasticity of solids. Hooke’s law states that stress s is linearly proportional to strain ε, that is (Fig. 5.4 and Fig. 5.6(a)): s ¼ Eε where E is the elasticity modulus.

ð5:4Þ

64 Constitutive laws

4

Figure 5.5 One-dimensional illustration of Newton’s law

Figure 5.6 Simple rheological models

5.1.1.5

Newton’s law

Newton’s law (Fig. 5.5 and Fig. 5.6(b)) is linear and given in the following form: s ¼ Zε_

ð5:5Þ

If this law is integrated over the time, it takes the following form with a condition, that is, ε ¼ 0 at t = 0: s ε¼ t Z

ð5:6Þ

5

Constitutive laws

65

If we assume that the strain rate is given in the following form: ε_ ¼

s Z

ð5:7Þ

The preceding equation can be written as: _ ε ¼ εt

ð5:8Þ

This has a similarity to the steady-state creep response. 5.1.1.6

Maxwell’s law

Substance in Maxwell’s law (Fig. 5.6(c)) is assumed to consist of elastic and viscous components connected in series. Therefore, total strain and its derivative are given as: ε ¼ εe þ εv and ε_ ¼ ε_ e þ ε_ v

ð5:9Þ

The constitutive relations for elastic and viscous responses are: εe ¼

s s and ε_ v ¼ E Z

ð5:10Þ

If s ¼ so for t  0 and ε ¼ εo with εo ¼ so =E, the preceding function becomes: ε¼

so so þ t E Z

ð5:11Þ

This equation also has a similarity to the steady-state creep response. 5.1.1.7

Kelvin-Voigt law

Substance in the Kelvin-Voigt law (Fig. 5.6(d)) is assumed to be elastic, and viscous components are connected in parallel. Therefore, total stress is given as: s ¼ Eε þ Zε_

ð5:12Þ

If stress is applied so at t = 0 with ε ¼ 0 and is sustained thereafter, the following relation is obtained: ε¼

so E ð1  et=tr Þ with tr ¼ Z E

ð5:13Þ

It is interesting to note that the preceding response is similar to the transient creep stage. Figure 5.7 shows the creep strain responses for different simple rheological models. 5.1.1.8

Generalized Kelvin model

The model (Fig. 5.8(a)) has a Hookean element and a Kelvin element connected in series. The total strain of the model is: ε ¼ εh þ εk

ð5:14Þ

66 Constitutive laws

6

Figure 5.7 Creep strain response of simple rheological models

Figure 5.8 More complex rheological models

The stress relations of each element are given as: εh ¼

s and s ¼ Ek εk þ Zε_ k Eh

Thus, one gets the following equation:

 s s_ _ s¼Z ε þ Ek ε  Eh Eh

ð5:15Þ

ð5:16Þ

If stress so is applied at t = 0 with ε ¼ 0 and εe ¼ so =Eh and is sustained thereafter, the following relation is obtained: ε¼

so so Z þ ð1  et=tr Þ with tr ¼ Ek Eh Ek

ð5:17Þ

7

Constitutive laws

67

As noted from this relation, instantaneous strain due to elastic response and transient creep stage can be modeled. 5.1.1.9

Zener model

The Zener model (Fig. 5.8(b)) is also known as the standard linear solid model, and it consists of a Hooke element and Maxwell element connected to each other in parallel. Total stress may be given in the following form: s ¼ sh þ sm

ð5:18Þ

The constitutive laws of Hooke and Maxwell elements are: sh ¼ Eh ε; ε ¼ εs þ εd ; ε_ ¼ ε_ s þ ε_ d ; ε_ s ¼

s s_ m ; ε_ ¼ m Em d Z m

Thus, one can easily get the following differential equation:
 dε 1 Eh Em 1 ds Em þ  þ s ε¼ dt Zm Eh þ Em Eh þ Em dt Zm

ð5:19Þ

ð5:20Þ

If stress is applied so at t = 0 with and εo ¼ so =ðEh þ Em Þ and is sustained thereafter, the following relation is obtained:   s Em E þ Eh et=tr with tr ¼ Zm m ð5:21Þ ε¼ o 1 Eh Eh þ Em Em Eh The creep response to be determined from this model involves the instantaneous strain and transient creep. 5.1.1.10

Burgers model

Burgers model (Fig. 5.8(c)) consists of Maxwell and Kelvin elements connected to each other in series. The constitutive relations for each element can be given as: ε_ m ¼

s s_ þ and s ¼ Ek εk þ Zε_ k Em Zm

ð5:22Þ

The total strain is given by: ε ¼ εm þ εk

ð5:23Þ

If stress is applied so at t = 0 with ε ¼ 0 and εm ¼ so =Em and is sustained thereafter, the following relation is obtained: ε¼

so so s Z þ ð1  et=tk Þ þ o t with tk ¼ k Em Ek Zm Ek

ð5:24Þ

As noted, this model can simulate the instantaneous strain due to elastic response and transient and steady state creep stages. Figure 5.9 shows and compares the creep strain responses for different more complex rheological models.

68 Constitutive laws

8

Figure 5.9 Creep responses from more complex rheological models

Figure 5.10 Comparison rheological models with experimental responses

Figure 5.10 compares the experimental responses with those from intuitive and rheological models. As noted from these figures, each model has its own merits and drawbacks (Aydan, 2016). 5.1.2 5.1.2.1

1-D nonlinear constitutive laws for solids Elasto-plastic law

In the following discussion, a constitutive law is derived based on the concepts of classical plasticity theory. The elasto-plastic behavior of materials is illustrated in Figure 5.11. The classical plasticity theory is based upon the following assumptions. •

Yield function is of the following form: Fðs;kÞ ¼ f ðsÞ  KðkÞ ¼ 0

•

ð5:25Þ

Flow rule, given here, holds: dεp ¼ l

@G @s

ð5:26Þ

9

Constitutive laws

69

Figure 5.11 Illustration of elasto-plastic behavior and some fundamental parameters

•

Prager’s consistency condition, given here, holds: dF ¼

•

@F @F @k p ds þ dε ¼ 0 @s @k @εp

ð5:27Þ

Strain increment is a linear sum of elastic and plastic increments: dε ¼ dεe þ dεp

•

ð5:28Þ

For elastic component, Hooke’s law holds: ds ¼ De dεe

ð5:29Þ

Inserting Equation (5.26) into Equation (5.27) and denoting the following by h: h¼

@F @k @G @k @εp @s

ð5:30Þ

yields the following between plastic strain increment dεp and stress increment ds as dεp ¼

1 @G @F ð dsÞ ¼ C p ds h @s @s

ð5:31Þ

The preceding relation is called Melan’s formula. The following can be written: 1 @G @F ð dsÞ ds ¼ De dε  De h @s @s Multiplying the preceding expression by @F De dε @F @s ds ¼ 1 þ 1h @F @s ðDe @G Þ @s @s

ð5:32Þ @F yields: @s

ð5:33Þ

70 Constitutive laws

10

Utilizing the preceding relation in Equation (5.32) yields the incremental elasto-plastic law as:
 @F De @G De @s @s dε ð5:34Þ ds ¼ De  h þ @F ðDe @G Þ @s @s

5.1.2.2

Visco-plastic models

(A) BINGHAM MODEL-ELASTIC PERFECTLY VISCO-PLASTIC MODEL

The visco-plastic model of the Bingham type assumes that the material behaves elastically below the yield stress level and visco-plastically above the yield stress level, given as: ε¼

s if s < so E

ð5:35Þ

ε¼

s  so s t þ if s > so E Z

ð5:36Þ

This equation corresponds to the perfectly visco-plastic material if so corresponds to yield threshold value of stress. Furthermore, the fluidity coefficient is defined as: g¼

1 Z

ð5:37Þ

(B) ELASTIC-VISCO-PLASTIC MODEL OF HARDENING TYPE (PERZYNA TYPE)

Elastic-visco-plastic model of hardening type (Perzyna type) (Fig. 5.12) assumes that the material behaves elastically below the yield stress level and visco-plastically above the yield stress level sY . The yield strength of visco-plastic material in relation to the viscoplastic strain of hardening type can be written as: Y ¼ sY þ Hεvp

Figure 5.12 Elastic-visco-plastic model

ð5:38Þ

11

Constitutive laws

71

Furthermore, total strain is assumed to be a sum of elastic strain and visco-plastic strain as: ε ¼ εe þ εvp

ð5:39Þ

Thus the stress–strain relations are given in the following form: sp ¼ s ¼ Eε if sp < Y

ð5:40Þ

sp ¼ sY þ Hεvp if sp > Y

ð5:41Þ

Total stress at any time can be written as: s ¼ sp þ s d

ð5:42Þ

The viscous component of stress is related to the visco-plastic strain rate as follows: sd ¼ Cp

dεvp dt

ð5:43Þ

Thus, one can obtain the following differential equation for visco-plastic response: s ¼ sY þ Hεvp þ Cp

dεvp dt

ð5:44Þ

Replacing the visco-plastic strain with the use of total strain and elastic strain in the preceding equation, one can easily obtain the following relation: H Eε þ

1 dε 1 ds E ¼ H s þ Eðs  sY Þ þ Cp dt Cp dt

ð5:45Þ

Let us assume that a constant stress s ¼ sA is applied and kept constant (creep test). The preceding differential equation is reduced to the following form: dε H H 1 þ ε¼ sA þ ðsA  sY Þ dt E Cp E Cp

ð5:46Þ

The solution of the differential equation is obtained as follows: ε ¼ Ce

CH t p

þ

1 1 sA þ ðsA  sY Þ E H

ð5:47Þ

when t = 0, ε ¼ εe ¼ sA =E. The final form of the preceding equation becomes: ε¼

 sA ðsA  sY Þ  H t 1  e Cp þ H E

ð5:48Þ

Figure 5.13 shows the elastic-visco-plastic strain response for visco-plastic hardening and Bingham-type visco-plastic behaviors.

72 Constitutive laws

12

Figure 5.13 Responses obtained from elastic-visco-plastic models (C) ELASTO-VISCO-PLASTIC MODEL OF HARDENING TYPE

Instead of using the elasticity model for linear (recoverable) response, some of the rheological models described in the previous section can be adopted. For the nonlinear (permanent) response, the models described can be utilized. For example, if the linear response is modeled using the Kelvin-Voigt–type model, the following relation would hold for the linear part (s < sy ): εr ¼

s and s ¼ Eεr þ Z ε_ r with ε ¼ εr E

ð5:49Þ

As for the nonlinear (permanent) part s  sy , the following can be written: s ¼ sY þ Hεp þ Cp

dεp dt

ð5:50Þ

Total strain is assumed to consist of linear (recoverable) and nonlinear (permanent) components: ε ¼ εr þ εp

ð5:51Þ

(D) AYDAN-NAWROCKI–TYPE ELASTO-VISCO-PLASTIC CONSTITUTIVE LAW

In an analogy to the derivation of an incremental elasto-plastic constitutive law, we start with the following: •

Yield function Fðs;kp ;kv Þ ¼ f ðsÞ  Kðkp ;kv Þ ¼ 0

•

ð5:52Þ

It should be noted here that the yield function is a function of permanent plastic and visco-hardening parameters (Fig. 5.14). Flow rule dεp ¼ l

@G ; @s

@G @G_ dε p ¼ l_ þl @s @s

ð5:53Þ

13

Constitutive laws

73

Figure 5.14 The elasto-visco-plastic model for one-dimensional response

•

Prager’s consistency condition dF ¼

•

@F @F @kp @F @kv  ds þ  dεp þ  dεp ¼ 0 @s @kp @ε @kv @εp

Linear decomposition of the strain increment dε and strain rate increment dε_ into their reversible and permanent components dεr and dεp dε ¼ dεr þ dεp ;

•

ð5:54Þ

dε_ ¼ dε_ r þ dε_ p ;

ð5:55Þ

Voigt-Kelvin’s law ds ¼ Dr dεr þ C r dεr

ð5:56Þ

where s is the stress tensor, Kðkp ; kv Þis the hardening function, G is the plastic potential, l is the proportionality coefficient, kp is the plastic hardening parameter, kv is the viscos hardening parameter, ε is the strain tensor, εr is the reversible strain tensor, εr is the reversible strain rate tensor, εp is the permanent strain tensor, εp is the permanent strain rate tensor, Dr is the elasticity tensor, Cr is the viscosity tensor, and () denotes the dot product. In elastic-visco-plastic formulations of the Perzyna type, flow rule is always assumed to be of the following form: @G dεp ¼ l_ @s

ð5:57Þ

The preceding equation implies that any plastic straining is always time dependent.

74 Constitutive laws

14

Here, permanent strain rate increment, given by Equation (5.53), is simplified to the following form by assuming that l_ ¼ 0: dεp ¼ l

@G_ @s

ð5:58Þ

The preceding equation implies that the plastic potential function shrinks (or expands) in time domain while keeping its original form in stress space.1 Substituting Equations (5.53) and (5.58) into Equation (5.54) and rearranging the resulting expression, together with the denotation of its denominator by hrp (hardening modulus): " # @F @kp @G @F @kv @G_ þ ð5:59Þ   hrp ¼  @kp @εp @s @kv @εp @s we have l as: l¼

1 @F  ds hrp @s

ð5:60Þ

Now let us insert the preceding relation into Equations (5.53) and (5.58). We have the constitutive relations between the permanent strain increment dεp , the permanent strain rate increment dε_ p , and the stress increment ds as: dεp ¼

1 @G @F 1 @G @F ð  dsÞ ¼  Þds ð hrp @s @s hrp @s @s

ð5:61Þ

dε_ p ¼

1 @G_ @F 1 @G_ @F ð  dsÞ ¼  Þds ð hrp @s @s hrp @s @s

ð5:62Þ

where ðÞ denotes the tensor product. The inverse of the preceding relations could not be obtained, that is, whether the plastic potential G is of the associated or nonassociated type. _ Therefore, the following technique is used to establish the relation between ds and dε, dε. Using the relations (5.55), (5.56), (5.61) and (5.62), one can write the following: ds ¼ Dr dε  Dr

1 @G @F 1 @G_ @F ð  dsÞ þ C r dε  C r ð  dsÞ hrp @s @s hrp @s @s

ð5:63Þ

Taking the dot products of the both sides of the preceding expression by @F=@s yields: @F @F  ðDr dεÞ þ @s  ðCr dεÞ @F @s    ds ¼ @F @F @s þ h1rp @s  Dr @G  Cr 1 þ h1rp @s @s

@G_ @s



ð5:64Þ

Substituting the preceding relation in (5.63) gives the incremental elasto-visco-plastic constitutive law as: ds ¼ Drp dε þ C rp dε

ð5:65Þ

1 It should be noted that permanent strain rate increment consists of time-independent and time-dependent parts.

15

Constitutive laws

75

where Drp ¼ Dr 

@G

@F

@G_

@F

Dr @s  @s Dr
   @F @G @F @G_ r r hrp þ  D @s þ @s  C @s @s

C

rp

C e @s  @s Cr
 ¼ C    @F @G @F @G_ r r hrp þ  D @s þ @s  C @s r

@s

5.2

Multidimensional constitutive laws

5.2.1

Fourier’s law

Fourier’s law states that heat flux qi is linearly proportional to the gradient of temperature T, that is: qi ¼ Kij

@T @xi

ð5:66Þ

where Kij is called the thermal heat conductivity tensor 5.2.2

Fick’s law

Fick’s law is essentially the same as that of Fourier’s, and it is used in diffusion problems. Fick’s law states that mass flux qi is linearly proportional to the gradient of mass concentration C, that is: qi ¼ Dij

@C @xi

ð5:67Þ

where Dij is called the diffusion coefficient tensor 5.2.3

Darcy’s law

Darcy’s law is essentially the same as that of Fourier’s, and it is used in groundwater seepage problems. Darcy’s law states that seepage flux qi is linearly proportional to the gradient of groundwater pressure p, that is: qi ¼ Kij

@p @xi

ð5:68Þ

where Kij is called the hydraulic conductivity tensor. 5.2.4

Hooke’s law

When rock or rock mass behaves linearly without any rate dependency, the simplest constitutive law is Hooke’s law. This law is written in the following form: sij ¼ Dijkl εkl where sij ; εkl and Dijkl are stress, strain and elasticity tensors, respectively.

ð5:69Þ

76 Constitutive laws

16

If material is homogeneous and isotropic, Equation (5.69) may be written as: sij ¼ 2mεij þ ldij εkk

ð5:70Þ

Where dij is Kronecker delta tensor, l and m are Lamé coefficients, which are given in terms of elasticity (Young’s) modulus (E) and Poisson’s ratio (u) as: l¼ 5.2.5

Eu E ;m ¼ ð1 þ uÞð1  2uÞ 2ð1 þ uÞ

ð5:71Þ

Newton’s law

Newton’s law is used in fluid mechanics. Newton’s law states that stress sij is linearly proportional to strain rate ε_ kl , that is: sij ¼ Cijklε_ kl

ð5:72Þ

where Cijkl is the viscosity tensor. 5.2.6

Kelvin-Voigt’s law

Kelvin-Voigt’s law is used in the field of visco-elasticity. Voigt’s law states that stress sij is linearly proportional to strain εij and strain rate ε_ ij , that is: sij ¼ Dijkl εkl þ Cijklε_ kl

ð5:73Þ

where ε_ kl and Cijkl are the strain rate and viscosity tensors, respectively. If material is homogeneous and isotropic, Equation (5.73) may be written in analogy to Equation (5.70) as: sij ¼ 2mεij þ ldij εkk þ 2m  ε_ ij þ l  dijε_ kk

ð5:74Þ

Coefficients l and m may be called viscous Lamé coefficients. 5.2.7

Navier-Stokes law

Navier-Stokes constitutive law can be visualized as a simple case of Kelvin-Voigt’s law. If the material behavior is associated with volumetric strain (εv ) and pressure (p) under the steady-state condition without any shear resistance like fluids, Equation (5.74) can be rewritten as: sij ¼ pdij þ 2m  ε_ ij þ l  dijε_ kk

ð5:75Þ

If the coefficient l ¼ 0, the preceding relation reduces to the following form: sij ¼ pdij þ 2m  ε_ ij or sij ¼ pdij þ m  g_ij

ð5:76Þ

The preceding constitutive law corresponds to the constitutive law commonly used in fluid mechanics known as Navier-Stokes law. There are different visco-elasticity models in literature. As all the models cannot be covered here, the reader is advised to consult the available literature on the topic of

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77

visco-elasticity (e.g. Farmer, 1983; Jaeger and Cook, 1979; Mirza, 1978; Owen and Hinton, 1980; Serata et al., 1968).

5.3

Nonlinear behavior (elasto-plasticity and elasto-visco-plasticity) for solids

Every material in nature starts to yield under a certain stress or strain state and rock or rock mass is a no exception. The terms used to describe the material behavior such as “elasticity” and “visco-elasticity” are replaced by the terms of “elasto-plasticity” or “elasto-visco-plasticity” as soon as material behavior deviates from linearity. The relation between total stress and strain or strain rate can no longer be used and every relation must be written in incremental form. For example, if the conventional plasticity models are used, the elasto-plastic constitutive law between incremental stress and strain tensors takes the following form: 5.3.1

Elasto-plastic law

The derivation of an incremental elasto-plastic constitutive law, based on the conventional elasto-plastic theory, starts with the following: •

Yield function Fðσ;kÞ ¼ f ðσÞ  KðkÞ ¼ 0

•

Flow rule dεp ¼ l

•

@G @σ

@F @F @k  dσ þ  dεp ¼ 0 @σ @k @εp

ð5:79Þ

Linear decomposition of the strain increment dε into its elastic and plastic components dεe and dεp dε ¼ dεe þ dεp

•

ð5:78Þ

Prager’s consistency condition (Drucker and Prager, 1952) dF ¼

•

ð5:77Þ

ð5:80Þ

Hooke’s law dσ ¼ De dεe

ð5:81Þ

where σ is the stress tensor, KðkÞ is hardening function, G is plastic potential, l is proportionality coefficient, k is hardening parameter, dε is strain tensor, dεe is elastic strain tensor, dεp is plastic strain tensor, De is elasticity tensor, and () denotes the dot product. Substituting Eqn. (5.78) into Eqn. (5.79) and rearranging the resulting expression together with the denotation of its denominator by h (hardening modulus): h¼

@F @k @G  @k @εp @σ

ð5:82Þ

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18

we have l as: l¼

1 @F  dσ h @σ

ð5:83Þ

Now, let us insert the preceding relation into Eqn. (24). We have the constitutive relation between the plastic strain increment dεp and the stress increment dσ, which is also known as Melan’s formula, as: dεp ¼

1 @G @F 1 @G @F ð  dσÞ ¼ ð  Þdσ ¼ Cp dσ h @σ @σ h @σ @σ

ð5:84Þ

where ðÞ denotes the tensor product. The inverse of the preceding relation cannot be obtained as the determinant of the plasticity matrix is jCp j ¼ 0j, irrespective of whether the plastic potential G is of the associated or nonassociated type. Therefore, the following technique is used to establish the relation between dε and dσ. Using the relations (5.82), (5.83) and (5.84), one can write the following: dσ ¼ De dε  De

1 @G @F ð  dσÞ h @σ @σ

ð5:85Þ

Taking the dot products of the both sides of the preceding expression by @F=@σ yields: @F

 ðDe dεÞ @F @σ    dσ ¼ 1@F @G @σ 1þ  De h@σ

ð5:86Þ

@σ

Substituting the preceding relation in (5.85) gives the incremental elasto-plastic constitutive law as: 0

1 @F e e @G dD  D @σ  @σ Adε dσ ¼ @De  @F @G h þ @σ  De @σ

ð5:87Þ

The hardening modulus h is generally determined as a function of a hardening parameter k by employing either a work-hardening model or a strain-hardening model. The hardening parameter k is defined for both cases as follows: Z k ¼ Wp ¼

σ  dεp work-hardening

ð5:88Þ

Z k¼

jjdεp jj

strain hardening

ð5:89Þ

where W p is plastic work. The materials (i.e. steel, glass fibers) exhibit a nondilatant plastic behavior and isotropically harden. Therefore, a work-hardening model is generally used together with the

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Constitutive laws

effective stress–strain concept, defined as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 p ðs  sÞ dεe ¼ ðdεp  dεp Þ se ¼ 2 3

79

ð5:90Þ

where s is the deviatoric stress tensor. As the volumetric plastic strain increment d εpv ¼ 0, together with the coaxiality of the stress and plastic strain, the hardening parameter k of work-hardening type can be rewritten in the following form: dk ¼ dW p ¼ σ  dεp ¼ s  dep ¼ se dεpe

ð5:91Þ

where dep is the deviatoric strain increment. The hardening modulus h for this case takes the following form with the use of Euler’s theorem2 by taking a homogeneous plastic potential G of order m: h¼

@F @k @G @K @G @K ¼ ¼m  σ G p p @k @ε @σ @W @σ @W p

ð5:92Þ

If F ¼ G and f ðσÞ ¼ se , then the hardening modulus h becomes: h¼m

@K @s f ðσÞ ¼ m e @W p @εe

ð5:93Þ

The hardening modulus can then be easily obtained from a gradient of the plot of a uniaxial test in s1 and εp1 space, since the effective stress and strain in the uniaxial state become: se ¼ s1 5.3.2

εpe ¼ εp1

ð5:94Þ

Elastic-visco-plasticity

These approaches assume that the materials are assumed to be elastic before yielding and behave in a visco-plastic manner following yielding. In visco-plastic evaluations, ep is replaced by evp. 5.3.2.1

Power-type models

When Norton-type constitutive law is used for creep response, the visco-plastic strain rate (evp ¼ εvp ) is expressed as follows:
n dεvp seq @seq ¼ ð5:95Þ dt so @σ Perzyna-type elastic-visco-plastic laws are used for representing nonlinear rate dependency involving plasticity: dεvp ¼ ls dt 2 x@f/@x = mf.

ð5:96Þ

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20

where l is proportionality coefficient and is interpreted as the fluidity coefficient. This parameter is obtained from uniaxial creep experiments as: l¼

5.3.2.2

ε_ c s

ð5:97Þ

Elasto-visco-plasticity

Another approach was proposed by Aydan and Nawrocki (1998), in which the material behavior is visco-elastic before yielding and becomes visco-plastic after yielding. The derivation of this constitutive law involves the following: Yield function Fðσ; kp ; kv Þ ¼ f ðσÞ  Kðkp ; kv Þ ¼ 0

ð5:98Þ

It should be noted that the yield function is a function of permanent plastic and visco-hardening parameters (Fig. 5.14). Flow rule dεp ¼ l

@G @G @G_ ; dε_ p ¼ l_ þl @σ @σ @σ

ð5:99Þ

Prager’s consistency condition dF ¼

@F @F @kp @F @kv  dσ þ  dεp þ  dε_ p ¼ 0 @σ @kp @εp @kv @ε_ p

ð5:100Þ

_ into Linear decomposition of the strain increment (dε) and strain rate increment (dε) their reversible (dεr ) and permanent components (dεp ) dε ¼ dεr þ dεp ; dε_ ¼ dε_ r þ dε_ p

ð5:101Þ

Incremental Kelvin-Voigt law dσ ¼ Dr dεr þ Cr dε_ r

ð5:102Þ

where σ is the stress tensor, ε is the strain tensor, Kðkp ; kv Þ is the hardening function, G is the plastic potential, λ is the proportionality coefficient, kp is the plastic hardening parameter, kv is the viscos hardening parameter, dεr is the reversible incremental strain tensor, dε_ r is the reversible incremental strain rate tensor, dεp is the permanent incremental strain tensor, dε_ p is the permanent incremental strain rate tensor, Dr is the elasticity tensor, Cr is the viscosity tensor, and ðÞ denotes dot product. In elastic-visco-plastic formulations of the Perzyna type, the flow rule is assumed to be of the following form: @G dε_ p ¼ l_ @σ

ð5:103Þ

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Constitutive laws

81

The flow rule implies that any plastic straining is time dependent. Aydan and Nawrocki (1998) have suggested the following form: dε_ p ¼ l

@G_ @σ

ð5:104Þ

This Aydan and Nawrocki (1998) flow rule implies that the plastic potential function shrinks (or expands) in time domain while keeping its original form in stress space and that the permanent strain increment consists of time-dependent and time-independent parts. Substituting Equation (5.104) in Equation (5.100) and rearranging the resulting equations yields the following: dF ¼

1 @F  dσ hrp @σ

ð5:105Þ

where hrp is called the hardening modulus and is given specifically as follows: " # @F @kp @G @F @kv @G_ hrp ¼  þ   @kp @εp @σ @kv @ε_ p @σ

ð5:106Þ

Inserting these relations into Equation (5.105) yields the constitutive relations of permanent strain increment and permanent strain rate increment in relation to stress increment as

 1 @G @F 1 @G @F  dσ ¼   dσ ð5:107Þ dεp ¼ hrp @σ @σ hrp @σ @σ

dε_ p ¼



 1 @G_ @F 1 @G_ @F  dσ ¼   dσ hrp @σ @σ hrp @σ @σ

ð5:108Þ

where () denotes the tensor product. Similar to the argument associated with Equation (5.84), the inverse of these relations cannot be determined. Therefore the following technique is used to establish the relation between stress increment and strain and strain rate increments. Using the preceding relations, one can write the following: dσ ¼ Dr dε  Dr



 1 @G @F 1 @G_ @F  dσ þ Cr dε_  Cr  dσ hrp @σ @σ hrp @σ @σ

ð5:109Þ

Taking the dot products of the both sides of the preceding expression by @F=@σ yields: @F _  ðDr dεÞ þ @F  ðCr dεÞ @F @σ @σ   r @G_  dσ ¼ r @G 1 @F 1 @F @σ 1 þ h @σ  D @σ þ h @σ  C @σ rp

ð5:110Þ

rp

Substituting the preceding equation in Equation (5.102) gives the incremental elasto-viscoplastic constitutive law as: dσ ¼ Drp dε þ Crp dε_

ð5:111Þ

82 Constitutive laws

22

where @G

Drp ¼ Dr 

@F

Dr @σ  @σDr
   _ @F @F r @G r @G hrp þ @σ  D @σ þ @σ  C @σ @G_

@F

Cr @σ  @σCr
 C ¼C    _ @F @F r @G r @G hrp þ @σ  D @σ þ @σ  C @σ rp

ð5:112Þ

r

ð5:113Þ

Figure 5.14 illustrates the elasto-visco-plastic model for a one-dimensional response.

5.3.3

Yield/failure criteria

Nonlinear behavior requires the existence of yield functions. These yield functions are also called failure functions at the ultimate state when rocks rupture. For a two-dimensional case, it is common to use the Mohr-Coulomb yield criterion given by: t ¼ c þ sn tan or s1 ¼ sc þ qs3

ð5:114Þ

where c, φ and sc are cohesion, friction angle and uniaxial compressive strength. sc and q are related to cohesion and friction angle in the following form: sc ¼

2ccos 1 þ sin and q ¼ 1  sin 1  sin

ð5:115Þ

Since the intermediate principal stress is indeterminate in the Mohr-Coulomb criterion and there is a corner-effect problem during the determination of incremental elasto-plasticity tensor, the use of Drucker-Prager criterion (Drucker and Prager, 1952) is quite common in numerical analyses, which is given by: pffiffiffiffi aI1 þ J2 ¼ k ð5:116Þ where I1 ¼ sI þ sII þ sIII ; J2 ¼

1 2 2 2 ðsI  sII Þ þ ðsII  sIII Þ þ ðsIII  sI Þ 6

Nevertheless, it is possible to relate the Drucker-Prager yield criterion with the MohrCoulomb yield criterion. On the π-plane, if the inner corners of the Mohr-Coulomb yield surface are assumed to coincide with the Drucker-Prager yield criterion, the following relations may be derived (Fig. 5.15): 2sin 6ccos a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ð3 þ sinÞ 3ð3 þ sinÞ where c; φ are cohesion and friction angle, respectively.

ð5:117Þ

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Constitutive laws

83

Figure 5.15 Illustration of yield criteria in principal stress space Source: From Owen and Hinton, 1980

In rock mechanics, a recent yield criterion is Hoek-Brown’s criterion (1980), which is written as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5:118Þ s1 ¼ s3 þ msc s3 þ ss2c where m and s are some coefficients. While the value of s is 1 for intact rock, the values of m and s change when they are used for rock mass. Aydan (1995) proposed a yield function for the thermo-plasticity yielding of rock as given by: s1 ¼ s3 þ ½S1  ðS1  sc Þeb1 s3 eb2 T

ð5:119Þ

where S1 is the ultimate deviatoric strength while coefficients b1 ; b2 are empirical constants. A number of examples of applications of yield (failure) criteria to actual experimental results involving igneous, metamorphic and sedimentary rocks are described and compared with one another as well as with experimental results. (a)

Sedimentary rocks

A series of uniaxial and triaxial compression and Brazilian tensile tests were carried out on Oya tuff, which is a well-known volcanic sedimentary rock in Japan (Seiki and Aydan, 2003). Figure 5.16 shows the failure criteria of Mohr-Coulomb, Hoek and Brown, and Aydan applied to experimental results. The best fits with experimental results were obtained for Aydan’s criterion and Mohr-Coulomb criterion for s3 > 0. However, if the yield criterion is required to evaluate both uniaxial compressive strength and tensile strength, the criteria of Hoek-Brown and Mohr-Coulomb cannot evaluate the triaxial strength of rocks. The next application is concerned with very weak sandstone from Tono mine in Central Japan Aydan et al. (2006). Figure 15.17 shows the fitted failure criteria of Mohr-Coulomb, Hoek and Brown, and Aydan applied to experimental results of sandstone of Tono. As seen from the Figure 5.4, the best fits to experimental results were obtained for Aydan’s criterion

84 Constitutive laws

24

Figure 5.16 Comparisons of yield criteria for Oya tuff (dry)

Figure 5.17 Comparisons of yield criteria for experimental results on Tono sandstone

and the Hoek and Brown criterion. Nevertheless, the criterion of Hoek-Brown deviates from experimental results when the confining pressure is greater than 1 MPa. A series of uniaxial and triaxial compression and Brazilian tests carried out on samples from a limestone formation in which Gökgöl karstic cave in Zonguldak province, Turkey, is located (Aydan et al., 2012). Triaxial compression experiments were carried under confining pressures up to 40 MPa. Figure 5.18 shows the experimental results together with several fitted failure criteria of bilinear Mohr-Coulomb, Hoek-Brown, and Aydan. The best fits to experimental results were obtained from the applications of the bilinear Mohr-Coulomb criterion and Aydan’s criterion. If the Hoek-Brown criterion is required to represent both the tensile strength and the uniaxial compression strength, it is well fitted to the lower-bound values. However, the estimated curve by Aydan’s criterion can better represent the bilinear Mohr-Coulomb criterion as well as all experimental results.

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Constitutive laws

85

Figure 5.18 Comparisons of yield criteria for experimental results on limestone

Figure 5.19 Comparisons of yield criteria for experimental results on Inada granite

(b)

Igneous rocks

Inada granite is a well-known igneous hard rock, and its uniaxial compressive strength is generally greater than 100 MPa. Figure 5.19 shows the fitted failure criteria of Mohr-Coulomb, Hoek and Brown, and Aydan applied to experimental results carried out under confining pressures up to 100 MPa. For confining pressures up to 50 MPa, the best fits to experimental results are those of Aydan and Mohr-Coulomb. If the parameters of the Hoek-Brown criterion are determined to represent tensile strength and uniaxial compressive strength, the estimated triaxial strengths for high confining pressures are entirely different from experimental results. Hirth and Tullis (1994) reported the results of triaxial experiments on quartz aggregates under very high confining pressures, which is almost 6 times its uniaxial compressive strength. The best fit to the experimental results is obtained for Aydan’s criterion, as seen in Figure 5.20. Up to a confining pressure of 1000 MPa, the estimation by the MohrCoulomb criterion is better than that by the Hoek-Brown criterion. Again, very high discrepancy is observed among the estimations by Hoek-Brown criterion and experimental results.

86 Constitutive laws

26

Figure 5.20 Comparisons of yield criteria for experimental results on quartz

(c)

Metamorphic rocks

The yield (failure) criterion of metamorphic rocks must consider the anisotropy caused by the orientation of schistosity in relation to the applied principal stresses during experiments. Jaeger and Cook (1979) developed a procedure involving yield conditions for shearing along a schistosity plane and shearing through rock, based on the Mohr-Coulomb yield (failure) criterion. There have been some attempts by several researchers to express the dependency of yield function to the schistosity orientation (McLamore and Gray, 1967; Donath, 1964; Nasseri et al., 2003). When the yield (failure) criterion is evaluated, one should take into account the effects of characteristics of rocks as well as the schistosity orientation with respect to applied stresses. Nevertheless, the overall functional form of yield criterion should be similar for a given orientation except the specific values of their parameters. Nasseri et al. (2003) reported extensive experimental research on the failure characteristics of Himalayan schists subjected to confining pressures up to 100 MPa. They also reported the tensile strength characteristics of all rock types. Figure 5.21 shows the failure criteria of bilinear Mohr-Coulomb, Hoek-Brown, and Aydan applied to experimental results of Himalayan chlorite schist for the orientation angle of 90 degrees. In fitting relations of the criteria of Hoek-Brown and Aydan, the failure criteria are required to represent uniaxial compressive strength and tensile strength. We note that experimental results are well represented by the bilinear Mohr-Coulomb criterion. Once again the Hoek-Brown criterion deviates from triaxial compressive experimental results if it is required to represent the tensile strength and compressive strength. Aydan’s criterion provides a best continuous fit to experimental results as noted in Figure 5.21. Waversik and Fairhurst (1970) presented results of triaxial compressive tests on Tennessee marble, while Haimson and Fairhurst (1970) reported the tensile and uniaxial compressive strength of the same rock. Again we fitted the failure criteria of bilinear Mohr-Coulomb, Hoek-Brown, and Aydan to a combined set of experiments on Tennessee marble with the requirement of representing its tensile and uniaxial compressive strength as shown in Figure 15.22. The overall tendency of fitted criteria to experimental results remains the same. Particularly, the requirement of representing both tensile and compressive strength

27

Constitutive laws

87

Figure 5.21 Comparisons of yield criteria for experimental results on chlorite schist

Figure 5.22 Comparisons of yield criteria for experimental results on Tennessee marble

by the criterion of Hoek-Brown result in different values of parameter m, as also noted by Betournay et al. (1991). The value of m is roughly equal to the ratio of uniaxial compressive strength to tensile strength, which is known as the brittleness index. (d)

Application to thermal triaxial compression experiments

In geomechanics, there is almost no yield (failure) criterion–incorporating effect of temperature on yield (failure) properties of rocks, although there are some experimental researches

88 Constitutive laws

28

Figure 5.23 Experimental results of Hirth and Tullis (1994) on quartz for three different ambient temperatures

Figure 5.24 The reduction of deviatoric strength of quartz as a function of temperature for a confining pressure of 1.17–1.2 GPa

(Hirth and Tullis, 1994). Aydan’s criterion is the only criterion known to incorporate the temperature, and it was used to study the stress state of the Earth (Aydan, 1995). Figure 5.23 shows the experimental results for three different values of ambient temperature reported by Hirth and Tullis (1994), while Figure 5.24 shows the reduction of strength with temperature for a given confining pressure of 1.17–1.2 GPa. Aydan’s yield (failure) criterion is applied to experimental results, as shown in Figures 5.23 and 5.24, and results are shown in Figure 5.25.

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89

Figure 5.25 Three-dimensional representation of Aydan’s failure criterion for experimental results of Hirth and Tullis (1994)

5.4

Equivalent models for discontinuum

Rock masses in nature contain numerous discontinuities in the form of cracks, joints, faults, bedding planes and the like. Therefore, various continuum equivalent models of discontinuum have been proposed and used since the beginning of 1970 to assess the stability of rock tunnels (e.g. Budiansky and O’Connel, 1976; Hill, 1963; Kachanov, 1958). Discontinuum is distinguished from continuum by the existence of contacts or interfaces between the discrete bodies that comprise the system. Relative sliding or separational movements in such localized zones present an extremely difficult problem in mechanical modeling and numerical analysis. The formulation for representing contacts is very important when a system of interacting blocks is considered, and it has been receiving a considerable interest among researchers. The main characteristics of the models are described in this section. 5.4.1

Equivalent elastic compliance model (EECM) (Singh’s model)

Singh’s model (1973) is based on the theory proposed by Hill (1963) for composite materials. The elastic constitutive law of the rock mass is obtained by making the following assumptions (Fig. 5.26): 1 2

Discontinuities are distributed in sets in the rock mass. The geometry of discontinuities (area and orientation) are known.

90 Constitutive laws

30

Figure 5.26 Singh’s model

3

4 5 6

The constitutive law of discontinuities is expressed in terms of relative normal and shear displacements and applied shear and normal stresses and the shear and normal stiffnesses are used to express the behavior of discontinuities. The stress tensor acting on discontinuities is related to that acting on the representative volume through a tensor called stress-concentration tensor. The strain tensor of the representative element is a linear sum of the strain tensor of the intact rock and the additional strain tensor due to discontinuities. The volume of discontinuities is assumed to be negligible as compared with that of the rock so that the stress tensor acting on the representative volume of rock mass is the same as that on the intact rock.

The constitutive law derived in a local coordinate system is then transformed to that in the global coordinate system. The formulations given by Goodman and Amadei and by Goodman are the simplified form of Singh’s model. Application of these models to tunnels in jointed media with a cross-continuous pattern and intermittent pattern is given and compared with a discrete model. This model is the first equivalent model to be applied to rock engineering problems. 5.4.2

Crack tensor model (CTM)

This model proposed by Oda (1982) for rock masses follows basically the same steps of Singh’s model in order to obtain the elastic constitutive law of the rock mass. The main differences are as follows: 1 2

Constitutive law is directly derived in a global coordinate system. The geometry of discontinuities (area and orientation) are represented by a series of even-order tensors (up to fourth-order tensors).

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Constitutive laws

91

Figure 5.27 Crack tensor model Source: Oda et al., 1993

3

The additional strain tensor due to discontinuities is determined from a procedure utilizing an analogy to the theoretical solutions for penny-shaped or elliptical inclusions embedded in an elastic medium used in linear elastic fracture mechanics (Fig. 5.27).

The application of this model to tunnels, particularly branching tunnels, is described in a recent paper by Oda et al. (1984, 1993). 5.4.3

Damage model (DM)

The damage model is based upon the theory proposed by Kachanov (1958) for creeping metals. It is elaborated by Murakami (1985) by introducing a second-order tensor called the damage tensor, and it is applied to a rock mass by Kyoya (1989) and Kawamoto et al. (1988). Assumptions 1 and 2 of Singh are also the same in this model. However, this model differs from other models, and it is based on the following additional assumptions: 1 2

The discontinuities are assumed to be not transmitting any stress across, implying the discontinuities have no stiffness at all. The stresses are assumed to be acting only on the intact parts, implying the parallel connection principle for the stress field. The average stress (Cauchy stress) is related to the stress (net stress or intensified stress) on the intact part through the second-order damage tensor, which represents a tensorial area reduction in the mass (Fig. 5.28).

Figure 5.28 Damage model Source: Kyoya, 1989

33

3 4

Constitutive laws

93

The strain tensor of the representative elementary volume is the same as that of the intact rock. The constitutive law is introduced between the net stress tensor and the strain tensor.

It should be noted that this model could not be directly used for thoroughgoing discontinuity sets because of Assumption 1. Nevertheless, Kyoya (1989) introduces some coefficients for normal and shear responses to differentiate the behavior under tension and compression. The applications of this model to tunnels and underground caverns are described by Kyoya (1989) and Kawamoto et al. (1988). Swoboda and Ito (1992) extended this model to model crack propagations in jointed media and gave several examples of its application. 5.4.4

Micro-structure models

Aydan et al. (1992, 1994,1996) proposed two models for discontinuous rock masses based on the microstructure theory of mechanics (Jones, 1975). Although the first assumption of Singh is the same as that in this approach, this model differs from others. The fundamental differences are as follows: 1 2

3 4

Discontinuities have a finite volume that enables one to model a wide range of discontinuities from joints to faults or fractured zones. The constitutive law of discontinuities is expressed in the conventional sense of mechanics. In other words, the constitutive law is expressed in terms of stresses and strains, and it is uniquely defined. The constitutive law is not restricted to elasticity, and it can be of any kind that can describe the mechanical response of discontinuities. Stress and strain fields of each constituent are related to one another using two concepts: Globally Series and Locally Parallel Model (GSLPM) and Globally Parallel and Locally Series Model (GPLSM) (Fig. 5.29).

5.4.5

Homogenization technique

The homogenization technique was mainly used to obtain the equivalent characteristics of composites (Bakhvalov and Panasenko, 1984; Sanchez-Palencia, 1980) and has been recently applied to soil (Auriault, 1983) and rocks (Fig. 5.30). Assumptions 1 and 3 of the micro-structure model also hold for this technique. Stress and strain fields of constituents are obtained from a perturbation of the displacement field. An influence tensor, which is a gradient of six vectorial functions called characteristic deformation functions for a given representative elementary volume (unit cell), is used to establish relations between the homogenized elasticity tensor and those of its constituents. Except for very simple cases, the equivalent parameters are obtained using a numerical method such as FEM.

Figure 5.29 Micro-structure model Source: From Aydan et al., 1992

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95

Figure 5.30 Homogenization model

References Auriault, J.L. (1983) Homogenisation: Application to porous saturated media. Summer School on Two-Phase Medium Mechanics, Gdansk. Aydan, Ö. (1995) The stress state of the earth and the earth’s crust due to the gravitational pull. The 35th US Rock Mechanics Symposium, Lake Tahoe. pp. 237–243. Aydan, Ö. (2011) Some issues in tunnelling through rock mass and their possible solutions. First Asian Tunnelling Conference, ATS11. pp. 33–44. Aydan, Ö. & Nawrocki, P. (1998) Rate-dependent deformability and strength characteristics of rocks. Int. Symp. On the Geotechnics of Hard Soils-Soft Rocks, Napoli, 1. pp. 403–411. Aydan, Ö., Tokashiki, N., Seiki, T. & Ito, F. (1992) Deformability and strength of discontinuous rock masses. Int. Conf. Fractured and Jointed Rock Masses, Lake Tahoe. pp. 256–263. Aydan, Ö., Seiki, T., Jeong, G.C. & Tokashiki, N. (1994) Mechanical behaviour of rocks, discontinuities and rock masses. Proc. of International Symposium Pre-failure Deformation Characteristics of Geomaterials, Sapporo, 2. pp. 1161–1168. Aydan, Ö., Tokashiki, N. & Seiki, T. (1996) Micro-structure models for porous rocks to jointed rock masses. Proc. 3rd Asia-Pacific Conf. on Computational Mechanics, Seoul.

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Aydan, Ö., Daido, M., Tano, H., Nakama, S. & Matsui, H. (2006) The failure mechanism of around horizontal boreholes excavated in sedimentary rock. 50th US Rock Mech. Symp., Paper No. 06130, Golden, Colorado. Aydan, Ö., Tokashiki, N. & Geniş, M. (2012) Some considerations on yield (failure) criteria in rock mechanics ARMA 12–640. Proc, of 46th US Rock Mechanics/Geomechanics Symposium, Chicago, 10p, (on CD). Bakhvalov, N. & Panasenko, G. (1984) Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht. Betournay, M.C., Gorski, B., Labrie, D., Jackson, R. & Gyenge, M. (1991) New considerations in the determining of Hoek and Brown material constants. Proc. 7th Int. Cong. on Rock Mechanics, Aachen, 1. pp. 195–200. Budiansky, B. & O’Connel, R.J. (1976) Elastic moduli of a cracked solid. International Journal of Solids and Structures, 12, 81–97. Donath, F.A. (1964) Strength variation and deformational behavior in anisotropic rock. In: Judd, W.R. (ed.) State of Stress in the Earth’s Crust. Elsevier, New York. pp. 281–297. Drucker, D.C. & Prager, W. (1952) Soil mechanics and plastic analysis for limit design. Quarterly of Applied Mathematics, 10(2), 157–165. Farmer, I. (1983) Engineering Behaviour of Rocks, 2nd edition, Chapman and Hall, London. Haimson, B.C. & Fairhurst, C. (1970) Some bit penetration characteristics in pink Tennessee marble. Proc. 12th US Rock Mechanics Symp. pp. 547–559. Hill, R. (1963) Elastic properties of reinforced solids: Some theoretical principles. Journal of Mechanics and Physics of Solids, 11, 357–372. Hirth, G. & Tullis, J. (1994) The brittle-plastic transition in experimentally deformed quartz aggregates. Journal of Geophysical Research, 99, 11731–11747. Hoek, E. & Brown, E.T. (1980) Empirical strength criterion for rock masses. Journal of Geotechnical Engineering Division, ASCE, 106(GT9), 1013–1035. ISRM (2007) The complete ISRM suggested methods for rock characterization, testing and monitoring: 1974–2006. In: Ulusay, R. & Hudson, J.A. (eds.) Suggested Methods Prepared by the Commission on Testing Methods, International Society for Rock Mechanics, Compilation Arranged by the ISRM Turkish National Group, Ankara, Turkey. Jaeger, J.C. & Cook, N.G.W. (1979) Fundamentals of Rock Mechanics, 3rd edition. Chapman & Hall, London. pp. 79, 311. Jones, R.M. (1975) Mechanics of Composite Materials. Hemisphere Pub. Co., New York. Kachanov, L.M. (1958) The Theory of Creep (English transl. by A.J. Kennedy). National Lending Library, Boston. Kawamoto, T., Ichikawa Y. & Kyoya, T. (1988) Deformation and fracturing behaviour of discontinuous rock mass and damage mechanics theory. International Journal for Numerical and Analytical Methods in Geomechanics, 12(1), 1–30. Kyoya, T. (1989) A Fundamental Study on the Application of Damage Mechanics to the Evaluation of Mechanical Characteristics of Discontinuous Rock Masses. Ph.D. Thesis, Nagoya University, Nagoya, Japan. McLamore, R. and Gray, K.E. 1967. The mechanical behaviour of anisotropic sedimentary rocks. Transactions of the American Society of Mechanical Engineers Series B, 62–76. Mirza, U.A. (1978) Investigation into the Design Criteria for Underground Openings in Rocks Which Exhibit Rheological Behaviour. PhD thesis, University of Newcastle upon Tyne., Newcastle upon Tyne, UK Murakami, S. (1985) Anisotropic damage theory and its application to creep crack growth analysis. Proc. Int. Conf. Constitutive laws for Engineering Materials: Theory and Applications, Elsevier, Amsterdam. pp. 535–551. Nasseri, B.M.H., Rao, K.S. & Ramamurthy, T. (2003) Anisotropic strength and deformational behaviour of Himalayan Schists. International Rock Mechanics and Mining Science, London, 40(1), 3–23.

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Oda, M. (1982) Fabric tensor for discontinuous geological materials. Soil and Foundations, Tokyo 22 (4), 96–108. Oda, M., Suzuki, K. & Maeshibu, T. (1984) Elastic compliance for rock-like materials with random cracks. Soil and Foundations, Tokyo, 24(3), 27–24. Oda, M., Yamabe, T., Ishizuka, Y., Kumasaka, H., Tada, H. & Kimura, K. (1993) Elastic stress and strain in jointed rock masses by means of crack tensor analysis. Rock Mechanics and Rock Engineering, Vienna, 26(2), 89–112. Owen, D.R.J. & Hinton, E. (1980) Finite Element in Plasticity: Theory and Practice. Pineridge Press Ltd, Swansea. Perzyna, P. (1966) Fundamental problems in viscoplasticity. Advances in Applied Mechanics, 9(2), 244–368. Sanchez-Palencia, E. (1980) Non-homogenous media and vibration theory. Lecture Note in Physics, No. 127, Springer, Berlin. Seiki, T. & Aydan, Ö. (2003) Deterioration of Oya Tuff and its mechanical property change as building stone. Proc. of Int. Symp. on Industrial Minerals and Building Stones, Istanbul, Turkey. pp. 329–336. Singh, B. (1973) Continuum characterization of jointed rock masses: Part I*The constitutive equations. Int. J. Rock Mech. Min Sci., 10, 311–335. Swoboda, G. & Ito, F. (1992) Two-dimensional damage failure propagation of jointed rock mass. Int. Symp. On Computational Mechanics, Balkema. Waversik, W.R. & Fairhurst, C. (1970) A study of brittle rock fracture in laboratory compression experiments. Int. J. Rock Mech. Min. Sci., 7, 561–575. Wawersik, W.R. (1983) Determination of steady state creep rates an activation parameters for rock salt. In: High Pressure Testing of Rock, Special Technical Publication of ASTM, STP86972-91.

Chapter 6

Laboratory and in-situ tests

Mechanical, seepage, heat, and diffusion properties related to constitutive laws described in Chapter 5 require tests on rocks and/or rock mass appropriate to physical and environmental conditions. In this chapter, the fundamental principles of available testing techniques used in the field of rock mechanics and rock engineering are explained. It should be noted that the details of each technique may require additional information about equipment and processing, which are explained in the suggested methods (SMs) published by ISRM (Brown, 1981; Hudson and Ulusay, 2007; Ulusay, 2012). Furthermore, size, shape, and environmental conditions are described in related ISRM SMs.

6.1

Laboratory tests on mechanical properties

Laboratory tests on mechanical properties of rocks may be determined from uniaxial tensile and compression tests, triaxial compression experiments, three/four bending experiments, direct shear and Brazilian experiments. Some of these experiments are illustrated in Figure 6.1. The fundamentals of these testing techniques are described in the following subsections.

Figure 6.1 Illustration of some testing techniques for determining mechanical properties

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6.1.1

2

Uniaxial compression tests

Specimens from drill cores are prepared by cutting them to the specified length and are thereafter grinded and measured. There are high requirements on the flatness of the end surfaces in order to obtain an even load distribution. The recommended ratio of height/diameter of the specimens is between 2 and 3. Strains and stress in uniaxial compression tests are defined as follows: Axial strain is calculated from the equation εa ¼

Dl lO

ð6:1Þ

where lO is the original measured axial length, and Dl is the change in measured axial length (defined to be positive for a decrease in length). Diametrical strain can be determined either by measuring the changes in the diameter of the specimen or by measuring the circumferential strain. In the case of measuring changes in the diameter, the diametric strain is calculated from the equation: εd ¼

Dd dO

ð6:2Þ

where d0 is the original undeformed diameter of the specimen, and Dd is the change in diameter (defined to be negative for an increase in diameter). In the case of measuring the circumferential strain, εc , the circumference is C ¼ pd; thus the change in circumference is DC ¼ pDd. Consequently, the circumferential strain εc is related to the diametric strain εd by: εc ¼

DC Dd ¼ C dO

ð6:3Þ

so that εc ¼ εd , where C and dO are the original circumference and diameter of the specimen, respectively. The compressive axial stress in the test specimen sa is calculated by dividing the compressive load P on the specimen by the initial cross-sectional area AO . sa ¼

P AO

ð6:4Þ

where compressive stresses and strains are considered to be positive in this test procedure. For a given stress level, the volumetric strain εv is calculated from the equation: εv ¼ εa þ 2εd

ð6:5Þ

The specimens are loaded axially up to failure or any other prescribed level whereby the specimen is deformed and the axial and the radial deformation can be measured using some equipment as shown in Figure 6.2. There are a tremendous number of studies on the stress and strain distributions induced in uniaxial compression experiments with the consideration of boundary conditions imposed in the experiments. Generally, the stiffness and Poisson’s ratio of the platens and specimen are different from each other. Both theoretical and numerical analyses such as FEM indicate that the stress and strain are not uniform within the specimen. Particularly, the distributions are highly nonuniform near the end of the specimen, as shown in Figure 6.3. The nonuniformity

Figure 6.2 A view of experimental setup and instrumentation in uniaxial compression experiment at Nagoya University and University of the Ryukyus

Figure 6.3 Finite element method simulation of axisymmetric rock sample (One-quarter of the specimen is used in view of symmetry.)

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4

of strain–stress distributions strongly depends upon the differences of stiffness and Poisson’s ratio of platens and specimen and their geometry. Stress and strain become uniform when the stiffness and Poisson’s ratio of platens and specimen are the same, which is not the common case in many experimental studies. An entirely wrong interpretation of nonuniformity stress distributions is caused by the differences of the stiffness and Poisson’s ratio as frictional effect. Frictional effect may come into action after relative slip occurs between the platens and the ends of the specimen. In practice, Vaseline oil, Teflon sheets or brushtype platens are used to deal with this issue.

6.1.2

Direct and indirect tensile strength tests (Brazilian tests)

Similar to the uniaxial compression tests, direct tensile tests are used to determine tensile strength and some deformability properties. The definitions of stress and strains are fundamentally the same except for the sign of stresses and strains. The specimens are bonded to the loading platens, or dog-shaped specimens are used to obtain the strain–stress relations. As the preparation and procedures are quite cumbersome, indirect tensile stress tests are generally preferred. The common procedure is to load a solid or hollow cylindrical specimen under compression, which results in tensile stresses within the samples. A cylindrical specimen is loaded diametrically across the circular cross section. This testing technique is known as the Brazilian tensile strength test. The loading causes tensile stresses perpendicular to the loading direction, which results in a tensile failure. Tensile stress induced in a solid cylinder of rock is theoretically given by the following equation: st ¼

2F pDt

ð6:6Þ

where F, D and t are applied load, diameter and thickness of the rock sample, respectively. The nominal strain of the Brazilian tensile test sample may be given as (see Hondros, 1959; Jaeger and Cook, 1979 for details): h is p d ð6:7Þ εt ¼ 2 1  ð1  uÞ t with εt ¼ 4 D E For most rocks, this formula may be simplified to the following form: εt ¼ 0:82

st E

ð6:8Þ

A plane stress finite element analysis was carried out for the Brazilian test. The properties of the platen were assumed to be those of aluminium with an elastic modulus of 70 GPa. Uniform compressive pressure with an intensity of 20 kgf cm2 was applied on the platens, and boundary conditions are shown in Figure 6.4. The maximum tensile stress occurs in the vicinity of the center of the sample, and its value is 1.08 kgf cm2. This is slightly greater than the theoretical estimation of 0.8 kgf cm2. This is probably due to the slight difference in the application of load boundary conditions. The computed radial displacement of the sample just below the platen was about 0.001 mm, which is almost equal to that estimated from Equation 6.7. Therefore, it is possible to determine the elastic modulus besides the tensile strength of rocks. Furthermore, the strain response in experiments should be similar to those of the uniaxial compression

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5

103

Figure 6.4 Boundary conditions and computed stress distributions

experiments provided that deformability characteristics remain the same under both tension and compression. Bending tests are also used to determine the tensile strength and deformability of rocks. The maximum tensile stress and maximum flexural strain of rock beam under a three-point bending configuration with a concentrated load (F) and rectangular prismatic shape (b, t) and maximum deflection (d) may be given in the following form: sf ¼

3FL 6dt sf FL3 ;A ¼ b  t ; ε ¼ ; ¼ 2bt2 f L2 εf 4Ad

ð6:9Þ

Four-point bending is generally recommended due to uniform stress distributions in the area between two applied loads points. Figure 6.5 illustrates stress distribution in a three-point bending test in a photo-elastic test.

6.1.3

Triaxial compression tests

Specimens from drill cores are prepared by cutting them to the specified length and are thereafter ground and measured to obtain the required the flatness of the end surfaces. The recommended ratio of height/diameter of the specimens is between 2 and 3. A membrane is mounted on the surface of the specimen in order to seal the specimen from the

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6

Figure 6.5 Stress distribution in a beam subjected to three-point bending condition

surrounding pressure media. Deformation measurement equipment is mounded on the specimen, and the specimen is inserted into the pressure cell, whereupon the cell is closed and filled with oil. A hydrostatic pressure is applied in the first step. The specimen is then further loaded by increasing the axial load under constant or increasing cell pressure up to failure or any other predefined load level. A test setup and equipment, triaxial cell, and instrumentation used in Nagoya University is shown in Figure 6.6. 6.1.4

Postfailure behavior in uniaxial and triaxial compression tests

The postfailure characteristics of rocks are quite important in rock engineering, when rock failure could not be prevented. For this purpose, servo-control testing devices developed for investigating postfailure characteristics of rocks (e.g. Rummel and Fairhurst, 1970; Waversik and Fairhurst, 1970; Hudson et al., 1972; Kawamoto et al., 1980). The servo-control devices try to provide sufficient support to measure the intrinsic properties of rocks during the postpeak response. Such a support system is provided through the loading system, which utilizes either servo-controlled oil-based jacks or wedge-like solid support. Figure 6.7 shows the principle of the device based on wedge-like solid support concept at Nagoya University, and Figure 6.8 illustrates how wedge-like support system activated during the deformation process. Figure 6.9(a) shows a general strain–stress responses for a typical rock sample, while Figure 6.9(b) illustrates the brittle and ductile behavior of rocks. The stress drops rapidly when rock exhibits brittle behavior. On the one hand, the stress gradually decreases in the postfailure regime of ductile rocks. Figure 6.10 shows the true response of the triaxial behavior of Ryukyu limestone under different confining pressures. Figure 6.11 shows views of a granite sample tested by the Nagoya University servocontrol testing machine. As noted from Figure 6.11, a macroscopic fracture zone is observed in the final postfailure stage. Figure 6.12 shows an actual image and X-ray CT tomographic image of another granite sample. As noted from the figure, a macroscopic fracture zone and a zone of fine cracks are seen in X-ray CT tomographic image (Aydan et al., 2016c) Figure 6.13 illustrates five different idealized fracturing situations in a given rock sample during the complete strain-–stress response (Aydan et al., 1993).

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7

105

Figure 6.6 Triaxial (a) compression device, (b) cell, and (c) instrumented sample at Nagoya University Rock Mechanics Laboratory

During stages 4 and 5, microscopic fractures coalesces into macroscopic shear bands. It should be noted that there is an argument if the strain–stress response in stages 4 and 5 should be considered a part of constitutive law or not. 6.1.5

Direct shear tests

Direct shear test devices can also be used to obtain the shear strength properties of intact rock or rock discontinuities. There are different types of direct shear test devices. Figure 6.14 shows a shear testing machine named OA-DSTM (Fig. 6.15), designed and built originally in 1991 for direct shear testing under three loading conditions: conventional direct shear loading, direct shear creep loading, and direct shear cyclic loading at Nagoya University (Aydan et al., 1994; Aydan et al., 2016a Both the shear and the normal loads on the shearing plane are designed to be 200 kN, and the system is displacement controlled. Furthermore, the direct shear box was vertical in order to eliminate dead loads on the shearing plane. The normal load is first imposed on the sample, and then shear loading is applied

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8

Figure 6.7 Servo-control testing device at Nagoya University

Figure 6.8 The illustration of the wedge-like solid support activation

through vertical jack. However, the system was needed to be upgraded for operational purposes as well as for dealing with dynamic loading conditions. Aydan et al. (2016a) have recently upgraded the shear testing machine by adding the dynamic shear loading option. The size of direct shear samples can be 100 × 100 × 100 mm or 150 × 75 × 75 mm. The original design size was 150 × 75 × 75 mm with the purpose of eliminating the rotational effects on the sample. This can be achieved if the ratio of sample length over sample height is greater than 2. The shear load and displacement and the normal load are directly recorded in computers using the outputs from the system. The outputs are real-time values of shear displacement in millimeters, and shear and normal loads in kilonewtons. Several examples

Figure 6.9 (a) Complete strain–stress relation of rocks, (b) illustration of brittle and ductile behavior

Figure 6.10 Strain–stress relation of Ryukyu limestone under different confining pressures

Figure 6.11 Views of a granite sample before (left) and after testing (right)

Figure 6.12 Actual (left) and X-ray CT (right) images of a granite sample in postfailure stage

Figure 6.13 Idealized five different fracturing situations in a given rock sample during the complete strain–stress response Source: From Aydan et al., 1993

Figure 6.14 View of dynamic shear testing machine OA-DSTM.

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12

of various direct shear tests on rock discontinuities and soft rocks are explained. Rock discontinuities are planar (polished and saw-cut of marbles). Soft rocks are sandy Ryukyu limestone (Awa-ishi), coral stone and Oya tuff. (a)

Conventional direct shear testing

Figure 6.15 show the responses measured during the direct shear experiment on the shear response of coral stone (honeycomb-like coral limestone). Once peak load is exceeded, the deformation rate increases as noted in Figure 6.15. An example of conventional direct shear test results on the polished planar surface of marble is shown in Figure 6.16(a). As seen from the figure, the yielding friction coefficient is about 0.27 (15 degrees), and hardening is observed. When the friction coefficient exceeds 0.52 (27.5 degrees), the relative shear displacement starts to increase. The ultimate friction angle coefficient is about 0.62 (31.8 degrees), while the residual friction coefficient is 0.58 (30.1 degrees). Figure 6.16(b) shows the view of interface after direct shear testing. As noted from the figure, the contacts on the planar surfaces were quite small despite the fact that the surfaces were polished and planar. Furthermore, striations occurred on the surface parallel to sliding direction, which would be commonly observed on the fault surface and slickenslides

Figure 6.15 Shear displacement–shear load relation of coral stone

Figure 6.16 (a) Relative displacement, shear, normal and friction coefficient of polished interface between blocks, (b) views of the sheared surfaces

Laboratory and in-situ tests

13

111

surfaces. In addition, this may also imply that it is practically very difficult to prepare exact planar surfaces that result in full contact of block surfaces. Tilting tests are carried out on the original blocks of polished and saw-cut surfaces. The results are given in Table 6.1 together with those from obtained direct shear experiment under a normal stress of 5 MPa. From the comparison of the table, the friction angle of polished surfaces seems to be nonrepresentative frictional property of planar discontinuities. On the other hand, the friction angle saw-cut surfaces are closer to the intrinsic friction of planar discontinuities. Nevertheless, it must be noted that the traces of saws on the surfaces would cause the friction angle to be directional (Aydan et al., 1996). Direct shear test results are plotted together with the results from tilting experiments in Figure 6.17. In the same figure, some failure criteria for rock discontinuities (i.e. Barton and Choubey, 1977; Aydan, 2008; Aydan et al., 1996) are also plotted. It is interesting to note that the shear strength of polished surface under high normal pressure is within the bounds obtained from the friction angle tests determined from tilting tests. The best fit to experimental results is obtained from the failure criterion of Aydan (Aydan, 2008; Aydan et al., 1966). The failure criterion of Barton and Choubey (1977) is close to the upper-bound strength envelope. Table 6.1 Friction angles of interfaces of marble blocks Condition Polished Saw-cut

Tilting Test

Direct Shear Test (NS: 5 MPa) Initial

Flow

Peak

Residual

16–19 28–35

15 23

27 28

31.8 31.0

30.1 29.0

Figure 6.17 Plot of experimental results for polished marble contacts together with some failure criteria for rock discontinuities

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(b)

14

Multistage direct shear testing

A multistage (multistep) direct shear test on a saw-cut surface of sandy Ryukyu limestone sample, which consists of two blocks with dimensions of 150 × 75 × 37.55 mm, was varied out. The initial normal load was about 17 kN and increases to 30, 40, 50, 60 and 70 kN during the experiment. Figure 6.18 shows the shear displacement and shear load responses during the experiment. As noted from the figure, the relative slip occurs between blocks at a constant rate after each increase of normal and shear loads. This experiment is likely to yield shear strength of the interface two blocks under different normal stress levels. Figure 6.19 shows the peak and residual levels of shear stress for each level of normal

Figure 6.18 Shear stress–shear load response of the interface of sandy limestone blocks during the multistage(step) direct shear experiment.

Figure 6.19 Comparison of shear strength envelope for the interface of sandy limestone blocks with experimental results from the multistage(-step) direct shear experiment.

Laboratory and in-situ tests

15

113

stress increment. Tilting tests were carried out on the same interface, and the apparent friction angles ranged between 35.4 and 39.6 degrees. Tilting test results and direct shear tests are plotted in Figure 6.19 together with shear strength envelopes using the shear strength failure criterion of Aydan (Aydan, 2008; Aydan et al., 1996). As noted from the figure, the friction angles obtained from tilting tests are very close to the initial part of the shear strength envelopes. However, the friction angle becomes smaller as the normal stress level increases. In other words, the friction angle obtained from tilting tests on saw-cut surfaces cannot be equivalent to the basic friction angle of planar discontinuities and interfaces of rocks. The basic friction angle of the planar interface of sandy limestone blocks is obtained as 27.5 degrees for the range of given normal stress levels. 6.1.6

Tilting tests

Tilting test technique is one of the cheapest techniques to determine the frictional properties of rock discontinuities and interfaces under different environmental conditions (Barton and Choubey, 1977; Aydan et al., 1995; Aydan, 1998). This technique can be used to determine the apparent friction angle of discontinuities (rough or planar) under low stress levels. It definitely gives the maximum apparent friction angle, which would be one of the most important parameters to determine the shear strength criteria of rock discontinuities as well as various contacts. Therefore, the data for determining the parameters of the shear strength criteria for rock discontinuities should utilize both tilting test and direct shear experiment. (a)

Theory of tilting tests

Let us assume that a block is put upon a base block with an inclination a as illustrated in Figure 6.20(a). The dynamic force equilibrium equations for the block can be easily written as follows: For s-direction Wsin a  S ¼ m

d2 s dt2

ð6:10Þ

Figure 6.20 (a) Mechanical model for tilting experiments, (b) loading path in tilting experiments and constitutive relation

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16

For n-direction Wcos a  N ¼ m

d2n dt2

ð6:11Þ

Let us further assume that the following frictional laws holds at the initiation and during the motion of the block (Aydan and Ulusay, 2002), as illustrated in Figure 6.20(b): At initiation of sliding S ¼ tan s N

ð6:12Þ

During motion S ¼ tan d N

ð6:13Þ

At the initiation of sliding, the inertia terms are zero so that the following relation is obtained: ð6:14Þ

tan a ¼ tan s

The preceding relation implies that the angle of inclination (rotation) at the initiation of sliding should correspond to the static friction angle of the discontinuity. If the normal inertia term is negligible during the motion and the frictional resistance is reduced to dynamic friction instantaneously, one can easily obtain the following relations for the motion of the block: d2 s ¼A dt2

ð6:15Þ

where A ¼ gðsin a  cos a tan d Þ. The integration of differential Equation (6.15) will yield the following: s¼A

t2 þ c1 t þ c2 2

ð6:16Þ

Since the following holds at the initiation of sliding: s¼0

and

v¼0

at

t ¼ Ts

ð6:17Þ

Equation (6.16) takes the following form: s¼

A 2 ðt  Ts Þ 2

ð6:18Þ

Coefficient A can be obtained either from a given displacement sn at a given time tn with the condition, that is: tn > Ts A¼2

sn 2 ðtn  Ts Þ

ð6:19Þ

Laboratory and in-situ tests

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115

or from the application of the least squares technique to measured displacement response as follows: n X

si ðti  Ts Þ

A ¼ 2 i¼1 n X

2

ð6:20Þ ðti  Ts Þ

4

i¼1

Once constant A is determined, the dynamic friction angle is obtained from the following relation:   1 A ð6:21Þ d ¼ tan1 tan a  cos a g

(b)

Tilting device and setup

An experimental device consists of a tilting device operated manually. During experiments, the displacement of the block and rotation of the base are measured through laser displacement transducers produced by KEYENCE, while the acceleration responses parallel and perpendicular to the shear movement are measured by a three-component accelerometer (Tokyo Sokki) attached to the upper block and WE7000 (Yokogawa) data acquisition system. The measured displacement and accelerations are recorded onto laptop computers. The weight of the accelerometer is about 98 gf. Figure 6.21 shows the experimental setup.

Figure 6.21 View of experimental setup for tilting device

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18

Figure 6.22 Responses of rough discontinuity of granite during a tilting test

A series of tilting tests are carried out on some discontinuities. Responses of some of these experiments are described as examples (Aydan, 2019). The measured responses during a tilting test on a rough discontinuity plane is shown in Figure 6.22 as an example. Figure 6.23 shows views of the tilting test on rough discontinuity plane of granite. As noted from the responses of rotation angle, relative displacement, and acceleration shown in Figure 6.23, fairly consistent results are observed. The static and dynamic friction coefficients of the interface were calculated from measured displacement response and weight of the upper block using the tilting testing equipment shown in Figure 6.22. The static and dynamic friction coefficients were estimated at 32.3–37.6 degrees and 30.3–35.6 degrees, respectively. Similarly experimental results on saw-cut discontinuity planes of Ryukyu limestone samples are shown in Figure 6.24, which shows responses measured during a tilting experiment on a saw-cut plane of Ryukyu limestone. The static and dynamic friction coefficients of the interface were calculated from measured displacement responses explained in the previous section, and they were estimated at 28.8–29.6 degrees and 24.3–29.2 degrees, respectively.

Figure 6.23 Views of tilting experiment on rough discontinuity plane of granite

Figure 6.24 Responses of saw-cut discontinuity planes of Ryukyu limestone samples during a tilting test

Figure 6.25 Views of some creep testing apparatuses

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6.1.7

20

Experimental techniques for creep tests

The methods for creep tests described herein are concerned with the creep characteristics of rocks under the indirect tensile stress regime of Brazilian test, uniaxial and triaxial compression tests and direct shear tests with the consideration of available creep testing techniques used in the rock mechanics field as well as other disciplines of engineering under laboratory conditions. (a)

Apparatuses

Apparatuses for creep tests can be of the cantilever type or load/displacement-controlled type (Fig. 6.26). Although the details of each testing machine may differ, the required features of apparatuses for creep tests are described herein. Cantilever-type apparatus has been used in creep tests since early times. It is, practically, the most suitable apparatus for creep tests, as the load level can be easily kept constant in time-space (Fig. 6.26(a)). The severest restrictions of this type of apparatus are the level of applicable load, which depends upon the length of the cantilever arm and its oscillations

Figure 6.26 Schematic illustration of cantilever type apparatuses for creep tests

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21

119

during the application of the load. Cantilever-type apparatus utilizing a multiarm lever overcomes load limit restrictions, and up to 500 kN loads can be applied to samples. The oscillation problem is also technically dealt with. Load is applied onto samples by attaching deadweights to the lever, which may be done manually for low-stress creep tests or mechanically for high-stress creep tests. In triaxial experiments, special load cells are required, and the confining pressure is generally provided through oil pressure. The utmost care must be taken for keeping the confining pressure constant with the consideration of continuous power supply for the compressor of the confining pressure system. (b)

Brazilian creep tests

The loading jigs and procedure used in the suggested method for Brazilian tests by ISRM should be followed unless the size of the samples differs from the conventional size (Fig. 6.26(b)). The displacement should be measured continuously or periodically, as suggested in the SM. The load application rate may be higher than that used in the SM. Once the load reaches the designated load level, it should be kept constant thereafter. If experiments are required to be carried out under a saturated condition, the jigs and sample should be put into a water-filled special cell. (c)

Uniaxial compression creep tests

Displacements are measured continuously or periodically as suggested in the SM. The load application rate may be higher than that used in the SM. Once the load reaches the designated load level, it should be kept constant thereafter. If experiments have to be carried out under saturated condition, the sample should be put into a water-filled special cell. (d)

Triaxial compression creep tests

The displacement is measured continuously or periodically as suggested in the SM. The load application rate may be higher than that used in the SM. Once the load reaches the designated load level, it should be kept constant thereafter. If experiments are required to be carried out under saturated condition, the sample should be put into a water-filled special cell. (e)

Impression creep test as an index test

The impression creep test technique utilizes the indenter, which is a cylinder with a flat end. The indenter makes a shallow impression on the surface of the specimen, and it is therefore named impression creep. There may be two different loading schemes during this experiment, namely, direct application of the deadweight (Fig. 6.27) or load by a cantilever frame (Fig. 6.26(c)). The potential use of this technique for the creep characteristics of rocks was explored by (Aydan et al., 2016; Rassouli et al., 2010). The critical issue with this technique is the definition of strain and stress, which can be associated with conventional creep experiments. (f)

Direct shear test device

The servo-control shear testing device shown in Figure 6.14 can be also used for creep tests on rock discontinuities.

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22

Figure 6.27 Impression creep apparatus utilizing deadweight

SHEAR LOAD (kN) NORMAL LOAD (kN)

80

Mortar Interface (ISRM-RPN4)

2

Shear Load

60 Shear Displacement

40

1

20 Normal Load: 50 kN

0

100

200 TIME(s)

300

0 400

SHEAR DISPLACEMENT (mm)

Source: Rassouli et al., 2010

Figure 6.28 Direct shear creep test on a mortar interface with a surface roughness profile number 4

An example of direct creep test on mortar sample having a surface roughness profile number RPN4 of ISRM is shown in Figure 6.28. As noted from experimental responses, the testing device can keep shear load constant on the sample, which is the major problem when servo-control testing machines are used. Although the duration of the test is short, the creep behavior of the interface indicates an almost linear response.

Laboratory and in-situ tests

23

6.2 6.2.1

121

In-situ mechanical tests Conventional mechanical tests

It is generally expensive to carry out experiments on large rock samples in-situ if rock mass is considered to be the equivalent continuum. Such experiments were carried out during the construction of Kurobe Dam in Japan (Nose, 1962), and the in-situ testing techniques are illustrated in Figure 6.29. The diameter of plate-bearing tests generally ranges between 30 and 60 cm. Shear strength samples are generally about 60 cm long and 30–40 cm high. Four tests are carried out to determine the peak cohesion and friction angle of rock masses. In addition, tests are repeated on sheared samples to determine residual strength parameters. The common size of triaxial samples is about 100 cm. However, the largest size of the triaxial test at Kurobe dam was 280 cm. Historically, uniaxial compression tests on rock masses were probably first undertaken in South Africa (i.e. Bieniawski, 1974; Van Heerden, 1975) using coal pillars. However, the first triaxial compressive strength tests were undertaken at the Kurobe Dam site by Kansai Electric Power Company (Nose, 1962). However, triaxial compression tests are not carried out due to their high cost and the huge difference between strength values obtained from triaxial compression tests and in-situ shear strength tests as seen in the Kurobe Dam project. This problem was pointed out by Hibino (2007), who was actively involved in the large powerhouse and dam construction projects. In addition, the natural underground openings and steep cliffs associated with Ryukyu limestone present some stability problems to the superstructures on the ground surface. Figure 6.30 shows an in-situ plate-loading test and rock shear test at the construction site of Minami-Daitojima fishing port in Ryukyu archipelago. 6.2.2

In-situ creep test

Results of an in-situ creep test method are used to predict time-dependent deformation characteristics of rock mass resulting from loading. This test method may be useful in structural design analysis where loading is applied over an extensive period. This test method is normally performed at ambient temperature, but equipment can be modified or substituted for operations at other temperatures. There are applications of this test

Figure 6.29 Illustration of in-situ testing techniques in Japan Source: Aydan et al., 2014, based on the original drawings by Hibino, 2007

122 Laboratory and in-situ tests

24

Figure 6.30 Views of in-situ experiments on rock mass in Minami Daitojima fishing port, Japan: (a) plate loading test, (b) rock shear test Source: Tokashiki and Aydan, 2012

technique in pillars of rock salt mines. In-situ creep tests are generally plate-bearing tests, with the direct shear creep test using the setups shown in Figure 6.29. (a)

Plate-bearing creep tests

The diameter of platens used in plate-bearing tests is 300 mm, and the maximum load is about 500 kN. The maximum nominal pressure is about 7.2 MPa. The deformation modulus is obtained from the following relation based on Boussinesq’s solution: E0 ¼

1  u2 F  d0 D

ð6:22Þ

where E; u; d0 ; D and F are deformation modulus, Poisson’s ratio, instantaneous settlement, diameter of platen and applied load. Poisson’s ratio is generally assumed to be 0.2 or 0.25. The total displacement is the sum of initial displacement and delayed creep displacement, given as: dt ¼ d0 þ dc

ð6:23Þ

The creep displacement is given as a fraction of the total displacement using a five-element generalized Voigt-Kelvin model (Fig. 6.31):      E1 E1 E2 E2 1  expð tÞ þ 1  expð tÞ ð6:24Þ dt ¼ d0 1 þ E0 Z1 E0 Z2 Thus, the creep displacement would be given as:      1  u2 F E1 E1 E2 E2  1  expð tÞ þ 1  expð tÞ dc ¼ D Z1 E0 Z2 E0 E0

ð6:25Þ

Figure 6.32 shows examples of plate-bearing creep tests on a rhyolite foundation of a dam site in Central Japan. Rock mass classified as CL and CM in the rock mass classification

Laboratory and in-situ tests

25

123

Figure 6.31 Five-element generalized Voigt-Kelvin model

Figure 6.32 Creep displacement of rhyolite rock mass at a dam site in Central Japan Table 6.2 Values of parameters for creep responses measured in plate-bearing test Rock Class DENKEN

RMQR

CL CM

20–40 40–60

δc =δe

E0 (MPa)

E1 =E0

E2 =E0

E1 =η1 (1 min1)

E2 =η2 (1 min1)

0.08 0.03

534 3332

0.4 0.5173

0.6 0.4828

0.238 0.300

0.009 0.008

system (DENKEN) of the Central Research Institute of Electric Power Companies of Japan. The values of parameters for creep responses shown in Figure 6.32 are listed in Table 6.2. (b)

Direct rock shear creep tests

In-situ direct shear rock test setup shown in Figure 6.29(b) is utilized for creep tests. The shearing area is 600 × 600 mm, and the height of the sample is 300 mm. Direct shear creep

124 Laboratory and in-situ tests

26

experiments are done in two stages. The first stage is called primary creep stage, and the specimen is loaded at a level of one-third of the ultimate peak shear strength at a given normal load. The duration of the primary creep stage is generally more than 90 minutes. The second stage is called the secondary creep stage, and the specimen is loaded at a level of two-thirds of the ultimate peak shear strength at a given normal load. The duration of the primary creep stage is generally more than 120 minutes. However, the duration of the creep tests may be several days to months depending on the importance of the structure. A generalized Voigt-Kelvin model having three elements is generally used as a model for the primary and secondary creep stages: wt ¼ we þ wc ¼ we ð1 þ að1  expðbtÞÞÞ; a ¼

Ke K ;b ¼ v Kv Zv

ð6:26Þ

where t; Ke ; Kv and Zv are applied shear stress, Hookean stiffness, Kelvinean stiffness and Kelvinean viscosity. Figure 6.33 shows examples of responses during direct shear creep tests on a rhyolite foundation of a dam site in Central Japan. Rock mass is classified as CM in the rock mass classification system (DENKEN) of the Central Research Institute of Electric Power Companies of Japan. The values of parameters for creep responses shown in Figure 6.33 are listed in Table 6.3.

Figure 6.33 Direct rock shear creep displacement of rhyolite rock mass at a dam site in Central Japan Table 6.3 Values of parameters for creep responses measured in direct shear test Stages

τ (MPa)

αwe (mm)

Ke MPa mm1

we (mm)

α ¼ KKev

β ¼ Kηvv (1 min1)

Primary creep Secondary Creep

3.2 7.4

0.062 0.165

5.839 4.888

0.548 1.594

0.114 0.109

0.12 0.16

Laboratory and in-situ tests

27

6.3

125

Thermal properties of rocks and their measurements

Thermal properties such as specific heat, heating or cooling coefficient and thermal conductivity are important to assess the heat transport through solids as noted from Equation (4.1). There are many techniques to measure thermal properties such as specific heat coefficient, thermal conductivity, thermal diffusion and thermal expansion coefficient. The details of such techniques can be found in various publications and textbooks (i.e. Clark, 1966; Somerton, 1992). Specific heat coefficient is commonly measured using the calorimeter tests. The earlier and common technique for thermal conductivity measurement is the divided bar technique (Birch, 1950) based on the steady-state heat flow assumption, and it is illustrated in Figure 6.34. This technique utilizes reference materials with well-known thermal properties. There are also techniques for measuring thermal conductivity utilizing transient heat flow (i.e. Carslaw and Jaeger, 1959; Popov et al., 1999, 2016; Sass et al., 1984). These techniques utilize line or plane sources, and temperature variations are measured by either contact sensor or infrared camera. These techniques utilize the analytical solutions developed by Carslaw and Jaeger (1959) An experimental technique using a device similar to a calorimeter-type apparatus is described in this section in order to measure thermal properties of rock materials from a single experiment, and its applications are given. Let us consider a solid (or group of solids) is enveloped by fluid (i.e. water (w)) as illustrated in Figure 6.35. It is assumed that solid and fluid have different thermal properties and temperature. For theoretical modeling, the following parameters are defined as follows: Q: heat; r: density; k: thermal conductivity; c: specific heat coefficient; T: temperature; m: mass;

Figure 6.34 Key components of a divided-bar apparatus Source: Popov et al., 2016A – pivot point, B – brass disks, C – reference material, D – rock specimen, E – hot plate, F – cold plate, G – heat source (concealed Peltier device), H – heat sink, I – holes for the insertion of temperature sensors, J – thermal insulation

126 Laboratory and in-situ tests

28

Figure 6.35 Illustration of physical and thermomechanical model

h:cooling coefficient; V : volume; As : surface area of solid; l : thermal expansion coefficient. The heat of a body is given in the following form: Q ¼ m  c  T ¼ rV  c  T Its unit is the joule (J ¼ N  m).

ð6:27Þ

Laboratory and in-situ tests

29

127

Assuming that mass and specific heat coefficient are constant, the heat rate (heat flux) is given in the following form: dQ dT ¼q¼mc dt dt

ð6:28Þ

The Newton cooling law is written in the following form: q ¼ h  As  DT

ð6:29Þ

where DT is the temperature difference between solid and enveloping fluid, and its unit is the watt (W ¼ J=s). In this particular model, the temperature of the surrounding fluid is assumed to be higher than the solid enveloped by the fluid. Furthermore, there is no heat flow from the system outward. In other words, it is thermally isolated. The heat flux from fluid can be given as: qw ¼ rw  cw  Vw

dTw dt

ð6:30Þ

The heat from fluid into solid should be equal to the use of the Newton cooling law: rw  cw Vw

@Tw ¼ h  As ðTw  Ts Þ @t

ð6:31Þ

Similarly, the heat change of the solid should be equal to that supplied from fluid as given by: rs  cs  Vs

@Ts ¼ h  As ðTw  Ts Þ @t

ð6:32Þ

It should be noted that the sign of heat flux is a plus sign (+). Rewriting Equation (6.31) yields the following: Ts ¼ Tw þ

rw cw vw @Tw  h  AS @t

ð6:33Þ

If the derivation of Equation (6.32) with respect to time is inserted into Equation (6.30), one easily gets the following: @ 2 Tw @T þa w ¼0 @t2 @t

ð6:34Þ

where a ¼ h  As

rw  cw  Vw þ rs  cs  Vs rw  cw  Vw  rs  cs  Vs

ð6:35Þ

The solution of Equation (6.34) is obtained as follows: Tw ¼ C1 þ C2 eat

ð6:36Þ

128 Laboratory and in-situ tests

30

Integral coefficients C1 and C2 of Equation (6.36) are obtained from the following conditions: Tw ¼ Ti

at

t¼0

and

T1 ¼ Tf

at

t¼1

ð6:37Þ

as C1 ¼ Tf ; C2 ¼ Ti  Tf

ð6:38Þ

Using integral constants given by Equation (6.38), Equation (6.32) becomes: Tw ¼ Tf þ ðTi  Tf Þeat ;

@Tw ¼ aðTi  Tf Þeat @t

ð6:39Þ

The average temperature of solid is obtained by inserting Equation (6.39) into Equation (6.33) as: Ts ¼ Tf 

rw  cw  Vw ðT  Tf Þeat r s  Cs  Vs i

ð6:40Þ

As Ts ¼ To at time t ¼ 0, Equation (6.40) can be rewritten as: Tf  To rw  cw  Vw ¼ Ti  Tf rs  cs  Vs

ð6:41Þ

Inserting Equation (6.41) into Equation (6.40) yields: Ts ¼ Tf  ðTf  To Þeat

ð6:42Þ

The temperature difference between the solid and enveloping fluid can be obtained from Equations (6.40) and (6.42) as: DTws ¼ ðTw  Ts Þ ¼ ðTi  To Þeat

ð6:43Þ

Therefore, if the values of To , Tf , Ti , rw , rs , Vw , Vs , cw are known, the specific heat coefficient of solid can be easily obtained. For example, the specific heat coefficient of water is 4.1783–4.2174 J g/K1 for a temperature range of 0–90°C. As the thermal properties of water remain almost constant for the given temperature range, the water would be used as fluid in the experimental setup. After obtaining the specific heat coefficient, the coefficient a is obtained from Equation (4.34), (4.36) or (4.37) using the curve-fitting technique to experimental response. Then using the value of coefficient a, the value of cooling coefficient is obtained from Equation (4.30). For determining the thermal conductivity coefficient(k), the following approach is used. Fourier law may be written for a one-dimensional situation as: q ¼ kA

@T @x

ð6:44Þ

Laboratory and in-situ tests

31

129

Assuming the specimen has a length (L) and using the Newton’s cooling law, we may write the following relationship: hADT ¼ kA

DT L

ð6:45Þ

Equation (6.44) can be rewritten, and the following relation holds between cooling coefficient and thermal conductivity: k ¼hL

ð6:46Þ

The characteristic length of a solid sample can be obtained from the volume of the solid from the following relationship: pffiffiffiffiffi L ¼ 3 Vs ð6:47Þ Linear thermal expansion coefficient (l) is defined as: l¼

1 dL  L dT

ð6:48Þ

dL where L is the length of sample. dT is the variation of length of sample with respect to temperature variation, and it is determined under the unstrained condition or 100 gf load on the sample. If the variation of length of sample at the equilibrium state with respect to the initial length before the commencement of the experiment is measured, it is straightforward to obtain the linear expansion coefficient. Similarly, width or diametrical changes can be also measured, and thermal expansion coefficients can be evaluated from the variation of side length or diameter for a given temperature difference. The technique described in this subsection is unique and quite practical considering the labor required in other techniques. The device for determining the thermal properties of geo-materials consists of a thermostat cell equipped with temperature sensors. The fundamental features of this device are illustrated in Figure 6.36. In the experiments, the temperature of sample, water, air and thermostat are measured. The method utilizes the thermal properties of water, whose properties remain to be the same up to 90ºC, to infer the thermal properties of geo-material substances. If the continuous measurements of temperatures are available, one can easily infer the thermal properties from the following equations as follows. Specific heat of geo-material

cs ¼

rw  cw  Vw Ti  Tf rs  Vs Tf  To

ð6:49Þ

where rw is the density of water, cw is the specific heat coefficient of water, Vw is the volume of water, rs is the density of sample, cs is the specific heat coefficient of sample, Vs is the volume of sample, Ti is the initial temperature of water, To is the initial temperature of sample, and Tf is the equilibrium temperature. The heat conduction coefficient (a) is obtained from fitting experimental results to the following equation: DTws ¼ ðTw  Ts Þ ¼ ðTi  To Þeat

ð6:50Þ

130 Laboratory and in-situ tests

32

Figure 6.36 Illustration of experimental setup

If the heat conduction coefficient (a) is determined, then Newton’s cooling coefficient is determined from the following equation: h¼

a rw  cw  Vw  rs  cs  Vs  As rw  cw  Vw þ rs  cs  Vs

ð6:51Þ

Finally, thermal conductivity coefficient is obtained from the following equation: k ¼hL

ð6:52Þ

where L is the characteristics sample side length. If the sample temperature can be measured, it will be very easy to determine the specific heat coefficient of the sample and subsequent properties. This is possible for granular materials since the temperature sensor can be embedded in the center of the sample. However, it is quite difficult to determine the equilibrium temperature Tf for solid samples. Therefore, the following procedure is followed for this purpose: • •

Step 1: Determine the heat conduction coefficient (a). Step 2: Plot the following equation in time space: Ts ¼ Tw  ðTi  To Þeat

• •

ð6:53Þ

Step 3: Determine the peak value from Equation (6.52), and assign it as equilibrium temperature Tf . Step 4: Then proceed to determine the rest of thermal properties using the procedure previously described.

TEMPERATURE (°C)

80

70

INADA GRANITE

t=2.63, T=69.2

Tcell Tthermos Tsample-top-surface Tair Tcell-T thermos

70 60

t=7.5, T=69.4

50

50 Tdif=(T w0-T t0)*exp(- 0.00867 * t)

40

40 30 20

60

30

Ts=T w-(T w-T t)*exp(-b*t)

0

20

40 TIME (minute)

60

20

131

TEMPERATURE DIFFERENCE ( °C)

Laboratory and in-situ tests

33

Figure 6.37 Application of the procedure to Inada granite sample

Figure 6.37 shows the application of the method to a cylindrical Inada granite sample. The temperature of the sample at the top was also measured. As noted in the figure, the temperature at the top of the sample achieves the peak value before the computed response. Since the temperature of the sample is averaged over the total volume of the sample at the proposed temperature, the computed sample temperature achieves its peak value later then that at the top surface of the sample. Thermal conductivity could be measured using the TK04 system described by Blum (1997). This system employs a single-needle probe (Von Herzen and Maxwell, 1959), heated continuously, in a half-space configuration for hard rock. The needle probe is a thin metal tube that contains a thermistor and a heater wire. The needle is assumed to be approximately an infinitely long, continuous medium; the temperature near the line source is measured as a function of time. If it is assumed that the sediment or rock sample to be measured can be represented as a solid in a fluid medium, it is then possible to determine a relationship between thermal diffusivity and thermal conductivity. With this assumption, the change in temperature of the probe as a function of time is given to a good approximation by (Von Herzen and Maxwell, 1959): q  a  ð6:54Þ TðtÞ ¼ ln 4 t 2 4k Ba where T = temperature (°C), q = heat input per unit time per unit length (W m1), k = thermal conductivity of the sediment or rock sample (W m°C1), t = time after the initiation of the heat (s), α= thermal diffusivity of the sample (m2 s1), B = a constant (1.7811), and a = the probe radius (m).

132 Laboratory and in-situ tests

34

This relationship is valid when t is large compared with a2/α. A plot of T versus ln(t) yields a straight line, the slope of which determines k (Von Herzen and Maxwell, 1959).

6.4

Tests for seepage parameters

6.4.1

Falling head tests

When rock is quite permeable, falling head tests, which utilize the deadweight of fluid, are also used for determining the permeability of rocks and discontinuities. In this subsection, analytical solutions for falling head tests for longitudinal flow and radial flow conditions are derived. (a)

Longitudinal falling head test method

Experimental setup used for this kind of test is shown in Figure 6.38 (Aydan et al., 1997). As seen from the figure, two manometers having cross sections a are assumed to be attached to both ends of the sample. During a test, the change of pressure and velocity of flow can be measured through these manometers. The level h2 of water at the lower tank is assumed to be constant in the following formulation. When an experiment starts, flow rate inside the pipe can be given as: vp ¼ a

@h1 @t

ð6:55Þ

where h1 is the level of water inside the manometer (1). At a given time, flow rate through the cross-section area A of the specimen is given by: vt ¼ vA

Figure 6.38 Illustration of longitudinal falling head test

ð6:56Þ

Laboratory and in-situ tests

35

133

It is assumed that flow rate through the specimen should be equal to the flow rate of the pipe. Then the pressure gradient in the specimen can be given in the following form: @p ðh  h2 Þ  rg 1 @x L

ð6:57Þ

where r is density, and g is gravitational acceleration. Substituting Equation (6.56) into Equation (6.55) and equalizing the resulting equation to Equation (6.54) yields the following differential equation for the change of water height h1 : @h1 kArg @t ¼ LaZ h1  h2

ð6:58Þ

where L is sample length. Solution of the preceding differential equation is: h1 ¼ h2 þ Ceat

ð6:59Þ

where a¼

kA rg La Z

If initial conditions are given by h1 ¼ h10

at

t¼0

where h10 is water height at manometer 1 at t ¼ 0. Thus the integration coefficient C is obtained as follows: C ¼ h10  h2

ð6:60Þ

Inserting the preceding integration coefficient in Equation (6.58) yields the following: Dh at ¼ lnð Þ Dho

ð6:61Þ

where Dh ¼ h1  h2 ; Dho ¼ h10  h2 . If a is substituted in the preceding equation, the following expression for permeability is obtained: h10 h2

k¼ (b)

La lnð h1 h2 Þ Z A rg t

ð6:62Þ

Radial falling head test method

Experimental setup used for this of kind test is shown in Figure 6.39 (Aydan et al., 1997). As seen from the figure, a manometer is placed on the top of the cylindrical hole drilled in the middle of test specimen. The cross-section area of this manometer is denoted by Ah. During the test, the change of pressure and velocity of flow can be measured with this manometer. The level h2 of water at the outer container is assumed to be constant. When experiment starts, flow rate inside the manometer can be given as: q ¼ rgAh

@h1 @t

ð6:63Þ

134 Laboratory and in-situ tests

36

Figure 6.39 Illustration of radial free-fall test

where h1 is the level of water inside the manometer. At a given time, the flow rate through a cross-section area of hole (Ap ) inside the test specimen is given by: vt ¼ vAp

ð6:64Þ

It is assumed that flow rate through the hole perimetry should be equal to the flow rate of the pipe. The pressure gradient in the specimen may be given in the following form: @p @ @ðh1  h2 Þ ðh  h2 Þ   ðrgðh1  h2 ÞÞ ¼ rg ¼ rg 1 @r @r @r rlnðro =ri Þ

ð6:65Þ

Substituting Equation (6.65) into Equation (6.64) and equalizing the resulting equation to Equation (6.63) yields the following differential equation for the change of water height h1 : @h1 k Ap 1 @t ¼ Z Ah ri lnðro =ri Þ h1  h2

ð6:66Þ

Laboratory and in-situ tests

37

135

Solution of the preceding differential equation is: h1 ¼ h2 þ Ceat

ð6:67Þ

where a¼

Ap 1 k ri lnðro =ri Þ Ah Z

Introducing the following initial conditions: h1 ¼ h10 at

t¼0

yields the integration constant C as: C ¼ h10  h2

ð6:68Þ

If integration constant is inserted into Equation (6.67), the following equation is obtained: at ¼ lnð

h1  h2 Þ h10  h2

ð6:69Þ

If a is substituted into the preceding equation, the following expression for permeability is obtained: k ¼ Zri lnðro =ri Þ

(c) (1)

Ah h10  h2 1 lnð Þ Ap h1  h2 t

ð6:70Þ

Transient pulse test method LONGITUDINAL FLOW TESTS

Brace et al. (1968) proposed a transient pulse method for longitudinal flow tests. In this method, the following assumptions are made (Aydan, 1998): • • • •

Fluid flow obeys Darcy’s law. The change of fluid density inside pores with respect time is negligible. The volume of reservoirs (V1 ; V2 ) is constant. The relation between pressure and volumetric strain of fluid is linear.

Permeability is obtained from pressure changes, which are applied to the ends of a specimen, with respect to time (Fig. 6.40). During experiments, flow rate is not measured. The volumetric strain of fluid inside reservoirs V1 ve V2 can be written as follows: ε1V 

DV1 ; V1

ε2V 

DV2 V2

ð6:71Þ

Similarly, for volumetric strain rate of fluid, the following relations can also be written as: ε_ 1V 

DV_ 1 DV_ 2 ; ε_ 2V  V1 V2

ð6:72Þ

136 Laboratory and in-situ tests

38

Figure 6.40 Illustration of transient pulse-method for longitudinal flow

or DV_ 1 ¼ ε_ 1V V1 ;

DV_ 2 ¼ ε_ 2V V2

ð6:73Þ

If the following relations exist between the volumetric strain of fluid and pressure in reservoirs: ε1V ¼ cf p1 ;

ε2V ¼ cf p2

ð6:74Þ

and, the compressibility coefficient (cf ) is constant, for volumetric strain rate, the following relations can be also written: ε_ 1V ¼ cf p_ 1 ; ε_ 2V ¼ cf p_ 2

ð6:75Þ

Flow rates may be defined as: vt1 ¼ DV_ 1 ;

vt2 ¼ DV_ 2

ð6:76Þ

Using Equations (6.72), (6.75) and (6.76), flow rates can be rewritten in the following form: vt1 ¼ cf V1

@p1 @p ; vt2 ¼ cf V2 2 @t @t

ð6:77Þ

Introducing the following boundary conditions: p ¼ p1

at

x ¼ 0;

p ¼ p2

at

x¼L

and using Darcy law, the following relations can be obtained for flow rates:     kA dp1 kA dp2 ; vt2 ¼  vt1 ¼  Z dx x¼0 Z dx x¼L

ð6:78Þ

where A is the cross-section area of the sample, and L is the length of the sample. Pressure gradients in the preceding equations are as follows: dp1 ðp  p2 Þ ;  1 L dx

dp2 ðp  p1 Þ  2 L dx

Laboratory and in-situ tests

39

137

Inserting the preceding equation into Equation (6.78) and equating the resulting equation to Equation (6.77) yields the following set of equations: @p1 1 ¼ b ðp1  p2 Þ V1 @t

ð6:79Þ

@p2 1 ¼ b ðp1  p2 Þ V2 @t

ð6:80Þ

where b¼

kA cf ZL

Brace et al. (1968) solved similar equations by using the Laplace transformation technique. Herein, the method of elimination will be used for solving the preceding set of equations (Kreyszig, 1983). Equation (6.79) can be rearranged as follows: p2 ¼ p1 þ

V1 @p1 b @t

ð6:81Þ

Taking the time derivative of these equations, the following expression is obtained: @p2 @p1 V1 @ 2 p1 ¼ þ @t @t b @t2

ð6:82Þ

Substituting Equations (6.81) and (6.82) into Equation (6.80) and rearranging the resulting equation yields the following homogeneous differential equation: @ 2 p1 @p þa 1 ¼0 2 @t @t

ð6:83Þ

where a¼b

V1 þ V2 V1 V2

The general solution of this differential equation is: p1 ¼ C1 þ C2 eat

ð6:84Þ

Introducing the following initial conditions: p1 ¼ pi

at

t ¼ 0;

p1 ¼ pf

at

t¼1

where pi is the applied initial pressure at Reservoir 1 (V1 ), pf is final pressure at the end of the test, yielding the integration constants C1 and C2 as: C1 ¼ pf ; C2 ¼ pi  pf

ð6:85Þ

Inserting these integration constants into Equation (6.84 (3.51)) gives the following equation: p1 ¼ pf þ ðpi  pf Þeat

ð6:86Þ

138 Laboratory and in-situ tests

40

Taking the time derivative of the preceding equation: @p1 V þ V2 at ¼ ðpi  pf Þb 1 e @t V1 V2

ð6:87Þ

Substituting Equations (6.86) and (6.87) into Equation (6.79) and rearranging yields the following equation: p2 ¼ pf  ðpi  pf Þ

V1 at e V2

ð6:88Þ

For the following initial condition for P2 : p2 ¼ p0

at

t¼0

Equation (6.87) takes the following form: ðpi  pf Þ ¼ ðpf  p0 Þ

V2 V1

ð6:89Þ

The preceding equation can be rewritten in a different way for pi  p0 as follows: ðpi  pf Þ ¼ ðpi  p0 Þ

V2 V1 þ V2

ð6:90Þ

Inserting this equation into Equation (6.85) and rearranging yields the following: p1  pf V1 þ V2 at ¼ lnð Þ ð6:91Þ pi  p0 V2 where a¼

kA V1 þ V2 cf LZ V1 V2

From the preceding equations, one gets the following equation to compute permeability: k¼

Zcf L V1 V2 Dp V2 1 lnð o Þ Dp V1 þ V2 t A V1 þ V2

ð6:92Þ

where Dp ¼ p1  pf ; Dpo ¼ pi  po . When gas is used as a permeation fluid, p1 and p2 are replaced with U1 ð¼ p21 Þ and U2 ð¼ p22 Þ, and permeability can be calculated using the same relation previously given. If the volume of Reservoir 2 (V2 ) is much greater than the volume of Reservoir 1 (V1 ), (V2 ?V1 ) (for instance, outer side of specimen is open to air) p0 ve pf given in the preceding equation will be equal to atmospheric pressure (pa ). For this particular case, Equation (6.91) takes the following form: k¼

Zcf LV1 pi  pa 1 lnð Þ A p1  pa t

ð6:93Þ

For different values of a, the relations between the normalized pressure change and time and natural logarithm of the normalized pressure change and time for transient pulse tests were computed and are shown in Figure 6.41. As seen in Figure 6.41, there is a linear

Laboratory and in-situ tests

41

139

Figure 6.41 Computed time–pressure relations

relation between the natural logarithm of normalized pressure change and time. In both figures, time is taken as a unitless parameter. However, unit depends on the description of the problem. For instance, it can be year, day or second. Theoretical and experimental curves for a transient pulse test on a rock salt (halite) specimen are shown in Figure 6.42. As seen from this figure, there is a slight difference between the curve obtained from the test and the curve from the theory, particularly in the initial stages. In order to obtain permeability value, the linear part of normalized pressure change and time relation is generally used to compute permeability for the selected range.

(2)

RADIAL TRANSIENT PULSE METHOD

The transient pulse method is also extended to radial flow by Aydan et al. (1997) This method is fundamentally very similar to that for longitudinal flow (Fig. 6.43). The only differences are associated with the pressure gradient and surface area at the inner and outer radii. Volumetric strain of fluid inside reservoirs V1 ve V2 can be written as follows: ε1V 

DV1 ; V1

ε2V 

DV2 V2

ð6:94Þ

Figure 6.42 An experimental result on rock salt Source: From Aydan and Üçpirti, 1997

Figure 6.43 Illustration of the radial transient pulse test

Laboratory and in-situ tests

43

141

Similarly, for volumetric strain rate of fluid, the following relations can also be written as: ε_ 1V 

DV_ 1 DV_ 2 ; ε_ 2V  V1 V2

ð6:95Þ

or DV_ 1 ¼ ε_ 1V V1 ;

DV_ 2 ¼ ε_ 2V V2

ð6:96Þ

If the following relation exists between volumetric strain of fluid and pressure: ε1V ¼ cf p1 ;

ε2V ¼ cf p2

ð6:97Þ

and, compressibility coefficient (cf ) is constant, for volumetric strain rate, the following relation can be also written: ε_ 1V ¼ cf p_ 1 ; ε_ 2V ¼ cf p_ 2

ð6:98Þ

Flow rate may be given as: vt1 ¼ DV_ 1 ;

vt2 ¼ DV_ 2

ð6:99Þ

Using Equations (6.96), (6.98) and (6.99), flow rate can be rewritten in the following form: vt1 ¼ cf V1

@p1 ; @t

vt2 ¼ cf V2

@p2 @t

ð6:100Þ

Introducing the following boundary conditions: p ¼ p1

at

r ¼ r1 ;

p ¼ p2

at

r ¼ r2

and using Darcy’s law, then the following relation can be obtained for flow rate:     kAp1 dp1 kAp2 dp2 vt1 ¼  ; vt2 ¼  ð6:101Þ Z dr r¼r1 Z dr r¼r2 where Ap1 is the surface area of the pressure injection hole, and Ap2 is the area of the pressure release surface. Pressure gradients in the preceding equations are as follows: dp1 1 ðp1  p2 Þ  ; r1 lnðr2 =r1 Þ dr

dp2 1 ðp2  p1 Þ  r2 lnðr2 =r1 Þ dr

ð6:102Þ

Inserting the preceding equation into Equation (6.101) and equalizing the resulting equation to Equation (6.100 (3.67)) yields the following set of equations: Ap1 ðp1  p2 Þ @p1 ¼ b @t V1 r1 lnðr2 =r1 Þ

ð6:103Þ

Ap2 ðp1  p2 Þ @p2 ¼b @t V2 r2 lnðr2 =r1 Þ

ð6:104Þ

142 Laboratory and in-situ tests

44

where b¼

k cf Z

Equation (6.102 (3.70)) can be rearranged as follows: p2 ¼ p1 þ

V1 r1 lnðr2 =r1 Þ @p1 bAp1 @t

ð6:105Þ

Taking the time derivative of the preceding equation, the following expression is obtained: @p2 @p1 V1 r1 lnðr2 =r1 Þ @ 2 p1 ¼ þ bAp1 @t @t @t2

ð6:106Þ

Substituting Equations (6.105) and (6.106) into Equation (6.104) and rearranging the resulting equation yields the following homogeneous differential equation: @ 2 p1 @p þa 1 ¼0 2 @t @t

ð6:107Þ

where a¼b

V2 r2 Ap1 þ V1 r1 Ap2 lnðr2 =r1 ÞV1 V2 r2 r1

General solution of this differential equation is: p1 ¼ C1 þ C2 eat

ð6:108Þ

Introducing the following initial conditions: p1 ¼ pi

at

t ¼ 0;

p1 ¼ pf

at

t¼1

yields the integration constants C1 and C2 as: C1 ¼ pf ; C2 ¼ pi  pf

ð6:109Þ

Inserting these integration constants into Equation (6.108) gives the following equation: p1 ¼ pf þ ðpi  pf Þeat

ð6:110Þ

Taking the time derivative of the preceding equation: V2 r2 Ap1 þ V1 r1 Ap2 at @p1 ¼ ðpi  pf Þb e @t V2 r2 V1 r1 lnðr2 =r1 Þ

ð6:111Þ

Laboratory and in-situ tests

45

143

Substituting Equations (6.110) and (6.111) into Equation (6.103) and rearranging yields the following equation: p2 ¼ pf  ðpi  pf Þ

V1 r1 Ap2 at e V2 r2 Ap1

ð6:112Þ

For the following initial condition for p2 : p2 ¼ p0

at

t¼0

Equation (6.111) takes the following form: ðpi  pf Þ ¼ ðpf  p0 Þ

V2 r2 Ap1 V1 r1 Ap2

ð6:113Þ

The preceding equation can be rewritten in a different way for pi  p0 as follows: ðpi  pf Þ ¼ ðpi  p0 Þ

V2 r2 Ap1 V1 r1 Ap2 þ V2 r2 Ap1

ð6:114Þ

Inserting this equation into Equation (6.110), and rearranging yields the following: p1  pf V1 r1 Ap2 þ V2 r2 Ap1 at ¼ lnð Þ pi  p0 V2 r2 Ap1

ð6:115Þ

where a¼

k V2 r2 Ap1 þ V1 r1 Ap2 cf Z V2 r2 V1 r1 lnðr2 =r1 Þ

From the preceding equations, one can use the following equation to compute permeability: k ¼ Zcf

V2 r2 Ap1 V2 r2 V1 r1 lnðr2 =r1 Þ Dpo 1 lnð Þ V2 r2 Ap1 þ V1 r1 Ap2 Dp V2 r2 Ap1 þ V1 r1 Ap2 t

ð6:116Þ

When gas is used as a permeation fluid, p1 and p2 are replaced with U1 ð¼ p21 Þ and U2 ð¼ p22 Þ, and permeability can be calculated using the same relation as previously given. If the volume of reservoir 2 (V2 ) is much greater than the volume of reservoir 1 (V1 ), (V2 ?V1 ) (for instance, the outer side of specimen is open to air) p0 ve pf given in the preceding equation will be equal to atmospheric pressure (pa ). For this particular case, Equation (6.116 (3.83)) becomes: k ¼ Zcf

V1 r1 lnðr2 =r1 Þ pi  pa 1 lnð Þ p1  pa t A p1

ð6:117Þ

144 Laboratory and in-situ tests

46

Figure 6.44 An experimental result on granite

Theoretical and experimental curves for a transient pulse test on the granite specimen are shown in Figure 6.44. As seen from this figure, there is a slight difference between the curve obtained from the test and the curve from the theory, particularly in the initial stages. In order to obtain permeability value, the linear part of normalized pressure change and time relation is generally used to compute permeability for the selected range.

References Aydan, Ö. (1998) Finite element analysis of transient pulse method tests for permeability measurements. The 4th European Conf. on Numerical Methods in Geotechnical EngineeringNUMGE98, Udine. pp. 719–727. Aydan, Ö. (2016a) Time Dependency in Rock Mechanics and Rock Engineering. CRC Press, Taylor and Francis Group. p. 241. Aydan, Ö. (2016b) Considerations on Friction Angles of Planar Rock Surfaces with Different Surface Morphologies from Tilting and Direct Shear Tests. ARMS2016, Bali. Aydan, Ö. (2008) New directions of rock mechanics and rock engineering: Geomechanics and Geoengineering. 5th Asian Rock Mechanics Symposium (ARMS5), Tehran, 3–21. Aydan, Ö. & Üçpırtı, H. (1997) The theory of permeability measurement by transient pulse test and experiments. Journal. of the School of Marine Science and Technology, Tokai University, 43, 45–66.

47

Laboratory and in-situ tests

145

Aydan, Ö. & Ulusay, R. (2002) Back analysis of a seismically induced highway embankment during the 1999 Düzce earthquake. Environmental Geology, 42, 621–631. Aydan, Ö., Akagi, T. & Kawamoto, T. (1993) Squeezing potential of rocks around tunnels; theory and prediction. Rock Mechanics and Rock Engineering, 26(2), 137–163. Aydan, Ö., Ulusay, R. & Tokashiki, N. (2014) A new rock mass quality rating system: Rock Mass Quality Rating (RMQR) and its application to the estimation of geomechanical characteristics of rock masses. Rock Mech Rock Eng, 47, 1255–1276. Aydan, Ö., Akagi, T., Okuda, H. & Kawamoto, T. (1994) The cyclic shear behaviour of interfaces of rock anchors and its effect on the long term behaviour of rock anchors. Int. Symp. on New Developments in Rock Mechanics and Rock Engineering, Shenyang. pp. 15–22. Aydan, Ö., Shimizu, Y. & Kawamoto, T. (1995) A portable system for in-situ characterization of surface morphology and frictional properties of rock discontinuities. Field Measurements in Geomechanics, 4th International Symposium. Bergamo, pp. 463–470. Aydan, Ö., Ücpirti, H. & Turk, N. (1997) Theory of laboratory methods for measuring permeability of rocks and tests (in Turkish). Bulletin of Rock Mechanics, Ankara, 13, 19–36. Aydan, Ö., Ito, T. & Rassouli, F. (2016a) Chapter 11: Tests on creep characteristics of rocks. CRC Press., London, pp. 333–364. Aydan, Ö., Tokashiki, N., Tomiyama, J., Iwata, N., Adachi, K. & Takahashi, Y. (2016b) The Development of a Servo-control Testing Machine for Dynamic Shear Testing of Rock Discontinuities and Soft Rocks. EUROCK2016, Ürgüp. pp. 791–796. Aydan, Ö., Tokashiki, N. & Edahiro, M. (2016c) Utilization of X-Ray CT Scanning technique in Rock Mechanics Applications. ARMS2016, Bali. Aydan, Ö., Ohta, Y., Kiyota, R. & Iwata, N. (2019) The evaluation of static and dynamic frictional properties of rock discontinuities from tilting and stick-slip tests. 46th Rock Mechanics Symposium of Japan. Tokyo, pp. 105–110. Barton, N.R. & Choubey, V. (1977) The shear strength of rock joints in theory and practice. Rock Mechanics, 10, 1–54. Bieniawski, Z.T. (1974) Geomechanics classification of rock masses and its application in tunnelling. Third Int. Congress on Rock Mechanics, ISRM, Denver, IIA, 27–32. Birch, F. (1950) Flow of heat in the Front Range, Colorado, Geological Society of America Bulletin, 61, 567–630. Blum, P. (1997) Physical properties handbook. ODP Technical Note, 26. Brace, W.F., Walsh, J.B. & Frangos, W.T. (1968) Permeability of granite under high pressure. Journal of Geophysical Research, 73(6), 2225–2236. Brown, E.T. (1981) Suggested Methods for Rock Characterization, Testing, Monitoring. Pergamon Press, Oxford. Carslaw, H. & Jaeger, J. (1959) Conduction of Heat in Solids. 2nd edition. Oxford University Press, Oxford, 510pp. Clark, S.P., Jr. (1966) Handbook of physical constants. Geological Society of America Memoir, 97, 587p. Hibino, S., 2007. Necessary Knowledge of Rock Mass for Engineers (Gijutsusha ni hitsuyo na ganban no chishiki), Kajima Pub. Co., Tokyo (in Japanese). Hondros, G. (1959) The evaluation of Poisson’s ratio and the modulus of materials of low tensile resistance by the Brazilian (indirect tensile) tests with particular reference to concrete. Australian Journal of Applied Sciences, 10, 243–268. Hudson, J.A. & Ulusay, R. (2007) The Complete ISRM Suggested Methods for Rock Characterization, Testing, Monitoring, 1974–2006, ISRM. Ankara. Hudson, J.A., Crouch, S.L. & Fairhurst, C. (1972) Soft, stiff and servo-controlled testing machines: A review with reference to rock failure. Engineering Geology, 6, 155–189. Jaeger, J.C. & Cook, N.G.W. (1979) Fundamentals of Rock Mechanics, 3rd edition. Chapman & Hall, London. pp. 79, 311.

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48

Kawamoto, T., Tokashiki, N. & Ishizuka, Y. (1980) On uniaxial compression test of rock-like materials using a new type of high stiff testing machine Japan. Material Science Journal (Zairyo), 30 (322), 517–523. Kreyszig, E. (1983) Advanced Engineering Mathematics. John Wiley & Sons, New York. Popov, Y., Pribnow, D., Sass, J., Williams, C. & Burkhardt, H. (1999) ISRM Suggested Methods for determining thermal properties of rock samples. Characterisation of rock thermal conductivity by high-resolution optical scanning. Geothermics, 28, 253–276. Popov, Y., Beardsmore, G., Clauser, C. & Roy, S. (2016) ISRM suggested methods for determining thermal properties of rocks from laboratory tests at atmospheric pressure. Rock Mechanics and Rock Engineering. Rassouli, F.S., Moosavi, M., & Mehranpour, M.H. (2010) The effects of different boundary conditions on creep behavior of soft rocks. The 44th U.S. Rock mechanics Symposium & 5th U.S. Canada symposium, Salt Lake City, Utah. Rummel, F. & Fairhurst, G. (1970) Determination of the post-failure behaviour of brittle rock using a servo-controlled testing machine. Rock Mechanics, 2, 189–204. Sass, J., Stone, C. & Munroe, R. (1984) Thermal conductivity determinations on solid rock – A comparison between a steady-state divided-bar apparatus and a commercial transient line-source device. Journal of Volcanology and Geothermal Research, 20(1–2), 145–153. Somerton, W.H. (1992) Thermal properties and temperature-related behavior of rock/fluid systems. Developments in petroleum science, 37. Elsevier Science Publishers B.V., Amsterdam, Netherlands, 257p. Tokashiki, N. & Aydan, Ö. (2012). Estimation of Rockmass Properties of Ryukyu Limestone. Asian Rock Mechanics Symposium, Seoul, 725–734. Ulusay, R. (2012) The ISRM Suggested Methods for Rock Characterization, Testing, Monitoring, 2007–2014. Springer, Vienna. Van Heerden, W.L. 1975. In-situ complete stress-strain characteristics of large coal specimens. J.S. Afr. Min. Metall., 75, 207–217. Von Herzen, R.P. & Maxwell, A.E. (1959) The measurement of thermal conductivity of deep-sea sediments by a needle-probe method. Journal Geophysical Research, 69, 1557–1563. Waversik, W.R. & Fairhurst, C. (1970) A study of brittle rock fracture in laboratory compression experiments. International Journal Rock Mechanics and Mining Science, 7, 561–575.

Chapter 7

In-situ stress estimation, measurement and inference methods

The stress state of the Earth is of paramount importance in geomechanics and geophysics. Particularly, the virgin stress state in the Earth’s crust is of great interest in mining and civil engineering since the stability of excavations is very much influenced by that. Geophysicians are also concerned with the stress state of the crust in association with understanding the earthquake mechanism and predicting earthquakes. Many in-situ stress inference techniques are classified into broadly direct or indirect techniques (Amadei and Stephansson, 1997). Direct techniques are generally costly, and they are only utilized for some important structures. The direct techniques utilize boreholes and assume that the surrounding rock behave elastically. However, the acoustic emission (AE) method can be used as a direct stress measurement method as it is less costly and it can be performed under well controlled conditions in laboratory once sampling is done. Indirect stress inference techniques utilizing borehole breakouts, fault striations and earthquake focal mechanism solutions are also proposed and used (Zoback and Healy, 1992; Angellier, 1984; Aydan, 2000a; Aydan and Kim, 2002). Recently a new stress inference technique utilizing damage zone around blast holes was proposed by Aydan (2012), and it was applied to several sites in Japan and Turkey.

In-situ stress estimation methods

7.1 7.1.1

Empirical approaches

The empirical approaches to estimate the crustal stresses are based on some empirical formulas utilizing stress measurements in various engineering projects and some earthquake prediction studies. The measurements of in-situ stresses in various engineering projects were carried out in South Africa (Hast, 1969) first and later in other countries. Measurements indicated that horizontal stresses could be several times vertical stresses in the shields such as the Canadian or Scandinavian shields (Herget, 1986; Stephanson et al., 1986). Furthermore, many measurements in association with the excavations in mining and civil engineering fields and earthquake prediction projects stress measurements in the Earth crust have been undertaken over last 50 years. However, most of the measurements are restricted to a depth below 5000 m. Brown and Hoek (1978) proposed some empirical relations, given here, to estimate the ratio of the horizontal stress to normalized vertical stress. l ¼ 1:0 þ

2 h

ð7:1Þ

148 In-situ stress estimation, measurement and inference methods

7.1.2

2

Analytical approaches

Many proposals for the stress state of the Earth have been made in this century, and these proposals may be classified on the basis of their main characteristics.

7.1.2.1

Approaches assuming the Earth in liquid state

Jeffreys and Bullen (1940) considered that the stress state of the Earth was hydrostatic, and they calculated the pressure of the Earth by considering the variation of density and gravitational acceleration. Anderson and Hart (1976) modified this approach by considering recent findings. Nadai (1950) also derived the following formula by assuming that the density of the Earth is constant and that the gravitational acceleration varies linearly with depth:
2 ! rm go Ro r sr ¼ sy ; sr ¼ ð7:2Þ 1 R0 2 where σr is radial stress, σθ is tangential stress, R0 is the radius of the Earth, r is radial distance, ρm is the mean density of the Earth, and g0 is gravitational acceleration at the Earth’s surface.

7.1.2.2

Approaches assuming the Earth in solid state

Terzaghi and Richart (1952) considered a vertical column and formulated the stress state of the Earth by assuming that lateral strains are zero and the medium is elastic as: sy ¼

v s ; s ¼ rgH 1v r r

ð7:3Þ

where v is Poisson’s ratio, H is the depth from ground surface, ρ is density, and g is gravitational acceleration. Salustowicz (1968) derived the following formula by assuming that the density of the Earth is constant and that it consists of a homogeneous elastic material together with the consideration of a linearly decreasing gravitational acceleration:
2 ! r sr ¼ rm go Ro a 1  ; ð7:4Þ R0
2 ! r ; sy ¼ rm go Ro a 1  b R0 where a¼

3v 1 þ 3v ;b ¼ 10ð1  vÞ ð3  vÞ

ð7:5Þ

In-situ stress estimation, measurement and inference methods

3

7.1.2.3

149

Approaches based on the assumption that the crust and mantle are in solid state and the core is in liquid state

Aydan (Aydan, 1993; Aydan and Kawamoto, 1994) proposed two models, namely: • •

Two-layered model (TLM) Multilayer model (MLM)

Two-layered model (TLM) (Fig. 7.1(a)): By considering that the gravitational acceleration remains constant up to the interface between the lower mantle and the outer core and thereafter decreases linearly (Fowler, 1990), the Earth was modeled as a simple spherical body consisting of a core and a mantle. By introducing the averaged values of physical and Lamé’s constants for each zone, the following formula were developed for radial and tangential stresses in the mantle and the core: Mantle sr ¼



4  R3i R3o 1 1 l þ m Ro  R4i Ro  Ri l þ m Pi  r g  rgr þ  r3 R3o R3o  R3i l þ 2m o o R3o R3i r3 l þ 2m o o ð7:6Þ

sy ¼



4  R3i R3o 1 1 l þ m Ro  R4i Ro  Ri 2l þ m r Pi þ r g þ rg þ  2r3 R3o R3o  R3i l þ 2m o o R3o R3i 2r3 l þ 2m o o 2 ð7:7Þ

Figure 7.1 (a) Double-layer model, and (b) multilayer model

150 In-situ stress estimation, measurement and inference methods

4

Core

sr ¼ pi þ


 ri go Ri r2 1  2 ; sy ¼ sr 2 Ri

ð7:8Þ

where λ, μ, λ*, μ* are average Lamé’s constants for the outer and inner zones, respectively, and Ri is the radius of the inner zone: h i 2 R3i R3o l þ m R4 R4 R2i ri go Ri o go r g 1 Ro Ri Ri o i  lrþ2m  4  l 15 R3o R3i l þ 2m o o 4m R2i 3l þ2m R3o R3i h i ð7:9Þ Pi ¼ R3i Ro3 Ri Ri 1 1 1 þ 3l þ2m  R3 þ 3l R3 R3 4m R2 o

i

i

o

Multilayered model (MLM) (Fig. 7.1(b)): Since physical and Lamé’s constants vary with depth, the preceding approach was extended to simulate a multilayered structure of the Earth in order to have a better solution for its stress state. The solutions for each layer were essentially similar to those for the two-layered model. The displacement for a layer numbered i in the core is obtained by assuming the Earth consists of n layers having constant physical and mechanical parameters in each respective layer as: ui ¼ Ai1 r þ Ai2

1 ri go r3  r2 10li Rmci

ð7:10Þ

By introducing the following conditions at the interface between layer i and layer i + 1: sr ¼ Pi;iþ1 at r ¼ Ri;iþ1 the integration constants Ai1 and Ai2 are obtained as Ai1 ¼

  R2i;iþ1 1 Pi;iþ1 þ ri go ; Ai2 ¼ 0 3li 2Rmc

where Rmc is the radius of the interface between the mantle and the core. Similarly, for a typical layer numbered k in the mantle, displacement uk can be obtained in the following form: uk ¼ Ak1 r þ Ak2

1 rk go r2  r2 lk þ 2mk 4

ð7:11Þ

By introducing the following conditions at the interface between layer k and layer k + 1 and the interface between layer k and layer k  1: sr ¼ Pk;k1 at r ¼ Rk;k1 sr ¼ Pk;kþ1 at r ¼ Rk;kþ1

In-situ stress estimation, measurement and inference methods

5

151

the integration constants Ak1 and Ak2 can be obtained as:  k
4   R3k;k1 R3k;kþ1 Rk;kþ1  R4k;kþ1 Pk;kþ1 R3k;kþ1  Pk;k1 R3k;k1 1 l þ mk r g Ak1 ¼ k þ k o R3k;k1 R3k;k1 R3k;k1 R3k;kþ1 3l þ 2mk R3k;kþ1  R3k;k1 lk þ 2mk  k 
4  Rk;kþ1  R4k;kþ1 1 R3k;k1 R3k;kþ1 l þ mk rg þ ðpk;kþ1  Pk;k1 Þ A ¼ k 3 4m Rk;kþ1  R3k;k1 lk þ 2mk k o R3k;k1 R3k;k1 k 2

Introducing the continuity condition of displacement, that was, ui  ui+1 = 0 at each interface r = Ri,i+1 yielded a linear equation system for interlayer pressures P1,2, … Pi,i1,Pi,i+1, Pn1,n. The solution of this equation system together with the condition of σr = 0 at the Earth’s surface gives the pressures on both sides of layers, from which the displacement and stresses of each layer could be easily obtained. 7.1.2.4

Approaches assuming that the crust is in solid state and plastic

Possible stress state in the crust associated with normal and thrust faulting were first discussed by Hubbert (1951) and Anderson (1951). This approach assumes that the crust is in plastic state and that the materials obey a Mohr-Coulomb–type yield criterion. The stress states associated with various faulting regimes are summarized and discussed in a textbook by Jaeger and Cook (1979) as follows: Normal faulting regime sv ¼ qsh þ sc ; q ¼

1 þ sin 1  sin

ð7:12Þ

Thrust faulting regime sH ¼ qsv þ sc

ð7:13Þ

Strike-slip faulting regime sH ¼ qsh þ sc

ð7:14Þ

where σv = ρgH,  is the friction angle, σc is uniaxial strength of crust, σH is the maximum horizontal stress, and σh is the minimum horizontal stress. As noted from the preceding equations, one of the stress components is always indeterminate. Parameters q and σc may be temperature and/or confining pressure dependent. This fact is also considered in some publications (i.e. Brace and Kohlstedt, 1980; McGarr, 1980; Shimada, 1993). 7.1.2.5

Comparisons

The approaches summarized in this subsection are compared with one another, and their main characteristics are discussed herein. Figure 7.2 shows plots of the distributions of

152 In-situ stress estimation, measurement and inference methods

6

Figure 7.2 Comparison of various approaches

radial and tangential stresses and the ratio of tangential stress to radial stress in the Earth obtained from various models. Nadai’s and Salustowicz’s models underestimate the radial stress distributions. While the lateral stress coefficient remains 1 for Nadai’s model, it is remarkable to note that the coefficients predicted by the TLM, MLM and Salustowicz’ model increase exponentially near the Earth’s surface and goes to infinity. The TLM predicts the radial stress distribution, which is very close to the pressure distribution reported by Anderson and Hart (1976). While the tangential stress by the TLM is equal to the radial one throughout the core, its distribution varies within the mantle. At the core–mantle interface, there is a discontinuity between tangential stresses of the core and the mantle, and the tangential stress is less than the radial stress in the mantle. However, it becomes greater than the radial stress, and it has a finite value at the Earth’s surface while the radial stress is nil there, which implies that the ratio of the horizontal stress to the vertical stress should be infinite at the Earth’s surface. The vertical stress distribution obtained from Terzaghi’s proposal is close to that of the radial stress, but it becomes very large in the core, which is quite different from those by other solutions. The lateral stress distribution is also different from the tangential stress distribution obtained from other solutions, and it could not explain that why lateral stresses should be large at the Earth’s surface. Amadei and Savage (1985) proposed a model to explain large horizontal stresses near the Earth’s surface by introducing the anisotropy of elastic constants of rock mass while using this one-dimensional column model. However,

In-situ stress estimation, measurement and inference methods

7

153

it must be noted that the large horizontal stresses near the Earth’s surface can also be obtained by taking into account the sphericity of the Earth even the rock mass remains isotropic. A better solution for the stress state in the Earth is provided by the multilayered model (Fig. 7.2). In the calculations, the Earth is assumed to be consisted of nine concentric spherical layers. While the radial stress distribution almost coincides with those by the twolayered model and pressure model, the tangential stress distribution is remarkably different from that by the two-layered model. Since the mechanical and physical properties used in the analysis differ in each layer, the tangential stress at each interface becomes discontinuous. Furthermore, the value of the tangential stress is much less than that predicted by the two-layered model or Salustowicz’s model. 7.1.3 7.1.3.1

Numerical approaches The approach assuming that the Earth is in fluid state

The original approach is based on the assumption that the Earth is in fluid state. This approach was proposed by Jeffrey and Bullen (1940), and Anderson and Hart (1976) used a better model for the density distribution of the Earth and utilized the finite different technique and spherical symmetry and obtained the pressure distribution in the Earth. 7.1.3.2

The approach assuming that the Earth is a spherical symmetric thermo-elasto-plastic body

Triaxial tests on rocks were undertaken by many experimenters in the fields of geomechanics and geophysics under high confining pressure and high temperature regimes (i.e. von Karman, 1911; Edmonton and Paterson, 1972; Byerlee, 1978; Shimada, 1993; Hirth and Tullis, 1994; etc.). These experiments showed that the mechanical behavior of rocks changes from a brittle behavior to a ductile behavior as the confining pressure increases (Fig. 7.3(a)). Furthermore, the strength decreases as temperature increases, as shown in Figure 7.3(b). Taking into account these facts, Aydan (1995) proposed a thermo-plastic yield criterion: s1  s3 ¼ ½S1  ðS1  S0 Þeb1 s3 eb2 T

ð7:15Þ

where S0 is the uniaxial compressive strength at room temperature, S1 is the ultimate deviatoric compressive strength at room temperature, T is temperature, b1,b2 are physical constants, σ1 is the maximum principal stress, and σ3 is the minimum principal stress (confining pressure). Figure 7.3 shows a plot of the preceding yield criterion in the space of temperature T and confining pressure σ3. Aydan (1995) assumed that the Earth is a spherical symmetric body so that the governing equilibrium equation is given by: @sr s  sy ¼ rg þ2 r @r r

ð7:16Þ

where ρ and g are density and gravitational acceleration, which may vary with depth. The constitutive law for elastic behavior is written as: ( ) " #( ) sr 2m þ l 2l εr ¼ or fsg ¼ ½Dfεg ð7:17Þ sy εy l 2ðm þ lÞ

154 In-situ stress estimation, measurement and inference methods

8

Figure 7.3 Triaxial compression tests on quartz Source: Data from Hirth and Tullis, 1994

It should be noted that Lamé’s constants in the preceding equation can vary with depth. Using the general procedure of finite element discretization and taking variations on δu, with the use of Gauss divergence theorem and the following conditions: du ¼ 0 on Gu ; and ^t ¼ 0 on Gt the weak form of Equation (7.16) takes the following form: Z

Z O

ðsr dεr þ 2sy dεy ÞdO ¼

rgdudO

ð7:18Þ

O

where dΩ = 4πr2dr. Let us assume that the displacement in a given element is approximated by the following expression: u ¼ ½NfUg

ð7:19Þ

where [N] is shape function, and {U} is nodal displacement vector. Using the preceding approximate form and the constitutive law (7.17), the following finite element form is obtained for a typical finite element: ½KfUg ¼ fFg

ð7:20Þ

where Z ½K ¼

½B ½D½BdO T

Oe

fFg ¼

Z rg½N dO T

Oe

In-situ stress estimation, measurement and inference methods

9

155

If a linear type shape function is chosen as given here: Ni ¼

rj  r j r  ri ; L ¼ rj  r ;N ¼ L L

ð7:21Þ

the stiffness matrix [K] and load vector {F} given in Equation (7.20) are specifically obtained as follows: " # ( 3 ) 2 2 3 4p K11 K12 rgp rj þ rj ri þ rj ri  3ri ; fFg ¼ ð7:22Þ ½K ¼ 2 L K21 K22 3 3rj3 þ rj2 ri þ rj ri2  ri3 where D1 þ 2D2 þ D1 3 ðrj  ri3 Þ  ðD2 þ D1 Þrj ðrj2  ri2 Þ þ D1 rj2 ðrj  ri Þ 3 D þ 2D2 þ D1 3 ðrj  ri3 Þ  ðD2 þ D1 Þri ðrj2  ri2 Þ þ D1 ri2 ðrj  ri Þ K22 ¼ 1 3 rj þ ri 2 D þ 2D2 þ D1 3 K12 ¼ K21 ¼  1 ðrj  ri3 Þ  ðD2 þ D1 Þ ðrj  ri2 Þ þ D1 ri rj ðrj  ri Þ 3 2

K11 ¼

D1 ¼ 2m þ l; D2 ¼ 2l

D1 ¼ 4ðm þ lÞ

In analyses, it was assumed that the stress–strain response of rocks constituting the Earth exhibit an elastic–perfectly plastic behavior. The elastic constants of rocks were taken from a report by Anderson and Hart (1976). The initial stiffness technique was chosen as the iteration technique to deal with the plastic behavior of rocks in finite element analyses (Owen and Hinton, 1980). The temperature distribution of the Earth was input as known by using a distribution reported in a textbook (p. 248) by Fowler (1990) (Fig. 7.4). A series of case studies, given here, were carried by Aydan (1995) (Fig. 7.5): CASE 1: The Earth was in liquid state (hydrostatic). CASE 2: The crust and mantle were elastic solids, and the core was in liquid state (nonhydrostatic). CASE 3: The crust and mantle were elasto-plastic solids, and the core was in liquid state (isothermic: room temperature). CASE 4: The crust and mantle were thermo-elasto-plastic solids, and the core was in liquid state (nonisothermic). Parameters b1 ; b2 ; So ; S1 of the yield criterion were chosen as 0.2 GPa1 , 0.0014 o C 1 , 0 MPa, 5 GPa, respectively. Figure 7.6 shows the distributions of radial and tangential stresses and the ratio of tangential stress to radial stress in the lithosphere for each case. The elasto-plastic analyses showed that the whole crust and mantle became plastic. Large tangential stresses seen in Case 2 were dissipated when the plastic behavior of the crust and mantle was considered. The tangential stresses dissipated in the mantle increased both radial and tangential stresses in the core. The consideration of the thermo-elasto plastic behavior of the mantle and the crust of the Earth further decreased the deviatoric stresses in the mantle and in the crust.

Figure 7.4 Illustration of spherical model of the Earth and material properties Source: Arranged from Aydan, 1994; Fowler, 1990

Figure 7.5 The stress state of the Earth Source: From Aydan, 1995

In-situ stress estimation, measurement and inference methods

11

157

Figure 7.6 Computed radial and tangential stresses in lithosphere

The lateral stress coefficient, which has a value of infinity at the ground surface for CASE 2, also decreased in magnitude when the plastic behavior was considered. In the next series of parametric studies, the uniaxial compressive strength parameter So was varied from 0 to 40 MPa in increments of 20 MPa, while the other parameters were kept the same as those in the previous thermo-elasto-plastic analysis. The computed distributions of radial and tangential stresses and the ratio of tangential stress to radial stress, together with in-situ observations (Aydan and Paşamehmetoğlu, 1994) and predictions by Shimada (1993) for a depth of 4 km in the Earth’s crust, are plotted in Figure 7.7. The radial stress (vertical stress) almost coincided with each other (Fig. 7.7(a)). They are almost equal to the vertical stress calculated from rgH (Shimada, 1993), and it is a good fit to in-situ measurements. The tangential stresses (horizontal stress) for each value of So fit fairly well to in-situ measurements for a depth of 1 km from the ground surface (Fig. 7.7(b)). However, finite element computations overestimate the tangential stresses at depths greater than 1 km. The ratio of tangential stress to radial stress for each value of So provides upper bounds for in-situ measurements (Fig. 7.7(c)). The discrepancy between computed and measured values may be eliminated if a thermo-elasto-visco-plastic constitutive law is employed in finite element analysis.

In-situ stress measurement methods

7.2 7.2.1

Stress relief (overcoring) method

The overcoring method has been used for many decades. This method is based on the principle of releasing the stress state of instrumented rock core inside a borehole (Fig. 7.8) by overcoring. It is also a widely used technique all over the world. There are two variations of this technique: • •

Leeman’s method or doorstopper method, which utilizes a flat-ended borehole Sugawara-Obara’s method, which utilizes a semi-spherical-ended borehole. Another variation to this approach is proposed by Kobayashi and Mizuta, who used a conicalended borehole (conical-ended borehole overcoring (CCBO).

158 In-situ stress estimation, measurement and inference methods

12

Figure 7.7 Comparison of computed and measured in-situ stresses Source: From Aydan and Kawamoto, 1997

The method involves creating a circular hole, placing some device for measuring strains or displacements into the hole, and then drilling over the top of that hole to relieve the stress and thus cause a deformation change. The measured deformation, together with the rock modulus and Poisson’s ratio measured in the laboratory, is used to calculate the magnitude and directions of the stresses existing in the rock by considering the geometry and elasticity of surrounding rock. 7.2.2

Flat jack method

This method is based on the principle of releasing the stress state of the cavity first and then restoring it by pressurizing the flat jack (Fig. 7.9). It is also a widely used technique all over the world. 7.2.3

Hydro-fracturing and sleeve fracturing method

The hydro-fracturing method of stress measurement involves pressurizing a fluid into a section of borehole bounded by two packers sufficient to fracture the rock around the

13

In-situ stress estimation, measurement and inference methods

159

Figure 7.8 Various stress measurement techniques Source: Arranged from Sugawara and Obara, 1993; Obert et al., 1962; Leeman, 1964

hole. The fracture is intended to occur parallel to the axis of the borehole. The method assumes that one of the principal stresses (generally vertical stress) is parallel to the borehole axis, and then the maximum and minimum stresses in the plane perpendicular to the borehole axis are obtained from pressure readings (Fig. 7.10). In addition, the tensile strength of the surrounding rock is necessary for computing stress components. The sleeve-fracturing method applies the internal pressure in the borehole through mechanical jacks instead of pressurized fluid. 7.2.4

Acoustic emission (AE) Method

The acoustic emission method utilizes the Kaiser effect for inferring the stress state. Since the stress tensor is a symmetric second-order tensor, it has six independent components. As a result, it is necessary to perform uniaxial or triaxial tests in six different directions. According to the Kaiser effect (Kaiser, 1953), it is expected that the acoustic emission response will differ when the (deviatoric) stress level exceeds the one that material was previously subjected to, as illustrated in Figure 7.11. The fundamental complexity in rock

160 In-situ stress estimation, measurement and inference methods

14

Figure 7.9 Illustration of stress measurement by the flat jack method Source: Arranged from ISRM, 1986

mechanics is that the Earth’s crust has a stress history. In actual experiments, one may find one or several stress levels. The question is how to select or to define the one, which reflects the current stress level that the rock was subjected to before unloading. The acoustic emission method (AEM) was first suggested by Kanagawa et al. (1976, 1981) for inferring the in-situ stresses. Since then, many attempts were made to measure the stress state by this method. Nevertheless, the cost of acoustic emission measurement equipment was quite high in the past, and it could not become a widely accepted method. Recently, the cost of equipment becomes less, and the experiments can be easily performed

Figure 7.10 Illustration of stress measurement by (a) the hydrofracturing and (b) sleeve fracturing methods Source: Arranged from Haimson, 1987; Stephanson et al., 1986

Figure 7.11 Illustration of stress measurement by AE method and Kaiser effect Source: Arranged from Daido et al., 2003

162 In-situ stress estimation, measurement and inference methods

16

Figure 7.12 Acoustic emission response of mortar during cyclic loading

under laboratory conditions. As a result, there is a growing interest in stress measurements by this method (Holcomb, 1993; Hughson and Crawhord, 1987; Seto et al., 1999; Wang et al., 2000; Watanabe and Tano, 1999; Watanabe et al., 1994, 1999). The validity of the AE method is always questioned as other methods of in-situ stress measurements were already carried out at a given site. Daido et al. (2003) carried out some fundamental experiments, and they found that the Kaiser effect is a sound concept to infer the stress state to which rock has been previously subjected. However, there may be some deviations depending upon the stress level with respect to the strength of the rock (Fig. 7.12). Particularly, if the stress level is greater than the unstable crack threshold value defined by Bieniawski (1967), the inferred stresses may be less than the actual level that rock was subjected to. On the other, the inferred stress levels may be greater than the actual ones if the stress level is lower than the unstable crack propagation threshold stress level. Although many researchers associate AE events in rocks with micro-cracking, it would be better to utilize the concept of permanent straining associated with the plastic behavior of materials, macroscopically as also seen in Figure 7.12. If such a concept is used, the fundamental of the acoustic emission method would be based on the firm principles of mechanics.

7.3 7.3.1

In-situ stress inference methods Borehole breakout method

When induced stresses exceed the strength of rock around the borehole, the methods based on the theory of elasticity cannot be used. Borehole breakouts are formed by spalling of fragments of the wellbore in a direction parallel to the minimum stress. Borehole spalling occurs during drilling and progresses with time. The identification and analysis of borehole breakouts as a technique for in-situ measurement of stress orientation and magnitude and for identifying orientation of both naturally occurring and induced fractures have received a great deal of attention. The inference of dimensions of borehole breakout is based on the Kirsch solution for circular holes with the use of the Mohr-Coulomb yield criterion. In practice, the dimensions of borehole breakout can be measured by borehole camera, laser scanning or mechanical caliper. This method is based on the principle of determining the stress state by observing the borehole breakouts in boreholes (Fig. 7.13). Zoback

17

In-situ stress estimation, measurement and inference methods

163

Figure 7.13 Borehole breakout method for stress estimation

applied the Kirsch solution together with Mohr-Coulomb yield criterion to estimate the shape, extent and orientation of breakouts to estimate the stress state. Although this technique is attributed to Zoback, the main principle was already proposed by Kastner (1962) for circular tunnels and was used by many (Talobre, 1957). 7.3.2

Fault type method

Possible stress states in the crust associated with faulting were first discussed by Hubbert (1951) and Anderson (1951). This approach assumes that the crust is in a plastic state and obeys a Mohr-Coulomb–type yield criterion. The stress states associated with various faulting regimes are outlined and discussed in a textbook by Jaeger and Cook (1979). When this technique is used, one of the stress components is always indeterminate. However, the vertical stress sz for strike–slip faulting regime is assumed to be equal to ðsH þ sh Þ=2 (Sibson, 1974). Parameters of the Mohr-Coulomb yield criterion may be temperature and/or confining pressure dependent. This fact is also considered in some publications (i.e. Brace and Kohlstedt, 1980; Shimada, 1993). 7.3.3

Fault striation method

Aydan (2000a) proposed a new method to infer the crustal stresses from the striations of the faults or other structural geological features, which may be quite useful in studying the

164 In-situ stress estimation, measurement and inference methods

18

Figure 7.14 View of fault striation at Tanyeri near Erzincan and illustrations for notation for fault striation method

stress state associated with past and current earthquakes. Figure 7.14 illustrates the notation used in this method, and the final expression to infer the stress state is: 2

q l12 6 b l2 6 2 6 6 1 l32 6 6 0 ll 6 1 2 6 4 0 l1 l3 0 l2 l3

m21 m22 m23 m1 m2 m1 m3 m2 m3

2l1 m1 2l2 m2 2l3 m3 ðl1 m2 þ l2 m1 Þ ðl1 m3 þ l3 m1 Þ ðl2 m3 þ l3 m2 Þ

2l1 n1 2l2 n2 2l3 n3 ðl1 n2 þ l2 n1 Þ ðl1 n3 þ l3 n1 Þ ðl2 n3 þ l3 n2 Þ

9 8 9 38 NIII > 2m1 n1 n21 > > > > > > > > > > > 7> > > > > Nxx > 2m2 n2 n22 > > > > > 7> > > > > > > > > 7< 7 Nyy = < n23 = 2m3 n3 7 ¼ > ðm1 n2 þ m2 n1 Þ 7 n1 n2 > > Nxy > > > > 7> > > > > > > > 7> > > > N > > > > ðm1 n3 þ m3 n1 Þ 5> xz > > n1 n3 > > > > > : : ; > ; Nyz ðm2 n3 þ m3 n2 Þ n2 n3

ð7:23Þ

where


9 8  > 31 > > > cos 45  > sy sz > > 2 > <
= ny nz 7 ;  5 > > cos 45 þ > > > > by bz 2 > > ; : 0 8
9  > > > > 2 3 1 > cos 135  8 9 > > sx sy sz > > 2 > < < l3 =
 = 6n n n 7 sIII ¼ m3 ¼ 4 x  y z5 > > : ; cos 45  > > > bx by bz n3 2 > > > > > ; : 0

8 9 2 sx < l1 = 6 sI ¼ m 1 ¼ 4 n x : ; bx n1

n ¼ fnx ; ny ; nz g; s ¼ fsx ; sy ; sz g; nx ¼ sin p sin d; ny ¼ sin p cos d; nz ¼ cos p sin i sx ¼ cos p sin d sin i; sy ¼  cos p cos d sin i; sz ¼ cos p sin i b ¼ s  n; b ¼ fbx ; by ; bz g; bx ¼ sy nz  sz ny ; by ¼ sz nx  sx nz ; bz ¼ sx ny  sy nx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi syy syz s 1 s s b ¼ II ¼ ðb  b2  4cÞ; NIII ¼ III ; Nxx ¼ xx ; Nyy ¼ ; Nzz ¼ 1; Nxy ¼ ; sIII 2 sz sz sz sz syz s ð1 þ 6a2 Þðq þ 1Þ Nyz ¼ ; Nxz ¼ xz ; sz ¼ gh; b ¼ ; 1  3a2 sz sz ð1  3a2 Þðq2 þ 1Þ  qð1 þ 6a2 Þ 2sin 1 þ sin ;q ¼ c¼ ; a ¼ pffiffiffi 2 1  3a 1  sin 3ð3 þ sinÞ

19

In-situ stress estimation, measurement and inference methods

165

and p is plunge, d is dip direction, i is striation angle, h is overburden, γ is unit weight,  is friction angle, and c is cohesion. 7.3.4

Focal plane solution method

Aydan (Aydan and Kim, 2002; Aydan, 2003; Aydan and Tokashiki, 2003) recently advanced this method to infer the stress state of the Earth crust from focal plane solutions. The focal plane solutions used in geoscience are derived by assuming that the pure-shear condition holds. As a result of this assumption, one of the principal stresses is compressive while the other one is tensile in focal plane solutions. This condition may also imply that the friction angle of the fault is nil. Therefore, the principal stresses are inclined at an angle of 45 degrees with respect to the normal of slip direction. This condition is used to determine the P-axis and T-axis in focal plane solutions. Each focal plane solution involves the fault plane on which the sliding takes place and the auxiliary plane. The normal of the auxiliary plane corresponds to the slip vector, and it is orthogonal to the neutral plane on which the P-axis and T-axis exist. 7.3.5

Core-disking method

When drilling is done in brittle rock under high stress, it is often reported that core disking occurs. Jaeger and Cook (1963) was first to investigate the stress state resulting in coredisking (Fig. 7.15a). The stress state is very complicated during the drilling operation as the compressive and torsional tractions are imposed on the rock stubs by the drill bit in the close vicinity of the borehole end. Numerical analyses clearly indicate local hightensile stresses in the closed vicinity of the borehole end. Nevertheless, the ridges on the surface of disking fractures are indicative of maximum in-situ stress perpendicular to the borehole axis (Fig. 7.15b). 7.3.6

Blast hole damage method

Aydan (2013) proposed a method to infer the in-situ stresses from the damage zones around the blast holes in this chapter, the blast hole-damage method (BDM). The several

Figure 7.15 Core disking at Kaore Underground Powerhouse site

166 In-situ stress estimation, measurement and inference methods

20

applications of the method to several sites where in-situ stress states are obtained by using direct or indirect techniques, and its validity is discussed in view of the measurements from other methods. The stress state around a circular cavity in an elastic medium under biaxial far-field stresses are first obtained Kirsch (1898). These solutions are modified to incorporate the effect of uniform internal pressures (Jaeger and Cook, 1979). In a polar coordinate system, the radial, tangential and shear stresses around the circular cavity can be written in the following forms:


a2  s  s
 a 2 a4  a2 s10 þ s30 10 30  cos2ðy  bÞ þ pi sr ¼ þ3 ð7:24aÞ 1 14 2 2 r r r r
a2  s  s
a4  a2 s þ s30 30 þ 10 cos2ðy  bÞ  pi ð7:24bÞ sy ¼ 10 1þ 1þ3 2 2 r r r
a2 a4  s10  s30 sin2ðy  bÞ ð7:24cÞ þ3 try ¼ 14 2 r r

where σ10, σ30 is far-field principal stresses, a is radius of hole, r is radial distance, β is inclination of σ10 far-field stress from horizontal, θ is angle of the point from horizontal, and Pi is internal pressure applied onto the hole perimetry.

The yield criteria available in rock mechanics are: Mohr-Coulomb s1 ¼ sc þ qs3

ð7:25aÞ

Drucker-Prager aI1 þ

pffiffiffiffi J2 ¼ k

ð7:25bÞ

Hoek and Brown (1980) s1 ¼ s3 þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi msc s3 þ ss2c

ð7:25cÞ

Aydan (1995) s1 ¼ s3 þ ½S1  ðS1  sc Þeb1 s3 eb2 T

ð7:25dÞ

where 1 2 2 2 I1 ¼ sI þ sII þ sIII ; J2 ¼ ððsI  sII Þ þ ðsII  sIII Þ þ ðsIII  sI Þ 6 2sin 6ccos 1 þ sin ; k ¼ pffiffiffi ; c is cohesion;  is friction angle; q ¼ ; a ¼ pffiffiffi 1  sin 3ð3 þ sinÞ 3ð3 þ sinÞ σ1 is ultimate deviatoric strength, T is temperature, and m, s, b1, b2 are empirical constants.

In-situ stress estimation, measurement and inference methods

21

167

Figure 7.16 Estimated yield zones around the blast hole

Mohr-Coulomb and Drucker-Prager yield criterion are a linear function of confining or mean stress, while the criteria of Hoek-Brown and Aydan are of the nonlinear type (Fig. 7.16). Furthermore, Aydan’s criterion also accounts for the effect of temperature. However, the effect of temperature is omitted in Figure 7.16 for the sake of comparison. If the yield criterion is chosen to be a function of minimum and maximum principal stresses, they can be given in the following form in terms of stress components given by Equation (7.24): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s  s 2 sy þ sr y r 2 ð7:26aÞ þ þ try s1 ¼ 2 2 s þ sr  s3 ¼ y 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s  s 2 y r 2 þ try 2

ð7:26bÞ

The damage zone around the blast hole under high internal pressure can be estimated using one of the yield criteria just listed. It should be noted that the yielding is induced by the high internal pressure in the blast hole, which is essentially different from in-situ stressinduced borehole-breakout. In other words, there will always be a damage zone around the blast hole perimeter when the blasting technique is employed. The blast hole pressure depends on the characteristics of the surrounding medium, the amount, layout and type of explosive, blasting velocity and the geometry of the blast hole. The blast hole pressure ranges from 100 to 10 GPa (i.e. Jaeger and Cook, 1979; Brady and Brown, 1985). First we assume that the properties of surrounding rock have the values as given in Table 7.1, and the blast hole (internal) pressure has a value of 400 MPa. The rock chosen roughly corresponds to an igneous rock such as granitic rock. The maximum farfield stress is inclined at an angle of 30 degrees from horizontal, and the lateral stress

168 In-situ stress estimation, measurement and inference methods

22

Table 7.1 Values of in-situ stress parameters and properties of yield functions σ10 (MPa)

k

σc (MPa)

σt (MPa)

ϕ (o)

m

σ1 (MPa)

b

50.0

4.0

100.0

5.0

60

20

400

31.39

Figure 7.17 Illustration of computation of strain and stress increments using the crustal deformations Source: Arranged from Aydan, 2000b

coefficient has a value of 4. Figure 7.16 shows the example of computation for the given conditions. The largest yield zone is obtained for Hoek-Brown (HB) criterion while the tension cutoff criterion (T) results in smaller yield zone. The criterion of the Aydan estimates a slightly larger yield zone than the Mohr-Coulomb (MC) criterion. It is very interesting to note that the yielding propagates in the direction of maximum far-field stress. In other words, the elongation direction of the yield zone would be the best indicator of the maximum far-field stress in the plane of the blast hole. 7.3.7

Global positioning system method

Although it is difficult to obtain total stresses from GPS measurements, it may be possible to determine principal stress variations and their orientations in the tangential plane to the Earth surface from the processing of GPS measurements as proposed by Aydan (2000b, 2004) and Aydan et al. (2011, 2013c). However, if the coaxiality condition between stress rate and total stress in the crust holds, it may be also possible to infer the orientation of maximum and minimum horizontal stresses in the crust. It should be noted that it is geometrically possible to compute strain rate components in a plane tangential to the Earth’s surface from the variation of positions of stations at a given time interval, although it is unlikely to obtain the components of strain rate components on other planes. For this purpose, a mesh and interpolation technique similar to those in the finite element method are utilized, and the nodes are selected to be corresponding to the location of GPS antennas (Fig. 7.17). The element type can be selected as desired with the consideration of the GPS network. Displacement increments of each node for a given time interval are used to compute the strain increments first, and then stress increments were computed with the incorporation of constitutive models of the Earth’s crust. Aydan (2000b) used Hooke’s law as a preliminary approach to compute the strain and stress increments.

In-situ stress estimation, measurement and inference methods

23

7.4

169

Comparisons

In this section, the applications of the method are described briefly, together with direct stress measurements.

7.4.1 7.4.1.1

Location-based comparison Yucca mountains (USA)

The first application was done to the site, the Yucca mountains Nuclear Waste Disposal Site. It should be noted that none of the stress relief methods can be applied since the rock around the hole becomes plastic at such depths. The same methods were also used at the Yucca mountain site for the same reasons (Stock et al., 1992). Faults in this site are generally normal faults, and their average dip directions and dip are 285 degrees and 76 degrees, respectively. The striation direction was chosen as 90 degrees. An earthquake occurred near Yucca mountain on June 14, 2002. Computed results for the fault striation (FSM) and focal mechanism parameters (FMS) of that earthquake are given Table 7.2, together with in-situ stress measurements (Meas.). From the comparison of computed and measured results, it can be said that the computations are quite close to the measurements. 7.4.1.2

Underground powerhouse of Okumino pumped storage scheme (Japan)

The next example of application was done to the site called Okumino in Gifu Prefecture of Japan. The Neodani fault exists at a distance of 5 km in the southwest of Okumino Powerhouse. In 1891, an earthquake with a magnitude 8 occurred along this fault. The maximum horizontal displacement of the fault was 8 m, and the northeast side of the fault was downthrown except one location called Midori, where the northeast side was thrown 6 m upward and 4 m laterally. Matsuda (1974) suggested that such a movement occurs as a result of compressive forces at the fault bend as seen in the Midori site. Therefore, this fault is regarded to be a left-lateral strike-slip fault, and the striation direction was taken as 0 degrees. Stress measurements were carried out by Chubu Electric Power Company during the construction of a powerhouse (Tsuchiyama et al., 1993). Computed results, together with in-situ stress measurements, are given in Table 7.3. From the comparison of computed and measured results, it can be said that the computations are once again quite close to the measurements. Table 7.2 Comparison of inferred and measured in-situ stresses at Yucca Mountain Nuclear Waste Site Method

FSM Meas. FMS

σ1

σ2

σ3

σ1 σv

d1

P1

σ2 σv

d2

P2

σ3 σv

d3

P3

1.05 – 1.14

285 – 164

74 – 58

0.73 – 0.79

195 – 27

0 – 25

0.35 – 0.38

105 – 288

16 – 19

σh σv

σH σv

dσ H

0.41 0.51 0.45

0.73 0.81 0.86

15 20–30 12

170 In-situ stress estimation, measurement and inference methods

24

Table 7.3 Comparison of computed results with measurements for Okumino Method

FSM Meas.

σ1

σ2

σ3

σ1 σv

d1

P1

σ2 σv

d2

P2

σ3 σv

d3

P3

1.47 1.50

110 104

5 4

1.01 1.07

230 204

80 73

0.49 0.29

20 13

9 17

σh σv

σH σv

dσ H

0.50 0.11

1.46 1.50

290 283

σh σv

σH σv

dσ H

1.44 1.67 0.99 1.43 0.94

2.21 2.86 2.18 2.70 1.96

84 104 91 88 102

Table 7.4 Comparison of in-situ stress measurements by various methods Method

σ1

σ2

σ3

σ1 σv

σ2 σv

d1

P1

90 105 89 89

16 16 25 1

1.59 1.69 1.01 1.59

d2

P2

σ3 σv

187 197 353 179

23 6 13 28

0.76 0.83 0.66 0.84

d3

P3

329 307 239 358

62 73 62 63

FSM BDM CBT HFM AEM

2.30 3.02 2.49 2.70

7.4.1.3

Planned underground powerhouse of Kaore pumped storage scheme (Japan)

An extensive in-situ stress measurement program was carried out at this site since the preliminary in-situ stress measurement program by the overcoring method yielded unusually high in-situ stresses that were not observed previously at other powerhouse construction sites in Japan (Ishiguro et al., 1997). In-situ stress measurements involve using overcoring methods, such as the borehole deformation method (BDM), conically ended borehole method (CBT), hydrofracturing method (HFM) and acoustic emission method (AEM). In addition to these methods, a stress inference method called fault striation method (FSM) (Aydan, 2000a) based on fault striations was also used. The dominant fault on the site was denoted as F-1. Its dip direction and dip were 297 and 51, respectively. The rake angle (striation angle) was 24 degrees. The friction angle of the fault was set to 30 degrees for the stress inference computations. Table 7.4 compares the inferred in-situ stress with the measurements obtained from various methods. 7.4.1.4

Antique underground city of Derinkuyu (Turkey)

The fourth example of application was associated with Derinkuyu Underground City. The author and his research group carried out stress measurements for Derinkuyu using the AE method, GPS method and FSM (Aydan et al., 1999; Watanabe et al., 1999; Aydan and Ulusay, 2013). A normal fault with a 10–20 cm thick brecciate zone at the seventh floor of Derinkuyu Underground City was observed. The faults in this site are generally normal faults, and their average dip directions and dip are 290 degrees and 83 degrees. The striation direction was chosen as 85 degrees. Computed results, together with in-situ stress measurements, are given in Table 7.5. From the comparison of computed and measured results, it can be once again said that the computations are quite close to the measurements. These results are confirmed by those obtained from the GPS method (Aydan et al., 1999; Aydan, 2000b). Figure 7.18 shows computer output for this site, together with a view

In-situ stress estimation, measurement and inference methods

25

171

Table 7.5 Comparison of computed results with measurements for Derinkuyu Method

FMS FSM AEM

σ1

σ2

σ3

σ1 σv

σ2 σv

σ3 σv

1.04 1.12 1.12

d1

P1

84 301 305

77 67 62

0.71 0.77 0.68

d2

P2

181 199 183

1 5 17

0.35 0.37 0.33

d3

P3

271 107 85.0

13 23 23

σh σv

σH σv

dσ H

0.48 0.38

0.77 1.04

15 18

Figure 7.18 View of the fault in the antique Derinkuyu Underground City and inferred stress state

of the fault. Furthermore, this example of application indicates that this technique could be useful in archeological underground structures. Similar studies were also done on the stability assessment of underground tombs of pharaohs in the Luxor region of Egypt by the author and his group (Aydan et al., 2008; Aydan and Geniş, 2004). 7.4.2 7.4.2.1

Application to regional crustal stress estimations Turkey

The crustal stresses were measured at only three locations in Turkey until 1996, and two of these measurements were done by the Middle East Technical University of Turkey. Three additional measurements by the acoustic emission (AE) method were performed through an international joint research study on the living environment of the Derinkuyu Underground City of Turkey, which was supported by the Monbusho grant-in-aid (09044154). There is an urgent necessity to clarify the present stress state of Turkey and to monitor its variation in time and in space thereafter. Aydan and his colleagues from Nihon University, Hacettepe University, and Istanbul Technical University have been continuing stress measurements by using the acoustic

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Figure 7.19 (a) View of stress measurement by the AE method at Hacettepe University, (b) sampling locations in Turkey for the AE method

emission technique since1998 (Aydan et al., 1999). The acoustic emission method was chosen because it is quite cheap to perform under laboratory conditions. At the initial stage of the research, the samples gathered from several locations in Turkey were brought to Japan, and stress measurements were carried out at Nihon University (Watanabe et al., 1999). Later, Watanabe and Tano (1999) Tano and Watanabe (1998) developed a portable system, and the tests were carried out at the rock mechanics laboratories of Hacettepe University (HU) and Istanbul Technical University (ITU) in Turkey (Fig. 7.19). The number of locations is presently 28, and the locations are shown in Figure 7.19. Table 7.6 summarizes stress measurements at several locations by the AE method. Figure 7.20 compares the stress measurements by the AE method, together with available in-situ stress measurements worldwide (Aydan, 1995) and some empirical relations for horizontal stress ratio over the overburden pressure developed by Hoek and Brown (1980) and Aydan and Paşamehmetoğlu (1994), as well as those from the thermo-elasto-plastic finite element analyses by Aydan (1995). It is interesting to note that the stress measurements by the AE method for Turkey are quite consistent with measurements using other methods and empirical formulas (Aydan and Kawamoto, 1997). Since the stress inference from the AE method is often questioned, the stress state inferred from focal plane solutions of large earthquakes, fault striations and GPS measurements are compared to check the validity of AE stress measurements (Aydan et al., 2011). Figure 7.21 compares the maximum horizontal stress directions inferred from different methods. So far, the in-situ stress inference results from the AE method reasonably agree with those inferred from other methods. It seems that the AE may be a quite useful tool for both engineers and geoscientists to infer crustal stresses. Furthermore, the study carried out by the author and his colleagues may be quite unique in which there is no applications of the AE method of this scale in any country. It would be challenging and interesting to compare the results reported in this chapter with the measurements by other direct in-situ stress measurements to be performed in the future. The North Anatolian Fault (NAF) and North-East Anatolian Fault (NEAF) run almost parallel to the shoreline of the Black Sea. The NAF is known to be 1500 km long and

Table 7.6 Principal stresses, maximum and minimum horizontal stresses and their directions for the different sampling locations in Turkey by the AE method Location

σ1 (MPa)

P1 (°)

d1 (°)

σ2 (MPa)

P2 (°)

d2 (°)

σ3 (MPa)

P3 (°)

d3 (°)

H (m)

Eynez İzmir Çayırhan Küre Dodurga Zonguldak İstanbul Ankara Bayburt Denizli Kırşehir Sivas Bigadiç Kestelek Eskişehir Seydişehir (Doğankuzu) Seydişehir (Mortaş) Ordu Emet Kayseri Orhaneli Demirbilek Marmara Gebze Avanos Derinkuyu

4.82 3.72 5.81 4.80 5.00 12.7 1.89 0.43 1.25 0.92 1.24 1.41 2.31 1.83 0.55 5.14 3.47 9.96 1.66 0.62 3.91 2.93 1.70 1.21 0.37 0.45

93 146 297 58 10 141 342 297 15 188 91 283 249 260 185 70 341 357 34 336 352 133 92 95 256 305

69 50 36 67 68 12 52 59 6 4 4 16 36 11 39 14 26 29 38 17 54 50 49 78 79 62

2.79 2.03 4.11 4.00 4.48 12.0 1.28 0.33 0.88 0.79 0.96 1.33 1.66 1.68 0.34 3.20 2.40 5.06 1.42 0.61 2.99 2.62 1.53 0.90 0.25 0.27

3 45 32 177 254 265 75 55 130 283 184 176 147 357 279 180 241 104 141 075 101 349 327 359 50 183

8 9 6 14 10 68 1 16 75 33 40 42 16 35 5 53 21 27 20 25 18 36 21 3 4 17

1.58 0.50 3.40 3.26 4.18 9.47 0.70 0.16 0.64 0.68 0.80 0.92 1.00 0.72 0.18 2.11 2.31 3.46 0.54 0.58 1.44 2.10 0.46 0.47 0.10 0.13

270 308 130 278 161 047 165 153 284 089 357 029 038 155 015 331 117 229 252 216 210 246 232 269 321 85

20 39 54 18 10 24 38 26 14 56 50 43 50 53 51 33 56 48 45 59 29 20 34 12 4 23

177 80 223 81 252 505 60 10 15 23 28 25 70 50 10 100 90 200 45 20 100 110 45 50 18 20

Source: From Watanabe et al., 1999, 2003; Aydan et al., 1999

Figure 7.20 Comparison of stress measurements by the AE method with those from other methods

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Figure 7.21 Comparison of maximum stress directions obtained from the AE method with those from other direct measurement methods and indirect stress techniques for Turkey

splays into several branches in western Turkey. It produces very large earthquakes and presents a great seismic danger to Turkey. The NEAF is less active and produces lesser earthquakes, which are smaller in magnitude as compared with those of the NAF. The Trakya fault (TF), which is thought to be an extension of the Tuzgölü and Eskişehir fault zone in Trakya (Thrace), is also a less active fault, and it produced the Edirne earthquake (Ms 5.2) in 1953. Despite the great importance in the seismicity of the NAF, there are very few stress measurement studies in the vicinity of these three faults. The AE method as a direct stress measurement technique and focal mechanism solutions method (FMSM), fault striation method (FSM) and GPS method were used as indirect stress inference technique to infer the stress state of the northern part of Turkey, which includes the major active faults NAF, NEAF and TF and is delimited by latitudes 39–43 and longitudes 25–45. Figure 7.22 shows the computed and measured results of maximum crustal horizontal stress direction for the northern part of Turkey. Figure 7.23 shows the maximum and minimum horizontal stress ratios normalized by the vertical stress for the same region. Although the data for the AE method and fault striations are still limited for the region, the results are promising to infer the stress state in a regional scale.

In-situ stress estimation, measurement and inference methods

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175

Figure 7.22 Directions of the maximum horizontal stress in Northern Anatolia

Figure 7.23 Comparison of measured and inferred maximum and minimum horizontal stress ratios normalized by the vertical stress

7.4.2.2

Japan

The approach used for Northern Turkey has been also applied to sites in Japan, where insitu stress measurements are available, and their validity has been checked (Aydan, 2013b). The details of the data used in computations can be found in Aydan (2013b). The friction angle of the faults is assumed to be 30 degrees by considering the experimental data of Byerlee (1978). Figure 7.24 shows the stress states associated with the 1891 Nobi-Beya earthquake, which is the greatest intraplate earthquake, and the 2011 Great East Japan earthquake, which is the greatest subduction earthquake, respectively.

176 In-situ stress estimation, measurement and inference methods

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Figure 7.24 Stress states associated with the greatest intraplate and interplate earthquakes in Japan: (a) 1891 Nobi-Beya earthquake, (b) 2011 Great East Japan earthquake

Figure 7.25 Computed maximum and minimum horizontal stress ratios normalized by vertical stress using the focal mechanism solutions Source: From Aydan, 2013b

The maximum lateral stress ratio ranged between 1.3 and 3, while the minimum lateral stress coefficients range between 0.48 and 2.05. It was interesting to note that these values are in accordance with values assumed in the design of large caverns in Japan (Hibino, 2007). Figure 7.25 shows the contours of maximum and minimum lateral stress coefficients, respectively. It seems that lateral stress coefficients are higher in the close vicinity of the subduction zones of major plates in the close vicinity of Japan. From the figure, one

31

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Figure 7.26 Active faults of Japan and the inference of directions and magnitudes of inferred maximum horizontal stresses from the proposed method Source: Arranged from Aydan (2003, 2013b)

can infer that the lateral stresses are quite high in Northern Japan. These results are also in accordance with inferences from reported previously active faults of Japan (Aydan, 2003). Figure 7.26 compares the directions of maximum horizontal stresses obtained from this study with those reported by Saito (1993) and Sugawara and Obara (1993). It is very interesting to note that the computed results are quite similar to those of the in-situ stress measurements. Furthermore, these results confirm the huge difference between the stress states of Euro-Asian and North American plates. 7.4.2.3

Ryukyu Archipelago

There are almost no in-situ stress measurements in the Ryukyu Islands except the location of the Yanbaru pumped storage scheme, although the Ryukyu Islands and their close vicinity are subjected to very large earthquakes from time to time. The authors have been recently investigating fault outcrops in the Ryukyu Islands and its close vicinity (Fig. 7.27a). In this subsection, the crustal stresses in the Ryukyu Islands from the striations of faults, and focal plane solutions are inferred and compared with reported stress measurements. The epicenters of earthquakes obtained by HARVARD (Seismological Division) along the Ryukyu arc are shown in Figure 7.27(c), and the parameters of the earthquakes nearby the Ryukyu Islands were chosen such that they satisfy the conditions that they have a focal depth less than of 30 km and moment magnitude greater Mw 6.0. The normalized stress tensor components by the vertical stress from the fault striations were obtained by using the method described previously. The inferred maximum horizontal stress and its direction from the fault striations and focal mechanism solutions of earthquakes are shown in Figure 7.27, together with the in-situ stress measurement. Figure 7.28 shows the ratio of the maximum horizontal stress ratio as a function of orientation from north. As seen from the figures, the directions and magnitudes of the maximum horizontal stress indicate almost the same tendency. Particularly, the direct stress measurements are very close to those inferred from the focal

Figure 7.27 (a) Views of sampling and fault striation, (b) measured and inferred stresses for Motobu, (c) inferred directions of the maximum horizontal stress and its normalized value Source: Arranged from Aydan and Tokashiki, 2003

3

N

E

S

W

N

3

Focal Mechanism

2

2

In-situ Measurement

σH / σv

σH / σv

Fault Striation

1

0

1

90

180

270

0 360

ORIENTATION FROM NORTH Figure 7.28 Comparison of the variation of measured and inferred maximum horizontal stress ratio with orientation

33

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179

plane solutions and fault striations in the same vicinity. This conclusion is quite similar to those in other sites by the first author (Aydan, 2000a; Aydan and Kim, 2002; Aydan et al., 2002; Watanabe et al., 1999, 2003). The compressive horizontal stress is high in the vicinity of the Ryukyu trench, while its magnitude decreases along Okinawa trough.

7.5

Integration of various direct measurement and indirect techniques for in-situ stress estimation

As mentioned in previous sections, there are several techniques to measure or to infer the stress state in the Earth’s crust. Each method or technique has its own merits and drawbacks. Furthermore, it is very difficult to obtain the true values of in-situ stresses, which are generally scattered. Therefore, the evaluation of in-situ stresses must be crosschecked using several methods or techniques. In any site, it is very likely to encounter several geological features. If the fault striation measurements are possible, the striations should be documented and ordered from new events to old events, and stress inferences should be carried out as a first step. The second step is to search the focal plane solutions of earthquakes with a hypocenter depth less than 33 km deep. The most difficult aspect is how to select the plane, which is associated with the earthquake. If the earthquake caused some surface disturbance, those surface disturbances would yield information to determine the causative fault. Furthermore, the aftershock activity (if it is available) could provide very useful information for selecting the causative fault and determining its geometry (i.e. Aydan et al., 1998). If this information is not available, it must be selected using some available documents on the geology, which may be slightly biased. The third step is the utilization of direct stress measurement method. If the in-situ stress measurement is critical for the structure, then one or several of the direct stress measurement methods should be selected on the basis of economic, environmental and technological considerations. The depth of structure and accessibility to rock block sampling would be quite important factors. As mentioned previously, the acoustic emission (AE) method could be used for this purpose. The utilization of borehole breakouts, fracturing in the vicinity of boreholes, damage around blast holes (Zoback et al., 1985; Aydan, 2012, 2013a; Aydan et al., 2013) could be used as supplementary information for obtaining or checking the results of direct and indirect in-situ stress evaluations. Furthermore, the stress variations utilizing the crustal deformation obtained from GPS measurements with the assumption that increments are coaxial with principal stresses could be one of the important tools to check the in-situ stress evaluations. The final step would be designating the in-situ stress state of the site on the basis of information from direct and indirect techniques. If one or several direct stress measurement method are used, their results should be selected in view of the information from indirect stress inference techniques. If direct stress measurements are not available, the information from the active or recently moved faults and or focal mechanism solutions should be used to evaluate the in-situ stresses. It should be born in mind that in-situ stress state would not remain the same in time (Aydan et al., 2012; Aydan, 2015, 2016). This may be very important issue for long and large engineering structures such as tunnels, underground powerhouses, suspension bridges and arch and gravity dams. The variations may be estimated from the computation

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34

of stress rates using the GPS measurements and/or displacements from the processing of acceleration records by the EPS method. In addition, the inferences from the use of focal plane solutions may be used for the variations of the regional stress state with time.

7.6

Crustal stress changes

The GPS method may be used to monitor the deformation of the Earth’s crust continuously with time. From these measurements, one may compute the strain rates and probably the stress rates. The stress rates derived from the GPS displacement rates can be effectively used to locate the areas with high seismic risk as proposed by Aydan (2000a, 2006, 2013a). Thus, daily variations of derived strain–stress rates from dense, continuously operating GPS networks in Japan and the United States may provide high-quality data to understand the behavior of the Earth’s crust preceding earthquakes. This approach is applied to the 2011 Great East Japan earthquake using the area shown in Figure 7.29 and an element consisting of Oshika, Wakuya and Rifu GPS stations of the GEONET (Aydan, 2013a). Figure 7.30a shows the computed principal stresses, maximum shear stress or disturbing stress and the orientation of the principal stress change. As noted from the figure, there is a release of the principal stress up to 5 MPa. The orientation of maximum principal stress change component was 269.2 degrees, and the orientation was 270.8 degrees after the earthquake while the coseismic change was less than 2 degrees. These changes should be added to those of the original stress field, which is not usually known unless in-situ stress measurements are carried out at the given area.

Figure 7.29 Locations of GPS stations used in computations

In-situ stress estimation, measurement and inference methods

35

181

Sakaguchi et al. (2013) carried out in-situ stress measurements at a depth of 290 m at Kamaishi mine 170 km away to the northeast of the epicenter. The measured in-situ stress were almost twice the ones measured before the earthquake. This implies that the changes of crustal stresses were tremendous. The stress measurement attempt by Lin et al. (2013) above the epicentral area of the earthquake using the borehole breakout method (BBM) did not yield decisive results. Nevertheless, the maximum horizontal stress direction inferred from the BBM was 319 ± 23 degrees. Aydan (2013b) recently computed the stress state of four major earthquakes in the vicinity of the epicentral area, including that the 2011 Great East Japan earthquake from their focal plane solutions using the method of Aydan (Aydan, 2000b; Aydan and Kim, 2002; Aydan et al., 2002). The results of computation are given in Table 7.7, and the associated stereo projection of principal stresses is shown in Figure 7.30(b). The maximum horizontal stress orientations inferred from the focal mechanism solutions are also close to the results reported by Lin et al. (2013). The deviations of measured orientation reported by Lin et al. (2013) from those computed by Aydan (2013b) may be related to the heterogeneity of rock units encountered during the drilling.

Figure 7.30 (a) Variations of stress changes computed from GPS measurements, (b) stress state inferred from the focal plane solution Source: From Aydan, 2013

Table 7.7 Inferred crustal stresses in Japan Method

σ1 σv

1978 Miyagi-oki 2003 Miyagi-hokubu 2008 Iwate-Miyagi 2011 Great East Japan

2.79 2.89 2.83 2.68

d1

P1

σ2 σv

d2

292.3 10.5 1.92 23.2 277.4 8 1.99 7.6 109.4 8.4 1.95 200.4 287.1 14.1 1.84 17.7

P2 4.7 1.8 6.9 2.2

σ3 σv

0.93 0.96 0.94 0.89

d3

P3

σh σv

137.1 110.6 329.3 116.3

78.4 81.8 79.1 75.7

1.92 1.99 1.94 1.84

σ1 σv

2.7 2.8 2.8 2.6

θ 291.2 277.1 288.2 286.4

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Shimada, M. (1993) Two types of brittle fracture of silicate rocks and scale effect on rock strength: Their implications in the earth crust. Proc. Scale Effects in Rock Masses, Lisbon, 55–62. Sibson, R.H. (1974) Frictional constraints on thrust, wrench and normal faults. Nature, 249, 542–544. Stephanson, O., Sarkka, P. & Myrvang, A. (1986) State of stress in Fennoscandia. Int. Symp. on Rock Stress and Rock Stress Measurements, Stockholm, 21–32. Stock, J.M., Healy, J.H., Hickman, S.H. & Zoback, M.D. (1992) Hydraulic fracturing stress measurements at Yucca Mountain, Nevada, and relationship to regional stress field. Journal. Geophysical Research, 90, 8691–8706. Sugawara, K. & Obara, Y. (1993) Measuring rock stress. In Comprehensive Rock Engineering, Chapter 21, Volume 3, Pergamon Press. pp. 533–552. Talobre, J. (1957) The Mechanics of Rocks, Dunod (in French), Paris. Terzaghi, K. & Richart, F.E. (1952) Stresses in rock about cavities. Geotechnique, 3, 57–90. Tsuchiyama, S., Aydan, Ö. & Ichikawa, Y. (1993) Deformational behaviour of a large underground opening and its back analysis. Int. Symp. Assessment and Prevention of Failure Phenomena in Rock Engineering, Istanbul. Pp. 865–870. Watanabe, H. & Tano, H. (1999) In-situ stress estimation of Cappadocia Region using the increment of AE event count rate. Journal of College of Engineering, Nihon University, 41(1), 35–42 (in Japanese). Watanabe, H., Tano, H. & Akatsu, T. (1994) Fundamental study on pre-stress measurement of triaxial compressed rock. Journal of College of Engineering, Nihon University, 35(A), 11–19 (in Japanese). Watanabe, H., Tano, H., Ulusay, R., Yüzer, E., Erdoğan, E. & Aydan, Ö. (1999) The initial stress state in Cappadocia. In: K. Matsui & H. Shimada (eds) Proc of the ’99 Japan-Korea Joint Symposium on Rock Engineering, Fukuoka, Japan. Pp. 249–260. Watanabe, H., Tano, H., Aydan, Ö., Ulusay, R. & Bilgin, A.H., Seiki, T. (2003) The measurement of the in-situ stress state by Acoustic Emission (AE) method in weak rocks. RS-Kumamoto, Int. Symp. On Rock Stress, Kumamoto. Zoback, M.D., et al. (1985) Well bore breakouts and in-situ stress. Journal Geophysical Research, 90, 5523–5530. Zoback, M.D. & Healy, J.H. (1992) In-situ stress measurements to 3.5 km depth in the Cajon Pass scientific research borehole: Implications for the mechanics of crustal faulting. Journal Geophysical Research, 97, 5039–5057. Zoback, M.L. & Zoback, M.D. (1980) State of stress in the conterminous United States. 85, 6113– 6156, 1980.

Chapter 8

Analytical methods

The solution of governing equations of coupled or uncoupled motion, mass transportation and energy transport phenomena requires certain methods, which may be analytical and numerical. When the resulting equations, including initial and/or boundary conditions, are simple to solve, the analytical methods are preferred. As the resulting equations and initial and/or boundary conditions are generally complex in many rock engineering problems, the use of numerical methods such as finite difference, finite element or boundary element methods becomes necessary.

8.1

Basic approaches

The fundamental governing equations presented in Chapter 4 are in the form of ordinary differential or partial differential equations for mass, momentum, energy conservation laws with chosen constitutive laws. The analytical methods basically attempt to solve the equations in their original form, and they can be categorized as intuitive or separation of variable techniques. 8.1.1

Intuitive function methods

This method is fundamentally based on choosing a function intuitively, which satisfy boundary/initial conditions. For example, the well-known Kirsch’s solutions for circular opening under biaxial far-field stresses are based on this approach, and it utilizes Airy’s stress function. The readers are advised to several books such as Jaeger and Cook (1979) on this aspect. 8.1.2

Separation variable method

This method assumes that the solution consists of the convolution of several functions of the independent variables (e.g. Kreyszig, 1983). The insertion of these functions to partial differential equations results in separated ordinary differential equations. Then the integral coefficients are determined so that the boundary and/or initial conditions are satisfied. Particularly, the determination of integral coefficients may be quite cumbersome. This method is also known as Fourier’s method.

188 Analytical methods

8.1.3

2

Complex variable solution

This method is a very powerful tool for the solution of many problems in elasticity. The method was originally devised by Kolosov (1909), and it is further expanded by several SSCB mathematicians. For example, Muskhelishvili (1962) provided a comprehensive textbook on this solution method. Similar textbooks by Milne-Thomson (1960), Green and Zerna (1968) and England (1971) can be found in literature. The well-known textbook by Kreyszig (1993) describes this method. The method is quite powerful in solving the partial differential equations subjected to very complex far-field boundary stress conditions for anisotropic elastic materials. Several applications of this method are described in the textbook by Jaeger and Cook (1979). Gerçek (1996) provided the application of the method for stress concentration around cavities having different shapes under biaxial farfield stresses, in which the integral constants are obtained numerically. The textbook by Verruijt (1970) describes the utilization of this method in seepage problems.

8.2

Analytical solutions for solids

In this section, several specific applications of the analytical solutions are described.

8.2.1

Visco-elastic rock sample subjected to uniaxial loading

Aydan (1997) proposed a method to model the dynamic response of rock samples during loading. In this subsection, this method and several examples of its application to some typical situations are presented. (a)

Theoretical formulation

Let us consider a sample under uniaxial loading as shown in Figure 8.1(a). The force equilibrium of such a sample can be written in the following form (Fig. 8.1(b)): s ¼ Dr ε þ Crε_ þ rH u€

ð8:1Þ

where Dr , Cr , r, H are elastic modulus, viscosity coefficient, density and sample height. If acceleration u€ is uniform over the sample and its strain ε is defined as: ε¼

u H

ð8:2Þ

Equation (8.1) becomes: s ¼ Dr ε þ Drε_ þ rH 2ε€

ð8:3Þ

Let us assume that stress is applied to the sample in the following form (Fig. 8.2): for 0  t  T0 s¼

s0 t T0

ð8:4Þ

Figure 8.1 (a) Uniaxial compression test, (b) its mechanical model

Figure 8.2 Time-history of uniaxial loading

190 Analytical methods

4

for t  T0 s ¼ s0

ð8:5Þ

The solutions of this ordinary differential equation are: Case 1: Roots are real. ε ¼ C1 el1 t þ C2 el2 t þ εp

ð8:6Þ

where l1 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ðCr þ Cr2  4Dr rH 2 Þ;l2 ¼ ðCr  Cr2  4Dr rH 2 Þ 2rH 2 2rH 2

for 0  t  T0 εp ¼

s0 1 ½l l t þ ðl1 þ l2 Þ rH 2 T0 l21 l22 1 2

for 0  t  T0 εp ¼

s0 1 rH 2 l1 l2

Case 2: Roots are same. ε ¼ ½C1 þ C2 telt þ εp

ð8:7Þ

where l¼

Cr 2rH 2

for 0  t  T0 εp ¼

1 s0 l3 rH 2 T0

for 0t  T0 ε¼

1 s0 l2 rH 2

Case 3: Roots are complex. ε ¼ ept ½Acosqt þ B sin qt þ εp

ð8:8Þ

5

Analytical methods

191

where p¼

Cr ; 2rH 2

q¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Dr rH 2  Cr2

for 0  t  T0 εp ¼

1 s0 ½ðp2 þ q2 Þt þ 2p 2 ðp2 þ q2 Þ rH 2 T0

for 0t  T0 εp ¼

1 s0 p2 þ q2 rH 2

Integration constants C1 and C2 can be determined from the following initial conditions: for 0  t < T0 ε¼0

at

t¼0

ε_ ¼ 0

at

t¼0

ð8:9Þ

for t  T0 ε ¼ ε0

at t ¼ T0

ε_ ¼ ε_ 0

at t ¼ T0

ð8:10Þ

Integration constants can be easily obtained for these conditions for each case. However, their specific forms are not presented as they are too lengthy. (b)

Applications

Several applications of the theoretical relations derived in the previous section are given herein to investigate the effects of viscosity coefficient, elasticity coefficient, loading rate, and sample height. 1

2

The effect of viscosity coefficient: Figure 8.3 shows the effect of the viscosity coefficient on the deformation responses of a sample. It is of great interest that when rock is elastic, an oscillating behavior must be observed. Furthermore, stress–strain relation is not linear, and it also oscillates as the applied stress is linearly increased. However, this oscillating behavior is suppressed as the viscosity coefficient increases. The effect of elasticity coefficient: Figure 8.4 shows the effect of elasticity coefficient on the deformation responses of a sample with a viscosity coefficient of 0 GPa  s. The amplitudes of the oscillating part and stationary part of strain decrease as the value of elasticity coefficient increases. Nevertheless, the oscillating behavior is apparent for each case.

192 Analytical methods

6

Figure 8.3 Effect of viscosity coefficient on dynamics response of a sample subjected to uniaxial compression

3

4

The effect of loading rate: Figure 8.5 shows the effect of loading rate on the deformation responses of a sample with a viscosity coefficient of 0 GPa  s. While the amplitude of the stationary part of strain remain the same, the amplitude of the oscillating part of strain decreases as the loading rate decreases. Although the oscillating behavior could not be suppressed, the effect of oscillation tends to become smaller. The effect of sample height: Figure 8.6 shows the effect of sample height on the deformation responses of a sample with a viscosity coefficient of 0 GPa  s. The amplitudes of the oscillating part and stationary part of strain remain the same while the period of oscillations becomes larger as the value of sample height increases.

8.2.2

Visco-elastic layer on an incline

A semi-infinite slab on an incline is considered and is assumed to be subjected to instantaneous gravitational loading (Fig. 8.7(a)). The original formulation was developed by Aydan (1994), and it is adopted herein for assessing the dynamic response of a semi-infinite layer on an incline, which is a very close situation to the slope stability assessment of Terzaghi (1960).

7

Analytical methods

193

Figure 8.4 Effect of elastic coefficient on dynamics response of a sample subjected to uniaxial compression

Let us assume that the deformation is purely due to shearing under gravitational loading and that the slab behaves in a visco-elastic manner of Kelvin-Voigt type given by (Fig. 8.7(b)): t ¼ Gg þ Zg_

ð8:11Þ

where G is the elastic shear modulus and Z is the viscos shear modulus. This model is known as the Voigt-Kelvin model (Eringen, 1980). When G ¼ 0, then it simply corresponds to a Newtonian fluid. On the other hand, when Z ¼ 0, it corresponds to a Hookean solid.

194 Analytical methods

8

Figure 8.5 Effect of loading rate on dynamics response of a sample subjected to uniaxial compression

Let us consider an infinitesimal element within an inclined infinitely long layer as illustrated in Figure 8.8. The governing equation takes the following form by considering the equilibrium of the element by applying Newton’s second law (Eringen, 1980): @t @p  þ rg sin a ¼ r€ u @y @x

ð8:12Þ

If the thickness of the liquified layer does not vary with x and the medium consists of the same material, then @p=@x ¼ 0, and the preceding equation becomes: @t þ rg sin a ¼ r€ u @y

ð8:13Þ

9

Analytical methods

195

Figure 8.6 Effect of sample height on dynamics response of a sample subjected to uniaxial compression

Figure 8.7 (a) Illustration of semi-infinite slope, (b) constitutive model Source: (a) modified from Terzaghi, 1960)

196 Analytical methods

10

Figure 8.8 Mechanical model for shearing of semi-infinite slope

(a)

Closed-form solutions

Assuming that shear strain and shear strain rate can be defined as: g¼

@u @u_ ; g_ ¼ @y @y

ð8:14Þ

and introducing the constitutive law given by Equation (8.11) into Equation (8.13) yields the following partial differential equation:   @2u @ 2 @u @2u þ G 2 ¼ rg sin a r 2 þZ 2 ð8:15Þ @t @y @y @t Let us assume that the solution of this partial differential equation using the separation of variable technique is given as (i.e. Kreyszig, 1983; Zachmanoglou and Thoe, 1986): uðy;tÞ ¼ Y ðyÞ  TðtÞ

ð8:16Þ

As a particular case based on intuitive approach, Y ðyÞ is assumed to be of the following form by considering an earlier solution of the equilibrium equation without the inertial term for semi-infinite slab with free-surface boundary conditions (Fig. 8.9(a)): y ð8:17Þ Y ðyÞ ¼ yðH  Þ 2 Inserting this relation into Equation (8.15), we have: r

@2T y @T þ GT ¼ rg sin a yðH  Þ þ Z @t2 2 @t

ð8:18Þ

Integrating the preceding equation with respect to y for bounds y ¼ 0 and y ¼ H results in the following second-order nonhomogeneous ordinary differential equation: @2T 3Z @T 3G 3g sin a þ  T¼ 2 2 2 @t rH @t rH H2

ð8:19Þ

11

Analytical methods

197

Figure 8.9 Boundary conditions

The solutions of this differential equation are: Case 1: Roots are real. T ¼ C1 el1 t þ C2 el2 t þ

1 3g sin a l1 l2 H 2

ð8:20Þ

where l1 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ð3Z þ 9Z2  12GrH 2 Þ; l2 ¼ ð3Z  9Z2  12GrH 2 Þ 2 2 2rH 2rH

Case 2: Roots are same. T ¼ ½C1 þ C2 telt þ

1 3g sin a l2 H 2

ð8:21Þ

where l¼

3Z 2rH 2

Case 3: Roots are complex. T ¼ ep

1mmt

½Acosqt þ B sin qt þ

1 3g sin a p2 þ q2 H 2

where p¼

3Z ; 2rH 2

q¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12GrH 2  9Z2

ð8:22Þ

198 Analytical methods

12

Integration constants C1 and C2 can be determined from the following initial conditions: uðy;tÞ ¼ 0

at t ¼ 0

_ uðy;tÞ ¼0

at t ¼ 0

ð8:23Þ

For the preceding initial conditions, the integration constants for each case are: Case 1: Roots are real. C1 ¼ 

1 3g sin a 1 3g sin a ; C2 ¼ 2 l1 ðl2  l1 Þ H l2 ðl2  l1 Þ H 2

ð8:24Þ

Case 2: Roots are same. C1 ¼ 

1 3g sin a ; l2 H 2

C2 ¼

1 3g sin a l2 H 2

ð8:25Þ

Case 3: Roots are complex. C1 ¼ 

1 3g sin a p 1 3g sin a ; C2 ¼  2 2 2 p þq H q p þ q2 H 2 2

ð8:26Þ

Integration constants for the constrained boundary conditions can be obtained in a similar manner. This situation may be quite relevant to the response of soft layers sandwiched between two relatively rigid layers in underground excavations and trapdoor experiment used particularly for underground openings in soil (Terzaghi, 1946). Figure 8.10 compares the solutions obtained from the closed-form solution and FEM for a 4 m thick semi-infinite slab. Using the solutions presented in this section, one may compare the expected responses under different circumstances. Such a comparison has been already done by (Aydan, 1994–1998). The negligence of the inertia component in Equation (8.15) results in a parabolic partial differential equation. Neglecting the viscous effect in the resulting equation would result in a differential equation of elliptical form. Figure 8.10 compares the responses obtained for three situations of the differential equation. As noted from the figure, all solutions converge to the solution obtained from the elliptical form (static case). The inertia component implies that displacement as well as resulting stresses and strains responses would be greater than those of the elliptical form.

8.2.2

One-dimensional bar

The equation of motion for the axial responses of rock bolts and rock anchors, together with the consideration of inertia component including mass proportional damping, can be written in the following form (Fig. 8.11): r

@ 2 ub @u @s 2 þ t þ ha b ¼ @x rb b @t2 @t

ð8:27Þ

The analytical solutions for Equation (8.27) are extremely difficult for the given constitutive laws, boundary and initial conditions. However, it is possible to obtain solutions for simple cases, which may be useful for the interpretation of results of site investigations.

Figure 8.10 (a) Dynamic response of 4 m thick semi-infinite slab under instantaneously applied gravitation load, (b) comparison of responses obtained for hyperbolic, parabolic and elliptical forms of the differential equation for a 4 m thick slab under instantaneous gravitational loading

Figure 8.11 Modeling of dynamic axial response of tendons

200 Analytical methods

14

Equation (8.27) may be reduced to the following form by omitting the effect of damping and interaction with surrounding rock as: @ 2 ub @2u ¼ Vp2 2b 2 @t @x

ð8:28Þ

where

sffiffiffiffiffi Eb Vp ¼ r

The general solution of partial differential Equation (8.28), which is also known as the D’Alambert solution, may be given as: ub ¼ hðx  Vp tÞ þ Hðx þ Vp tÞ For a very simple situation, the solution may be given as follows: 2p ub ¼ Asin ðx  Vp tÞ L

ð8:29Þ

ð8:30Þ

Where L is tendon length. Thus, the eigenvalues of tendon may be obtained as follows: 1 fp ¼ n Vp ; n ¼ 1; 2; 3 ð8:31Þ 2L Similarly, the eigenvalues of traverse vibration of the tendon under a given prestress may be obtained as follows: 1 ð8:32Þ fT ¼ n VT ; n ¼ 1; 2; 3 2L where VT ¼

8.2.3

rffiffiffiffiffi so r

Circular cavity in elastic rock under far-field hydrostatic stress

An analytical solution is herein presented in the case of a circular underground opening excavated in a hydrostatic state of stress. To start the derivations, the following equations are set: Then, the governing equations for bolted and unbolted sections are (Fig. 8.12): dsr sr  sy þ ¼0 dr r

ð8:33Þ

where r is the distance from opening center, sr is the radial stress, and sy is the tangential stress. The constitutive law between stresses and strains of rock is of the following form: 2 nr 3( ) ( ) 1  εr 1  nr 7 sr 1  n2r 6 7 6 ¼ 5 s 4 n Er εy y  r 1 1  nr

15

Analytical methods

201

Figure 8.12 Notations Source: From Aydan et al., 1993

or (

sr sy

2

) ¼

6 Er ð1  nr Þ 6 ð1 þ vr Þð1  2nr Þ 4 

1 nr 1  nr



n r 3( ) 1  nr 7 εr 7 5 ε y 1

ð8:34Þ

where Er is the elastic modulus of rock, εr is the radial strain, εy is the tangential strain in u: εr ¼

du ; dr

εy ¼

u r

ð8:35Þ

The stresses and displacements in the bolted and unbolted sections can be obtained by solving the governing equations (Equation (8.33)) together with the constitutive law (Equation (8.34)) and the boundary conditions. Substituting the constitutive law (Equation (8.34))

202 Analytical methods

16

in governing equations (Equation (8.33)), together with the relations (Equation (8.35)), results in the following differential equations: r2

d2u du þr u¼0 dr2 dr

ð8:36Þ

The general solutions of the preceding differential equations are of the following forms: u ¼ A1 r þ A2

1 r

ð8:37Þ

By introducing the following boundary conditions for each section: •

Unbolted section sr ¼ Pb

at

r¼b

sr ¼ s0

at

r¼1

integration constants A1 and A2 are obtained as: A1 ¼

ð1 þ nr Þð1  2nr Þ s0 ; Er

A2 ¼

ð1 þ nr Þ 2 b ðs0  Pb Þ: Er

Inserting the preceding constants in Equations (8.37) result in the following expressions after some manipulations:   1 þ nr b2 ð1  nr Þs0 r þ ðs0  Pb Þ 2 u¼ ð8:38Þ Er r The preceding displacement fields are for a body that is initially unstressed. As the initial displacement field has already taken place in the rock mass before the excavation of openings, this initial displacement field has to be subtracted from the preceding expressions. The initial displacement field is the state when Pb was equal to the far-field stress s0 . Inserting these identities in the preceding expressions yields: u0 ¼

1 þ nr ð1  nr Þs0 r Er

ð8:39Þ

Finally, one obtains the following expressions for displacement fields due to the excavation of the opening: ue ¼

1 þ nr b2 ðs0  Pb Þ 2 Er r

ð8:40Þ

where subscript e denotes excavation. Variations in displacement fields will, in turn, bring about variations in radial and tangential stresses. These variations are obtained from the preceding relations together with Equation (8.35) and the constitutive law (Equation (8.34)) as: sre ¼ ðs0  Pb Þ sye ¼ ðs0  Pb Þ

b2 r2

b2 r2

ð8:41Þ ð8:42Þ

17

Analytical methods

203

The stress state in the surrounding medium is therefore defined as the sum of variation in the postexcavation state and the preexcavation stress state given as: sr ¼ s0 þ sre ¼ s0  ðs0  Pb Þ

b2 r2

ð8:43Þ

sy ¼ s0 þ sye ¼ s0 þ ðs0  Pb Þ

b2 r2

ð8:44Þ

Note: The preceding relations for stresses can also be directly obtained from the governing equations (Equation (8.33)) with the use of following identities: sr þ sy ¼ 2s0

ð8:45Þ

and the same boundary conditions. 8.2.4 (a)

Unified analytical solutions for circular/spherical cavity in elasto-plastic rock

General solution

CONSTITUTIVE LAWS

Generalized form of the constitutive law between stresses and strains of rock in the elastic region for radially symmetric problem (cylindrical and spherical openings) can be given as: (

sr sy

)

" ¼

l þ 2m

nl

l

nl þ 2m

#(

εr

)

εy

ð8:46Þ

where n is the shape coefficient and has a value of 1 for a cylindrical opening and 2 for a spherical opening; sr is radial stress; sy is tangential stress; εr is radial strain; εy is tangential strain. l and m are Lamé constants and are given as: l¼

En E ; m¼ ð1 þ nÞð1  2nÞ 2ð1 þ nÞ

ð8:47Þ

Where E is the elastic modulus of rock, and n is Poisson’s ratio of rock. EQUILIBRIUM EQUATION

When the problem is radially symmetric, the momentum law for static case takes the following form: dsr s  sy þn r ¼0 dr r where r is distance from opening center.

ð8:48Þ

204 Analytical methods

18

COMPATIBILITY CONDITION

The compatibility condition between strain components for radially symmetric openings is given as: dεy εy  εr þ ¼0 ð8:49Þ dr r Relations between strain components and radial displacement (u) are: du u εr ¼ ; εy ¼ dr r

ð8:50Þ

BEHAVIOR OF ROCK MATERIAL

An elastic-perfect-residual plastic model as shown in Figure 8.13 approximates the behavior of rock. Although it is possible to consider the strain-softening behavior, it is extremely difficult to obtain closed-form solutions, and numerical techniques would be necessary. Rock was assumed to obey the Mohr-Coulomb yield criterion. Although it is possible to derive solutions for the Hoek-Brown criterion, it is not intentionally done as the generalized Hoek-Brown criterion violates the Euler theorem used in the classical theory of plasticity for constitutive modeling of rocks. Failure zones about radially symmetric openings excavated in rock mass for elastic-perfect-residual plastic behavior and yield functions for each region are illustrated in Figure 8.14 and given by: s1 ¼ qs3 þ sc ; s1 ¼ q s3 þ sc ;

1 þ sin 1sin 1 þ sin q ¼ 1sin

q¼

perfectly plastic region

ð8:51aÞ

residual plastic region

ð8:51bÞ

Figure 8.13 Mechanical models for rock mass

19

Analytical methods

205

Figure 8.14 States about an opening and notations (Gravity is considered in dotted zone.)

where s1 is the maximum principal stress, s3 is the minimum principal stress, sc is the uniaxial compressive strength of intact rock, sc is the uniaxial compressive strength of broken rock,  is the internal friction angle of intact rock and  is the internal friction angle of broken rock. Relations between total radial and tangential strains in plastic regimes are assumed to be of the following form: εr ¼ f εy

for perfectly plastic region



εr ¼ f εy for residual plastic region

ð8:52aÞ ð8:52bÞ

where f and f  are physical constants obtained from the tests. These constants may be interpreted as plastic Poisson’s ratios (Aydan et al., 1993).

(I)

STRESS AND STRAIN FIELD AROUND OPENING

(1)

RESIDUAL PLASTIC ZONE ða  r  Rpb Þ

Inserting the yield criterion Equation (8.52b) into the governing Equation (8.48) with s3 ¼ sr and s1 ¼ sy yields: dsr s s þ nð1  q Þ r ¼ n c dr r r

ð8:53Þ

The solution of the preceding differential equation is: sr ¼ Crnðq

 1Þ



sc q 1 

ð8:54Þ

206 Analytical methods

20

The integration constant C is obtained from the boundary condition sr ¼ pi at r ¼ a as:   sc 1 ð8:55Þ C ¼ pi þ  nðq1Þ q 1 a where pi is internal or support pressure. Thus, the stresses now take the following forms:   nðq 1Þ sc r s ð8:56Þ   c sr ¼ pi þ  q 1 a q 1   nðq 1Þ sc r s  ð8:57Þ   c sy ¼ q pi þ  q 1 a q 1 Solving the differential equation obtained by inserting the relation given by Equation (8.52b) in Equation (8.50) yields: A εy ¼ f  þ1 ð8:58Þ r The integration constant A is determined from the continuity of the tangential strain at perfect-residual plastic boundary r ¼ Rpb as: 

f þ1 A ¼ εpb y Rpb

ð8:59Þ

εpb y in Equation (8.59) is the tangential strain at perfect-residual plastic boundary (r ¼ Rpb ) and it is specifically given by: εsf ep ð8:60Þ εpb y ¼ Zsf εy Zsf ¼ εe where Zsf is the tangential strain level at the perfect-residual plastic boundary (Fig. 8.13); εep y is the tangential strain at the elastic-perfect plastic boundary as: 1þn ðp0  srp Þ εep ð8:61Þ y ¼ nE srp in Equation (8.61) is the radial stress at elastic-perfect plastic boundary. As a result, the tangential strain in the surrounding rock becomes:  f  þ1 Rpb 1þn ðp0  srp Þ Zsf εy ¼ ð8:62Þ nE r (2)

PERFECTLY PLASTIC ZONE ðRpb  r  Rpp Þ

Inserting the yield criterion Equation (8.51a) into the governing Equation (8.48) with s3 ¼ sr and s1 ¼ sy gives: dsr s s þ nð1  qÞ r ¼ n c dr r r

ð8:63Þ

The solution of the preceding differential equation is: sr ¼ Crnðq1Þ 

sc q1

ð8:64Þ

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Analytical methods

207

The integration constant C is obtained from the boundary condition sr ¼ srp at r ¼ Rpp as:   sc 1 ð8:65Þ C ¼ srp þ nðq1Þ q  1 Rpp Thus, the stresses now take the following forms: !nðq1Þ   sc r s sr ¼ srp þ  c Rpp q1 q1 !nðq1Þ   sc r s  c sy ¼ q srp þ Rpp q1 q1

ð8:66Þ

ð8:67Þ

Since the derivation of the tangential strain is similar to the previous case, the final expression takes the following form:  f þ1 Rpp 1þn εy ¼ ðp0  srp Þ ð8:68Þ nE r The relation between the plastic zone radii is also found from the requirement of the continuity of tangential strain at r ¼ Rpb and relation Equation (8.62) as: 1 Rpp ¼ Zsff þ1 Rpb

(3)

ð8:69Þ

ELASTIC ZONE ðRpp  rÞ

The derivation of stresses and displacement expressions for a cylindrical opening was previously given in detail with the consideration of initially stressed elastic medium by a farfield hydrostatic in-situ stress (p0 ). The final forms of the expressions for radially symmetric openings are of the following forms:  nþ1 Rpp ð8:70Þ sr ¼ p0  ðp0  srp Þ r  nþ1 Rpp 1 sy ¼ p0 þ ðp0  srp Þ n r

ð8:71Þ

 nþ1 Rpp 1þn ðp0  srp Þ εy ¼ nE r

ð8:72Þ

The specific form for srp is obtained from the continuity condition of tangential stresses at r ¼ Rpp by equality Equation (8.68) and Equation (8.72) as: srp ¼

p0 þ nðp0  sc Þ 1 þ nq

ð8:73Þ

208 Analytical methods

22

(II) PLASTIC ZONES RADIUS AROUND OPENING (1)

PERFECTLY PLASTIC-RESIDUAL PLASTIC ZONE BOUNDARY RADIUS (Rpb )

The perfectly plastic-residual plastic zone boundary radius is found from the requirement of the continuity of radial stresses, i.e. by the equality of Equations (8.73) and (8.67), at r ¼ Rpb as: 1 9nðq1Þ 8 nð1qÞ  ð1þnÞ½ðq1Þþa a þ qa1= Rpb < ð1þnqÞðq1Þ ðZsf Þ f þ1  q1 ¼  ; : a b þ qa1

ð8:74Þ

where b is the support pressure normalized by overburden pressure and given as: b¼

pi p0

ð8:75Þ

and a is also the competency factor as: a¼ (2)

sc p0

ð8:76Þ

PERFECTLY PLASTIC AND ELASTIC ZONE BOUNDARY RADIUS (Rpp )

The perfectly plastic and elastic zone boundary radius is also found by inserting srp given by Equation (8.73) in the radial stress Equation (8.68) with sr ¼ pi at r ¼ a as:

1 Rpp ð1 þ nÞ½ðq  1Þ þ a nðq1Þ ¼ ð8:77Þ ð1 þ nqÞ½ðq  1Þb þ a a

(III) NORMALIZED OPENING WALL STRAINS (1)

ELASTIC STATE

The tangential strain at the opening wall can be obtained as: εay ¼

1þn ðp0  pi Þ nE

ð8:78Þ

say ¼

nþ1 1 p0  pi n n

ð8:79Þ

If the opening is strained to its elastic limit, then say ¼ sc for pi ¼ 0. Thus, we have the elastic strain limit as: εey ¼

1 þ n sc  E nþ1

ð8:80Þ

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Analytical methods

209

Using the preceding relation in Equation (8.78), one obtains the normalized opening wall strain (x) as: x¼ (2)

  εay n þ 1 1  b 1 ¼ n a εey

ð8:81Þ

PERFECTLY-PLASTIC STATE

The tangential strain at the opening wall can be obtained as:  f þ1 Rpp 1þn ðp0  srp Þ εay ¼ nE a

ð8:82Þ

The elastic limit is given as: εey ¼

1þn ðp0  srp Þ nE

ð8:83Þ

Using the preceding relation in Equation (8.82), one obtains the normalized opening wall strain as: εa x ¼ ye ¼ εy (3)



ð1 þ nÞ½ðq1Þ þ a ð1 þ nqÞ½ðq1Þb þ a

f þ1

nðq1Þ

ð8:84Þ

RESIDUAL PLASTIC STATE

The tangential strain at the opening wall can be obtained as:  f  þ1 Rpb 1þn ðp0  srp ÞZsf εay ¼ nE a

ð8:85Þ

Using Equations (8.81) and (8.83), one obtains the normalized opening wall strain as:

x¼

8 9 f þ1 nð1qÞ sc ) q q    ðq1Þ  2ðq1Þ  1=2 ua 1 þ n Rt pc Rc R pi R pi R pt pt þ ¼ 1 c ; c¼ ; c¼ 2E a a 2E a Rt pc a pc Rt pc

ð8:116Þ Furthermore, the applied pressure is equal to radial pressure on the walls of a spherical body in view of the equivalence of work done by the pressure of the indenter to that induced by the wall of the spherical body on the surrounding medium as: pp ¼ pi

ð8:117Þ

Assuming that the volume of the hemispherical body beneath the indenter remains for a given impression displacement (d), the outward displacement (ua ) of the hemispherical cavity wall can be easily related to the impression displacement (d) as follows: d ¼ 2ua

ð8:118Þ

35

8.2.6

Analytical methods

221

Two-dimensional analytical methods

It is generally difficult to derive closed-form solutions for surface and underground excavations with complex geometry and complex material behavior. Most solutions would be limited to one-dimension in space, and time may be incorporated in certain solutions. For deformation-stress analyses, it is rare to find closed-form solutions for surface structures while there are some closed-form solutions for underground openings in twodimensional elastic space. The most famous solution is probably that of Kirsch (1898) for a circular hole under biaxial stress state. Solutions for arbitrary shape openings are also developed, and the reader is advised to consult the textbooks by Timoshenko and Goodier (1951 Savin (1965), Muskhelishvili (1962), Jaeger and Cook (1979). It becomes more difficult to obtain analytical solutions when the surrounding media start to behave in an elasto-plastic manner. The simple yet often used closed solutions are for openings with a circular geometry excavated in elasto-plastic media. Several solutions are developed using different yield criteria and postyielding models (e.g. Talobre, 1957; Terzaghi, 1946). There are some solutions for underground openings in elasto-plastic rock supported by rock bolts, shotcrete and steel ribs (Hoek and Brown, 1980; Aydan et al., 1993). Galin (1946) was first to obtain closed-form solutions around circular openings enclosed completely by a plastic zone under bi-axial stress state in Tresca-type perfectly plastic materials. Detournay (1986) attempted to obtain solutions for the same situation with the MohrCoulomb yield criterion and discussed several cases using the same solution technique. There are also limited number of analytical solutions for the problems of seepage, heat flow and diffusion since most solutions would be valid for the given boundary and initial conditions. The analytical solutions for displacement, strain and stress field around cavities exhibiting nonlinear behavior under nonhydrostatic conditions are generally difficult to obtain. However, some analytical solutions were obtained by Kirsch (1889) for circular cavities, by Ingliss (1913) for elliptical cavities and by Mindlin for circular cavities in gravitating media when the surrounding medium behaves elastically. Muskhelishvili Muskhelishvili (1962) devised a general method based on complex variable functions for arbitrary shape cavities. The stress state around a circular cavity in an elastic medium under a biaxial far-field stresses were first obtained by Kirsch (1898) using Airy’s stress function. These solutions are modified to incorporate the effect of uniform internal pressures (Jaeger and Cook, 1979). In a polar coordinate system, the radial, tangential and shear stresses around the circular cavity can be written in the following forms (Fig. 8.24):  a2  s  s  a2 a4  a2 s10 þ s30 10 30 1 14 þ3  cos2ðy  bÞ þ pi sr ¼ r r r r 2 2  a2  s  s  a4  a2 s10 þ s30 10 30 1þ 1þ3 sy ¼ þ cos2ðy  bÞ  pi r r r 2 2

ð8:119Þ

 a2 a4  s10  s30 try ¼ 14 þ3 sin2ðy  bÞ r r 2 where s10 ; s30 are the far-field principal stresses, a is the radius of hole, r is radial distance, b is the inclination of s10 far-field stress from horizontal, y is the angle of the point from horizontal, and pi is the internal pressure applied onto the hole perimetry.

222 Analytical methods

36

Figure 8.24 Stress tensor components and far-field stresses around a circular hole

The yield criteria available in rock mechanics are: Mohr-Coulomb s1 ¼ sc þ qs3

ð8:120aÞ

Drucker-Prager aI1 þ

pffiffiffiffi J2 ¼ k

ð8:120bÞ

Hoek and Brown (1980) s1 ¼ s 3 þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi msc s3 þ ss2c

ð8:120cÞ

Aydan (1995) s1 ¼ s3 þ ½S1  ðS1  sc Þeb1 s3 eb2 T

ð8:120dÞ

where 1 2 2 2 I1 ¼ sI þ sII þ sIII ; J2 ¼ ððsI  sII Þ þ ðsII  sIII Þ þ ðsIII  sI Þ 6 2 sin  6c cos  1 þ sin  ; k ¼ pffiffiffi ; c is cohesion;  friction angle; q ¼ a ¼ pffiffiffi 1  sin  3ð3 þ sin Þ 3ð3 þ sin Þ where s1 is the ultimate deviatoric strength, T is the temperature, and m,s, b1 ; b2 are empirical constants. Mohr-Coulomb and Drucker-Prager yield criteria are a linear function

37

Analytical methods

223

Figure 8.25 Comparison of yield criteria

of confining or mean stress, while the criteria of Hoek-Brown and Aydan are of the nonlinear type. Furthermore, Aydan’s criterion also accounts for the effect of temperature. Figure 8.25 compares several yield criteria for different rocks. When Aydan’s criterion is used, the effect of temperature is omitted in Figure 8.25 for the sake of comparison. It should be also noted that the criterion of Hoek and Brown often fails to represent triaxial strength data if it is required to represent tensile and compressive strength contrary to common belief, that is, the best yield criterion for rocks (Aydan et al., 2012). If the yield criterion is chosen to be a function of minimum and maximum principal stresses, they can be given in the following form in terms of stress components given by Equation (8.121): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s  s 2 s þ sr y r s1 ¼ y ð8:121aÞ þ þ t2ry 2 2 s þ sr  s3 ¼ y 2

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s  s 2 y r 2 þ try 2

ð8:121bÞ

The damage zone around the blast hole under high internal pressure can be estimated using one of these yield criteria. It should be noted that the yielding is induced by the high internal pressure in the blast hole, which is essentially different from in-situ stressinduced borehole breakout. In other words, there will always be a damage zone around the blast hole perimeter when the blasting technique is employed. The blast hole pressure depends on the characteristics of the surrounding medium; the amount, layout and type of explosive; the blasting velocity; and the geometry of the blast hole. The blast hole pressure ranges from 100 MPa to 10 GPa (i.e. Jaeger and Cook, 1979; Brady and Brown, 1985). First we assume that the properties of surrounding rock have the values as given in Figure 8.26 and that the blast hole (internal) pressure has values of 400 and 450 MPa

224 Analytical methods

38

Figure 8.26 Estimated yield zones around the blast hole

for the Takamaruyama tunnel and Kaore Powerhouse, respectively. The rock chosen roughly corresponds to an igneous rock such as granitic rocks. The maximum far-field stress is slightly inclined at an angle of 10 degrees from horizontal and lateral stress coefficient has a value of 4 for Kaore Powerhouse. Figure 8.26 shows the example of computation for the given conditions. The largest yield zone is obtained for Hoek-Brown (HB) criterion while the tension cutoff criterion (T) results in smaller yield zone. The criterion of Aydan estimates a slightly larger yield zone than the Mohr-Coulomb (MC) criterion. It is very interesting to note that the yielding propagates in the direction of maximum far-field stress. In other words, the elongation direction of the yield zone would be the best indicator of the maximum far-field stress in the plane of the blast hole. In the next example, the far-field stress state is assumed to be isotropic while keeping the values of parameters the same. Except the tension cutoff criterion, all yield criteria estimate almost the same-size yield zones. It should be, however, noted that all yield criteria satisfy the same values of tensile and compressive strength. Therefore, the similarity of the size of yield zones should not be surprising. If the strength of surrounding rock is anisotropic, the yield functions considering the effect anisotropy should be used. If the elastic constants of the rock are anisotropic, closed-form solutions capable of representing anisotropy should be used instead of those given by Equation (8.121). It should be also noted that the plastic zone developed using the actual elasto-plastic analyses would be larger than those estimated from the elastic solutions (Equation (8.27)). If such discrepancies are expected to be larger, it would be better to use the elasto-plastic finite element method. Nevertheless, the basic conceptual model would be the same. Gerçek (1996 1997) proposed a seminumerical technique to obtain the integration constant stress functions based on Muskhelishvil’s method. Galin (1946, see Savin, 1961 for English version) was first to develop analytical solutions for circular holes in Trescatype material under nonhydrostatic initial stress state. His solution was extended to Mohr-Coulomb materials by Detournay (1986). He further discussed problems of the

39

Analytical methods

225

Figure 8.27 Comparison of approximately estimated and exact yield zone Source: From Aydan, 2018 1987

nonenveloping yield zone around the circular hole. Kastner (1962) proposed a method for estimating the approximate yield zone around circular cavities under nonhydrostatic stress condition using Kirsch’s solutions. This method is also employed by Zoback et al. (1980) to estimate the shape of borehole breakouts, which was used to infer the in-situ stress state. Gerçek (1993) also used the same concept for arbitrarily shaped cavities to estimate the extent of possible yield zone. Although this method estimates the extent of yield zone smaller than the actual one as shown by the first author (Aydan, 1987) for circular cavities under hydrostatic stress state, as shown in Figure 8.27, it yielded the estimated shape of yield zone similar to that by exact solutions. Furthermore, it may also provide some rough guidelines for the anticipated zone for reinforcement by rockbolts. Aydan and Geniş (2010) extended the same concept to estimate overstressed zones about cavities of arbitrary shape based on the stress state computation method proposed by Gerçek (1993) and using strain energy, distortion energy, extension strain and no-tension criteria in addition to Mohr-Coulomb yield criteria. Figure 8.28 shows the overstressed zones around a tunnel subjected to the hydrostatic initial stress state at different stages of excavations and the contours of maximum principal stress. The most critical stress state is during the excavation of the top heading, and the stress state becomes more uniform as the excavation approaches a circular shape. Furthermore, the extent of the tensile stress zone gradually decreases in size as the excavation progresses. An interesting yield zone developed around a circular opening excavated in a granodioritic hard rock at a level of 420 m in Underground Research Laboratory (URL) in Winnipeg, Canada. Figure 8.29 shows the prediction of an overstressed zone around the circular opening at Underground Research Laboratory (URL). Except for the shearing and tensionyielding model, all methods estimated the most likely location of yield zone. In addition to the estimations by the approximate approach, FEM analyses incorporating the yield criteria

226 Analytical methods

40

Figure 8.28 Overstressed zones around a tunnel subjected to a hydrostatic initial stress state at different stages of excavations and the contours of maximum principal stress

adopted in the approximate method were performed (Fig. 8.29). As noted from the figure, it seems that the distortion energy concept yields results close to the observations as shown in Figure 8.29.

8.2.7

Three-dimensional analytical solutions

The solutions for three-dimensional situations are quite rare except for a very few solutions. Boussinesq (1885) derived solutions for the distribution of stresses in a half-space resulting from surface loads is largely used in various applications. Kelvin considered a half-space

41

Analytical methods

227

Figure 8.29 View of the opening and estimated yield zones by different methods

problem with a point load of P (Fig. 8.30). The final expressions are given in the following form: Displacement components

  Pzr P 2ð1  2nÞ 1 z2 þ þ 3 ur ¼ ; u ¼ 0; uz ¼ 4pGð1  nÞR3 y 4pGð1  nÞ R R R

ð8:122Þ

Stress components   P 2ð1  2nÞz 3r2 z Pð1  2nÞz  ; sy ¼ 2pð1  nÞ R3 R5 2pð1  nÞR3   P ð1  2nÞz 3z3 sz ¼  5 2pð1  nÞ R3 R   P ð1  2nÞz 3rz2 try ¼  5 ; try ¼ tzy ¼ 0 2pð1  nÞ R3 R sr ¼

where R¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi z2 þ r 2

ð8:123Þ

228 Analytical methods

42

Figure 8.30 Notation for half space under a point load P

Analytical solutions for fluid flow through porous rocks

8.3 8.3.1

Some considerations on Darcy’s law for rocks and discontinuities

Darcy’s law (Equation 5.3) is generally used as a constitutive model for the fluid flow through porous rock and rock discontinuities together with the assumption of laminar flow. A brief description of Darcy’s law is presented in this subsection. Darcy performed a series of experiments on a porous column in 1856. From these experiments, he found that the volume discharge rate Q is directly proportional to the head drop h2  h1 and to the cross-sectional area A but that it is inversely proportional to the length difference l2  l1 . Calling the proportionality constant K as the hydraulic conductivity, Darcy’s law is written as: Q ¼ KA

h2  h1 l2  l1

ð8:124Þ

The negative sign signifies the groundwater flows in the direction of head loss. Darcy’s law is now widely accepted and used in modeling fluid flow in porous or fractured media. It is elaborated and written in a differential form, which is given here for the one-dimensional case as: v ¼ K

@h @x

ð8:125Þ

43

Analytical methods

229

This law is analogous to Fourier’s law in heat flow presented in Chapter 5. Darcy’s law is theoretically derived for tube-like pores and slit-like discontinuities in this subsection (Aydan et al., 1997; Üçpırtı and Aydan, 1997). (a)

Darcy’s law for rock with cylindrical pores

Equilibrium equation for x-direction is given as:

X 2 2 Fx ¼ pp½ðr þ DrÞ  r2   ðp þ DpÞp½ðr þ DrÞ  r2  þ ðt þ DtÞ2pðr þ DrÞDx  t2prDx ¼ 0 ð8:126Þ

Rearranging the resulting expression and taking the limit and omitting the second-order components yields:

dp dt t   ¼0 dx dr r

ð8:127Þ

Assuming that the flow is laminar and a linear relationship holds between shear stress and strain rate g_ as: du_ dv du t ¼ Zg_ g_ ¼ ¼ v ¼ u_ ¼ ð8:128Þ dr dr dt Now, let us insert the preceding relation into Equation (8.129). We have the following partial differential equation: dp d 2 v Z dv Z 2  ¼0 dx dr r dr

ð8:129Þ

Integrating the preceding partial differential equation for r-direction yields the following: v¼

1 dp r2 þ C1 ln r þ C2 Z dx 4

ð8:130Þ

Introducing the following boundary conditions as: D v ¼ v0 as r ¼ 2 t¼0

as r ¼ 0

yields the integration constants C1 and C2 as: C1 ¼ 0; C2 ¼ v0 

1 dp D2 Z dx 16

where D is the diameter of the pore. If velocity v0 is given in the following form: v0 ¼ a

1 dp D2 Z dx 16

the integration coefficient C2 can be obtained as follows: C2 ¼ ð1 þ aÞ

1 dp D2 Z dx 16

ð8:131Þ

230 Analytical methods

44

The flow rate q passing through the discontinuity for a unit time is: Z 2p Z y¼D2 q¼ vrdrdy

ð8:132Þ

r¼0

0

The explicit form of q is obtained as: q¼

p D4 dp Z 128 dx

ð8:133Þ

If the flow rate q is redefined in terms of an average velocity v over the pore area as q ¼  vp

D2 4

ð8:134Þ

we have the following expression: v ¼ ð1 þ aÞ

1 D2 dp Z 32 dx

ð8:135Þ

This relation is known as Hagen-Poiseuille for a ¼ 0. As an analogy to Darcy’s law, we can rewrite the preceding expression as: v ¼ 

k dp Z dx

ð8:136Þ

where k ¼ ð1 þ aÞ

D2 32

or

k ¼ ð1 þ aÞ

a2 ; 8

a¼

D 2

This is known as the actual permeability of the pores. Let us assume that the ratio (porosity) n of the area of pores over the total area is given by (Fig. 8.31a): n¼

N 1X D2 p i ; or At i¼1 4

n¼

2 NpD 4At

Figure 8.31 Geometrical models for Darcy’s law

ð8:137Þ

45

Analytical methods

231

Figure 8.32 Illustration of logitudinal and radial flows

Then, the apparent permeability ka is related to the actual permeability as: ka ¼ nk (b)

ð8:138Þ

Darcy’s law for slit-like discontinuities

For x-direction, force equilibrium equation for fluid can be given as follows (Fig. 8.31b): X ð8:139Þ Fx ¼ pðxÞ Dy  pðxþDxÞ Dy þ tðyþDyÞ Dx  tðyÞ Dx ¼ 0 where p is pressure and t is shear stress. Equation (8.140) takes the following partial differential form by taking Taylor expansions of p and t as: dp dt  ¼0 dx dy

ð8:140Þ

Assuming that flow is laminar and that the relation between shear stress t and shear strain rate g_ is linear: _ g_ ¼ t ¼ Zg;

du_ dv ¼ ; dy dy

v ¼ u_ ¼

du dt

ð8:141Þ

where Z is viscosity and u_ is deformation rate. Substituting the preceding relations into Equation (8.141) yields the following partial differential equation: dp d2v Z 2 ¼0 dx dy

ð8:142Þ

Integrating the preceding equation for the y-direction yields the following expression for flow velocity v: v¼

1 dp y2 þ C1 y þ C2 Z dx 2

ð8:143Þ

232 Analytical methods

46

Introducing the following boundary conditions in Equation (8.144): v ¼ vo

h y¼ ; 2

as

t¼0

as y ¼ 0

yields the integration constants C1 and C2 as: C1 ¼ 0; C2 ¼ vo 

1 dp h2 Z dx 8

ð8:144Þ

where h is the aperture of discontinuity. If it is assumed that the following relation exists for vo : vo ¼ a

1 dp h2 Z dx 8

ð8:145Þ

then, the integration constant C2 can be written as: C2 ¼ ð1 þ aÞ

1 dp h2 Z dx 8

ð8:146Þ

Total flow rate vt through the discontinuity at a given time is: Z vt ¼ 2

y¼h2

vdy

ð8:147Þ

y¼0

The explicit form of vt is obtained as: vt ¼ ð1 þ aÞ

1 h3 dp Z 12 dx

ð8:148Þ

For a ¼ 0, the preceding equation is well-known as a cubic law equation in groundwater hydrology (Snow, 1965), and it is introduced to the field of geomechanics by Polubarinova-Kochina in 1962. Let us redefine the flow rate vt in terms of an average velocity v and the discontinuity aperture h as: vt ¼ vh

ð8:149Þ

Inserting this equation into Equation (8.148) yields the following: v ¼ 

1 h2 dp Z 12 dx

ð8:150Þ

As an analogy to Darcy’s law, the preceding equation may be rewritten as: v ¼ 

kd dp Z dx

ð8:151Þ

where kd ¼ ð1 þ aÞ

h2 12

47

Analytical methods

233

where kd is called the permeability of discontinuity. If discontinuity porosity nd is defined as XN Ai of discontinuities to total area At (Fig. 8.31b): i¼1 d

the ratio of total area nd ¼

N 1X Ai At i¼1 d

ð8:152Þ

the following relation between apparent permeability kda and actual permeability kd is obtained as: kda ¼ nd kd 8.3.2 8.3.2.1

ð8:153Þ

Permeability tests based on steady-state flow Pressure difference and flow velocity method

During an experiment, pressures, which are applied at the ends of a test specimen, and flow velocity are measured. If the change of density of fluid with respect to pressure is negligible, permeability can be obtained from the following equation: k¼

vt lnðr2 =r1 ÞZ 2pr1 H p1  p2

ð8:154Þ

where vt is flow rate and H is specimen length. If a gas is used as a permeation fluid, the preceding equation will have the following form: k¼

8.3.2.2

vt lnðr2 =r1 Þ pZ pH p21  p22 1

ð8:155Þ

Pressure difference method 1

For this kind of test, gas is used as a permeation fluid. If the change of pressure with respect to time is linear, the velocity of gas will also change linearly. The relationship between the mass compressibility coefficient and the volumetric compressibility coefficient is given by: c ¼ rc

ð8:156Þ

For a compressed gas in a reservoir with a constant volume, the variation of the gas mass for a unit time can be written as: q ¼ rc V

dp1 dt

ð8:157Þ

For a given time, the mass passing through a specimen with a cross-section area A may be given by: q ¼ rA v 1 This test is valid if pressure rate remains constant with time.

ð8:158Þ

234 Analytical methods

48

Equating Equations (8.158) and (8.159), using Equation (4.80) and rearranging the resulting expression yields the final equation of permeability (Aydan et al., 1997; Aydan, 2016):2 k¼

V Z lnðr2 =r1 Þ dp1 pL ðp21  p22 Þ dt

ð8:159Þ

where V is the applicable volume (the sum of the volume of the supplemental reservoir, the volume of the pressure injection tubing, and the sample injection hole), Z is the gas viscosity, r2 is the radial distance from the center of the gas injection hole to the periphery of the specimen, r1 is the radius of the gas injection hole, dp1 =dt is the time rate change of the injection pressure, L is the length of the gas injection hole, and p1 and p2 are pressure at reservoirs 1 and 2. 8.3.3 8.3.3.1

Permeability tests based on non-steady-state flow (transient flow tests) Falling head test method (based on dead weight of liquid)

Experimental setup used for this kind test is shown in Figure 8.33 (Aydan et al., 1997). As seen from the figure, a pipe is placed on the top of the cylindirical hole drilled in the middle of test specimen. The cross-section area of this pipe is denoted by Ah. During the test, the change of pressure and velocity of flow can be measured. A height of water at the outside surface of the test specimen (h2 ) is assumed to be constant. When the experiment starts, flow rate inside the pipe can be given as: q ¼ rgAh

@h1 @t

ð8:160Þ

Figure 8.33 Longitudinal transient falling head test

2 This relation was also derived by Zeigler (1976). However, how he derived Eq. (8.160) is not known to the author.

49

Analytical methods

235

where h1 is the height of water inside the pipe. At a given time, flow rate through a crosssection area of hole (Ap ) inside test specimen is given by: vt ¼ vAp

ð8:161Þ

It is assumed that flow rate through the hole perimetry should be equal to the flow rate of the pipe. Then the pressure gradient in the specimen can be given in the following form: @p @ @ðh1  h2 Þ ðh  h2 Þ  ðrgðh1  h2 ÞÞ ¼ rg ¼ rg 1 @r @r @r rlnðro =ri Þ

ð8:162Þ

Substituting Equation (8.162) together with Equation (5.10) into Equation (8.161) and equalizing the resulting equation to Equation (8.160) yields the following differential equation for the change of water height h1 : @h1 k Ap 1 @t ¼ Z Ah ri lnðro =ri Þ h1  h2

ð8:163Þ

The solution of the preceding differential equation is: h1 ¼ h2 þ Ceat

ð8:164Þ

where a¼

Ap 1 k ri lnðro =ri Þ Ah Z

Introducing the following initial conditions: t ¼ 0 at

h1 ¼ h10

yields the integration constant C as: C ¼ h10  h2

ð8:165Þ

If the integration constant is inserted into Equation (8.165), the following equation is obtained: h  h2 at ¼ lnð 1 Þ ð8:166Þ h10  h2 Now, if a is substituted into the preceding equation, the following expression for permeability is obtained: A h  h2 1 ð8:167Þ k ¼ Zri lnðro =ri Þ h lnð 10 Þ Ap h1  h2 t 8.3.3.2

Transient pulse method for radial flow

Brace et al. (1968) proposed a transient pulse method for longitudinal flow tests. Aydan et al. (1997) and Aydan (2016) proposed a permeability test for radial flow. Their method is explained in detail herein. This method is fundamentally very similar to Brace’s method (Fig. 8.34). The volumetric strain of fluid inside reservoirs V1 ve V2 can be written as follows: ε1V

DV1 ; V1

ε2V

DV2 V2

ð8:168Þ

236 Analytical methods

50

Figure 8.34 Transient pulse radial flow setup for intact rock and discontinuities

Similarly, for the volumetric strain rate of fluid, the following relations can also be written as: ε1V

DV1 ; V1

DV2 V2

ð8:169Þ

DV_ 2 ¼ ε_ 2V V2

ð8:170Þ

ε2V

or DV_ 1 ¼ ε_ 1V V1 ;

If the following relation exists between the volumetric strain of fluid and pressure: ε1V ¼ cf P1 ;

ε2V ¼ cf P2

ð8:171Þ

and the compressibility coefficient (cf ) is constant, for volumetric strain rate, the following relation can be also written: ε_ 1V ¼ cf P_ 1 ; ε_ 2V ¼ cf P_ 2

ð8:172Þ

Flow rate may be given as: vt1 ¼ DV_ 1 ;

vt2 ¼ DV_ 2

ð8:173Þ

Using Equations (8.171), (8.172) and (8.173), flow rate can be rewritten in the following form: vt1 ¼ cf V1

@P1 ; @t

vt2 ¼ cf V2

@P2 @t

ð8:174Þ

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Analytical methods

237

Introducing the following boundary conditions: r ¼ r1

as P ¼ P1 ;

r ¼ r2

as P ¼ P2

and using Equation (8.174), then the following relation can be obtained for flow rate vt 1 ¼ 

kAp1 dP1 ð Þ ; Z dr r¼r1

vt2 ¼ 

kAp2 dP2 ð Þ Z dr r¼r2

ð8:175Þ

where Ap1 is the surface area of pressure injection hole, and Ap2 is the area of pressure release surface. Pressure gradients in the preceding equations are as follows: dP1 1 ðP1  P2 Þ  ; r1 lnðr2 =r1 Þ dr

dP2 1 ðP2  P1 Þ  r2 lnðr2 =r1 Þ dr

ð8:176Þ

Inserting the preceding equation into Equation (8.175) and equalizing the resulting equation to Equation (8.174) yields the following set of equations: Ap1 ðP1  P2 Þ @P1 ¼ b @t V1 r1 lnðr2 =r1 Þ

ð8:177Þ

Ap2 ðP1  P2 Þ @P2 ¼b @t V2 r2 lnðr2 =r1 Þ

ð8:178Þ

where b¼

k cf Z

Equation (8.177) can be rearranged as follows: P2 ¼ P1 þ

V1 r1 lnðr2 =r1 Þ @P1 bAp1 @t

ð8:179Þ

Taking the time derivative of the preceding equation, the following expression is obtained: @P2 @P1 V1 r1 lnðr2 =r1 Þ @ 2 P1 ¼ þ bAp1 @t @t @t2

ð8:180Þ

Substituting Equations (8.180) and (8.181) into Equation (8.179) and rearranging the resulting equation yields the following homogeneous differential equation: @ 2 P1 @P þa 1 ¼0 2 @t @t

ð8:181Þ

where a¼b

V2 r2 Ap1 þ V1 r1 Ap2 lnðr2 =r1 ÞV1 V2 r2 r1

The general solution of this differential equation is: P1 ¼ C1 þ C2 eat

ð8:182Þ

238 Analytical methods

52

Introducing the following initial conditions: t ¼ 0 as P1 ¼ Pi ;

t¼1

as P1 ¼ Pf

yields the integration constants C1 and C2 as: C1 ¼ Pf ;

C2 ¼ Pi  Pf

ð8:183Þ

Inserting these integration constants into Equation (8.182) gives the following equation: P1 ¼ Pf þ ðPi  Pf Þeat

ð8:184Þ

Taking the time derivative of the preceding equation: V2 r2 Ap1 þ V1 r1 Ap2 at @P1 ¼ ðPi  Pf Þb e @t V2 r2 V1 r1 lnðr2 =r1 Þ

ð8:185Þ

Substituting Equations (8.184) and (8.185) into Equation (8.179) and rearranging yields the following equation: P2 ¼ Pf  ðPi  Pf Þ

V1 r1 Ap2 at e V2 r2 Ap1

ð8:186Þ

For the following initial condition for P2 : t ¼ 0 as P2 ¼ P0 Equation (8.187) takes the following form: ðPi  Pf Þ ¼ ðPf  P0 Þ

V2 r2 Ap1 V1 r1 Ap2

ð8:187Þ

The preceding equation can be rewritten in a different way for Pi  P0 as follows: ðPi  Pf Þ ¼ ðPi  P0 Þ

V2 r2 Ap1 V1 r1 Ap2 þ V2 r2 Ap1

ð8:188Þ

Inserting this equation into Equation (8.185) and rearranging yields the following: at ¼ lnð

P1  Pf V1 r1 Ap2 þ V2 r2 Ap1 Þ Pi  P0 V2 r2 Ap1

ð8:189Þ

where a¼

k V2 r2 Ap1 þ V1 r1 Ap2 cf Z V2 r2 V1 r1 lnðr2 =r1 Þ

From the preceding equations, one get the following equation to compute permeability: k ¼ Zcf

V2 r2 Ap1 V2 r2 V1 r1 lnðr2 =r1 Þ P1  Pf 1 lnð Þ V2 r2 Ap1 þ V1 r1 Ap2 Pi  P0 V2 r2 Ap1 þ V1 r1 Ap2 t

ð8:190Þ

When gas is used as a permeation fluid, P1 and P2 are replaced with U1 ð¼ P21 Þ and U2 ð¼ P22 Þ, and permeability can be calculated using the same relation previously given. If the volume of reservoir 2 (V2 ) is greater than the volume of reservoir 1 (V1 ), (V2 ?V1 ) (for

53

Analytical methods

239

instance, the outer side of the specimen is open to air), then P0 ve Pf given in the preceding equation will be equal to atmospheric pressure (Pa ). For this particular case, the permeability of a specimen can be obtained from the following equation: V r lnðr2 =r1 Þ P1  Pf 1 ð8:191Þ lnð Þ k ¼ Zcf 1 1 Ap1 Pi  P0 t 8.3.3.3

Theory of interface or discontinuity permeability in radial flow tests

For the cylindrical coordinate system, the force equilibrium equation for the fluid can be given as follows (Fig. 8.34) (Aydan et al., 1997; Aydan, 2016): ðtðzþDzÞ  tðzÞ ÞrDrDy þ srðrÞ rDyDy  srðrþDrÞ ðr þ DrÞDyDy þ 2sy Dr sin ð

Dy Þ¼0 2

ð8:192Þ

with the use of the following relation: sin ð

Dy Dy Þ 2 2

The preceding equation can be rearranged. Then, if Taylor expansion is used for t and sr in Equation (8.192), the following partial differential is obtained: dsr sr  sy dt   ¼0 dz dr r

ð8:193Þ

For hydrostatic case (sr ¼ sy ¼ p), Equation (8.193) becomes: dsr dt  ¼0 dz dr

ð8:194Þ

Assuming that flow is laminar and the relation between shear force and shear strain rate g_ is linear: _ g_ ¼ t ¼ Zg;

du_ dv ¼ ; dz dz

sr ¼ p; v ¼ u_ ¼

du dt

ð8:195Þ

Substituting the preceding relations into Equation (8.194) yields the following partial differential equation: dp d2v Z 2 ¼0 dr dz

ð8:196Þ

Integrating the preceding partial differential equation for y-direction yields the following expression for flow velocity v: v¼

1 dp z2 þ C1 z þ C2 Z dr 2

ð8:197Þ

Introducing the following boundary conditions in Equation (8.197): z¼

h 2

as v ¼ 0;

z¼0

as t ¼ 0

240 Analytical methods

54

yields the integration constants C1 and C2 as: C1 ¼ 0; C2 ¼ 

1 dp h2 Z dr 8

Total flow rate vt through the discontinuity at a given time is: Z 2p Z z¼h2 vrdzdy vt ¼ 2

ð8:198Þ

ð8:199Þ

z¼0

0

The explicit form of vt is obtained as: vt ¼ 

2pr h3 dp Z 12 dr

ð8:200Þ

Let us redefine the flow rate vt in terms of an average velocity v over the discontinuity aperture area as: vt ¼ 2pr vh

ð8:201Þ

Inserting this equation into Equation (8.200), the following expression is obtained: v ¼ 

1 h2 dp Z 12 dr

ð8:202Þ

In an analogy to Darcy’s law, the preceding equation may be rewritten as: v ¼ 

kd dp Z dr

ð8:203Þ

where kd ¼

h2 12

or

h¼

pffiffiffiffiffiffiffiffiffi 12kd

and kd is called intrinsic permeability of discontinuity. Inserting Equation (8.203) into Equation (8.199), the flow rate vt can be obtained as: vt ¼ 2prhkd

dp dr

ð8:204Þ

The preceding equation can be rewritten in the following form: dr 2phkd dp ¼ r vt The integral form of the preceding equation may be written as: Z ro Z po dr 2phkd ¼ dp r vt ri pi

ð8:205Þ

ð8:206Þ

If integration is carried out, the permeability of discontinuity (kd ) can be found after some manipulations as follows: kd ¼ ½

vt lnðrro Þ 2=3 pffiffiffi i  4p 3ðpi  po Þ

ð8:207Þ

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Analytical methods

241

Figure 8.35 Pressure responses of a sandstone sample in a transient pulse test

Figure 8.35 shows pressure responses of reservoirs 1 and 2 in a transient pulse test on a sandstone sample. Despite some scattering of pressure responses, the variations of reservoir pressures tend to decrease with time and become asymptotic to a stabilizing pressure. The permeability of the sandstone sample was 3.1 × 10–12 m2.

8.4

Analytical solutions for heat flow: temperature distribution in the vicinity of geological active faults

As a first case, the geological fault is assumed to be sandwiched between two nonconductive rock slabs, and closed-form solutions are derived for temperature rises within the fault due to shearing. Then a more general case is considered such that a seismic energy release takes place within the fault, and the adjacent rock is conductive. The solution of the governing equation for this case is solved with the use of the finite element method (Aydan, 2016). Several examples were solved by considering some hypothetical energy release functions, and their implications are discussed. If a geological fault and its close vicinity may be simplified to a one-dimensional situation, as shown in Figure 8.36, by assuming that mechanical energy release is due purely to shearing with no heat production source. Thus, Equation (4.5) may be reduced to the following form: rc

@T ¼ rq þ tg_ @t

ð8:208Þ

Let us assume that the heat flux obeys Fourier’s law, which is given by: q ¼ k

@T @x

ð8:209Þ

Inserting Equation (8.209) into Equation (8.208) yields the following equation: rc

@T @2T ¼ k 2 þ tg_ @t @x

ð8:210Þ

The solution of the preceding equation will yield the temperature variation with time.

242 Analytical methods

56

Figure 8.36 Fault model

The energy release during earthquakes is a very complex phenomenon. Nevertheless, some simple forms relevant for overall behavior may be assumed in order to have some insight to the phenomenon. Two energy release rate functions of the following form are assumed: t E_ ¼ tg_ ¼ Atey

ð8:211Þ

E_ ¼ tg_ ¼ A e

ð8:212Þ

 yt

Constants A and A* depend on the shear stress and shear strain rate history with time and fault thickness. Constants y and y are time history constants. For situations illustrated in Figure 8.36, constants A and A* will take the following forms:For Equation (8.210): to uf ð8:213Þ A¼ 2 hy For Equation (8.211): to uf A ¼  hy

ð8:214Þ

Where uf ; h are the final relative displacement and thickness of the fault. to is the shear stress acting on the fault, and it is assumed to be constant during the motion. Two specific situations are analysed: • •

Creeping fault Fault with hill-shaped seismic energy release rate

In the case of creeping fault, the energy release rate is almost constant with time. The geometry of the fault is assumed to be one-dimensional as shown in Figure 8.37. Figures 8.38 and 8.39 show the computed temperature differences at selected locations with time and temperature difference distribution throughout the whole domain at selected time steps. In the computations, the energy release rate is assumed to be taking place within the fault zone only. The increase of temperature difference is parabolic, and they keep increasing as time goes by. Nevertheless, the temperature difference increases are about one-tenth of those of the fault sandwiched between nonconductive rock mass slabs.

CREEPING FAULT

16 12

Distance from fault 0.0 m 1.0 m 5.0 m 10.0 m 15.0 m 2

8

1.0

Energy Release

0.8 0.6

o

k=0.15 m /day/ C

0.4 4

0

0.2 20

40

60

TIME (DAYS)

80

0 100

ENERGY RELEASE RATE (m 2/day/oC)

o

TEMPERATURE INCREASE ( C)

Figure 8.37 Faulting models and energy release types

Figure 8.38 Temperature difference variations for a fault sandwiched between conductive rock mass slabs for creeping condition

244 Analytical methods

TEMPERATURE INCREASE ( o C)

Fault

20

58

Rock Mass CREEPING FAULT

15 Time (days)

10

5 10 20 40 70 100

5 0

5

10

15

20

DISTANCE FROM THE CENTRE OF FAULT (m)

TEMPERATURE INCREASE ( o C)

30

20 Energy Release Function: A*t*exp(-t/θ)

25 15 20

Distance 0.0 m 1.0 m 5.0 m 10.0 m 15.0 m

15

10

10 5 5 0

20

40

60

80

0 100

ENERGY RELEASE RATE (m 2 /day/ o C)

Figure 8.39 Temperature distributions at different time steps for a fault sandwiched between conductive rock mass slabs for creeping condition

TIME (days) Figure 8.40 Temperature difference variations for a fault sandwiched between conductive rock mass slabs for hill-shaped energy release function

Figures 8.40 and 8.41 show the computed temperature differences at selected locations with time and temperature difference distribution throughout the whole domain at selected time steps for a fault with a hill-like energy release rate. In the computations, the energy release rate is assumed to be taking place within the fault zone only. The increase of

Analytical methods

TEMPERATURE INCREASE (o)

59

Fault

25

245

Rock Mass

20 15

Time (days) 5 10 20 40 70 100

10 5 0

5

10

15

20

DISTANCE FROM THE CENTRE OF FAULT (m)

Figure 8.41 Temperature distributions at different time steps for a fault sandwiched between conductive rock mass slabs for hill-shaped energy release function

temperature difference is parabolic. Temperature differences increase at first, and then they tend to decay in a similar manner to the assumed seismic energy release rate function. This situation will be probably quite similar to the actual situation in nature. The temperature difference increases are about one-tenth of those of the fault sandwiched between nonconductive rock mass slabs. These results indicate that the observation of ground temperatures may be a very valuable source of information in the prediction of earthquakes because atmospheric temperature measurements near the ground surface may be quite problematic in interpreting the observations. However, the observation of hot-spring temperature, which reflects the actual ground temperature, may be very good tool for such measurements without any deep boring.

8.5

Analytical solutions for diffusion problems

8.5.1

Drying testing procedure

Let us consider a sample with volume V dried in air with infinite volume as shown in Figure 8.42 (Aydan, 2003). Water-contained Q in a geo-material sample may be given in the following form: Q ¼ rw yw V

ð8:215Þ

where rw ; yw and V are water density, water content ratio and volume of sample, respectively. Assuming that water density and sample volume remain constant, the flux q of water content may be written in the following form: q¼

dQ dy ¼ rw V w dt dt

ð8:216Þ

246 Analytical methods

60

Figure 8.42 Physical and mechanical models for water migration during drying process

Air is known to contain water molecules of 6 g m3 when relative humidity is 100%. When the relative humidity is less than 100%, water is lost from geo-materials to air. If such a situation presents, the water lost from the sample to air may be given in the following form using a concept similar to Newton’s cooling law in thermodynamics: q ¼ rw As hDy ¼ rw As hðyw  ya Þ

ð8:217Þ

Where h and As are the water loss coefficient and surface area of sample. Requiring that the water loss rate of sample should be equal to the water loss into air on the basis of the mass conservation law, one can easily write the following relation: rw As hðyw  ya Þ ¼ rw V

dyw dt

ð8:218Þ

The solution of differential Equation (8.217) is easily obtained in the following form yw ¼ ya þ Ceat

ð8:219Þ

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Analytical methods

247

where a¼h

As V

The integration constant may be obtained from the initial condition, that is: yw ¼ yw0

at

t¼0

ð8:220Þ

as follows: C ¼ yw0  ya

ð8:221Þ

Thus the final expression takes the following form: yw ¼ ya þ ðyw0  ya Þeat

ð8:222Þ

If the water content migration is considered a diffusion process, Fick’s law in one dimension may be written as follows: q ¼ rw D

@yw @x

ð8:223Þ

Requiring that the water loss rate given by Equation (8.222) to be equal to that given by Equation (8.222) yields the following relation: V D¼h ð8:224Þ As If surface area As and volume V of sample are known, it is easy to determine the water migration diffusion constant D from drying tests, provided that the coefficient a and subsequently h are determined from experimental results fitted to Equation (8.221). If samples behave linearly, water migration characteristics should remain the same during the swelling and drying processes. Recent technological developments have made it quite easy to measure the weight of samples and the environmental conditions such as temperature and humidity. Figure 8.43 shows an automatic weight and environmental conditions monitoring system developed for such tests. It is also possible to measure the volumetric variations (shrinkage) during the drying process using noncontact-type displacement transducers (i.e. laser transducers). Physical and mechanical properties of materials can be measured using the conventional testing machines such as wave velocity measurements, uniaxial compression tests, elastic modulus. Tuff samples used in the tests were from Avanos, Ürgüp and Derinkuyu of the Cappadocia region in Turkey and Oya in Japan. The samples from the Cappadocia region are gathered from historical and modern underground rock structures. They represent the rocks in which historical and modern underground structures were excavated. These tuff samples bear various clay minerals as given in Table 8.2 (Temel, 2002; Aydan and Ulusay, 2003). As noted from the table, the clay content is quite high in Avanos tuff, and most of the clay minerals are smectite. In drying experiments, the samples that underwent swelling were dried in a room with an average temperature of 23°C and relative humidity of 65–70. Figures 8.44, 8.45 and 8.46 show the drying test results for some tuff samples from the Cappadocia region in Turkey. As seen from the figures, it takes a longer time for the tuff sample from Avanos compared with Ürgüp and Derinkuyu samples. The Derinkuyu sample dries much rapidly than the

248 Analytical methods

62

Figure 8.43 Experimental setup for measuring water content during drying Table 8.2 XRD results from the samples of Ürgüp (Kavak tuff) and Avanos Specimen Number

Clay Percentage

UR-1 (Ürgüp) UR-2 (Ürgüp) AV-1(Avanos) AV-2 (Avanos)

74 60 94 82

Clay Fraction Smectite

Kaolin

Illite

83 67 84 95

14 25 13 5

3 8 3 T

T: Trace amount

others. Each sample was subject to drying twice. Once again, it is noted that the drying period increases for Avanos tuff after each run, whereas Derinkuyu tuff tends to dry much rapidly in the second run. From these tests, it may be also possible to determine the diffusion characteristics of each tuff. The theory derived in the previous section could be applied to the experimental results shown in Figures 8.44, 8.45 and 8.46. To obtain the constants of water migration model, Equation (8.221) may be rewritten as follows:   y  ya ln w ¼ at yw0  ya

ð8:225Þ

The plot of experimental results in the semilogarithmic space first yields the constant a, from which constant h and diffusion coefficient D can be computed subsequently. The results are shown in Figures 8.44, 8.45 and 8.46. The unit of parameters a; h and D are 1 h1, cm h1 and cm2 h1, respectively. The computed values of parameters a; h and D are also shown in the same figures.

RELATIVE WATER CONTENT (%)

Drying Test on Avanos Tuff

2

10

α=0.01655; h=0.0571; D=0.1968

1

10

α=0.0332; h=0.1144; D=0.3944

0

10

o

T = 23 C; RH=65 (%)

-1

10

0

AV1-13 D=53mm H=99mm Wd=215 gf Ws=275 gf

50 100 TIME (hours)

150

RELATIVE WATER CONTENT (%)

Figure 8.44 Determination of constants for relative water content variation during drying of Avanos tuff

Drying Test on Ürgüp Tuff

102

101 α=0.05; h=0.652; D=0.47

100

10-1 0

U2-2 D=46mm H=92mm Wd=200 gf Ws=230,248 gf

10

T = 23 oC; H=67 (%)

20 30 40 TIME (hours)

50

60

Figure 8.45 Determination of constants for relative water content variation during drying of Ürgüp tuff

250 Analytical methods

64

Figure 8.46 Determination of constants for relative water content variation during drying of Derinkuyu tuff

Figure 8.47 Experimental setups: (a) top surface unsealed, (b) top-surface sealed

8.5.2

Saturation testing technique

Initially dry samples can be subjected to saturation, and water migration characteristics may be obtained. The sides of samples can be sealed and subjected to saturation from the bottom. The top surface may be sealed and unsealed, as illustrated in Figure 8.47. Samples can be isolated against water migration from the sides by sealing while the bottom surface of the samples can be exposed to saturation by immersing in water up to a given depth. There may be two conditions at the top surface, which could be either

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Analytical methods

251

exposed to air directly or sealed. When the top surface is sealed, the boundary value would be changing with time. The water migration coefficient can be determined from the solution of the following diffusion equation: @yw @2y ¼ D 2w @t @x

ð8:226Þ

When the top surface is unsealed, the top boundary condition (x=H) is: yw ¼ ya

ð8:227Þ

On the other hand, if the top surface is sealed, the boundary condition is time dependent, and it can be estimated from the following condition: qx¼H ¼ q^n ðtÞ

ð8:228Þ

For some simple boundary conditions, the solution of partial differential Equation (8.226) can be easily obtained using the technique of separation of variables (i.e. Keryszig, 1983). In the general case, it would be appropriate to solve it using finite difference technique or finite element method (i.e. Aydan, 2003, 2016).

8.6

Evaluation of creep-like deformation of semi-infinite soft rock layer

The simplified analytical model introduced in this section is based the theoretical model developed by Aydan (1994, 1998). The momentum conservation law for an infinitely small element of ground on a plane with an inclination of α for each respective direction can be written in the following form (Fig. 8.48):

Figure 8.48 Modeling of a layer subjected to shearing Source: From Aydan, 1994, 1998

252 Analytical methods

66

x-direction @t @p ¼  rg sin a @y @x

ð8:229Þ

y-direction @p ¼ rg sin a @y

ð8:230Þ

where τ, p,ρ, g are shear stress, pressure, density and gravitational acceleration, respectively. The variation of pressure along the x-direction is given by: @p @h ¼ rg cos a @x @x

ð8:231Þ

If shear stress related to shear strain is linearly as given in the following form: t ¼ Gg; g ¼

@u @y

one can easily obtain the solution given as: t ¼ rg cos aðtan a 

@h Þðh  yÞ @x

ð8:232Þ

If the variation of ground surface height (h) is neglected, the resulting equation for shear stress and displacement takes the following form: t ¼ rg sin aðh  yÞ; u ¼

rg sin a  y y h G 2

ð8:233Þ

As is well-known, rainfall induces groundwater level fluctuations. However, these fluctuations are not as high as presumed in many limiting equilibrium approaches to analyzing the failure of slopes. In other words, the whole body, which is prone to fail, does not become fully saturated. However, the monitoring results indicate that a certain thickness of layer becomes saturated. In view of experimental results, the deformation modulus would become smaller during the saturation process and recover its original value upon drying. The deformation modulus during saturation may be assumed to be the plastic deformation modulus (Gp), and the displacement induced during the saturation period may be viewed as the plastic (irrecoverable) deformation (Fig. 8.49). With the use of this concept and the analytical model previously presented, one can easily derive the following equation for deformation induced by saturation as: us ¼

 rg sin a  y  y h t Gs 2

ð8:234Þ

where t is the thickness of saturated zone in a given cycle of saturation drying. The plastic deformation would be the difference between displacements induced under saturated and

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Analytical methods

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Figure 8.49 Constitutive modeling of cyclic softening-hardening of marl layer

Figure 8.50 Comparison of measured and computed displacements

dry states, and it will take the following form:     1 1 y    h t up ¼ rg sin ay Gs Gd 2

ð8:235Þ

where Gd and Gs are shear modulus for dry and saturated states, respectively. Thus the equivalent shear modulus may be called the plastic deformation modulus (Gp) in this chapter and can be written as: Gp ¼

Gs Gd Gd  Gs

ð8:236Þ

The time for saturation and drying of marls is very short (say, in hours). With this observational fact and experimental results, the analysis presented is based on the day unit. Figure 8.50 compares the computed displacement and displacement measured at

254 Analytical methods

68

monitoring station No.1 of the Gündoğdu district of Babadağ town with the consideration of thickness of the saturation zone (Kumsar et al., 2016). Despite some differences between computed and measured responses, the analytical model can efficiently explain the overall response of the landslide area of the Gündoğdu district of the town of Babadağ.

References Aydan, Ö. (1987). Approximate estimation of plastic zones about underground openings. Interim report (unpublished), Nagoya University, 8p. Aydan, Ö. (1989). The stabilisation of rock engineering structures by rockbolts. Doctorate Thesis, Nagoya University. Aydan, Ö. (1994) The dynamic shear response of an infinitely long visco-elastic layer under gravitational loading. Soil Dynamics and Earthquake Engineering, Elsevier, 13, 181–186. Aydan, Ö. (1995) Mechanical and numerical modelling of lateral spreading of liquified soil. The 1st Int. Conf. on Earthquake Geotechnical Engineering, IS-TOKYO’95, Tokyo. pp. 881–886. Aydan, Ö. (1997) Dynamic uniaxial response of rock specimens with rate-dependent characteristics. SARES’97. pp. 322–331. Aydan, Ö. (1998). A simplified finite element approach for modelling the lateral spreading of liquefied ground. The 2nd Japan-Turkey Workshop on Earthquake Engineering, Istanbul. Aydan, Ö. (2003) The moisture migration characteristics of clay-bearing geo-materials and the variations of their physical and mechanical properties with water content. 2nd Asian Conference on Saturated Soils, UNSAT-ASIA. pp. 383–388. Aydan, Ö. (2016). Time Dependency in Rock Mechanics and Rock Engineering. CRC Press, Taylor and Francis Group, London, 241p. Aydan, Ö. (2018). Rock Reinforcement and Rock Support. CRC Press, Taylor and Francis Group, London, 486p. Aydan, Ö. & Geniş, M. (2010) Rockburst phenomena in underground openings and evaluation of its counter measures. Journal of Rock Mechanics, Turkish National Rock Mechanics Group, (Special Issue 17), 1–62. Aydan, Ö. & Nawrocki, P. (1998) Rate-dependent deformability and strength characteristics of rocks. In Proceedings of Symposium on the Geotechnics of Hard Soils-Soft Rock, Napoli, 1. pp. 403–411. Aydan, Ö. & Ulusay, R. (2003) Geotechnical and geoenvironmental characteristics of man-made underground structures in Cappadocia, Turkey. Engineering Geology, 69, 245–272. Aydan, Ö., Ersen, A., Ichikawa, Y. & Kawamoto, T. (1985) Temperature and thermal stress distributions in mass concrete shaft and tunnel linings during the hydration of concrete (in Turkish). The 9th Mining Science and Technology Congress of Turkey, Ankara. pp. 355–368. Aydan, Ö., Güloğlu, R. & Kawamoto, T. (1986) Temperature distributions and thermal stresses in tunnel linings due to hydration of cement (in Japanese). Tunnels and Underground, 17(2), 29–36. Aydan, Ö., Akagi, T. & Kawamoto, T. (1993) Squeezing potential of rocks around tunnels: Theory and prediction. Rock Mechanics and Rock Engineering, 26(2), 137–163. Aydan, Ö., Seiki, T., Jeong, G.C. & Tokashiki, N. (1994). Mechanical behaviour of rocks, discontinuities and rock masses. Int. Symp. Pre-failure Deformation Characteristics of Geomaterials, Sapporo, 2, 1161–1168. Aydan, Ö., Akagi, T., Ito, T., Ito, J. & Sato, J. (1995a) Prediction of deformation behaviour of a tunnel in squeezing rock with time-dependent characteristics. Numerical Models in Geomechanics, NUMOG, V, 463–469. Aydan, Ö., Akagi, T., Ito, T. & Sezaki, M. (1995b) The design of supports of tunnels in squeezing rocks (in Japanese). The 25th Rock Mechanics Symposium of Japan, JSCE, 51–55.

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Aydan, Ö., Akagi, T. & Kawamoto, T. (1996) The squeezing potential of rock around tunnels: theory and prediction with examples taken from Japan. Rock Mechanics and Rock Engineering, 29(3), 125–143. Aydan, Ö., Üçpırtı, H. & Türk, N. (1997) Theory of laboratory methods for measuring permeability of rocks and tests. Kaya Mekaniği Bülteni, 13, 19–36. Aydan, Ö., Watanabe, S. & Tokashiki, N. (2008) The inference of mechanical properties of rocks from penetration tests. 5th Asian Rock Mechanics Symposium (ARMS5), Tehran, 213–220. Aydan, Ö., Ohta, Y., Tano, H. (2010). Multi-parameter response of soft rocks during deformation and fracturing with an emphasis on electrical potential variations and its implications in geomechanics and geoengineering. The 39th Rock Mechanics Symposium of Japan, Tokyo, 116–121. Aydan, Ö., Uehara, F. & Kawamoto, T. (2012) Numerical study of the long-term performance of an underground powerhouse subjected to varying initial stress states, cyclic water heads, and temperature variations. International Journal of Geomechanics, ASCE, 12(1), 14–26. Boussinesq, J. (1885) Applications des potentiels à l’étude de l’équilibre et mouvement des solides elastiques. Gauthier-Villard, Paris. Brace, W.F., Walsh, J.B. & Frangos, W.T. (1968) Permeability of granite under high pressure. Journal of Geophysical Research, 73, 2225–2236. Bieniawski, Z.T. (1970) Time-dependent behaviour of fractured rock. Rock Mechanics, 2, 123–137. Brady, B.H.G. & Brown, E.T. (1985) Rock Mechanics for Underground Mining. Kluwer Academic Publications, New York, Boston, London, Moscow. 527p. Detournay, E. (1986) An approximate statical solution of the elastoplastic interface for the problem of Galin with a cohesive-frictional material. International Journal of Solids and Structures, Elsevier, 22, 1435–1454. England, A.H. (1971) Complex variable methods in elasticity. Wiley-Interscience, 181p. Eringen, A.C. (1980) Mechanics of Continua. R. E. Krieger Pub. Co., New York. Fenner, R. (1938) Researches on the notion of ground stress (in German). Glückauf, 74, 681–695. Galin, L.A. (1946) Plane elastic-plastic problem: Plastic regions around circular holes in plates and beams. Prikladnaia Matematika i Mechanika, 10, 365–386. Gerçek, H. (1993) Qualitative prediction of failures around non-circular openings. In: Paşamehmetoğlu, A.G. et al. (eds.) Proc. Int. Symp. on Assessment and Prevention of Failure Phenomena in Rock Engineering. A.A. Balkema, Rotterdam. pp. 727–732. Gerçek, H. (1996) Special elastic solutions for underground openings. Milestones in Rock Engineering: The Bieniawski Jubilee Collection, Balkema, Rotterdam. pp. 275–290. Gerçek, H. (1997) An elastic solution for stresses around tunnels with conventional shapes. International Journal of Rock Mechanics and Mining. Science, 34(3–4), paper No. 096. Green, A.E. & Zerna, W. (1968) Theory of elasticity, Clarendon Press, Oxford. Hoek, E. & Brown, E.T. (1980) Underground Excavations in Rock. Inst. Min. & Metall., 251, 21–26. London. Inglis, C.E. (1913) Stresses in plates due to the presence of cracks and sharp corners. Transactions of the Institute of Naval Architects, 55, 219–241. Jaeger, J.C. & Cook, N.G.W. (1979) Fundamentals of Rock Mechanics, 3rd edition. Chapman & Hall, London. pp. 79, 311. Kastner, H. (1961) Statik des Tunnel- and Stollenbaues, (“Design of. Tunnels”), 2nd edition. Springer-Verlag, Berlin. Kirsch, G. (1898) Die theorie der elastizitat und die bedürfnisse der festigkeitslehre. Veit Ver. Deut. Ing., 42, 797–807. Kolosov, G.V. (1909) An application of the theory of functions of a complex variable to a planar problem in the mathematical theory of elasticity. Dorpat (Yuriev) University, Doctoral Thesis, 187p. Kreyszig, E. (1983) Advanced Engineering Mathematics. John Wiley & Sons, New York.

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Kumsar, H., Aydan, Ö., Tano, H., Çelik, S.B. & Ulusay, R. (2016) An integrated geomechanical investigation, multi-parameter monitoring and analyses of Babadağ-Gündoğdu creep-like landslide. Rock Mechanics and Rock Engineering, Special Issue on the Deep-seated landslides. DOI:10.1007/s00603-015-0826-7. Ladanyi, B. (1974) Use of the long-term strength concept in the determination of ground pressure on tunnel linings. Proc. of 3rd Congr. Int. Soc. Rock Mech., Denver, 2B, 1150–1165. Lama, R.D. & Vutukuri, V.S. (1978) Handbook on Mechanical Properties of Rocks. Trans Tech Publications, Clausthal, Germany. Milne-Thomson, L.M. (1960) Plane elastic systems. Springer-Verlag, Berlin, Heidelberg. Muskhelishvili, N.I. (1962) Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen. Polubarinova-Kochina, P.YA. (1962) Theory of Groundwater Movement. Princeton University Press, Princeton. Sezaki, M., Aydan, Ö. & Yokota, H. (1994) Non-destructive testing of shotcrete for tunnels. Int. Conf. on Inspection, Appraisal, Repairs & Maintenance of Buildings & Structures, Bangkok, 209–215. Snow, D.T. (1965) A Parallel Plate Model of Fractured Permeable Media. PhD Dissertation, University of California, Berkeley. Talobre, J. (1957) The Mechanics of Rocks, Dunod (in French), Paris. Temel, A. (2002) Personal Communication. Hacettepe University, Geological Engineering Department, Ankara, Turkey. Terzaghi, K. (1925) Erdbaumechanik auf bodenphysikalischer Grundlage. F. Deuticke’s Verlag, Leipzig, Vienna. Terzaghi, K. (1946) Rock defects and loads on tunnel support. Introduction to rock tunnelling with steel supports. R.V. Proctor & T.L. White (eds.). Commercial Sheering & Stamping Co., Youngstown, Ohio, U.S.A., 271p. Terzaghi, K. (1960) Stability of steep slopes on hard unweathered rock. Geotechnique, 12, 251–270. Timoshenko, S. & Goodier, J.N. (1951) Theory of elasticity. McGraw Hill Book Company, New York, 519p. Üçpırtı, H. & Aydan, Ö. (1997) An experimental study on the permeability of interface between sealing plug and rock. The 28th Rock Mechanics Symposium of Japan. pp. 268–272. Verruijt, A. (1970) Theory of Groundwater Flow. MacMillian, London, UK. Zachmanoglou, E.C. & Thoe, D.W. (1986) Introduction to Partial Differential Equations with Applications. Dover Pub. Inc., New York. Zeigler, T.W. (1976). Determination of rock mass permeability. U.S. Army Corps of Engineers Waterways Experiment Station, Vicksburg, Miss. Tech. Rept. S-76-2. 112pp. Zoback, M.D., Tsukahara, H. & Hickman, S.H. (1980) Stress measurements at depth in the vicinity of the San Andreas Fault: Implications for the magnitude of shear stress at depth. Journal of Geophysical Research, 85(B11), 6157–6173.

Chapter 9

Numerical methods

In this chapter, the first part is related to the solution of fundamental governing equations using the finite element method. Nevertheless, an illustrative example is given in the introduction to explain the similarity and dissimilarity of various numerical methods as well as exact solutions. Although formulations for multidimensional situations are not presented, they can be easily extended to such situations by just selecting shape functions for multidimensional situations as the general forms of equations would remain the same. In the second part, some numerical procedures developed for rock masses involving discontinuities are presented, and several examples are given.

9.1

Introduction

There are three approximate methods: • • •

Finite difference method (FDM) Finite element method (FEM) Boundary element method (BEM)

The characteristics of the closed-form and approximate methods are briefly discussed through solving the following ordinary differential equation: d2 u
u¼0 dx2

ð9:1Þ

The boundary conditions are as follows: u¼0 u¼1 9.1.1

at at

x¼0 x¼1

Closed-form solution

If the solution of Equation (9.1) is a series of exponential functions elx , the characteristics equation can be obtained as: l2
1 ¼ 0

ð9:2Þ

258 Numerical methods

2

Hence, the roots are: l1 ¼ 1;

l2 ¼
1

ð9:3Þ

Thus, the solution is of the following form: u ¼ C1 ex þ C2 e
x

ð9:4Þ

The integration constants are obtained from the boundary conditions as: C1 ¼ 0:4254589 9.1.2

and

C2 ¼
0:4254589

Finite Difference Method (FDM)

The finite difference method is the earliest approximate method, and it is called a strong form approximate solution. It utilizes the Taylor expansion of dependent variable to discretize the governing equation. Let us assume that the domain is discretized into n segments with equal interval Dx. Equation (9.1) at a node j may be rewritten as:
2  du
uj ¼ 0 ð9:5Þ dx2 x¼xj The Taylor expansions of function u at nodes i, j and k may be written as:

2  du Dx du Dx2 ui ðxj
DxÞ ¼ uj
þ
03 dx x¼xj 1! dx2 x¼xj 2! uj ðxj Þ ¼ uj

ð9:6Þ ð9:7Þ





2  du Dx du Dx2 þ þ 03 uk ðxj þ DxÞ ¼ uj þ dx x¼xj 1! dx2 x¼xj 2! From the preceding relations, one gets the following relation:
2  uk
2uj þ ui du ¼ dx2 x¼xj Dx2

ð9:8Þ

ð9:9Þ

Thus the finite difference form of the preceding equation takes the following form:   1 2 1 u
ð þ 1Þu þ u ¼0 ð9:10Þ j Dx2 i Dx2 Dx2 k This simultaneous equation system for a domain divided into n segments will result in: ½KfUg ¼ fFg

ð9:11Þ

where matrix ½K has n
1 rows and n þ 1 columns, vector fUg has n þ 1 rows and vector fFg has n
1 rows. However, if the boundary conditions are introduced, it yields the following simultaneous equation system: ½K  fU  g ¼ fF  g

ð9:12Þ

3

Numerical methods

259

The resulting matrix ½K   has n
1 rows and n
1 columns. Similarly, the resulting vectors fU  g and fF  g have n
1 rows. Therefore, it becomes possible to solve this simultaneous equation system. Example 1: Let us assume that we have two segments and three nodes. Accordingly, u1 ¼ 0, u3 ¼ 1, Dx ¼ 0:5. From Equation (9.12), we obtain unknown u2 as: u2 ¼ 0:444444444444

ð9:13Þ

Example 2: Let us assume that we have four segments and five nodes. Accordingly, u1 ¼ 0, u5 ¼ 1, Dx ¼ 0:25. From Equation (9.12), we get the following equation system for unknown fu g: 2


33

6 4 16 0

16
33 16

9 38 9 8 0 > u > = = > < < 2> 7 0 16 5 u3 ¼ > > ; ; > : : >
16 u4
33 0

ð9:14Þ

The solution of this simultaneous equation system yields the following: u2 ¼ 0:215114752376; u3 ¼ 0:443674176776; u4 ¼ 0:69963237225 9.1.3

Finite Element Method (FEM)

The finite element method is relatively new, but it is the most widely used method in engineering and science as compared with FDM or other methods. The governing equation is first integrated over the domain, and then the resulting integral equation is discretized. Therefore, it is called a weak form solution as there is a possibility that the solution may be different from the actual one. (a)

Weak formulation

Taking a dot product of Equation (9.1) by a trial function dv and integrating it yields the following: Z 1 Z 1 d2 u dv  2 dx
dv  udx ¼ 0 ð9:15Þ dx 0 0 Introducing the integral by parts for the first term gives: Z 1 Z 1 ddv du 1  dx þ dv  udx ¼ ½dv  ^t0 dx dx 0 0

ð9:16Þ

where ^t ¼

du n dx

and n is the unit normal vector. Let us assume that the trial function v is the same as the function u, which is generally called the Galerkin approach in finite element formulation.

260 Numerical methods

(b)

4

Discretization

The domain is discretized into subdomains called elements. The function u is approximated by a chosen function in an element, and it is summed up for the whole domain. For this particular problem, let us choose a linear function of the following form: u ¼ ax þ b

ð9:17Þ

Let us assume that the function u at nodes i and j are known. Thus we can write the following: " #( ) ( ) ui xi 1 a ¼ ð9:18Þ xj 1 uj b Taking the inverse of the preceding relation, one gets coefficients a and b. Inserting these coefficients in Equation (9.17) yields the following: u ¼ Ni ui þ Nj uj

ð9:19Þ

where Ni ¼

xj
x ; xj
xi

Nj ¼

x
xi xj
xi

The preceding equation may be rewritten in a compact form as: u ¼ ½NfUe g

or u ¼ NUe

ð9:20Þ

where ½N ¼ ½Ni ; Nj , fUe g ¼ fui ; uj g.1 The derivative of the preceding equation takes the following form: T

dNj du dNi ¼ ui þ u dx dx dx j

ð9:21Þ

The preceding relation is rewritten in a compact form as: du ¼ ½BfUe g dx

or

du ¼ BUe dx

ð9:22Þ

where ½B ¼ ½Bi ; Bj , Bi ¼
1=Le ; Bj ¼ 1=Le ; Le ¼ xj
xi . The dot product of two vectors are presented in the following form in the finite element method: c ¼ a  b ! c ¼ fag fbg T

ð9:23Þ

Equation (9.16), which holds for the whole domain, must also hold for each element as: Z xj Z xj du du x du  udx ¼ ½du  ^txji ð9:24Þ  dx þ xi dx dx xi 1 should be noted that the horizontally written vector is defined as the transpose of the vector.

5

Numerical methods

261

Inserting relations given by Equations (9.20) and (9.22) into Equation (9.24) and using the finite element convention for dot product (Equation (9.23)) yields the following: 
 Z xj  Z xj T T T xj ^ fdUe g ½B ½Bdx þ ½N ½Ndx fUe g
½½N txi ¼ 0 ð9:25Þ xi

xi

The preceding relation implies the following: ½Ke fUe g ¼ fFe g

ð9:26Þ

where Z ½Ke  ¼

xj

Z

xj

½B ½Bdx þ T

xi

½N ½Ndx; T

xi

T x fFe g ¼ ½½N ^txji

For a typical element, one gets the preceding relations specifically for a shape function given by Equation (9.17): " # " # ( ) ( ) ^ti ui 1 1
1 L 2 1 þ ; fUe g ¼ ; fFe g ¼ ½Ke  ¼ L
1 1 6 1 2 ^tj uj The sum-up of the preceding relation for the whole domain is: ½KfUg ¼ fFg

ð9:27Þ

where ½K ¼

n X

½Ke ;

fUg ¼

k¼1

n X k¼1

fUe g;

fFg ¼

n X fFe g k¼1

Example 1: Let us assume that we have two elements and three nodes. Accordingly, u1 ¼ 0, u3 ¼ 0, xj
xi ¼ 0:5. From Equation (9.27), we obtain unknown u2 as: u2 ¼ 0:4423076

ð9:28Þ

Example 2: Let us assume that we have four elements and five nodes. Accordingly, u1 ¼ 0, u5 ¼ 0, xj
xi ¼ 0:25. The solution of the simultaneous equation system (Equation (9.27)) yields the followings: u2 ¼ 0:214787576025; u3 ¼ 0:443140650725; u4 ¼ 0:699481489062

9.1.4

Comparisons

Solutions obtained from the approximate methods for the example chosen are compared with that by the closed-form solution (CFS). Figures 9.1(a) and (b) show comparisons of computations for two element (three nodes) and four element (five nodes) discretizations of the domain by the FDM and FEM with that by the CFS, respectively. As seen from both figures, the approximate solutions almost coincide with the exact ones at nodal

262 Numerical methods

6

Figure 9.1 Comparison of computations by FDM and FEM with that by CFS

points. Increasing the number of nodes results in better solutions, and errors caused by discretization decreases.

9.2

1-D hyperbolic problem: equation of motion

As shown in Chapter 3, the equation of momentum for 1-D problems can be written as: @s @2u þb¼r 2 @x @t

ð9:29Þ

Let us assume that this equation is subjected to following boundary and initial conditions Boundary conditions uð0;tÞ ¼ 0

as x ¼ 0

tð0;tÞ ¼ t0

as x ¼ L

ð9:30Þ

Initial conditions uðx; 0Þ ¼ 0 as t ¼ 0 _ 0Þ ¼ 0 uðx;

as t ¼ 0

ð9:31Þ

Let us further assume that the material is of Kelvin-type as given here: s ¼ Eε þ Cε_

ð9:32Þ

Displacement–strain and strain rate are given as: ε¼

@u ; @x

ε_ ¼

@u_ @x

ð9:33Þ

7

Numerical methods

263

where u_ ¼

@u @t

We utilize finite element method to solve this equation system in the following section. 9.2.1

Weak form formulation

Taking a variation on displacement field du, the integral form of Equation (9.29) becomes: Z Z Z @s @2u dV þ du  bdV ¼ rdu  2 dV du  ð9:34Þ @x @t V V V where dV ¼ dAdx. Applying the integral by parts to the first term on the LHS with respect to x yields the following: Z Z Z Z @du @2u L  sdV þ rdu  2 dV ½du  t0 dA þ du  bdV ¼ ð9:35Þ @t A V V @x V Where t ¼ s  n. The preceding equation is called the weak form of Equation (9.29). Inserting the constitutive relation given by Equation (9.32) into Equation (9.35), we obtain the following: Z Z Z Z Z @du @u @du @u @2u L  dV þ C  dV þ rdu  2 dV ½du  t0 dA þ du  bdV ¼ E @x @x @x @x @t A V V V V ð9:36Þ

9.2.2

Discretization

For this particular problem, a linear function of the following form for the space is chosen: uðtÞ ¼ ax þ b

ð9:37Þ

Let us assume that the function u at nodes i and j are known. Thus we can write: " #( ) ( ) ui xi 1 a ð9:38Þ ¼ uj xj 1 b Taking the inverse of the preceding relation, one gets coefficients a and b. Inserting these coefficients in Equation (9.37) yields the following: u ¼ Ni ui þ Nj uj

ð9:39Þ

where Ni ¼

xj
x ; xj
xi

Nj ¼

x
xi xj
xi

264 Numerical methods

8

The preceding equation may be rewritten in a compact form as: u ¼ ½NfUe g

or u ¼ NUe

ð9:40Þ

where ½N ¼ ½Ni ; Nj , fUe g ¼ fui ; uj g. The derivative of the preceding equation takes the following form: T

dNj du dNi ¼ ui þ u dx dx dx j

ð9:41Þ

The preceding relation is rewritten in a compact form as: du ¼ ½BfUe g dx

du ¼ BUe dx

or

ð9:42Þ

Where ½B ¼ ½Bi ; Bj , Bi ¼
1=Le ; Bj ¼ 1=Le ; Le ¼ xj
xi . Equation (9.36), which holds for the whole domain, must also hold for each element: Z Z Z Z Z @2u @du @u @du @u x  dV þ E  dV ¼ rdu  2 dV þ C ½du  txji dA þ du  be dV @t @x @x @x @x Ve Ve Ve Ae Ve ð9:43Þ The discretised form of the preceding equation becomes: Z Z Z T T T € _ r½N ½NfU e gdV þ C½B ½BfU e gdV þ E½B ½BfUge dV ¼ Ve

Ve

ð9:44Þ

Ve

Z

Z ½Nk  te xji dA þ T

Ae

½N be dV

x

T

Ve

The preceding equation may be written in a compact form as: ½Me fU€ e g þ ½Ce fU_ e g þ ½Ke fUe g ¼ fFe g

ð9:45Þ

where Z

Z

½Me  ¼

r½N ½NdV ;

½Ce  ¼

T

Ve

Z C½B ½BdV ; T

Ve

Z

½Ke  ¼

E½B ½BdV ; T

Ve

Z

fFe g ¼

½½N te xji dA þ T

Ae

½N be dV

x

T

Ve

For the total domain, we have the following: € þ ½CfUg _ þ ½KfUg ¼ fFg ½MfUg

ð9:46Þ

where ½M ¼

n n n n n X X X X X ½Me k ; ½C ¼ ½Ce k ; ½K ¼ ½Ke k ; fFg ¼ fFe gk ; fUg ¼ fUe gk k¼1

k¼1

k¼1

k¼1

k¼1

9

Numerical methods

265

Equation (9.46) could not be solved as it is. For a time step m, we can rewrite Equation (9.46) as: _ m þ ½KfUgm ¼ fFgm € m þ ½CfUg ½MfUg

ð9:47Þ

Therefore, we discretize displacement field {U} for time-domain using the Taylor expansion as it is in the finite difference method as: fUgm
1 ¼ fUgm


@fUgm Dt @ 2 fUm g Dt2 þ
03 @t2 2! @t 1!

fUgm ¼ fUgm

ð9:48Þ ð9:49Þ

fUgmþ1 ¼ fUgm þ

@fUm g Dt @ 2 fUm g Dt2 þ þ 03 @t 1! @t2 2!

ð9:50Þ

From the preceding relations, one easily gets the following: _ m¼ fUg

1 ðfUgmþ1
fUgm
1 Þ Dt

ð9:51Þ

€ m¼ fUg

1 ðfUgmþ1
2fUgm þ fUgm
1 Þ Dt2

ð9:52Þ

Inserting these relations into Equation (9.47), we get the following: ½M  fUgmþ1 ¼ fF  gmþ1

ð9:53Þ

where 

1 1 ½M þ ½C ½M  ¼ Dt2 Dt





   2 1 1 ¼ ½M
½K fUgm
½M
½C fUgm
1 þ fFgm Dt2 Dt2 Dt 



fF gmþ1

9.2.3

Specific example

For a typical two-noded element, the followings are obtained: " # " # " # rLe Ae 2 1 CAe 1
1 EAe 1
1 ½Me  ¼ ; ½Ce  ¼ ; ½Ke  ¼ ; 6 L
1 1 L
1 1 1 2

fFe g ¼

( ) ti tj

b þ 2

( ) 1 1

( ;

fUe g ¼

ui uj

)

266 Numerical methods

10

If the space is discretized into two elements, we have the following simultaneous equation system: 2  1 38 9 8 9  1 ðK11 Þ ðK12 Þ 0 F U1 > > > = = < < 1> 6 7  6 ðK  Þ1 ðK  Þ1 þ ðK  Þ2 ðK  Þ2 7 U2 F ¼ 2 22 11 12 4 21 5> > > ; : ; : >  2  2 F U 0 ðK21 Þ ðK22 Þ 3 3 mþ1 mþ1

9.2.4

1-D Parabolic problem: creep problem

If the inertia term in Equation (9.29) is negligible, and the constitutive law is of the Kelvin type, then the finite element form of Equation (9.29) becomes: _ þ ½KfUg ¼ fFg ½CfUg

ð9:54Þ

For a time step m, we get the following equation using the Taylor expansion: ½C fUgmþ1 ¼ fF  gmþ1

ð9:55Þ

where

  1 1  ½C
½K fUgm þ fFgm ½C  ¼ ½C; fF gmþ1 ¼ Dt Dt 

9.2.5

1-D elliptic problem: static problem

If the inertia term in Equation (9.29) is negligible, and the constitutive law is of the Hookean type, then the finite element form of Equation (9.29) becomes: ½KfUg ¼ fFg

9.2.6

ð9:56Þ

Computational examples

In the first example, the dynamic response of a layer of infinite length and 1 m thick is analysed. The body force of the layer is assumed to be applied suddenly, and the selected viscosity coefficient (V) is 0.2 and 0.5. Figure 9.2 shows the computed displacement response of some points with time. It is interesting to note that fluctuations occur as the viscosity coefficient decreases in magnitude. In the second example, the same problem is analysed using hyperbolic and parabolic formulations. The results are shown in Figure 9.3 As seen from the figure, the computed responses of both hyperbolic and parabolic selected points converge to those, which could be obtained from the elliptical formulation.

Figure 9.2 Comparison of hyperbolic and parabolic solutions (V = 0.2)

Figure 9.3 Comparison of hyperbolic and parabolic solutions (V = 0.5)

268 Numerical methods

12

Parabolic problems: heat flow, seepage and diffusion

9.3 9.3.1

Introduction

The governing equation for heat flow, fluid flow, seepage and diffusion problems takes exactly the same form except the physical meaning of variables are different. In the followings, a finite element formulation of such a governing equation and its discretisation are given. Furthermore, some sample computations are carried out. 9.3.2

Governing equation

As shown in Chapter 4, the laws of mass conservation law and heat flow of nonconvective or nonadvective type take the following form for 1-D problems:


@q @T þ g ¼ rc @x @t

ð9:57Þ

where T can be temperature, water head or mass concentration. Let us assume that this equation is subjected to following boundary and initial conditions: Boundary conditions Tð0;tÞ ¼ 0

at

x¼0

qn ð0;tÞ ¼ q0

at

x¼L

ð9:58Þ

Initial conditions Tðx; 0Þ ¼ 0 _ 0Þ ¼ 0 Tðx;

at

t¼0

at

t¼0

ð9:59Þ

Let us further assume that the material obeys a linear type of constitutive law between flux q and dependent variable T: q ¼
k

9.3.3

@T @x

ð9:60Þ

Weak form formulation

Taking a variation on variable δT, the integral form of Equation (9.57) becomes: Z Z Z @q @T
dT  dV þ dT  gdV ¼ rcdT  dV @x @t V V V

ð9:61Þ

where dV=dAdx. Applying the integral by parts to the first term on the LHS with respect to x yields the following: Z Z Z Z @dT @T L  qdV þ rcdT  dV ð9:62Þ
½dT  qn 0 dA þ dT  gdV ¼
@x @t A V V V

13

Numerical methods

269

where qn = q  n. The preceding equation is called the weak form of Equation (9.57). Inserting the constitutive relation given by Equation (9.60) into Equation (9.62), we obtain the following: Z Z Z Z @dT @T @T L  dV þ rcdT  dV ð9:63Þ
½dT  qn 0 dA þ dT  gdV ¼ k @x @x @t A V V V

9.3.4

Discretization

For this particular problem, let us choose a linear function of the following form for the space: TðtÞ ¼ ax þ b

ð9:64Þ

Let us assume that the function T at nodes i and j are known. Thus we can write the following: " #( ) ( ) xi 1 Ti a ¼ ð9:65Þ Tj xj 1 b Taking the inverse of the preceding relation, one gets coefficients a and b. Inserting these coefficients in Equation (9.64) yields the following: T ¼ Ni Ti þ Nj Tj

ð9:66Þ

where Ni ¼

xj
x ; xj
xi

Nj ¼

x
xi xj
xi

The preceding equation may be rewritten in a compact form as: T ¼ ½NfTe g or

T ¼ NTe

ð9:67Þ

where [N] = [Ni, Nj],[N] {Te}T = {Ti,Tj}. The derivative of the preceding equation takes the following form: dNj dT dNi ¼ Ti þ T dx dx dx j

ð9:68Þ

The preceding relation is rewritten in a compact form as: dT ¼ ½BfTe g or dx

dT ¼ BTe dx

ð9:69Þ

where [B] = [Bi,Bj], Bi =
1/Le, Le = xj
xi. Equation (9.63), which holds for the whole domain, must also hold for each element: Z Z Z Z @T @dT @T x dV þ k  dV ¼ rcdT  ½dT  qen xji dA þ dT  ge dV ð9:70Þ @t @x @x Ve Ve Ae Ve

270 Numerical methods

14

The discretized form of the preceding equation becomes: Z Z Z Z  xj T T T T rc½N ½NfT_ e gdV þ k½B ½BfUge dV ¼ ½Nk  qen x dA þ ½N ge dV Ve

Ve

i

Ae

ð9:71Þ

Ve

The preceding equation may be written in a compact form as: ½Me fT_ e g þ ½Ke fTe g ¼ fFe g where Z T ½Me  ¼ rc½N ½NdV ;

ð9:72Þ Z

½Ke  ¼

Ve

Z



k½B ½BdV ; fFe g ¼ T

Ve

Ae



x e j n xi

½N q T

Z dA þ

½N ge dV . T

Ve

For the total domain, we have: _ þ ½KfTg ¼ fFg ½MfTg

ð9:73Þ

where ½M ¼

n X ½Me k ;

½K ¼

k¼1

n X ½Ke k ;

fFg ¼

k¼1

n X fFe gk ; k¼1

fTg ¼

n X fTe gk k¼1

Equation (9.73) could not be solved as it is. For a time step m, we can rewrite Equation (9.73) as: _ ðmþyÞ þ ½KfTgðmþyÞ ¼ fFgðmþyÞ ½MfTg

ð9:74Þ

Therefore, we discretisz dependent variable {T} for time-domain for using the Taylor expansion as it is in the finite difference method: fTgm ¼ fTgðmþyÞ
y ¼ fTgðmþyÞ


@fTgðmþyÞ yDt @ 2 fTgðmþyÞ y2 Dt2 þ
03 1! @t @t2 2!

fTgmþ1 ¼ fTgðmþyÞþð1
yÞ ¼ fTgðmþyÞ þ

ð9:75Þ

@fTgðmþyÞ ð1
yÞDt @ 2 fTgðmþyÞ ð1
yÞ2 Dt2 þ þ 03 1! @t @t2 2! ð9:76Þ

From preceding relations, one easily gets: fTgðmþyÞ ¼ yfTgmþ1 þ ð1
yÞfTgm

_ fTg ðmþyÞ ¼

fTgmþ1
fTgm Dt

fFgðmþyÞ ¼ yfFgmþ1 þ ð1
yÞfFgm

ð9:77Þ ð9:78Þ

Inserting these relations into Equation (9.74), we get: ½M  fTgmþ1 ¼ fF  gmþ1

ð9:79Þ

15

Numerical methods

where

271



 1 ½M þ y½K ½M  ¼ Dt 

 fF  gmþ1 ¼

9.3.5

 1 ½M
ð1
yÞ½K fTgm þ yfFgmþ1 þ ð1
yÞfFgm Dt

Steady-state problem

When the time variation of dependent variable T is negligible, then the problem is called a steady-state problem. This type of special case corresponds to the elliptical problem. The final finite element form of the discretized governing equation becomes: ½KfTg ¼ fFg

ð9:80Þ

where ½K ¼

n X

½Ke k ;

fFg ¼

k¼1

9.3.6

n X fFe gk ; k¼1

fTg ¼

n X

fTe gk

k¼1

Specific example

For a typical two noded element, the followings are obtained: " # " # rcLe Ae 2 1 kAe 1
1 ½Me  ¼ ½Ke  ¼ 6 L
1 1 1 2 If the space is discretized into two elements, we have the following simultaneous equation system: 2  1 38 9 8 9  1 ðK11 Þ ðK12 Þ 0 F T1 > > > = = < < 1> 6 7  6 ðK  Þ1 ðK  Þ1 þ ðK  Þ2 ðK  Þ2 7 T2 F ¼ 2 22 11 12 4 21 5> > > ; : ; : >  2  2 F3 mþ1 T3 mþ1 0 ðK21 Þ ðK22 Þ 9.3.7

Example: simulation of a solid body with heat generation

A specific example is given here by simulating the temperature variation in a solid body for the following three different conditions: 1 2 3

Heat generation only Heat flux input only Heat generation + heat flux

The heat generation function is assumed to be of the following form: g ¼ At exp
t=t

ð9:81Þ

272 Numerical methods

16

Figure 9.4 Comparison of temperature responses of some selected points for various conditions

where A and τ are physical constants determined from heat generation tests of a solid, and t is time. Figure 9.4 compares the results of computations for three different conditions. When the heat generation only condition is considered, temperature first increases and then decreases in a similar manner to the heat generation function. When the heat flux only condition is considered, temperature increases monotonically and tends to be asymptotic. When the heat generation + heat flux condition is considered, temperature first increases and then decreases. Finally, it becomes asymptotic to those computed for the heat flux only condition.

9.4

Finite element method for 1-D pseudo-coupled parabolic problems: heat flow and thermal stress; swelling and swelling pressure

9.4.1

Introduction

In the following sections, a finite element formulation for a coupled heat flow and stress problem (thermo-mechanics) and its discretisation are given. Although the stress field is coupled with the heat flow, the heat flow field is uncoupled. Therefore, such a problem may be called as a pseudo-coupled problem.

9.4.2

Governing equations

9.4.2.1

Governing equation for a heat flow

As shown in Chapter 4, the laws of mass conservation law and heat flow of nonconvective or nonadvective type takes the following form for 1-D problems:


@q @T þ g ¼ rc @x @t

ð9:82Þ

17

Numerical methods

273

where T is temperature. Let us assume that this equation is subjected to following boundary and initial conditions: Boundary conditions Tð0;tÞ ¼ 0

as x ¼ 0

qn ð0;tÞ ¼ q0

as x ¼ L

ð9:83Þ

Initial conditions Tðx; 0Þ ¼ 0 _ 0Þ ¼ 0 Tðx;

as t ¼ 0 as t ¼ 0

ð9:84Þ

Let us further assume that the material obeys a linear type of constitutive law between flux q and dependent variable T: q ¼
k

9.4.2.2

@T @x

ð9:85Þ

Governing equation for stress field

As shown in Chapter 4, the equation of momentum for 1-D problems without inertia term can be written as: @s þb¼0 @x

ð9:86Þ

Let us assume that this equation is subjected to the following boundary conditions: uð0;tÞ ¼ 0

as x ¼ 0

tð0;tÞ ¼ t0

as x ¼ L

ð9:87Þ

Let us further assume that the material is of the Hookean type: s ¼ Eε

ð9:88Þ

The displacement–strain relation is given as: ε¼

@u @x

ð9:89Þ

The incremental form of Equation (9.86) takes the following form if the body force remains constant with time: @s_ ¼0 @x

ð9:90Þ

Similarly, the constitutive law and displacement rate and strain rate relations may be rewritten as: s_ ¼ Eε_ @u_ ε_ ¼ @x

ð9:91Þ ð9:92Þ

274 Numerical methods

9.4.3

18

Coupling of heat and stress fields

If the mechanical energy rate is slow, the effect of a heat field on stress field is coupled through volumetric strain caused by temperature variation. For one-dimensional situations, this is written as: ε_ t ¼ ε_
ε_ 0

ð9:93Þ

where ε_ 0 ¼ aDT, α is the thermal expansion coefficient, and ΔT is the temperature variation of a given point per unit time.

9.4.4 9.4.4.1

Weak form formulation Weak formulation for heat flow field

Taking a variation on variable δT, the integral form of Equation (9.4.1) becomes: Z Z Z @q @T
dT  dV þ dT  gdV ¼ rcdT  dV ð9:94Þ @x @t V V V where dV ¼ dAdx. Applying the integral by parts to the first term on the LHS with respect to x yields: Z Z Z Z @dT @T L
½dT  qn 0 dA þ dT  gdV ¼
 qdV þ rcdT  dV ð9:95Þ @x @t A V V V where qn = q  n. The preceding equation is called the weak form of Equation (9.82). Inserting the constitutive relation given by Equation (9.85) into Equation (9.95), we obtain the following: Z Z Z Z @dT @T @T L
½dT  qn 0 dA þ dT  gdV ¼ k  dV þ rcdT  dV ð9:96Þ @x @x @t A V V V

9.4.4.2

Weak formulation for stress field

_ the integral form of Equation (9.4.9) Taking a variation on displacement rate field du, becomes: Z @s_ ð9:97Þ du_  dV ¼ 0 @x V where dV ¼ dAdx. Applying the integral by parts to the LHS with respect to x yields: Z Z @du_ L _  sdV ½du_  t_0 dA ¼ A V @x

ð9:98Þ

19

Numerical methods

275

where t_ ¼ s_  n. The preceding equation is called the weak form of Equation (9.90). Inserting the constitutive relation given by Equation (9. 91) and Equation (9. 93) into Equation (9.98), we obtain: Z Z Z @du_ @du_ @u_ L DTe dV ¼ E  dV ð9:99Þ ½du_  t_0 dA þ Ea @x @x @x A V V

9.4.5

Discretization

9.4.5.1

Discretization of heat flow field

For this particular problem, let us choose a linear function of the following form for the space: TðtÞ ¼ ax þ b

ð9:100Þ

Let us assume that the function T at nodes i and j are known. Thus we can write: " #( ) ( ) xi 1 Ti a ¼ ð9:101Þ Tj xj 1 b Taking the inverse of the preceding relation, one gets coefficients a and b. Inserting these coefficients in Equation (9.100) yields: T ¼ Ni Ti þ Nj Tj

ð9:102Þ

where Ni ¼

xj
x ; xj
xi

Nj ¼

x
xi xj
xi

The preceding equation may be rewritten in a compact form as: T ¼ ½NfTe g or

T ¼ NTe

ð9:103Þ

where [N] = [Ni,Nj], {Te}T = {Ti,Tj}. The derivative of the preceding equation takes the following form: dNj dT dNi ¼ Ti þ T dx dx dx j

ð9:104Þ

The preceding relation is rewritten in a compact form as: dT ¼ ½BfT_ e g dx

or

dT ¼ BT_ e dx

ð9:105Þ

where [B] = [Bi,Bj], Bi =
1/Le,Bj = 1/L = xj
xi. Equation (9.106), which holds for the whole domain, must also hold for each element: Z Z Z Z @T @dT @T x dV þ k  dV ¼ rcdT  ½dT  qen xji dA þ dT  ge dV ð9:106Þ @t @x @x Ve Ve Ae Ve

276 Numerical methods

20

The discretized form of the preceding equation becomes: Z Z Z Z T T T e xj T _ rc½N ½NfT e gdV þ k½B ½BfUge dV ¼ ½½N qn xi dA þ ½N ge dV Ve

Ve

Ae

Ve

ð9:107Þ The preceding equation may be written in a compact form: ½Me fT_ e g þ ½Ke fTe g ¼ fFe g

ð9:108Þ

where Z

Z

½Me  ¼

rc½N ½NdV ; ½Ke  ¼

Z k½B ½BdV ; fFe g ¼

T

T

Ve

Ve

Z ½½Nk  qen xji dA þ T

Ae

½N ge dV

x

T

Ve

For the total domain, we have the following: _ þ ½KfTg ¼ fFg ½MfTg

ð9:109Þ

where ½M ¼

n X ½Me k ;

½K ¼

k¼1

n X

½Ke k ;

fFg ¼

k¼1

n X

fFe gk ;

k¼1

fTg ¼

n X fTe gk k¼1

Equation (9.109) could not be solved as it is. For a time step m, we can rewrite Equation (9.4.28) as: _ ðmþyÞ þ ½KfTgðmþyÞ ¼ fFgðmþyÞ ½MfTg

ð9:110Þ

Therefore, we discretise dependent variable {T} for time-domain for using the Taylor expansion as it is in the finite difference method: fTgm ¼ fTgðmþyÞ
y ¼ fTgðmþyÞ


@fTgðmþyÞ yDt @ 2 fTgðmþyÞ y2 Dt2 þ
03 1! @t @t2 2!

fTgmþ1 ¼ fTgðmþyÞþð1
yÞ ¼ fTgðmþyÞ þ

ð9:111Þ

@fTgðmþyÞ ð1
yÞDt @ 2 fTgðmþyÞ ð1
yÞ2 Dt2 þ þ 03 1! @t @t2 2! ð9:112Þ

From preceding relations, one easily gets the following: fTgðmþyÞ ¼ yfTgmþ1 þ ð1
yÞfTgm ;

_ fTg ðmþyÞ ¼

fTgmþ1
fTgm Dt

fFgðmþyÞ ¼ yfFgmþ1 þ ð1
yÞfFgm

ð9:113Þ ð9:114Þ

Inserting these relations into Equation (9.110), we get the following: ½M  fTgmþ1 ¼ fF  gmþ1

ð9:115Þ

21

Numerical methods

where

277



   1 1  ½M  ¼ ½M þ y½K ; fF gmþ1 ¼ ½M
ð1
yÞ½K fTgm þ yfFgmþ1 þ ð1
yÞfFgm Dt Dt 

9.4.5.2

Discretization of stress field

For this particular problem, let us choose a linear function of the following form for the space: u_ ¼ ax þ b

ð9:116Þ

Let us assume that the function u_ at nodes i and j are known. Thus we can write the following: " #( ) ( ) xi 1 u_ i a ¼ ð9:117Þ u_ j xj 1 b Taking the inverse of the preceding relation, one gets coefficients a and b. Inserting these coefficients in Equation (9.116) yields the following: u_ ¼ Niu_ i þ Nju_ j

ð9:118Þ

where Ni ¼

xj
x ; xj
xi

Nj ¼

x
xi xj
xi

The preceding equation may be rewritten in compact form as: u_ ¼ ½NfU_ e g or u_ ¼ NU_ e

ð9:119Þ

where [N] = [Ni,Nj], fU_ e g ¼ fu_ i ; u_ j g. The derivative of the preceding equation takes the following form: T

dNj du_ dNi ¼ u_ i þ u_ dx dx dx j

ð9:120Þ

The preceding relation is rewritten in a compact form: du_ ¼ ½BfU_ e g dx

or

du_ ¼ BU_ e dx

ð9:121Þ

where [B] = [Bi,Bj],Bi =
1/Le,Bj = 1/Le,Le = xj
xi. Equation (9.119), which holds for the whole domain, must also hold for each element: Z Z Z @du_ @u_ @du_ x  dV ¼ DTe dV þ ½du_  t_xji dA E Ea ð9:122Þ @x @x @x Ve Ve Ae The discretized form of the preceding equation becomes: Z Z Z T T T x _ e dV ¼ E½B ½BfUg Ea½B DTe dV þ ½N t_e xji dA Ve

Ve

Ae

ð9:123Þ

278 Numerical methods

22

The preceding equation may be written in compact form as: ½Ke fU_ e g ¼ fF_ e g where

Z

ð9:124Þ

E½B ½BdV ; fF_ e g ¼

½Ke  ¼

Z

T

Ve

x ½Nk  t_e xji dA þ

Z Ea½B DTe dV

T

Ae

T

Ve

For the total domain, we have: _ ¼ fFg _ ½KfUg

ð9:125Þ

where ½K ¼

n X ½Ke k ;

_ ¼ fFg

k¼1

9.4.6

n X

fF_ e gk ;

k¼1

_ ¼ fUg

n X

fU_ e gk

k¼1

Specific example

For a typical two-noded element for heat flow field, the following is obtained: " # " # ( ) ( ) ( ) qi Ti rcLe Ae 2 1 kAe 1
1 ge 1 ; ½Ke  ¼ ; fFe g ¼ ; fTe g ¼ þ ½Me  ¼ 6 L
1 1 2 1 qj Tj 1 2 Similarly for a typical two-noded element for stress field, the following is obtained: ( ) ( ) " # ( ) u_ i t_i
1 EAe 1
1 EaA DT e e þ ; fF_ e g ¼ ; fU_ e g ¼ ½Ke  ¼ L
1 1 L u_ j t_j 1 9.4.7

Example: simulation of heat generation and associated thermal stress

A specific example is given herein by simulating the temperature variation in a solid with a length of 1 m and associated stress field for the following conditions. (a)

Heat generation + heat flux

The heat generation function is assumed to be of the following form: g ¼ Ag te
t=tg

ð9:126Þ

where Ag and τg are physical constants determined from heat generation tests, and t is time. Furthermore, the variation elastic modulus of hardening solid with time is assumed to be of the following form: EðtÞ ¼ Ae ð1
e
t=te Þ

ð9:127Þ

where Ae and τe are physical constants determined from uniaxial tests of hardening solid, and t is time.

23

Numerical methods

279

Figure 9.5 Temperature and stress response of some selected points

Figure 9.5 shows the variation of temperature and associated stress of some selected points.

9.5

Hydromechanical coupling: seepage and effective stress problem

9.5.1

Introduction

In the following sections, a finite element formulation for a coupled seepage and stress problem and its discretization are given. This is a fully coupled problem, and it is generally called a consolidation problem in the geotechnical engineering field. 9.5.2 9.5.2.1

Governing equations Governing equation for seepage field

As shown in Chapter 4, the volumetric variation of porous media takes the following form for 1-D problems: @ε @v ¼
@t @x

ð9:128Þ

where v is relative velocity. Let us assume that this equation is subjected to the following boundary and initial conditions: Boundary conditions vð0;tÞ ¼ 0

at

x¼0

qn ð0;tÞ ¼ q0

at

x¼L

ð9:129Þ

280 Numerical methods

24

Initial conditions pðx; 0Þ ¼ 0 at

t¼0

_ 0Þ ¼ 0 pðx;

t¼0

at

ð9:130Þ

Let us further assume that the seepage obeys a linear type of constitutive law (Darcy’s law) between velocity v and dependent variable pressure p: v ¼
k

9.5.2.2

@p @x

ð9:131Þ

Governing equation for stress field

As shown in Chapter 4, the equation of momentum for 1-D problems without the inertia term can be written as: @s þb¼0 @x

ð9:132Þ

Let us assume that this equation is subjected to the following boundary conditions: uð0;tÞ ¼ 0

at x ¼ 0

tð0;tÞ ¼ t0

at x ¼ L

ð9:133Þ

Displacement–strain relation is given as: ε¼

@u @x

ð9:134Þ

Let us assume that the total stress σ may be related to the effective stress law of Terzaghi through the following relation: s ¼ s0
p

ð9:135Þ

The incremental form of Equation (9.132) together with the effective stress law takes the following form if the body force remains constant with time: _ @s0 @p_
¼0 @x @x

ð9:136Þ

Let us further assume that the material is of Hookean type: s0 ¼ Eε

ð9:137Þ

25

Numerical methods

281

Similarly, the constitutive law and displacement rate and strain rate relations may be rewritten as: s_0 ¼ Eε_ ε_ ¼

9.5.3

ð9:138Þ

@u_ @x

ð9:139Þ

Weak form formulation

9.5.3.1

Weak form formulation for seepage field

Taking a variation on variable δP, the integral form of Equation (9.128) becomes: Z

@ε dp  dV ¼
@t V

Z dp  V

@v dV @x

ð9:140Þ

where dV = dAdx. Applying the integral by parts to the RHS with respect to x yields: Z dp  V

@ε dV
@t

Z V

@dp  vdV ¼
@x

Z ½dp  qn 0 dA L

ð9:141Þ

A

where qn ¼ v  n. The preceding equation is called the weak form of Equation (9.128). Inserting the constitutive relation given by Equation (9.121) into Equation (9.131), we obtain the following: Z

@p dp  dV þ @t V

9.5.3.2

Z

@dp @p  dV ¼
k @x @x V

Z ½dT  qn 0 dA L

ð9:142Þ

A

Weak form formulation for stress field

_ the integral form of Equation (9.136) Taking a variation on displacement rate field du, becomes: Z du_  V

_ _
pÞ @ðs0 dV ¼ 0 @x

ð9:143Þ

where dV ¼ dAdx. Applying the integral by parts to the LHS with respect to x yields: Z

½du_  t_0 dA ¼

Z

L

A

V

@du_ _ _
pÞdV  ðs0 @x

ð9:144Þ

282 Numerical methods

26

_
pÞ _  n. The preceding equation is called the weak form of Equation (9.135). where t_ ¼ ðs0 Inserting the constitutive relation given by Equation (9.137) and Equation (9.138) into Equation (9.144), we obtain: Z

½du_  t_0 dA ¼

Z

L

E

A

9.5.4 9.5.4.1

V

@du_ @u_  dV
@x @x

Z E V

@du_ _  pdV @x

ð9:145Þ

Discretization Discretization for physical space

(A) INTERPOLATION (SHAPE) FUNCTION FOR PRESSURE FIELD

For this particular problem, let us choose a linear function of the following form for the space in a local coordinate system whose origin is at a nodal point t: pðtÞ ¼ ax þ b;

x ¼ x
xi ;

dx ¼ dx

ð9:146Þ

Let us assume that the function P at nodes i and k are known. Thus we can write: " #( ) ( ) 0 1 a Pi ¼ ð9:147Þ Pk L 1 b Taking the inverse of the preceding relation, one gets coefficients a and b. Inserting these coefficients in Equation (9.146) yields the following: p ¼ Ni Pi þ Nk Pk

ð9:148Þ

where x Ni ¼ 1
; L

Nk ¼

x L

The preceding equation may be rewritten in a compact form: p ¼ ½Np fPe g

or

p ¼ Np Pe

ð9:149Þ

where ½Np ¼ ½Ni ; Nk , fPe g ¼ fPi ; Pk g. The derivative of the preceding equation takes the following form: T

dp dp dNi dN ¼ ¼ P þ k Pk dx dx dx i dx

ð9:150Þ

The preceding relation is rewritten in a compact form: dp dp ¼ ¼ ½Bp fPe g dx dx

or

dp dp ¼ ¼ Bp Pe dx dx

where ½B ¼ ½Bi ; Bk , Bi ¼
1=L; Bk ¼ 1=L; L ¼ xk
xi .

ð9:151Þ

27 (B)

Numerical methods

283

INTERPOLATION (SHAPE) FUNCTION FOR DISPLACEMENT FIELD

For this particular problem, we have to choose a quadratic function of the following form for the space discretisation of displacement field if the shape function for the pressure field is linear in a local coordinate system whose origin is at a nodal point i: uðtÞ ¼ a þ bx þ cx2 ;

x ¼ x
xi ;

dx ¼ dx

ð9:152Þ

Let us assume that the function u at nodes i, j and k are known. Thus we can write the following: 2 3 1 0 0 8a9 8 U 9 > > > = < i> 2 7< = 6 6 1 L L 7 b ¼ Uj ð9:153Þ 4 5 2 4 > ; ; > : > : > Uk c 1 L L2 Taking the inverse of the preceding relation, one gets coefficients a, b and c. Inserting these coefficients in Equation (9.152) yields the following: u ¼ Ni Ui þ Nj Uj þ Nk Uk

ð9:154Þ

where Ni ¼ ð1


2x x Þð1
Þ; L L

Nj ¼

4x x ð1
Þ; L L

x 2x Nk ¼
ð1
Þ L L

The preceding equation may be rewritten in a compact form: u ¼ ½Nu fUe g or

u ¼ Nu U e

ð9:155Þ

where ½Nu ¼ ½Ni ; Nj ; Nk , fUe g ¼ fUi ; Uj ; Uk g. The derivative of the preceding equation takes the following form: T

dNj du du dNi dN ¼ ¼ Ui þ Uj þ k Pk dx dx dx dx dx

ð9:156Þ

The preceding relation is rewritten in a compact form: du du ¼ ¼ ½Bu fUe g dx dx

or

du du ¼ ¼ Bu Ue dx dx

ð9:157Þ

where ½B ¼ ½Bi ; Bj ; Bk , Bi ¼

1 ð4x
3LÞ; L2

Bj ¼

4 ðL
2xÞ; L2

Bk ¼

1 ð4x
LÞ; L2

L ¼ xk
xi

Equation (9.142) for seepage field, which holds for the whole domain, must also hold for each element: Z Z Z @ε @dp @p L  dV ¼
½dp  qen 0 dA dp  dV þ k ð9:158Þ @t @x @x Ve Ve Ae

284 Numerical methods

28

The discretized form of the preceding equation becomes: Z Z Z T T  T qe L dA ½Np ½Bu dV fU_ e g þ k½Bp ½Bp dV fPge dV ¼
½½N p n 0 Ve

Ve

ð9:159Þ

Ae

The preceding equation may be written in a compact form: ½Ce pu fU_ e g þ ½Ke pp fPe g ¼ fQe g

ð9:160Þ

where Z ½Ce pu ¼

Z ½Np ½Bu dV ; T

Z

½Ke pp ¼

k½Bp ½Bp dV ; T

Ve

 qe  dA ½½N p n 0

fQe g ¼


Ve

T

L

Ae

For the total domain, we have: _ þ ½K fPg ¼ fQg ½Cpu fUg pp

ð9:161Þ

where ½Cpu ¼

n X k ½Ce pu ;

½Kuu ¼

k¼1

n X k ½Ke pp ;

fQg ¼

k¼1

n X fQe gk ; k¼1

fTg ¼

n X

fTe gk

k¼1

Similarly Equation (9.145) for stress field, which holds for the whole domain, must also hold for each element: Z Z Z @du_ @u_ @du_ L _  dV þ E  pdV ¼ E ½du_  t_0 dA ð9:162Þ @x @x @x Ve Ve Ae The discretised form of the preceding equation becomes: Z Z Z T T _ _  T t_e L dA E½Bu ½Bu dvfUge ½Bu ½Np dvfPge ¼ ½½N 0 Ve

Ve

ð9:163Þ

Ae

The preceding equation may be written in a compact form: ½Ke uu fU_ e g
½Ce up fP_ e g ¼ fF_ e g

ð9:164Þ

where Z ½Ke uu ¼

T E½Bu ½Bu dV ; fF_ e g ¼

Ve

Z

 T t_e L dA ½N 0 Ae

For the total domain, we have: _
½C fPg _ ¼ fFg _ ½Kuu fUg up

ð9:165Þ

29

Numerical methods

285

where ½Kuu ¼

n n n n n X X X X X k k _ ¼ _ ¼ _ ¼ ½Ke uu ; ½Cup ¼ ½Ce up ; fFg fF_ e gk ; fUg fU_ e gk ; fPg fP_ e gk k¼1

k¼1

k¼1

k¼1

Above equations (9.161) and (9.145) can be written in a compact form: " #( ) " #( ) ( ) 0 0 Kuu
Cup U F_ U_ þ ¼ 0 Kpp Cpu 0 P Q P_

k¼1

ð9:166Þ

The preceding equation may be rewritten in a more compact form: _ þ ½HfTg ¼ fY g ½MfTg where

"

½M ¼

9.5.4.2

Kuu


Cup

Cpu

0

#

ð9:167Þ "

; ½H ¼

0

"

#

0

0 Kpp

_ ¼ ; fTg

" # " # # _ U U_ U ; ½Y  ¼ ; fTg P P_ P_

Discretization for time domain

Equation (9.167) could not be solved as it is. For a time step m, we can rewrite Equation (9.167): _ ½MfTg ðmþyÞ þ ½HfTgðmþyÞ ¼ fY gðmþyÞ

ð9:168Þ

Therefore, we discretize dependent variable {T} for time-domain by using the Taylor expansion as it is in the finite difference method: fTgm ¼ fTgðmþyÞ
y ¼ fTgðmþyÞ


@fTgðmþyÞ yDt þ 02 1! @t

fTgmþ1 ¼ fTgðmþyÞþð1
yÞ ¼ fTgðmþyÞ þ

@fTgðmþyÞ ð1
yÞDt þ 02 1! @t

ð9:169Þ

ð9:170Þ

From preceding relations, one easily gets the following: fTgðmþyÞ ¼ yfTgmþ1 þ ð1
yÞfTgm ;

_ ðmþyÞ ¼ fTg

fTgmþ1
fTgm Dt

fY gðmþyÞ ¼ yfY gmþ1 þ ð1
yÞfY gm

ð9:171Þ ð9:172Þ

Inserting these relations into Equation (9.168), we get the following ½M  fTgmþ1 ¼ fY  gmþ1

ð9:173Þ

286 Numerical methods

where

30



 1 ½M þ y½H ½M  ¼ Dt   1  fY gmþ1 ¼ ½M
ð1
yÞ½H fTgm þ yfY gmþ1 þ ð1
yÞfY gm Dt 

9.5.5

Specific example

For a typical element, the followings are 2 7E 8E E 5
6 3L 3L 3L 6 6 6 8E 16E 8E 2 6


6 3L 3L 3L 3 6 6 E 8E 7E 1 6 ½Me  ¼ Ae 6

6 3L 3L 3L 6 6 6 5 2 1 6
0 6 6 3 6 6 4 1 2 5

0 6 3 6 8 _ 9 8 _ 9 Fi > Ti > > > > > > > > > > > > > > > > > > > _ _ > > > > F T j j > > > > > > > > > > > > > > > < F_ = < T_ > = k k ; fT_ e g ¼ ; fYe g ¼ > > > > > > P_ i > Qi > > > > > > > > > > > > > > > > > > > > Qk > > > P_ k > > > > > > > > > : ; : > ;

9.5.6

obtained: 3 1 6 7 7 2 0 2 7 7 7 6 3 7 60 7 kAe 6 57 6 60
7; ½He  ¼ L 6 67 7 60 7 4 7 0 7 0 7 5 0

0

0

0

0

0

0

0

0

0

0

0

1

0

0
1

0

3

7 0 7 7 7 0 7 7
1 7 5 1

8 9 Ti > > > > > > > > > > > > T > j > > > > > >

= k fTe g ¼ > Pi > > > > > > > > > > > > > P > k> > > > : > ;

Example: simulation of settlement under sudden loading

A specific example is given herein by simulating the settlement and pore pressure variation in the ground by considering the order of approximation function for pressure and displacement field. Figures 9.6 and 9.7 show the variation of settlement and pore pressure at some selected nodes. The results for both situations are almost exactly the same.

9.6 9.6.1

Biot problem: coupled dynamic response of porous media Introduction

In the following subsections, a finite element formulation for a coupled seepage and stress problem for dynamic responses saturated porous media and its discretization is given. This is a fully coupled problem, generally called Biot’s problem in the geotechnical engineering field.

31

Numerical methods

287

Figure 9.6 Settlement of ground under rapid load for different shape functions

Figure 9.7 Pore pressure of ground under rapid load for different shape functions

9.6.2 9.6.2.1

Governing equations Governing equation for fluid phase

As shown in Chapter 4, the equation of motion for fluid phase of porous media takes the following form for 1-D problems: @sf rf @ 2 w 1 @w @2u þ rf g ¼ rf 2s þ þ @x @t n @t2 K @t

ð9:174Þ

288 Numerical methods

32

where w is relative displacement. Let us assume that this equation is subjected to following boundary and initial conditions: Boundary conditions wð0;tÞ ¼ 0

at

x¼0

t ð0;tÞ ¼ t

at

x¼L

f n

f 0

ð9:175Þ

Initial conditions wðx; 0Þ ¼ 0

at t ¼ 0

_ 0Þ ¼ 0 wðx;

at t ¼ 0

ð9:176Þ

Let us further assume that the fluid obeys a linear type of constitutive law (Biot’s law) in terms of fluid and skeleton strains and dependent variable fluid pressure p:
p ¼ sf ¼ Mðaεs þ εf Þ 9.6.2.2

ð9:177Þ

Governing equation for total stress system

As shown in Chapter 4, the equation of motion for 1-D problems takes the following form: @s @2u @2w þ rg ¼ r 2s þ rf 2 @x @t @t

ð9:178Þ

Let us assume that this equation is subjected to the following boundary conditions: us ð0;tÞ ¼ 0

at

x¼0

ts ð0;tÞ ¼ t0s

at

x¼L

ð9:179Þ

Displacement–strain relation is given as: εs ¼

@us ; @x

εf ¼

@w @x

ð9:180Þ

Let us assume that the total stress σ may be related to the effective stress law of Biot through the following relation: s ¼ s0
ap

ð9:181Þ

Let us further assume that the skeleton obeys the following constitutive law: s ¼ ð2m þ lÞεs
ap ¼ ð2m þ l þ a2 MÞεs þ aMεf

ð9:182Þ

33

Numerical methods

9.6.3

289

Weak form formulation

9.6.3.1

Weak form formulation for fluid phase

Taking a variation on variable δw, the integral form of Equation (9.169) becomes: Z

Z Z rf @ 2 w @2u 1 @w dV dw  rf 2s dV þ dw  dV þ dw  2 K @t @t n @t V V V Z Z @p ¼
dw  dV þ dw  rf gdV @x V V

ð9:183Þ

where dV ¼ dAdx. Applying the integral by parts to the RHS with respect to x yields the following: Z

Z Z Z rf @ 2 w @ 2 us 1 @w @dw dV
 pdV ¼ dw  rf 2 dV þ dw  dV þ dw  2 @t K @t @t n V V V V @x Z Z f L
½dw  tn 0 dA þ dw  rf gdV A

ð9:184Þ

V

where tnf ¼ p  n. The preceding equation is called the weak form of Equation (9.184). Inserting the constitutive relation given by Equation (9.177) into Equation (9.184), we obtain: Z dw  rf V

Z M V

9.6.3.2

@ 2 us dV þ @t2

Z dw  V

@dw @w  dV ¼
@x @x

rf @ 2 w dV þ n @t2

Z

Z

dw  V

1 @w dV þ K @t

Z aM V

@dw @us  dV þ @x @x

Z ½dw  tnf 0 dA þ

dw  rf gdV

L

A

ð9:185Þ

V

Weak form formulation for total stress field

Taking a variation on displacement rate field dus , the integral form of Equation (9.6.5) becomes: Z Z Z Z @s @2u @2w dV þ dus  rgdV ¼ dus  r 2s dV þ dus  rf 2 dV dus  ð9:186Þ @x @t @t V V V V where dV ¼ dAdx. Applying the integral by parts to the LHS with respect to x yields the following: Z Z Z Z Z @dus @2u @2w L  sdV þ dus  r 2s dV þ dus  rf 2 dV ½dus  t0 dA þ dus  rgdV ¼ @t @t A V V @x V V ð9:187Þ

290 Numerical methods

34

where ts ¼ ss  n. The preceding equation is called the weak form of Equation (9.178). Inserting the constitutive relation given by Equation (9.182) into Equation (9.187), we obtain the following: Z Z Z @dus @us L 2  dV ½dus  t0 dA þ dus  rgdV ¼ ð2m þ l þ a MÞ @x A V V @x Z þ aM V

9.6.4 9.6.4.1

@dus @w dV þ  @x @x

Z

@2u dus  r 2s dV þ @t V

Z dus  rf V

@2w dV @t2

ð9:188Þ

Discretization Discretization for physical space

(A) INTERPOLATION (SHAPE) FUNCTION FOR RELATIVE DISPLACEMENT FIELD

For this particular problem, let us choose a linear function of the following form for the space in a local coordinate system whose origin is at a nodal point i: wðtÞ ¼ ax þ b;

x ¼ x
xi ;

dx ¼ dx

ð9:189Þ

Let us assume that the function w at nodes i and j are known. Thus we can write: " #( ) ( ) wi 0 1 a ¼ ð9:190Þ wj L 1 b Taking the inverse of the preceding relation, one gets coefficients a and b. Inserting these coefficients in Equation (9.189) yields the following: w ¼ Ni Wi þ Nj Wj

ð9:191Þ

where x Ni ¼ 1
; L

Nj ¼

x L

The preceding equation may be rewritten in a compact form as w ¼ ½NfWe g

or

w ¼ NWe

ð9:192Þ

where ½N ¼ ½Ni ; Nj , fWe g ¼ fWi ; Wj g. The derivative of the preceding equation takes the following form: T

dNj dw dw dNi Wi þ W ¼ ¼ dx dx j dx dx

ð9:193Þ

The preceding relation is rewritten in a compact form: dw dw ¼ ¼ ½BpfWe g dx dx

or

dw dw ¼ ¼ BWe dx dx

where ½B ¼ ½Bi ; Bj , Bi ¼
1=L; Bj ¼ 1=L; L ¼ xj
xi .

ð9:194Þ

35 (B)

Numerical methods

291

INTERPOLATION (SHAPE) FUNCTION FOR SKELETON DISPLACEMENT FIELD

As for displacement field of skeleton, we chose also a linear interpolation function. The resulting expressions will be similar so that it will not be presented herein. Equation (9.180) for fluid phase, which holds for the whole domain, must also hold for each element: Z Z Z Z rf @ 2 w @2u 1 @w @dw @us dV þ  dV dw  rf 2s dV þ dw  dV þ dw  aM 2 @t K @t @x @x @t n Ve Ve Ve Ve Z þ

M Ve

@dw @w  dV ¼
@x @x

Z

Z ½dw  tnf 0 dA þ

dw  rf gdV

L

Ae

ð9:195Þ

Ve

The discretised form of the preceding equation becomes: Z Z Z rf 1 T T T € e dV þ ½N ½NfW_ ge dV ½N ½NfWg rf ½N ½NfU€ e gdV þ Ve Ve n Ve K Z

Z

Z

aM½B ½BfUgdV þ

þ

 T te L dA þ ½½N n 0

M½B ½BfWgdV ¼


T

T

Ve

Ve

Ae

Z rf g½N dV T

Ve

ð9:196Þ The preceding equation may be written in a compact form: ½Me 1 fU€e g þ ½Me 1 fW€e g þ ½Ce 1 fW_ e g þ ½Ke 1 fUe g þ ½Ke 1 fWe gþ ¼ fFe g1 s

f

f

s

r½N ½NdV ;

½Me 1 ¼

f

ð9:197Þ

where Z

Z

½Me 1 ¼ s

rf T ½N ½NdV ; Ve n Z f T ½Ke 1 ¼ M ½B ½BdV

T

f

Ve

Z ½Ke 1 ¼ aM

½B ½BdV ;

s

T

Ve

Z fFe g1 ¼


Z ½Ce 1 ¼ f

Ve

1 T ½N ½NdV ; K

Ve

 te  dA þ ½½N n 0 T

Z rf g½N dV

L

Ae

T

Ve

For the total domain, we have: s € f € þ ½Cf fW_ g þ ½Ks fUg þ ½Kf fW gþ ¼ fFg1 þ ½M1 fWg ½M1 fUg 1 1 1

where ½M1 ¼ s

n X s ½Me 1 ; k¼1

n X s s ½Ke 1 ; ½K1 ¼ k¼1

½M1 ¼ f

n X

½Me 1 ; f

½C1 ¼ f

k¼1 n X f f ½K1 ¼ ½Ke 1 ; k¼1

n X f ½Ce 1 ; k¼1

fFg1 ¼

n X k¼1

fFe g1

ð9:198Þ

292 Numerical methods

36

Similarly Equation (9.183) for stress field, which holds for the whole domain, must also hold for each element: Z Z Z @dus @us L  dV ½dus  t0 dA þ dus  rgdV ¼ ð2m þ l þ a2 MÞ @x Ae Ve Ve @x Z þ aM

@dus @w dV þ  @x Ve @x

Z dus  r Ve

@ 2 us dV þ @t2

Z dus  rf Ve

@2w dV @t2

ð9:199Þ

The discretized form of the preceding equation becomes: Z Z Z T € e rf ½NT ½NdvfW € ge þ ð2m þ l þ a2 MÞ½BT ½BfUgdV r½N ½NdvfUg Ve

Ve

Ve

Z

Z

þ

T

Ve

Z

 T t_e L dA þ ½½N 0

aM½B ½BfW gdV ¼

rg½N dV

ð9:200Þ

T

Ae

Ve

The preceding equation may be written in a compact form: s € f € þ ½Ke s fUg þ ½Ke f fW gþ ¼ fFg2 ½Me 2 fUg þ ½Me 2 fWg 2 2

ð9:201Þ

where Z

Z

½Me 2 ¼

r½N ½NdV ;

s

½Me 2 ¼

T

rf ½N ½NdV ;

f

Ve

T

Ve

Z

Z

½Ke 2 ¼ ð2m þ l þ a2 MÞ

½B ½BdV ;

s

T

½Ke 2 ¼ aM

Ve

Z

 T te L dA þ ½½N n 0

fFe g1 ¼
Ae

½B ½BdV ;

f

T

Ve

Z rf g½N dV T

Ve

For the total domain, we have the following: s € f € þ ½Ks fUg þ ½Kf fWgþ ¼ fFg ½M2 fUg þ ½M2 fWg 2 2 2

ð9:202Þ

where ½M2 ¼ s

n X k¼1

½Me 2 ; ½M2 ¼ s

f

n X k¼1

½Me 2 ; ½K2 ¼ f

s

n n n X X X s f f ½Ke 2 ; ½K2 ¼ ½Ke 2 ; fFg2 ¼ fFe g2 k¼1

k¼1

Equations (9.193) and (9.197) can be written in a compact form: #( ) " s " s #( ) " #( ) ( ) € U K1 Kf1 M1 Mf1 0 Cf1 U_ U F_ 1 þ þ ¼ _ € W Ms2 Mf2 Ks2 Kf2 0 0 W W F2

k¼1

ð9:203Þ

37

Numerical methods

293

The preceding equation may be rewritten in a more compact form: € þ ½CfTg _ þ ½HfTg ¼ fY g ½MfTg

ð9:204Þ

where " ½M ¼

Ms1

Mf1

Ms2

Mf2

( € ¼ fTg

9.6.4.2

) € U ; € W

"

# ;

½C ¼ (

_ ¼ fTg

0 Cf1 0

) U_ ; _ W

0

#

" ;

½H ¼ (

fTg ¼

U W

Ks1

Kf1

Ks2

Kf2

#

) ;

( fY g ¼

F1

)

F2

Discretization for time domain

Equation (9.199) could not be solved as it is. For a time step m, we can rewrite Equation (9.199): _ þ ½HfTg ¼ fY g € þ ½CfTg ½MfTg m m m m

ð9:205Þ

Therefore, we discretize field {T} for time-domain using the Taylor expansion as it is in the finite difference method: fTgm
1 ¼ fTgm


@fTgm Dt @ 2 fTgm Dt2 þ
03 @t 1! @t2 2!

fTgm ¼ fTgm

fTgmþ1 ¼ fTgm þ

ð9:206Þ

ð9:207Þ @fTgm Dt @ 2 fTgm Dt2 þ þ 03 @t 1! @t2 2!

ð9:208Þ

From preceding relations, one easily gets the following: _ m¼ fTg

1 ðfTgmþ1
fTgm
1 Þ Dt

ð9:209Þ

€ m¼ fTg

1 ðfTgmþ1
2fTgm þ fTgm
1 Þ Dt2

ð9:210Þ

Inserting these relations into Equation (9.200), we get the following: ½M  fTgmþ1 ¼ fY  gmþ1

ð9:211Þ

294 Numerical methods

38

where  ½M   ¼ fY  gmþ1

9.6.5

 1 1 ½C ½M þ Dt2 Dt     2 1 1 ¼ ½M
½H fTgm
½M
½C fTgm
1 þ fY gm Dt2 Dt2 Dt

Specific example

For a typical element, the 2 rL rL 6 3 6 6 6 rL rL 6 6 3 6 6 6 ½Me  ¼ Ae 6 rf L rf L 6 6 3 6 6 6r L r L 6 f f 4 6 3 2

following is obtained: r L r L3 f

3 rf L 6 rf L 3n rf L 6n

aM 6 L 6 6 aM 6
6 L 6 6 ½He  ¼ Ae 6 2m þ l þ a2 M 6 6 L 6 6 2m þ l þ a2 M 6
4 L 8 9 F_ 1i > > > > > > > > > 1 > > > F > j > > > > > > < F1 > = i fYe g ¼ > > Fi2 > > > > > > > > > 2 > > > F > > j > > > > : ; 9.6.6

f

6 7 7 rf L 7 7 7 3 7 7 rf L 7; 7 6n 7 7 rf L 7 7 5 3n

2

6 60 Ae L 6 6 ½Ce  ¼ 60 6K 6 60 4

aM L aM L 2m þ l þ a2 M
L 2m þ l þ a2 M L


0

M L M
L aM L aM
L

0 2

1

0 1

2

0 0

0

0 0

0

3 7 7 7 7 7; 7 7 5

3 M L 7 7 M 7 7 L 7 7 7 aM 7; 7
L 7 7 aM 7 7 5 L


8 9 8 9 8 9 > > U€ i > U_ i > Ui > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > € _ > > > > > > U U U > j= > > > < < j> = < j> = € i fT_ e g ¼ W_ i fTe g ¼ Wi 2mmfT€e g ¼ W > > > > > > > > > > > > > > > > > > > > > €j > Wj > > > > > > > W W_ j > > > > > > > > > > > > > > > > > : ; : ; : ;

Example: simulation of dynamic response of saturated porous media

A specific example is given herein by simulating the response of a half space under sinusoidal cyclic loading. No drainage is allowed at the bottom, and the fluid phase is free of

39

Numerical methods

295

traction at the ground surface. Figure 9.8 shows the variation of displacement, velocity and acceleration responses of some selected nodes.

9.7

Introduction of boundary conditions in simultaneous equation system

9.7.1

Formulation

The simultaneous equation system results in a [m×m] square matrix: ½KfUg ¼ fFg

ð9:212Þ

It should be noticed that matrix [K] is a square symmetric matrix and that its determinant jKj is zero. Thus, its inverse is not possible due to the singularity problem. The solution becomes possible if the boundary conditions are introduced. The introduction of boundary conditions associated with vectors {U} and {F} can be partitioned into two unknown and known parts: ( fUg ¼

fUgu

)

fUgn

(

; and fFg ¼

fFgn

)

fFgu

ð9:213Þ

Accordingly, Equation (9.207) may be rewritten as: "

½Kuu

½Kun

½Knu

½Knn

#(

fUgu fUgn

)

( ¼

fFgn

)

fFgu

ð9:214Þ

As the solution of fUgu is required, one can write the following equation: ½Kuu fUgu þ ½Kun fUgn ¼ fFgn

ð9:215Þ

Rearranging Equation (9.215), we have the following relations ½Kuu fUgu ¼ fFgn
½Kun fUgn

ð9:216aÞ

or
1

fUgu ¼ ½Kuu ðfFgn
½Kun fUgn Þ

ð9:216bÞ

If the unknown part of vector {F} is required, Equation (9.216) results in the following relation: ½Knu fUgu þ ½Knn fUgn ¼ fFgu

ð9:217Þ

One can easily obtain fFgu from Equation (9.217) by inserting unknown vector fUgu obtained from Equation (9.216) in the previous stage.

Figure 9.8 Acceleration, velocity, and displacement responses

41

Numerical methods

9.7.2

297

Actual implementation and solution of Equation (9.211b)

Rearranging Equation (9.207) in the form of Equation (9.208) and solving using the relation given by (9.211b) require some extra computational time. Instead of rearrangement in the form of Equation (9.208), the following procedure is implemented. For each boundary condition Uk , the force vector is modified by changing i from 1 to m except row i: Fk ¼ Fk
Kki Uk Then 1 is assigned to Kkk , and the other components of the row and column matrix [K] are assigned to 0. The value of vector Fk is assigned to Uk . The actual implementation of this procedure in the FEM program coded in True BASIC programming language is given here: ************************************************************************* ! ! ************** Displacement Boundary condition is implemented *************** ! FOR IB=1 TO NBS IBNI=IBS(IB) FOR I=1 TO NODE GFS(I)=GFS(I)-GKS(IBNI,I)* UDIS(IB) IF IIBNI THEN GKS(IBNI,I)=0. GKS(I,IBNI)=0. ELSE GKS(IBNI,IBNI)=1.0 GFS(IBNI)=UDIS(IB) END IF NEXT I NEXT IB ! *************************************************************************

9.8

Rayleigh damping and its implementation

The final forms of the discretized form of the equation of motion (Equation 9.47) irrespective of method of solution (FDM, FEM, BEM) and continuum or discontinuum, depending upon the character of governing equation, may be written in the following form: _ þ ½Kfφg ¼ fFg ½Mf€ φg þ ½Cfφg

ð9:218Þ

The specific forms of matrices [M], [C], [K] and vector {F} in the preceding equation will only differ depending upon the method of solution chosen and dimensions of physical space. Viscosity matrix [C] is associated with the rate dependency of geomaterials. However, in many dynamic solution schemes, viscosity matrix [C] is expressed in the following form using Rayleigh damping approach ½C ¼ a½M þ b½K

ð9:219Þ

298 Numerical methods

42

Figure 9.9 Illustration of numerical techniques to deal with nonlinearity

where α, β are called proportionality constants. This approach becomes very convenient in large-scale problems if central finite difference technique and mass lumping are used. However, it should be also noted that it is very difficult to determine these parameters from experiments. Again, in nonlinear problems, the deformation moduli of rocks are reduced in relation to straining using an approach commonly used in soil dynamics. In such approaches, the reduction of moduli is determined from cyclic tests. Nevertheless, it should be noted that the validity of such an approach for rock and rock masses is quite questionable.

9.9

Nonlinear problems

If material behavior involves nonlinearity, the preceding equation system must be solved iteratively with the implementation of required conditions associated with the constitutive law chosen. The iteration techniques may be broadly classified as the initial, secant or tangential stiffness method (e.g. Owen and Hinton, 1980). See Figure 9.9.

9.10

Special numerical procedures for rock mass having discontinuities

The existence of discontinuities in rock mass has special importance on the stability of rock engineering structures, directional seepage, diffusion or heat transport, and its treatment in any analysis requires a special attention. Various types of finite element methods with joint or interface elements, discrete element method (DEM), displacement discontinuity analysis (DDA), discrete finite element method (DFEM) and displacement discontinuity method (DDM) have been developed so far. Although these methods are mostly concerned with the solution of equation of motion, they can be used for seepage, heat transport or diffusion

43

Numerical methods

299

problems. The fundamental features of the available methods are described in a recent review on these methods by Kawamoto and Aydan (1999). (a)

No-tension finite element method

The no-tension finite element method is proposed by Valliappan in 1968 (Zienkiewicz et al., 1968). The essence of this method lies with the assumption of no tensile strength for rock mass since it contains discontinuities. In the finite element implementation, the tensile strength of media is assumed to be nil. It behaves elastically when all principal stresses are compressive. The excess stress is redistributed to the elastically behaving media using a similar procedure adopted in the finite element method with the consideration of elastic-perfectly plastic behavior. (b)

Pseudo-discontinuum finite element method

This method was first proposed by Baudendistel et al. in 1970. In this method, the effect of discontinuities in the finite element method is considered through the introduction of directional yield criterion in elasto-plastic behavior. Its effect on the deformation characteristics of the rock mass is not taken into account. If there is any yielding in a given element, the excess stress is computed, and the iteration scheme for elastic-perfectly plastic behavior is implemented. If there is more than one discontinuity set, the excess stress is computed for the discontinuity set that yields the largest value. (c)

Smeared crack element

The smeared crack element method within the finite element method was initially proposed by Rashid (1968) and adopted by Pietruszczak and Mroz (1981) in media having weakness planes or developing fracture planes. This method evaluates the equivalent stiffness matrix of the element and allows the directional plastic yielding within it. (d)

Discrete finite element method (DFEM)

Finite element techniques using contact, joint or interface elements have been developed for representing discontinuities between blocks in rock masses. The simplest approach for representing joints is the contact element, which was originally developed for bond problems between steel bars and concrete. The contact element is a two-noded element having normal and shear stiffnesses. This model has been recently used to model block systems by Aydan and Mamaghani (e.g. Aydan et al., 1996) by assigning a finite thickness to the contact element and employing an updated Lagrangian scheme to deal with large block movements. The contact element can easily deal with sliding and separation movements. (e)

Finite element method with joint or interface element (FEM-J)

Goodman et al. (1968) proposed a four-noded joint element for joints. This model is simply a four-noded version of the contact element of Ngo and Scordelis (1967), and it has the

300 Numerical methods

44

following characteristics. In a two-dimensional domain, joints are assumed to be tabular with zero thickness. They have no resistance to the net tensile forces in the normal direction, but they have high resistance to compression. Joint elements may deform under normal pressure, especially if there are crushable asperities. The shear strength is presented by a bilinear Mohr-Coulomb envelope. The joint elements are designed to be compatible with solid elements. Ghaboussi et al. (1973) proposed a four-noded interface element for joints. This model is a further improvement of the joint element by assigning a finite thickness to joints. (f)

Displacement discontinuity method (DDM)

This technique is generally used together with the boundary element method (BEM). The discontinuities are modeled as a finite length segment in an elastic medium with a relative displacement. In other words, the discontinuities are treated as internal boundaries with prescribed displacements. As an alternative approach to the technique of Crouch and Starfield (1983), Crotty and Wardle (1985) use interface elements to model discontinuities, and the domain is discretized into several subdomains. (g)

Discontinuous deformation analysis method (DDA)

Shi (1988) proposed a method called discontinuous deformation analysis (DDA). Intact blocks were assumed to be deformable and are subjected to constant strain and stress due to the order of the interpolation functions used for the displacement field of the blocks. In the original model, the inertia term was neglected so that damping becomes unnecessary. For dynamic problems, although damping is not introduced into the system, large time steps are used in the numerical integration in time-domain results in artificial damping. It should be noted that this type of damping is due to the integration technique for time-domain and has nothing to do with the mechanical characteristics of rock masses (i.e. frictional properties). Although the fundamental concept is not very different from Cundall’s model, the main difference results from the solution procedure adopted in both methods. In other words, the equation system of blocks and its contacts are assembled into a global equation system in Shi’s approach. Recently Ohnishi et al. (1995) introduced an elasto-plastic constitutive law for intact blocks and gave an application of this method to rock engineering structures. (h)

Discrete element method (DEM)

Distinct element method (rigid block models) for jointed rocks was developed by Cundall in 1971. In Cundall’s model, problems are treated as dynamic ones from the very beginning of formulation. It is assumed that the contact force is produced by the action of springs that are applied whenever a corner penetrates an edge. Normal and shear stiffness were introduced between the respective forces and displacements in his original model. Furthermore, to account for slippage and separation of block contacts, he also introduced the law of plasticity. For the simplicity of calculation of contact forces due to the overlapping of the block, he assumed that the blocks do not change their original configurations. To solve the equations of the whole domain, he never assembled the equilibrium equations of blocks into a

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large equation system but solved them through a step-by-step procedure, which he called marching scheme. His solution technique has two main merits: 1 2

The storage memory of computers can be small (note that computer technology was not so advanced during the late 1960s); therefore, it could run on a microcomputer. The separation and slippage of contacts can be easily taken into account since the global matrix representing block connectivity is never assembled. If a large assembled matrix is used, such a matrix will result in zero or very nearly zero diagonals, subsequently causing singularity or ill conditioning of the matrix system.

As the governing equation is of the hyperbolic type, the system could not become stabilized even for static cases unless damping is introduced into the equation system. In recent years, he has improved the original model by considering the deformability of intact blocks and their elasto-plastic behavior. Cundall’s model has been actively used in rock engineering structures design by the NGI group in recent years (e.g. Barton et al., 1986).

References Aydan, Ö., Mamaghani, I.H.P. & Kawamoto, T. (1996) Application of discrete “finite element method (DFEM) to rock engineering”. North American Rock Mechanics Symp., Montreal, 2, 2039–2046. Barton, N., Harvik, L., Christianson, M. & Vik, G. (1986) Estimation of joint deformations, potential leakage and lining stresses for a planned urban road tunnel. Int. Symp. on Large Rock Caverns. Helsinki, pp. 1171–1182. Baudendistel, M., Malina, H. & Müller, L. (1970) Einfluss von Discontinuitaten auf die Spannungen und Deformationen in der Umgebung einer Tunnelröhre. Rock Mechanics, Vienna, 2, 17–40. Crotty, J.M. & Wardle, L.J. (1985) Boundary integral analysis of piece-wise homogenous media with structural discontinuities. International Journal. Rock Mechanics and Mining Science, 22(6), 419– 427. Crouch, S.L. & Starfield, A.M. (1983) Boundary Element Methods in Solid Mechanics. Allen & Unwin, London. Cundall, P.A. (1971) The Measurement and Analysis of Acceleration in Rock slopes. PhD Thesis, University of London, Imperial College. Ghaboussi, J., Wilson, E.L. & Isenberg, J. (1973) Finite element for rock joints and interfaces. Journal Soil Mechanics and Foundation Engineering. Division, ASCE, 99(SM10), 833–848. Goodman, R.E., Taylor, R. & Brekke, T.L. (1968) A model for the mechanics of jointed rock. Journal Soil Mechanics and Foundation Engineering. Division, ASCE, 94(SM3), 637–659. Kawamoto, T. & Aydan, Ö. (1999) A review of numerical analysis of tunnels in discontinuous rock masses. International Journal of Numerical and Analytical Methods in Geomechanics, 23, 1377– 1391. Ngo, D. & Scordelis, A.C. (1967) Finite element analysis of reinforced concrete beams. Journal of American Concrete Institute, 152–163. Ohnishi, Y., Sasaki, T. & Tanaka, M. (1995) Modification of the DDA for elasto-plastic analysis with illustrative generic problems. 35th US Rock Mechanics Symposium, Lake Tahoe. pp. 45–50. Owen, D.R.J. & Hinton, E. (1980) Finite Element in Plasticity: Theory and Practice. Pineridge Press Ltd, Swansea. Pietruszczak, S. & Mroz, Z. (1981) Finite element analysis of deformation of strain-softening materials. International Journal of Numerical Methods in Engineering, 17, 327–334.

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Rashid, Y.R. (1968) Ultimate strength analysis of prestresses concrete pressure vessels. Nuclear Engineering and Design, 7, 334–344. Shi, G.H. (1988) Discontinuous Deformation Analysis: A New Numerical Model for the Statics and Dynamics of Block Systems. PhD Thesis, Department of Civil Engineering, University of California, Berkeley. 378p. Zienkiewicz, O.C., Valliappan, S. & King, I.P. (1968) Stress analysis of rock as a “no-tension” materials. Geotechnique, 18, 56–66.

Chapter 10

Ice mechanics and glacial flow

Ice can be treated as a rock-like material, and some sessions at the Second ISRM Congress in Belgrade in 1970 were devoted to it. However, almost no papers appeared in the following congresses and events on ice and its mechanics. This chapter is devoted to ice, its mechanics and related engineering problems.

10.1

Physics of ice

All materials in nature can be in solid, fluid or gaseous form depending upon the temperature and pressure. Ice is the solidified phase of water under a temperature less than 0°C. Ice-sheets cover the north and south polar regions of the Earth and mountains and plateaus greater than 2500 m above the sea level in other regions. The thickness of ice-sheets is measured to be more than 2700 m in Greenland, and the thickness varies depending upon location. Figures 10.1–10.4 show views of several ice-sheets and glaciers in Northern Hemisphere.

Figure 10.1 Views of ice-sheets and glaciers in Alaska

Figure 10.2 Views of ice-sheets and glaciers in Canada

Figure 10.3 Views of ice-sheets and glaciers in Norway

Figure 10.4 Views of ice-sheets and glaciers in Switzerland

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Mechanical properties of ice

Polycrystalline ice is generally treated as an isotropic solid. The p-wave and s-wave velocities of ice are more than 3890 m s
1 and 1800 m s
1. However, the wave velocities may be slightly anisotropic. The Young modulus and shear modulus of ice are 9.5 and 3.5 GPa, while the Poisson ratio is about 0.33 (i.e. Truffer, 2013). Ice is generally viewed as a visco-plastic material in terms of constitutive law. As the overall unit weight of ice is less than water, it floats in water.

10.3

Glaciers and ice domes/sheets

Besides the southern and northern poles, glaciers are found in mountainous regions of the Earth such as the Alps and Scandinavian Peninsula in Europe, the Andes in South America, the Rockies in North America, the Taurus, North Anatolian and Elbruz mountains in West Asia, the Himalayas in Asia and even in New Zealand. There are also traces of glaciers even in England and Turkey. A typical glacier is visualized to be consisted of two zones: an accumulation zone and an ablation zone, as illustrated in Figure 10.5 (e.g. Bennet and Glasser, 2009; Truffer, 2013). They are mainly driven by gravitational forces, and they flow like a visco-plastic fluid. Ice domes/sheets are found on relatively flat lithospherical areas and mainly in the north and south poles such as Greenland and Antarctica. The motion of the ice domes/sheets is also gravity driven from the center toward their flanks. Nevertheless, their motion entirely depends upon pressure distribution related to the geometry of ice domes/sheets (Fig. 10.6). When they reach open seas or are floating in the seas, they may be broken into pieces called icebergs, and their motion would then be governed by Coriolis forces resulting from the rotation of the Earth. Velocity and shear stress distribution of a typical longitudinal section of a glacier are illustrated in Figure 10.7. If the ice is modeled as a Newtonian material, the velocity (v) and shear stress distribution (τ) are represented by the following equations:   g z z ð10:1Þ v ¼ vo þ z sin a H
; t ¼ gH sin a 1
2Z 2 H

Figure 10.5 Illustration of main components of a glacier

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Figure 10.6 Illustration of an ice dome/sheet and its motion

Figure 10.7 Velocity and shear stress distribution along a longitudinal section of a glacier

Figure 10.8 Velocity distributions in a glacier in a channel

However, if the shear stress at the base exceeds the frictional resistance between glacier and bedrock, some relative slip will occur. Similarly, the velocity distributions in the direction of the flow will appear as shown in Figure 10.8 for no-slip and slip conditions. Ice in the upper section of glaciers is brittle, and it may be fractured depending upon the geometry and thickness of the glaciers. They may have crevasses as well as fractures going through its entire thickness. Figure 10.9 shows examples of crevasses and fractures in the Athabasca glacier in Canada. In the ablation zone, some surface and undersurface

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Figure 10.9 Crevasses and fractures in the ablation zone of Athabasca glacier in Canada

Figure 10.10 Surface rivers and a sinkhole in Gorner glacier near Zermatt in Switzerland

rivers may form. These rivers may also lead to sinkholes within the glaciers, as seen in Figure 10.10. Striations on bedrocks are caused by glacier flow in relation to their direction of motion. Figure 10.11 shows an example of striations on the bedrock observed in Athabasca glacier.

10.4

Cliff and slope failures induced by glacial flow

It is well-known that the glaciers apply shear stresses on the bedrock of the channel. As a result, erosion takes place at contact areas, which may result in overhanging configurations on the sidewalls of the channel. Depending upon the resulting geometry and

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Figure 10.11 Striations observed on the bedrock of the Athabasca glacier

Figure 10.12 Toe erosions of sidewalls at Gorner glacier in Switzerland

discontinuities and characteristics of rock mass, cliff and/or slope failure may occur as shown in Figure 10.12. As the climatic conditions can be quite harsh and sedimentary rocks are quite vulnerable to degradation, slope failure may take place from time to time. The governing mechanism is quite similar to the cliff failures caused by the toe erosion phenomenon. Figure 10.13 shows an example of a slope failure in the ablation zone at the northern side of the Athabasca glacier in Canada. Rocks were shale, sandstone and mudstone at this locality. Ice-sheets may also slide down especially during thawing seasons. These failures may be similar to planar slide initially. Figure 10.14 shows examples of ice-sheet slides in the ablation zones. Huge avalanches of ice-sheets, together with debris material, may occur when they are subjected to seismic forces. Some examples of such failures have been reported in the past. The 1962 and 1970 Huascaran snow, ice and rock avalanches in Peru, Mt. Cook ice-rock avalanche in New Zealand, and 2015 Langtang ice-rock avalanche in Nepal due to the 2015

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Figure 10.13 Slope failure at the northern side of the Athabasca glacier in Canada

Gorkha earthquake were all quite devastating. Figure 10.15 shows the satellite images of the Langtang ice-rock avalanche before and after the event. The Gorkha earthquake also caused the failure of an ice-cliff between Lingtren and Pumori peaks with Khumbu glacier on the mountain of Everest, which resulted in 18 casualties in a base camp about 800 m below. The ice-sheet failure occurred on the western side of the glacier at an elevation of approximately 6300–6400 m. The receding glaciers reduce the pressure at the sidewalls and base. The variation of normal pressure at the sidewalls particularly induces stress relaxation in the rock mass of slopes, which results in the opening and/or propagation of existing discontinuities and in the increase of permeability and seepage characteristics of the rock mass. These changes result in instability problems at the sidewall slopes of glaciers. For example, a huge rock slope failure occurred in Randa near Zermatt in Switzerland on April 18, 1991 (Götz and Zimmermann, 1993). The Randa slope failure interrupted the railway line connecting Zermatt to the Rhône Valley (Fig. 10.16). The railway line was buried for 800 m and the road for 200 m. The fallen rock mass dammed the Vispa river. A channel was excavated before a potential catastrophic failure of the dam. A huge rockfall occurred on the east face of Eiger mountain in Grindelwald glacier. In early June of 2006, an 18 cm wide crack appeared at the top of the rock block, and the crack grew at a rate of 90 cm a day. It finally failed when the crack was roughly 4.8 m feet (Fig. 10.17).

Figure 10.14 Examples of ice-sheet slides and falls

Figure 10.15 Satellite images of Langtang ice-rock avalanche before and after the event caused by the 2015 Gorkha earthquake Source: Processed by Immerzel and Kraaijenbrink, 2015

Figure 10.16 Views of Randa slope failure

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Figure 10.17 Views of rock slope failure on the east face of Eiger mountain

Figure 10.18 Formation and cracking of glacier caves

10.5

Glacial cave failures

Glacial caves develop within glaciers due to seepage of melting water through crevasses (Fig. 10.18). Most glacier caves are located near the contact zone with bedrock. These caves may cause some sinkholes within the glacier, as seen in Figure 10.10. However, the most catastrophic failures may occur at the tip of the glaciers in the ablation zones when they meet glacial lakes or open seas. The failure of caves may result in some tsunami-like events in the lakes and open seas.

10.6

Moraine lakes and lake burst

Glaciers create moraine lakes in the vicinity of their tips. The material originates from debris from the failure of sidewall slopes of the channels and the erosion of bedrock. The receding glaciers also result in increases in the size of lakes. The lakes may fail due to piping phenomenon or overtopping. These may expose great danger to settlements and engineering facilities downstream of the valleys. See Figure 10.19.

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Figure 10.19 Examples of moraine lakes

Figure 10.20 Ice thinning and calving.

10.7

Calving and iceberg formation

When ice domes, ice-sheets or glaciers reach to open seas, they start to break down into blocks, which is known as calving, as illustrated in Figures 10.20 and 10.21. As a result of calving, icebergs are formed. These icebergs float in the seas and move mainly as a result of Coriolis forces due to the Earth’s rotation. Ice calving results in the toppling of ice blocks into the sea, which may create some small-scale tsunamis (Fig. 10.22). They may expose great danger to nearby settlements as well as ship traffic and offshore platforms. The famous Titanic ship accident was due to the collision of the ship with an iceberg.

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Figure 10.21 Calving of glaciers

Figure 10.22 Toppling of calved ice-slabs and subsequent tsunami-like waves

References Bennet, M.R. & Glasser, N.F. (2009) Glacial Geology: Ice Sheets and Landforms. Wiley-Backwell, Chichester, West Sussex, 385p. Götz, A. & Zimmermann, M. (1993) The 1991 rock slides in Randa: Causes and consequences. Landslide News, 7(3), 22–25. Immerzel, W.W. & Kraaijenbrink, P.D.A. (2015) Landsat 8 Reveals Extent of Earthquake Disaster in Langtang Valley. http://mountainhydrology.org/nepal-quake/. Truffer, M. (2013) Ice Physics. University of Alaska, Fairbanks. 120p.

Chapter 11

Extraterrestrial rock mechanics and rock engineering

Mankind is now exploring the ways to find out the characteristics of other planets and the possibility of exploiting their mineral resources. The United States and Russia (former USSR) sent several spacecrafts to the Moon, Venus and Mars, and some of them landed on the Moon, Venus and Mars (NASA, 2008). Some of most impressive images from the Apollo Program of NASA is that of a human next to a lunar rock mass and hitting lunar rock with a geologist’s hammer (Figure 11.1). The images from recent Mars exploration rovers showed striking similarities between rocks on Earth and those of Mars. One can easily notice layered rock masses, jointing, weathering effects and some toppling-type rock slope stability problems (Fig. 11.2). This chapter is entirely based on information obtained from images mostly released by NASA and the interpretations of the author, together with some further information and interpretations by others. Although the environmental conditions on Mars and other planets are different from those on Earth, the principles governing mechanical and engineering aspects of rocks on other planets should be quite similar to those developed for the rocks of Earth. Therefore, the next generations of our discipline would definitely see its extension to the rocks of other planets.

Figure 11.1 Images from the Moon involving humankind from the Earth Source: Images by NASA

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Figure 11.2 Some images of rocks and rock slope on Mars Source: Arranged from images of NASA

Figure 11.3 Comparison of Venus, Earth and Mars

11.1

Solar system

The solar system consists of eight planets: Mercury, Venus, Earth, Mars, Saturn, Jupiter, Uranus, Neptune (Watters, 1995). On the basis of computations, the existence of a planet between Mars and Jupiter was estimated. At this location, an asteroid belt now exists that may have corresponded to the missing planet. Initially, Pluto was found in 1930, and it was considered to be the ninth planet until 2006. On the basis of the definition of planets of our solar system, it was then disqualified as the ninth planet. The densities of planets differ as a function of distance from the Sun and decrease as the distance increases. Venus, Earth and Mars are considered to be quite similar to one another in terms of size and geology (Fig. 11.3). Venus has no natural satellites, while the Earth has the Moon as a natural satellite. Mars has two natural satellites called Deimos and Phobos.

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Moon

The Moon is about 284400 km away from the Earth and is 3474.2 km in diameter. The average density is 3.34 g cm
3 with a gravitational acceleration of 1.62 m s
2 at the surface, which is about one-sixth that of the Earth. The Moon is assumed to be a part of the Earth initially. The Moon is the most extensively investigated solar body by the United States and Russia. Recently, Japan and China sent some explorers to the Moon. However, the most extensive explorations and investigations have been carried out by NASA of the United States so far. 11.2.1

Surface structure

The surface topography of the Moon has been measured by the methods of laser altimetry and stereo image analysis as shown in Figure 11.4. The surface topography of the Moon on the far side is very different from that on the near side. The far side of the Moon is about 1.9 km higher than the near side. The surface of the Moon has been greatly shaped by the impact of meteorites. The south pole Aitken basin has the lowest elevation while the highest elevations are found just to the northeast of this basin, and it has been suggested that this area might represent thick ejecta deposits on the Moon. The difference between lowest and highest elevations is about 13 km. Although there are no tectonic plates in the Moon, huge rift valleys are found, and they are covered by numerous ancient volcanic plains.

Figure 11.4 Surface topography of the Moon Source: From NASA

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11.2.2

4

Inner structures

During the NASA Apollo program, some passive-type seismometers were installed on the Moon. During this program from 1969 to 1972, Apollo 12, 14, 15 and 16 seismometers were installed (Fig. 11.5). These seismometers recorded moonquakes, which were initially unexpected (Fig. 11.6). The records from the moonquakes revealed an inner structure of the Moon like that of the Earth (e.g. Irvine, 2002; Latham et al., 1971; Nakamura, 2003; Toksöz et al., 1977). It was found that the earthquakes were categorized as deep earthquakes resulting from the tidal force interaction between the Earth and the Moon, shallow earthquakes due to tectonic forces and thermal stress variations and other earthquakes due to

Figure 11.5 Seismometers and lunar seismic network

Figure 11.6 Records of moonquakes and lunar impacts (time marks 10 minutes)

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meteorite and lunar module impacts. The s of the shallow earthquakes could be up to 5.5, while the magnitudes of the deep earthquakes were much smaller, generally less than 2. The shallow earthquakes may pose a hazard to a lunar habitat, and they may last for more than 10 minutes. The deep earthquakes occur at depths of 700 to 1000 km and occur periodically at 14 and 206 days, implying the causes of the events are terrestrial and solar tidal forces. It was found that the vibrations caused by impacts can last for more than 1 hour while deeper earthquakes last only 10–20 minutes. The reason for such a long seismic vibration is considered to be due to the existence of a thick elastic crust (Fig. 11.7). As the viscosity of the crust is small, the vibrations last longer for surface impacts. On the basis of these investigations, the inner structure of the Moon is well understood, and it resembles that of the Earth except for the thickness of each layer (Fig. 11.8). Particularly, the outer core is quite thin, and inner core is small.

Figure 11.7 Illustration of the response of the Moon crust to impacts Source: Irvine, 2002

Figure 11.8 (a) Inner structure, (b) seismic wave velocity and density distributions

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6

Geology and rocks of the Moon

The geological history of the Moon has been organized into six major epochs starting about 4.5 billion years ago. It was initially in molten state. On the basis of the knowledge of the Earth, it was assumed that olivine and pyroxene formed first, followed by crystallized anorthositic plagioclase feldspar, forming an anorthositic crust about 50 km in thickness. From the Apollo missions, 382 kg rock samples were brought to the Earth from the Moon. Figure 11.9 shows views of several examples of Moon rocks such as olivine, basalt and anorthositic plagioclase. Other rock samples are tuff breccias associated probably with meteorite impacts. Figure 11.10 shows larger rock blocks. One can easily notice bedding planes and discontinuities and fracture planes, which are quite similar to those of the Earth. The mean lunar surface temperature varies from 107°C at day down to
153°C at night. The temperature difference is thus 260°C. This large temperature difference is very likely to create large thermal stresses, which may impose cyclic contraction–expansion as well as new crack extension in the upper crust of the Moon. It is noted that some thrust-type, extensiontype huge cracks/fractures and lava flows exist on the surface of the Moon (Fig. 11.11). The active seismic experiments on the Moon during the Apollo program of NASA revealed that the top 20 km of the Moon’s crust is highly heterogeneous and disturbed.

Figure 11.9 Views of some Moon rocks Source: Original images from NASA

Figure 11.10 Large-scale in-situ rock blocks of Moon

Figure 11.11 Large-scale surface fractures on the Moon surface

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Figure 11.12 Craters and large-scale mass movements on the Moon

The Moon has been heavily bombarded by meteorites of different sizes, creating craters (Fig. 11.12). The surface topography of the Moon is greatly affected and shaped by the impacts of meteorites. In some of large craters, melted rocky material has been noted. In addition, large-scale mass movements exist within the craters.

11.3 11.3.1

Mars Inner structure

Many models for the inner structure and temperature distribution of Mars have been proposed by various researchers (e.g. Sohl and Spohn, 1997; Johnston and Toksöz, 1977; Toksöz and Hsui, 1978; Steinberger et al., 2010). Particularly, Sohl and Spohn (1997) utilized two meterorites from Mars for estimating the internal structure of that planet. Although all models are based on concepts developed for Earth, they are fundamentally estimations as there is no seismic network for Mars yet. NASA has deployed a three-component seismometer through the Viking program in 1976 (Anderson et al., 1977). In 2018, the second seismometer was installed on Mars as a part of an InSight instrument, which has been equipped with many sensors. The Viking seismometer recorded numerous quakes during its operation. The InSight seismometer recorded the first marsquake on April 6, 2019. Nevertheless, the models for the inner structure of Mars still needs some validation. The radius of Mars is about 3389 km with a crustal thickness of 10–50 km. The mantle is divided mainly into two layers, which may be called the upper and lower mantles. The core and mantle interface is estimated to be at a depth of 2000 km with a 150 km transition zone. Mars is estimated to be chemically composed of a silicate crust, a Fe-Mg silicate mantle and a metallic Fe core. The average density is about 3.933 g cm
3. The gravitational

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Figure 11.13 Comparison of the distributions of density, wave velocities and gravity of Mars

acceleration is 0.377 g. Figure 11.13 shows the estimated density, wave velocities and gravitational acceleration of Mars. 11.3.2

Geology and tectonics

The surface geology of Mars was mapped by the United States Geological Survey (USGS) and Scott and Carr (1978), utilizing also Mars Orbiter Laser Altimeter (MOLA) data of NASA. The surface geology fundamentally involves volcanics, basalt, breccia, sedimentary deposits (Aydan, 2016). The tectonic activity of Mars is not very active compared to that of Earth. Figure 11.14 shows a map of the tectonics of Mars prepared by the USGS. Nevertheless, a very large-scale Vallis rift zone, Tharsis volcanic chain, depression zones, fracture zones, faults, folding, metamorphism, discordant sedimentation, columnar jointing are found on Mars (Figs. 11.15–11.17). 11.3.3

Rocks of Mars

It is very likely that the rocks of Mars would be quite close to those of Earth. Therefore, the classification of rocks should be igneous, metamorphic and sedimentary on the basis of the images from many NASA Mars explorers. The images and drilling operations performed by Mars explorers (Opportunity, Sprit, Curiosity) on some selected rocks and rock blocks have indicated that the rocks are quite similar to those of the Earth, although it still needs some clarification whether rocks seen in Earth fully exist on Mars. However, the sedimentary rocks (such as conglomerate, sandstone, siltstone, mudstone, sulphate deposits), metamorphic rocks (phyllite, shale, schists) and extrusive rocks (basalt, breccia) of Mars are clearly similar to those of Earth (Figs. 11.18–11.20). Furthermore, flow planes, bedding planes and schistosity planes, which are intrinsic to each rock class, are distinctly observed. 11.3.3.1

Rock discontinuities of Mars

As explained in Chapter 2, there are various causes for the formation of discontinuities in rocks. The causes can be grouped into (1) tension discontinuities, (2) shear discontinuities,

Figure 11.14 Topography and tectonics of Mars

Figure 11.15 Large-scale faulting, shearing, folding, and metamorphism structures

Figure 11.16 Large-scale fractures, rift zones

Figure 11.17 Sedimentation, tilting, folding and volcanic eruptions and columnar jointing

Figure 11.18 Views of extrusive volcanic rocks

Figure 11.19 Views of metamorphic rocks

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Figure 11.20 Views of sedimentary rocks

(3) discontinuities due to intrinsic properties of rocks (sedimentation, metamorphism, magma flow). Without any exception, discontinuities in the rocks/rock masses of Mars are fundamentally similar to those observed in Earth as seen in Figure 11.21 (Aydan, 2016). Furthermore, the surface morphology and filling situations of rock discontinuities are quite similar to those seen in Earth (Fig. 11.22).

11.3.3.2

Weathering of rocks of Mars

From the surface morphology of Mars, it is well-known that the environment of Mars was quite different from the present. There were lakes, rivers and oceans, and climatic conditions were different from today. Images from Mars explorers clearly show that the weathering of the rocks of Mars is quite similar to that on Earth, as seen in Figure 11.23. As seen in the images of Figure 11.23, differential weathering, solution and oxidation-type weathering can be easily noticed. Compared to the daily temperature difference on the Moon, the daily temperature difference in Mars is about 80°C. If the rock is not saturated, such a temperature difference would not be of great significance except for soft rocks, as seen in Figure 11.23.

Figure 11.21 Discontinuities in rocks/rock masses of Mars

Figure 11.22 Some images of discontinuity filling of rocks on Mars

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Figure 11.23 Weathering of rocks of Mars

11.4

Venus

Surface observations of Venus have been difficult in the past, due to its extremely dense atmosphere, which is composed primarily of carbon dioxide with a small amount of nitrogen. The first attempts to explore Venus were carried out by the Soviets in the 1960s through the Venera Program. Venera lander missions took place until the early 1980s. Venera 13 and Venera 14 landed on the planet and sent the first color photographs of the surface. Venera 15 and Venera 16 conducted mapping of the Venusian terrain with synthetic aperture radar. The United States launched the Mariner 1 in 1962 and subsequently NASA’s Magellan spacecraft in 1989 to map the surface of Venus with radar. The Magellan provided the most high-resolution images to date of the planet and was able to map 98% of the surface and 95% of its gravity field (Fig. 11.24). Venus, like Uranus, rotates clockwise, which is known as a retrograde rotation. Due to the slow rotation on its axis, it takes 243 Earth-days to complete one rotation. The orbit of the planet takes 225 Earth-days. Venus is totally enshrouded by clouds of more than 50 km in height. Due to the continuous global cloud cover and greenhouse effect, surface temperatures on Venus approach 735 K (462°C), and it is the hottest planet in our solar system. As a result of these high temperatures, water is vaporized. The atmospheric pressure at the surface of Venus is 90 times that found at the surface of Earth.

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Figure 11.24 Topography of Venus

11.4.1

Interior

It is quite likely that Earth and Venus have similar concentrations of the major elements such as iron, magnesium, calcium, silicon and aluminum. This notion is supported by density measurements determined from the motions of passing spacecraft. The bulk density of Venus (5.24 g/cm3) is only slightly less than that of Earth (5.52 g cm
3). Venus has a metallic core composed predominantly of iron, a mantle of dense iron and magnesium silicates, and a crust of lighter silicate minerals possibly enriched in aluminum, alkalis, and the radioactive elements uranium and thorium. Similar to the other terrestrial planets, Venus’s interior is essentially composed of three layers: a crust, a mantle and a core. It is believed that Venus’s crust is 8–40 km thick depending upon the assumed model, its mantle 3000 km thick, and the core is about 3000 km thick. However, there is an argument about the Venusian interior as to whether or not the planet’s core is liquid or solid, as the planet’s lacks a substantial magnetic field due to its slow rotation. Venus is presumed to have emerged about 4.6 billion years ago by the accretion of tiny objects. During the bombardment phase, enough heat was produced to melt the whole proto planet. After a certain cooling period, the molten mass developed a crust, a mantle and a core. Convection in the mantle caused by internal heating deformed the outer crust,

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which is thinner in low-lying areas (corresponding to seas on the Earth) and thicker in the highlands (corresponding to continents on the Earth). High mountain regions developed by uplifting and outflowing. 11.4.2

Tectonics

The tectonic activity of Venus is quite different from that of Earth (Fig. 11.25). As the surface temperature of Venus is greater than 460°C, the crustal rocks are likely to be softer and weaker than those of Earth and to behave in a ductile manner. As a result of these features of crustal rocks, the stress buildup in the crust and upper mantle of the Venus would be much smaller than that in the Earth. Consequently, the subduction of the soft plastic crust into the upper mantle is not deep. Compared with the subduction of Earth’s crust into the upper mantle, which is up to 670 km in depth, the subduction of Venus’s crust is estimated to be 250–300 km deep. In view of these facts, the size and the drag of the mantle convection in Venus should be weaker than on Earth so that the tectonic motion of Venus is less pronounced. Venus’s surface appears to have been shaped by extensive volcanic activity. Venus also has 167 large volcanoes that are over 100 km across. The highest point on Venus is Maxwell Montes. Magellan gathered much finer details of the surface topography of Venus, and approximately 1000 impact craters were found. Interestingly, none of the craters seen were smaller than 2 km in diameter. This suggests that any meteoroid small enough to create a crater having a diameter of less than 2 km would have broken apart and burned up during its passage through the dense Venusian atmosphere. The Ishtar Terra mountains formed as a result of the uplift of the plateau and formation of the Maxwell mountains, which rise 6 km above Lakshmi Planum. Low-lying areas and craters became filled with lava. The shield in the Beta Regis area was formed before the mountain ridges along the fault that broke the surface of the shield (Fig. 11.23). The surface of Venus has shown that a wide variety of tectonic features are ubiquitous on the planet. Its surface is strongly deformed at a variety of scales. Its lithosphere has been extended and compressed, domed and depressed. Broad crustal domes and rift valleys are common tectonic features on Venus. Beta Regio is a large domical upland about 2500 km

Figure 11.25 Comparison of tectonic model of Venus with that of Earth Source: From Christiansen and Spilker, 2018

332 Extraterrestrial rock

18

across that is crisscrossed by many faults. The gentle rise is about 4 km high and is crossed by a central trough. A multitude of nearly parallel linear scarps show that the depression is a fault-bounded rift valley, formed as the dome was pulled apart by the extension of the lithosphere. Extensional tectonism has produced long belts of deformation marked by abundant fractures and grabens. These belts persist over hundreds of kilometers. The region around Beta Regio and Atla Regio has fracture belts of diverse orientation. Because of the extensional nature and abundance in the uplands, it is inferred that they formed in response to the uplift of the lithosphere over mantle plumes. 11.4.3

Rocks

The Soviets have successfully landed several spacecraft in the plains regions. Seven of the landers conducted chemical analyses of rocks, which indicate a composition similar to that of terrestrial basaltic volcanic rocks. Venera Spacecraft landed on Venus and took some images of rocks of Venus (Fig. 11.26). The thin, plate-like slabs of rock could be due to molten lava that cooled and cracked. The composition and texture of these rocks is similar to terrestrial basaltic lava. The weathering processes of rocks on Venus are due to the extremely high temperature and pressure at the surface, as well as the composition of the atmosphere. Each of these factors exerts some control on chemical reactions between gas and rock. Minerals that crystallized at high temperatures in lava flows and that are exposed to CO2 and SO2 gas in the Venusian atmosphere are inherently unstable and decompose to form new minerals. The weathered zone probably consists of a mixture of incompletely reacted minerals and newly formed weathering products. For example, it is predicted that, as weathering

Figure 11.26 Images of rocks and soil in landing sites of USSR Venera spacecraft landed on Venus

19

Extraterrestrial rock

333

decomposes basaltic lavas, iron oxides and sulfur-rich minerals will form as iron silicates are destroyed. However, carbonate minerals, which could remove significant amounts of carbon dioxide from the atmosphere, are not stable on the hot, dry surface of Venus. Weathering varies with altitude on Venus. At high altitudes and low temperatures, sulfur in the atmosphere reacts with lava flows to form the mineral pyrite (FeS2). Pyrite is highly reflective of radar waves and may explain why mountain peaks and high plateaus of Venus are radar bright. At lower altitudes and higher temperatures, magnetite (Fe3O4) and anhydrite (CaSO4) may be the stable minerals produced by weathering. Magnetite is not as reflective as pyrite.

11.5 11.5.1

Issues of rock mechanics and rock engineering on the Moon, Mars and Venus In-situ stress

In-situ stress state is one of the most important items if rock mechanics and rock engineering are to be utilized on the Moon and other planets. Particularly, the engineering design requires data on in-situ stress states as well as quake-related topics. Figure 11.27 shows the density, gravity and hydrostatic pressure distributions with depth in Venus and Mars together with those of Earth. The in-situ stress measurements and estimations by one or several empirical, analytical and numerical methods can be used as explained in Chapters 7, 8 and 9. However, proper knowledge and information on the characteristics of materials constituting extraterrestrial objects are necessary. For example, the elastic constants obtained from wave velocities, density and gravitational distributions may be used for stress state in the extraterrestrial object. Such a preliminary elastic analysis is carried out for the stress state of Mars under spherical symmetric conditions, and results are shown in Figure 11.28. However, it should be noted that the actual stress conditions are likely to be different from the actual ones as it is very likely that the behavior would involve thermo-elasto-plastic behavior rather than elastic behavior. In other words, experiments on rocks and materials constituting the other extraterrestrial objects are definitely necessary for better evaluations of their in-situ stress states. 11.5.2

Slope stability

As on Earth, slope stability issues would be of great significance in other extraterrestrial objects. Although the gravitational acceleration and environmental conditions are different from those of Earth, slope stability issues are likely to be similar to those of Earth. Figure 11.29 shows some examples of slope failures observed on Mars. The slope failures in Mars involve curved shear failure, planar/wedge sliding, toppling failure, bending failure, although their scales may be different. Similar types of slope failures are also noted in the Moon. Nevertheless, such failures are likely to be seen in other planets such as Venus, as seen in Figure 11.26. Figure 11.30 shows a classification of failure modes of rock slopes on Earth (Aydan, 1989). Therefore, the methods developed for rock slopes can be utilized with due considerations of environmental and gravitational differences using the principles of rock mechanics. Figure 11.31 shows an application of the integrated slope stability assessment proposed by Aydan et al. (1991) to rock slopes of Mars. Lines in the figure represent the relations

Figure 11.27 Comparison of the density, gravity and hydrostatic pressure distributions with depth in Venus and Mars together with those of Earth Source: From Steinberger et al., 2010

Figure 11.28 Preliminary elastic analysis carried out for the stress state of Mars under spherical symmetric condition

21

Extraterrestrial rock

335

Figure 11.29 Views of slope failures observed in Mars

between the bedding plane and lower slope angle for the stability of rock slopes for jointed rock mass. The friction angles used in this graph are for those of basaltic rock joints on Earth. 11.5.3

Impact- and vibration-induced mass movements

Including Earth, planets and the Moon are bombarded by meteorites of various sizes from time to time. Furthermore, quakes (earthquakes, moonquakes, marsquakes) occur from time to time. These events result in shock waves and vibrations. Figure 11.32 show some examples of recent impacts on Mars and the displaced rock blocks in the Moon and Mars. A rock block displaced on Mars shown in Figure 11.33 is considered, and the conditions for its motion are analyzed herein. The travel path length was 675 cm, and the inclination of the path was almost 23.5 degrees. As rocks at the site were inferred to be basaltic, the result of a dynamic friction experiment on a saw-cut discontinuity of basalt from Mt. Fuji was utilized (Fig. 11.34).

Figure 11.30 Classification of rock slope failures Source: From Aydan, 1989

Figure 11.31 Relation between inclination of bedding plane and lower slope angle for rock slopes on Mars

Figure 11.32 Images of impacts and rock block movements

FRICTION COEFFICIENT(S/N) NORMALIZED ACCELERATION(a/g)

Figure 11.33 Displaced rock block on Mars

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0

Mt.Fuji Basalt Saw-cut Surface S/N

Upper Block Acceleration

10

20 TIME(sec)

Figure 11.34 Dynamic friction test on a basaltic saw-cut discontinuity

30

25

Extraterrestrial rock

339

The simple mechanical considerations imply that the maximum acceleration and velocity to displace the rock block may be derived as given here: • •

Maximum acceleration a > gm tanðφ
aÞ

ð11:1Þ

Maximum velocity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vmax > 2gm tanðφ
aÞd

ð11:2Þ

As seen from Figure 11.34, the friction angle of basaltic saw-cut surface from Mt. Fuji is greater than 30 degrees. If these values are used, the conditions to displace the rock block shown Figure 11.33 are estimated to be as: • •

Maximum acceleration: 42.13 cm s
2 Maximum velocity: 238.5 cm s
1

As understood from this simple example, rock dynamics would be necessary for the Moon and other planets. 11.5.4

Properties of rocks and discontinuities

As discussed in previous subsections, rocks from the Moon and other planets are quite similar to those of Earth. In rock mechanics and rock engineering, numerous experiments on rock, rock discontinuities and rock masses have been carried out, and many mechanical and engineering properties have been determined. Nevertheless, it is needless to say that we need to carry out similar experiments for extraterrestrial rocks probably in their environment. During the Apollo program, rock samples of more than 300 kg have been brought to Earth. However, no mechanical experiments have been carried out yet. The Mars rovers provided many images of rocks, rock masses and discontinuities. Back analyses of many images may provide some information on the mechanical analyses. For example, one may easily infer the frictional properties of stable and unstable blocks shown in Figure 11.35 with their base rocks. Furthermore, the back analyses of failed rock slopes and rock cliffs may further provide information on the strength properties of rock masses. Some of these analyses are reported by several researchers (e.g. Brunetti, 2014; Conway et al., 2011; Crosta et al., 2014; Lucchitta, 1979; Schultz, 2002; Neuffer and Schultz, 2006). The force and rate of advance during drilling operations by the Curiosity may also provide other data on the mechanical properties of intact rocks. 11.5.5

Sinkholes

Sinkholes are a severe problem on Earth. This problem has been often observed in karstic terrains, evaporitic formations and volcanic regions on Earth. The images from the Moon and Mars, as seen in Figure 11.36, clearly showed that sinkholes also formed on both the Moon and Mars. These sinkholes have been found in basaltic flow areas. Some of sinkholes may also be formed by impacts of meteorites.

Figure 11.35 Some images to infer frictional properties of discontinuities on Mars

Figure 11.36 Sinkholes on the Moon and Mars

27

Extraterrestrial rock

11.6

341

Conclusions and future studies

Although the environmental, fluids and gravitational conditions on Mars and other planets are different from those on Earth, the principles governing mechanical and engineering aspects of rocks on other planets should be quite similar to those developed for rocks and rock discontinuities of Earth. Therefore, the next generations of our discipline will definitely see its extension to the rocks, rock discontinuities, rock masses and rock engineering aspects of other planets. Future studies are necessary to explain: 1 2 3 4 5

The stress state of planets by empirical, analytical and computational methods, as well as in-situ measurements. Why mountains are higher in Mars and lower in Venus compared with those on Earth. Why tectonism is less pronounced on Mars and Venus. The properties of rocks, discontinuities and rock masses of the Moon and other planets. The effect of weathering on each planet.

Acknowledgments The author gratefully acknowledges NASA and the people involved in the development and operation of the Moon, Mars and other planets exploration programs and processing and releasing their images on related websites.

References Anderson, D.L., Miller, W.F., Latham, G.V., Nakamura, Y., Toksöz, M.N., Dainty, A.M., Duennebier, F.K., Lazarewicz, A.R., Kovach, R.L. & Knight, T.C.D. (1977) Seismology on Mars. Journal of Geophysical Research, 82(28), 4524–4546. Aydan, Ö. (1989) The Stabilisation of Rock Engineering Structures by Rockbolts. Doctorate Thesis, Nagoya University, Faculty of Engineering. Aydan, Ö. (2016) Some thoughts about rock mechanics aspects of Mars. UNSW, 3rd Off Earth Mining Forum, 2017OEMF. Aydan, Ö., Ichikawa, Y., Shimizu, Y. & Murata, K. (1991) An integrated system for the stability of rock slopes. The 5th Int. Conf. on Computer Methods and Advances in Geomechanics, Cairns, 1. pp. 469–465. Brunetti, M. (2014) Statistics of Terrestrial and Extraterrestrial Landslides. Doctoral Thesis, Universita Degli Studi di Perugia. DOI:10.13140/2.1.4107.3444. 109p. Christiansen, E.H. & Spilker, B. (2018) Exploring the Planets. Published by Prentice Hall in 1990, 1995. http://explanet.info. Conway, S.J., Balme, M.R., Lamb, M.P., Towner, M.C. & Murray, J.B. (2011) Enhanced runout and erosion by overland flow under subfreezing and low pressure conditions: Experiments and application to Mars. Icarus, 211(1), 443–457. Crosta, G.B., Utili, S., Blasio, F.V. & Riccardo Castellanza, R. (2014) Reassessing rock mass properties and slope instability triggering conditions in Valles Marineris, Mars. Earth and Planetary Science Letters, 388, 329–342. Irvine, T. (2002) Moonquakes. March, Newsletter. Vibrationdata.com. pp. 4–13. Johnston, D.H. & Toksöz, M.N. (1977) Internal structure and properties of Mars. Icarus, 32, 73–84. Latham, G., Ewing, M., Press, F., Sutton, G., Dorman, J., Nakamura, Y., Lammlein, D., Duennebier, F. & Toksöz, N. (1971). Moonquakes, Science, 174, 687–692.

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Lucchitta, B.K. (1979) Landslides in Vallis Marineris, Mars. Journal of Geophysical Research, 84, 8097–8113. Nakamura, Y. (2003) New identification of deep moonquakes in the Apollo lunar seismic data. Physics of Earth and Planetary Interiors, 139, 197–205. NASA (2008) Exploration: NASA’s Plans to Explore the Moon, Mars and Beyond. www.nasa.gov/. NASA: Images. https://mars.nasa.gov/mer/gallery/images.html (Accessed October 2019). Neuffer, P.D. & Schultz, R.A. (2006) Mechanisms of slope failure in Valles Marineris, Mars. Quarterly Journal of Engineering Geology and Hydrogeology, 39(3), 227–240. Schultz, R.A. (2002) Stability of rock slopes in Valles Marineris, Mars. Geophysical Research Letters, 29, 1932. Scott, David H., and Carr, Michael H. (1978). Geologic Map of Mars: U.S. Geological Survey Investigations Series I-1083, scale 1:25000000, http://pubs.er.usgs.gov/publication/i1083 Sohl, F. & Spohn, T. (1997) The interior structure of Mars: Implications from SNC meteorites. Journal Geophysical Research and Planets, 102(E1), 1613–1635. Steinberger, B., Werner, S.C. & Torsvik, T.H. (2010) Deep versus shallow origin of gravity anomalies, topography and volcanism on Earth, Venus and Mars. Icarus, 207, 564–577. Toksöz, M.N. & Hsui, A.T. (1978) Thermal history and evolution of Mars. Icarus, 34(3), 537–547. Toksöz, M.N., Goins, N.R. & Cheng, C.H. (1977) Moonquakes: Mechanisms and relation to tidal stresses. Science, 196, 979–981. Watters, T.R. (1995) Planets: A Smithsonian Guide. MacMillan, New York, USA. 256p.

Appendices

Appendix 1

Definitions of scalars, vectors and tensors and associated operations

A1.1

Scalar

Scalar is a quantity having a magnitude, and it remains the same irrespective of direction. It is defined as a rank-0 tensor. Examples of scalar quantities are volume, density, mass, temperature, energy, pressure. It can be easily added, subtracted, multiplied and divided.

A1.2

Vector

Scalar is a quantity having both magnitude and direction, as illustrated in Figure A1.1. It is defined as a rank-1 tensor. Examples of vectors are force, velocity, displacement, acceleration, moment.

A1.3 A1.3.1

Vector operations Addition and subtraction

The addition and subtraction of vectors obey the geometrical parallelogram rule, and they are illustrated as shown in Figure A1.2 and are mathematically expressed as follows: c¼aþb

ðA1:1Þ

d¼ab

ðA1:2Þ

Figure A1.1 Illustration of vectors

346 Appendices

4

Figure A1.2 Geometrical illustration of addition and subtraction of vectors

Figure A1.3 Illustration of cross product of two vectors

A1.3.2

Dot product

The dot product of two vectors is defined as follows: a  b ¼ kakkbk cos y

ðA1:3Þ

The preceding quantity is geometrically interpreted as the projection of vector ðaÞ in the direction of vector ðbÞ. A1.3.3

Cross product

The cross product of two vectors is defined as (Fig. A1.3). ab ¼ c

ðA1:4Þ

The magnitude of this quantity corresponds to the area bounded by vectors ðaÞ and ðbÞ, and the direction of resulting vector (c) is perpendicular to the plane constituted by vectors ðaÞ and ðbÞ: kck ¼ kakkbk siny A1.3.4

ðA1:5Þ

Unit vector

Unit vector is defined as a vector whose magnitude is 1. This vector is utilized to define the vectorial quantities.

5

Appendices

A1.3.5

347

Coordinate systems and base vectors

Fundamentally there are different coordinate systems such as Cartesian, cylindrical, spherical coordinate systems. (See Figure A1.4.) Here, an orthogonal coordinate system (x1, x2, x3) is considered. Three unit vectors (e1, e2, e3) are assumed to be parallel to the coordinate axes, and they are called base-vectors. For the chosen coordinate system, the following relations hold for dot product operations: e1  e1 ¼ 1;

e2  e2 ¼ 1;

e3  e3 ¼ 1;

e1  e2 ¼ 0;

e2  e3 ¼ 0;

e3  e1 ¼ 0

ðA1:6Þ

As for the cross-product, the relations hold among base vectors: e1  e1 ¼ 0 ;

e2  e2 ¼ 0 ;

e1  e2 ¼ e3 ;

e2  e3 ¼ e1 ;

e2  e1 ¼  e3 ;

e3  ε3 ¼ 0 ; e3  e1 ¼ e2 ;

e3  e2 ¼  e1 ;

ðA1:7Þ

e1  e3 ¼  e2

In the chosen coordinate system, vector a can be expressed as follow a ¼ a1 e1 þ a2 e2 þ a3 e3 ¼

3 X

ai ei ¼ ai ei

ði ¼ 1; 2; 3Þ

ðA1:8Þ

ai ¼ a  ei

ðA1:9Þ

i¼1

The component of vector a can be give as follows: a1 ¼ a  e1 ;

A1.4 A1.4.1

a2 ¼ a  e2 ;

a3 ¼ a  e3 ;

Vector operations on a Cartesian coordinate system Addition and subtraction

Addition and subtraction vectors may be given in the following form using base vectors and their components: a  b ¼ ða1  b1 Þ e1 þ ða2  b2 Þ e2 þ ða3  b3 Þ e3 ¼

3 X ðai  bi Þ ei i¼1

¼ ðai  bi Þ ei ði ¼ 1; 2; 3Þ

A1.4.2

ðA1:10Þ

Dot product

The dot product of two vectors may be given in the following form using base vectors and their components: a  b ¼ a1 b1 þ a2 b2 þ a3 b3 ¼

3 X ai bi ¼ ai bi i¼1

ði ¼ 1; 2; 3Þ

ðA1:11Þ

348 Appendices

6

Figure A1.4 Illustration of some of coordinate systems

A1.4.3

Cross product

The cross product of two vectors may be given in the following form using base vectors and their components: a  b ¼ ða2 b3  a3 b2 Þ e1 þ ða3 b1  a1 b3 Þ e2 þ ða1 b2  a2 b1 Þ e3

A1.5

ðA1:12Þ

Tensors of rank n

Generally, tensors of rank n have magnitude and directional components. Although it is difficult to visualize geometrically, examples are stress, strain, elasticity, viscosity relations. A1.5.1

Definition of tensors of rank n

The rank or order of a tensor is defined in terms of the number of independent (nonrepeated) base vectors and magnitude. Examples are as follows: 1

Tensor of second order D ¼ D11 e1 e1 þ D12 e1 e2 þ D13 e1 e3 þ D21 e2 e1 þ D22 e2 e2 þ D23 e2 e3 þ D31 e3 e1 þ D32 e3 e2 þ D33 e3 e3 or

(A1.13a)

D ¼ D11 e1 e1 þ D12 e1  e2 þ D13 e1  e3 þ D21 e2  e1 þ D22 e2  e2 þ D23 e2  e3 þ D31 e3  e1 þ D32 e3  e2 þ D33 e3  e3 (A1.13b)

7

Appendices

349

or D ¼

3 X 3 X

Dij ei ej ¼

i¼1 j¼1

3 X 3 X Dij ei  ej ¼ Dij ei ej ¼ Dij ei  ej

ðA1:13cÞ

i¼1 j¼1

ði ¼ 1; 2; 3Þ; ðj ¼ 1; 2; 3Þ

2

where  is called the tensor product, and ei ej ¼ ei  ej are fundamentally the same and are called dyad. Kronecker Delta Tensor The Kronecker delta tensor is known as the identity tensor, whose normal components have the value of 1 and are given as follows: I ¼ dij ei ej

3

dij ¼ 1i ¼ j dij ¼ 0i 6¼ j

Tensor of rank 3 C ¼

3 X 3 X 3 X

Cijk ei ej ek ¼

i¼1 j¼1 k¼1

3 X 3 X 3 X

Cijk ei  ej  ek

ðA1:15Þ

i¼1 j¼1 k¼1

C ¼ Cijk ei ej ek ¼ Cijk ei  ej  ek 4

ðA1:14Þ

ði ¼ 1; 2; 3Þ; ðj ¼ 1; 2; 3Þ; ðk ¼ 1; 2; 3Þ

ðA1:16Þ

Tensor of rank 4 E ¼

3 X 3 X 3 X 3 3 X 3 X 3 X 3 X X Eijkl ei ej ek el ¼ Eijkl ei  ej  ek  el i¼1 j¼1 k¼1 l¼1

ðA1:17Þ

i¼1 j¼1 k¼1 l¼1

E ¼ Eijkl ei ej ek el ¼ Eijkl ei  ej  ek  el

ði ¼ 1; 2; 3Þ; ðj ¼ 1; 2; 3Þ;

ðA1:18Þ

ðk ¼ 1; 2; 3Þ; ðl ¼ 1; 2; 3Þ

A1.5.2 A1.5.2.1

Tensor operations Multiplication of a tensor with a scalar

The multiplication of a tensor with scalar results in a tensor with the same order, whose components are magnified by the value of scalar value as given here: E ¼ lD A1.5.2.2 1

ðA1:19Þ

Operation of tensor with vectors

The dot product of vector and tensor of rank 2 can be given in the following form: c ¼ a  D ¼ ðai ei Þ  ðDij ei ej Þ ¼ ai Dij ðei  ei Þej ¼ ai Dij ej

ðA1:20Þ

d ¼ D  a ¼ ðDij ei ej Þ  ðaj ej Þ ¼ Dij aj ðei Þej  ej ¼ Dij aj ei

ðA1:21Þ

350 Appendices

2

8

Fundamentally, the resulting products are vectors, having different quantities unless the tensor of rank 2 is symmetric. The dot product of two tensors of rank 2 is written in the following form: C ¼ D  E ¼ ðDij ei ej Þ  ðEjk ej ek Þ ¼ Dij Ejk ei ðej  ej Þek ¼ Dij Ejk ei ek

3

The rank of the resulting tensorial quantity is 2. The double dot product of two tensors of rank 2 is written in the following form: W ¼ D : E ¼

4

ðA1:22Þ

ðDij ei ej Þ : ðEij ei ej Þ ¼ Dij Eij ðei  ei Þðej  ej Þ ¼ Dij Eij

ðA1:23Þ

The resulting quantity is a scalar. Tensor product of two tensors of rank 2 The resulting tensorial quantity of the tensor product of two tensors of rank 2 are written in the following form, and its rank is 4. F ¼ DE ¼

A1.6

ðDij ei ej Þ  ðEkl ek el Þ ¼ Dij Ekl ei ej ek el ¼ Fijkl ei ej ek el

ðA1:24Þ

Matrix representation of tensors

Base vectors may not be utilized when vectors or tensors are expressed. Herein, matrix operations are introduced to represent some vectorial or tensorial operations. A1.6.1

Matrix representation of vectors

A vector can be represented using either a horizontal (1 × 3) or a vertical matrix (3 × 1) as given here: 2 3 a1 6 7 ðA1:25Þ a ¼ ½a1 ; a2 ; a3 ; a ¼ 4 a2 5 a3 The dot product of two vectors may be represented using the matrix operations as given here: 2 3 b1 6 7 ðA1:26Þ a  b ¼ ½a1 ; a2 ; a3 4 b2 5 b3 A1.6.2

Matrix representation of tensors

The tensor of rank 2 may be given in a matrix form as: 2

A11

6 A ¼ 4 A21 A31

A12

A13

3

A22

7 A23 5

A32

A33

ðA1:27Þ

9

Appendices

351

Similarly, the Kronecker delta tensor can be written as: 2

1

6 I ¼ 40 0

0 0

3

7 1 05

ðA1:28Þ

0 1

The dot product of the vector and tensor of rank 2 is expressed using the matrix operations as follows: 2

A11

6 a  A ¼ ½a1 ; a2 ; a3 4 A21 A31 2

A11

6 A  a ¼ 4 A21 A31

A1.7

A12

A13

A12

A13

3

A22

7 A23 5

A32

A33

32

3

a1

A22

76 7 A23 54 a2 5

A32

A33

ðA1:29Þ

ðA1:30Þ

a3

Coordinate transformation 0

0

Let us consider the position of point P in two coordinates systems (oxy, ox y ). (See Figure A1.5.) From the geometry, the following relations can be obtained as follows: " 0 # " #" # cos y sin y x1 x1 ¼ ðA1:31Þ 0 sin y cos y x2 x2 The inverse of the preceding relation can be shown to be: " # " #" 0 # x1 cos y sin y x 1 ¼ 0 x2 sin y cos y x2

ðA1:32Þ

Let us replace the components of the matrix in the following manner: b11 ¼ cos y;

b12 ¼ sin y;

b21 ¼ sin y;

b22 ¼ cos y

ðA1:33Þ

Equation (A1.31) and Equation (A1.32) may be rewritten as follows: 0

x i ¼ bij xj

ðA1:34Þ

0

ðA1:35Þ

xj ¼ bjk x k

If Equation (A1.35) is inserted into Equation (A1.34), the following relation is obtained: 0

0

x i ¼ bij bjk xk0 ¼ dik x k

ðA1:36Þ

352 Appendices

10

Figure A1.5 Coordinate systems

The matrix operation bij bjk ¼ dik fundamentally corresponds to identity tensor and specifically is given as: "

cos y

sin y

sin y

cos y

A1.8

#"

cos y

sin y

sin y

cos y

"

# ¼

1 0

#

0 1

ðA1:37Þ

Derivation

A1.8.1

Derivative of a scalar function

The derivative of a scalar function is given here and is called directional derivation: df ¼

@f @f @f dx1 þ dx2 þ dx @x1 @x2 @x3 3

ðA1:38Þ

This expression can also be written as: df ¼ ð

@f @f @f e þ e þ e Þ  ðdx1 e1 þ dx2 e1 þ dx3 e3 Þ @x1 1 @x2 2 @x3 3

ðA1:39Þ

with the following definitions: r ¼

@ @ @ e þ e þ e @x1 1 @x2 2 @x3 3

dx ¼ dx1 e1 þ dx2 e1 þ dx3 e3

ðA1:40Þ ðA1:41Þ

11

Appendices

353

Equation (A1.39) may also be written in the following form: df ¼ ðrf Þ  dx

A1.8.2

ðA1:42Þ

Divergence

The dot product between the directional derivation operator ðrÞ and a vector ðaÞ results in the following form and is interpreted as the divergence of vector ðaÞ: div a ¼ ðr  a ¼

@ @ @ @a @a @a e1 þ e2 þ e3 Þ  ða1 e1 þ a2 e1 þ a3 e3 Þ ¼ 1 þ 2 þ 3 @x1 @x2 @x3 @x1 @x2 @x3 ðA1:42Þ

A1.8.3

Rotation

The cross product between directional the derivation operator ðrÞ and a vector ðaÞ results in the following form and is interpreted as the rotation of vector ðaÞ:       @a3 @a2 @a1 @a3 @a2 @a1    ðA1:43Þ e þ e þ e curl a ¼ r  a ¼ @x2 @x3 1 @x3 @x1 2 @x1 @x2 3

A1.8.4

Gradient of a vector: second-order tensor

The tensor product between the directional derivation operator ðrÞ and a vector ðaÞ can be carried out in two ways and results in the following form. It is interpreted as the gradient of vector ðaÞ: D ¼ ra ¼

@ @ @ e þ e þ e Þða e þ a2 e1 þ a3 e3 Þ @x1 1 @x2 2 @x3 3 1 1

¼

@a1 @a @a @a @a @a @a @a @a e e þ 2e e þ 3e e þ 1e e þ 2e e þ 3e e þ 1e e þ 2e e þ 3e e @x1 1 1 @x1 1 2 @x1 1 3 @x2 2 1 @x2 2 2 @x2 2 3 @x3 3 1 @x3 3 2 @x3 3 3

¼

@aj e e ¼ aj;i ei ej @xi i j

ðA1:44aÞ

@ @ @ E ¼ a r ¼ ða1 e1 þ a2 e1 þ a3 e3 Þð e þ e þ eÞ @x1 1 @x2 2 @x3 3 @a1 @a @a @a @a @a @a @a @a e e þ 1e e þ 1e e þ 2e e þ 2e e þ 2e e þ 3e e þ 3e e þ 3e e @x1 1 1 @x2 1 2 @x3 1 3 @x1 2 1 @x2 2 2 @x3 2 3 @x1 3 1 @x2 3 2 @x3 3 3

¼

@ai e e ¼ ai;j ei ej @xj i j

ðA1:44bÞ

354 Appendices

A1.8.5

12

Divergence of a tensor (second-order tensor)

The dot product between the directional derivation operator ðrÞ and a tensor ðDÞ of rank 2 results in the following form and is interpreted as the divergence of a tensor ðDÞ of rank 2: f ¼ ðr  D ¼

@ @ @ e þ e þ e Þ  ðD11 e1 e1 þ D12 e1 e2 þ D13 e1 e3 þ @x1 1 @x2 2 @x3 3

D21 e2 e1 þ D22 e2 e2 þ D23 e2 e3 þ D31 e3 e1 þ D32 e3 e2 þ D33 e3 e3 Þ

ðA1:45aÞ

or ð

@Dji @ ej Þ  ðDji ej ei Þ ¼ e ¼ Dji;j ei @xj @xj i

ðA1:45bÞ

Appendix 2

Stress analysis

A2.1

Definition of stress vector

Stress vector is defined as the limit of an infinitesimal force acting over an infinitesimal area with a unit normal vector as given here (Fig. A2.1(a)): tðnÞ ¼ lim

DS!0

Df df ¼ DS dS

ðA2:1Þ

Newton’s action-and-reaction law requires at a given surface within a body of equilibrium (Fig. A2.1(b)): tðnÞ  tðnÞ ¼ 0 The stress vector is also known as the traction vector.

Figure A2.1 Illustration of (a) traction vector, (b) unit normal vectors

ðA2:2Þ

356 Appendices

14

Figure A2.2 Illustration of stress tensor components

A2.2

Stress tensor

A stress vector acting on a cubic body results in nine components of the second-order stress tensor (Fig. A2.2). In two-dimensional space, the stress tensor has four components. It is given as using tensorial notation: σ ¼ sij ei ej

ðA2:3Þ

The first and second subscripts of a component of stress tensor correspond to the surface and the axis, respectively.

A2.3

Relationship between stress vector and stress tensor: Cauchy relation

Cauchy states that the stress vector and stress tensor can be related to each other in the following form: t ¼ n  σ ¼ sji nj ei

ðA2:4Þ

To derive this relation, let us consider a two-dimensional body in equilibrium. The effect of the top part is taken into account as the traction acting on the plane with surface area Δs (see Figure A2.2.). The unit normal and traction vectors and stress tensor can be given as: n ¼ n1 e1 þ n2 e2

ðA2:5Þ

t ¼ t1 e1 þ t2 e2

ðA2:6Þ

σ ¼ s11 e1 e1 þ s12 e1 e2 þ s21 e2 e1 þ s22 e2 e2

ðA2:7Þ

where n1 ¼ cos y; n2 ¼ cosð90  yÞ ¼ sin y

15

Appendices

357

Figure A2.3

The force equilibrium in directions x1 and x2 can be written as: x1-direction X F x ¼ t1 Ds  s11 n1 Ds þ s21 n2 Ds ¼ 0; t1 ¼ s11 n1 þ s21 n2

ðA2:8aÞ

x2-direction X F x ¼ t2 Ds  s12 n1 Ds þ s22 n2 Ds ¼ 0; t2 ¼ s12 n1 þ s22 n2

ðA2:8bÞ

1

2

Utilizing Equations (A2.5) to (A2.7) together with Equation (A2.8), the following relation can be written: t ¼ ðn1 e1 þ n2 e2 Þ  ðs11 e1 e1 þ s12 e1 e2 þ s21 e2 e1 þ s22 e2 e2 Þ

ðA2:9aÞ

t ¼ n  σ ¼ sji nj ei ð j ¼ 1; 2; i ¼ 1; 2Þ

ðA2:9bÞ

or

Thus Equation (A2.9) corresponds to Cauchy’s relation. Normal and shear components of traction vector on the plane can be given by the following relations: sN ¼ t  n

ðA2:11Þ

sS ¼ t  s

ðA2:12Þ

Where s ¼ s1 e1 þ s2 e2 ; s1 ¼ sin y; s2 ¼ cos y

ðA2:13Þ

358 Appendices

A2.4

16

Stress transformation

Stress transformation becomes necessary when stress tensors are to be related to each other in different coordinate systems. This transformation law may be derived by requiring the stress vector to be the same independent of the coordinate system. Let us introduce coordinate systems ox1 x2 x3 and ox1 0 x2 0 x3 0 together with stress tensors (sij and s0km ), as shown in Figure A2.4. Thus, the following relation may be written: t ¼ sji nj ei ¼ s0km n0k e0m

ðA2:14Þ

Taking the dot products of the both sides of Equation (A2.14) with e0m yields: sji nj ðe0m  ei Þ ¼ s0km n0k

or

sji nj bmi ¼ s0km n0k

ðA2:15Þ

The unit normal vector on an undashed coordinate system can be related to a dashed coordinate system through the transformation law: nj ¼ bkj n0k

ðA2:16Þ

Inserting Equation (16) into Equation (15) yields the following relation: ðsji bkj bmi  s0km Þn0k ¼ 0

ðA2:17Þ

As n0k is arbitrary chosen, Equation (17) requires the following identity: s0km ¼ sji bkj bmi

ðA2:18Þ

Equation (18) may also be represented in the matrix form as follows: ½s0  ¼ ½b½s½b

T

Figure A2.4 Stress components in two coordinate systems

ðA2:19Þ

17

Appendices

359

Using a similar procedure, the stress tensor in undashed system can be related to that in dashed system as follows: sij ¼ s0km bjk bim

ðA2:20Þ

In matrix form, it is written as: ½s ¼ ½b ½s0 ½b

ðA2:21Þ

T

A2.5

Principal stresses, stress invariants

The stress tensor can be represented by three orthogonal components as shear stress components disappear at the planes on which principal stresses act. Thus, principal stresses can be related to the stress tensor on a given coordinate system by requiring the stress vectors for both situations to be equivalent: t ¼ sji nj ei ¼ sni ei

or

ðsji nj  sni Þei ¼ 0

ðA2:22Þ

with the following relation: ni ¼ dij nj

ðA2:23Þ

Equation (A2.22) can be rewritten as: ðsji  sdij Þnj ei ¼ 0

ðA2:24Þ

Equation (A2.24) requires the following condition to be satisfied: jsji  sdij j ¼ 0

ðA2:25Þ

Taking the determinant given in Equation (A2.25) yields the following relation, which yields three roots that correspond to the values of principal stresses: s3  I1 s2 þ I2 s  I3 ¼ 0

ðA2:26Þ

where I1 ¼ sii ¼ trðσÞ ¼ s11 þ s22 þ s33 ; I2 ¼ 12 ðsii sjj  sij sij Þ; I3 ¼ jsij ¼ detðσÞj ði ¼ 1; 2; 3; j ¼ 1; 2; 3Þ

A2.6

Representation of stress tensor on Mohr Circle for 2-D condition

Mohr (1882) devised a method to represent graphically the stress components using the Mohr-Circle method (Fig. A2.5): " 0 # " #" 0 #" # s11 s012 s11 s012 b11 b12 b11 b21 ¼ ðA2:27Þ s021 s022 s021 s022 b21 b22 b12 b22

360 Appendices

18

Figure A2.5 Stress components in two-coordinate systems

where "

b11

b12

b21

b22

#

" ¼

c s

s

#

1 ; c ¼ cosy; s ¼ siny; cos2 y ¼ ð1 þ cos 2yÞ 2 c

1 sin2 y ¼ ð1  cos 2yÞ; sin2y ¼ 2cosy siny 2

ðA2:28aÞ

ðA2:28bÞ

Carrying out the matrix operation given in Equation (A2.27) together with Equation (A2.28) and the symmetry property of the stress tensor, one easily gets: s011 ¼ s11 c2 þ s22 s2 þ 2s12 cs

or

s011 ¼

s022 ¼ s11 s2 þ s22 c2  2s12 cs

or

s022 ¼

s012 ¼ ðs22  s11 Þcs þ s12 ðc2  s2 Þ

or

s11 þ s22 s11  s22 þ cos2y þ s12 sin2y 2 2 ðA2:29aÞ s11 þ s22 s11  s22  cos2y  s12 sin2y 2 2 ðA2:29bÞ s012 ¼ 

s11  s22 sin2y þ s12 cos2y 2 ðA2:29cÞ

As the shear stress should disappear in order to obtain principal stresses, the angle of rotation should take the following form from Equation (A2.29c):   1 2s12 y ¼ tan1 2 s11  s22

ðA2:30Þ

19

Appendices

361

Figure A2.6 Graphical representation of stress tensor components on Mohr’s circle

Furthermore, one can also derive the following relations for trigonometric relations in terms of stress tensor components: 1 s  s22 s12 ffi ffi 11 cos 2y ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sin 2y ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s11 s22 2   2 s11 s22 2 2 2 þ s þ s 12 12 2 2

ðA2:31Þ

Using Equation (A2.31) in Equations (A2.29a) and (A2.29b), one can easily obtain the following relations for principal stresses: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r s11 þ s22 s11  s22 2 0 2 s11 ¼ sI ¼ þ þ s12 ; s022 ¼ sII 2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r s11 þ s22 s11  s22 2 2  ¼ þ s12 ðA2:32Þ 2 2 Accordingly, the maximum shear stress is obtained using Equation (A2.32) as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s  s 2 sI  sII 11 22 2 tmax ¼ ¼ þ s12 2 2

ðA2:33Þ

Figure A2.6 shows the graphical presentation of the components of stress tensor and principal stresses on Mohr’s circle for the 2-D condition.

Appendix 3

Deformation and strain

A3.1

Preliminaries

Let us consider a body in the space of two coordinate systems represented by OX1 X2 and ox1x2, as shown in Figure A3.1. Coordinate system OX1X2 is introduced at time step (t = A) before the deformation of the body. On the other hand, the coordinate system ox1x2 is introduced at time step (t = B) after the deformation of the body. Let us also introduce base vectors Ei, ei associated with each coordinate system. If we describe the deformation of the body using the coordinate system OX1X2, it is called a Lagrangian description. On the other hand, if we describe the deformation of

Figure A3.1 Coordinate system and notations

364 Appendices

22

the body using the coordinate system, ox1x2, it is called a Eulerian description. Herein, the derivation of strain tensors is derived in a two-coordinate system. Let us introduce the following preliminary relations: Lagrangian description Position vectors X ¼ X1 E1 þ X2 E2

ðA3:1Þ

x ¼ x1 E1 þ x2 E2

ðA3:2Þ

Displacement vector u ¼ u1 E1 þ u2 E2

ðA3:3Þ

Eulerian description Position vectors X ¼ X1 e1 þ X2 e2

ðA3:4Þ

x ¼ x1 e1 þ x2 e2

ðA3:5Þ

Displacement vector u ¼ u1 e1 þ u2 e2

ðA3:6Þ

The relation position vectors before and after deformation may be given as: x¼X þ u

ðA3:7Þ

dx ¼ dX þ du

ðA3:8Þ

Tensorial operation ðabÞ  c ¼ aðb  cÞ

A3.2

ðA3:9Þ

Derivation of strain tensor using Lagrangian description

In this subsection, strain tensor is derived using the Lagrangian description. The position vector of an arbitrary point in the body after deformation using the coordinate system OX1X2 may be given as follows: x1 ¼ x1 ðX1 ; X2 Þ;

x2 ¼ x2 ðX1 ; X2 Þ

ðA3:10Þ

Points P and Q in the body before deformation move to a new position denoted by P0 and Q0 . The power of length before and after deformation can be written as: dS 2 ¼ dX  dX ¼ dX1 dX1 þ dX2 dX2 ¼ dXi dXi ; ds2 ¼ dx  dx ¼ dx1 dx1 þ dx2 dx2 ¼ dxi dxi ;

i ¼ 1; 2

i ¼ 1; 2

ðA3:11Þ ðA3:12Þ

23

Appendices

365

One may write the following relation for length change: ds2  dS 2 ¼ dx  dx  dX  dX

ðA3:13aÞ

Or ds2  dS 2 ¼ ðdx1 dx1 þ dx2 dx2 Þ  ðdX1 dX1 þ dX2 dX2 Þ ¼ dxi dxi  dXi dXi ;

i ¼ 1; 2 ðA3:13bÞ

The length vector after deformation can be given in Lagrangian description as follows:   @x1 @x1 @x2 @x2 EE þ EE þ EE þ E E  ðdX1 E1 þ dX2 E2 Þ dx ¼ rX x  dX ¼ @X1 1 1 @X2 1 2 @X1 2 1 @X2 2 2 ðA3:14Þ or  dx ¼ F  dX ¼

   @x1 @x1 @x2 @x2 dX þ dX E þ dX þ dX E @X1 1 @X2 2 1 @X1 1 @X2 2 2

ðA3:15Þ

where F ¼

@x1 @x @x @x E1 E1 þ 1 E1 E2 þ 2 E2 E1 þ 2 E2 E2 @X1 @X2 @X1 @X2

ðA3:16Þ

F is called a deformation gradient. Variation of the displacement vector is similarly given as: du ¼ rX u  dX   @u1 @u1 @u2 @u2 ¼ EE þ EE þ EE þ E E  ðdX1 E1 þ dX2 E2 Þ @X1 1 1 @X2 1 2 @X1 2 1 @X2 2 2 ðA3:17Þ or  du ¼ H  dX ¼

   @u1 @u @u2 @u dX1 þ 1 dX2 E1 þ dX1 þ 2 dX2 E2 @X1 @X2 @X1 @X2

ðA3:18Þ

where H¼

@u1 @u @u @u E E þ 1E E þ 2E E þ 2E E @X1 1 1 @X2 1 2 @X1 2 1 @X2 2 2

ðA3:19Þ

H is called the displacement gradient tensor. If the preceding relation is inserted into Equation (A3.13), the following relation may be written: ds2  dS 2 ¼ ð F  dXÞ  ð F  dXÞ  dX  dX ¼ ðdX  Fc Þ  ð F  dXÞ  dX  dX ðA3:20Þ

366 Appendices

24

provided that: Fc ¼ FT ;

Ic ¼ I ;

dX ¼ I  dX

ðA3:21Þ

Equation (A3.20) may be rewritten as: ds2  dS 2 ¼ dX  ðFc  FÞ  dX  dX  ðIc  IÞ  dX ¼ dX  ðFc  F  Ic  IÞ  dX ðA3:22Þ If we use the following relations: dx ¼ dX þ du ¼ I  dX þ H  dX ¼ ðH þ I Þ  dX

ðA3:23Þ

Fc  F ¼ FT  F ¼ ðH þ IÞ  ðH þ IÞ

ðA3:24Þ

ðH þ IÞ ¼ HT þ IT ¼ HT þ I

ðA3:25Þ

FT  F ¼ HT  H þ HT  I þ I  H þ I  I ¼ HT  H þ HT þ H þ I

ðA3:26Þ

T

T

Lagrangian Strain tensor is defined as: 1 1 L ¼ ½FT  F  I ¼ ½H þ HT þ HT  H 2 2

ðA3:27aÞ

1 T T L ¼ ½ðrX uÞ þ ðrX uÞ þ ðrX u Þ  ðrX u Þ 2

ðA3:27bÞ

or

The Lagrangian strain tensor is also known the Green strain tensor. It is interpreted as a finite strain tensor. In index notation, it is expressed as follows: " # 1 @uj @uk @ui @ui 1 ¼ ½uj;k þ uk;j þ ui;j ui;k  þ þ ðA3:28Þ Ljk ¼ 2 @Xk @Xj @Xj @Xk 2

A3.3

Derivation of strain tensor using Eulerian description

In this subsection, strain tensor is derived using Eulerian description. The position vector of an arbitrary point in the body after deformation using the coordinate system ox1x2 may be given as follows: X1 ¼ X1 ðx1 ; x2 Þ;

X2 ¼ X2 ðx1 ; x2 Þ

ðA3:29Þ

The length vector before deformation can be given in Eulerian description as follows:   @X1 @X1 @X2 @X2 ee þ ee þ ee þ e e  ðdx1 e1 þ dx2 e2 Þ dX ¼ rx X  dx ¼ @x1 1 1 @x2 1 2 @x1 2 1 @x2 2 2 ðA3:30Þ

25

Appendices

367

or  dX ¼ J  dx ¼

   @X1 @X1 @X2 @X2 dx þ dx e þ dx þ dx e @x1 1 @x2 2 1 @x1 1 @x2 2 2

ðA3:31Þ

where J ¼

@X1 @X @X @X e1 e1 þ 1 e1 e2 þ 2 e2 e1 þ 2 e2 e2 @x1 @x2 @x1 @x2

ðA3:32Þ

J is denoted as the deformation gradient tensor. The variation of displacement is given using Eulerian description as follows:  du ¼ rx u  dx ¼

@u1 @u @u @u e1 e1 þ 1 e1 e2 þ 2 e2 e1 þ 2 e2 e2 @x1 @x2 @x1 @x2

  ðdx1 e1 þ dx2 e2 Þ ðA3:33Þ

or  du ¼ K  dx ¼

   @u1 @u1 @u2 @u2 dx þ dx e þ dx þ dx e @x1 1 @x2 2 1 @x1 1 @x2 2 2

ðA3:34Þ

where K ¼

@u1 @u @u @u e1 e1 þ 1 e1 e2 þ 2 e2 e1 þ 2 e2 e2 @x1 @x2 @x1 @x2

ðA3:35Þ

K is the denoted displacement gradient in Eulerian description. If it is inserted into Equation (A3.13), the following relation may be written as: ds2  dS 2 ¼ dx  dx  ðJ  dx Þ  ðJ  dx Þ ¼ d x  dx  ðdX  Jc Þ  ðJ  dXÞ ðA3:36Þ provided that Jc ¼ JT ; Ic ¼ I ;

dx ¼ I  dx

ðA3:37Þ

Equation (A3.36) can be rewritten as: ds2  dS 2 ¼ dx  ðIc  IÞ  dx  dx  ðJc  JÞ  dx ¼ dx  ð Ic  I  Jc  J Þ  dx ðA3:38Þ

368 Appendices

26

Introducing the following relations: dX ¼ dx  du ¼ I  dx  K  dx ¼ ð I  KÞ  dx

ðA3:39Þ

Jc  J ¼ JT  J ¼ ðI  KÞ  ðI  KÞ

ðA3:40Þ

ðI  KÞ ¼ IT  KT ¼ I  KT

ðA3:41Þ

JT  J ¼ KT  K  KT  I  I  K þ I  I ¼ KT  K  KT  K þ I

ðA3:42Þ

T

T

the Eulerian strain tensor is defined as: 1 1 E ¼ ½I  JT  J  ¼ ½K þ KT  KT  K  2 2

ðA3:43aÞ

1 T T E ¼ ½ðrx uÞ þ ðrx u Þ  ðrx uÞ  ðrx u Þ 2

ðA3:43bÞ

or

The Eulerian strain tensor is also known the Almani strain tensor, and it is a finite strain tensor used for the large deformation of materials. In index notation, it is rewritten as: " # 1 @uj @uk @ui @ui 1 ¼ ½uj;k þ uk;j  ui;j ui;k  þ  Ejk ¼ 2 @xk @xj @xj @xk 2

A3.4

ðA3:44Þ

Relation between small strain theory and finite strain theory

As noted from the strain definitions given by (A3.28) and (A3.44), the term corresponding to the power of strain components is noted. Therefore, the finite strain tensors are geometrically nonlinear, and their use in practice becomes troublesome. When the strain is small, say, less than 10%, the nonlinear components may be omitted. Furthermore, if coordinate systems are assumed to be the same, say, x = X, strain tensors given by (A3.28) and (A3.44) reduced to the following form: " # 1 @uj @uk 1 ¼ ½uj;k þ uk;j  þ Ejk ¼ Ljk ¼ 2 @xk @xj 2

ðA3:45Þ

27

Figure A3.2 One-dimensional normal deformation

Figure A3.3 Simple shear deformation

Appendices

369

370 Appendices

A3.5 A3.5.1

28

Geometrical interpretations of strain tensor Uniaxial deformation

Let us consider a body deformed uniaxially as shown in Figure A3.2. Length change for this example may be written as: ds2  dS 2 ¼ 2L11 dX12

ðA3:46Þ

Provided that: ds2 ¼ ðdX1 þ du1 Þ þ dX22 ; dS 2 ¼ dX12 þ dX22 2

dX1 du1 þ du21 ¼ 2L11 dX12

ðA3:47Þ

and du21  0, one can obtain the following: du1 ¼ L11 dX1 A3.5.2

ðA3:48Þ

Simple shear deformation

Let us consider a body deformed in simple shear as shown in Figure A3.3. Length change for this example may be written as: ds2  dS 2 ¼ 2L12 dX1 dX2

ðA3:49Þ

Provided that: ds2 ¼ ðdX1 þ du1 Þ þ dX22 ; dS 2 ¼ dX12 þ dX22 2

dX1 du1 þ du21 ¼ 2L12 dX1 dX2

ðA3:50Þ

and du21  0, the following relation may be written: du1 ¼ L12 dX2 This relation can be easily interpreted as angle variation

ðA3:51Þ

Appendix 4

Gauss divergence theorem

A4.1

One-dimensional (1-D) Gauss theorem

Gauss theorem is written as for 1-D case: Z Z @f dO ¼ fn dG O @x G

ðA4:1Þ

The preceding expression is explicitly written as: Z

@f ðxÞ DxDyDz ¼ O @x

Z ðf  nÞjx¼a DyDz x¼b

G

ðA2:2Þ

As na ¼ cos 180 ¼ 1, nb ¼ cos 0 ¼ 1 and f ðx ¼ aÞ ¼ fa , f ðx ¼ bÞ ¼ fb , the preceding expression is rewritten as: Z Z Z @f ðxÞ DxDyDz ¼ fb DyDz  fa DyDz ðA3:3Þ O @x G G Figure A4.1 illustrates the geometrical interpretation of the Gauss divergence theorem.

Figure A4.1 Geometrical illustration of Gauss divergence theorem

372 Appendices

A4.2

30

Three-dimensional (3-D) Gauss theorem

To get 3-D version of Equation (A4.1), let us introduce the replacements: @ @ @ @ ! r ¼ ex þ ey þ ez @x @x @y @z

ðA4:4Þ

n ! n ¼ nx ex þ ny ey þ nz ez

ðA4:5Þ

where nx ¼ cos a, ny ¼ cos b and nz ¼ cos g. If the integrand is a scalar function ( f), Equation (A4.1) takes the following form in the 3-D case: Z Z rfdO ¼ f ndG ðA4:6Þ O

G

If the integrand is a vector (v), Equation (A4.1) takes the following form in the 3-D case: Z Z r  vdO ¼ v  ndG ðA4:7Þ O

G

If the integrand is a tensor (σ), Equation (A4.1) takes the following form in the 3-D case: Z Z r  σdO ¼ σ  ndG ðA4:8Þ O

G

Appendix 5

Geometrical interpretation of Taylor expansion

A scalar function  at a given coordinate x + Δx can be expressed using the Taylor expansion as: φxþDx ¼ φx þ

@φ @2φ @ ðnÞ φ Dx þ 2 Dx2 þ    ðnÞ Dxn þ    @x @ x @ x

ðA5:1Þ

The first term on the right-hand side is the value of function  at position (x). The second term involves the gradient of function  at position (x) multiplied by the position increment Δx, which corresponds to Δ, which is the increment of function . As noted from the figure, there is deviation between the exact value at x + Δx. If the higher terms of function  are possible, the use of higher-order derivatives is expected to yield better estimations. However, the linear term is often utilized in the derivation of governing equations in many applications of mechanics. Therefore, the mechanics are called linear mechanics.

Figure A5.1

Appendix 6

Reynolds transport theorem

Inserting the time derivation operator into the integral operator: d dt

Z

Z

dðÞ dO þ ðÞdO ¼ O O dt

Z

dðdOÞ ¼ ðÞ dt O

Z  O

 dðÞ þ ðÞr  v dO dt

ðA6:1Þ

Equation (A6.1) is also known the Reynolds transport theorem. The time derivative of infinitely small volume dΩ takes the following form in 3-D and 1-D: In 3-D dO ¼ JdOo ¼ ðrxo  xÞdOo

ðA6:2Þ

where J is Jacobian. In 1-D dx ¼

@x dx ¼ Jdxo @xo o

ðA6:3Þ

Detailed supplementary explanation for Equation (A6.3)     dðdxÞ d @x d @x @x dðdxo Þ ¼ dxo ¼ dxo þ dt dt @xo dt @xo @xo dt

ðA6:4Þ

The time derivative of the second term is nil as the initial control element length (dxo) is constant. Thus we have:     dðdxÞ d @x @ dx @v @v @x @v ¼ dxo ) dx ) dx ) dx dxo ) dt dt @xo @xo dt @xo o @x @xo o @x

ðA6:5Þ

376 Appendices

34

Therefore, we may write the following relations given in 1-D and generalized to 3-D versions as follow: 1-D to 3-D @ )r @x

ðA6:6aÞ

v)v

ðA6:6bÞ

dx ) dO

ðA6:6cÞ

@v dx ) ðr  vÞdO @x

ðA6:6dÞ

Index

action 4, 14, 16, 102, 300; chemical 4; fluid 16; infiltrating water 14; physical 4; pressure 16; relative slip 102; springs 300; temperature 16 airflow 54 analytical solution 125, 132, 188, 198, 200, 203, 215, 221, 224, 226, 228, 241, 245; circular hole 224; diffusion problems 245; equation 198; fluid flow 228; heat flow 241; solids 188, 198 atmospheric 138, 143, 146, 239, 245, 329; pressure 138, 143, 146, 239, 329; temperature 245 axisymmetric 53, 101; radial flow 53; rock sample 101 bar 125, 146, 198, 211, 299; divided 125, 146; one-dimensional 198; radius 211; steel 299 Barton, N. 29, 41, 111, 113, 145, 301 behavior 3, 4, 38, 47, 61, 68, 96, 104, 107, 120, 146, 153, 157, 162, 180, 191, 204, 213, 215, 220, 221, 242, 298, 299, 301, 333; brittle 107, 153; creep 120, 146; ductile 107, 153; elastic 153, 220, 333; elasto-plastic 68, 155, 157, 162, 213, 220, 299, 301, 333; hydromechanical 47; mechanical 3, 4, 61, 153, 157, 162; post-failure 104; time dependent 215; visco-elastic 214 Bieniawski, Z. 121, 145, 162, 183, 214, 255 blasting 167, 223 body force 44, 45, 51, 58, 213, 266, 273, 280 borehole 36, 37, 38, 40, 60 breakout 147, 162, 163, 167, 179, 181, 185, 223, 225 Brekke, T.L. 301 Brown, E.T. 3, 5, 83–87, 96, 99, 145, 147, 166– 168, 172, 183, 204, 213, 221–224, 255 cavern 2, 93, 176, 182, 301 cavity 158, 166, 200, 203, 219–221; circular 166, 200, 203, 221; spherical 40, 203, 219, 220

chemical reaction 332 condition 2, 4, 9, 16, 19, 35, 41, 49, 55, 57, 64, 69, 73, 76, 77, 80, 86, 99, 102–106, 111, 113, 118, 119, 128, 129, 132, 133, 135–138, 141–143, 146, 147, 150, 151, 154, 162, 165, 168, 172, 177, 187, 188, 191, 196–198, 201–204, 206–207, 212, 215, 216, 221, 224, 225, 229, 232, 235, 237- 239, 243–244, 247, 250–251, 257–258, 262, 268, 271–273, 278–280, 288, 295, 297–298, 301, 306, 308, 315, 327, 333–335, 339, 341, 359, 361; alkaline 9; boundary 102, 103, 136, 141, 146, 187, 196–198, 201–203, 206–207, 212, 215, 221, 229, 232, 237, 239, 251, 257–258, 262, 268, 273, 279, 288, 295, 297; climatic 4, 308, 327; compatibility 204; consistency 64, 73, 77, 80; continuity 151, 207; creeping 243, 244; drained 9; dynamic 2, 50, 106; environmental 19, 99, 113, 247, 315, 333–334; flow 132; gravitational 341; ill-conditioning 301; initial 133, 135, 137, 138, 142, 143, 191, 198, 221, 235, 238, 247, 262, 268, 273, 279–280, 288; laboratory 118, 162, 172; loading 105, 106, 215; pure-shear 165; saturated 119; slip 306; spherical-symmetric 333; steady-state 76; stress 188, 215, 225, 333; undrained 49; unstrained 129; three-point bending 104; yield 86 Cook, N.G.W. 3, 5, 77, 86, 96, 102, 145, 151, 163, 165–167, 184, 187–188, 219, 221, 223, 255, 308 creep 65–68, 71, 79–80, 91, 96–97, 105, 117– 124, 145–146, 214, 251, 256, 266; Brazilian 119; compression 119; device 105, 119, 120; experiment 80; failure 214; impression 119–120; in-situ 121–122; plate-bearing 122; primary 124; secondary 124, 214; shear 105, 119–120, 124; steady-state 65, 67; strength 214; tertiary 214; test 71, 117–119, 123, 214; transient 65, 67, 214

378 Index dam 309 damping 198, 200, 297, 300, 301 degradation 9, 308 diffusion 43, 51, 54, 247–248, 251, 268, 298; coefficient 54, 248; constant 247; equation 251; flux vector 51; phenomenon 51; problem 53; process 247 discontinuity 19, 20–23, 26–29, 34–36, 38–39, 93, 114, 116–117, 152, 230, 232–233, 240, 298–300, 335, 338; aperture 232, 240; filling 328; formation 19; friction angle 114; natural 47–49; persistency 27; plane 116– 117; orientation 26; permeability 233, 240; porosity 233; saw-cut 116, 335, 338; set 27, 38–39; shear 29; spacing 27, 39; surface morphology 28, 36 displacement transducer 115, 247 drilling 158, 162, 165, 181, 323, 339 earthquake 5, 145, 147, 164, 169, 172, 174– 184, 242, 245, 254, 309, 311, 314, 318–319, 335; 1891 Nobi-Beya 175–176, 184; 1953 Edirne 174; 1998 Adana-Ceyhan 183; 2003 Miyagi-Hokubu 182; 2015 Gorkha 309, 311, 314; focal mechanism 147; Great East Japan Earthquake 174–176, 181–184; prediction 147; subduction 175 effect 5, 36, 86–87, 102, 145, 159, 161–162, 166–167, 183–185, 191–195, 198, 200, 209, 212–213, 219, 221, 223–224, 274, 299, 315, 329, 341, 356; anisotropy 224; Body forces 213; corner 82; damping 200; discontinuity 299; elasticity 191, 193; filling 36; frictional 102; green-house 329; Kaiser 159, 161–162, 183–184; loading rate 192, 194; pressure 212; reinforcement 209; rotational 106; sample height 192, 195; scale 185; temperature 87, 167, 223; Tensile yielding 219; viscosity 191–192, 198; weathering 315, 341 element 4, 10, 43, 47, 59, 65–67, 90, 97, 101–102, 122–124, 144, 154–157, 168, 172, 180, 187, 194, 224, 241, 251, 254, 257–302, 330, 375; boundary 187, 257, 300–301; cubic 43, 47; finite 59, 97, 101–102, 144, 154–157, 168, 172, 184, 187, 224, 241, 251, 257–302; Hookean 65, 67; infinitesimal 194, 251, 254; Kelvin 65, 67; Maxwell 67; representative 90, 93 equation of motion 54, 102, 198, 262, 287–288, 297–298 Eringen, A.C. 43, 52, 60, 193–194, 255 Ersen, A. 254, 298, 320 experiment 4, 35, 61, 68, 80, 83–89, 97, 99– 103, 110–113, 115–122, 124–125, 128–130, 132–133, 135, 139, 140, 144, 146, 153, 160, 162, 183, 198, 217, 219, 228, 233–234,

378 247, 248, 335; bending 99; Brazilian 99; compression 84, 86–87, 97, 99–103, 146; creep 80, 118–119; drying 247; dynamic friction 335; impression 217, 219; in-situ 122; laboratory 183; seismic 320; shear 110– 113; tilting 111, 113, 116–117; trapdoor 198 fault 19–21, 89, 93, 110, 147, 163–165, 169–179, 182–185, 241–245, 256, 323–324, 331–332 Fick's law 53, 61–62, 75, 247 flow rule 68, 72–73, 77, 80–81 foundation 3, 19, 23, 97, 122, 124, 183, 217, 301 friction angle 23, 82, 110–115, 121, 144, 151, 165–166, 170, 175, 205, 222, 335, 339 Gauss divergence theorem 154 geoengineering 5, 59, 144, 182, 255 Goodier, J.N. 217, 221, 256 Hinton, E. 77, 83, 97, 155, 184, 298, 302 Hoek, E. 3, 5, 83–87, 96, 147, 166–168, 172, 183, 204, 213, 221–224, 255 hydraulic conductivity 50 Ichikawa, Y. 23, 40, 96, 185, 254, 341 Ikeda, K. 38, 41 Inglis, C.E. 221, 255, 256 iteration scheme 299 Ito, F. 5, 145, 183, 255 Jaeger, J.C. 3, 5, 77, 86, 96, 102, 125, 145, 151, 163, 165–167, 184, 187–188, 219, 221, 223, 255 Kirsch, G. 162, 163, 166, 184, 187, 221, 225, 255 Kreyszig, E. 137, 146, 187–188, 196, 256 Kumsar, H. 2, 5, 183, 254, 255, 256 Ladanyi, B. 215, 256 laser 7, 29, 31, 115, 162, 247, 317, 323; profiling 29; scanning 27, 31, 162; technology 27, 31; transducer 115, 247, 317, 323 law 43–53, 55–56, 73, 75, 122–124, 128, 187, 193, 228, 266, 268, 272–273, 280, 298, 300, 305, 355, 358; constitutive 46–48, 54, 266, 268, 280–281, 288, 298, 300, 305, 355, 358; Darcy 48, 50, 52 61; Effective stress 49, 280; Energy conservation 45, 47, 52, 59; Fick 53, 61, 75, 247; Fourier 46, 61–62, 75, 128, 187, 228, 241; friction 2, 4; mass conservation 43, 45–49, 51–53, 55, 268, 272–273; momentum conservation 44, 51, 55–59; transformation 358; Voigt-Kelvin 73, 73, 122–124 lining 254, 256, 301; design 268–269

233 measurement 26, 38, 40, 115–116, 145, 157, 162, 178, 182, 247, 266, 296, 319; displacement response 115–116, 266, 296; field 40, 145; ın-situ 157, 162, 178, 182; wave velocity 26, 38, 247, 319 method 257–258, 265, 270, 59, 101, 168, 224, 241, 251, 257–260, 265, 270, 272, 276, 285, 293, 298–299; discrete finite element 298–299; finite difference 257–258, 265, 270, 276, 285, 293; finite element 59, 101, 168, 224, 241, 251, 257, 259–260, 263, 272, 298– 299, 301; secant 298; tangential stiffness 298 minerals 4, 7–10; calcite 10; clay 8–9; dolomite 10; evaporate 10; feldspar 9; gypsum 10; halite 10; pyrite 10; Rock-forming 4, 10; silicate 7–9; Non-silicate 7, 9 modulus 102, 122, 188, 193, 201, 203, 211, 219, 247, 252–253, 278, 298, 305; deformation 122, 219, 252–253, 298; elastic 102, 188, 201, 203, 211, 219, 247, 278; shear 193, 211, 253, 305 mudstone 15, 17–18, 53, 308, 323 Müller, L. 3, 5, 301 Nawrocki, P. 72, 80–81, 95, 214, 254 nuclear waste 3, 169 numerical analysis 89, 301 overburden 165, 172, 208 Owen 77, 83, 97, 155, 184, 298, 302 Oya tuff 83–84, 97, 110 permeability 50, 52, 59, 62, 132–133, 135, 138–139, 143–145, 230–235, 238–241, 255–256, 308 prismatic block 21 ratio 76, 100, 102, 122, 145, 148, 158, 203, 205, 219; Poisson's 76, 100, 102, 122, 145, 148, 158, 172, 174–178, 203, 205, 219; stress 172, 174–178 relaxation 20, 214, 309 Reynolds transport theorem 55–56, 375 rock 7–8, 11–18, 21, 23, 83, 85–86, 96, 167, 183, 224, 308, 323, 326–327; igneous 7, 11–13, 18, 21, 85, 167, 224; metamorphic 7–8, 16–18, 23, 86, 323, 326; sedimentary 14–18, 21, 60, 83, 96, 183, 308, 323, 327 rock anchor 198 rockbolt 23, 225, 254, 341 rock reinforcement 23, 254 rock support 23, 254 rock salt 10, 16, 97, 122, 139–140 Ryukyu limestone 104, 107, 110, 112, 116–117, 121, 146

Index

379

sandstone 14, 17–18, 53, 83–84, 241, 308, 323 shale 17–18, 308, 323 shape function 154, 155, 257, 261, 282, 283, 287, 290, 291 shear 112, 124, 166, 180, 184, 193, 196, 211, 221, 229, 231, 239, 242, 252–253, 256, 305–307, 359–361; modulus 193, 211, 253, 305; strain 196, 231, 239, 242, 252; stress 112, 124, 166, 180, 184, 221, 229, 231, 242, 252, 256, 305–307, 359–361 simultaneous equation system 258–259, 261, 266, 271, 295 softening 204, 217, 253, 302 specific heat coefficient 59, 125, 127–130 strength 82–87, 102–103, 153, 157, 159, 215–216, 219, 299; tensile 83–87, 102– 103, 159, 219, 299; triaxial 83, 85, 223; uniaxial compressive 82–87, 153, 157, 205, 215–216, 219 Taylor expansion 44, 231, 239, 258, 265–266, 270, 276, 285, 293, 373 Terzaghi, K. 3, 6, 49, 148, 152, 185, 192, 195, 198, 221, 256, 280 test 79, 100, 102–105, 110–113, 116–118, 121, 144–146, 153–154, 189, 247, 278; direct shear 105, 110–113, 118, 119, 124, 144; stick-slip 145; tilting 111, 113, 116–117; triaxial compression 103–104, 118, 121, 153–154, 159; uniaxial compression 79, 100, 102, 121, 146, 189, 247, 278 thermal conductivity 125, 128–131, 146 Timoshenko, S. 217, 221, 256 toppling 23, 313–315, 333 tunnel 1, 3, 6, 40–41, 89–91, 93, 95, 145, 163, 179, 210, 215, 217, 224–226, 254–256, 301; circular 163; irrigation 1; support 6; Takamurayama 224 Ulusay, R. 2, 5, 40, 96, 99, 114, 145–146, 170, 182–183, 185, 247, 254, 256 updated Lagrangian scheme 299 Vardar, M. 19, 23 yield criterion 82–87, 151, 162–163, 166–168, 204, 221–222, 224–225, 227, 300; Aydan 222; Drucker-Prager 222; Hoek-Brown 83– 87, 167–168, 204 222–224; Mohr-Coulomb 82–86, 151, 162–163, 166–167, 221–222, 224–225, 300 yield zone 167–168, 224–225, 227

Rock Mechanics and Rock Engineering

Rock Mechanics and Rock Engineering

Volume 2: Applications of Rock Mechanics – Rock Engineering

Ömer Aydan Department of Civil Engineering, University of the Ryukyus, Nishihara, Okinawa, Japan

Cover photograph description: Eurasia subsea rock tunnel beneath the Bosporus in Istanbul

CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2020 Taylor & Francis Group, London, UK Typeset by Apex CoVantage, LLC All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/ or the information contained herein. Library of Congress Cataloging-in-Publication Data Applied for Published by: CRC Press/Balkema Schipholweg 107C, 2316 XC Leiden, The Netherlands e-mail: [email protected] www.crcpress.com – www.taylorandfrancis.com Volume 1 ISBN: 978-0-367-42162-5 (Hbk) ISBN: 978-0-367-82229-3 (eBook) DOI: https://doi.org/10.1201/9780367822293 Volume 2 ISBN: 978-0-367-42165-6 (Hbk) ISBN: 978-0-367-82230-9 (eBook) DOI: https://doi.org/10.1201/9780367822309 Two-volume set ISBN: 978-0-367-02935-7 (Hbk) ISBN: 978-0-429-00123-9 (eBook) DOI: https://doi.org/10.1201/9780429001239

Contents

Prefaceix  1 Introduction

1

  2 Applications to surface rock engineering structures 2.1 Cliffs with toe erosion 2.2 The dynamic response and stability of slopes against wedge sliding 2.3 Complex shearing, sliding and buckling failure of an open-pit mine 2.4 Dynamic response of reinforced rock slopes against planar sliding 2.5 Bridge foundations 2.6 Masonry structures 2.7 Reinforcement of dam foundations 2.8 Cylindrical sockets (piles)

3 3 8 10 19 23 35 37 42

  3 Applications to underground structures 3.1 Stress concentrations around underground openings 3.2 Dynamic excavation of circular underground openings 3.3 Evaluation of tunnel face effect 3.4 Abandoned room and pillar lignite mines 3.5 Karstic caves 3.6 Stability analyses of tomb of Pharaoh Amenophis III 3.7 Retrofitting of unlined tunnels 3.8 Temperature and stress distributions around an underground opening 3.9 Waterhead distributions around a shallow underground opening

47 47 50 53 54 61 63 64 66 66

  4 Rock mass classifications and their engineering utilization 4.1 Introduction 4.2 Rock Mass Rating (RMR) 4.3 Q-system (rock Tunneling Quality Index) 4.4 Rock Mass Quality Rating (RMQR) 4.5 Geological Strength Index classification 4.6 Denken’s classification and modified Denken’s classification 4.7 Estimations of engineering properties

71 71 71 76 81 89 89 90

vi Contents

  5 Model testing and photo-elasticity in rock mechanics 5.1 Introduction 5.2 Model testing and similitude law 5.3 Principles and devices of photo-elasticity 5.4 1G models 5.5 Base-friction model test 5.6 Centrifuge tests 5.7 Dynamic shaking table tests

103 103 103 106 108 110 111 112

  6 Rock excavation techniques 6.1 Blasting 6.2 Machine excavations 6.3 Impact excavation 6.4 Chemical demolition

155 155 198 199 200

  7 Vibrations and vibration measurement techniques 7.1 Vibration sources 7.2 Vibration measurement devices 7.3 Theory of wave velocity measurement in layered medium 7.4 Vibrations by shock waves for nondestructive testing of rock bolts and rock anchors

203 203 203 204

  8 Degradation of rocks and its effect on rock structures 8.1 Degradation of major common rock-forming minerals by chemical processes 8.2 Degradation by physical/mechanical processes 8.3 Hydrothermal alteration 8.4 Degradation due to surface or underground water flow 8.5 Biodegradation 8.6 Degradation rate measurements 8.7 Needle penetration tests for measuring degradation degree 8.8 Utilization of infrared imaging technique for degradation evaluation 8.9 Degradation assessment of rocks by color measurement technique 8.10 Effect of degradation process on the stability of rock structures

229

  9 Monitoring of rock engineering structures 9.1 Deformation measurements 9.2 Acoustic emission techniques 9.3 Multiparameter monitoring 9.4 Applications of monitoring system 9.5 Principles and applications of drone technology 9.6 Applications to maintenance monitoring 9.7 Monitoring faulting-induced deformations

271 271 277 278 280 291 298 298

222

230 230 241 241 243 246 249 250 251 253

Contents vii

10 Earthquake science and earthquake engineering 301 10.1 Introduction 301 10.2 Earthquake occurrence mechanics 301 10.3 Causes of earthquakes 308 10.4 Earthquake-induced waves 312 10.5 Inference of faulting mechanism of earthquakes 317 10.6 Characteristics of earthquake faults 319 10.7 Characterization of earthquakes from fault ruptures 321 10.8 Strong motions and permanent deformation 325 10.9 Effects of surface ruptures induced by earthquakes on rock engineering structures 346 10.10 Response of Horonobe underground research laboratory during the 20 June 2018 Soya region earthquake and 6 September 2018 Iburi earthquake352 10.11 Global positioning method for earthquake prediction 359 10.12 Application to Multi-parameter Monitoring System (MPMS) to earthquakes in Denizli basin 367 Index

381

Preface

Rock is the main constituent of the crust of the Earth, and its behavior is the most complex one among all materials in geosphere to be dealt by humankind. Furthermore, it contains various discontinuities, which make the thermo-hydro-mechanical behavior of rocks more complex. These simply require higher level of knowledge and intelligence in the Rock Mechanics and Rock Engineering (RMRE) community. Furthermore, the applications of the principles or rock mechanics to mining, civil and petroleum engineering fields, as well as earthquake science and engineering, are diverse, and it constitutes rock engineering. Recently, the International Society for Rock Mechanics (ISRM) added “Rock Engineering” in 2017 to its name while its acronym remains ISRM. Rock mechanics is concerned with the theoretical and applied science of the mechanical behavior of rock and rock masses, and it is one of branches of mechanics concerned with the response of rock and rock masses to their physical-chemical environment. Rock engineering is concerned with the application of the principles of mechanics to physical, chemical and electromagnetic processes in the uppermost part of the Earth and the design of the rock structures associated with mining, civil and petroleum engineering. This book is intended to be a fundamental book for younger generations and newcomers, as well as a reference book for experts specialized in rock mechanics and rock engineering. The book is divided into two volumes, due to the wide spectra of rock mechanics and rock engineering, titled Rock Mechanics and Rock Engineering: Fundamentals of Rock Mechanics and Rock Mechanics and Rock Engineering: Applications of Rock Mechanics – Rock Engineering. In the first volume, the fundamental concepts, theories, analytical and numerical techniques and procedures of rock mechanics and rock engineering, together with some emphasis on new topics, are described as concisely as possible while keeping the mathematics simple. The second volume is concerned with the applications of rock mechanics and rock engineering in practice. It ranges from classical rock classifications, the response and stability of surface and underground structures, to model testing, monitoring, excavation techniques and rock dynamics. Particularly, earthquake science and engineering, vibrations and nondestructive techniques are presented as a part of rock dynamics. Although Rock Mechanics and Rock Engineering consists of two volumes, each volume is complete in its content, and it should serve the purposes of educators, students, experts as well as practicing engineers. It is strongly hoped that these two volumes would fulfill the expectations and would serve further advances in rock mechanics and rock engineering.

Chapter 1

Introduction

Rock engineering is concerned with the applications of the principles of rock mechanics in practice. These applications involve the construction of transportation facilities such as tunnels, high-cut rock slopes, foundations of large bridges, nuclear power plants, dams, storage of oil and natural gases in caverns in civil engineering, exploitation of natural sources such as metallic minerals, coals in the form of open-pits or underground mines in mining engineering, extraction of oil and gas in petroleum engineering and utilization of geothermal energy. Some of the practical applications of empirical, analytical and numerical methods involve the evaluation of deformation and stress state of surface, semi-underground and underground rock engineering structures in the short and long term, as well as under static and dynamic loading conditions. Some fundamental examples of applications to surface and underground structures are explained with the consideration of practical conditions. The model testing technique in rock mechanics and rock engineering has been an important tool for engineers for understanding the response of rock engineering structures as well as obtaining design parameters if the similitude law is properly established for a given structure. With the development of numerical methods, response and stability of rock engineering structures could be evaluated under very complicated initial and boundary conditions as well as rock mass behavior. Nevertheless, model testing is still a useful yet powerful technique to have an insightful view of what is taking place with regard to a given structure under the given conditions. They may also provide a clear visual yet quantitative picture of the phenomenon, which may be quite difficult to evaluate by numerical methods. From earlier times, many rock classifications have been proposed, and some of them provide quantitative characterization of rock masses with or without their applications to certain rock engineering structures. These classifications and their utilization for estimating the mechanical properties of rock masses are presented. Model tests have been used in engineering for thousands of years, and it is still widely used in many engineering applications. The similitude law used in model testing is described, and some specific examples are given. Principles of various model testing techniques under static and dynamic conditions are explained, and various specific examples of model tests are described in order to illustrate their use in rock mechanics and rock engineering. Humankind has devised many excavation techniques to create underground openings, construct foundations of rock engineering structures and pass through steep valleys. Blasting is still the most commonly used excavation technique, and its principles are explained. Besides the excavation, the positive and negative characteristics of the blasting technique are presented. Furthermore, the principles of machine-based excavation techniques and expansive chemical agents are described.

2 Introduction

Vibrations in rock mechanics and rock engineering result from different processes such as blasting, machinery, impact hammers, earthquakes, rockburst, bombs (including missiles), traffic, winds, lightning, weight drop and meteorites. Vibrations may also be induced by impact hammers, blasting with small explosives, Tunnel Boring Machine (TBM), and they may be used to evaluate wave velocity characteristics of rocks and rock masses for assessing the rock mass properties for design purposes. The related chapter describes the devices for measuring vibrations and their utilization for characterization of rock mass conditions such as existence of weak/fracture zones, cavities and their properties. Furthermore, they are used to infer some yielding or loosening around rock structures. Various field examples of its utilization in rock mechanics and rock engineering are presented. Degradation of rock masses subjected to atmospheric conditions and/or gas/fluids percolating through rocks and rock discontinuities is quite well-known. This issue is explained, and the causes of degradation such as the alteration of minerals, weakening of particle bonds and/or solution of particles are explained. Furthermore, the effects of degradation processes on the properties of rocks and rock engineering structures are described. The monitoring of rock mass movements, as well as of their responses to various environmental conditions, has become quite common in the construction of various rock engineering structures. This chapter is devoted how to measure deformation responses utilizing direct and space-borne optical and laser techniques as well as variations of various parameters using the multiparameter monitoring technique, which may include temperature, water level, acoustic emissions, electric potential, infrared imaging technique. Various laboratory and field examples are described. Earthquakes are often encountered in many long-term rock engineering projects. Therefore, understanding the behavior of rock mass during shaking as well as various rock engineering structures during earthquakes is of great importance. Chapter 10 is devoted to the science and engineering aspects of earthquakes and their effect on rock engineering structures. In addition, some specific examples are given for evaluating ground motions caused by earthquakes on the basis of principles of rock mechanics and how to design rock engineering structures against the motions caused by earthquakes. Furthermore, the possibility of earthquake prediction on the basis of principles of rock mechanics is studied and discussed.

Chapter 2

Applications to surface rock engineering structures

2.1  Cliffs with toe erosion 2.1.1  Analytical approach As pointed out in the previous section, the toe erosion of rock cliffs results in overhanging rock blocks. If the overhanging part of the cliff is continuously connected to the rest of rock mass, these rock blocks may be modeled as cantilever beams. However, depending upon the erosion type, their configuration may change from a rectangular prism to triangular prism. If the bending theory is employed, one can easily derive the following set of equations by assuming that cliffs are subjected to gravitational and seismic loads as illustrated in Figure 2.1 for a unit thickness. Equivalent beam thickness  x h = hb 1− (1− α)  (2.1)  L Shear force  x Q = Vo − (1 + kv )γ hb x 1− (1− α)  (2.2)  2L  Bending moment 1 x M = M o + Vo x − (1 + kv )γ hb x 2  − (1− α)  (2.3)  2 6L  Bending stress at the outer fiber 1 + α  M + 6 2 (2.4) σ = kh γ hb L   2  h where α=

 (1− α)  hs , Vo = (1 + kv )γ hb L 1−  (2.5a)  hb 2 

 1 (1− α)  M o = −(1 + kv )γ hb L2  −  (2.5b)  2 3 

4  Surface rock engineering structures

Figure 2.1  Modeling of overhanging cliffs

Figure 2.2 Comparison of distribution of bending stress at the outermost fiber of beam with different configurations

hb, hs, γ and L are beam height at the base and at the far end, unit weight of rock mass and erosion depth, respectively. kh, and kv are horizontal and vertical seismic coefficients. Figure 2.2 shows the bending stress distributions along the outermost fiber of the beam for different geometrical configurations. The severest condition occurs when the beam has a rectangular shape and the value of the bending stress is much higher for the rectangular configuration. Tensile stress is also the largest at the base of the cantilever beam. As discussed by Aydan and Kawamoto (1992), the cantilevers fail immediately once the tensile

Surface rock engineering structures  5

stress exceeds the tensile strength of rock mass. Furthermore, the seismic loads in addition to gravitational load would make the cliffs more vulnerable to failure during earthquakes. While the consideration of seismic loads is based on the seismic coefficient method in this section, one may assess any amplification from a response analysis if the frequency content of earthquake waves is of great importance for a given earthquake record using the following formula for the natural frequency (first mode) of cantilever beams as a first approximation: f1 =

1.8752 2π

EI (2.6) mL4

where L is erosion depth, E is elastic modulus, m is mass per unit length, and I is the inertia moment of area. 2.1.2  Numerical analyses Finite element method (FEM) is one of the powerful numerical techniques to analyze the stability of the cliffs. This technique was employed by Kawamoto et al. (1992) to backanalyze the failure of an overhanging cliff at Echizen along the Japan seashore, which killed 15 people in 1984. One of the main purposes of the finite element method was to check the application limits of the bending theory presented in the previous section. The stress state of an overhanging block having different shapes was analyzed using a two-dimensional elastic finite element method under gravitational and seismic loading to illustrate stress changes during the erosion process and compare with estimations from the bending theory of cantilever beams presented in previous section for material properties given in Table 2.1. The reason for using elastic finite element analysis was that, when cantilever structures start to rupture in tension in brittle rock mass, it immediately results in total failure. Figure 2.3 shows the finite element model for simulating the erosion process for a rectangular cliff model with a height of 10 m for three erosion depths (4 m, 8 m and 12 m), together with assumed boundary conditions. The horizontal stress distributions along the vertical sections, namely ES1 (4 m), ES2 (8 m) and ES3 (12 m) for a rectangular shape under gravitational loading, are shown in Figure 2.4. As noted from the figure, the stress distribution is not linear as expected from the bending theory of cantilevers. Figure 2.5(a) shows the horizontal stress distribution near the top surface. As expected, tensile stress develops parallel to the top surface of the cliff model, and the amplitude of tensile stress increases as the erosion depth increases. When the ratio of erosion depth to overhanging beam thickness exceeds 1, it is also interesting to note that the maximum tensile stress occurs near the surface projection of the erosion tip as expected from the bending theory of cantilevers. However, the value of tensile stress is lower than that obtained from

Table 2.1  Properties used in elastic finite element analyses Unit Weight (kN m−3)

Elastic Modulus (GPa)

Poisson’s Ratio

25

10

0.25

6  Surface rock engineering structures

Figure 2.3  Finite element mesh and assumed boundary conditions

Figure 2.4 Horizontal stress distribution at the base of cantilever beam for various erosion depths computed by FEM

cantilever theory. This is considered to be due to the difference between the finite element model and cantilever models. Furthermore, the vertical stress shown in Figure 2.5(b) in the vicinity of the erosion depth is very high and compressive. However, the amplitude of the compressive stress is about 2.5 times the maximum tensile stress. This simply implies that the possibility of the failure of overhanging blocks is much more likely in tension than in

Surface rock engineering structures  7

Figure 2.5 (a) Horizontal stress distribution near the top surface the FEM model for various erosion depths, (b) vertical stress distribution at the erosion level of the cliff for various erosion depths

Figure 2.6 (a) Horizontal stress distribution at the base of cantilever beam for various erosion depths computed by FEM, (b) horizontal stress distribution near the top surface of the cliff for various values of horizontal seismic load coefficient

compression when the actual ratio (10–20) of the compressive strength of rocks to their tensile strength is taken into account. Horizontal stress distributions along the vertical section emanating from the erosion tip computed from the finite element method for two different Poisson’s ratios (0.00 and 0.25) are compared with the distribution from the bending theory of cantilever beams as shown in Figure 2.6(a). The stress distributions are not influenced by the variation of Poisson’s ratio. The stress distribution is not linear as expected from the bending theory of cantilever beams. However, the distribution of tensile part is relatively linear as compared with compressive

8  Surface rock engineering structures

stresses. The maximum value of tensile stress computed from the finite element method is about 75% of that computed from the bending theory. Nevertheless, the work done is the same for both computational methods. Furthermore, the amplitude of the compressive part is about 2.5 times the maximum tensile stress. Once again, this simply implies that the possibility of the failure of overhanging blocks is much more likely in tension rather than in compression when the actual ratio (10–20) of the compressive strength of rocks to their tensile strength is taken into account. The effect of seismic loads on the stress state of cliffs is simulated using the seismic coefficient method. Figure 2.6(b) shows the horizontal stress distribution near the top surface of the model for horizontal seismic load coefficients of 0.0, 0.3 and 0.6. As expected, the stress components increase as the seismic load coefficient increases. However, this increase will be linear, which may be inferred from the theoretical formulation.

2.2 The dynamic response and stability of slopes against wedge sliding The authors have advanced the method of stability assessment proposed by Kovari and Fritz (1975) for wedge failure of rock slopes under different loading conditions and confirmed its validity through experiments (Kumsar et al., 2000). Aydan and Kumsar (2010; Aydan, 2017) extended to evaluate sliding responses of rock wedges under dynamic loading conditions under submerged conditions with viscous resistance. Let us consider a wedge subjected to dynamic and water loading as shown in Figure 2.7. One can easily write the following

Figure 2.7  Illustration of mathematical model for wedge failure

Surface rock engineering structures  9

dynamic equilibrium conditions for the wedge during sliding motion on two basal planes in a coordinate system Osnp shown in Figure 2.7. ∑ Fs = (W − Ev ) sin ia − Ei cos ia − S = m

d 2s (2.7a) dt 2

∑ Fn = (W − Ev ) cos ia − Ei sin ia − N = m ∑ Fp = −N1 cos ω1 − N 2 cos ω2 − E p = m

d 2n (2.7b) dt 2

d2 p (2.7c) dt 2

where N = N1 sin ω1 + N 2 sin ω2, W is weight of wedge, Ev is dynamic vertical load, Ei is dynamic force in the direction of intersection line, and Ep is dynamic load perpendicular to intersection line. Other parameters are shown in Figure 2.7. Although the dynamic vectorial equilibrium equation is written in terms of its component, they correspond to a very general form for wedge sliding along the intersection line while being in contact with two basal planes. Furthermore, the earthquake force is decomposed to its corresponding components in the chosen coordinate system. One can easily obtain the following identity from Equation (2.7c) by assuming that there are no motions upward and perpendicular to the intersection line: N1 + N 2 = (W − Ev ) cos ia − Ei sin ia  λi − E pλ p (2.8) where λi =

cos ω1 + cos ω2 sin ω1 − sin ω2 , λp = (2.9) sin (ω1 + ω2 ) sin (ω1 + ω2 )

If the resistance is assumed to obey the Mohr-Coulomb criterion (Aydan and Ulusay, 2002; Aydan et al., 2008) one may write the following: T = ( N1 + N1 ) µ, µ = tan φ (2.10a) Following the initiation of sliding, the friction angle can be reduced to the kinetic friction angle as given here: µ = tan φr (2.10b) where ϕr are residual cohesion and friction angle. Under frictional condition, it should be noted that normal force (N1 + N1) cannot be negative (tensile). If such a situation arises, normal force (N1 + N1) should be set to 0 during computations. Let us introduce the following parameters: ηv =

Ep ap Ev av E a = = , ηi = i = i , η p = (2.11) W g W g W g

10  Surface rock engineering structures

where av, ai, ap are acceleration components resulting from dynamic loading. The following dynamic equilibrium equation must be satisfied during the sliding motion of the wedge: S = T(2.12) If the relations given by equations are inserted in Equation (2.7a), one can easily obtain the following differential equation: s =

d 2s = g (1− ηv ) A + ηi B + η p C  (2.13) dt 2

where A = (sin ia − cos ia µλi ) , B = (cos ia + sin ia µλi ); C = µλ p

Since dynamic loads are very complex in the time domain, the solution of Equation (2.13) is possible only through numerical integration methods. The time-domain problems in mechanics are generally solved by finite difference techniques. For this purpose, there are different finite difference schemes. In this article, the solution of Equation (2.13) based on the linear acceleration finite difference technique (i.e. Aydan and Ulusay, 2002; Aydan et al., 2008). One can write the velocity (ṡ) and displacement of wedge (s) for a time step n + 1 as follows: sn+1 = sn +

s sn ∆t + n+1 ∆t (2.14) 2 2

sn+1 = sn +

s sn s ∆t + n ∆t 2 + n+1 ∆t 2 (2.15) 1 3 6

Provided that resulting dynamic shear force exceeds the shear resistance of the wedge at time (t = ti = iΔt), one can easily incorporate the variation of shear strength of discontinuities from peak state (µ = tan ϕp) to residual state (µ = tan ϕr).

2.3 Complex shearing, sliding and buckling failure of an open-pit mine 2.3.1  A limiting equilibrium method Ulusay et al. (1995) proposed a limiting equilibrium analysis method for a failure mechanism of an open-pit mine in Eastern Turkey as shown in Figure 2.8. The interslice forces were evaluated using the approach of Aydan et al. (1992). It is assumed that failure takes place through shearing of intact layers at the back of the slope and sliding along a bedding plane. For the pit-floor layer, there may be three possible modes: MODE 1: compressive failure, MODE 2: buckling failure, and MODE 3: combined compressive and buckling failure (Aydan et al., 1996b).

Surface rock engineering structures  11

Figure 2.8  Failure mechanism Source: Proposed by Ulusay et al. (1995)

Figure 2.9  Mechanical models for stability assessment

For the sliding and shearing part, the force system acting on a typical block may be modeled as shown in Figure 2.9(a). Note that a lateral force is also assumed to act in order to consider the lateral stresses. The equilibrium equations for the chosen coordinate system can be written as: s-direction −Ti + Wi sin αi + H i cos αi + Fi−1 cos(αi − θi−1 ) − Fi cos(αi − θi ) + (U is−1 −U is ) cosαi = 0

(2.16)

n-direction N i + U ib −Wi cos αi + H i sin αi + Fi−1 sin (αi − θi−1 ) − Fi sin (αi − θi ) + (U is−1 −U is ) sin αi = 0

(2.17)

12  Surface rock engineering structures

Assuming that the rock obeys the Mohr-Coulomb yield criterion and that the ratio of the horizontal force to the weight of the slice is given in the following forms: Ti =

ci Li + N i tan ϕi , H i = λWi (2.18) SF

One easily obtains an equation for interslice force Fi, which can be solved step by step to obtain the force Fn together with the condition of F0 = 0. The resistance of the pit-floor against compressive failure would be similar to the thrusttype faulting. Therefore, no equation is given here. As for the buckling failure of the coal seam, the following nonhomogeneous differential equation holds (Figure 2.9(b)): q d 2 u Fn + u = o ′ x( L − x) (2.19) 2 EI dx 2 EI where E is elastic modulus, I is second areal inertia moment, u is displacement, and qo’ is effective distributed load. Solution of the preceding equation is: u = A cos kx + B sin kx +

qo ′ F 2 ( Lx − x 2 + 2 ); k 2 = n (2.20) 2 EI 4 EIk k

If u = 0 and du/dx = 0 at the ends of the layer, the integration constants A and B are obtained as follows: A=−

q L qo ′ , B = − o ′ (2.21) Fn k Fn k 2

Assuming that du/dx = 0 at x = L/2, the critical buckling load is obtained as: 2

 8.99  Fn = EI  (2.22)  L  Introducing I = bt3/12 and Fn = σobt, the critical axial stress for buckling is obtained as follows: 2

t  σocr = 6.735 E   (2.23)  L  where t is layer thickness, and L is span. An application of the preceding approach is shown in Figure 2.10. Figure 2.10(a) was obtained from force Fn = σobt by considering the combined shearing and sliding failure for SF = 1 by varying lateral stress coefficient λ. Figure 2.10(b) was obtained from buckling analysis by assuming that E/σC is 65 (continuous line) and 26 (broken line). Since it is more likely that the peak strength values hold, the slope may become unstable and the pit-floor fails in compression provided that the lateral stress coefficient is 0.13 for a uniaxial strength of 596 kPa. As for buckling failure, the lateral stress coefficient failure should be greater than 0.1 and less than 0.13 in view of the actual range of L/t at the time of failure. The uncovered span of the lignite seam was 113 m at the time failure. If the thin gyttja formation just above the lignite seam near the toe of the slope is neglected, the effective span is about 153 m. For a 5 m thick lignite seam, the value of L/t for compressive failure is found to be 21.95 from Figure 2.10(b). If the effect of the gyttja

Surface rock engineering structures  13

Figure 2.10  Computed stability chart for the lignite seam

formation on L/t ratio is taken into account, the L/t is 22.6. On the other hand, if its effect is neglected, L/t is 30.6. Considering these numbers, it has been contemplated that the lignite seam would likely be buckling rather than failing in compression. If the thickness of the seam involved in failure is greater than 5 m, the possibility of failure by buckling increases more rapidly as compared with that by compression. 2.3.2  Discrete finite element analyses The discrete finite element method proposed by Aydan-Mamaghani (Aydan et al., 1996a; Mamaghani et al., 1999) was chosen to simulate the failure process of the slope. This method is based entirely on the finite element method and can simulate very large deformations of jointed media. Material properties used in the analyses are given in Table 2.2. Figure 2.11 shows the finite element mesh and boundary conditions. First, a series of elastic analyses was carried out by varying lateral stress coefficient κ to see the magnitude of the axial stress of lignite seam at the location adjacent to the sliding benches. Figure 2.12 shows the relation between lateral stress coefficient κ and the axial stress in the lignite seam. As seen from the figure, the lateral stress coefficient must be greater than 0.58 and less than 0.78 to cause the buckling of the seam. Otherwise, the lignite

Table 2.2  Geomechanical parameters used in discrete finite element method (DFEM) analysis Unit

Loam, marl, blue clay Lignite Base layer Weak clay Fracture plane

λ

µ

c

ϕ

(MPa)

(MPa)

(kPa)

(°)

408 710 670 3.2 3.2

5.4 9.4 38 1.1 1.1

60 161 161 23 23

25 33 33 6 20

λ, µ are Lamé coefficients; c is cohesion; ϕ is internal friction angle.

Figure 2.11  Finite element mesh used in analyses

Figure 2.12  Computed stability chart for buckling failure

Surface rock engineering structures  15

Figure 2.13  Deformed configurations at various pseudo time steps

seam must fail in compression, which is contradictory to field evidence. The results further indicate that if the lateral stress coefficient is less than 0.58, the axial stress in the seam may be tensile. By setting the lateral stress coefficient κ as 0.7, an elasto-plastic analysis was carried out. Figure 2.13 shows the deformed configuration of the open-pit for each respective

16  Surface rock engineering structures

pseudo time step. As seen from this figure, the sliding of the benches on the left-hand side and buckling of the lignite seam at pit-floor are well simulated. It should be noted that if the analysis becomes non-convergent in finite element analysis, this may be taken as the indication of failure of the structure, and each iteration step can be regarded as pseudo time step. With this concept in mind, the displacement responses of open-pit at selected point shown in Figure 2.13 are plotted in Figures 2.14 and 2.15. The

Figure 2.14  Pseudo time step vs displacement for point A

Figure 2.15  Pseudo time step vs displacement for point B

Surface rock engineering structures  17

displacement response of point A, which is located at the center of the pit-floor, corresponds to the heaving of the floor. The heaving of the floor proceeds at a constant rate up to pseudo time step 6 and thereafter increases with an increasing rate, and the floor buckles. The displacement response of point B, which is selected at the rear top of the sliding benches, corresponds to the horizontal displacement of the sliding body. This response is very similar to the measured response shown in Figure 2.15. 2.3.3  Estimation of postfailure deformation The method used for estimating postfailure motions of the failed body is based on the earlier proposals by Aydan et al. (2006, 2008), Aydan and Ulusay (2002) and Tokashiki and Aydan (2010, 2011). Let us consider a landslide body consisting of N number of blocks sliding on a slip surface as shown in Figure 2.16. If interslice forces are assumed to be nil as assumed in the simple sliding (Fellenius-type) model, one may write the following equation of motion for the sliding body: n

∑ ( Si − Ti ) = m

i =1

d 2s (2.24) dt 2

where m, s, t , n, Si and Ti are total mass, travel distance, time, number of slices, shear force and shear resistance, respectively. Shear force and shear resistance may be given in the following forms, together with Bingham-type yield criterion: b

Si = Wi (1 +

 ds  aH ) sin αi , Ti = ci Ai + ( N i −U i ) tan φi + ηWi  i  (2.25)  dt  g

where Wi, Ai, Ni, Ui, αi, αv, αH, ci, ϕi, η and b are weight, basal area, normal force, uplift pore water force, basal inclination, vertical and horizontal earthquake acceleration, cohesion, friction angle of slice i, Bingham-type viscosity and empirical coefficient, respectively. If normal force and pore water uplift force related to the weight of each block as given here: N i = Wi (1 +

aV ) cos αi, U i = ruWi (2.26) g

Figure 2.16  Mechanical model for estimating postfailure motions

18  Surface rock engineering structures

One can easily derive the following equation with the use of Equations 2.25–2.26: b

 ds  d 2s + η   − B (t ) = 0 (2.27)  dt  dt 2 where B(t ) =

a cA a g  n ∑ mi (sin αi (1 + H ) − (cos αi (1 + V ) − ru ) tan φi ) − i i  (2.28) m i=1 g g g 

In the derivation of Equation (2.28), the viscous resistance of the shear plane of each block is related to the overall viscous resistance in the following form: b

b n  ds   ds  η mg   = ∑ η mi g  i  (2.29)  dt   dt  i =1

Equation (2.29) can be solved for the following initial conditions together with the definition of the geometry of basal slip plane. At time t = t0: s = s0 and v = v0(2.30) There may be different forms of constitutive laws for the slip surface (i.e. Aydan et al., 2006, 2008; Aydan and Ulusay, 2002). The simplest model is elastic-brittle plastic to implement. If this model is adopted, the cohesion will exist at the start of motion, and it will disappear thereafter. Therefore, cohesion component introduced in Equation (2.29) may be taken as nil as soon as the motion starts. Thereafter, the shear resistance will consist of mainly frictional component together with some viscous resistance. The method just explained was applied to the Kitauebaru landslide in Okinawa (Japan) involving bedding plane and fault plane. This method utilizes the Bingham-type visco-plastic yield criterion. Although the assumed geometry of the open-pit mine is slightly different from the actual one, it was applied to the failure in the Kışlaköy open-pit mine (Ulusay et al., 2019). Figure 2.17 shows the displacement response during failure. The actual displacement

Figure 2.17  Displacement response of the mass center of the failed body

Surface rock engineering structures  19

Figure 2.18  Deformed configurations of the failed body

of the failed body was about 34 m. The estimated displacement of the failed body is about 33 m. The material properties are shown in Figures 2.17 and 2.18, which are based on those given in Table 2.2. Figure 2.18 shows the deformation configuration of the failed body in space with time. Despite some difference between the assumed and actual geometries of the failed body, the estimations are very close to the actual ones.

2.4 Dynamic response of reinforced rock slopes against planar sliding As the deformation of rock slope occurs mainly due to slippage along the failure surface, a dynamic limiting equilibrium method developed originally by Aydan and Ulusay (2002) and elaborated by Aydan et al. (2008) and Aydan and Kumsar (2010) was used to simulate the slip of the model slope on the failure surface (Aydan et al., 2018). Figure 2.19 shows a view of the mechanical model for the dry condition with the consideration of the experimental fact; that is, the unstable part moves like a monolithic body irrespective of layered or single body. One can easily write the following limiting equilibrium equations for s- and n-directions, respectively, as follows: W sin α + E cos α − T cos (α ± β ) − S = m

d 2s (2.31a) dt 2

W cos α + E sin α − T sin (α ± β ) − N = m

d 2n (2.31b) dt 2

where W, E, T, S, N and m are weight, seismic load, force provided by rock bolts/rock anchors, shear and normal forces and mass of the unstable body, respectively. α and β are the inclination of failure plane and rock bolts/rock anchors from the horizontal. s and n are the amount of shear and normal displacement of the unstable body. (−) is signed used when the acute angle between the bolt force and horizontal is positive, which is denoted as Case 1. (+) sign is used the acute angle between the bolt force and horizontal is negative, which is denoted as Case 2.

20  Surface rock engineering structures

Figure 2.19  Mechanical model for reinforced rock slope

Let us assume that the inertia force for n-direction during sliding is negligible and the resistance of the failure plane is purely frictional as given here: S = tan (φ ) (2.32) N One can easily obtain the following equation for the rigid body motion of the sliding rock burst body: m

d 2s = Aw + AE − AT (2.33) dt 2

where Aw = W (sin α − cos α tan φ ) AE = E (cos α + sin α tan φ ) AT = T (cos (α ± β ) + sin (α ± β ) tan φ) As the earthquake or shaking-induced force E will be proportional to the mass of the sliding body, it can be related to ground shaking in the following form: E=

ag (t ) g

W (2.34)

where ag is base acceleration, and g is gravitational acceleration. The reinforcement effect (T) of rock bolts is due to resolved components and the dowel effect. The bar/tendon may be pre-tensioned at the time of installation. Therefore, the total

Surface rock engineering structures  21

acting force (Tt) may be contemplated as the sum of pretension force (Tp) and deformationinduced force (Td) as given here: Tt = Tp + Td (2.35) The deformation-induced axial force in a rock bolts during the shearing process may be given in the following form (Aydan, 2018): Td = Bδ b (2.36) where b and B are empirical constants, and δ is the displacement of rock bolt. It is experimentally well-known that the rock bolts crossing a discontinuity plane are bent during shearing and that there is an effective length of rock bolts mobilized during the shearing process. Thus, the extension of rock bolt would be given by: 2

δ =  −  o = ( o + δh ) + δv2 −  o (2.37) where ℓo, δh and δv are the effective length of rock bolt mobilized at the failure plane and horizontal and vertical movement of the sliding unstable body. The force in the rock bolt can then be obtained by inserting the extension value from Equation (2.37) into Equation (2.36). The small amount of deformation may cause the yielding of rock bolts/rock anchors. In such cases, their axial force may be assumed to be equivalent to their yielding value. If there is a hardening type of response, then the following type constitutive may be adopted: Tt = TY + H δ (2.38) where TY and H are yielding strength and hardening modulus, respectively. The mathematical model described in this section can be used for both unreinforced and reinforced rock slopes. For the unreinforced case, the resistance provided by rock bolts/rock anchors is neglected. An application of the dynamic limiting equilibrium method is shown in Figure 2.19. Three different values are used for the kinetic (residual) friction angle. In this particular simulation, if the slip stops, the peak friction angle is used for the initiation of slip in the next cycle. Although it is not reported here, the slip becomes much larger if the friction angle is assumed to be equal to residual value once the slip is initiated. The values used in computations are also shown in Figure 2.20. The initiation of the slip occurred slightly at a higher friction angle than that determined from tilting tests. Figure 2.21 compares the slip responses for three different values of residual friction angle. When the residual friction angle is equal to the peak friction angle, which corresponds to perfectly plastic behavior, the amount of slip is quite smaller than those for a lower residual friction angle. When the residual friction angle is 0.625 times the peak friction angle, it corresponds to the kinetic friction angle determined from tilting tests. For this particular situation, the slip is largest. When the residual friction angle is 0.725 times the peak friction angle, the slip at the first stage is equal to that measured in the experiment as seen in Figure 2.21 It is very likely that if the peak friction angle is reduced as a function of the slip cycles, it is quite possible to get better estimations of the measured responses. It

22  Surface rock engineering structures

Figure 2.20 Comparison of measured slip response with the estimated responses for different values of residual friction angle

Figure 2.21 Expanded comparison of measured slip response with the estimated responses for different values of residual friction angle, shown in Figure 2.20

should be noted that the slip of the potentially unstable block is restrained to 12.3 mm, and the displacements exceeding this value are not considered. The theoretical approach is applied to model tests shown in Figure 2.19 by selecting that the friction angle is 39 degrees. The computed results are shown in Figure 2.22. The reinforcement effect (TB) of rock bolts are normalized by the weight (WB) of the unstable body. As noted from the figure, the computed results are quite similar to experimental results both quantitatively and qualitatively. However, the computations indicate that the yielding should started a bit later than the measured results. The discrepancy may result from the complexity

Surface rock engineering structures  23

Figure 2.22  Comparison of computed responses with measured responses

of actual frictional behavior of the sliding surface. Nevertheless, the theoretical model is capable of modeling the dynamic response of the support system.

2.5  Bridge foundations 2.5.1 Back-analysis of the constitutive law parameters of the foundation rock The application involves a back-analysis of the constitutive law parameters of the foundation rock of the pier 3P of the Akashi suspension bridge in western Japan (Aydan, 2016, 2018). The rock consists of Kobe tuff, which is a relatively soft rock. The diameter of the foundation was 40 m and its height was 80. Following the lowering of the caisson foundation to the sea bottom, it was filled with concrete, which increased the load on the foundation. First the filling of the inner ring was completed, and then the outer ring was filled with some time lag. The deformation of the ground was measured during the filling stages. It was required to obtain the time-dependent characteristics of foundation formation by considering the loading associated with the construction procedure. The constitutive law of the foundation rock was assumed to be of Kelvin type. The problem was considered to be an axisymmetric problem; the finite element mesh used in the back analyses is shown in Figure 2.23. The elastic modulus and viscosity coefficient of Kelvin model of foundation rock were 833 MPa and 3.3 GPa ⋅ day , respectively (Figures 2.24 and 2.25). Figure 2.26 compares the computed response with the measured response for the loading condition shown in the same figure. Figure 2.27 shows the displacement of the pier for about 4 years. The expected creep displacement is about 108 mm. 2.5.2  Settlement and stress state and circular rigid foundation The example is concerned with settlement and stress state beneath foundations subjected to surcharge loads through a relatively rigid foundation with a diameter of 3 m, and it is modeled as an axisymmetric problem. Figure 2.28 shows the computed settlement and pressure

Figure 2.23  Finite element mesh used in the back-analyses

Figure 2.24 Time response of applied load and measured displacement

Figure 2.25  Back-analysis of measured displacement by Kelvin model

Figure 2.26  Comparison of measured and computed displacement

Figure 2.27  Estimated creep displacement of the pier

Figure 2.28  Settlement and pressure contours beneath a circular foundation

26  Surface rock engineering structures

contours beneath the foundation. The estimated settlement and pressure contours are generally in agreement with theoretical solutions by Timoshenko and Goodier (1951). 2.5.3 Analysis of tunnel-type anchorage of suspension bridge under tension Numerical methods such as the finite element method (FEM) is generally used to check the local straining, stresses and local safety factors in linear analyses or yielding in nonlinear analyses. A specific example is described here. The analyses reported in this subsection are for the examination of the design and to predict the post-construction performance of the tunnel-type anchorage of a 1570 m long suspension bridge. The analyses are carried out to investigate the stability of the concrete anchorage body and surrounding rock upon the suspension load of the bridge applied on the anchorage by considering two loading intensity conditions using the finite element method: 1 2

The design loading condition Loading three times the design loading condition

The purpose of considering the first loading condition was to see the state of stress in concrete and in the surrounding rock and the possibility of any plastification in the anchorage and the rock for a given material properties and a yield criterion under the normal loading (i.e. design load) conditions including earthquake forces. The second loading condition, on the other hand, was to see the effect of unexpectedly high loads upon the performance of the anchorage body and the response of the surrounding rock. For each loading condition, two possible cases are investigated: 1 2

Full bonding No tension slit just under the loading plane

The first case corresponds to an actual situation. The second case, on the other hand, visualizes a tensile crack just under the loading plane in a concrete body. By this, it is intended to see the effect of such a tensile crack on the stability of the concrete anchorage body and surrounding rock. The rock at the site under consideration is mainly granite. However, there is an almost vertical dioritic volcanic intrusion that seems to have disturbed the surrounding rock, and it is highly weathered. The main portion of the anchorage is situated in granitic rock. The granitic rock is classified as CH class rock in the classification of DENKEN (Ikeda, 1970). The remaining part of the anchorage is located in the dioritic volcanic intrusion (dyke) and is classified as DH class rock in the same classification. The geological cross section of the rock mass along the anchorage is shown in Figure 2.29. The material properties used in the analyses were determined from in-situ shear tests and plate-bearing tests in other near construction sites with similar geology and are given in Table 2.3. The material properties listed in Table 2.3 is highly conservative and represents the lowest values of the respective tests. Poisson’s ratio for every rock class is assumed to be 0.2. The elastic modulus, cohesion and tensile strength is 0.6 times the undisturbed rock mass The anchorage is modeled as an axisymmetric body considering the geological formations shown in Figure 2.29. Besides the modeling of geologic formations mentioned in the

Figure 2.29  Geological cross section Table 2.3  Material properties Rock Class

Esb MPa

Soil-like DH

80 80–150

75 180

150 350

CL

150–300

330

560

CM

300–600

600

1200

CH

600–1200

1280

2400

CONCRETE

Es MPa

25000

Ed MPa

Cohesion MPa

ϕ (°)

σC MPa

σt MPa

γ (kN m−3)

0 0.1 0.07 0.5 0.4 0.8 0.7 1.2 1.0 4.97

35 37 32 40 40 40 40 45 45 45

0 0.4 0.25 2.14 1.54 3.43 3.00 5.79 4.83 24

0 0.04 0.02 0.21 0.15 0.34 0.30 0.58 0.48 15

19 20.5 19.0 22.0 20.0 23.5 21.0 24.5 22.0 24

28  Surface rock engineering structures

previous section, a loosening zone of 2 m wide, which may be caused during the excavation of the anchorage tunnel, was assumed to exist in the finite element model. The material properties of loosening zones are assumed to be 0.6 times those of the respective geological formations. In all the analyses reported herein, a finite element program considering an elasto-plastic behavior of rock mass was used. The element type used in the analyses is a 4-noded isoparametric element. The dimensions of the analysis domain were taken as two times the total anchorage length vertically and horizontally, as shown in Figure 2.30. The gravity in the analyses, was taken into account as follows: γ* = γ sin α, σ z = γ * h, σ r = K o σ z , σθ = K o σ z (2.39) where γ is unit weight or rock or concrete, h is depth form surface, Ko is lateral initial-stress coefficient, α is inclination of the anchorage axis from horizontal, σz is vertical stress, σr is radial stress, and σθ is tangential stress. The value of the lateral stress coefficient Ko was taken as 1 in view of the ground-stress measurements in the near vicinity of the anchorage site. The results in terms of the following items: 1 2 3

Maximum principal compressive stress in concrete and in rock Maximum principal tensile stress in concrete and in rock Minimum safety factor against shear failure (SFS) for concrete and for rock

Figure 2.30  Finite element mesh

Surface rock engineering structures  29

Minimum safety factor against shearing (SFS) is defined as:

SFS =

σ1 + σ3 sin φ 2 (2.40) σ1 − σ3 2

c cos φ +

Note that the sign of the compressive stress is taken as negative (−). 4

Minimum factor of safety against tensile failure (SFT) for concrete and for rock,

which is defined as follows: SFT =

σt (2.41) σ1 or σ3

First the calculated results are presented for the ordinary design load condition for fully bonded and with no-tension slit cases and are compared and summarized. Figure 2.31 shows

Figure 2.31  Comparison of principal stress distributions

30  Surface rock engineering structures Table 2.4  Comparison of maximum compressive and tensile stresses Material Type

Fully Bonded

No-tension Slit

Max. Comp. Stress MPa

Max.Ten. Stress MPa

Max. Comp. Stress MPa

Max.Ten. Stress MPa

DH*

3.59 0.44 2.34 0.44

1.24 – – 0.004

6.46 0.44 2.34 0.44

0.35 – – 0.004

C H*

1.217

–

1.221

–

Concrete DH CH

principal stress distributions. As noted from the stress distributions, the magnitude of compressive principal stresses are higher in the fully bonded case than those in the no-tension slit case as the existence of the tension slit tends to create higher tensile stresses, which result in the reduction in the magnitude of initial in-situ compressive stresses. As expected, the notension slit relieves the concrete block just below the loading plate, which manifests itself as the principal stress directions become more vertical as compared with the fully bonded case. The maximum compressive stress and maximum tensile stress are summarized in Table 2.4. When this table is carefully examined, except concrete, there seems to be no difference between two cases. However, as noted from the overall principal stress distribution, there is a remarkable change in the overall distributions. On the other hand, the comparison of the safety factors may be more relevant as they will reflect the local changes more clearly. Figures 2.32 and 2.33 show the safety factor distributions against shearing and tensile failures, respectively. The minimum safety factors are listed in Table 2.5. The existence of no-tension slit has a very remarkable effect on the safety factor of concrete against tensile failure as it results in lower tensile stresses in concrete. On the contrary, the safety factors for rock formations tend to decrease in magnitude as expected. The safety factor in rock formation CH is about 5. The Gauss point at which these values are observed is very near the surface and is next to the anchorage body, as the given value for this rock formation corresponds to a very blocky rock mass with a very low value of tensile strength. In conclusion, it may be stated that the stress state at this point is not a representative for the overall anchorage body, and it could have no effect on the stability of the anchorage body. Next, we summarize and discuss the calculated results for the case of three times the design loading condition. The principal stress distributions are shown in Figure 2.34. Tables 2.6 and 2.7 compare the maximum compressive and tensile stresses and minimum safety factors in concrete and rock formations. Although the values of safety factors and maximum compressive and tensile stresses differ from the design load cases, the overall behavior and conclusions are same. The minimum safety factor distributions in concrete and rock are shown in Figures 2.35 and 2.36. When the minimum compressive stresses given in Table 2.8 for two loading conditions are compared, it seems that the increase in the intensity of load has almost no effect on the increment of compressive stress in rock. As pointed out in previous sections, the superimposed initial state of stress causes such an impression. When the superimposed in-situ stresses are

Figure 2.32  Comparison of safety factor distributions (concrete)

Figure 2.33  Comparison of safety factor distributions (rock)

Table 2.5  Comparison of minimum safety factors Material Type

Fully Bonded

No-tension Slit

SFS

SFT

SFS

SFT

DH*

2.96 15.43 21.12 5.34

1.03 9999 9999 5.51

2.69 15.21 20.66 5.25

2.70 9999 9999 5.18

C H*

10.29

9999

10.26

9999

Concrete DH CH

Figure 2.34  Comparison of principal stress distributions

Table 2.6  Comparison of maximum compressive and tensile stress Material Type

Concrete DH CH

DH* C

* H

Fully Bonded

No-tension Slit

Max. Comp. Stress MPa

Max.Ten. Stress MPa

Max. Comp. Stress MPa

Max.Ten. Stress MPa

8.62 0.44 2.34 0.44

5.85 0.015

17.24 0.019 2.34 0.44

3.21 – – 0.031

1.262

–

0.0296

1.275

–

Table 2.7  Comparison of minimum safety factors Material Type

Fully Bonded

No-tension Slit

SFS

SFT

SFS

SFT

DH*

0.375 4.72 6.76 1.34

0.23 2.17 9999 0.54

1.11 4.65 6.60 1.31

0.40 2.12 9999 0.53

C H*

3.13

9999

3.12

9999

Concrete DH CH

Figure 2.35  Comparison of safety factor distributions (concrete)

34  Surface rock engineering structures

Figure 2.36  Comparison of safety factor distributions (rock) Table 2.8  Comparison of stress increment in rock for two loading conditions Element No

324 344 324 324 324

Average Int. Stress

0.966 0.940 0.920 0.890 0.860

With Int. Stress 1T

3T

1.16 1.13 1.16 1.10 1.07

1.38 1.34 1.45 1.33 1.28

σ13T / σ11T

1.190 1.185 1.250 1.210 1.190

Without Int. Stress 1T

3T

0.194 0.190 0.240 0.210 0.210

0.414 0.400 0.530 0.440 0.420

σ13T / σ11T

2.13 2.11 2.21 2.10 2.00

excluded from the calculated results, the effect of such a load increase will be apparent. To show this, a calculation was carried out using the material properties given in Table 2.5. The FEM analyses indicated that the anchorage of a suspension bridge based on the pure gravitational and frictional concept is highly safe for the given geometry and material properties, which are determined very conservatively. The anchorage is even safe for the loads

Surface rock engineering structures  35

applied three times the design load. When the rock has a few discontinuities and a relatively high tensile strength, the occurrence of tensile crack in the concrete just under the loading plane does not cause any serious problem. On the other hand, when the rock has a number of discontinuities and low tensile strength, the concrete must be reinforced to behave as a monolithic body in order to reduce high tensile stresses in the rock, which may cause the failure of surrounding rock in the shape of a cone rather than as a cylindrical failure form.

2.6  Masonry structures 2.6.1  Shuri Castle arch gate A series of discrete finite element method (DFEM) analyses were performed on the masonry arch gates of Shuri Castle in Okinawa island, Japan. Figures 2.37 and 2.38 show a computational example on the arch gate under a horizontal seismic load and gravity. As noted, the failure occurs due to the sliding of the sidewall and subsequent rotation and fall of the arch blocks.

Figure 2.37  Mesh used in the DFEM analysis

Figure 2.38  Deformed configuration of the arch under horizontal seismic load

36  Surface rock engineering structures

2.6.2  Iedonchi masonry arch bridge Some numerical analyses by the discrete finite element method have been performed to investigate the effect of horizontal shaking using the seismic coefficient method. Figure 2.39 shows an example of the computation for a horizontal seismic coefficient of 0.2. Figure 2.40 shows the deformed configuration of the masonry arch bridge under gravitational load and the removal of one of the blocks on the left abutment. This computation was carried out to investigate the effect of configurations of the abutment of the bridge. As noted from the computed results, the removal of the rock block reduces the shear resistance of the abutment, the bridge starts to exhibit large deformation and it fails.

Figure 2.39 Deformed configuration of the arch bridge under horizontal seismic load (k = 0.2; ϕ = 10 degree)

Figure 2.40 Deformed configuration of the arch bridge under gravitational load with lesser abutment block (ϕ = 10 degree)

Surface rock engineering structures  37

2.7  Reinforcement of dam foundations The failure of the foundation of the Malpasset Arch Dam in France in 1959 is one of the most catastrophic events in rock engineering. This event had a profound effect on the design of dams on rock foundation with a strong emphasis on the effect of major structural weaknesses existing in the rock mass under high water pressures. Figure 2.41 shows some illustrations of possible modes of failure of gravity dams. The analyses of the foundations of dams can be done through some limiting equilibrium approaches for simple conditions. However, the use of numerical analyses with the consideration of discontinuities in rock mass would be necessary for complex conditions. 2.7.1  Limit equilibrium method for foundation design A dam subjected to base-shearing and reinforced by rock anchors shown in Figure 2.42 is considered. The safety factors of the dam against base shearing and overturning about point O may be easily obtained, respectively, as given here: Safety factor against base shearing SFs =

cLb + N tan ϕ (2.42) U s − Ta cos α

Figure 2.41 Possible failure modes of gravity dams: (a) base shearing, (b) planar sliding along a thoroughgoing discontinuity plane, (c) flexural or block toppling failure, (d) buckling failure Source: Modified and redrawn from U.S. Army Corps of Engineers (1994)

38  Surface rock engineering structures

Figure 2.42  Possible force conditions acting on gravity dam foundations Table 2.9  Properties of blocks, discontinuity and rock anchor used in analyses Blocks

Discontinuity

Rock Anchor

λ (MPa)

µ (MPa)

En (MPa)

GS (MPa)

h mm

σt (MPa)

C (MPa)

ϕ (o)

σt (MPa)

Ab (m2)

Gg (GPa)

100

100

5

1

10

0

0

25

200

0.1

3

Safety factor against overturning about point O SFm =

W ( Lb − xc ) + Ta ( ya cos α + ( Lb − xa ) sin α) (2.43) Hw 2 U s + LbU b + Nen 3 3

where N = W + Tα sin α ‒ Ub 2.7.2  Discrete finite element analyses of foundations A series of analysis using the discrete finite element method (DFEM) on a simple foundation model consisting of lower and upper blocks with a discontinuity plane between two blocks was carried out by changing the inclination of the rock anchor with respect to the normal of discontinuity plane and its yielding strength. Material properties used in the analyses are given in Table 2.9. The upper block was assumed to be subjected to a uniform 10 MN

Surface rock engineering structures  39

compressive normal load, while a 7 MN uniformly distributed shear load was applied over the discontinuity plane. Five different cases were analyzed (Figures 2.43–2.47) Case 1: No rock anchor (Figure 2.43) Case 2: Rock anchor inclined at an angle of 45 degrees with respect to the normal of discontinuity (Figure 2.44) Case 3: Rock anchor inclined at an angle of +45 degrees with respect to the normal of discontinuity (Figure 2.45) Case 4: Rock anchor inclined at an angle of 0 degree with respect to the normal of discontinuity and rock anchor behaves elastically (Figure 2.46) Case 5: Rock anchor inclined at an angle of 0 degree with respect to the normal of discontinuity and rock anchor yields (Figure 2.47). Tensile strength of rock anchor is reduced to 100 MPa. When the friction angle of the discontinuity plane was set to 40 degrees, the behavior was elastic and no relative slip occurred. The limiting friction angle for elastic behavior is 35 degrees, and if the friction angle of the discontinuity plane is less than 35 degrees, the relative slip is likely. The friction angle of the discontinuity plane was reduced from 40 degrees to 25 degrees. The computation results for this case (Case 1) is shown in Figure 2.43. The nonlinear analysis was based on the secant method. As noted from Figure 2.43, the relative slip of the upper block accelerates after each computation step. The installation of a rock anchor at different inclinations with respect to the normal of the discontinuity plane restricts the relative slip of the upper block. Particularly, the effect of the rock anchor is largest among all cases when its inclination is +45 degrees. When the relative slip of the discontinuity plane is considered, the largest relative slip occurs when the inclination is −45 degrees. On the other hand, the relative slip is smallest when the inclination of the rock anchor is +45 degrees.

Figure 2.43  Illustration of the DFEM model and deformed configurations (Case 1)

Figure 2.44  Illustration of the DFEM model and deformed configurations (Case 2)

Figure 2.45  Illustration of the DFEM model and deformed configurations (Case 3)

Figure 2.46  Illustration of the DFEM model and deformed configurations (Case 4)

Surface rock engineering structures  41

Figure 2.47  Illustration of the DFEM model and deformed configurations (Case 5)

Figure 2.48  Relative slip response of the upper block for different inclinations of rock anchor

When the inclination of the rock anchor is 0 degree, the relative slip is just between those for Case 2 and Case 3. However, if the rock anchor yields, the relative slip gradually increase after its yielding. The computational results are compared in Figure 2.48 as a function of computation step. It should be noted that the analysis is a pseudo dynamic type, and if the behavior of the analyzed domain remains elastic, no further deformation takes place, and it remains constant with respect to the increase of computation step number. This series of analyses clearly illustrates that rock anchors/rock bolts can be quite effective in reinforcing the foundations.

42  Surface rock engineering structures

2.8  Cylindrical sockets (piles) If the socket and surrounding ground behaves elastically, the solution of the resulting equation would take the following form (Aydan, 1989, 2018): wb = A1eα z + A2 e−α z (2.44) where α=

Gr 2Kr . , Kr = rb ln (r0 / rb ) Eb rb

rb, Eb, Gr are radius, elastic modulus of socket, and shear modulus of rock mass. Let us introduce the following boundary conditions for a socket with an end-bearing also (see Figure 2.49): σz = σ0   at  z = 0 (2.45a) wb = we  at  z = L(2.45b) Thus, integration constants A1, A2 of Equation (2.44) can be obtained as given here: A1 =

  1  w − σ0 e−αL  ,  −α L  e Eb α e + e   αL

A2 =

  1  w + σ0 eαL    −α L  e Eb α e + e   αL

Figure 2.49  Illustration of boundary conditions for a socket with an end bearing

(2.46)

Surface rock engineering structures  43

The axial displacement and axial stress of the socket and shear stress at the interface between the socket and surrounding rock can be expressed using the integration constants as follows: Axial displacement wb =

  1  w eα z + e−α z ) + σ0 (eα ( L− z ) − e−α ( L− z ) )   −α L  e ( Eb α e + e   αL

(2.47a)

Axial stress σb =

1 (−Ebαwe (eαz − e−αz ) + σ0 (eα( L−z ) + e−α( L−z ) ))  eαL + e−αL

(2.47b)

Interface shear stress τb =

  Kr  w eα z + e−α z ) + σ0 (eα ( L− z ) − e−α ( L− z ) ) (2.47c)  −α L  e (  Eb α e + e  αL

If the end of the socket is rigidly supported so that: we = 0

(2.48)

For this particular case, the axial displacement and axial stress of the socket and shear stress at the interface between the socket and surrounding rock can be expressed using the integration constants as follows: Axial displacement wb =

σ0 eα ( L− z ) − e−α ( L− z )  Eb α eαL + e−αL

(2.49a)

Axial stress σb = σ 0

eα ( L− z ) + e−α ( L− z )  eαL + e−αL

(2.49b)

Interface shear stress τb =

σ0 K r eα ( L− z ) − e−α ( L− z ) (2.49c) Eb α eαL + e−αL

However, it is very unlikely that the ends of the sockets would be rigidly supported. Therefore, the displacement of the surrounding ground beneath the socket tip may be approximated using the following equations proposed by Timoshenko and Goodier (1951, 1970), Aydan et al. (2008) and Aydan (2016): Timoshenko and Goodier (1951/1970) we =

π (1− ν r2 ) rb pe (2.50a) 2 Er

44  Surface rock engineering structures

Aydan et al. (2008), Aydan (2016) we =

(1 + ν r ) rb pe (2.50b) Er

where Er, υr are elastic modulus and Poisson’s ratio of rock mass, and pe is the pressure acting at the tip of the socket. pe is fundamentally unknown. One needs to obtain it, by requiring the continuity of displacement of the socket tip and rock mass beneath at z = L using an iterative technique such as Runga-Kutta or Newton-Raphson method. This method was applied to a reinforced concrete socket for a foundation of a bridge. The socket was 20 m long and had a diameter of 1.2 m. Table 2.10 gives the material properties. Table 2.10  Properties of blocks, discontinuity and rock anchor used in analyses Rock Burst

Socket

Er (MPa)

Vr

Es (GPa)

rs (m)

L m

σ0 (MPa)

720

0.25

20

0.6

20

2.476

Figure 2.50  Axial stress, displacement and shear stress distribution along the socket

Surface rock engineering structures  45

Figure 2.50 shows the distributions of axial stress, displacement and shear stress for two cases: For CASE 1, the end was rigidly supported, and for CASE 2, the socket displacement obeys Equation (2.47). The dotted lines correspond to CASE 1 while continuous lines correspond to CASE 2. As the socket is relatively long, the stress at the socket tip is only 7.53% of that at the socket top.

References Aydan, Ö. (1989) The Stabilisation of Rock Engineering Structures by Rockbolts. Doctorate Thesis, Nagoya University. Aydan, Ö. (2016) Time Dependency in Rock Mechanics and Rock Engineering, London, CRC Press, ISRM Book Series, No. 2, 246 pages, 9781138028630. Aydan, Ö. (2017) Rock Dynamics. CRC Press, Taylor and Francis Group, 462p, ISRM Book Series No. 3, ISBN 9781138032286. Aydan, Ö. (2018) Rock Reinforcement and Rock Support. CRC Press, Taylor and Francis Group, 486p, ISRM Book Series, No. 6, ISBN 9781138095830. Aydan, Ö., Daido, M., Ito, T., Tano, H. & Kawamoto, T. (2006) Prediction of post-failure motions of rock slopes induced by earthquakes. 4th Asian Rock Mechanics Symposium, Singapore, Paper No. A0356 (on CD). Aydan, Ö. & Kawamoto, T. (1992) The stability of slopes and underground openings against flexural toppling and their stabilisation. Rock Mechanics and Rock Engineering, 25(3), 143–165. Aydan, Ö. & Kumsar, H. (2010) An experimental and theoretical approach on the modeling of sliding response of rock wedges under dynamic loading. Rock Mechanics and Rock Engineering, 43(6), 821–830. Aydan, Ö., Mamaghani, I.H.P. & Kawamoto, T. (1996a) Application of discrete finite element method (DFEM) to rock engineering structures. NARMS’96. pp. 2039–2046. Aydan, Ö, Shimizu, Y. & Kawamoto, T. (1992) The stability of rock slopes against combined shearing and sliding failures and their stabilisation. Int. Symp. on Rock Slopes, New Delhi. pp. 203–210. Aydan, Ö. & Ulusay, R. (2002) Back analysis of a seismically induced highway embankment during the 1999 Düzce earthquake. Environmental Geology, 42, 621–631. Aydan, Ö., Ulusay, R. & Atak, V.O. (2008) Evaluation of ground deformations induced by the 1999 Kocaeli earthquake (Turkey) at selected sites on shorelines. Environmental Geology, Springer Verlag, 54, 165–182. Aydan, Ö., Ulusay, R., Kumsar, H. & Ersen, A. (1996b) Buckling failure at an open-pit coal mine. EUROCK’96. pp. 641–648. Aydan, Ö., Takahashi, Y., Iwata, N., Kiyota, R. & Adachi, K. (2018) Dynamic response and stability of un-reinforced and reinforced rock slopes against planar sliding subjected ground shaking. Journal of Earthquake and Tsunami, 12(4), 1841001. Ikeda, K. (1970). Classification of rock conditions for tunnelling. 1st Int. Congress on Engineering Geology, IAEG, Paris, 1258-1265. Kawamoto, T., Aydan, Ö., Shimizu, Y. & Kiyama, H. (1992) An investigation into the failure of a natural rock slope. The 6th Int. Symp. Landslides, ISL 92, 1, 465–470, Christchurch. Kovari, K. & Fritz, P. (1975). Stability analysis of rock slopes for plane and wedge failure with the aid of a programmable pocket calculator. 16th US Rock Mech. Symp., Minneapolis, USA, 25-33. Kumsar, H., Aydan, Ö. & Ulusay, R. (2000) Dynamic and static stability of rock slopes against wedge failures. Rock Mechanics and Rock Engineering, 33(1), 31–51. Mamaghani, I.H.P, Aydan, Ö. & Kajikawa, Y. (1999) Analysis of masonry structures under static and dynamic loading by discrete finite element method. JSCE Geotechnical Journal (626), 1–12. Timoshenko, S. & Goodier, J.N. (1951/1970) Theory of Elasticity. McGraw Hill Book Company, New York, 519 pages. Tokashiki, N. & Aydan, Ö. (2010) Kita-Uebaru natural rock slope failure and its back analysis. Environmental Earth Sciences, 62(1), 25–31. Tokashiki, N. & Aydan, Ö. (2011) A comparative study on the analytical and numerical stability assessment methods for rock cliffs in Ryukyu islands. The 13th International Conference of the

46  Surface rock engineering structures International Association for Computer Methods and Advances in Geomechanics, Melbourne, Australia. pp. 663–668. Ulusay, R., Ersen, A. & Aydan, Ö. (1995) Buckling failure at an open-pit coal mine and its back analysis. The 8th Int. Congress on Rock Mechanics, ISRM, Tokyo. pp. 451–454. Ulusay, R., Aydan, Ö. & Ersen, A. (2019) Assessment of a complex large slope failure at Kışlaköy open pit mine, Turkey. Proceedings of 2019 Rock Dynamics Summit in Okinawa. pp. 45–52. United States Army Corps of Engineers (1994) Engineering and Design: Rock Foundations. EM11101-2908, Washington, 121 pages.

Chapter 3

Applications to underground structures

3.1  Stress concentrations around underground openings The stability of underground openings during and after excavation is always of great concern to engineers. The size and location of possible yielding or failure zones are always necessary for providing support and reinforcement for safety (Aydan, 1989, 2018). For a quick assessment of approximate size and locations, elastic solutions are used together with some yield criteria. Common yield criteria follow. 3.1.1  Criteria for stability assessment (a)  Energy methods Energy methods have been used in mining for a long time, and it is based on the linear behavior of materials. One energy method is called strain energy, and it is expressed in terms of principal strain and stress components as follows (e.g. Jaeger and Cook, 1979; Aydan and Kawamoto, 2001): 1 Ws = [σ1ε1 + σ 2 ε2 + σ3 ε3 ] (3.1) 2 If Hooke’s law is introduced, Equation (3.1) takes the following form: Ws =

1  2 σ1 + σ12 + σ12 − 2ν (σ1σ 2 + σ 2 σ3 + σ3σ1 ) (3.2) 2 E 

For the uniaxial condition, that is, σ1 = σc, σ2 = 0, σ3 = 0, strain energy at the time of yielding is reduced to the following form: Wsu =

1  2 σc (3.3) 2 E  

In addition, the distortion energy concept or shear strain concept is introduced instead of the energy concept. The distortion energy is given in the following form: Wd = Ws −Wv (3.4) The volumetric strain energy may be written as: 1 Wv = [σv εv ] (3.5) 2

48  Underground structures

where 1 1 σv = (σ1 + σ 2 + σ3 ), εv = (ε1 + ε2 + ε3 ) (3.6) 3 3 Accordingly, the distortion energy may be rewritten as follows: Wd =

1  (σ1 − σ 2 ) 2 + (σ 2 − σ3 ) 2 + (σ3 − σ1 ) 2  (3.7) 4G 

Under the uniaxial condition, that is, Equation (3.6) reduces to the following form: Wdu =

1  2 σc (3.8) 2G  

However, it should be noted that it becomes difficult to define the energy when the material behavior becomes nonlinear. (b)  Extensional strain method Stacey (1981) proposed the extensional strain method for assessing the stability of underground openings in hard rocks. He stated that it was possible to estimate the spalling of underground cavities in hard rocks through the use of his extensional strain criterion. The extensional strain is defined as the deviation of the least principal strain from linear behavior (Figure 3.1). This definition actually corresponds to the definition of initial yielding in the theory of plasticity. This initial yielding is generally observed, at 40–60% of the deviatoric strength of materials.

Figure 3.1  Illustration of extensional strain concept Source: Aydan et al. (2001, 2004)

Underground structures  49

(c)  Elasto-plastic method In the elasto-plastic method, there are several models to model the strength of rock using the criteria of Mohr-Coulomb, Drucker-Prager, Hoek-Brown, Aydan. The simplest one is the brittle plastic model, in which the strength is abruptly reduced from the peak strength to its residual value. Aydan et al. (2001) recently combined both squeezing and rockbursting phenomena, and a more general strength reduction model is proposed as a function of strain level. This model, at least, treats both phenomena in a unified manner. 3.1.2 Stress distributions around underground openings in biaxial stress state The stress state around underground openings is evaluated using the semi-numerical technique proposed by Gerçek (1996). Figure 3.2 show principal stress distribution of an underground openings with different shapes subjected to biaxial far field. The largest far-field stress is inclined at an angle of 20 degrees. As expected, stress concentrations occur at

Figure 3.2  Principal stress distribution around underground openings

50  Underground structures

Figure 3.3  Potential yield/failure zone around underground openings

the lower-right and upper-left corners of the openings. Assessment of potential yield/failure zones around the openings are shown in Figure 3.3. Uniaxial compressive and tensile strength of intact rock was 218 and 8.0 MPa, respectively.

3.2 Dynamic excavation of circular underground openings The excavation of tunnels is done through drilling-blasting or mechanically such as TBM and/or excavators. The most critical situation on stress state is due to the drilling-­blasting– type excavation since the excavation force is applied almost impulsively. The dynamic response of circular tunnels during excavations under hydrostatic in-situ stress conditions can be given as (Figure 3.4): ∂σ r σθ − σ r ∂ 2u + = p 2 (3.9) r ∂r ∂t

Underground structures  51

Figure 3.4 Illustration of dynamic excavation of a circular opening under hydrostatic in-situ stress condition

where σr, σθ, u, ρ and r are radial, tangential stresses, radial displacement, density and distance from the center of the circular cavity, respectively. Let us assume that the surrounding rock behaves in a visco-elastic manner of Kelvin-Voigt type, which is specifically given as: σ r   D1   =  σθ   D2

D2  εr   C1  +  D1  εθ  C2

C2  εr     (3.10) C1  εθ 

where εr, εθ and εr , εθ are radial and tangential strain and strain rates, respectively. The strain and strain rates are related to the radial displacement in the following form: εr =

∂ε ∂ε u ∂u ; εθ = and εr = r ; εθ = θ (3.11) ∂t ∂t r ∂r

Eringen (1961) developed closed-form solution for Equation (3.9) under blasting loads. In order to deal with more complex boundary and initial conditions and material behavior, Equation (3.9) is preferred for solving using a dynamic finite element code. The discretized finite element form of Equation (3.9) together with the constitutive law given by Equation (3.10) takes the following form:  + CU + KU = F (3.12) MU Equation (3.12) has to be discretized in time-domain, and the resulting equation would take the following form: K * U n+1 = F *n+1 (3.13)

52  Underground structures

The specific form of matrices in Equation (3.13) may change depending upon the method adopted in the discretization procedure in time-domain. For example, if the central difference technique is employed, the final forms would be the same as those given in Subsection 10.2. A finite element code has been developed by the author and used in the examples presented in this subsection. In this subsection, the dynamic response of a circular tunnel under the impulsive application of excavation force is presented. The results were initially reported in Aydan (2011). Figure 3.5 shows the responses of displacement, velocity and acceleration of the tunnel surface with a radius of 5 m. As noted from the figure, the sudden application of the excavation force, in other words, and the sudden release of ground pressure result in 1.6 times the static ground displacement at the tunnel perimeter, and shaking disappears almost within 2 s. As time progresses, it becomes asymptotic to the static value, and velocity and acceleration disappear. The resulting tangential and radial stress components near the tunnel perimeter (25 cm from the opening surface) are plotted in Figure 3.6 as a function of time. It is of great interest

Figure 3.5  Responses of displacement, velocity and acceleration of the tunnel surface

Figure 3.6 Responses of radial and tangential stress components nearby the tunnel surface (25 cm away from the perimeter)

Underground structures  53

that the tangential stress is greater than that under static condition. Furthermore, very high radial stress of tensile character occurs near the tunnel perimeter. This implies that the tunnel may be subjected to transient stress state, which is quite different from that under static conditions. However, if the surrounding rock behaves elastically, they will become asymptotic to their static equivalents. In other words, the surrounding rock may become plastic even though the static condition may imply otherwise.

3.3  Evaluation of tunnel face effect Advancing tunnels utilizing support systems consisting of rock bolts, shotcrete, steel ribs and concrete lining are three-dimensional complex structures and is a dynamic process. However, tunnels are often modeled as one-dimensional axisymmetric structures subjected to hydrostatic initial stress state as a static problem. The effect of tunnel face advance on the response and design of support systems is often replaced through an excavation stress release factor determined from pseudo three-dimensional (axisymmetric) or pure threedimensional analyses as given here: f =

e−bx / D (3.14) 1 / B + e−bx / D

where x is distance from tunnel face, and the values for coefficients B and b suggested by Aydan (2011) are 2.33 and 1.7, respectively. Figure 3.7a illustrates an unsupported circular tunnel subjected to an axisymmetric initial stress state. The variation of displacement and stresses along the tunnel axis were computed using the elastic finite element method. The radial displacement at the tunnel wall is normalized by the largest displacement and is shown in Figure 3.7b. As seen from the figure, the radial displacement takes place in front of the tunnel face. The displacement is about 28–30% of the final displacement. Its variation terminates when the face advance is about +2-D. Almost 80% of the total displacement takes place when the tunnel face is about +1-D. The effect of the initial axial stress on the radial displacement is almost negligible. Figure 3.7c shows the variation of radial, tangential and axial stress around the tunnel at a depth of 0.125R. As noted from the figure, the tangential stress gradually increases as the distance increases from the tunnel face. The effect of the initial axial stress on the tangential stress is almost negligible. The radial stress rapidly decreases in the close vicinity of the tunnel face, and the effect of the initial axial stress on the radial stress is also negligible. The most interesting variation is associated with the axial stress distribution. The axial stress increases as the face approaches, and then it gradually decreases to its initial value as the face effect disappears. This variation is limited to a length of 1R(0.5 D) from the tunnel face. It is also interesting to note that if the initial axial stress is nil, even some tensile axial stresses may occur in the vicinity of tunnel face. Figure 3.7d shows the stress distributions along the r-axis of the tunnel at various distances from the face when the initial axial stress is equal to initial radial and tangential stresses. As noted from the figure, the maximum tangential stress is 1.5 times the initial hydrostatic stress, and it becomes twice the distance from the tunnel face is +5R, which is almost equal to theoretical estimations for tunnels subjected to the hydrostatic initial stress state. The stress state near the tunnel face is also close to that of the spherical opening subjected to the hydrostatic stress state. The stress state seems to change from spherical state to the cylindrical state (Aydan, 2011). It should be noted that it would be almost impossible to simulate exactly the same displacement and stress changes of 3-D analyses in the vicinity of

54  Underground structures

Figure 3.7 (a) Computational model for elastic finite element analysis, (b) normalized radial displacement of the tunnel surface, (c) normalized stress components along the tunnel axis at a distance of 0.125R, (d) the variation of stresses along the r-direction at various distances from the tunnel face

tunnels by 2-D simulations using the stress-release approach irrespective of the constitutive law of surrounding rock as a function of distance from the tunnel (Aydan et al., 1988; Aydan and Geniş, 2010).

3.4  Abandoned room and pillar lignite mines When abandoned lignite mines and quarries are of room and pillar type, their short-term and long-term stability may be evaluated using some simple analytical techniques. Roof stability is generally evaluated using beam theory and/or arching theory under gravitational, earthquake and point loading (i.e. Coates, 1981; Obert and Duvall, 1967; Aydan, 1989; Aydan and Tokashiki, 2011). The tributary area method is quite widely used in mining engineering for assessing the pillar stability. Aydan and his coworkers (Aydan and Geniş, 2007; Aydan and Tokashiki, 2011; Geniş and Aydan, 2013) extended the method to cover the effects of earthquake and point loading, in addition to gravitational loading, creep and degradation of geomaterials, in order to evaluate the stability of roof and pillars (Figure 3.8).

Underground structures  55

Figure 3.8  Models for roof and pillars Source: From Aydan and Genis¸ (2007)

Figure 3.9 Illustration of finite element models with boundary conditions for roof of natural underground openings

Aydan and Tokashiki (2011) performed a series of finite element analyses for assessing the stability of roofs of shallow natural underground openings. Basically, two different situations were considered (Figure 3.9): 1 Beam with or without cracks 2 Beam on soft and rigid elastic foundations with or without cracks at abutments and center of the beam

56  Underground structures

All models are assumed to be subjected to body forces together with a distributed load due to the dead weight of 5 m thick topsoil with a unit weight of 22 kN m−3. The material properties used in the analyses are given in Table 3.1. As the problem is symmetric, half of the region is modeled in finite element analyses (Figure 3.9). 3.4.1  Beams with and without cracks Figure 3.10 shows the principal stresses in roof beam with and without cracks. Cracks were assumed to propagate to the half thickness of the roof beam at abutments and at the center. As seen in Figure 3.10a, the principal stress distribution is almost the same as what could be obtained from the bending theory of beams. The tensile stress is highest at the uppermost fiber of the roof beam at abutments. When the crack occurs, tensile stresses are drastically reduced in the roof beam. Nevertheless, high tensile stresses occur near the tips of cracks. The compressive stresses become higher at abutments and the top center of the cracked beam, as expected. Definitely, the arch action is induced in the cracked beam.

Table 3.1  Material properties used in analyses Layer

Elastic Modulus (GPa)

Poisson’s ratio

Unit weight (kN m−3)

Roof Foundation Abutment

0.6 6 60

0.25 0.25 0.25

23 23 23

Figure 3.10  Principal stress distributions in beams with and without cracks

Underground structures  57

3.4.2  Beams on elastic foundations with overburden soil Figure 3.11 compares the principal stress distributions of beams with or without topsoil on soft elastic foundations, whose elastic modulus is 10 times that of the beam. As noted from the figure, stresses are lower than those shown in Figure 3.10. This is due to the stresses being distributed over a large area and the consideration of Poisson’s ratio in finite element analyses. Furthermore, the highest tensile stress occurs at the center of the lowermost fiber of the beam. The highest tensile stress occurs near the ground surface slightly farther away from the abutment. These results are somewhat different from those of the beam rigidly supported at abutments. The existence of the topsoil increases the amplitude of principal stresses. Nevertheless, the overall response remains basically the same. 3.4.3 Roof beams supported by rigid abutments with or without cracks The elastic modulus of the ground and beam beyond abutments was increased to 100 times that of the roof beam. Furthermore, the existence of cracks was also considered in the analyses. Figures 3.12 and 3.13 show the principal stress distributions and deformed configurations of the models with or without cracks. If the rigidity of the abutments increases, the stress state approaches those of the single beam. Similarly, the presence of cracks causes a stress state close to those of the cracked beam shown in Figure 3.10(b). As shown, simple analytical models and computations from the two-dimensional elastic finite element method yield very similar results. Nevertheless, stresses computed from the FEM in the roof are less than those computed from the beam theory with built conditions. On the other hand, stresses computed from the FEM in pillars are slightly higher than those computed from the tributary area method. However, the stress state in the roof would be quite different if the opening depth increases. In such cases, the effects of gravitational load in the stress state of the roof should be also taken into account. Nevertheless, the stability of

Figure 3.11  Principal stress distribution of beams on elastic foundations

58  Underground structures

Figure 3.12  Deformed configurations and principal stress distributions (without cracks)

Figure 3.13  Deformed configurations and principal stress distributions (with cracks)

pillars become more important than the roof itself under such conditions, and the tributary area method would yield quite reasonable values for the stress state in pillars for stability assessment. 3.4.4 Stress state in an abandoned room and pillar mine beneath Kyowa Secondary School under static and earthquake loading Figure 3.14 shows the distribution of minimum principal stress (tension is positive) for an abandoned room and pillar mine beneath Kyowa Secondary School in Mitake Town of Gifu Prefecture, Japan, under the static loading condition. Although the maximum pillar stresses

Underground structures  59

Figure 3.14  Contours of minimum principal stress beneath the Kyowa Secondary School Table 3.2  Material properties of layers Layer

γ kN m−3

E MPa

v

c MPa

ϕ (o)

Topsoil Upper Mst-Sst Lignite Lower Mst-Sst Chert

19 19 14 19 19

270 750 400 1073 3647

0.35 0.3 0.3 0.3 0.3

0.0 0.7 0.66 1.00 3.00

38 25 45 45 45

are slightly higher than those computed from the tributary area method, the quick stability assessment using the tributary area method should be quite acceptable. This area would be subjected to the anticipated Nankai-Tonankai-Tokai earthquake in the future, and there is a great concern about it. That authors have been involved with the stability assessment of the abandoned lignite mine beneath Kyowa Secondary School during the anticipated Nankai-Tonankai-Tokai earthquake (Aydan et al., 2012; Geniş and Aydan, 2013). Material properties of investigated ground are given in Table 3.2. The authors carried out 1-D, 2-D and 3-D dynamic simulations for an estimated base ground motion data at Mitake Town obtained from the method of Sugito et al. (2001). Figure 3.15 illustrates the numerical model of the ground and abandoned lignite mine beneath the Kyowa Secondary School. Figure 3.16 shows the computed responses from 1-D and 3-D numerical analyses. It is interesting to note that responses from 1-D and 3-D analyses are quite similar to each other. A three-dimensional elasto-plastic numerical analysis of abandoned lignite mine beneath the Kyowa Secondary school (Figure 3.17) uses the estimated ground motion record (Figure 3.18), based on the methods developed by Sugito et al. (2000) and Aydan (2012) for the anticipated Nankai-Tonankai-Tokai mega earthquake. The Nankai earthquake terminates at 43 s, and the Tonankai earthquake starts and terminates at 75 s. The last earthquake is Tokai earthquake, and it terminates at about 125 s.

Figure 3.15  Illustration of models used in numerical analyses and selected section

Figure 3.16  Acceleration responses at selected section from 1-D and 3-D numerical analyses

Figure 3.17  3-D views of the numerical model and abandoned room and pillar mine

Figure 3.18  Input base acceleration record

Underground structures  61

Figure 3.19  Failure state of pillar of abandoned mine at different time steps

The failure of the pillars starts at the deepest site and propagates towards shallower parts as estimated from 2-D numerical analyses (Figure 3.19). The failure state of pillars can be broadly classified as total failure and partial failure, as illustrated. When the Nankai earthquake terminates, about 60% of the pillars were already totally or partially yielded. The Tonankai earthquake is the nearest one to the Mitake town, and all the pillars are in total failure state. The Tokai earthquake has no further effect on the failure state.

3.5  Karstic caves Karstic caves are quite common worldwide whenever limestone and evaporate deposits exist. In the coral limestone formation in the Ryukyu islands of Japan, there are many karstic caves, which present many geoengineering problems. The authors are involved with the stability assessment of some karstic caves in relation to some engineering projects or

62  Underground structures

preservation of some monumental structures (i.e. Tokashiki, 2011; Aydan and Tokashiki, 2011; Geniş et al., 2009). There is a huge karstic cave beneath the Himeyuri monument in Okinawa island. The enlargement of the monument was considered, and the authors were consulted on whether the karstic cave would be stable upon the enlargement. Figure 3.20 shows a view of the monument and the beam modeling of the overhanging part. Table 3.3 and Figure 3.21 compares the maximum tensile and compressive stresses computed from beam theory and FEM. Despite some slight differences, the results are quite similar.

Figure 3.20 View and beam modeling of overhanging part Table 3.3  Comparison of maximum compressive and tensile stresses from FEM and bending theory Loading Condition

Max.Tensile Stress (MPa)

Max. Compressive Stress (MPa)

FEM

Theory

FEM

Theory

Natural Present Planned-2

0.557 0.631 0.770

0.677 0.713 0.991

−1.363 −1.402 −1.478

−0.677 −0.713 −0.991

Figure 3.21  Comparison of stresses obtained from the bending theory and FEM

Underground structures  63

Figure 3.22  Computed maximum principal stress distributions from 3-D numerical analysis

Figure 3.23  Numerical model of the tomb and assumed in-situ stress state

A 3-D analysis of the vicinity of Himeyuri monument and the cave beneath it was carried out with the consideration of surface loading due to the deadweight of the monument structure (Aydan et al., 2011). The cave was considered to be circular in plain view. The maximum tensile stress was much smaller than that computed from the bending theory and 2-D FEM analysis. An additional axisymmetric FEM analysis was also performed, and it yielded similar results. However, the cave has an ovaloid shape in plan, and the actual stress state is expected to be closer to that of 2-D analyses. Furthermore, there are some cross-joints in the rock mass so that the actual stress state should be quite close to that of 2-D-FEM analyses (Figure 3.22).

3.6  Stability analyses of tomb of Pharaoh Amenophis III Three-dimensional elasto-plastic numerical analyses of the tomb were carried out for 12 different situations using the FLAC code developed by ITASCA (1997) (Figure 3.23). Only two cases, in which the properties of rock mass are assumed to be equivalent to those of

64  Underground structures Table 3.4  Considered rock mass properties. sc (MPa)

f

E (GPa)

C (MPa)

st (MPa)

n

g (kN m−3)

Remarks

20 5

35 30

2 0.8

5.2 1.44

2 0.5

0.25 0.25

20 20

Dry Wet

Figure 3.24  Stress distribution and plastic zone development around the tomb

intact rock under dry and wet conditions, as given in Table 3.4, are presented (Aydan et al., 2008, Egypt). The computational results indicated that if rock mass is assumed to be dry, there should be no plastic zone development in rock mass around the tomb (Figure 3.24). However, if rock becomes saturated, the computational results indicated that the damage in the walls between J-room and Jd-room has a great influence on the overall stability of the J-room and adjacent rooms. the saturation of rock mass from time to time due to floods may also have a negative effect on the stability of the tomb.

3.7  Retrofitting of unlined tunnels The Unten tunnel in Nakijin region of Okinawa island was an unlined single-lane roadway tunnel. Following the collapse of Toyohama tunnel in Hokkaido island in 1996, the authorities were ordered to check the safety of all roadway tunnels in Japan. The Unten tunnel was designated as unsafe after the checking procedure, and it was decided to close it to traffic. However, the strong demand by local residents to keep the tunnel open to traffic resulted in the reassessment of the stability of the tunnel and its retrofitting. The site investigations revealed several thoroughgoing discontinuities as shown in Figure 3.25. Model experiments using the base friction apparatus indicated that the tunnel might

Underground structures  65

be unstable if the frictional properties of discontinuities decrease with time. The reduction of friction angle of discontinuities was achieved by introducing the double-layer Teflon sheets along discontinuities in the model tests. The discrete finite element method (DFEM) was used to assess the stability of the tunnel (see Tokashiki et al., 1997, for the details of numerical analyses). The DFEM analyses also indicated that the tunnel might become unstable if the frictional properties of discontinuities were drastically reduced. For retrofitting the tunnel, glass-fiber rock bolts were installed, and reinforced concrete lining was constructed. Figure 3.26 shows the deformed configurations

Figure 3.25  Distribution of major discontinuities around Unten tunnel

Figure 3.26  Deformation of surrounding rock mass with/without the measures of retrofitting

66  Underground structures

of the tunnel without and with retrofitting, respectively. As noted from the figures, the tunnel should be stable if the selected measures of retrofitting are employed.

3.8 Temperature and stress distributions around an underground opening The example given here is concerned with temperature fluctuation in rock mass in an underground opening subjected to temperature variation applied on the surface of the opening (see details by Aydan et al., 2008). The temperature fluctuation in the opening was based on actual measurements, and it was subjected to ±10 degrees yearlong sinusoidal temperature variation. Temperature distribution is computed from an FEM program based on the theory presented in Chapters 4 and 7 of the Volume 1 of this book. Thermo-mechanical properties used in computations are given in Table 3.5. Figure 3.27 shows the temperature distribution and associated maximum principal and maximum shear stress distributions in rock mass around the underground opening (Figure 3.28). Day 99 corresponds to the highest temperature, and Day 272 corresponds to the lowest temperature in the cavern. As noted from the figure, temperature and principal stress distributions are reversed while the maximum shear stress remains the same at both extreme values of temperature fluctuations.

3.9 Waterhead distributions around a shallow underground opening An example of waterhead distributions in rock mass around an underground opening was investigated. The finite element program was based on the program developed by Verruijt (1982). As expected, the groundwater in rock mass above the opening was close to drained condition (Figure 3.29). Table 3.5 Thermo-hydro-mechanical properties of surrounding rock mass Unit weight Elastic Modulus Poisson’s Cohesion Friction Thermal Diffusivity Thermal Expansion (kN m−3) (GPa) ratio (MPa) Angle (o) (m2 day−1) Coefficient (1 oC−1) 26

5–10

0.25

3

40

0.1

Figure 3.27  Applied temperature at the surface of the cavern

1.0 × 10 –5

Figure 3.28  Computed temperature stress distributions

Figure 3.29 FEM mesh and computed water head distribution in rock mass around underground opening

68  Underground structures

References Aydan, Ö. (1989) The Stabilisation of Rock Engineering Structures by Rockbolts. Doctorate Thesis, Nagoya University, Faculty of Engineering. Aydan, Ö. (2004) Damage to abandoned lignite mines induced by 2003 Miyagi-Hokubu earthquakes and some considerations on its possible causes. Journal of School of Marine Science and Technology, 2(1), 1–17. Aydan, Ö. (2011) Some issues in tunnelling through rock mass and their possible solutions. Proc. First Asian Tunnelling Conference, Tehran, ATS-15. pp. 33–44. Aydan, Ö. (2012) Ground motions and deformations associated with earthquake faulting and their effects on the safety of engineering structures. Encyclopedia of Sustainability Science and Technology, Springer, New York. R. Meyers (Ed.), pp. 3233–3253. Aydan, Ö. (2016) The state of art on large cavern design for underground powerhouses and long-term issues. In: The Second Volume of Encyclopedia on Renewable Energy. John Wiley and Sons. New York. pp. 467–487. Aydan, Ö., Daido, M., Owada, Y., Tokashiki, T. & Ohkubo, K. (2004) The assessment of rock bursting in rock engineering structures with a particular emphasis on underground openings. 3rd Asian Rock Mechanics Symposium, Kyoto, Vol. 1, pp. 531–536. Aydan, Ö. & Geniş, M. (2004) Surrounding rock properties and openings stability of rock tomb of Amenhotep III (Egypt). ISRM Regional Rock Mechanics Symposium, Sivas. Pp. 191–202. Aydan, Ö. & Geniş, M. (2007) Assessment of dynamic stability of an abandoned room and pillar underground lignite mine. Rock Mechanics Bulletin. Turkish National Rock Mechanics Group, ISRM, Ankara, No. 16. pp. 23–44. Aydan, Ö. & Geniş, M. (2010) A unified analytical solution for stress and strain fields about radially symmetric openings in elasto-plastic rock with the consideration of support system and longterm properties of surrounding. International Journal of Mining and Mineral Processing (IJMMP), 1(1–32). Aydan, Ö., Geniş, M., Akagi, T. & Kawamoto, T. (2001) Assessment of susceptibility of rockbursting in tunneling in hard rocks. International Symposium on Modern Tunnelling Science and Technology, IS-KYOTO, 1, 391–396. Aydan, Ö., Geniş, M., Sugiura, K. & Sakamoto, A. (2012) Characteristics and amplification of ground motions above abandoned mines. First International Symposium on Earthquake Engineering, JAEE, Tokyo, Vol. 1, pp. 75–84. Aydan, Ö. & Kawamoto, T. (2001) The stability assessment of a large underground opening at great depth. 17th International Mining Congress and Exhibition of Turkey, Ankara. pp. 277–278. Aydan, Ö., Kyoya, T., Ichikawa, Y., Kawamoto, T., Ito, T. & Shimizu, Y. (1988). Three-dimensional simulation of an advancing tunnel supported with forepoles, shotcrete, steel ribs and rockbolts. The 6th International Conference on Numerical and Analytical Methods in Geomechanics, Innsbruck, Austria. Vol. 2, pp. 1481–1486. Aydan, Ö., Ohta, Y., Daido, M., Kumsar, H. Genis, M., Tokashiki, N., Ito, T. & Amini, M. (2011) Chapter 15: Earthquakes as a rock dynamic problem and their effects on rock engineering structures. In Advances in Rock Dynamics and Applications, Editors Y. Zhou and J. Zhao, CRC Press, Taylor and Francis Group, London, pp. 341–422. Aydan, Ö. & Tokashiki, N. (2011) A comparative study on the applicability of analytical stability assessment methods with numerical methods for shallow natural underground openings. The 13th International Conference of the International Association for Computer Methods and Advances in Geomechanics, Melbourne, Australia. pp. 964–969. Aydan, Ö., Tsuchiyama, S., Kinbara, T., Uehara, F., Tokashiki, N. & Kawamoto, T. (2008) A numerical analysis of non-destructive tests for the maintenance and assessment of corrosion of rockbolts and rock anchors. The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG), Goa, India. pp. 40–45.

Underground structures  69 Coates, D.F. (1981) Rock Mechanics Principles. Canadian Government Pub Centre, Ottawa, 410 pages. Eringen, A.C. (1961) Propagations of elastic waves generated by dynamical loads on a circular cavity. Journal of Applied Mechanics, ASME, 28, 218–222. Geniş, M. & Aydan, Ö. (2013) A numerical study on the ground amplifications in areas above abandoned room and pillar mines and longwall old mines. The 2013 ISRM EUROCK International Symposium, Wroclaw, pp. 733–738. Geniş, M., Tokashiki, N. & Aydan, Ö. (2009) The stability assessment of karstic caves beneath Gushikawa Castle remains (Japan). EUROCK 2010, pp. 449–454. Gerçek, H. (1996) Special elastic solutions for underground openings. Milestones in Rock Engineering: The Bieniawski Jubilee Collection, Balkema, Rotterdam. pp. 275–290. ITASCA. 1997 FLAC3D-Fast Lagrangian Analysis of Continua (Version 2.0). 5 Vols. Minneapolis: Itasca Consulting Group, Inc. Jaeger, J.G. & Cook, N.G.W. (1979) Fundamentals of Rock Mechanics. 3rd Ed. Chapman and Hall, London. Obert, L. & Duvall, W.I. (1967) Rock Mechanics and the Design of Structures in Rock. John Wiley & Sons, New York. Sugito, M., Furumoto, Y. & Sugiyama, T. (2000) Strong motion prediction on rock surface by superposed evolutionary spectra, 12th World Conference on Earthquake Engineering, 2111/4/A, CD-ROM. Stacey, T.R. (1981) A simple extension strain criterion for fracture of brittle rock. International Journal Rock Mechanics and Mining Science & Geomechanics Abstracts, Oxford, 18, 469–474. Tokashiki, N. (2011) Study on the Engineering Properties of Ryukyu Limestone and the Evaluation of the Stability of its Rock Mass and Masonry Structures. PhD Thesis, 221p., Engineering and Science Graduate School, Waseda University. Tokashiki, N., Aydan, Ö., Shimizu, Y. & Kawamoto, T. (1997) The assessment of the stability of a very old tunnel by discrete finite element method (DFEM), Numerical Methods in Geomechanics, NUMOG VI, Montreal. pp. 495–500. Verruijt, A. (1982) Groundwater Flow. 2nd Ed. Macmillan Press Ltd., London, 144 pages.

Chapter 4

Rock mass classifications and their engineering utilization

4.1 Introduction Rock mass classifications are used for various engineering design and stability analyses, and they are initially proposed for the design of a given rock structure. However, this trend has been now changing, and the main objectives of rock mass classifications have become to identify the most significant parameters influencing the behavior of rock masses, to divide a particular rock mass formulation into groups of similar behavior, to provide the characterizations of each rock mass class, to derive quantitative data and guidelines for engineering design, and to provide a common basis for engineers and geologists. These are based on empirical relations between rock mass parameters and engineering applications like tunnels and other underground caverns. In the empirical methods, rock mass classification systems are extensively used for feasibility and predesign studies and often also for the design. Although the history of rock classifications for a given specific structure is old, the rock mass classification system proposed by K. Terzaghi (1946) for tunnels with steel set support has become the basis for the follow-up quantitative rock mass classifications. Currently, there are many classification systems in rock engineering, particularly in tunneling, such as Rock Mass Rating (RMR) (Bieniawski, 1974, 1989), Q system (Barton et al., 1974), RSR (Wikham et al., 1974), Rock Mass Quality Rating (RMQR) proposed by Aydan et al., 2014. In addition, rock mass classifications of NEXCO (known as DOROKODAN) and JR (KYU-KOKUTETSU) are commonly used to design tunnels in Japan. Nevertheless, the utilization of these systems for characterization of complex rock mass conditions is a challenge for engineers and is not always possible in some cases. In this chapter, several classification systems have been briefly explained, and computations have been done based on RMQR.

4.2  Rock Mass Rating (RMR) Bieniawski (1974) published the details of a rock mass classification called the Geomechanics Classification or the Rock Mass Rating (RMR) system. Over the years, this system has been refined as more case records have been examined, and the reader should be aware that Bieniawski (1989) has made significant changes in the ratings assigned to different parameters and that the 1989 version of the classification is suggested by Bieniawski (1989). The following six parameters are used to classify a rock mass using the RMR system (Table 4.1): 1 2 3 4 5 6

Uniaxial compressive strength (UCS) of rock material Rock Quality Designation (RQD) Spacing of discontinuities Condition of discontinuities Groundwater conditions Orientation of discontinuities

>250 MPa

Uniaxial comp. strength

5

4

3

2

>10 MPa

Point-load strength index

1

10

Damp

Completely dry 15

45

Cohesion of rock mass (kPa) Friction angle of rock mass (deg)

Weathering Ratings

Unweathered 6

E. Guidelines for Classification of Discontinuity conditions Discontinuity length (persistence) 20 m 0 > 5 mm 0 Slickensided 0 Soft filling >5 mm 0

DS ≥ 0.07 m 0.07 m > DS 24 m ≥ 0.3 m RQD= 100 100 > 75 > RQD ≥ 35 35 > RQD RQD2 ≥ 75 20 16 12 8 4 1–0 None Healed or Rough Relatively Slicken sided with thin Thick fill or separation (t > 10 mm) intermittent smooth infill or separation (t < and light 5 mm) 30 26 22 15 7 1 Or, alternatively, excluding “None” and “Healed or intermittent” classes Aperture or separation None 0.1–0.25 0.25–0.5 0.5–2.5 2.5–10 mm > 10 mm or Very mm mm mm tight, < 0.1 mm Rating (R DCA ) 6 5 4 3 2 1 Infilling None Surface Thin Thin filling Thick filling 60 > t Very thick filling staining coating < 1 1 < t 10 > 10 mm or shear zone t only mm mm > 60 mm 6 5 4 3 2 1 Rating (R DCI ) Roughness Descriptive Very Rough Smooth Smooth Slicken-sided Shear band/ rough undulating planar zone Profile No. in 10 9 8 7 6 5 4 3 2 1–0 ISRM (2007)

Fresh

Degradation degree (DD)

Table 4.5  Rating of parameters of RMQR rock classification system

6

7 Capillarity or electrically absorptive 5

Rating (R DCR) Damp

9 Non absorptive

Dry

*RQMR = RDD + RDSN + RDS + RDC + RGWSC + RGWAC

Groundwater seepage condition (GWSC) Rating (RGWSC ) Groundwater absorption condition (GWAC) Rating RGWAC )

9

4

5 Slightly absorptive

10 Wet

8

3

3 Moderately absorptive

7 6 Dripping

4

3

2

1 Highly absorptive

5 Rowing

1–0

0 Extremely absorptive

2 1–0 Gushing

84  Rock mass classifications

response of rock masses, weathering and/or the negative action of hydrothermal alteration may be accounted for as the degradation degree (DD) of intact rock. Groundwater is also an important parameter affecting the mechanical response of rock masses. There are also cases, where some rocks may absorb groundwater electrically or chemically, resulting in the drastic reduction of material properties and/or swelling. RMQR system incorporates important parameters of the available quantitative modern rock classifications, and Table 4.5 provides the descriptions of each parameter and their ratings. In the following subsections, first, the basic concepts involving each parameter and their ratings on the basis of knowledge gained in rock mechanics and rock engineering are briefly explained. (a)  Degradation Degree (DD) The degradation processes generally cause weakening of the bonds between particles or grains constituting rocks, and, physically, they cause the reduction of the strength and deformation modulus of intact rock and also influence the joint spacing and discontinuity filling material in the form of clay. Therefore, in RMQR, degradation degree, which is considered as one of the elements of the joint condition parameter in some previously developed classifications, is taken as one of the input parameters. (b)  Discontinuity set number (DSN) The rock mass structurally would have at least one discontinuity set associated with the surface shape of erosion. There may be some cases where rock mass is completely shattered and crushed. Therefore, the discontinuous nature of rock masses may be described through some adjectives, such as none, one set plus random, two sets plus random, three sets plus random, four sets or more, and crushed/shattered. It should be noted that, if the discontinuity set number is four or more, it would definitely imply that it was subjected to tectonic events in the past. (c)  Discontinuity spacing (DS) The modern rock mass classifications consider that the rock mass is massive when the discontinuity spacing is greater than 2–3 m. This definition may not be so important when the underground openings have a smaller size, say, less than 8–6 m in diameter or span. However, when one considers the present common size of major underground powerhouses and storage caverns for crude oil and gas, the rock mass around the underground opening would look very blocky. Therefore, the present discontinuity spacing definitions are not compatible with actual circumstances, and it needs some improvements with consideration of the actual size of underground structures. To describe the representative discontinuity spacing, RMQR includes six categories of discontinuity spacing, as given in Table 4.5 with their ratings. As understood from Table 4.3, RQD is not sensitive to the variation of discontinuity spacing greater than 1 m, and RQD should not be used to determine the rating for discontinuity spacing if the discontinuity spacing is greater than 1 m. However, by considering that RQD is a commonly used parameter, particularly in borehole cores, it is also included in Table 4.5 as an alternative parameter to discontinuity spacing, provided that it is inferred to be less than 1 m and free of drilling-induced disturbance.

Rock mass classifications  85

(d)  Discontinuity condition (DC) The causes of the formation of discontinuities in rock masses are various, and the condition of discontinuities is closely related to their genesis. The condition of discontinuities not involving tectonic events are generally favorable unless they are filled with clayey material or discontinuity walls are subjected to weathering. However, the tectonically induced fractures may be associated with relative shear displacement, and they may produce slickensided discontinuities with a certain thickness of clayey gouge. Such conditions would considerably reduce the shear strength of discontinuities, and they may be squeezed out under redistributed in-situ stress or washed away under high groundwater pressure. Table 4.6 describes the possible discontinuity conditions and the ratings suggested for visual observations. (e)  Groundwater condition The effects of groundwater on rock mass are described through adjectives such as dry, damp, wet, dripping, flowing and gushing. It is known that the strength and deformation modulus of weak rocks such as water-absorbing rocks decrease drastically with water content. It is also reported that even such properties of hard rocks may decrease with saturation. Rocks containing water-absorbing minerals have this feature, and the geomechanical properties of the surrounding rock mass may be drastically reduced. Furthermore, it may also show large volumetric changes (swelling, contraction) during excavation and cyclic groundwater changes and/or disintegration. Therefore, in addition to the seepage condition of groundwater (GWSC), the groundwater absorption characteristics of rocks (GWAC) are taken into account. The descriptions and their ratings for these two characteristics are determined from Table 4.5. The value of RMQR ranges between 0 and 100. Rock mass is divided into six classes and their rating ranges are given in Table 4.7.

Table 4.6 The possible discontinuity conditions and the ratings suggested for visual observations Aperture or separation

None or 0.1–0.25 0.25–0.5 mm 0.5–2.5 mm 2.5–10 mm >10 mm very tight, mm 10 mm

5 Rough

3 Smooth planar 5 4

2 Slickensided 3 2

1–0 Very thick filling or shear zones t > 60 mm 1–0 Shear band/ zone 1–0

5

3

1–0

9

8

4 Smooth undulating 7 6

9

8

7

6

4

2

86  Rock mass classifications

(f)  Application of RMQR to rock support design The design of support systems of tunnels in rock engineering is of great importance, as these structures are required to be stable during their service lifetime. Provided that the elements of support systems are resistant against chemical actions due to environmental conditions and their long-term behavior is satisfactory, the support systems must be designed against anticipated load conditions. As rock masses have many geological discontinuities and weakness zones, the load acting on support systems may be due to the deadweight of potential unstable blocks formed by rock discontinuities, which may be designated as structurally controlled or local instability modes and independent of the insitu stress state or inward displacement of rock mass due to elasto-plastic or elasto-visco plastic behavior induced by in-situ stresses (Figures. 4.2 and 4.3). Therefore, the main purpose of the design of support systems must be well established with due considerations of these situations. Aydan and Kawamoto (1999) developed a database system for large underground openings, and it was named CAVERN. This database system was modified recently to include the RMQR classification system. It was renamed as UGCAVERN database system and converted to the MS-ACCESS environment from the previous development environment dBasePLUS III. The system includes parameters related to the geometry, support system,

Table 4.7  Rock classes of RMQR classification system Rock Class

I

Description Solid or Rock of rock mass material RMQR 100 ≥ RMQR > 95 DENKEN A

II

III

IV

Very good

Good

Medium

V

Poor or Weak 95 ≥ RMQR 80 > RMQR 60 > RMQR 40 > RMQR > 80 ≥ 60 ≥ 40 ≥ 20 B CH CM CL

Figure 4.2 Instability modes of underground openings Source: Arranged from Aydan (1989)

VI Very poor/ very weak 20 > RMQR D or F

Rock mass classifications  87

Figure 4.3  Load conditions in a tunnel

rock classifications, in-situ stress and geomechanical properties of intact rock and rock mass and measured displacements of the large underground openings. It has about 110 entries of worldwide large underground openings. Aydan (Aydan et al., 1993, 1996) also developed two databases for tunnels through squeezing rocks as well as for their geomechanical properties named as SQTUN (103 entries) and SQUZROCK (171 entries), respectively, and they were developed originally on dBasePLUS III environment. The both databases have been converted to their equivalents on MS-ACCESS environment. Recently, Aydan and Genis (2010) expanded SQTUN to case histories of rockbursting and renamed as SQROCKBURST (146 entries). These databases originally include RMR and

88  Rock mass classifications

Q system as two rock classification systems, namely Japan Railway Classification (JRAC) and Japan Roadway Classification (JROC). The interrelations among several parameters can be explored using a code developed in True-BASIC programming language. The competency implies that the intrinsic rock material does not yield under an induced stress state, and the ratio of the UCS of the intact rock over the major in-situ stress is generally more than 4. Tunnels, which are also becoming large in recent years (width is up to 14 m), are relatively smaller in size (10–11 m wide, 7–9 m high) and long linear structures. There is rich worldwide experience in tunneling under diverse rock conditions. Tunnels may be excavated in various rock masses, which may be subjected to squeezing, rockbursting and structural failure. Even all these failure modes may be experienced in a single tunnel. Except new large tunnels, the support system of tunnels generally consists of rock bolts, shotcrete, steel ribs as primary support members and concrete lining to smooth the airflow, to prevent direct seepage of groundwater into the tunnel and auxiliary extra safety measure against rock loads after the introduction of the New Austrian Tunneling Method (NATM). When rock mass is not competent against stress-induced yielding, they may be lined with the invert concrete liner. When tunnels are excavated by TBMs, rock bolts and shotcrete may be totally disappeared. Using these databases and adopting the approach of Aydan and Kawamoto (1999), several interrelations have been established for the dimensions of support members and related size parameters of the underground openings with the consideration of structurally controlled and stress-induced instability modes using the databases mentioned before. However, the interrelations could not be presented in this article due to lack of space. The design of support systems is relatively simple once the modes of structural instability, which may be also categorized as local instability modes by Aydan (1989), are defined. The procedures described by Aydan (1989, 1994) and Kawamoto et al. (1991) can be easily adopted for such a purpose. As for the design of support members against stress-induced instability modes, the use of past experiences, analytical and numerical methods (i.e. Bieniawski, 1989; Barton et al., 1974; Barton and Grimstad, 1993 Wickham et al., 1974; Aydan, 1989, 1994; Aydan et al., 1992, 1993, 1996) using the geomechanical properties of rock mass, which may be obtained with the use of RMQR and intrinsic properties of intact rock, is necessary, together with in-situ stress state and geometry of underground openings. Aydan et al. (2014) suggested Tables 4.8 and 4.9 for the empirical design of support systems for tunnels in competent rock, which may be subjected to even stress-induced failure modes such as squeezing and

Table 4.8  Support system for cavern in competent discontinuous rock mass (S: 20–25 m, H: 40–50 m) RMQR range

Bolts

Anchors

Roof

100 ≥ RMQR > 95 95 ≥ RMQR > 80 80 ≥ RMQR > 60 60 ≥ RMQR > 40

Sidewall

Shotcrete

Roof

Sidewall

Roof

Sidewall

L (m)

e (m)

L (m)

e (m)

L (m)

e (m)

L (m)

e (m)

t (mm)

t (mm)

– 3 4 5

– 2.5 2.2 1.9

– 5 6 7

– 3.0 2.7 2.3

– 8 10 12

– 4.0 3.7 3.3

– 10 12 15

– 5.0 4.3 3.6

– 100 150 200

– 80 120 150

L is length; e is spacing; t is thickness; bolt is 200 kN; anchor is 400 kN; UCS of shotcrete: 10 MPa; S is span (width); H is Height)

Rock mass classifications  89 Table 4.9  Support system for tunnels (D and B; 10 m span) RMQR Range

100 ≥ RMQR > 95 95 ≥ RMQR > 80 80 ≥ RMQR > 60 60 ≥ RMQR > 40 40 ≥ RMQR > 20 20 ≥ RMQR

Rock bolts

Shotcrete Steel Ribs

Lb

eb

ts

(m)

(m)

(mm)

None 2–3 3–4 4–5 5–6 6–7

None 2.5 2.0 1.5 1.0 0.5

None 50 100 150 200 250

Wire Mesh Lining (mm) Concrete Invert

ti (mm) Bolt None None Light Medium Heavy Very heavy

None None Yes Yes Yes Yes

None None 200 300 500 800

None None None 300 500 800

– – – – 5–6 6–7

rockbursting, respectively. The numbers in Tables 4.8 and 4.9 are based on the databases previously mentioned, together with the considerations of past experiences as well as some empirical, analytical and numerical methods (i.e. Aydan and Kawamoto, 1999; Aydan and Ulusay, 2014 Aydan et al., 1993, 1996, 2000; Aydan, 2011; Kawamoto et al., 1991). Nevertheless, Tables 4.8 and 4.9 should actually be sufficient for the design of the support system of many tunnels. Engineers generally employ empirical methods to estimate the properties of rock mass for the stability assessment of structures and feasibility studies since in-situ tests are usually expensive to perform. For such purposes, rock mass classification systems such as RMR and Q system are often used. Besides the utilization of rock classification for preliminary structural design, some empirical relations among RMR, Q-value, GSI and the like and the rock mass properties such as unit weight, deformation modulus, uniaxial compressive strength, friction angle, elastic wave velocity are proposed and used in practice. In the following section, a brief history of rock classifications and modern classifications and some recent trends are presented.

4.5  Geological Strength Index classification The Geological Strength Index (GSI) was introduced by Hoek (1994). The rock mass characterization is straightforward, and it is based upon the visual impression of the rock structure, in terms of blockiness, and the surface condition of the discontinuities indicated by joint roughness and alteration. The combination of these two parameters provides a practical basis for describing a wide range of rock mass types, with diversified rock structure ranging from very tightly interlocked strong rock fragments to heavily crushed rock masses. Based on the rock mass description the value of GSI is estimated from the contours given in his table. Hoek also proposed establishing some empirical relations between his GSI numbers and RMR.

4.6 Denken’s classification and modified Denken’s classification An example of rock classification, which is called Denken’s classification in Japan, is briefly described. This classification system is widely used in Japan, and it constitutes the basis for other rock classifications proposed for some specific structures. The evaluation of rock mass

90  Rock mass classifications

based on visual impressions of intact rock, jointing, weathering, the sound of geologic hammer and rock classes is denoted as A, B, CH, CM, CL, D and F in descending order of rock quality.

4.7  Estimations of engineering properties There are presently four engineering approaches to assess the strength of rock masses: 1 2 3 4

In-situ testing Empirical relations based on the elastic wave velocity of rock masses Empirical relations based on indices of rock mass classifications Reduction factor using the elastic wave velocity of intact rock and of rock mass and properties

Although several rock classifications are used in many countries, it seems that RMR, Q-­system and recently RMQR are the most widely known rock classifications (Bieniawski, 1974; Barton et al., 1974; Aydan et al., 2014). Hoek and Brown (1980, 1988) tried to establish some relations between the parameters of their empirical yield criterion and RMR. Bieniawski and his coworkers (Bieniawski, 1974; Kalamaras and Kalamaras, 1995; Jasarevic and Kovacevic, 1996; Serafim and Pereira, 1983; Aydan et al. (1997; Aydan and Dalgıç, 1998) also tried to establish some empirical relations between RMR-value and deformation modulus and the compressive uniaxial strength values of rock masses (Table 4.10). 4.7.1  Elastic modulus Bieniawski (1978) proposed the following function between RMR-value and elastic modulus of rock masses: Em = 2 * RMR −100 (4.6) The unit of Em is GPa. However, this function could not be applied to rock masses having a RMR value less than 50. Serafim and Pereira (1983) put forward the following function in view of experimental data from dam sites: Em = 10(( RMR−10 )/ 40 ) (4.7) The unit of Em is GPa in the preceding equation. Jasarevic and Kovacevic (1996) have come up with the following empirical relation between RMR value and elastic modulus of rock masses by taking into account experiments performed on Adriatic limestone: Em = e( 4.407+0.081*RMR ) (4.8) Recently, Aydan et al. (1997) developed the following function by using experimental data from sites in Japan: Em = 0.0097 RMR 3.54 (4.9) The unit of Em is MPa in Equations (4.5) and (4.6).

Rock mass classifications  91

4.7.2  Uniaxial compressive strength The following relation between RMR value and the uniaxial strength of rock masses was proposed by Aydan et al. (1997): σcm = 0.0016 RMR 2.5 (4.10) The unit is MPa. Although this function is applicable to rock masses, it could not cover all experimental data. The scattering is likely to be associated with the strength of intact rock. It is more desirable to establish a reduction coefficient as a function of RMR value between the mass strength and intact rock strength, which may handle the scattering due to the intrinsic strength of rocks. Hoek and Brown (1980) proposed such a relation as given here: σcm = σci e( RMR−100 )/ B (4.11) Hoek and Brown (1980) initially suggested the value of 6 for constant B, and they remodified the value of constant B as 9 later. On the other hand, Kalamaras and Bieniawski (1995) suggested the following formula between the uniaxial strength of intact rock and that of rock mass as a function of RMR value: σcm = 0.5

RMR −15 σci (4.12) 85

Aydan and Dalgıç (1998) put forward the following function for estimating the mass strength of squeezing rocks: σcm =

RMR σci (4.13) RMR + 6(100 − RMR)

4.7.3  Friction angle In the literature, it is difficult to find any relation between friction angle of rock mass and RMR value. Aydan and Dalgıç (1998) suggested the use of an empirical relation proposed by Aydan et al. (1993) after obtaining the mass strength. Aydan and Kawamoto (1999) recently proposed the following function: φm = 20 + 0.05 * RMR (4.14) Although no direct relation between RMR value and the cohesion of rock mass was proposed, the cohesion of rock masses may be obtained from the following theoretical relation among cohesion, uniaxial compressive strength and friction angle once the uniaxial compressive strength and friction angle are known. cm =

σcm 1− sin φm (4.15) 2 cos φm

92  Rock mass classifications

4.7.4 Relation between rock mass properties and Rock Mass Quality Rating (RMQR) The design of many geoengineering structures is based on the equivalent properties of rock masses. For this purpose, in-situ tests on the strength properties of rock masses are carried out using uniaxial and triaxial compression, direct shear and plate loading tests (e.g. Nose, 1992; Lama, 1974, Archambault and Ladanyi, 1970, 1972; Brown and Trollope, 1970; Cording et al. 1972; Einstein et al. 1969; Goldstein et al. 1966; Kawamoto 1970; Protodyakonov and Koifman, 1964, Van Heerden, 1975; Vardar 1977; Ulusay et al. 1993, Walker 1971). However, it is very rare to carry out in-situ triaxial compression experiments due to their cost. Using the available experimental data, some empirical direct relations among different mechanical properties and some rock mass classification parameters are proposed by various researchers. Most of these relations are concerned only with elastic modulus and rock mass strength, except those by the authors. As discussed by Aydan et al. (1997), the scattering of experimental data and rock classification indexes is very large, and such approaches generally fail when intact rock itself is a soft rock. Therefore, the property of intact rock and rock mass classification indexes must be involved in such evaluations. The recent tendency is to obtain mass properties from the utilization of properties of intact rock and rock mass classification indexes (i.e. Hoek and Brown, 1980, 1988; Hoek, 1994; Aydan and Kawamoto, 2000). There are several proposed relations between the normalized properties of rock mass by those of intact rock and rock mass classification indexes as listed in Table 4.10. Aydan and Dalgıç (1998) proposed an empirical relation between RMR and rock mass strength in terms of strength of intact rock. This relation was extended to other geomechanical properties of rock mass by Aydan and Kawamoto (2000). Recently Aydan and Ulusay (2014 provided relations for six different mechanical properties of rock mass using the proposed relation by Aydan and Kawamoto (2000). In this study, RMR is replaced by RMQR, and it is given in the following form for any mechanical properties of rock mass in terms of those of intact rock.

α = α0 − (α0 − α100 )

RMQR (4.16) RMQR + β (100 − RMQR)

where á0 and á100 are the values of the function at RMQR = 0 and RMQR = 100 of property á, and â is a constant to be determined by using a minimization procedure for experimental values of given physical or mechanical properties. The authors proposed some values for these empirical constants with the consideration of in-situ experiments carried out in Japan as given in Table 4.11. When a representative value of RMQR is determined for a given site, the geomechanical properties of rock mass can be obtained using Equation (4.16) together with the values of constants given in Table 4.10 and the values of intact rock for a desired property. The empirical relations for normalized properties presented in the previous section are compared with the experimental results from in-situ tests carried out at various large projects (underground power houses, dams, nuclear power plants and underground crude oil and gas storage caverns) in Japan. Figure 4.4 compares the experimental results for elastic modulus and Poisson’s ratio of rock mass. The experimental results on normalized elastic modulus of

Table 4.10 Empirical relations between rock mass classification and normalized properties of rock mass Property

Relation

Proposed by

Deformation modulus, E M

Em = 0.009 e RMR / 22.82 + 0.000028 RMR 2 Ei

Nicholson and Bieniawski 1990

Em RMR = Ei RMR + β(100 − RMR )

Aydan and Kawamoto (2000)

(1− 0.5D) Em = 0.02 + 1+ e(( 60 +15D−GSI )/11) Ei

Hoek and Diederichs (2006)

1 Em RMR  = 1− cos(π  2  100  Ei

Mitri et al. (1994)

Em = 10 0.0186 RQD−1.91 Ei

Zhang and Einstein (2004)

Em = e( RMR−100 )/ 36 Ei

Galera et al. (2005)

σcm = s σci ( s = e( RMR−100 )/ 9 ) σcm = e( RMR−100 )/ 24 σci

Hoek and Brown (1980)

Uniaxial compressive strength, σcm

σcm = Cohesion, cm

cm =

Kalamaras and Bieniawski (1995) Aydan and Dalgıç (1998)

RMR σci RMR + 6(100 − RMR )

Aydan et al. (2012)

RMR ci RMR + 6(100 − RMR )

Friction angle, ϕm

ϕm RMR = 0.3 + 0.7 ϕi RMR + β(100 − RMR )

Aydan and Kawamoto (2000)

Poisson’s ratio, vm

υm RMR = 2.5 −1.5 υi RMR + (100 − RMR )

Aydan et al. (2012)

Tensile strength, σm

σtm RMR = σti RMR + 6(100 − RMR )

Tokashiki (2011)

s, a are rock mass constants, ci is cohesion of intact rock, f is the friction angle of intact rock, and ni is Poisson’s ratio of intact rock.

Table 4.11 Values of α0, α100 and β for various properties of rock mass Property (α)

α0

α100

β

Deformation modulus Poisson’s ratio Uniaxial compressive strength Tensile strength Cohesion Friction angle

0.0 2.5 0.0 0.0 0.0 0.3

1.0 1.0 1.0 1.0 1.0 1.0

6 0.3 6 6 6 1.0

94  Rock mass classifications

Figure 4.4 Comparison of experimental data for (a) deformation modulus, (b) Poisson’s ratio of rock mass with empirical relation (Equation 4.16) together with values of parameters given in Table 4.11

rock mass are closely represented by the empirical relation (4.16), together with the values given in Table 4.11, and they are clustered around the curve with the value of coefficient β as 6. It should be noted that experiments on the Poisson’s ratio of rock masses are quite rare. In this particular comparison, Poisson’s ratio of rock mass in tunnels through squeezing rocks correlated with RMQR using the approach proposed by Aydan and Dalgıç (1998) and Aydan et al. (2000) is also included. The data for RMQR value less than 50 is mainly from those of rock masses exhibiting squeezing behavior (Aydan et al., 1993, 1996). The measured data is well enveloped by the empirical relation with the values of coefficient β ranging between 0.1 and 3. The authors suggest that the values of α0, α100 and β should be 2.5, 1.0 and 1, respectively as given in Table 4.11. Figure 4.5 compares experimental results with empirical relations for normalized uniaxial compressive strength and tensile strength of rock masses by those of intact rock. The uniaxial compressive strengths of rock masses plotted in this figure are mostly obtained using rock shear test together with the Mohr-Coulomb failure criterion. The experimental results generally confirm the empirical relation given in Equation (4.16) in analogy to that proposed by Aydan and Dalgıç (1998). In literature, there is almost no in-situ experimental procedure or experimental results for the tensile strength of rock mass to the knowledge of the authors. However, there is a possibility of utilizing plate loading tests, large-scale water chamber experiments and borehole jacking tests for indirect inference of tensile strength of rock mass from the measured responses. The authors investigated the Ryukyu limestone cliffs along the shores of Okinawa, Miyako, Kurima, Ikema, Ishigaki, Ikejima, Heianza and Miyagi and Iriomote islands of Japan for inferring the tensile strength of rock masses. The authors also back-analyzed the stable and unstable (failed) cliffs using a theory based on the cantilever theory (Tokashiki

Rock mass classifications  95

Figure 4.5 Comparison of experimental data for (a) uniaxial compression and (b) tensile strengths of rock masses with empirical relations (Equation 4.16), together with values of parameters given in Table 4.11

Figure 4.6 Comparison of (a) cohesion and (b) friction angle of rock mass with empirical relation (Equation 4.16), together with values of parameters given in Table 4.11

and Aydan, 2010). Tokashiki and Aydan (2011a, 2011b) fitted the inferred tensile strength of the rock mass normalized by that of intact rock using the empirical relation of Aydan and Kawamoto (2000). In this study, such evaluations were revisited, and RMQR values were recalculated. The results are plotted in Figure 4.6 by varying the value of the empirical constant β between 5 and 7. As the ratio of the uniaxial compressive strength of rock to its tensile strength is within the range of 10–20 and remains constant for the same rock type, it is found that the value of empirical constant β could be designated as 6 in view of the inferred

96  Rock mass classifications

tensile strength of rock mass. It is interesting to notice that the values of empirical constant β for elastic modulus, uniaxial compressive and tensile strength of rock masses are the same. The Mohr-Coulomb yield criterion is one of the most commonly used in rock engineering. Although the Hoek-Brown criterion (Hoek et al., 2002) was claimed to be the best criterion for rocks and rock mass by some, the recent paper by Aydan et al. (2012) clearly demonstrated that the validity of such a claim is found to be false through comparisons of experimental results on all rock types with the Hoek-Brown criterion. The linear MohrCoulomb yield criterion can be safely used for a possible stress state encountered in actual engineering projects. The authors again utilize the empirical relation (Equation 4.16), together with the values of parameters given in Table 4.11, for comparing with experimental results as shown in Figure 4.6. The data used in this comparison are directly from rock shear tests carried out on Ryukyu limestone and on rock masses in other sites in Japan. The experimental results generally confirm the empirical relation (Equation 4.16) based on the formula of Aydan and Kawamoto (2000) and Aydan et al. (2012). The major issue in using Equation (4.15) to obtain the geomechanical properties of rock mass in terms of those of intact rock is how to select the value of constant β. For practical applications, the authors strongly suggest the use of the values given in Table 4.11. Aydan et al. (2012) recently proposed a procedure to evaluate the direct shear tests on rock masses with the use of both Mohr-Coulomb and Aydan yield/failure criteria (Aydan, 1995), together with the use of the unified formula of Aydan and Kawamoto (2000). The same approach can be adopted herein, replacing RMR with RMQR in the respective equations. The specific form of the Mohr-Coulomb yield criterion in shear stress and normal stress space may be written as: τ = cm + σn tan ϕm(4.17) where cm =

 RMQR  RMQR ci , φm = 0.3 + 0.7  φi (4.18)  100  RMQR + 6(100 − RMQR)

Similarly, Aydan’s criterion is written in terms of shear stress and normal stress by neglecting the effect of temperature as:   c  τ = cm∞ 1− 1− mo  e−bm σn + σ n tan φm∞ (4.19)   cm∞  where cm∞ =

RMQR RMQR c∞ , cmo = co , RMQR + 6 (100 − RMQR) RMQR + 6 (100 − RMQR)

  RMQR  bi (4.20) bm = 0.3 + 0.7  RMQR + (100 − RMQR) 

Rock mass classifications  97

The procedure of Aydan et al. (2012) and both yield/failure criteria just described briefly were also applied to rock shear experiments carried out at Minami Daitojima island, and the results are shown in Figure 4.7. The RMQR values of rock mass at the site of in-situ experiments ranged between 69 and 79. The uniaxial compressive strength and friction angle of intact rock were 88 MPa and 61 degrees, respectively. As noted from Figure 4.7, a good fitting to experimental results is obtained for the criteria of Mohr-Coulomb and Aydan (1995) according to the procedure adopted from that proposed by Aydan et al. (2012). Figure 4.8a shows an application of the approach previously described to in-situ shear experiments carried out on andesite together with a fitted relation to experimental results. RMQR values of rock mass at the adits, where in-situ experiments were carried out, ranged between 37 and 61. The uniaxial compressive strength and friction angle of intact rock were

Figure 4.7 Comparison of in-situ rock shear experiments with yield/failure criteria of Mohr-Coulomb and Aydan (1995)

Figure 4.8 Comparison of in-situ rock shear experiments with Mohr-Coulomb and Aydan (1995) yield/ failure criteria

98  Rock mass classifications

90 MPa and 60 degrees, respectively. Using the approach proposed Aydan et al. (2012), the fitted relations to Mohr-Coulomb and Aydan’s yield/failure criteria functions to experimental results shown in Figure 4.8a and estimations are shown in Figure 4.8b. As noted from Figure 4.8b, a good fit with experimental results is obtained. As pointed out in the previous section, most of the empirical relations available in literature are related to the deformation modulus of rock mass. Following the publication of data on the uniaxial strength of rock mass by Aydan and Dalgıç (1998), one can see a number of equations thereafter. However, the direct comparison of empirical relations for the deformation modulus and uniaxial compressive strength of rock mass with estimations from Equation (4.16) is not possible unless they are related to RMQR using the relations among RMQR, RMR and Q-value. Furthermore, the comparisons of relations based on GSI and RQD are not possible with the estimations by Equation (4.16) for the aforementioned properties. Figures 4.9 and 4.10 show the comparison of estimations from Equation (4.16) with those from some of the available empirical relations. In the same figure, experimental results are also plotted. As noted from both figures, the experimental results are scattered as the rocks vary from sedimentary origin to igneous rocks. Additional reason may be the difference between the actual and assigned RMQR values; it is difficult to reach original geological reports for data points. Almost all experimental results are enveloped by Equation (4.16) for three different values of constant β. The empirical relation proposed by Nicholson and Bieniawski (1990) is quite close to Equation (4.16) with β = 6, while estimations, for example, by Galera et al. (2005) and Mitri et al. (2004), are quite far from the experimental results.

Figure 4.9 Comparison of various empirical relations with experimental results for normalized elastic modulus of rock mass

Rock mass classifications  99

Figure 4.10 Comparison of estimations from various empirical relations with experimental results for normalized uniaxial compressive strength of rock mass

Regarding the uniaxial compressive strength of rock mass, the estimations from Equation (4.16) envelopes all experimental results. The empirical relation proposed by Kalamaras and Bieniawski (1995) is quite close to Equation (4.16) with β = 6, while estimations by Hoek and Brown (1980) are quite poor and underestimate the uniaxial compressive strength of rock mass. However, it should be noted that the empirical relation proposed by Kalamaras and Bieniawski (1995) has a value greater than zero.

References Aydan, Ö. (1989) The Stabilisation of Rock Engineering Structures by Rockbolts. Doctorate Thesis, Nagoya University, 204 pages. Aydan, Ö. (1990) The arch formation effect of rockbolts (in Turkish). Madencilik, 28(3), 33–40. Aydan, Ö. (1994) Rock reinforcement and support, chapter 7. In: Vutukuri, V.S. & Katsuyama, K. (ed.) Introduction to Rock Mechanics. Industrial Publishing and Consulting, Inc., Tokyo, 193–248. Aydan, Ö. (1995) The stress state of the earth and the earth’s crust due to the gravitational pull. The 35th US Rock Mechanics Symposium, Lake Tahoe, 237–243. Aydan, Ö. (2011) Some Issues in Tunnelling through Rock Mass and Their Possible Solutions. First Asian Tunnelling Conference, ATS11-15,33–44. Aydan, Ö., Akagi, T. & Kawamoto, T. (1993) The squeezing potential of rocks around tunnels: Theory and prediction. Rock Mechanics Rock Engineering, 26(2), Vienna, 137–163. Aydan, Ö., Akagi, T. & Kawamoto T. (1996) The squeezing potential of rock around tunnels: theory and prediction with examples taken from Japan. Rock Mechanics and Rock Engineering, 29(3), 125–143.

100  Rock mass classifications Aydan, Ö. & Dalgıç, S. (1998) Prediction of deformation of 3-lanes Bolu tunnels through squeezing rocks of North Anatolian Fault Zone (NAFZ). Regional Symposium on Sedimentary Rock Engineering, Taipei, 228–233. Aydan, Ö., Dalgıç, S. & Kawamoto, T. (2000) Prediction of squeezing potential of rocks in tunnelling through a combination of an analytical method and rock mass classifications. Italian Geotechnical Journal, Roma, 34(1), 41–45. Aydan, Ö. & Genis, M. (2010) A unified analytical solution for stress and strain fields about a radially symmetric openings in elasto-plastic rock with the consideration of support system and longterm properties of surrounding rock. International Journal of Mining and Mineral Processing, 1(1), 1–32. International Science Press. Aydan, Ö. & Kawamoto, T. (1992) The flexural toppling failures in slopes and underground openings and their stabilisation. Rock Mechanics and Rock Engineering, 25(3), 143–165. Aydan, Ö. & Kawamoto, T. (1999) A proposal for the design of the support system of large underground caverns according to RMR rock classification system (in Turkish). Mühendislik Jeolojisi Bülteni, 17, 103–110. Aydan, Ö. & Kawamoto, T. (2000) The assessment of mechanical properties of rock masses through RMR rock classification system. GeoEng2000, UW0926, Melbourne. Aydan, Ö. & Kawamoto, T. (2001) The stability assessment of a large underground opening at great depth. 17th International Mining Congress and Exhibition of Turkey, IMCET, Ankara, 1, 277–288. Aydan, Ö., Ohta, Y., Geniş, M., Tokashiki, N. & Ohkubo, K. (2010) Response and stability of underground structures in rock mass during earthquakes. Rock Mechanics and Rock Engineering, 43(6), 857–875. Aydan, Ö., Seiki, T., Jeong, G.C. & Tokashiki, N. (1994) Mechanical behaviour of rocks, discontinuities and rock masses. International Symposium Pre-failure Deformation Characteristics of Geomaterials, Sapporo, 2, 1161–1168. Aydan, Ö., Tsuchiyama, S., Kinbara, T., Uehara, F., Tokashiki, N. & Kawamoto, T. (2008) A numerical analysis of non-destructive tests for the maintenance and assessment of corrosion of rockbolts and rock anchors. The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG), Goa, India. pp. 40–45. Aydan, Ö., Uehara, F. & Kawamoto, T. (2012) Numerical study of the long-term performance of an underground powerhouse subjected to varying initial stress states, cyclic water heads, and temperature variations. International Journal of Geomechanics, ASCE, 12(1), 14–26. Aydan, Ö., Ulusay, R. & Kawamoto T. (1997) Assessment of rock mass strength for underground excavations. The 36th US Rock Mechanics Symposium, New York, 777–786. Aydan, Ö., Ulusay, R. & Tokashiki, N. (2014) A new rock mass quality rating system: Rock Mass Quality Rating (RMQR) and its application to the estimation of geomechanical characteristics of rock masses. Rock Mechanics and Rock Engineering, 47, 1255–1276. Barton, N., Lien, R. & Lunde, I. (1974) Engineering classification of rock masses for the design of tunnel supports. Rock Mechanics, 6(4), 189–239. Barton, N., Løset, F., Lien, R. & Lunde, J. (1980) Application of the Q-system in design decisions. In Subsurface Space (ed. M. Bergman) 2, 553–561. New York. Bieniawski, Z.T. (1974) Geomechanics classification of rock masses and its application in tunnelling. Third International Congress on Rock Mechanics, ISRM, Denver, IIA. pp. 27–32. Bieniawski, Z.T. (1978). Determining rock mass deformability: Experience from case histories. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 15, 237–247. Bieniawski, Z.T. (1989) Engineering Rock Mass Classifications. Wiley, New York. Brown, E.T. & Trollope, D.H. (1970) Strength of a model of jointed rock. Journal of Soil Mechanics and Foundation Division, ASCE, SM2, Washington, 685–704. Cording, E.J., Hendron, A.J. & Deere, D.U. (1972) Rock engineering for underground caverns. ASCE, Symposium on Underground Chambers, Washington, pp. 567–600.

Rock mass classifications  101 Einstein, H.H., Nelson, R.A., Bruhn, R.W. & Hirshfeld, R.C. (1969) Model studies of jointed rock behaviour. 11th US Rock Mechanics Symposium, Berkeley, pp. 83–103. Galera, J.M., Alvarez, M. & Bieniawski, Z.T. (2005) Evaluation of the deformation modulus of rock masses: Comparison of pressuremeter and dilatometer tests with RMR prediction. Proceedings of International Symposium on ISP5-PRESSIO 2005, Madrid, pp. 1–25. Goldstein, M., Goosev, B., Pyrogovsky, N., Tulinov, R. & Turovskaya, A. (1966) Investigation of mechanical properties of cracked rock. 1st ISRM Congress, Lisbon, 1. pp. 521–524. Grimstad, E. & Barton, N. (1993) Updating of the Q-System for NMT. Proceedings of the International Symposium on Sprayed Concrete—Modern Use of Wet Mix Sprayed Concrete for Underground Support, Fagernes, (Eds Kompen, Opsahl and Berg). Norwegian Concrete Association, Oslo. pp. 46–66. Hoek, E. (1994) Strength of rock and rock masses. ISRM News Journal, 2(2), 4–16. Hoek, E. & Brown, E.T. (1980) Empirical strength criterion for rock masses. Journal of Geotechnical Engineering Division, ASCE, Washington, 106(GT9), 1013–1035. Hoek, E. & Brown, E.T. (1988) The Hoek-Brown failure criterion: A 1988 update. 15th Canadian Rock Mechanics Symposium, Toronto, pp. 31–38. Hoek, E. & Diederichs, M.S. (2006) Empirical estimation of rock mass modulus. International Journal for Rock Mechanics and Mining Science, Oxford, 43(2), 203–215. Hoek, E., Carranza-Torres, C.T. & Corkum, B. (2002) Hoek-Brown failure criterion-2002 edition. Proceedings of the 5th North American Rock Mechanics Symposium, Toronto, 1. pp. 267–273. Kalamaras G.S. & Bieniawski, Z.T.A.(1995) Rock mass strength concept for coal seams incorporating the effect of time. 8th ISRM Congress, 1, 295–302. Kawamoto, T. (1970) Macroscopic shear failure of jointed and layered brittle media. 2nd ISRM Congress, Belgrade, 2. ISRM, pp. 215–221. Kawamoto, T., Aydan, Ö. & Tsuchiyama, S. (1991) A consideration on the local instability of large underground openings. International Conference GEOMECHANICS’91, Hradec. pp. 33–41. Lama, R.D. (1974) The Uniaxial Compressive Strength of Jointed Rock Mass. Prof. L. Müller Festschrift, Univ. Karlsruhe. pp. 67–77. Ladanyi, B. & Archambault, G. (1970) Simulation of shear behaviour of a jointed rock mass. 11th US Rock Mechanics Symposium, Berkeley. pp. 105–125. Ladanyi, B. & Archambault, G. (1972) Evaluation de la resistance au cisaillement d’un massif rocheux fragmente. 24th International Geological Congress, Montreal, Sec. 130. pp. 249–260. Løset, F. (1992) Support needs compared at the Svartisen Road Tunnel. Tunnels and Tunnelling, June. Mitri, H.S., Edrissi, R., & Henning, J. (1994). Finite element modelling of cable bolted slopes in hard rock ground mines. Proceedings of SMME Annual Meeting, New Mexico, Albuquerque, pp. 94–116. Nicholson, G.A. & Bieniawski, Z.T. (1990). A non-linear deformation modulus based on rock mass classification. International Journal of Mining and Geological Engineering, 8, 181–202. Nose, M. (1962) On the in-situ experiments of rock mass at the site of Kurobe IV Dam. First Rock Mechanics Symposium of Japan, Tokyo, Paper No. 11, 24pp. (in Japanese). Palmstrom, A. & Broch, E. (2006) Use and misuse of rock mass classification systems with particular reference to the Q-system. Tunnelling and Underground Space Technology, 21(6), 575–593. Protodyakonov, M.M. & Koifman, M.I. (1964) Uber den Masstabseffect bei Untersuchung von Gestein und Kohle. 5. Landertreffen des Internationalen Buros für Gebirgsmechanik, Deutsche Akademie der Wissenschaften, Berlin, 3, 97–108. Serafim, J.L. & Pereira, J.P. (1983) Considerations of the geomechanics classification of Bieniawski. II International Symposium Engineering and Geological Underground Constructions, Lisbon, 1, pp. 33–42. Terzaghi, K. (1946) Rock defects and loads on tunnel supports. In Rock Tunnelling with Steel Supports, (Eds R.V. Proctor and T. White). Commercial Shearing and Stamping Co., Youngstown, pp. 15–99. Tokashiki, N. (2011) Study on the engineering properties of Ryukyu limestone and the evaluation of the stability of its rock mass and masonry structures. PhD Thesis, Waseda University, 220 pages (in Japanese with English abstract).

102  Rock mass classifications Tokashiki, N. & Aydan, Ö. (2010) The stability assessment of overhanging Ryukyu limestone cliffs with an emphasis on the evaluation of tensile strength of rock mass. Journal of Geotechnical Engineering JSCE, 66(2), 397–406. Tokashiki, N. & Aydan, Ö. (2011a) A comparative study on the analytical and numerical stability assessment methods for rock cliffs in Ryukyu Islands, Proceedings of 13th International Conference of the International Association for Computer Methods and Advances in Geomechanics, Melbourne, Australia, pp. 663–668. Tokashiki, N. & Aydan, Ö. (2011b). Application of rock mass classification systems to Ryukyu limestone and the evaluation of their mechanical properties. Proceedings of the 40th Rock Mechanics Symposium of Japan, Tokyo, pp. 387–392 (in Japanese). Ulusay, R., Aksoy, H. & Ider, M.H. (1993) Geotechnical approaches for the design of a railway tunnel section in andesite. Engineering Geology, 34, 81–93. Van Heerden, W.L. (1975) In-situ complete stress-strain characteristics of large coal specimens. Journal of the South African Institute of Mining and Metallurgy, 75, 207–217. Vardar, M. (1977) Zeiteinfluss auf des Bruchverhalten des Gebriges in der Umgebung von Tunbeln. Veröff. D. inst. F. Bodenmech., University of Karlsruhe, Heft 72. Walker, P.E. (1971) The Shearing Behaviour of a Block Jointed Rock Model. Thesis, Queens University, Belfast. Wickham, G.E., Tiedemann, H.R. & Skinner, E.H. (1974) Ground support prediction model— RSR concept. Proceedings of 2nd Rapid Excavation Tunneling Conference. AIME, New York, pp. 691–670. Zhang, L. & Einstein, H.H. (2004) Using RQD to estimate the deformation modulus of rock masses. International Journal of Rock Mechanics and Mining Sciences, 41, 337–341.

Chapter 5

Model testing and photo-elasticity in rock mechanics

5.1 Introduction Model tests have been used in engineering for thousands of years and are still widely used in many engineering applications. The main purpose of the model tests may be classified as (e.g. Fumagalli, 1973; Egger, 1979; Aydan and Kawamoto, 1992; Everling, 1965; Erguvanlı and Goodman, 1972; Aydan et al. 1988): 1 2 3 4

Understanding the governing mechanism and the behavior of the structure, Determining design values through the use of similitude law, Validating analytical and/or numerical models, and Educating students, engineers and/or public.

Model tests were widely used in engineering applications when the computational tools are not advanced, and the similitude law play an important role on the design of many modern rock engineering structures (e.g. Fumagalli, 1973; Egger, 1979). There are many laboratories worldwide for different engineering applications (Figure 11.1). The recent tendency in model testing is to utilize the models to validate analytical and numerical models. Particularly, the failure process is of great concern when the stability of the structures is assessed. The model tests, together with advanced monitoring, observation and imaging tools, generally provide a clear picture and governing mechanism of the phenomenon investigated. Particularly, this type of utilization of the model tests is likely to be common from now on.

5.2  Model testing and similitude law When a given structure is modeled, its geometrical and mechanical scaling is of great importance. Although it is generally difficult, the reduction of material properties is desirable at the geometrical scaling. The similitude ratio η of a model is mathematically expressed as given here (Fumagalli, 1973): η=

ζ (5.1) Ψ

There are many geometrical and mechanical parameters for scaling, such as H is height, γ is unit weight, σ is stress, σn is normal stress, σc is uniaxial compressive strength, σt is tensile strength, τc is shear strength, c is cohesion, ϕ is friction angle, t is time, f is frequency, and

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g is gravitational acceleration. Letters p and m are used to denote prototype and model, respectively. Let’s denote strength similitude ratio ζ and stress similitude ratio as Ψ, expressed as: ζ=

σcp σtp , (5.2) σcm σtm

Ψ=

σp (5.3) σm

When the strength of discontinuities is considered, the strength ratio should be shear strength resistance. For example, the similitude ratio η of a slope model can be obtained as given here. The stress similitude ratio Ψ may be given as follows: Ψ=

σp γ pH p (5.4) = σm γm H m

Geometrical scaling ratio λ and unit weight ratio ρ are expressed as follows; λ=

Hp γp , = ρ (5.5) Hm γm

Hence, Equation (5.5.) may be rewritten as: Ψ = λ ⋅ ρ (5.6) If the Mohr-Coulomb failure criterion is used, the strength similitude ratio ζ may be written as: ζ=

σcp C p µ p (5.7) = ⋅ σcm C m µ m

It should be noted that the frictional resistance are expresses as given here: µ p = 2 cos ϕ p / (1− sin ϕ p ) µ m = 2 cos ϕ m / (1− sin ϕ m ) Thus, the model similitude ratio η given by Equation (5.1) can be obtained as follows: η=

C p µp γm H m ⋅ ⋅ ⋅ (5.8) C m µm γ p H p

Similitude ratio η for the shear strength of discontinuities may be obtained as described here. Let us introduce parameters α, β for stress components as a function of the location to express shear and normal stresses acting on discontinuities in prototype and models: τ p = αγ p H p , σ np = βγ p H p (5.9) τ m = αγ m H m , σ nm = βγ m H m (5.10)

Model testing and photo-elasticity  105

Accordingly, the stress similitude ratio Ψ may be written as: Ψ=

τ p αγ p H p γ p H p = = (5.11) τ m αγ m H m γ m H m

If we utilize the Mohr-Coulomb criterion, the strength similitude ratio ζ is given as: ζ=

C p + σ np tan ϕ p C p + βγ p H p tan ϕ p (5.12) = C m + σ nm tan ϕ m C m + βγ m H m tan ϕ m

The model similitude ratio η given in Equation (5.1) may be obtained as: η=

C p + βγ p H p tan ϕ p γ m H m ⋅ ⋅ (5.13) C m + βγ m H m tan ϕ m γ p H p

If cohesions (cp = cm = 0) of prototype and model are nil, the model similitude ratio is obtained as: η=

tan ϕ p (5.14) tan ϕ m

Thus, similitude ratio η becomes independent of geometrical scale, and it depends only on the friction angle. In other words, for cohesion-less discontinuities, the overall behaviors of prototype and model should be almost the same. On the other hand, if friction angles (ϕp = ϕm = 0) are nil, the similitude ratio η can be expressed as: η=

C p γm H m ⋅ ⋅ (5.15) Cm γ p H p

If the model is subjected to gravity, the similitude ratio for gravitational acceleration would be 1: gp = 1(5.16) gm Using Equation (5.16), the similitude ratio t for time can be given as: λ = 1 (5.17) t2 where t = tp/tm. The similitude ratio f for vibration frequency may be written as: f2 =

1 , λ

f =

1 λ

(5.18)

where f = f p/ f m. If Equation (5.18) is used, the frequency of the model can be assigned if the frequency of rock mass in nature is given.

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5.3  Principles and devices of photo-elasticity The photo-elastic phenomenon was discovered by Brewster (1815) and experimental frameworks were developed at the beginning of the 20th century by Coker and Filon (1930, 1957). Since then, many studies were carried out (e.g. Bieniawski and van Tonder, 1969). Photo-elastic experiments were extended to determine three-dimensional states of stress with advances of the technology. Parallel to developments in experimental technique, the advance in digital polariscopes made possible by light-emitting diodes and continuous monitoring of structures under load became possible. As a result, dynamic photo-elasticity is developed, which contributes greatly to the study of complex phenomena such as failure of materials and structures. The method relies on the property of birefringence exhibited by certain transparent materials. Birefringence is a phenomenon in which a ray of light passing through a given material experiences two refractive indices. Photo-elastic materials exhibit the property of birefringence, and the magnitude of the refractive indices at each point in the material is directly related to the state of stresses at that point when they are subjected to loading. In such

Figure 5.1  Principle of photo-elasticity testing

Figure 5.2 Stress distribution in continuum and layered media beneath relatively rigid foundations (model material: polyurethane)

Model testing and photo-elasticity  107

materials, maximum shear stress and its orientation are obtained from analyzing the birefringence with a polariscope. When a ray of light passes through photoelastic material, its electromagnetic wave components are resolved along the two principal stress directions. The difference in the refractive indices leads to a relative phase retardation between the two components. A simple polariscope consists of a light source, polarizer, photo-elastic model, and analyzer as illustrated in Figure 5.1(a). Figure 5.1(b) shows a simple implementation of the concept utilizing a polariscope and digital camera. Figures 5.2–5.4 show several examples of photo-elastic tests on model structures.

Figure 5.3  Stress distribution in stable and failing cliffs (model material: gelatin)

Figure 5.4 Stress distribution around single and double tunnels in continuum (model material: polyurethane)

108  Model testing and photo-elasticity

5.4  1G models The models are tested under gravitational action. As the load is generally small, only enough to fracture model material except under tension, this technique is generally used for studying the stability of models of foundations, slopes, and underground structures involving discontinuities in different patterns (Figure 5.5). To induce failure, the base of the model frame may be tilted to change the orientations of discontinuities as shown in Figure 5.6. It is also possible to investigate the effect of faulting on the stability and failure modes of rock slopes, foundations and underground structures (Aydan et al., 2011) under 1G model conditions. The author and his coworkers used some model setups to investigate the effects of faulting due to earthquakes on underground structures (Aydan et al., 2010, 2011). The orientation of faulting can be adjusted as desired. The maximum displacement of faulting of the moving side of the faulting experiments was varied between 25 and 100 mm. The base of the experimental setup can model rigid body motions of base rock, and it has a box 780 mm long, 250 mm wide and 300 mm deep.

Figure 5.5  Some examples of 1G model experiments using wooden blocks

Figure 5.6  1G gravitational model test examples

Model testing and photo-elasticity  109

Figure 5.7  Effect of thrust faulting on the model rock slopes Source: Aydan et al. (2011)

Figure 5.7 shows that a series of experiments were carried out on rock slope models with breakable material under a thrust faulting action with an inclination of 5.5 degrees. When layers dip toward valley side, the ground surface is tilted, and the slope surface becomes particularly steeper. As for layers dipping into the mountain side, the slope may become unstable, and flexural or columnar toppling failure occurs. Although the experiments are still insufficient to draw conclusions, they do show that discontinuity orientation has great effects on the overall stability of slopes in relation to faulting mode. These experiments clearly show that the forced displacement field induced by faulting has an additional destructive effect besides ground shaking on the stability of slopes. This experimental device was used to investigate the effect of forced displacement due to faulting on underground openings. Figure 5.6 shows views of some model experiments on shallow underground openings subjected to the thrust faulting action with an inclination of 5.5 degrees. Underground openings were assumed to be located on the projected line of the fault. In the experiment, three adjacent tunnels had been excavated. While one of the tunnels was situated on the projected line of faulting, the other two tunnels were located in the footwall and hanging wall side of the fault. As seen in Figure 5.8, the tunnel completely collapsed or was heavily damaged when it was located on the projected line of the faulting. When the tunnel was located on the hanging wall side, the damage was almost nil in spite of the close proximity of the model tunnel to the projected fault line. However, the tunnel in the footwall side of the fault was subjected to some damage due to the relative slip of layers pushed toward the slope. This simple example clearly shows that the damage state may differ depending upon the location of tunnels with respect to fault movement.

110  Model testing and photo-elasticity

Figure 5.8  Effect of faulting on underground openings Source: Aydan et al. (2011)

5.5  Base-friction model test Base friction model test was first contemplated by Erguvanlı and Goodman (1972). The principle of this model testing device is based on the frictional resistance between the basal surface and the model, which is restricted in the direction of motion. Erguvanlı and Goodman (1972) used a flour-and-oil-based material that permitted cracking within the model. In 1979, Egger (1979) presented an advanced base friction machine where a uniform pressure could be applied to increase the stresses on the model Furthermore, Egger (1979) introduced a mixture of BaSO5, ZnO and vaseline, which can be compacted under different pressures to develop materials with different mechanical properties. Bray and Goodman (1981) examined the mathematical basis for the base friction concept and its limitations. Under static conditions, the stress induced in the model is quite similar to the actual conditions, and the stress in the model is reduced through the friction coefficient between the moving base and the model. This model is very effective in studying the failure mechanism of various rock engineering structures, and it clearly illustrates the governing mechanism of the failure. For example, Aydan and Kawamoto (1987, 1992) developed their theoretical model to study the flexural toppling failure of rock slopes and underground

Model testing and photo-elasticity  111

Figure 5.9 Several examples of model tests of rock engineering structures using the base-friction apparatus

structures in layered rock mass models. Aydan (2019) developed a new tilting base-friction apparatus device that can be used to study the failure process quite similar to the actual conditions with slowed motions. Figure 5.9 shows several examples of model tests of rock engineering structures.

5.6  Centrifuge tests Bucky (1931) was the first to propose the utilization of centrifuge testing in rock mechanics related to mine stabilities. However, the use of centrifugal acceleration to simulate increased gravitational acceleration was first proposed by Phillips (1869). The high gravity is created by spinning models in a centrifuge. When the centrifuge rotates with an angular velocity of ω, the centrifugal acceleration at any radius r is given by: αc = rω2(5.19) The centrifugal acceleration is N times the gravitational acceleration: αc = Ng(5.20) The stress in the model acts linearly, and it is possible to create stress levels, which may lead the failure of the model. The similitude ratio is claimed to be 1, which implies that there is no scaling effect in model scale and prototype.

112  Model testing and photo-elasticity

Many centrifuge experiments were carried in relation to the slope stability and underground stability in South Africa, Sweden, Russia, Japan (e.g. Stephansson, 1971; Sugawara et al., 1983; Stacey, 2006)

5.7  Dynamic shaking table tests 5.7.1  Characteristics of shaking table The shaking table used for model tests was produced by AKASHI. Its operation system was recently updated by IMV together with the possibility of applying actual acceleration wave forms from earthquakes. Table 5.1 gives the specifications of the shaking table and monitoring devices. The size of the shaking table is 1000 × 1000 mm2. The maximum acceleration is 600 gals for a model with a weight of 100 kgf. The displacement response of models was monitored using laser displacement transducers, and the input acceleration of the shaking table and acceleration response of the retaining wall were measured using the two accelerometers. Two shaking test (ST) devices were used. The shaking table at Nagoya University (NU) was used for studying the response of models slopes for unbreakable material, and the shaking table at University of the Ryukyus (UR) was used to study the response of model slopes with breakable material. Main features of the shaking table apparatuses are given in Table 5.2. Figure 5.10 shows sketches of the devices together with the mounted model and instrumentations. Slope models were two-dimensional and were mounted on the table with metal frames. The metal frame at NU-ST was 1200 mm long and 800 mm high, and it was Table 5.1  Specifications of monitoring sensors and shaking table Shaking Table and Sensors

Specifications

Shaking Table – AKASHI

Frequency Stroke Acceleration Range Range Range

Accelerometers Laser displacement Transducer OMRON KEYENCE

1–50 Hz 100 mm 600 gals 10G 0–300 mm 0–100 mm

Table 5.2  Specifications of shaking tables Parameters

NU Shaking Table

UR Shaking Table

Vibration direction Operation method Table size Load Stroke Amplitude Wave form

Uniaxial Electro-oil servo 1300 × 1300 30 kN 150 mm 5G Harmonic, triangular, rectangular, random

Uniaxial Magnetic 1000 × 1000 6 kN 100 mm 0.6G Harmonic, triangular, rectangular, random

Model testing and photo-elasticity  113

Figure 5.10  Illustration of shaking tables and instrumentation

1000 mm long and 750 mm high at UR-ST. The frame width was 100 mm wide at the two shaking table experiments. Acceleration responses of the slope at several locations and input waves were measured using the accelerometers. 5.7.2  Applications to slopes and cliffs 5.7.2.1  Model materials (A)  NONBREAKABLE MATERIALS

Blocks with dimensions of 10 × 10 × 100 mm and 10 × 20 × 100 mm were made of wood and used to simulate the discontinuity sets in rock masses. Direct shear tests were carried out on discontinuities between wood blocks, and he results, together with shear strength envelopes, are shown in Figure 5.11.

Figure 5.11  Shear tests on interfaces between wood blocks

114  Model testing and photo-elasticity (B)  BREAKABLE MATERIALS

Breakable blocks are made of BaSO5, ZnO and Vaseline oil, which is commonly used in base friction experiments (Aydan and Kawamoto, 1992). The properties of the materials of blocks and layers are described in detail by Aydan and Amini (2009) and Egger (1979). Figure 5.12 shows the variation of the strength of the model material with respect to compaction pressure. The material can be powderized and reused after each experiment. The friction angle of interfaces between blocks are tested and shown in Figure 5.13.

Figure 5.12 Variation of tensile strength of model material

Figure 5.13  Shear tests on interfaces between breakable blocks

Model testing and photo-elasticity  115

5.7.2.2  Testing procedure The metal frames have some special attachments to generate different discontinuity patterns. The models were subjected to some selected forms of acceleration waves through a shaking table. The acceleration responses of model slopes were measured using accelerometers installed at various points in the slope. (A)  NONBREAKABLE BLOCKS

Model slopes were prepared by arranging wood blocks in various patterns to generate discontinuity sets with different orientations in space. Slope angles were 5.5, 63 and 90 degrees, and the height and base width of model slopes were 800 mm and 1200 mm, respectively. The intermittency angle ξ of cross joints were 0 and 5.5 degrees (Aydan et al., 1989), and one discontinuity set was always continuous as such sets in actual rock masses always do exist. The inclination of the thoroughgoing (continuous) set was varied from 0 to 180 degrees by 15 degrees. At some inclinations, model slopes were statically unstable, and at such inclinations no tests were done. Besides varying the inclination of the continuous set, the following cases were investigated: CASE 1: Frequency was varied from 2.5 Hz to 50 Hz while the amplitude of the acceleration was kept at 50 or 100 gal. CASE 2: The amplitude of the acceleration waves was varied until the failure of the slope occurred, while keeping the frequency of the wave at 2.5 Hz. (B)  BREAKABLE BLOCKS

The inclination of a thoroughgoing discontinuity set was selected as 0, 5.5, 60, 90, 120, 135 and 180 degrees. Before forcing the models to failure in each test, the vibration responses of some observation points in the slope were measured with the purpose of investigating the natural frequency of slopes and amplification through sweep tests with a frequency range between 3 and 5.0 Hz. Also, deflection of the slope surface was monitored by laser displacement transducers and acoustic emission sensors. 5.7.3  Model experiments Various parameters such as the effect of the frequency and the amplitude of input acceleration waves are investigated in relation to discontinuity patterns and their inclinations and to the slope geometry for the model slopes with nonbreakable and breakable models. The model slopes were finally forced to fail by increasing the amplitude of input acceleration waves, and the forms of instability were investigated. 5.7.3.1  Natural frequency of model slopes (A)  MODEL SLOPES WITH NONBREAKABLE BLOCKS

Figure 5.14 shows the amplification of waves measured at selected points in relation to the variation of input wave frequency. The inclination of the thoroughgoing set for both discontinuity patterns was 75 degrees. The letter on each curve indicates the selected points within

116  Model testing and photo-elasticity

Figure 5.14 Variation of amplification with respect to frequency of model slopes and measurement locations

Figure 5.15 Variation of natural frequency of model slopes with respect to the inclination of thoroughgoing discontinuity set

the model slopes. It is noted that if the natural frequency of the slopes exists, it varies with the spatial distributions of the sets and the structure of the mass. In the following, the frequency responses are discussed and compared for each respective inclination of the thoroughgoing discontinuity set for the point A (see Figure 5.15 for location) as shown in Figure 5.15. The slope angle was 63 degrees in the sweep tests shown in Figure 5.15. The results for each discontinuity set pattern are indicated in the figure for intermittent patterns (IP) and for cross-continuous pattern (CCP). Inclination 0: The natural frequency of the slope is 10 Hz for a cross-continuous pattern and 20 Hz for an intermittent pattern, respectively. Therefore, the natural frequencies of the

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slopes for an intermittent pattern and cross-continuous pattern are different, even though the slope geometry and intact material are same. This may be related to the resulting slender columnar structure of the mass in the case of a cross-continuous pattern. Inclinations 15, 30, 5.5: The slopes for these inclinations of the thoroughgoing set could not be tested as they were statically unstable for the slope angle of 60 degrees. Inclination 60: From the figure, the natural frequencies for both patterns coincide, and they have a value of 30 Hz. This may be attributed to the similarity of the structure of the mass for this inclination of the thoroughgoing discontinuity set. Inclination 75: Natural frequencies of the slopes for both patterns are almost the same, and they appear to have a value of 17.5 Hz. Similar reasoning as in the case of inclination of 60 can be stated for this case. Inclination 90: No tests for this inclination could be made. Inclination 120: Slopes having an intermittent pattern were only tested because slopes having a cross-continuous pattern could not be tested as they were statically unstable. For this inclination of the thoroughgoing set, the natural frequency of the slope has a value of 35 Hz. Inclination 150: The natural frequencies of the slopes for both patterns are almost the same and have a value of 35 Hz. Inclination 165: The natural frequency of the slope is 22.5 Hz for a cross-continuous pattern and 30 Hz for an intermittent pattern, respectively. In addition, the natural frequencies of the slopes for intermittent and cross-continuous patterns are different. (B)  MODEL SLOPES WITH BREAKABLE BLOCKS

Fundamentally, the vibration response of model slopes are quite similar to those of model slopes made with unbreakable blocks. Figure 5.16(a) shows the input and measured wave forms at selected two points on the slope. The amplification of the vibration response is highest at the slope crest and the amplification at the top-back (ACC-TB) are a bit smaller than that at the slope crest as seen in Figure 5.16(b). From this figure, we can clearly state that amplification of the acceleration waves increases towards the slope (free) surfaces. In addition to this, the amplifications are larger at the top and have the maximum value at the crest of the slope (ACC-TC) as noted in Figures 5.14 and 5.15 for model slopes made with nonbreakable blocks. 5.7.3.2  Stability of model slopes: failure tests When rock slopes are subjected to shaking, passive failure modes occur in addition to active modes (Aydan et al., 2009a, 2009b, 2011; Aydan and Amini, 2009). Figures 5.17 and 5.18 show examples of failures of some model slopes consisting of nonbreakable and breakable blocks and/or layers. The experiments also show that flexural toppling failure of passive type occur when layers (60 degrees or more) dip into valley-side. The records of base acceleration and deflection of the slope surface of the model are shown in Figure 5.19 for a layer inclination of 90 degrees as an example. The acoustic emissions are also shown in the figure. Acoustic emissions start to increase long before the displacement starts to increase. This observation may also have important implications for the monitoring of rock slopes. These responses were observed in experiments on layered and blocky model slopes made with breakable blocks.

Figure 5.16  Acceleration responses of selected points on model slopes made with breakable blocks

Figure 5.17  Failure modes of rock slope models with nonbreakable material

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Figure 5.18  Failure modes of rock slope models with breakable material

Figure 5.19  Acceleration, displacement and acoustic emission responses of a model slope

5.7.4 Shaking table model tests of slopes subjected to planar sliding 5.7.4.1  Device and models In order to understand the dynamic response and stability of rock slopes against planar sliding, several shaking table tests on rock slopes with a potentially unstable block on a plane dipping to the valley-side shown in Figure 5.20 were carried out. The shaking table used for model rock slopes under dry condition was produced by AKASHI, and its details are given in Subsection 5.7.1.

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Figure 5.20 Typical setup for model tests on the shaking table

Figure 5.20 shows the views of model tests on the shaking tables. The actual rock slope with a height of 50 m and having a potential plane of failure at an angle of 15 degrees is scaled down to a model with a scale of 1/500. The model material is Ryukyu limestone. The planar surfaces were saw-cut using a large diameter sawing blade. The input wave, the acceleration of upper block acceleration and relative displacement of the potentially unstable blocks were measured using accelerometers and laser displacement transducers as shown in Figure 5.20. A stopper was utilized to prevent breakage of model slope blocks upon failure. Therefore, the maximum sliding displacement was limited to 12–18 mm. The input base wave on the shaking table was horizontal. Experiments were carried out using sinusoidal acceleration waves to simulate earthquakes with a given frequency and a maximum base acceleration up to 600 gals for the model weight of 100 kgf. 5.7.4.2  Material properties 5.7.4.2.1  FRICTION ANGLE OF FAILURE PLANE

The rock block of the experiment model is made of Ryukyu limestone. In hard rock, the influence of rock deformation is small, and the effect of the slip surface becomes dominant. Therefore, the tilting test is conducted for checking frictional properties. Figure 5.21 shows a view of a tilting test. Figure 5.22 shows an example of the determination of the dynamic friction angle from the displacement response measured in a tilting test (see Aydan, 2016, 2019 Aydan et al., 2011 for details). The peak friction angle ranged between 39.9 and 5.0.8 degrees, while the kinetic friction angle ranged between 22.6 and 26.1 degrees on the basis of three experiments. 5.7.4.2.2  BONDING CAPACITY OF MODEL ROCK BOLTS

The width of rock bolt models was 10 mm and they were made of adhesive tapes. 14 pullout tests on the model rock bolts were carried out by changing the anchorage length. The results of a pull-out test for anchorage length of 35 mm and bond strength as a function of

Figure 5.21 View of tilting experiments

Figure 5.22 Typical tilting test result and determination of kinetic friction angle (αf is base rotation angle at failure, ϕs is static friction angle, ϕd is dynamic (kinetic) friction angle, A0 is coefficient of fitting function, t0 is time of sliding initiation, and t is time.)

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bolt length are shown in Figure 5.23. These model rock bolts may be visualized as fully grouted rock bolts. 5.7.4.3  Shaking table tests on model slopes 5.7.4.3.1  UNREINFORCED LAYERED ROCK SLOPES

A series of model tests on rock slopes using layered coral limestone was carried out. Before each failure test, a sweeping test was carried to check the natural frequency characteristics of model rock slopes with a frequency ranging between 1 and 50 Hz at a constant acceleration of 100 gals. Figure 5.24 shows the horizontal acceleration records of the accelerometers fixed to the shaking table (Shaking table (H)) and the top of the model slope (model top (H)) as shown in Figure 5.2(b). The results clearly indicate that the model rock slopes have some natural frequency characteristics. Figure 5.25 shows the views of layered rock slope models, while Figure 5.26 shows the measured acceleration and relative slip responses. In the experiments, it was observed that

Figure 5.23  Bond strength of model rock bolts

Figure 5.24 Horizontal acceleration records at the top of the model rock slope and applied base acceleration on shaking table, along with their Fourier spectra

Figure 5.25 Views of layered rock slope models before and after shaking

Figure 5.26  Acceleration and slip response of unreinforced layered rock slope model

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the unstable layered part of the slope moves as a monolithic body until it was restrained by the stopper. Then the upper unstable layers start to move individually. In other words, there is no essential difference regarding the overall slip behavior of unstable parts whether it is a monolithic body or layered. This fact is quite important when the stability assessment methods are developed. The critical acceleration to initiate the slip of the potentially unstable part ranged between 330 and 350 gals. 5.7.4.3.2  EXPERIMENTS ON REINFORCED ROCK SLOPE MODELS

A series of experiments were carried out to investigate the number and length of rock bolts on the layered rock slope models. Figure 5.27 shows views of the reinforced layered rock slope model before and after shaking. Figure 5.28 shows the measured responses. As the rock bolts are not initially prestressed, a small amount of slip occurs as seen in Figure 5.27. This value is almost the same as that for the unreinforced case when Figures 5.26 and 5.28 are compared with each other. However, rock bolts restrain the movement of potentially unstable part of the layered model slope after a slip of 1.2–1.6 mm relative slip (Figure 5.29). Some tests were carried out to check the effect of length of rock bolts on the layered rock slope models (Figure 5.30). Figure 5.31 shows the measured acceleration and slip response of the layered rock slope model with rock bolts not crossing the failure plane. In other words, the unstable part of the rock slope model was stitched to create a kind of monolithic block above the potential failure plane. Although the initiation of slip was slightly higher than that of the unreinforced layered rock slope model, the rock bolts did not act to restrain the movement of the potentially unstable part of the slope. This fact implies that if rock bolts are not anchored into the stable part below the potential failure, the effect of rock bolting or rock anchoring would be nil. Therefore, the short rock bolts installed in slopes would not have any major reinforcement effect on the stability of rock slopes prior to planar sliding except to prevent the relative sliding of small blocks above the potential failure surface.

Figure 5.27 Views of reinforced layered rock slope models before and after shaking

Figure 5.28  Measured acceleration and slip responses of reinforced layered rock slope model

Figure 5.29 Views of model rock bolts before and after shaking

Figure 5.30 Views of layered rock slope model with fully grouted rock bolt models

Figure 5.31 Measured acceleration and slip responses of reinforced layered rock slope model with rock bolts not crossing the failure plane

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Figure 5.32 Measured acceleration and slip responses of reinforced layered rock slope model with rock bolts crossing the failure plane with insufficient anchorage length

Next, the length of rock bolts was increased, and they had anchorage length in the stable part. However, the length was not sufficient to prevent the sliding failure after a given acceleration level. Figure 5.32 shows the measured acceleration and relative slip responses. The initiation of slip was almost the same as that for the unreinforced case; the large movement of the unstable part occurred at an acceleration level of 550 gals, and total collapse was induced at the acceleration level of 740 gals. These examples clearly showed that the rock bolts must have sufficient length, anchored in the stable part of the slope, and number to prevent the failure of the slope against planar sliding.

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5.7.5 Shaking table model tests of slopes subjected to wedge sliding 5.7.5.1  Preparation of models Six special molds were prepared to cast model wedges (Kumsar et al., 2000; Aydan and Kumsar, 2010). For each wedge configuration, three wedge blocks were prepared. Each base block had dimensions of 15.0 × 100 × 260 mm. Base and wedge models were made of mortar and their geomechanical parameters were similar to those of rocks. The composition of the mortar used for the preparation of the models is 1781 kgf m−3 of fine sand, 360 kgf m−3 of cement with a water-cement ratio of 0.5. The cement used in mortar was rapid hardening type, and samples were cured for about 7 days in a room with a constant temperature. The wedge angles and the initial intersection angles of wedge blocks are listed in Table 5.3. In addition, several mortar slabs were cast to measure the friction angle of sliding planes. A number of tilting tests were performed. The inferred friction angle measured in tilting tests ranged between 30 degrees and 35 degrees with an average of 32 degrees. Each wedge base block was fixed on the shaking table to receive same shaking with the shaking table during the dynamic test. The accelerations acting on the shaking table at the base and wedge blocks were recorded during the experiment and saved digitally in a data file (Figure 5.33). Furthermore, in the second series of dynamic tests, a laser displacement transducer was used to record the movement of the wedge block during the experiments. The reason for recording accelerations at three different locations is to determine the acceleration at the moment of failure as well as any amplification from the base to the top of the block. In fact, when the amplitude of input acceleration wave is increased, there is a sudden decrease on the wedge block acceleration records during the wedge failure, while the others are increasing. A barrier was installed at a distance of 20–30 mm away from the front of the wedge block to prevent their damage by falling off from the base block. Dynamic testing of the wedge models was performed in the laboratory by means of a one-dimensional shaking table, which moves along horizontal plane. The applicable waveforms of the shaking table are sinusoidal, sawtooth, rectangular, trapezoidal and triangle. The shaking table has a square shape with 1 m side length. The frequency of waves to be applicable to the shaking table can range between 1 Hz and 50 Hz. The table has a maximum stroke of 100 mm and a maximum acceleration of 6 m s−2 for a maximum load of 980.7 N.

Table 5.3  Geometric parameters of wedges Wedge Number

Intersection Inclination – ia (o)

Half-wedge Angle (o)

TB1(Swedge120) TB2(Swedge100) TB3(Swedge90) TB5.(Swedge70) TB5(Swedge60) TB6(Swedge5.5)

29 29 31 27 30 30

61.5 51.5 5.7.8 5.0.0 33.8 26.0

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Figure 5.33 View of a wedge model

Figure 5.34  Dynamic response of wedge model: (a) TB1, (b) TB2

5.7.5.2  Shaking table tests Three experiments were carried out on each wedge block configuration and dynamic displacement responses of the wedge blocks in addition to the acceleration responses were measured. Figures 5.35–5.36 shows typical acceleration time and displacement time responses for each wedge block configuration. As it is noted from the responses shown in Figures 5.35–5.36, the acceleration responses of the wedge block indicate some high-frequency waveforms on the overall trend of the acceleration imposed by the shaking table. When this type of waveform appears, the permanent displacement of the wedge block with respect to the base block takes place. Depending on the amplitude of the acceleration waves as well as its direction, the motion of the block may cease. In other words, a step-like behavior occurs. The motion of the block starts when the amplitude of the input wave acts in the direction of the downside and exceeds the frictional resistance of the wedge block. When the

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Figure 5.35  Dynamic response of wedge model: (a) TB3, (b) TB5

Figure 5.36  Dynamic response of wedge model: (a) TB5, (b) TB6

direction of the input acceleration is reversed, the motion of the block terminates after a certain amount of relative sliding. As a result, the overall displacement response is step-like. Another important observation is that the frictional resistance between the wedge block and base block limits the inertial forces acting on the wedge block and the base block, even though the base block may undergo higher inertial forces. The sudden jumps in the acceleration response of the wedge block as seen in Figures 5.35–5.36 are due to the collision of the wedge block with the barrier. The initiation of the sliding of the wedge blocks was almost the same as those measured in the first series of the experiments. 5.7.6  Model experiments on shallow underground openings The authors have been performing model experiments on underground openings for some time (Aydan et al., 1994; Geniş and Aydan, 2002). The first series of experiments on shallow underground openings in discontinuous rock mass using nonbreakable blocks were reported by Aydan et al. (1994), in which a limit equilibrium method was developed for assessing

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Figure 5.37  Failure modes of shallow tunnels adjacent to slopes with breakable material

their stability. These experiments have now been repeated using breakable material following the observations of damage to tunnels caused by the 2008 Wenchuan earthquake. The inclination of continuous discontinuity plane varied between 0 degrees and 180 degrees. Figure 5.37 shows views of some experiments. Unless the rock mass model itself failed, the failure modes were very similar to those of the model experiments using hard blocks. In some experiments with discontinuities dipping into the mountain side, flexural toppling of the rock mass model occurred. The comparison of the preliminary experimental results with the theoretical estimations based on Aydan’s method (Aydan, 1989, Aydan et al., 1994) are remarkably close to each other. 5.7.7  Monumental structures 5.7.7.1  Perry Banner Rock A physical model of the Perry Banner Rock was prepared using model material, which has almost the same density of the original rock at a scale of 1/5.0 and total model height of 300 mm. The model was equipped with two laser displacement transducers (LDT-F, LDT-B), one acoustic emission sensor (AE-sensor) and two accelerometers (Acc. Top, Acc. B), as shown in Figure 5.38. The model was fixed on the shaking table to receive same shaking during the dynamic test. The displacements, acoustic emissions (AE) and accelerations acting on the shaking table at the base and the model were recorded during the experiment and saved digitally in a data file.

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Figure 5. 38  General and close-up views of the model and its instrumentation

Figure 5.39 Views of orientations of shaking

As the shaking table can apply uniaxial accelerations, the model was shaken in three directions, namely 0, 45 and 90 degrees in order to investigate the effect of the inclination of the thoroughgoing discontinuity plane (Figure 5.39). The shaking direction angle is the acute angle between the strike of the thoroughgoing discontinuity plane and direction of shaking. Before the experiment leading to failure, the model was tested using the sweep testing procedure. The acceleration was fixed to 100 gals, and the frequency of wave was varied from 1 Hz to 50 Hz. The final experiment was concerned with the effect of grouting the gap in the model. (A)  MODELS TESTS WITHOUT COUNTERMEASURES

Figures 5.40, 5.41 and 5.42 show the acceleration, AE and displacement responses measured for three directions, respectively. Depending upon the direction of the shaking, the blocks starts to exhibit nonlinear behavior at about 100 gals for the direction of 90 degrees while it is about 230 gals for the direction of 0 degree. The blocks become unstable when the base acceleration exceeds 350 gals.

Figure 5.40  Responses for the shaking orientation of 0 degrees

Figure 5.41  Responses for the shaking orientation of 45 degrees

Figure 5.42  Responses for the shaking orientation of 90 degrees

134  Model testing and photo-elasticity (B)  MODELS TESTS WITH COUNTERMEASURES

Rock bolts and rock anchors are one of the effective ways of dealing with reinforcement issues in rock engineering (Aydan, 1989, 2018). However, the utilization of such countermeasures may not be attractive due to the disturbance of the appearance especially in archeological structures. As the resistance of the model was minimal for the orientation of 90 degrees, an experiment was carried out by introducing a bonding resistance to the thoroughgoing discontinuity plane. The bonding agent was double-sided bonding tape. Figure 5.43 shows the responses measured during the experiment. As noted from the figure and comparison with the response shown in Figure 5.42, the experimental results clearly indicated that the increase of the resistance was possible, and the overall seismic resistance of the block increases. This experimental finding was taken into account, and it was implemented during the remedial measures of the potentially unstable block. 5.7.7.2  Retaining walls of historical castles (A)  MODEL SETUP

An acrylic transparent box 630 mm in length, 300 mm in height and 100 mm in width was used, as shown in Figure 5.44. The wall thickness was 10 mm so that the box was relative rigid, and the frictional resistance of sidewalls was quite low. The blocks used were made of Ryukyu limestone with a size of 5.0 × 5.0 × 99.5 mm with the consideration of materials used for the retaining walls of historical castles in Ryukyu archipelago. Furthermore, the base block was such that the overall wall inclination can be chosen as 70, 83 and 90 degrees. The base block was fixed by two-sided tape to the base of the acrylic box. In addition, the Ryukyu limestone of the same size was laid over the base as seen in Figure 5.44. This was expected to provide a condition similar to the actual conditions observed in many historical castles in Ryukyu archipelago. The wall height was 240 mm, and the ratio of the height to width was 1/6. When the retaining wall inclination is 90 degrees without backfill material, the wall was expected to start rocking at an acceleration level of 167 gals.

Figure 5.43  Responses of the model with countermeasures for the shaking orientation of 90 degrees

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Figure 5.44  Illustration of model box

Figure 5.45 Views of backfill materials

(B)  BACKFILL MATERIALS AND THEIR PROPERTIES

Three different backfill materials were chosen (Figure 5.45). Glass beads were chosen to represent the lowest shear-resistant backfill material while the angular fragments of Motobu limestone was selected as the highest shear-resistant backfill material. The third backfill material was rounded river gravels having a shear resistance between those of the two other backfill materials. A special shear testing setup was developed to obtain the shear strength characteristics of backfill materials under low normal stress levels, which are quite relevant to the model tests to be presented in this study. Figure 5.46 shows the shear strength envelopes for three backfill materials. As noted from the figure, the shear strength of rounded river gravel is in between the shear strength envelopes of glass-beads and Motobu limestone gravel. The strength of backfill materials is frictional, and the friction angle of the glass beads is about 21.68 degrees.

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Figure 5.46  Shear strength envelopes for backfill materials Table 5.4  Friction angle between Ryukyu limestone and backfill materials Parameter

Glass-beads

Rounded River Gravel

Motobu Limestone Fragments

Friction angle

12.5–16.8

25.0–27.5

25.9–27.8

Another important factor for the stability of the retaining walls of historical castles as well as other similar structures is the frictional resistance between the backfill material and retaining wall blocks. For this purpose, tilting experiments were carried out. The backfill material contained in a box is put upon the Ryukyu limestone platens without any contact and tilted until it slides. This response of the backfill material contained in the box was measured using laser-displacement transducers. The inferred friction angles are given in Table 5.4. The lowest friction angle was obtained in the case of glass beads as expected. (C)  SHAKING TABLE TESTS ON RETAINING WALLS WITH GLASS BEADS BACKFILL

A series of sweep tests were carried before the failure tests. Regarding the glass beads backfill material, the retaining walls were statically unstable for 90 degrees while they failed during the sweep test on the retaining walls with an inclination of 83 degrees. Therefore, we could show one example for retaining walls for the inclination of 70 degrees (Figure 5.47(a)). Its Fourier spectra analysis is shown in Figure 5.47(b). The results indicated no apparent natural frequency was dominant. The situation was quite similar in all experiments. Therefore, more emphasis will be given to the failure experiments. Although the test on the retaining wall with an inclination of 83 degrees was intended for a sweep test, it resulted in failure. Figure 5.48 shows the displacement and base acceleration during the test. Failure tests on the retaining walls with an inclination of 70 degrees were

Model testing and photo-elasticity  137

Figure 5.47 (a) Acceleration records of the shaking table and top of the retaining wall, (b) Fourier spectra of acceleration records

Figure 5.48 Acceleration and displacement responses on the retaining wall with an inclination of 83 degrees

carried out by applying sinusoidal waves with a frequency of 3 Hz. The amplitude waves were gradually increased until the failure occurred. Figure 4.49 shows an example of failure. The yielding initiated at about 110 gals, and the total failure occurred when the input acceleration reached 215 gals. Figure 5.50 shows the retaining wall before and after the failure test. The retaining wall failed due to toppling (rotation) failure, although some relative sliding occurred with the block at the toe of the model retaining wall. (D)  SHAKING TABLE TESTS ON RETAINING WALLS WITH RIVER GRAVEL BACKFILL

A series of sweep tests were carried before the failure tests as explained in the previous section. Regarding the rounded river gravel backfill material, the retaining walls were statically unstable for 90 degrees while the sweep test on the retaining walls with an inclination

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Figure 5.49 Acceleration and displacement responses on the retaining wall with an inclination of 70 degrees

Figure 5.50 Views of the model retaining wall with an inclination of 70 degrees before and after the test

of 83 and 70 degrees could be carried. We show one example for retaining walls for the inclination of 83 degrees in Figure 5.51(a) and its Fourier spectra analysis in Figure 5.51(b). The results indicated there was no dominant natural frequency for the given range of frequency. The situation was quite similar in all experiments for 83 and 70 degrees retaining wall models. Failure tests on the retaining walls with inclinations of 83 and 70 degrees were carried out by applying sinusoidal waves with a frequency of 3 Hz. The amplitude waves were gradually increased until the failure occurred. Figures 5.52(a) and 5.52(b) show the acceleration and displacement responses of retaining walls with inclinations of 83 and 70 degrees as examples of failure tests. The yielding initiated at about 100 gals, and the total failure occurred when the input acceleration reached 210 gals for 83-degree retaining walls. On the other hand, the yielding initiated at 220 gals, and total failure occurred when the input acceleration was 430 gals for 70-degree retaining walls as seen in Figure 5.53. The retaining wall failed due to toppling (rotation) failure, although some relative sliding occurred with the block at the toe of the model retaining wall (Figure 5.53).

Figure 5.51 (a) Acceleration records of the shaking table and top of the 83 retaining wall with rounded river gravel backfill, (b) Fourier spectra of acceleration records

Figure 5.52 (a) Acceleration and displacement responses on the retaining wall with an inclination of 83 degrees, (b) acceleration and displacement responses on the retaining wall with an inclination of 70 degrees

Figure 5.53 Views of the model retaining wall with an inclination of 70 degrees before and after the test

140  Model testing and photo-elasticity (E) SHAKING TABLE TESTS ON RETAINING WALLS WITH MOTOBU LIMESTONE GRAVEL BACKFILL

A series of sweep tests were carried out before the failure tests as explained in the previous section. Regarding the angular Motobu limestone gravel backfill material, the retaining walls were statically unstable for 90 degrees with a height of 25.0 mm. However, they were stable when the height was reduced to 160 mm. The sweep test on the retaining walls with inclinations of 90, 83 and 70 degrees were carried out. We show one example for retaining walls for the inclination of 70 degrees in Figure 5.54(a) and its Fourier spectra analysis in Figure 5.54(b). Again, the results indicated there was no dominant natural frequency for the given range of frequency. The situation was quite similar in all experiments for 90-, 83- and 70-degree retaining wall models. Failure tests on the retaining walls with inclinations of 90, 83 and 70 degrees were carried out by applying sinusoidal waves with a frequency of 3 Hz. The procedure was the same as those in previous experiments. Figures 5.55, 5.56 and 5.57 show acceleration

Figure 5.54 (a) Acceleration records of the shaking table and top of the 70-degree retaining wall with rounded river gravel backfill, (b) Fourier spectra of acceleration records

Figure 5.55 Acceleration and displacement responses on the retaining wall with an inclination of 90 degrees

Model testing and photo-elasticity  141

and displacement responses of retaining walls with inclinations of 90, 83 and 70 degrees as examples of failure tests. The yielding initiated at about 110 gals, and the total failure occurred when the input acceleration reached 260 gals for 90-degree retaining walls. On the other hand, the yielding initiated at 130 gals, and the total failure occurred when the input acceleration was 300 gals for 83-degree retaining walls, as seen in Figure 5.58. The retaining walls failed due to toppling (rotation) failure. The retaining walls with a 70-degree inclination and height of 25.0 mm did not fail during the entire test up to 400 gals as seen in Figures 5.59 and 5.60. Although some relative sliding occurred with the block at the toe of the model retaining wall when the base acceleration reached the level of 300 gals (Fig. 5.60). However, some settlement of the backfill occurred, and the retaining wall was pushed into passive sliding mode.

Figure 5.56 Acceleration and displacement responses on the retaining wall with an inclination of 83 degrees

Figure 5.57 Acceleration and displacement responses on the retaining wall with an inclination of 70 degrees

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Figure 5.58 Views of the model retaining wall with an inclination of 90 degrees before and after the test

Figure 5.59 Views of the model retaining wall with an inclination of 83 degrees before and after the test

Figure 5.60 Views of the model retaining wall with an inclination of 70 degrees before and after the test

5.7.8  Model test on masonry arches There are many historical stone masonry structures in the Ryukyu islands. Ryukyu limestone blocks are generally used in the construction of the historical stone masonry structures. The major historical stone masonry structures are castles (i.e. Shuri Castle, Nakijin Castle,

Model testing and photo-elasticity  143

Nakagusuku Castle, Gushikawa Castle etc.), burial tombs (Yodore) and imperial gardens. The authors were asked to assess the static and dynamic stability of walls and arch gates of Shuri Castle, a high retaining wall at Yodore imperial tomb and the arch bridge of Iedonchi imperial garden near the Shuri Castle. Some of experimental studies as well as numerical stability analyses are briefly reported in this subsection. The Shuri Castle, whose ruins remain in Naha City of Okinawa Prefecture, is said to date back to the 12th century or earlier. The castle grounds and buildings were completely destroyed by the bombing of the U.S. Army during the Battle of Okinawa in 1945. Only some of castle and retaining walls remain with a certain degree of damage. The castle has been under restoration according to a map drawn during the Meiji period. In addition, the main buildings of the castle, castle walls, retaining walls and arches have been reconstructed. Since these structures are of masonry type without reinforcement, their seismic stability during earthquakes is of great concern. Five arch configurations are denoted as Type-A, -B, -C, -D and -E and four of which (Type-A, -B, -D, -E) are commonly used in Shuri Castle in Okinawa island, Japan. All arch configurations were tested (Figure 5.61). The remaining arch form (Type-C) is quite common almost all over the world. The arches of Shuri Castle generally consist of two monolithic blocks in the form of a semicircle or an ovaloid shape while the Type-C arch consists of several blocks and has a semicircular shape. As the shaking table was uniaxial, the effect of the direction of input acceleration wave was investigated by changing the longitudinal axis of the arches (Figure 5.62). The experimental results indicated that the common form of failure for all arch types for a shaking direction of 0 degrees is sliding at abutments and inward rotational fall of arch blocks subsequently. As for 90 degrees shaking, the arch failed in the form of toppling. The failure for 45 degrees shaking was a combination of sliding and toppling. The experiments clearly indicated that the amplitude of acceleration waves to cause failure was the lowest for 90 degrees shaking, while it was maximum for 0 degrees shaking.

Figure 5.61  Common arch types used in Shuri Castle

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Figure 5.62  Failure modes of arch Type-A

5.7.9  Model tests subjected to tsunami waves 5.7.9.1  Tsunami generation device Shimohira et al. (2019) performed a series of experiments using a model tsunami generation device at the University of the Ryukyus designed by the author, and it is named as OATGD2000X to study the tsunami waves due to thrust and normal faulting event shown as shown in Figure 5.63. The dimensions and characteristics of the device were quite similar to that used in Tokai University except the wave-induction system. A tank was lowered or raised through pistons with a given velocity to generate rising or receding tsunami waves. The pressure and wave velocity at specified locations were measured using pressure sensors, and the amount of tank movement was measured using laser transducers. Figure 5.64. shows an example of record during the movement of tanks inducing rising and receding tsunami waves.

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Figure 5.63 View of the tsunami generation device OA-TGD2000X

Figure 5.64 Water head response at specified locations in relation to the tank movement

5.7.9.2 Experiments 5.7.9.2.1  TRIANGULAR RYUKYU LIMESTONE BLOCKS

Triangular-shaped prismatic blocks shown in Figures 5.65 and 5.66 tested under the same condition. The longest side of the triangular prismatic block shown in Figure 5.65 was downward, while the longest side of the triangular prismatic block shown in Figure 5.66 was upward. While the downward triangular prismatic block was almost non-displaced, the upward triangular prismatic block was considerably displaced. One of the main reasons for such a big difference when they are subjected to the tsunami forces is that the tsunami wave applies a surging uplift force on the block. As for the downward triangular prism, the surging force increases the normal force on the block. We also put a rectangular prism of Ryukyu

146  Model testing and photo-elasticity

Figure 5.65 Views of the downward triangular prism block at different time steps

Figure 5.66 Views of the upward triangular prism block at different time steps

limestone next to the triangular prismatic block. The displacement of the rectangular block was quite small. 5.7.9.2.2  PLASTER BLOCKS

First a rectangular prismatic block made of plaster was subjected to rising tsunami waves as shown in Figure 5.67. The overall behavior is fundamentally similar to those tested in Tokai University. Nevertheless, the block was toppled toward the downstream side and displaced horizontally in the direction of receding tsunami waves as seen in Figure 5.67.

Model testing and photo-elasticity  147

Figure 5.67 Views of the plaster block test at different time steps

Next two plaster prismatic blocks were laid over the Ryukyu limestone blocks as shown in Figure 5.68. The density of the plaster blocks is almost half of that of the Ryukyu limestone blocks. As seen from images 2 and 3 in Figure 5.68, the plaster blocks thrown upward and displaced in the direction of the tsunami waves. This experiment clearly demonstrates the importance of the density and overhanging degree of blocks when they are subjected to tsunami waves in nature. 5.7.9.2.3  BREAKABLE OVERHANGING CLIFFS

The next series of experiments involve the breakable blocks. Finding appropriate material for breakable blocks under the forces induced by tsunami forces by the experimental device was quite cumbersome. Although the materials had a very small density as compared to those in nature, it provided an insight view on the mechanism of formation of tsunami boulders, which was the main goal of this study. Figures 5.69 and 5.70 show the images of the models at different time steps. The surging tsunami wave enters under the overhanging blocks and applies upward forces. As a result, the overhanging block starts to bend upward, and breaks after a certain amount of displacement. In other words, the failure of the overhanging blocks is quite close to cantilever beams. However, the failure of the overhanging blocks is against gravity. Once the block is broken, it is dragged by the overflowing tsunami waves. This observation is in accordance with the mechanism proposed by Aydan and Tokashiki (2019) for the formation of tsunami boulders. Our experiments clearly indicated that if

Figure 5.68 Top views of the plaster block overhanging the base Ryukyu limestone blocks at different time steps

Figure 5.69 Views of the experiments using a breakable overhanging block at different time steps

Model testing and photo-elasticity  149

Figure 5.70 Views of the experiments using a breakable overhanging block at different time steps

the inclination of the lower side of the overhanging block ranges is 10–20 degrees, it is quite vulnerable to failure. 5.7.10  Experiments on tunnel models Large rock samples of siliceous sandstone of Shizuoka third tunnel and Ryukyu limestone blocks having a circular hole with a diameter of 58 mm were tested. The height of the samples was 300 mm with a 200 mm width. In addition to acceleration measurements, multiparameter measurements in model tunnel experiments were performed on sandstone and limestone samples as illustrated in Figure 5.71. Figure 5.72 shows the acceleration responses measured on a sandstone sample (300 × 200 × 138 mm) having a circular hole with a diameter of 58 mm. The overall acceleration responses are quite similar to those observed in uniaxial compression experiments. However, multiple acceleration responses with growing amplitudes occurred before that during the final rupture state. This phenomenon may be related to the ejection of small fragments from the perimeter of the model tunnel before the final failure of the sample. The observations of the experiments on the model tunnel in a brittle hard rock resemble the rockburst phenomenon experienced in rock engineering. Figure 5.73 shows a view of ejection of the rock fragments from the perimeter of the model tunnel.

Figure 5.71  Layout of instruments of the samples

Figure 5.72  Acceleration response of a sandstone sample with a circular hole

Model testing and photo-elasticity  151

Large rock samples of Ryukyu limestone blocks having a circular hole with a diameter of 58 mm were tested. The samples were 270 mm high, 160 mm wide and 140 mm thick. In addition to acceleration measurements, multiparameter measurements in model tunnel experiments were performed on limestone samples as illustrated in Figure 5.71. Figure 5.74 shows the acceleration responses measured on a limestone sample denoted by BE-2–6-TUN. The overall acceleration responses are quite similar to those observed in

Figure 5.73 Views of the sample with a model tunnel during the experiment

Figure 5.74  Acceleration response of a limestone sample with a circular hole

152  Model testing and photo-elasticity

uniaxial compression experiments. The mobile part of the loading sample experiences larger accelerations. However, the amplitudes of acceleration responses of limestone sample are smaller than those measured in the experiment of the previous sandstone sample. The overall strength of limestone sample with a model tunnel is smaller than that of the sandstone sample, and the sandstone sample fails in a more brittle manner compared to the limestone sample. The compressive failure took place at sidewalls while tensile cracks appeared in the roof and floor of the model tunnel as expected. Furthermore, the failure of the sample was less violent as compared to that of the sandstone sample.

References Aydan, Ö. (1989). The stabilisation of rock engineering structures by rock bolts. Geotechnical Engineering Department, Nagoya University, Nagoya, Doctorate Thesis. Aydan, Ö. (2016). Issues on Rock Dynamics and Future Directions. Keynote, ARMS2016, Bali. Aydan, Ö. (2018). Rock Reinforcement and Rock Support. CRC Press, Taylor and Francis Group, 486p. Aydan, Ö. (2019). Some considerations on the static and dynamic shear testing on rock discontinuities. Proceedings of 2019 Rock Dynamics Summit in Okinawa, 7–11 May 2019, Okinawa, Japan, ISRM (Editors: Aydan, Ö., Ito, T., Seiki T., Kamemura, K., Iwata, N.), pp. 187–192. Aydan, Ö. & Amini, M. G. (2009) An experimental study on rock slopes against flexural toppling failure under dynamic loading and some theoretical considerations for its stability assessment. Journal of Marine Science and Technology, 7(2), 25–40. Aydan, Ö. and Geniş, M. (2010): Rockburst phenomena in underground openings and evaluation of its counter measures. Journal of Rock Mechanics, TNGRM, Special Issue, 17, 1–62. Aydan, Ö. & Kawamoto, T. (1987) Toppling failure of discontinuous rock slopes and their stabilisation (in Japanese). Journal of Mining and Metallurgy Institute of Japan, Tokyo, 103(597), 763–770. Aydan, Ö. & Kawamoto, T. (1992) The flexural toppling failures in slopes and underground openings and their stabilisation. Rock Mechanics and Rock Engineering, 25(3), 143–165. Aydan, Ö. & Kumsar, H. (2010). An experimental and theoretical approach on the modeling of sliding response of rock wedges under dynamic loading. Rock Mechanics and Rock Engineering, 43(6), 821–830. Aydan, Ö., Kyoya, T., Ichikawa, Y., Kawamoto, T. and Shimizu, Y. (1988). A model study on failure modes and mechanism of slopes in discontinuous rock mass. Proceedings of the 23 National Conference on Soil Mechanics and Foundation Engineering, JSSMFE, Miyazaki, 1, 1089–1092. Aydan, Ö., Ohta, Y., Amini, M. & Shimizu, Y. (2019) The dynamic response and stability of discontinuous rock slopes. Proceedings of 2019 Rock Dynamics Summit in Okinawa, 7–11 May 2019, Okinawa, Japan, ISRM (Eds Aydan, Ö., Ito, T., Seiki T., Kamemura, K., Iwata, N.), pp. 519–524. Aydan, Ö., Ohta, Y., Daido, M., Kumsar, H. Genis, M., Tokashiki, N., Ito, T. & Amini, M. (2011) Chapter 15: Earthquakes as a rock dynamic problem and their effects on rock engineering structures. Advances in Rock Dynamics and Applications (Eds Y. Zhou and J. Zhao), CRC Press, Taylor and Francis Group, pp. 341–422. Aydan, Ö., Ohta, Y., Geniş, M., Tokashiki, N. & Ohkubo, K. (2010) Response and earthquake induced damage of underground structures in rock mass. Journal of Rock Mechanics and Tunnelling Technology, 16(1), 19–45. Aydan, Ö., Shimizu, Y. & Ichikawa, Y. (1989). The effective failure modes and stability of slopes in rock mass with two discontinuity sets. Rock Mechanics and Rock Engineering, 22(3), 163–188. Aydan, Ö., Shimizu, Y. & Karaca, M. (1994). The dynamic and static stability of shallow underground openings in jointed rock masses. The 3rd International Symposium on Mine Planning and Equipment Selection, Istanbul, October, 851–858. Aydan, Ö. & Tokashiki, N. (2019) Tsunami boulders and their implications on a mega earthquake potential along Ryukyu Archipelago, Japan. Bulletin of Engineering Geology and Environment, 78(6), 3917–3925.

Model testing and photo-elasticity  153 Bieniawski, Z.T. & van Tonder, C.P.G. (1969) A photoelastic-model study of stress distribution and rock around mining excavations. Experimental Mechanics, 9(2), 75–81. Bray, J.W. & Goodman, R.E. (1981) The theory of base friction models. International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, Oxford, 18, 553–568. Brewster, D. (1815) Experiments on the depolarization of light as exhibited by various mineral, animal and vegetable bodies with a reference of the phenomena to the general principle of polarization, Philosophical Transactions of the Royal Society, 29–53. Bucky, P.B. (1931) The Use of Models for Study of Mining Problems. Technical Publication 525, American Institute of Mineral and Metallurgical Engineering, NewYork. Coker, E.G. & Filon, L.N.G. (1930/1957) Treatise on Photoelasticity. Cambridge Press, Cambridge. Egger, P. (1979) A new development in the base friction technique. Colloquium on Geomechanical Models, ISMES, Bergamo. pp. 67–81. Erguvanlı, K. & Goodman, R.E. (1972) Applications of models to engineering geology for rock excavations. Bulletin of the Association of Engineering Geologist, 9(1). Everling, G. (1965) Model tests concerning the interaction of ground and roof support in gate roads. International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, 1, 319–326. Fumagalli, E. (1973) Statical and Geomechanical Models. Springer-Verlag, Vienna. Geniş, M. & Aydan, Ö. (2002). Evaluation of dynamic response and stability of shallow underground openings in discontinuous rock masses using model tests. Korea-Japan Joint Symposium on Rock Engineering, Seoul, Korea, July, pp. 787–794. Kumsar, H., Aydan, Ö. & Ulusay, R. (2000): Dynamic and static stability of rock slopes against wedge failures. Rock Mechanics and Rock Engineering, 33(1), 31–51. Phillips, E. (1869) De l’equilibre des solides elastiques semblables. Comptes rendus de l'Académie des Sciences, Paris, 68, 75–79. Shimohira, K., Aydan, Ö., Tokashiki, N., Watanabe, K. & Yokoyama, Y. (2019) Proceedings of 2019 Rock Dynamics Summit in Okinawa, 7–11 May 2019, Okinawa, Japan, ISRM (Eds: Aydan, Ö., Ito, T., Seiki T., Kamemura, K., Iwata, N.), pp. 193–198. Stacey, T.R. (2006) Considerations of failure mechanisms associated with rock slope instability and consequences for stability analysis. The Journal of the South African Institute of Mining and Metallurgy, 106, 485–493. Stephanson, O. (1971) Stability of single openings in horizontally bedded rock. Engineering Geology, 5(1), 5–71. Sugawara, K., Akimoto, M., Kaneko, K. & Okamura, H. (1983) Experimental study on rock slope stability by the use of a centrifuge. Proceedings of the Fifth International Society for Rock Mechanics Congress, pp. C1–C4.

Chapter 6

Rock excavation techniques

6.1 Blasting 6.1.1 Background Blasting is the most commonly used excavation technique in mining and civil engineering applications. Blasting induces strong ground motions and fracturing of rock mass in rock excavations. The excavation of rocks in mining and civil engineering applications by blasting technique is the most common technique since chemical blasting agents were developed centuries ago (i.e. Hendron, 1977; Hoek and Bray, 1981). However, the development of modern blasting techniques is after the invention of dynamites. Blasting induces high ground motions and fracturing of rock mass adjacent to blast holes (i.e. Thoenen and Windes, 1942; Attewell et al., 1966; Siskind et al., 1980; Kutter and Fairhurst, 1971). Particularly, high ground motions may also induce some instability problems of rock mass and structures nearby (Northwood et al., 1963; Tripathy and Gupta, 2002; Kesimal et al., 2008; Hao, 2002; Ak et  al. 2009; Geniş et  al. 2013). Furthermore, it may cause some environmental problems due to noise as well as vibrations of structures near populated areas (i.e. Aydan et al., 2002). Models for the attenuation of ground motions induced by blasting are generally based on velocity-type attenuation following the initial suggestions pioneered by United States Bureau of Mines (USBM) (Thoenen and Windes, 1942), and many models follow the footsteps of the USBM model (i.e. Attewell et al., 1966; Tripathy and Gupta, 2002 etc.). These models are often used in an empirical manner to assess the environmental effects on structures and human beings (Northwood et al., 1963; Hendron, 1977; Siskind et al., 1980). Although it is mathematically possible to relate ground motion parameters with each other, it is not always straightforward to do so (Aydan et al., 2002). Depending upon the sampling intervals of ground motion records, as well as superficial effects particularly during the integration process, the ground motion parameters and records may be different if they are measured by velocity meters or accelerometers. However, it should be noted that the acceleration records are essential when they are used in stability assessments. 6.1.2  Blasting agents 6.1.2.1 Dynamite Dynamite is an explosive material of nitroglycerin, using diatomaceous earth or other absorbent substance such as powdered shells, clay, sawdust or wood pulp. The Swedish chemist and engineer Alfred Nobel invented dynamite in 1867. Dynamite is usually in the form of

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cylinders about 200 mm long and about 3.2 mm in diameter, with a weight of about 186 g. Dynamite is generally used in underground excavations as it produces less harmful gases during explosion. 6.1.2.2  Ammonium nitrate/fuel oil (ANFO) ANFO (ammonium nitrate/fuel oil) is a widely used bulk industrial explosive mixture. It consists of 94% porous prilled ammonium nitrate (NH4NO3) (AN) that acts as the oxidizing agent and absorbent for the fuel and 6% fuel oil (FO). ANFO is widely used in open-cast coal mining, quarrying, metal mining and civil construction as it is a low-cost and ease-of-use matter among other conventional industrial explosives. The initiation of blasting is achieved using primer cartridges. 6.1.2.3  Blasting pressure for rock breakage The detonation pressure is empirically related to the density (ρ0) and detonation velocity (D) of explosives, and the following formula is generally used for estimating the intrinsic pressure of explosives: p=

1 ρo D 2 (6.1) 1+ γ

where γ is parameter related to the intrinsic properties of explosives. Its value is generally 3 (Persson et al., 1994). Table 6.1 summarizes the detonation pressure of several explosives used in rock breakage. However, the actual blasting pressure acting on the wall of holes is much less than the detonation pressure due to the gap between explosives, thickness of the casing, deformability characteristics and fractures in rock mass. 6.1.3 Measurement of blasting vibrations in open-pit mines and quarries The author has conducted measurements of blasting-induced motions in open-pit mines and quarries. The results of measurements conducted by the author are briefly explained in this subsection. 6.1.3.1  Orhaneli open-pit lignite mine Measurements were carried out separately in Orhaneli open-pit and Gümüşpınar village (Aydan et al, 2002). Figure 6.1 shows a general view of open-pit mine and nearby

Table 6.1  Summary of basic parameters of commonly used explosives Explosive

Detonation Velocity (m s−1)

Density (g cm−3)

Detonation Pressure (GPa)

Dynamite ANFO

4500–6000 (7600) 2700–3600

1.3, 1.593 (1.51) 0.882–1.10

6–13.6 (22.0) 0.7–9.0

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Figure 6.1 General view of Orhaneli open-pit mine, Gümüşpınar village behind the pit and the major fault plan on the left side

Gümüşpınar village. The village is about 1 km from the open-pit mine. In the open-pit mine, a major normal fault and several minor strike-slip faults were observed. The minor strikeslip fault is observed near the ground surface, and they are limited to near-surface layers. The rock mass above the lignite seam consists of tuff, sandstone, breccia and marl. Figure 6.2 shows the layout of the instrumentation employed at the open-pit mine blasting test. The blasting hole is 9.65 m deep with a diameter of 25 cm. It is a single hole with two deck charges. Both upper and lower deck charges consist of 50 kg of ANFO and 1 kg of cap-sensitive emulsion explosive (dynamite). Initiation was done by nonelectric shock tube (NONEL) detonator with 25 ms delay. The upper deck was fired first. Figure 6.3 shows a view of blasting operation experiment. The distance to the blasting hole was 66 m from Acc-3. Figure 6.3 shows the records of accelerometer denoted as Acc-1 during blasting. As seen from the figure, the magnitude of the vertical component of the acceleration waves is the largest, while that of the traverse components is the least among other components. Furthermore, the peak of the vertical component appears a few milliseconds before the others. The reason for the difference between vertical and horizontal components may be related to the damage and weakening caused by the previous blasts to the top 1–1.5 m of the bench on which the instruments were located. Therefore the weakened top part of the bench can be regarded as a low-velocity layer as compared with the rest of the bench below. This lowvelocity (damaged) layer may cause the attenuation of horizontal components, while its effect on the vertical component is less since the vertical component wave travels mostly through undisturbed marl beds. Furthermore, the interface between coal and marl, which is just 8–10 m below the ground surface, may act as a good reflecting surface so that the vertical component is enhanced in amplitude.

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Figure 6.2 Layout of instrumentation employed in the open-pit mine blasting test

Figure 6.3 Acceleration records from Acc-1 accelerometer

Figure 6.4 shows the vertical acceleration records of accelerometers denoted as Acc-1, Acc-2 and Acc-3 during blasting. Although the magnitudes of initial peaks follow the order of distance to the blasting hole, the magnitude of the farthest accelerometer denoted as Acc-3 is larger than the others. This may be caused by some slight variation of fixation and ground

Rock excavation techniques  159

Figure 6.4 Vertical components from accelerometers denoted as Acc-1, Acc-2 and Acc-3

Figure 6.5 Acceleration components obtained from the numerical derivation of velocity records

conditions beneath the accelerometers. The peak value exceeds 500 gal, and the waves attenuate as time goes by within 0.2 s following the blasting. The ground near accelerometers was blasted previously. The accelerometer locations could be disturbed at varying degrees depending upon the distance to the previous blast holes. Therefore, the attenuation of records of accelerometer Acc-2 is greater than the others, and long-period waves become dominant. Figure 6.5 shows the acceleration obtained from the numerical derivation of velocity records using sampling interval of the velocity-meter (denoted Vel-3) next to the accelerometer Acc-3 (see Figure 6.2 for location). Although the records are very similar to each other, the accelerations computed from velocity meters are larger in amplitude as compared with those from accelerometers. For example, for the vertical component, the maximum

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amplitude of the acceleration wave was 525 gals from the accelerometer as compared with 609 gals computed from the velocity record of the velocity meter. The amplitude obtained from the velocity meter records is about 1.16 times that of the true acceleration records. The difference may arise from one or all of the following reasons: slight variation of fixation of instruments, the errors inherent in numerical derivation arising from digital wave forms or the frequency dependence of directly measured acceleration value. The validity of the first and second reasons must be checked by further measurements and analysis. If, especially, the second reason holds true, the empirical damage criteria based on the peak particle velocity of the ground could not be employed straightforwardly in the case integration of directly measured acceleration values, or vice versa. Hence a different damage criterion should be developed for the case of direct monitoring of acceleration. The amplitude of vertical component of acceleration waves caused by blasting is larger than that of other components. The amplitude of the acceleration waves is in the order of vertical, longitudinal (radial) and traverse (tangential). However, the response spectra imply that amplifications are in the reverse order. Fourier spectra of longitudinal, traverse and vertical components of the acceleration records of the accelerometer and velocity meter indicated that dominant frequencies of the waves observed at 8–10 Hz and 30–40 Hz account for the fundamental vibration mode. The results indicate that structures having a natural period less than 0.06 s could be very much influenced. The effect of blasting should be smaller for structures having natural periods greater than 0.1 s. 6.1.3.2  Demirbilek open-pit lignite mine Aydan et al. (2014a) performed ground motion measurements in the open-pit mine near Demirbilek village during blasting, in which the attenuation of ground motions and the effect of existing faults were investigated, using both velocity meters and accelerometers simultaneously. The YOKOGAWA WE7000 modular high-speed PC-based data-acquisition system was used. It can handle 16 channels simultaneously. The sampling interval was set to 10 ms during measurements. AR-10TF accelerometers with three components were used, and this device can measure accelerations up to 10 g. X and Y components of the accelerometer were aligned radially and in tangential directions with respect to the blasting location. Z-direction measured the up and down (UD) component of ground motions. Figure 6.6 shows one of the layouts for ground motion observations and a view of blasting.

Figure 6.6 Layout of ground motion observations and a view of blasting

Rock excavation techniques  161

Figure 6.7 Acceleration responses measured during the in-situ blasting experiment number 47

A typical blast hole in the lignite mines of Turkey is generally 8–10 m deep with a diameter of 25 cm. It consists of 50–75 kg of ANFO and 0.5–1 kg of cap-sensitive emulsion explosive (dynamite). One third of each blast hole stemmed with soil, and initiation was done by nonelectric shock tube detonator with a 25–50 ms delay. The characteristics of ground motions induced by blasting depends upon the amount of explosive, the layout of blast holes and benches, delays and geomechanical properties of rock mass. Figure 6.7 shows an example of acceleration responses during the blasting experiment numbered 47 with a 15.5 kgf ANFO explosive. The distance of the blasted hole was approximately 40 m away from the monitoring location on the same bench level. Typical rocks observed in the lignite mine are lignite itself, marl, sandstone, mudstone and siltstone. As mentioned in the introduction, most of the attenuation relations used in the evaluation of the effects of blasting are of the velocity type. There are very few attenuation relations for blasting-induced accelerations. Dowding (1985) proposed an empirical attenuation relation for maximum acceleration. Wu et al. (2003) also developed an empirical relation using the results of small-scale field blast tests involving soil and granite. However, the empirical relations particularly overestimate maximum accelerations within a distance of 100 m of the blast location. Nevertheless, the empirical relation proposed by Dowding performs better than that by Wu et al. (2003). Therefore, the authors develop their attenuation relations for ground conditions typical in the lignite mines of Turkey. The attenuation of ground acceleration may be given in the following form as a convolution of three functions F, G and H in analogy to the attenuation relation of ground motions induced by earthquakes (Aydan 2012; Aydan et al. 2011): amax = F (V p )G ( Re ) H (W ) (6.2) where VP, Re and W are elastic wave velocity, distance from the explosion location and weight of explosives. The units of VP, Re and W are m s−1, meter (m) and kilogram force (kgf) while the unit of acceleration is gal. In analogy to the spherical attenuation relation proposed

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Figure 6.8  Attenuation of maximum ground acceleration with distance for a single-hole blasting experiment

by Aydan (1997, 2001, 2007, 2012; Aydan and Ohta, 2011), the functions F, G and H may be assumed to be of the following forms: Vp / a

F (V p ) = Ao (e

−1) (6.3a)

W /b

−1) (6.3b)

H (W ) = (e

− Re / c

G ( Re ) = Ae

(6.3c)

The coefficients of a, b and c are found to be approximately 4600, 550 and 85, respectively, for the observation data obtained from single-hole blasting experiments at Demirbilek openpit mine as shown in Figure 6.8, together with the values of the coefficients of Equation (6.3) for each component. The values of coefficient Ao are found to be 3200, 2800 and 4000 for radial, tangential and vertical components, respectively. However, the value of coefficient c may be different for each component of ground acceleration in relation to explosive type. The value of coefficient c is applicable to ANFO explosives, which are commonly used in Turkish lignite mines. The empirical relation by Dowding (1985) follows a similar approach in order to take into account the effect of ground conditions in attenuation relations. 6.1.3.3  ELI I ş ıkdere open-pit mine (A)  2010 MEASUREMENTS

2010 measurements were carried out using a single accelerometer of G-MEN type. The blasting holes were 7 m deep, and two rows of the holes with a separation distance of 7 m were drilled parallel to the bench face. For each hole, the amount of ANFO was 50–75 kg, and stemming was 1/3 of the total depth of the hole. Each sag has 25 kg ANFO. The delay between the front and back rows of the holes was 25 ms. Blasted rock mass consists mainly of marl. Figure 6.9 shows views of blasts while Figure 6.10 shows the acceleration records for different blasts. As noted from the figure, the acceleration records are not symmetric with respect to time axis.

Figure 6.9 View of blasting on 23 August 2010 at ELI Işıkdere open-pit mine

Figure 6.10 View of blasting on 23 August 23 2010 at ELI Is˛ıkdere open-pit mine

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Figure 6.11 (a) Attenuation of radial, tangential and vertical accelerations, (b) the increase of the maximum acceleration with respect to sag number

Figure 6.11 shows the attenuation of radial, tangential and vertical accelerations and the increase of the maximum acceleration with respect to the sag number in accordance with functional forms of Equations (6.3b) and (6.3c). As expected, the maximum acceleration decreases as a function of distance while the amplitude of the maximum ground acceleration increases as the amount of ANFO increases. (B)  2011 MEASUREMENTS

The 2011 measurements were carried out using 7 QV3-OAM stand-alone accelerometers with trigger mode and 2 G-MEN-type accelerometers. The first series of investigations were aimed to see the attenuation of accelerometers. The nearest station to the blasting point was 10 m (Figure 6.12). Figure 6.13 shows some of the acceleration records.

Figure 6.12 Views of blasting experiments before and during blasts

Figure 6.13 Acceleration records for the first series of blasting experiments

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Figure 6.14 Attenuation of maximum ground acceleration with respect to distance

Figure 6.15 Rock fall-induced ground motions triggered by blasting

Figure 6.14 shows the attenuation of maximum acceleration as a function of distance. As indicated previously, the vertical component is the largest, while the tangential component is the smallest. However, the attenuations with distance are different from each other and tangential component attenuates gradually compared with the rapid attenuation of other components. Figure 6.15 shows an acceleration record in which ground motions are induced by rock falls seen in Figure 6.12(a) also triggered by blasting. 6.1.3.4  Motobu quarry A trial measurement was done at Motobu limestone quarry in Okinawa island (Figure 6.16). The monitoring of blasting-induced vibrations is done using four stand-alone accelerometers with trigger mode, whose locations with respect to blasting are shown in Figure 6.16. The first row consists of 6 holes of 13.5 m depth spaced at a distance while the second row, spaced at a distance of 3.5 m, was 7 holes. The hole was filled with 5 m high ANFO and

Rock excavation techniques  167

Figure 6.16 Several views of the blasting and location of instruments

8.6. m stemming material. However, one of the accelerometers did not function during the blasting. Figure 6.17 shows the acceleration records. The axial component was larger than the UD component, and they attenuate with distance from the blasting location. In the same records, some ground motions induced by the falling rock blocks are also noticed.

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Figure 6.17 Acceleration records at Motobu limestone quarry

6.1.4  Measurements at underground openings The author has conducted measurements of blasting-induced motions in several tunnels in Japan and Turkey. The results of measurements conducted by the author are briefly explained in this subsection.

Rock excavation techniques  169

6.1.4.1  Kuriko Tunnel The Kuriko Tunnel in Fukushima prefecture is excavated through Mt. Kuriko (Watanabe et al. 2013). A geology of the tunnel consists of granite, rhyolite, tuff, andesitic dykes and intercalated sedimentary rocks such as sandstone, mudstone and conglomerate. While granite is exposed in the Fukushima side (east), folded sedimentary rocks with folding axis aligned in north–south outcrops on Yonezawa side (west). Sedimentary rocks are covered with tuff and rhyolite and intruded with andesitic dykes (Figure 6.18). The length of blast hole rounds was 1.5 m, and a total of 156 kg dynamite was used at the location where measurement was done. Figure 6.19 shows the layout of blast holes,

Figure 6.18 Geological cross section of Kuriko evacuation tunnel beneath Mt. Kuriko Source: Modified from Haruyama and Narita (2009)

Figure 6.19 Typical layout and plan view of blast holes used at Kuriko Tunnel

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Figure 6.20 Layout of instrumentation at Kuriko Tunnel

while Figure 6.20 shows the layout of instrumentation and their views with respect to tunnel advance direction. Figure 6.21 shows the acceleration records. The axial and tangential components were larger than the radial component, and they attenuate with distance from the blasting location. Furthermore, several shocks are recorded in relation to blasting sequence. 6.1.4.2  Taru-Toge Tunnel Taru-Toge Tunnel is being constructed as a part of an expressway project connecting ShinTomei Expressway and Chuo Expressway at the boundary of Shizuoka and Yamanashi Prefectures in Central Japan (Imazu et al. 2014; Aydan et al. 2016). The tunnel passes through a series of mudstone, sandstone, conglomerate layers with folding axes aligned north–south. (A)  INSTRUMENTATION AND INSTALLATION

The total number of accelerometers was 13, and the accelerometers were fixed to the plates of the rock bolts or steel ribs (Figures 6.22 and 6.23). In addition, three more accelerometers were attached to the plates of rock bolts at the passage tunnel between the main tunnel and the evacuation tunnel for wave velocity measurements during the blasting operation at the main tunnel. The accelerometers can be synchronized, and they can be set to the triggering mode with the capability of recording pretrigger waves for a period of 0.5 s. The trigger threshold and the period of each record can be set to any level and chosen time as desired. The accelerometer is named QV3-OAM-SYC, has a storage capacity of 2 GB and is a stand-alone type. It can operate for two days using its own battery, and the power source can be solar light if appropriate equipment is used. In other words, it is an eco-friendly acceleration monitoring

Figure 6.21 Acceleration records for each component

Figure 6.22 View of the accelerometer and its fixation in the main tunnel

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Figure 6.23 Locations of blasting and accelerometers (Note that accelerometer S15 had a battery problem during blasting at the main tunnel.)

Figure 6.24 Acceleration records at measurements stations S17 (27 m) and PT06 (83 m)

system. Figure 6.23 shows the location of accelerometers triggered during each blasting operation. The blasting (Blasting-1) at the evacuation tunnel was done at 21:18 on 28 February 2014. The second blasting (Blasting-2) was carried out at 10:54 on 5 March 2014. (B)  CHARACTERISTICS OF GROUND VIBRATION DURING BLASTING

Blasting-1  The blasting (Blasting-1) at the evacuation tunnel was done at 21:18 on 28 February 2014, and the amount of explosive was 75 kgf with 10 rounds with a delay of 0.4–0.5 s. The threshold value for triggering was set to 10 gals and the total number of the triggered accelerometers was 9. The most distant accelerometer was PT06, and its distance from the blasting location was 83 m. The highest acceleration was recorded at the accelerometer denoted as S17, and its value was about 1000 gals. Figure 6.24 shows the acceleration records at the stations denoted S17 and PT05. As noted from the figure, the amplitude of accelerations decreases with distance as expected. Another interesting observations is that the acceleration records are not symmetric with respect to the time-axis. Furthermore, acceleration wave amplitudes differ depending upon the direction.

Rock excavation techniques  173

Figure 6.25 also shows the Fourier spectra of each component of acceleration waves. As noted, the figures for the Fourier spectra of the radial and axial components of the accelerometer S17 consist of higher-amplitude and higher-frequency content. As for the distant accelerometer PT05, the vice-versa condition is observed, as expected. Figure 6.26 shows the acceleration response spectra of acceleration records taken at S17 and PT05 for the respective directions. As noted from the figure, the acceleration response spectra have very short natural periods, as expected. Figure 6.27 shows the attenuation of maximum acceleration at all stations. As the wave forms are unsymmetric with respect to time-axis, peak values are plotted as positive peak (PP) and negative peak (NP) with the consideration of their position in relation to the blasting location. As noted from the figure, the attenuation is quicker at the unblasted side compared with those on the blasted side. The data is somewhat scattered, and this may be related to the existence of structural weakness zones in the rock mass.

Figure 6.25 Fourier spectra of acceleration records shown in Figure 6.24

Figure 6.26 Acceleration responses spectra of acceleration records shown in Figure 16.3.

Figure 6.27 Attenuation of maximum ground acceleration with distance for Blasting-1

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Blasting-2  The second blasting (Blasting-2) at the main tunnel was carried out at 10:54 on 5 March 2014, and the total amount of blasting was 30.4 kgf with 0.4–0.5 ms delays per round. Figure 6.28 shows the tunnel face before and after blasting of the lower bench. The threshold value for triggering was set to 10 gals, and the total number of the triggered accelerometers was 14. The most distant accelerometer was PT05 and its distance from the blasting location was 132.5 m. The highest acceleration was recorded at the accelerometer denoted as S20, and its value was about 1030 gals. Figure 6.29 shows the acceleration records at the stations denoted S20 and PT05. As also noted from the figure, the amplitude of accelerations decreases with distance as expected. Another interesting observations is that the acceleration records are not symmetric with respect to the time-axis. Furthermore, acceleration wave amplitude differs depending upon the direction of measurements. The comments for Fourier spectra basically would be the same except those for the tangential component. Lower-frequency content waves become dominant for the tangential component as seen in Figure 6.30. Figure 6.31 shows the acceleration response spectra of acceleration records taken at S20 and PT05 for the respective directions. As noted from the figure, the acceleration response spectra have very short natural periods, as expected.

Figure 6.28 Views of tunnel face before and after blasting

Figure 6.29 Acceleration records at measurements stations S20 (11.6 m) and PT05 (132.5 m)

Rock excavation techniques  175

Figure 6.30 Fourier spectra of acceleration records shown in Figure 6.29

Figure 6.31 Acceleration response spectra of acceleration records shown in Figure 6.29

Figure 6.32 Attenuation of maximum ground acceleration with distance for Blasting-2

Figure 6.32 shows the attenuation of maximum acceleration at all stations triggered. As the wave forms are unsymmetric with respect to time axis, peak values are plotted as positive peak (PP) and negative peak (NP) with the consideration of their position in relation to the blasting location. As the passage tunnel exists on the west side between the main tunnel and the evacuation tunnel, the maximum accelerations are somewhat smaller at the station S18. Despite some scattering of measured results, the attenuation of maximum acceleration decreases with the increase of distance exponentially. The data scattering may also be related to the existence of structural weakness zones in the rock mass.

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6.1.4.3  Zonguldak tunnels Several tunnels in association with rehabilitating the intercity roadways in Turkey have been excavated in Zonguldak and its close vicinity, using the drilling and blasting technique. Genis et al. (2013) have been monitoring the blasting-induced vibrations in several adjacent tunnels near at Sapça, Üzülmez ve Mithatpaşa tunnels. In this subsection, the outcomes of the monitoring of vibrations in tunnels and at the ground surface are briefly presented for assessing the effects of blasting on the adjacent structures. The Sapça tunnels are double-tube two-lanes tunnels. The main purposes of the measurements were to see the effect of blasting at a new tunnel on the adjacent tunnel and ground surface. Figure 6.33 shows the position of tunnels, blasting location and measurement locations of a tunnel excavated through intercalated sandstone, siltstone and claystone. Figure 6.33 also shows the measurements at ground surface (70 m) and at the tunnel face (32 m) of the adjacent tunnel. The amount of blasting was 6–12 kg for each round. Although the ground motions are less on the ground surface than those at the adjacent tunnel due to the distance from the location of blasting, the ground motions are relatively high regarding the UD component. The Üzülmez tunnels are also two-lanes double-tube tunnels. The pillar distance between the tunnels is about 11 m. Three accelerometers were installed at the pillar side of the adjacent tunnel and two accelerometers at the mountainside. One more accelerometer was installed at the tunnel where blasting was carried out. Figure 6.34 shows the measurement results. Despite the distance, the measurements were highest in the tunnel of blasting, and the accelerations were high at the pillar side. The Mithatpaşa tunnels fundamentally have similar geometrical features, while rock mass is limestone, and some karstic caves were encountered during excavation.

Figure 6.33 Position of tunnels, blasting location and measurement locations and measured responses

Figure 6.34 Position of tunnels, blasting location and measurement locations and measured responses

Figure 6.35 Position of tunnels, blasting location and measurement locations and measured responses

178  Rock excavation techniques Table 6.2 Values of constants for maximum ground velocity and ground acceleration for different situations Coefficient

A0 a b c

Inside Tunnels

Adjacent Tunnels

Ground Surface

Acceleration (gal)

Velocity (kine)

Acceleration (gal)

Velocity (kine)

Acceleration (gal)

Velocity (kine)

4000 4600 240 100

140 4600 240 120

2000 4600 240 120

80 4600 240 100

6000 4600 240 100

200 4600 240 120

Figure 6.35 shows the position of tunnels, blasting location and measurement locations and measured responses. The blasting was very close to the portal of the tunnel. The total amount of the explosives was 192 kg while the amount of explosive for rounds changed between 10 kg to 25 kg. The accelerations were high for UD and traverse components. Genis et al. (2013) extended Equation 6.2 to the estimation of maximum ground velocity in addition to the maximum ground acceleration and determined the constants of Equation 6.3 for tunnels from the measurements presented in this subsection. They are given in Table 6.2. The coefficients are slightly different from those for measurements at lignite mines. The reason may be the difference of explosives and confinement in underground excavations from those at ground surface. 6.1.5  Multiparameter monitoring during blasting The real-time monitoring of the stability of tunnels is of great importance when tunnels are prone to failure during excavation such as rockbursting or squeezing. It is also known that when rock starts to fail, the stored mechanical energy in rock tends to transform itself into different forms of energy. Experimental studies by the authors showed that rock indicates distinct variations of multiparameters during deformation and fracturing processes. These may be used for the real-time assessment of the stability of rock structures. The parameters measured involve electric potential variations, acoustic emissions, rock temperature, temperature and humidity of the tunnel in addition to the measurements of convergence and loads on support members during the face advance. An application of this approach was done to the third Shizuoka tunnel of the second Tomei Expressway in Japan (Aydan et al., 2005). The parameters measured involve electric potential variations, acoustic emissions, rock temperature, temperature and humidity of the tunnel in addition to the measurements of convergence and loads on support members during the face advance. The tunnel excavation of the tunnel was done through the drilling-blasting technique. Each blasting operation causes both dynamic and static stress variations around the tunnel and its close vicinity. Measurements were carried out at two phases. In the first phase, the effect of the face advance of the Nagoya-bound tunnel on the Tokyo-bound tunnel was investigated. Figure 6.36 shows the layout of instrumentation.

Rock excavation techniques  179

Figure 6.36 Layout of instrumentation and face advance

Figure 6.37 shows the measured AE and electric potential responses as a function of time. Vertical bars in the same figure indicate the blasting operations. After each blasting operation, distinct AE and EP variations were observed. These variations cease after a certain period of time. The electric potential increase simultaneously and tends to decrease as time goes by. AE response also showed the same type of response. When the tunnel is stable, it is expected that these variations should disappear on the basis of experimental observations and theoretical considerations by the author.

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Figure 6.37 AE and EP responses measured resulting from face advance during the period from 19 August to 26 August 2004

The second phase measurements were planned to see both the effect of face advance on AE and EP responses in the same tunnel as well as that of the adjacent tunnel. Furthermore, two new electric potential measurement devices with higher impedance were used in addition to low-impedance electric potential measurement devices. Figure 6.38 shows the layout of instrumentation and face advance schedule. The responses of AE and EP are shown in Figure 6.39, and they were basically similar to those of the first phase. Nevertheless, the amplitude of variations was much larger than those of the previous phase. This was thought to be the closeness of the instruments to the tunnel face where blasting operations were carried out. The electric potential measurements with low-impedance and high-impedance electric potential devices were almost the same. However, the high-impedance electric potential measurement devices are desirable. In addition to that, some problems were noted with the fixation of electrodes into the rock mass and the damage by flying rock fragments during the blasting operation. However, the electric potential variations may sometimes include the far-field effects from the deformation of the Earth’s crust associated with earthquakes. During the measurement of electric potentials, an additional device would be necessary outside the tunnel. In this particular case, such a device was installed about 2 km west of the tunnel. The same instrumentation was repeated at Tarutoge Tunnel. Fıgure 6.40 shows the variations of acoustic emissions (AE) and electric potential variations in relation to blasting at the main tunnel and evacuation tunnel. The overall responses are similar to those measured at Shizuoka Third Tunnel.

Rock excavation techniques  181

Figure 6.38 Layout of instrumentation and face advance schedule

A trial measurement was carried out on the groundwater pressure variation during blasting operation at Demirbilek open-pit lignite mine. Figure 6.41 shows the monitoring results during the blasting operation numbered Demirbilek-07-blasting. The 1500 mm deep borehole, with a diameter of 160 mm, was drilled and filled with water. A water pressure sensor was installed in the borehole. The water pressure was measured using the same monitoring system. Although a very slight time lag exists between the peaks of ground acceleration and groundwater, the water pressure fluctuation occurred in response to the ground acceleration fluctuation. Furthermore, the groundwater level deceases thereafter due to seepage into surrounding rock as well as the increased permeability due to the damage rock by blasting.

Figure 6.39 Measured AE and EP responses during face advance

Figure 6.40 AE and EP responses measured resulting from face advance during the period from 24 July to 26 July 2014

Figure 6.41 Water level fluctuation during blasting at Demirbilek open-pit mine

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6.1.6  The positive and negative effects of blasting In this section, the positive and negative effects of blasting and blasting operations are presented and discussed. 6.1.6.1  In-situ stress inference Aydan (2013) proposed a method to estimate the stress state from the damage zone around blasted holes. This method was applied to the damage zone around blasted holes, and some stress inferences were made for the tunnel face at Kuriko Tunnel and Taru-Toge Tunnel. The estimations are compared with those from other methods (Aydan 2000: Aydan 2003). In this section, these examples are briefly presented. Figure 6.42 shows the damage zone around two blasted holes and the inferred stress state.

Figure 6.42 Views of damage and estimated yield zones around blast holes

Rock excavation techniques  185 (A)  KURIKO TUNNEL

The damage around blast holes are shown in Figure 6.42. The fracture zone formation around the blasted holes are almost elliptical with the longitudinal axis almost horizontal. These results imply that lateral stress is higher than the vertical stress on the tunnel face plane. The lateral stress coefficient was taken as 2.2, and the inclination of the maximum principal stress was assumed to be 0 from horizontal by taking into account the actual shape of the damage zone around blast holes. The results are given in Table 6.3 for a blast hole pressure of 150 MPa; the inferred plastic zones for different yield criteria and actual plastic zone are shown in Figure 6.43. The estimated damage zones are quite close to those shown in Figure 6.4. The in-situ stress estimations are also similar to the in-situ stress measurements by the AE method, in which the lateral stress coefficient was found to be 1.7.

Table 6.3 Inferred in-situ stress parameters and yield function parameters used in computations σ10 (MPa)

k

σc (MPa)

σt (MPa)

ϕ(o)

m

S∞ (MPa)

b1 (1/MPa)

9.0

2.2

90

6

60

14.93

360

0.045

Figure 6.43 Estimated damage zones around blast holes for Kuriko Tunnel

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Figure 6.44 Estimated stress state from the damage zone around blasted holes (B)  TARU-TOGE TUNNEL

Imazu et al. (2014) utilized the fault striation method (Aydan, 2000) and blasted hole damage method (BHMD) (Aydan, 2013) at Taru-Toge Tunnel. The estimated lateral stress coefficient from the fault striation method at the tunnel cross section ranges between 1.6 and 1.7. The BHDM was applied to the damage zone around blasted holes, and some stress inferences were made for the tunnel face. Figure 6.44 shows the damage zone around two blasted holes and the inferred stress state. The estimations indicate that the lateral stress coefficient is about 1.8 and that it is inclined with an angle of 10 degrees to the west. It is also interesting to note that this ratio is also very close to that estimated from the fault striation method. 6.1.6.2 Rock mass property estimation from wave velocity using blastinginduced waves The mechanical properties of rock mass may be estimated using the elastic wave velocity. Direct relations or normalized relations exist in the literature (i.e. Ikeda, 1970; Aydan et al., 1993, 1997; Sezaki et al., 1990). The uniaxial compressive strength (UCS) of intact rock ranges between 40–70 MPa, and its P-wave velocity is in the range of 3–4 km s−1. The direct relations between the UCS of rock mass and elastic wave velocity proposed by Aydan et al. (1993, 2014), and Sezaki et al. (1990) are, respectively: σcm = 5 (V pm −1.4)

1.43

(6.4)

σcm = 1.67 (V pm − 0.33) (6.5) 2

2.7 σcm = 0.98V pm (6.6)

Rock excavation techniques  187

where σcm and VPm are uniaxial compressive strength (UCS) and P-wave velocity of rock mass. Another relation, which is called “rock mass strength ratio,” is as follows: 2

V pm   σ (6.7) σcm =   V pi  ci where σci and Vpi are uniaxial compressive strength (UCS) and P-wave velocity of intact rock. Aydan et al. (2016) developed a portable system to measure the P-wave and S-wave velocities of the surrounding rock mass. The measurement system consists of five accelerometers connected to one another with wire and operated through a “start-stop” switch. This system was first used on 5 March 2014 and 1–3 September 2016 at Tarutoge Tunnel. The distance between accelerometers in 1–3present system can be up to 6 m (measurement length is about 30 m), and the sampling interval is 50 μs. Figure 6.45 shows the installation of the device in Tarutoge Tunnel, and measured velocity responses are shown in Figure 6.46. The estimated wave velocity of rock mass in the instrumented zone is estimated to be 1.9–2.6, 2.0–2.7 and 1.8–2.4 km s−1 during 6 March 2015. The normalized UCS of rock mass is estimated to be about 0.2–0.46 times that of intact rock. If direct relations are used and normalized by that of intact rock (Aydan et al., 2014b), the normalized UCS of rock mass would be 0.1–0.2 times that of intact rock. The measurement during 1–3 September 2016 yielded that the P-wave velocity was about 2.7 km s−1. In view of the P-wave velocity of intact rock, the results are quite close to those

Figure 6.45 Fixation of an accelerometer for wave velocity measurement

188  Rock excavation techniques

Figure 6.46 Acceleration records during the arrival of P-waves induced by blasting

of 5 March 2016 measurements. The normalized UCS of rock mass is estimated to be about 0.46 times that of intact rock. 6.1.6.3  Instability problems (A) EVALUATION OF EFFECTS OF BLASTING ON BENCH STABILITY AND RESPONSES

The width and height of benches range between 16 to 46 m and 8 to 10m, respectively, at Demirbilek open-pit lignite mines (Aydan et al., 2014a). The slope angle of the benches also varies between 50 and 60 degrees depending upon the mining operations and layouts. The failure of benches was observed in the upper marl unit, and they were of planar type on the east slope of the open-pit mine. Figure 6.47 shows the failure of benches at different locations at the east side of the open-pit mine. As seen in the pictures, the planar sliding failure is the dominant failure mode. The engineers of the open-pit mine also reported that the stability issues become very important following heavy rains. On the other hand, benches on the west slope of the open-pit mine were more stable, although some cracking and opening were observed. The instability problems on the benches of the west slope were associated with normal fault. Unless the fault plane cut through the benches, there were no major slope stability problems on the west slope benches. Aydan and his coworkers (Aydan et al., 1996; Aydan and Ulusay, 2002) proposed a method to estimate the movements of slopes involving sliding failure. This method was further elaborated in subsequent studies (Aydan et al., 2006, 2008, 2009a, 2011; Tokashiki and Aydan, 2010; Aydan and Kumsar, 2010). The author utilized this technique for assessing the response and stability of the slope subjected to dynamic forces as well as gravity and pore water pressures. The distance of the blasting location is one of the most important parameters for analyzing the response and stability of bench slopes.

Rock excavation techniques  189

Figure 6.47 Views of stability problems on the benches of the east slope

First the effect of vibrations measured by a single blast hole experiment with a distance of 100 m and an ANFO explosive of 50 kgf on the benches consisting of the upper marl unit was investigated. The computations indicated that no movement would occur under dry and fully saturated conditions. Then, the number of blast holes was assumed to be 18, which is commonly used during blasting operations in the open-pit mining of lignite mines in Turkey. The amplitude of the accelerations was increased by 4.7 times that of the single blast hole with the consideration of 18 blast holes with an ANFO explosive of 50 kgf. In the computations, both horizontal and vertical accelerations were considered. If the rock mass is assumed to be dry, the computations indicated that no relative movement along bedding planes emanating from the toe of the benches would occur. If the groundwater coefficient is more than 0.76, relative movement along the bedding planes occurs. Figure 6.48 shows the computed relative displacement responses in relation to the horizontal base acceleration. We introduce a water force coefficient denoted as ru to count the effect of groundwater in the body subjected to slide. It is defined as the volume of water to the total volume of the body prone to sliding. As noted from the figure, if rock mass is fully saturated (ru = 1.0) or nearly fully saturated (ru = 0.8), some permanent displacement would occur after each blasting operation. The sliding body in benches would gradually be displaced, and the separation of the sliding body from the rest of benches would occur at the upper levels and relative offsets at the toe of the slope. When the relative displacement becomes more than the half of the typical block size, blocks from the benches would fall to the lower level benches (Aydan et al., 2009b). This process would repeat itself successively at a given time interval.

190  Rock excavation techniques

Figure 6.48  Relative displacement responses of the benches of the east slope for each blasting operation

Figure 6.49 Rock fall at the portal of Mithatpaşa Tunnel

It is well-known that the blasting may cause the individual blocks to topple or sliding. Figure 6.49 shows an example of rockfall at the portal of Mithatpaşa Tunnel described in Subsection 6.3.4. Such events may cause casualties as well as property damage. The stability of such blocks can be evaluated using some dynamic limiting equilibrium approaches already

Rock excavation techniques  191

Figure 6.50 Blasting-induced failure at Gökgöl cave

mentioned. Similar events can also be found at karstic caves as seen at Gökgöl karstic cave in Zonguldak (Figure 6.50). The major collapse of the hall at the cave was caused by uncontrolled blasting operations in 1960 for enlarging the national highway 6.1.6.4  Vibration effects on buildings Using the acceleration records of accelerometer Acc-1 and accelerations computed from the records of velocity meter Vel-3, a series of response analyses are carried out. Figure 6.51 shows the normalized acceleration response spectra of each component of acceleration records with damping coefficient (h) values of 0.000, 0.025, 0.050, respectively. The results indicate that structures having a natural period less than 0.06 s could be very much influenced. The effect of blasting should be smaller for structures having natural periods greater than 0.1 s. Figure 6.52 and 6.53 show the responses of a structure with a natural period of less than 0.06 s and damping coefficient of 0.05 (h= 0.05) for each acceleration component from the accelerometer measurements and velocity meter measurements. As understood from Figures 6.52 and 6.53, the absolute acceleration acting on structures should be less than the accelerations of input motion. It would be safe to assume that the induced accelerations caused by blasting should act on the ground and structure system without any reduction.

Figure 6.51 Normalized acceleration response spectra of the records of accelerometer Acc-1 and records of velocity meter Vel-3

Figure 6.52 Absolute vertical acceleration response of a structure system (Acc-1)

Rock excavation techniques  193

Figure 6.53 Absolute vertical acceleration response of a structure system using the acceleration computed from the velocity records of velocity meter Vel-3

6.1.6.5  Air pressure due to blasting A typical pressure–time profile for a blast wave in free air is shown in Figure 6.54. It is characterized by an abrupt pressure increase at the shock front, followed by a quasi exponential decay back to ambient pressure po and a negative phase in which the pressure is less than ambient. The pressure-time history of a blast wave is often described by exponential functions such as Frielander’s equation (Smith and Etherington, 1994)  t p = po + ps 1−  e−bt / Ts (6.8)  Ts  where ps is peak overpressure, Ts is duration of the positive phase, and is is specific impulse of the wave that is the area beneath the pressure–time curve from the arrival at time to to the end of the positive phase.

Figure 6.54 Air pressure variation during blasting

194  Rock excavation techniques

It is common to use the scaled distance to evaluate the effects of blasting given as: Z=

R n

W

(6.9)

where R is distance and W is mass of the explosive. When the value of n is 3, it is called Hopkinson blast scaling. USBM suggests the value of n as 2. The attenuation of air pressure is generally given in the following form: p = AZ −α (6.10) If W is given in kg and R in m, the unit of p is kPa, the value of A is generally about 186, and the power of α ranges between 1.2 and 1.5. Air pressure changes were measured at Takamaruyama, Kuriko and Tarutoge Tunnels using TR-73U produced by TANDD Corporation. Although the device may not be appropriate for very sensitive air pressure changes induced by blasting, the measurements are quite meaningful, as seen in Figures 6.55–6.57. The cross sections of the tunnels are almost the

Figure 6.55 Air pressure fluctuations at Takamaruyama Tunnel

Figure 6.56 Air pressure fluctuations at Kuriko Tunnel

Rock excavation techniques  195

Figure 6.57 Air pressure fluctuations at Tarutoge Tunnel

same. Takamaruyama and Kuriko Tunnels are single tubes, while the Taru-Toge Tunnel has a branch connecting to the evacuation tunnel. The air pressure decreases at the evacuation tunnel when the shock wave passes by the branch and then increases as the air pressure is confronted by the door at the tunnel portal. Furthermore, the amplitude of the waves decreases as the distance increases. 6.1.6.6  Flyrock distance Flyrocks are another major issue for the safety of people and damage to machinery and equipment when blasting is employed (Figure 6.58). The flyrock issue was caused by either inappropriate stemming, blasting sequence or charges or the existence of some weak zones in rock mass. It is reported that the flyrock may travel up to a distance of 900 m in worst cases. The ejection velocity may reach up to 300 m s−1. Flyrock distance in common practice is less than 50 m, and it may sometimes reach a distance of 95 m. There are some empirical relations to estimate the maximum distance of flyrocks from the blasting relations. One of the empirical relations is given by Lundborg (1981): Lmax = 260 D 2 / 3 (6.11) The unit of fly distance is m, and D is given in inches. The fly distance may be obtained from the simple physical laws of an object thrown with an initial angle (β0) and velocity (v0) at a given height (h0). The air resistance may be taken into account as a viscous drag Fd. The fundamental equations of the flying object may be written as: F F d2y d2x = − d ; 2 = −g − d (6.12) 2 m dt m dt

196  Rock excavation techniques

Fıgure 6.58 Flyrock during blasting operation at Işıkdere open-pit lignite mine

where g is gravitational acceleration. If the viscous resistance is given in the following form: Fd = η mv n (6.13) the governing equation becomes: 2 d2x n d y = −η v , = −g − ηv n (6.14) dt 2 dt 2

where η is viscous drag coefficient. When n = 2, it is called Stoke’s law. If n = 1, it is possible to solve the preceding differential equation. Otherwise the differential equations become nonlinear. The solution of Equation 6.14 without drag force would yield the trajectory of a flyrock with conditions illustrated in Figure 6.59: g x = vo cos βt and y = yo + vo sin βt − t 2 (6.15) 2

Rock excavation techniques  197

Figure 6.59 Trajectory of a flyrock and its conditions

The maximum fly distance of flyrock in air can be estimated from Equation (6.15) with condition of y = 0: X max =

vo cos β vo sin β + (vo sin β ) 2 + 2 yo g (6.16) g

(

)

When yo= 0, then the travel distance of a flyrock takes the following form: X max =

(vo ) 2 sin 2β (6.17) 2g

The maximum fly distance would be obtained when β = 450. If β = 00, the travel distance of a flyrock ejected from a given height (yo) can be obtained from Equation 6.16: X max = vo

2 yo (6.18) g

As noted from these relations, the initial conditions are very important for the fly distance of flyrocks. If drag forces are taken into account, the solution of Equation 6.12 yields the trajectory of the flyrock: x=

vo cos β 1 g g 1− e−ηt ) and y = yo + vo sin β + (1− e−ηt ) − t (6.19) (  η η η η

The solution of Equation 6.12 for n > 1 requires the utilization of numerical techniques because it is difficult to obtain the closed-form solutions. Figure 6.60 compares the effect of viscous drag resistance of air on the fly trajectory of the flyrock.

198  Rock excavation techniques

Figure 6.60 Comparison of the effect of viscous drag resistance of air on the fly trajectories of the flyrock

6.2  Machine excavations 6.2.1  Road headers A roadheader is a piece of excavation equipment consisting of a boom-mounted cutting head, a loading device usually involving a conveyor and a crawler traveling track to move the entire machine forward into the rock face (Figure 6.61). The cutting head can be a general purpose rotating drum mounted in line or perpendicular to the boom together with picks. Roadheaders were first developed for the coal mining industry in the early 1960s (Copur et al., 1998). In general, roadheaders can be divided into two types: milling (axial) type, with the cutter head rotating around the boom axis, and ripping (transverse) type, with the head rotating perpendicularly to the boom axis (Copur et al., 1998). 6.2.2  Tunnel boring machines (TBMs) A tunnel boring machine (TBM) is used to excavate tunnels, shafts and raise-bores with a circular cross section through various ground conditions. Tunnel diameters can range from 1 m to 17.6 meters to date (Figure 6.62). Tunnel boring machines are used as an alternative to drilling and blasting (D&B) methods in rock. The first successful application of the TBM was during the construction of the tunnels beneath Thames in 1825. TBMs have the advantages of limiting the disturbance to the surrounding ground and producing a smooth tunnel wall. This significantly reduces the cost of lining the tunnel. When surrounding rock is heavily fractured and sheared, the TBM may get stuck. Modern TBMs typically consist of the rotating cutting head, followed by a main bearing, a thrust system and trailing support mechanisms. The type of machine used depends on the particular geology of the project, the amount of groundwater present and other factors. TBMs can be shielded or open depending upon the ground condition. When the face is unstable or the groundwater condition is bad, earth-pressure balanced (EBP)–type TBMs are used.

Rock excavation techniques  199

Figure 6.61 Excavation of storage rooms by roadheaders at Cappadocia

Figure 6.62 Views of Hida and Pinglin Tunnels

6.3  Impact excavation Breaker is a powerful percussion hammer fitted to an excavator for breaking rocks. It is powered by an auxiliary hydraulic or pneumatic system from the excavator, which is fitted with a foot-operated valve for this purpose. They are generally used when blasting cannot be used due to safety or environmental issues (Figure 6.63).

Figure 6.63 Views of breakers

200  Rock excavation techniques

Figure 6.64 Trimming basaltic rock blocks at Iguassu Falls in Brazil.

6.4  Chemical demolition Chemical demolition is a technique in use since the 1970s. This technique is used where blasting could not be implemented due to safety and environmental concerns. The method itself is generally expensive compared to other methods. The basic principle is based on injecting expansive grout into the holes and splitting rock through the arrangement of holes in a given pattern and spacing. Figure 6.64 shows an example of chemical demolition to trim basaltic rock blocks at Iguassu Fall in Brazil.

References Ak, H., Iphar, M., Yavuz, M. & Konuk, A. (2009) Evaluation of ground vibration effect of blasting operations in a magnesite mine. Soil Dynamics and Earthquake Engineering, 29(4), 669–676. Attewell, P.B., Farmer, I.W. & Haslam, D. (1966) Prediction of ground vibration from major quarry blasts. The International Journal of Mining and Mineral Engineering, 621–626. Aydan, Ö. (1997) Seismic characteristics of Turkish earthquakes. Turkish Earthquake Foundation, TDV/TR 97–007, 41 pages. Aydan, Ö. (2000) A stress inference method based on structural geological features for the full-stress components in the earth’ crust, Yerbilimleri, Ankara, 22, 223–236. Aydan, Ö. (2001) Comparison of suitability of submerged tunnel and shield tunnel for subsea passage of Bosphorus. Geological Engineering Journal, 26(1), 1–17.

Rock excavation techniques  201 Aydan, Ö. (2003) The Inference of crustal stresses in Japan with a particular emphasis on Tokai region. International Symposium on Rock Stress. Kumamoto, 343–-348. Aydan, Ö. (2007) Inference of seismic characteristics of possible earthquakes and liquefaction and landslide risks from active faults (in Turkish). The 6th National Conference on Earthquake Engineering of Turkey, Istanbul, 1, 563–574. Aydan, Ö. (2012) Ground motions and deformations associated with earthquake faulting and their effects on the safety of engineering structures. Encyclopaedia of Sustainability Science and Technology, Springer, New York, R. Meyers (Ed.), 3233–326.3. Aydan, Ö (2013) In-situ stress inference from damage around blasted holes. Journal of Geo-System Engineering, Taylor and Francis, 16(1), 83–91. Aydan, Ö., Akagi, T. & Kawamoto, T. (1993) Squeezing potential of rocks around tunnels: Theory and prediction. Rock Mechanics and Rock Engineering, Vienna, 26(2), 137–163. Aydan, Ö., Bilgin, H.A. & Aldas, U.G. (2002) The dynamic response of structures induced by blasting. Int. Workshop on Wave Propagation, Moving load and Vibration Reduction Okayama, Japan, Balkema. pp. 3–10. Aydan, Ö., Daido, M., Tano, H., Tokashiki, N. & Ohkubo, K. (2005) A real-time multi-parameter monitoring system for assessing the stability of tunnels during excavation. ITA Conference, Istanbul. pp. 663–669. Aydan, Ö., Geniş, M. & Bilgin, H.A. (2014a) The effect of blasting on the stability of benches and their responses at Demirbilek open-pit mine. Environmental Geotechnics. ICE, 1(4), 240–248. Aydan, Ö. & Ohta, Y. (2011) A new proposal for strong ground motion estimations with the consideration of characteristics of earthquake fault. Seventh National Conference on Earthquake Engineering, Istanbul, Paper No. 66, 1–10 pages. Aydan, Ö., Ohta, Y., Daido, M., Kumsar, H., Genis, M., Tokashiki, N., Ito, T. & Amini, M. (2011) Chapter 16.: Earthquakes as a rock dynamic problem and their effects on rock engineering structures. In: Zhou, Y. & Zhao, J. (ed.) Advances in Rock Dynamics and Applications. CRC Press, Taylor and Francis Group, London. pp. 341–422. Aydan, Ö., Tano, H., Ideura, H., Asano, A., Takaoka, H., Soya, M. & Imazu M. (2016) Monitoring of the dynamic response of the surrounding rock mass at the excavation face of Tarutoge Tunnel, Japan. EUROCK2016, Ürgüp, 1261–1266. Aydan, Ö., Üçpırtı, H. & Kumsar, H. (1996) The stability of a rock slope having a visco-plastic sliding surface. Rock Mechanics Bulletin (Kaya Mekaniği Bülteni), 6, 39–49. Aydan, Ö., Ulusay, R. & Kawamoto, T. (1997) Assessment of rock mass strength for underground excavations, Proc. of the 36th US Rock Mechanics Symposium, New York. pp. 777–786. Aydan, Ö., Ulusay, R. & Tokashiki, N. (2014b) A new Rock Mass Quality Rating System: Rock Mass Quality Rating (RMQR) and its application to the estimation of geomechanical characteristics of rock masses. Rock Mechanics and Rock Engineering, Springer, Vienna, 47(4), 666–676. Copur, H., Ozdemir, L. & Rostami, J. (1998) Road-Header Applications in Mining and Tunnelling Industries. Society for Mining, Metallurgy and Exploration, Orlando, FL. Dowding, C.H. (1986.) Blast Vibration Monitoring and Control. Englewood Cliffs, NJ, Prentice-Hall. Genis, M., Aydan, Ö. & Derin, Z. (2013) Monitoring blasting-induced vibrations during tunnelling and its effects on adjacent tunnels. Proc. of the 3rd Int. Symp. on Underground Excavations for Transportation, Istanbul. pp. 210–217. Hao, H. (2002) Characteristics of non-linear response of structures and damage of RC structures to high frequency blast ground motion. Wave2002, Okayama. Hendron, A.J. (1977) Engineering of rock blasting on civil projects. In: Structural and Geotechnical Mechanics. Prentice-Hall, Englewood Cliffs, NJ. Ikeda, K. (1970) A classification of rock conditions for tunnelling. Proceedings of the 1st Int. Congr. on Engineering Geology, IAEG, Paris. pp. 6.6.8–6.66. Imazu, M., Ideura, H. & Aydan, Ö. (2014) A Monitoring System for Blasting-induced Vibrations in Tunneling and Its Possible Uses for The Assessment of Rock Mass Properties and In-situ Stress Inferences. Proceedings of the 8th Asian Rock Mechanics Symposium, Sapporo, 881–890.

202  Rock excavation techniques Kesimal, A., Ercikdi, B. & Cihangir, F. (2008) Environmental impacts of blast-induced acceleration on slope instability at a limestone quarry. Environmental Geology, 6.4(2), 381–389. Kutter, H.K., Fairhurst, C. (1971) On the fracture process in blasting. International Journal of Rock Mechanics Min. Sci., & Geomech., Abstr., Pergamon Press, 8., 181-188. Lundborg, N. (1981) The probability of flyrock damages. Swedish Detoni Research Foundation, Stockholm, D.S. 6., 39 pp. Northwood, T.D., Crawford, R. & Edwards, A.T. (1963) Blasting vibrations and building damage. The Engineer, 216. Persson, P.A., Holmberg, R. & Lee, J. (1994). Rock Blasting and Explosives Engineering. Boca Raton, FL: CRC Press, 540 pp. Sezaki, M., Aydan, Ö., Ichikawa, Y. & Kawamoto, T. (1990) Mechanical properties of rock mass for the pre-design of tunnels by NATM using a rock mass data-base (in Japanese). Journal of Civil Engineers of Japan, Construction Division, 421-VI-13, 6.6.–133. Siskind, D.E., Stagg, M.S., Koop, J.W. & Dowding, C.H. (1980) Structure response and damage produced by ground vibration from surface mine blasting. United States Bureau of Mines, Report of Investigations, No. 86.07. Smith, P.D. & Hetherington, J.G. (1994) Blast and Ballistic Loading of Structures. Butterworth-­ Heinemann Ltd, Great Britain. Thoenen, J.R. & Windes, S.L. (1942) Seismic effects of quarry blasting. U.S. Bureau of Mines Bulletin, 442. Tripathy, G.R. & Gupta, I.D. (2002) Prediction of ground vibrations due to construction blasts in different types of rock. Rock Mechanics and Rock Engineering, 36(3), 196–204. Watanabe, H., Aydan, Ö. & Imazu, M. (2013) An integrated study on the stress state of the vicinity of Mt. Kuriko. The 6th International Symposium on In-Situ Rock Stress (SENDAI). pp. 831–838. Wu, C., Hao, H., Lu, Y. & Zhou, Y. (2003) Characteristics of stress waves recorded in small-scale field blast tests on a layered rock-soil site. Geotechnique, 63(6), 687–699.

Chapter 7

Vibrations and vibration measurement techniques

7.1  Vibration sources Vibrations are caused by different processes such as blasting, machinery, impact hammers, earthquakes, rockburst, bombs (including missiles), traffic, winds, lightning, weight drop and meteorites. Vibrations induced by impact hammers, blasting with small explosives, TBM may be used to evaluate wave velocity characteristics of rocks and rock masses for assessing the rock mass properties for design purposes. In recent years, it is also used to assess the rock mass conditions such as the existence of weak/fracture zones or cavities. They are used to infer the rock mass properties as well as some yielding or loosening around rock structures. In addition, vibrations may be used to investigate the soundness of support or reinforcement members. Off-course, they are used to investigate the effect of shaking caused by earthquakes or other large-scale vibration sources.

7.2  Vibration measurement devices Vibration measurements devices may be of acceleration, velocity or displacement types. The devices may utilize piezoelectric sensors, servo-acceleration sensor, electro-kinetic velocity sensors and noncontact displacement sensor. The devices used for measuring vibrations are called accelerometers, velocity meters, displacement meters. A piezoelectric sensor utilizes piezoceramics or single crystals such as quartz or tourmaline. The basic concept is to convert the mechanical motion into electric signals. Servo-acceleration sensors utilize a displacement detector; a current is fed to the coil to get the pendulum mass back to the original position when it is subjected to a motion. The current will be proportional to the acceleration that is converted to an output voltage. The servo-type accelerometer has higher sensitivity, stability and more accurate phase responses in the lower frequency range than those of other vibration transducers. These sensors are commonly used in micro tremor measurements. Electro-kinetic sensors convert vibrations into electric signals through the measurement of a streaming potential induced by the passage of polar fluid through a permeable ­refractory-ceramic or fritted-glass member between two chambers.

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7.3 Theory of wave velocity measurement in layered medium 7.3.1 Principles Let us consider a two-layered medium as shown in Figure 7.1. S denotes the source geophone, and R denotes the receiver geophone. It is assumed that the wave velocity V1 of layer 1 is smaller than that of layer 2 (V2 > V1). For this particular situation, there will be numerous wave paths. Among them, three wave paths would be of particular importance. These wave paths are called direct wave (S-R), reflected wave path (S-C-R) and refracted wave path (S-A-B-R). If the medium is assumed to be elastic and its density remains the same within the layer, Snell’s law holds for incidence angle and refraction angle: sin i V1 = (7.1) sin r V2 The refraction angle r of the refracted wave path shown in Figure 7.1 is 90 degrees. Therefore, the critical incidence wave angle ic can be easily obtained from Equation (7.1) as follows: sin ic =

V1 (7.2) V2

One can easily write the following relation between distance and arrival time for the direct wave path: t=

X (7.3) V1

As for the reflected wave path, the relations between distance and arrival time is given by: t=

SC CR + (7.4) V1 V1

Figure 7.1 Wave paths in a two-layered medium

Vibrations and vibration measurement  205

As SC = CR and given by: SC = CR = H 2 +

X2 4

Equation (7.4) becomes: t=

2 X2 H2 + (7.5) V1 4

The relation between distance and arrival time for the refracted wave path shown in Figure 7.1 can be written as follows: t=

SA AB BR (7.6) + + V1 V2 V1

From the geometry of the path, one can write the following relations: SA = BR =

1 H ⋅ , AB = X − 2 H tan ic (7.7) V1 cos ic

as 2

V  cos ic = 1−  1  V2 

Equation (7.6) takes the following form: 2

t=

V  2H X 1−  1  + (7.8) V2  V1 V2

The thickness of layer 1 can be obtained by equating the arrival times of the direct wave and refracted wave if the critical distance Xc and wave velocities V1 and V2 are obtained from the records as follows: X H= c V2

1−

V1 V2 2

V  1−  1  V 

(7.9)

2

As an application of this theory, a computation example was carried out, and the results are shown in Figure 7.2, together with the assumed wave velocities and the thickness of layer 1. The technique described is generally used to identify the loosed zones, excavation-induced damage zone (EDMZ) around the excavations or velocity structure of the rock mass. Several practical applications of this method are given in the following subsections.

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Figure 7.2 Relations between distance and arrival time for different wave paths

Figure 7.3 Some views of measurements and instruments

7.3.2  Elastic wave velocity measurements 7.3.2.1  Measurements at Amenophis III pharaoh underground tomb The instrument called Mc SEIS III developed by Oyo Corporation of Japan was used for this purpose (Figure 7.3). The device has three geophones and one hammer equipped with

Vibrations and vibration measurement  207

Figure 7.4 Layouts of geophones and source

an accelerometer for triggering (Hamada et al. 2004; Aydan and Geniş, 2004; Aydan et al. 2008). Figure 7.4 shows some views of measurements. Instrumentation layouts are shown in Figure 7.4. Figure 7.5 shows some of records. The measurements on the floor yielded low elastic wave velocities less than those of intact rocks and pillars, which implies that some low-velocity zones exist beneath the floor as noted from Table 7.1. The pillars, on the other hand, yielded high elastic wave velocities. Furthermore, the wave velocity increases as the elevation increases.

Figure 7.5 Some examples of velocity records Table 7.1  Elastic wave velocity measurements Chamber

Direction

Lowest Vp (km s−1)

Highest Vp (km s−1)

Je Je J J Jd Jd I

N-S E-W N-S E-W N-S E-W E-W

0.862 1.200 1.000 1.087 1.042 0.980 1.049

1.139 1.471 2.090 1.800 1.139 1.050 1.220

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Chamber

Direction

Lowest Vp (km s−1)

Highest Vp (km s−1)

G G F F D-1 D-2 D-1 D-2 B B-1 B-2 Jd-Pillar-1 Jd-Pillar-2 Je-Pillar F-Pillar1–1 F-Pillar1–2 F-Pillar2–1

N-S E-W N-S E-W N-S N-S E-W E-W N-S E-W E-W Vertical Vertical Vertical Vertical Vertical Vertical

1.262 0.971 0.885 1.049 1.563 1.389 1.087 1.428 2.000 2.083 1.370 2.549 2.195 1.611 1.857 1.767 1.652

1.470 1.000 1.471 2.500 1,754 1.818 1.754 1.563 2.500 2.501 1.613 3.399 2.358 2.712 3.064 2.371 1.867

Figure 7.6 View of measurements at B7F in Derinkuyu Underground City

7.3.2.2  Measurements at Derinkuyu Underground City The in-situ characterization of the tuff at Derinkuyu Underground City and its variation with depth were also assessed in this study. For this purpose, geotomographic investigations (Figures 7.6–7.7) at its several floors were carried out. The in-situ P-wave velocity of rock mass ranges between 0.9 and 1.3 km s−1. However, the variation of Vp with depth was insignificant. Therefore, the rock mass is considered to be fairly uniform.

Figure 7.7 Example of vibrations records at B7F

Figure 7.8 Distance–travel time relations at each underground floor.

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7.3.2.3  Measurements at a Mitake abandoned lignite mite Elastic-wave velocity measurements were carried out at an abandoned lignite mine in Mitake town using McSEIS-III (Figure 7.9). Measurements were concerned with the characteristics of lignite pillars that support the overburden rock mass. In addition, some in-situ index tests such as Schmidt hammer rebound tests and needle penetration tests were performed at the abandoned mine. The results are summarized in Table 7.2. The -wave velocity of intact lignite samples ranges between 1.57 and 2.26 km s−1. 7.3.2.4  Geotomographic investigations at a fault zone Inoue and Hokama (2019) carried out some geotomographic investigations in a fault zone in Urasoe City in Okinawa island. This investigation is summarized here. (A) PROCEDURE

The method is based on P-wave velocity. The vibration source was a hammer, and vibrations were measured using geophones (Figure 7.10). McSeis-SX was used to monitor and process the vibration data. Figure 7.11 illustrates one of the diagrams between distance and

Figure 7.9  In-situ elastic wave velocity measurement Table 7.2  Results of in-situ index tests Rock

Rebound Number (R)

p-wave velocity (km s−1)

Needle penetration index (N/mm)

Lignite Sandstone Mudstone

18–44 14–28  7–18

1.5 – –

7.0–12.5 20.0–25.0 11.1–20.0

Figure 7.10 Illustration of the measurement technique

Figure 7.11 Example of distance–arrival time of P-waves

Vibrations and vibration measurement  213

arrival time of P-waves for a given measurement section. Measurements were done along five sections. The authors utilized the simultaneous iterative reconstruction technique (SIRT) method for geotomographic investigations (Figure 7.12). This method utilizes the least square principle. It is thought to be insensitive to the errors of the measurement data. It can be used to reconstruct high-quality images from even inaccurate data with much noise. Furthermore, it is always convergent. Because of these advantages, SIRT is a good algorithm for the reconstruction of geotomography. (B) RESULTS

The measurements along five lines were analyzed using the SIRT method. Figure 7.13 shows a 3-D view of the contours of the P-wave velocities along each measurement line. The velocity of elastic P-waves ranges between 0.4 to 3.4 km s−1. The elastic P-wave velocity of intact the Ryukyu limestone samples generally ranges between 3.5 and 4.9 km s−1. When this fact is taken into account, the condition of rock mass varies depending upon the location. In other words, the rock mass condition is highly influenced by the fracturing state due to faulting. Except the weathered surface layers, the wave velocity is expected to decrease in the fracture zones. Figures 7.14 and 7.15 show the p-wave velocity cons was quite low, in the range of 0.4– 1.8 km s−1. The elastic wave velocity of rock mass along the major faults ranged between 0.8 and 1.4 km/s. The elastic wave velocity of the undisturbed zone was more than 3.2 km/s. 7.3.2.5 Effect of impact vibrations at the ground surface and an underground arch structure One of the major concern was the impacts induced by the landing or takeoff of airplanes on an underground arch structure beneath the runway (Minei et al., 2019). Although it was difficult to do such monitoring using actual planes, some tests on the impact vibrations at the ground surface on arch structure were carried out using a 10 tf truck and 1 tf sandbag dropped from a height of 100–180 cm. the accelerations induced by the passing of the truck over a 10 cm high barrier was quite small and attenuated very rapidly. Therefore, it was decided to use a 1 tf sandbag falling from a height of 100 cm or 180 cm, as

Figure 7.12 Illustration of the concept of SIRT method

Figure 7.13 P-wave velocity contours along five measurement sections

Figure 7.14 P-wave velocity distribution along the rock-cut section 1

Vibrations and vibration measurement  215

Figure 7.15 P-wave velocity distribution along the rock-cut

shown in Figure 7.16. Besides accelerometers on the ground surface, five accelerometers were installed in the underground arch structure (Figure 7.17). As shown in Figure 7.16, a 1 tf sand bag was dropped at various locations projected on the ground surface, and induced vibrations were measured. Figure 7.19 shows the attenuation of maximum accelerations measured at various points in relation to the distance from the source point. The attenuation of maximum accelerations are fitted to the following equation: amax = 20000e0.8*W *h

1 (7.10) 1+ r b

where W (tf) is weight of dropped bag, h (m) is the drop height and r (m) is the distance from the drop location. The coefficient b is an empirical value, and it was found that it ranges between 2 (cylindrical) and 3 (spherical). Including the measurements in the underground arch structure, its value is 2.5 and fits the in-situ measurements. As noted from Figure 7.19, the vibrations were drastically reduced as a function of the distance from the source area. Therefore, the dynamic impact effects of the airplanes during landing or takeoff would be quite small on the actual underground structure.

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Figure 7.16 View of the in-situ experiment and setup of accelerometers

Figure 7.17 Locations of accelerometers in the underground arch structure

7.3.2.6 Measurements of vibrations induced by pile construction in Gushikawa bypass bridge next to underground tomb The assessment of ground conditions is of great importance to assess the response of the ground during construction and in-service of the piles, including the 400-year-old underground tomb, in relation to the construction of pile foundations of the Gushikawa Bypass

Vibrations and vibration measurement  217

Figure 7.18 Locations of measurement points and sandbag drop

Figure 7.19 Attenuation of maximum accelerations with distance and its comparison with the attenuation relation

Bridge. The authors utilized some dynamic vibration measurements to assess the ground conditions (Tomori et al. 2019). The initial vibration measurements were carried out using five QV3-OAM-EX/W portable accelerometers developed by Aydan et al. (2016), which were utilized for different purposes, a vibration measurements system consisting of five TOKYO SOKKI AR-10TF accelerometers, Yokogawa WE7000 measurements station and laptop computer, and TOKYO SOKUSHIN SPC-51 micro tremor device. As the source of vibration, a 1 tf sandbag was dropped from a height of 1 m above the ground (Figure 7.20(a)). As the bag was torn when it hit the ground, it was decided to use

218  Vibrations and vibration measurement

Figure 7.20 Views of vibration sources

the backhoe bucket as the vibration source (Figure 7.20(b)). Although it is difficult to adjust the vibration level as desired, it proved to be quite useful as the vibration source. Figure 7.21 shows the layout of accelerometer sensors A1-A5, EX/W1-EX/W5 and SPC51. The main purposes of vibrations were to measure the wave velocity of the ground (Figure 7.21) and the transmission and attenuation of the acoustic emission (AE) signals with distance (Figure 7.22). in particular, the attenuation of AE signals between Sensor AE-1 (top), Sensor AE-3 (above the tomb) and Sensor AE4 (outer sidewall of the tomb) was of great importance. The results indicated that the amplitude of the vibrations was drastically reduced as a function of distance as shown in Figure 7.23. Figure 7.24 shows the attenuation of AE count numbers at AE-1, AE-3 and AE4. The result indicated that AE signals were also drastically reduced with distance. 7.3.2.7 Effect of fault in vibration propagation at Demirbilek open-pit lignite mine It is well-known that blasting operations cause ground vibrations. The characteristics of the vibrations induced by blasting depend on the blasting geometry, the blasting material and blasting time pattern (Aydan et al., 2002, 2014). The results of measurements to be reported in this article were obtained at the Demirbilek open-pit mine in Western Turkey with an emphasis on the effect of fault on the vibration characteristics induced by blasting by using the same blasting geometry, blasting material and blasting time pattern except the location of the blasting hole with respect to the fault strike. Figure 7.25 shows the layout of the blasting experiment. The fault dips to SE with an inclination of 70 degrees, and the north side of the blasting area is very close to the crest of the open-pit mine. Ten blastings were performed.

Figure 7.21 Layout of sensors for measuring vibrations

Figure 7.22 Acceleration records induced by the backhoe bucket striking the ground

Figure 7.23 Attenuation of maximum acceleration with distance

Figure 7.24 Attenuation of AE counts

Figure 7.25 Layout of instrumentation for acceleration measurement during blasting at Demirbilek

Vibrations and vibration measurement  221

The blasting operations numbered from DB22 to DB26 involved the blasting on the hanging wall with increasing charge weight while the blasting operations numbered from DB27 to DB31 involved blasting on the footwall of the fault. Figures 7.26 and 7.27 show the acceleration responses at the observation points set on the footwall and hanging wall of the fault for blasting operations numbered DB26 and DB30. The DB26 and DB30 blasting operations were done on the hanging wall side and footwall side, respectively. The maximum ground acceleration at the footwall side was always less than that on the hanging wall side, even though the blasting was on the footwall side. This may be an indication of the effect of the fault and the free surface on the resulting wave propagations and their characteristics. Vibrations induced by three different phenomena, namely the fracturing of rocks, faulting and blasting near a fault have some similar characteristics. The maximum accelerations are always higher on the moving side or hanging wall side of the fault. The experimental observations are consistent with maximum ground acceleration measurements on the hangingwall and footwall side of the faults during earthquakes.

Figure 7.26 Comparison of induced acceleration responses of accelerometers numbered Acc-1 and Acc-2 for direction x for blasting operation numbered DB26

Figure 7.27 Comparison of induced acceleration responses of accelerometers numbered Acc-1 and Acc-2 for direction x for blasting operation numbered DB30

222  Vibrations and vibration measurement

7.3.2.8  Ground vibration due to lightning Vibrations may be caused by lightning in the air as well as in the ground. An interesting record was taken at the Nakagusuku Castle monitoring site at 18:41 on 15 August 2015. Figure 7.28 shows the accelerations records induced by the lightning. The maximum ground acceleration was 346 gals.

7.4 Vibrations by shock waves for nondestructive testing of rock bolts and rock anchors Vibrations by shock waves for nondestructive testing of rock bolts and rock anchors are briefly presented here. For details, readers are referred to Aydan (2018). Several bar-type tendons (1200 mm long, 36 mm in diameter) and cable-type tendons (1300 mm long and 6 wires with a diameter of 6 mm) with/without artificial corrosion under bonded and unbonded conditions were prepared (Figure 7.29 and Figure 7.30), and the responses of the tendons

Figure 7.28 Vibrations induced by lightning at Nakagusuku Castle monitoring site

Figure 7.29 Samples with bar-type and cable-type tendons embedded in rock and a typical ­experimental setup

Vibrations and vibration measurement  223

under single impact waves induced by a impact hammer or special Schmidt hammer–like device (ponchi) were measured. Three different sensors are used, and two of them had a center hole for inducing impact waves on tendons (Figure 7.31). The waves can be recorded as displacement, velocity or acceleration, and the device for monitoring and recording consists of an amplifier and a small handheld-type computer (Figure 7.32). The measurement

Figure 7.30 Bar-type and cable-type tendons with/without artificial corrosion used in experiments

Figure 7.31 Views of sensors

224  Vibrations and vibration measurement

can be done through a single person, and the Fourier spectra of recorded data can be stored in the small handheld computer and visualized in the site of measurement. In the following, only the results obtained are given without any reference to the sensor or hammer unless it is mentioned. Figure 7.33 shows the wave responses of a 1200 mm steel bar induced by an impact hammer and its numerical simulation using the numerical method described previously. The wave velocity inferred from acceleration records directly was about 5520 m s−1. The Fourier spectra of wave records induced by the impact hammer and a special Schmidt hammer–like device (ponchi) are shown in Figure 7.34. The frequency content interval is about 2300 Hz and the inferred velocity of the steel bar was 5520 m s−1. These results are consistent with each other. Figures 7.35 and 7.36 show the effect of a coupler on the steel bar with a length of 1000 mm and the acceleration response of a single 1000 mm bar also shown in the figures. As noted from the figures, two reflections occur in the 2 m long coupled bar. The main reflection is due to from the other end of the coupled bar and secondary reflection is due to the coupler.

Figure 7.32 Views of PC-pocket-type sampling and recording device

Figure 7.33 Impact hammer–induced wave response and its numerical simulation for a 1200 mm long steel bar

Figure 7.34 Normalized Fourier spectra of recorded acceleration records induced by impact hammer and a special Schmidt hammer–like device

Figure 7.35 Impact hammer–induced wave responses of single and two 1 m long bars connected to each other with a coupler

Figure 7.36 Normalized Fourier spectra of recorded acceleration records of single and two 1 m long bars connected to each other with a coupler

226  Vibrations and vibration measurement

Figure 7.37 shows the Fourier spectra of 1200 mm bar with and without an artificially induced area reduction to simulate corrosion. As expected from the numerical analysis highfrequency content would be generated by the partially reflected waves from the artificial corrosion zone. This feature is clearly observed in the computed Fourier spectra. Figure 7.38 shows the effect of bonding of the bar to the surrounding rock. Although the frequency of the bonded bar is slightly smaller than that of the unbonded rock anchor, the Fourier spectra for the first mode is quite close to each other. Nevertheless, the frequency content starts to change after the second or higher modes.

Figure 7.37 Normalized Fourier spectra of recorded acceleration records for bars with or without corrosion

Figure 7.38 Normalized Fourier spectra of recorded acceleration records of bonded and unbonded bars

Vibrations and vibration measurement  227

References Aydan, Ö. (2003) Actual observations and numerical simulations of surface fault ruptures and their effects engineering structures. The Eight U.S.-Japan Workshop on Earthquake Resistant Design of Lifeline Facilities and Countermeasures against Liquefaction. Technical Report, MCEER-03–0003. pp. 227–237. Aydan, Ö. (2018) Rock Reinforcement and Rock Support. CRC Press, Taylor and Francis Group, 486p, London, ISRM Book Series, No. 6. Aydan, Ö., Bilgin, H.A. & Aldas, U. G. (2002) The dynamic response of structures induced by blasting. International Workshop on Wave Propagation, Moving Load and Vibration Reduction. Okayama, Baklema, 3–10. Aydan, Ö., Daido, M., Tano, H., Tokashiki, N. & Ohkubo, K. (2005a). A real-time multi-parameter monitoring system for assessing the stability of tunnels during excavation. ITA Conference, Istanbul. pp. 1253–1259. Aydan, Ö. & Geniş, M. (2004) Surrounding rock properties and openings stability of rock tomb of Amenhotep III (Egypt). ISRM Regional Rock Mechanics Symposium, Sivas. pp. 191–202. Aydan, Ö., Geniş, M. & Bilgin, H.A. (2014) The effect of blasting on the stability of benches and their responses at Demirbilek open-pit mine. Environmental Geotechnics. ICE, 1(4), 240–248. Aydan, Ö., Sakamoto, A., Yamada, N., Sugiura, K. & Kawamoto, T. (2005b) The characteristics of soft rocks and their effects on the long term stability of abandoned room and pillar lignite mines. Post Mining 2005, Nancy. Aydan, Ö., Tano, H., Geniş, M., Sakamoto, I. & Hamada, M. (2008) Environmental and rock mechanics investigations for the restoration of the tomb of Amenophis III. Japan-Egypt Joint Symposium New Horizons in Geotechnical and Geoenvironmental Engineering, Tanta, Egypt. pp. 151–7.2. Aydan, Ö., Tano, H., Ideura, H., Asano, A., Takaoka, H., Soya, M. & Imazu, M. (2016). Monitoring of the dynamic response of the surrounding rock mass at the excavation face of Tarutoge Tunnel, Japan. EUROCK2016, Ürgüp, 1261–1266. Aydan, Ö., Tano, H., Imazu, M., Ideura, H. & Soya, M. (2007) The dynamic response of the Taru-Toge tunnel during blasting. ITA WTC 2007: Congress and 42st General Assembly, San Francisco, USA. Aydan, Ö., Ulusay, R., Hasgür, Z. & Hamada, M. (1999) The behavior of structures built on active fault zones in view of actual examples from the 1999 Kocaeli and Chi-Chi earthquakes. ITU International Conference on Kocaeli Earthquake, Istanbul. pp. 131–142. Hamada, M., Aydan, Ö. & Tano, H. (2004) Rock mechanical investigation: Environmental and rock mechanical investigations for the conservation project in the royal tomb of amenophis III. Conservation of the Wall Paintings in the Royal Tomb of Amenophis III. First and Second Phases Report, UNESCO and Institute of Egyptology, Waseda University. pp. 83–138. Haruyama H and Narita, A. (2008), On the characteristics of the modified open TBM used in the excavation of the Kuriko evacuation tunnel, Tohoku Technological Research Meeting, Sendai, JSCE, VI-5. Inoue, H. & Hokama, K. (2019) Assessment of rock mass conditions of Ryukyu Limestone formation for a rock-cut in Urasoe Fault Zone (Okinawa) by elastic wave velocity tomography. Proceedings of 2019 Rock Dynamics Summit in Okinawa. pp. 744–749. Minei, H., Nagado, Y., Ooshiro, Y., Aydan, Ö., Tokashiki, N. & Geniş, M. (2019) An integrated study on the large-scale arch structure for protection of karstic caves at New Ishigaki Airport. Proceedings of 2019 Rock Dynamics Summit in Okinawa. pp. 390–395. Tomori, T., Yogi, K., Aydan, Ö. & Tokashiki, N. (2019) An integrated study on the response of unsupported underground cavity to the nearby construction of piles of Gushikawa By-Pass Bridge. Proceedings of 2019 Rock Dynamics Summit in Okinawa. pp. 408–413.

Chapter 8

Degradation of rocks and its effect on rock structures

It is well-known that rocks surrounding rock engineering structures or constituting rock mass are prone to degradation when they are subjected atmospheric conditions and/or gas/fluids percolating through rocks and rock discontinuities. Degradation results from the alteration of minerals, the weakening of particle bonds and/or solution of particles. Figure 8.1 shows several examples of degradation situations of rocks. Properties of intact rock can be drastically changed due to degradation processes. In this chapter, major fundamental processes are explained, and the effects of degradation processes on the properties of rocks are described.

Figure 8.1 Examples of degradation situations of some rocks

230  Degradation of rocks Table 8.1  Degradation of some common minerals Original Minerals

Weathering Process

Resulting Minerals

Minerals containing Fe, Mg: olivine, pyroxene, amphibole Feldspars Biotite (micas) Calcite, gypsum

H 2CO3 (= H 2O + CO2 ) alteration, oxidation H 2CO3 alteration Hydrolysis Solution by water

Clay minerals, Fe-oxides Clay minerals Clay minerals None

8.1 Degradation of major common rock-forming minerals by chemical processes Table 8.1 summarizes some common resulting minerals due to degradation (weathering, oxidation, alteration processes). This subsection presents some of these processes. Feldspars are one of the common rock-forming minerals. Oxidation is the ionic reaction with some components in the original mineral. For example, the oxidation of iron (Fe) results in the hematite mineral: 4 Fe + 3 O2--> 2 Fe2 O3 (hematite) Hydrolysis is a chemical reaction of minerals with water containing dissolved CO2. For example, orthoclase transforms into kaolin as a result of the chemical reaction given by: 2 K Al Si3O6 + 6 H2O + CO2 ---> Al2 Si2 O5 (OH) 4 + 4 H2SiO4 + K2CO3 (orthoclase) (kaolinite) The resulting minerals from the chemical processes are generally clay minerals. It is generally known that dark color minerals are much more vulnerable to degradation compared with whitish or glassy color minerals. While plagioclase is more resistant to weathering, orthoclase is much more vulnerable to weathering when they are exposed to water or acid environments. It is well-known that muscovite is resistant to weathering. However, it may be easily disintegrated due to their platy structure in a freezing–thawing environment. On the other hand, biotite is quite vulnerable to weathering particularly in tropical regions. Calcite and evaporates can be easily solved in water containing dissolved CO2. This process is expressed in the following form: H2O + CO2 + CaCO3--> Ca++ + 2 HCO3

8.2  Degradation by physical/mechanical processes 8.2.1  Freezing–thawing process and its effects on rocks Freezing–thawing cycles are an important process in the degradation of rocks with time. This process repeats itself endlessly if atmospheric air temperature drops below zero during the cold seasons. There is a great concern in many countries to assess the behavior of

Degradation of rocks  231

Figure 8.2 Examples of freezing–thawing–induced failures in nature

engineering structures subjected to freezing–thawing cycles as this process may result in failures and fatalities. The collapse of tuffaceous rock mass above the portal of the Toyohama Tunnel in Hokkaido island of Japan killed more than 10 people traveling through the tunnel at the time of the collapse (Figure 8.2(a), Ishikawa and Fujii, 1997). Particularly rocks whose mechanical properties are influenced by water content are quite vulnerable to degradation and disintegration when they are subjected to freezing and thawing cycles. The Alps are the highest mountains in Europe, and their peaks are more than 4000 m high. The temperatures are generally below 0 degree when the altitude is greater than 2500 m. Temperature measurement in rock 4 m deep from the surface at the north face of the Sphinx Observatory of the Jungfraujoch was recorded to be fluctuating between +0.8oC and −9oC. Although the rock temperature would depend upon the depth from the surface, the observation data implies that the fluctuation of rock temperature may range between 8 and 10 oC. The measured air temperature difference descending from Gornergrat (3130 m) to Zermatt (1620 m) was about 15oC, while the elevation difference between two stations is about 1510 m. The limit altitude of trees in the vicinity of the Jungfraujoch and Gornergrat observatories is about 2500 m. The degradation of rocks above the tree limit altitude was quite severe, and the surface of mountains were covered with debris of disintegrated rock blocks as seen in Figure 8.3. The disintegration of rocks due to freezing and thawing process was particularly severe in schistose rocks, particularly green schist and micaschist as seen in Figure 8.3(c). One of the interesting observations was done near the Zermatt station of Zermatt-Rothorn Cable Car Route. Rock bolts were used to support the rock blocks in the vicinity of this station. The freezing and thawing process caused the disintegration of rock, which had fallen off. Nevertheless, the pieces of rocks underneath the rock bolt plate were still in place as seen in Figure 8.4. The rock was green-schist. Many antique and modern underground and semiunderground openings are excavated in tuffs in the Cappadocia region. The erosion and subsequent partial or total collapse of these structures take place from time to time, and the preservation of these antique underground or semiunderground remains as the assets of past civilizations is of paramount importance. The failures of these structures generally occur during thawing or after heavy rains as a result of the degradation due to freezing–thawing and decrease of strength properties of tuffs with saturation (Aydan and Ulusay, 2003, 2013).

Figure 8.3 Effect of freezing−thawing in the vicinity of Gornergrat Observatory

Figure 8.4 Disintegration of rocks in the vicinity of rock bolted rock surface in Zermatt

Degradation of rocks  233

(a)  Freezing–thawing experiments The general procedure used for freezing–thawing experiments involves freezing rocks under saturated conditions at −20–30 oC and thawing in an oven under the temperature of +20–100 oC. However, these conditions do not exist in nature, and the degradation of material properties such as uniaxial compressive strength drastically decreases. Therefore, the validity and applicability of current freezing–thawing procedures used on rocks are quite questionable. Furthermore, the fully saturated condition may not always be observed in nature. Rocks are tuff obtained from Oya in the Tochigi Prefecture and Asuwayama in the Fukui Prefecture of Japan and Derinkuyu and Zelve in the Cappadocia Region of Turkey. Table 8.2 gives physico-mechanical properties of tuffs used in the experiments. Two sets of samples were prepared. One set of samples were directly subjected to temperature cycles under dry conditions, while the samples of the other set were kept immersed in water in their containers, as seen in Figure 8.3(b). Samples were taken out from the climatic chamber at certain cycles to measure their physico-mechanical properties using nondestructive testing procedures such as P-wave velocity measurement, a needle-penetration device and weight change. The strain–stress responses of samples within the elastic range at the selected cycles were also measured, and their deformation modulus were determined. An environmental chamber LHU-113, whose temperature can be varied between – 20 oC and +85 oC produced by ESPEC, was used (Figure 8.5). The temperature cycles can be Table 8.2  Geomechanical properties of tuffs (dry) Property

Derinkuyu

Zelve

Oya

Asuwayama

Unit weight (kN m−3) UCS (MPa) Tensile strength (MPa) Elastic modulus (GPa) Poisson ratio Elastic wave velocity (km s−1)

13.7–15.9 7.9–10.3 0.5–0.9 1.5–3.6 0.2–0.26 1.5–2.2

10.9–15.1 0.8–3.6 0.26–0.46 0.5–2.2 0.27–0.31 1.1–1.8

13.5–15.7 8.7–24.8 0.8–1.5 0.9–3.0 0.25–0.3 1.6–1.8

20.1–22.0 28.6–37.7 2.1–3.8 3.2–4.5 0.15–0.21 2.8–3.2

Figure 8.5 Views of the environmental chamber and samples

234  Degradation of rocks

Figure 8.6 Monitored temperature and humidity in the environmental chamber

programmed as desired. As yearly temperature variations were generally between –10 oC and 30 oC in the areas of rock sampling, the temperature of the chamber was varied between –10 oC and 30 oC, and the total duration of each cycle was 3 h. The temperature and humidity of the chamber were monitored continuously as shown in Figure 8.6. Temperature measurements in Derinkuyu and Zelve (Aydan et al., 2014; Ulusay et al., 2013) indicated that temperature may decrease below the freezing point during the period of time starting from November to March, as seen in Figure 8.7. The area receives rainfall and snow during this period also. The effect of cyclic the freezing–thawing process on tuff samples of Derinkuyu and Zelve has been recently investigated by the authors. During these tests, samples were subjected to a temperature cycle ranging between –10oC and 20oC. Some of the samples were subjected to this temperature cycle under dry conditions, while the rest were subjected under saturated conditions. The experiments clearly indicated that the effect of the temperature cycle has no big influence on the properties of Cappadocia tuffs while they may destroy the bonding structure of rocks under saturated conditions. The effect of freezing–thawing on tuffs with larger water content absorption capacity is much higher than that on tuffs with less water absorption capacity. Figure 8.8(a) shows the views of dry and saturated tuff samples of Zelve subjected to the same temperature and humidity environment. While the saturated sample was heavily damaged by the freezing–thawing process, there was no visible damage to the dry sample. The same type of experiments carried out at the University of the Ryukyus in 2015 yielded very similar results for very soft rocks and mortar samples (Figure 8.8(b)). (b)  Effects of freezing–thawing on physico-mechanical properties A series of experiments were carried out on samples of tuffs of Zelve and Derinkuyu subjected to freezing–thawing cycles under dry and saturated conditions with high temperature variations, and changes in weight, elastic wave velocity, NPI and UCS were determined.

Degradation of rocks  235

Figure 8.7 Variations of temperature and rainfall in Derinkuyu and Zelve

Figure 8.8 Views of dry and saturated samples subjected to the same temperature environment

Figures 8.9 and 8.10 show the change in weight of dry and saturated samples, and stress– strain relationship in dry samples, respectively. Change in weight of the saturated samples indicated that occurrence of spalling is directly related with freezing–thawing cycles in nature. Except for mechanical effects resulting in volumetric change during the freezing– thawing test, samples in the dry state were not affected by freezing–thawing cycles.

236  Degradation of rocks

Figure 8.9 Effect of freezing–thawing cycle on (a) weight and (b) stress–strain relation of Zelve tuff

Figure 8.10 Effect of freezing–thawing cycles on uniaxial stress–strain responses and elastic wave velocity of Derinkuyu tuff

Figure 8.11 Variation of NPI with the cycle number of thawing–freezing for Oya tuff

The decrease in UCS of the Zelve tuff is about 40% at the end of the 40th cycle of the freezing–thawing tests. Except for mechanical effect resulting in volumetric change during the freezing–thawing test, samples in the dry state were not affected. The value of the needle penetration index (NPI) is expected to decrease as the cycle number of thawing–freezing increases. Figure 8.11 shows the variation of NPI with the cycle

Degradation of rocks  237

number of thawing–freezing for Oya tuff. As noted from the figure, the value of NPI drastically decreases with the increasing cycle number of thawing–freezing for saturated samples of Oya tuff. 8.2.3  Cyclic saturation and drying (slaking) (a)  Cyclic saturation and drying process on soft rocks It is well-known that some soft rocks may absorb or desorb water, resulting in volumetric changes, which may lead to their disintegration. An experimental procedure is described to measure the water content migration characteristics and associated volumetric variations. An experimental device illustrated in Figure 8.12 was developed (Aydan et al., 2006). The experimental setup consists of an automatic scale, an electric current inductor, electrodes, isolators, laser displacement transducer, voltmeter, rock sample and laptop computers to monitor and to store the measurement parameters and temperature-humidity unit consisting of sensors and logger. Rock samples collected from the Tono Underground Mine in Central Japan were first fully soaked with water for a certain period of time. Then they are put on the automatic scale and dried. During the drying process, the weight, height and voltage changes of the sample were continuously measured. The temperature and humidity changes of the drying environment were also continuously monitored. Figure 8.13(a) shows temperature, humidity, shrinkage strain, weight change and electrical resistivity variations on both fine-grain and coarse-grain sandstone samples under laboratory conditions. While the weight change (water content) of coarse-grain sandstone was slightly large than that of fine-grain sandstone, there was a remarkable difference between the shrinkage strains of samples. The shrinkage strain of fine-grain sandstone was more than twice that of coarse-grain sandstone. The electrical resistivity of samples increases as the samples lose their water content. The relation between water content and electrical

Figure 8.12 Illustration of experimental setup

238  Degradation of rocks

Figure 8.13 Multiparameter responses on water content migration tests

resistivity is studied and presented elsewhere (Kano et al., 2004). It is considered that if the electrical resistivity of surrounding rock could be measured continuously in-situ, it may be quite useful for evaluating the water content variations and associated volumetric variations. The same experimental setup and the same samples were used in the NATM adit in order to observe the same multiparameter responses under in-situ conditions. The measured responses are shown in Figure 8.13(b).

Degradation of rocks  239

Figure 8.14 Views of samples after the completion of the experiments

Figure 8.15 Multiparameter responses of a sandstone sample from Mitake

The temperature and humidity in the adit were almost constant. Even though the weight change (water content) of coarse-grain sandstone was twice that of fine-grain sandstone, the shrinkage strain of fine-grain sandstone sample, was almost 4 times that of the coarse-grain sandstone sample under in-situ conditions. These experimental results clearly indicated that fine-grain sandstone was prone to high volumetric strains in relation to water content variations. Tangential and axial cracks appeared on the outer surface of fine-grain sandstone samples while such cracks were not observed on coarse-grain sandstone samples (Figure 8.14). This experimental observation could be directly associated with the magnitude of shrinkage strains. Because of some technical problems, acoustic emission responses could not be measured. However, similar experiments were carried out on a sandstone sample from Mitake, which is near the Tono mine and belongs to the same geologic formation. The results are shown in Figure 8.15. As noted from the figure, there is a distinct acoustic emission response when the sample loses its water content. The largest acoustic emission activity corresponds to the

240  Degradation of rocks

largest shrinkage strain and water content variation rate. These acoustic emission activities are generally associated with cracking in samples. Therefore, the acoustic emissions in actual boreholes could not be directly related to cracking in the surrounding rock. (b)  Effect of cyclic saturation and drying on mechanical properties of rocks The cyclic saturation and drying tests were carried out in accordance with the procedures given by International Society for Rock Mechanics and Rock Engineering (ISRM (2007)) and American Society for Testing Methods (ASTM (2000)). The weight loss and uniaxial compressive strength (UCS) of the tuff samples from Avanos were determined at different cycles. Due to the fact that the samples subjected to durability tests would be broken during uniaxial compression tests at certain cycles, various specimens prepared from the blocks were used in these tests. The UCS was determined on dry samples following 2, 4, 9, 11 and 14 test cycles by applying this method. As seen from Figure 8.16, after wetting–drying tests, some change in weight is observed, and the weight loss at the end of 14 test cycles is about 4.5% (Fig. 8.16). Figure 8.16 suggests that the reduction in UCS between the 2nd and 14th cycles is about 1.05 MPa. This change in UCS means the total loss in strength is about 40%. In freezing–thawing tests, the specimens were subjected to a total of eight test cycles. The weight loss at the end of eight test cycles is about 6% (Fig. 8.16). Figure 8.16 indicates that the reduction in UCS between the 1st and 11th cycles is about 1.05 MPa. This change in UCS means the loss in strength is about 40%. Comparison of the wetting–drying and freezing–thawing test results indicates that freezing–thawing cycles are more effective on decrease in weight loss and UCS of the tuff studied. Based on the 4-cycle slake durability tests, Id values from the 1st to 4th cycles are between 81.5–98%, 69.3–94%, 56.7–88.3% and 40.7–77.1%, respectively. Particularly after the 2nd cycle, a considerable disintegration of the samples was observed. Based on the wetting–drying tests, the UCS of the Kavak tuff decreased by 4% and 22% at the end of 22nd and 40th cycles, respectively. The same tests on the Zelve tuff suggest that a 36.3% decrease in the UCS occurs at the end of 5th cycle.

Figure 8.16 Variation of loss of weight and UCS with number of cycles

Degradation of rocks  241

Figure 8.17 Variation of normalized elastic modulus and uniaxial compressive strength and needle penetration index (NPI) in relation to saturation

The mechanical properties of soft rocks prone to water absorption and desorption may depend upon the degree of saturation (Aydan and Ulusay, 2003, 2013). These property changes may also greatly affect their responses of rock engineering structures. Figure 8.17 shows an example of changes of the mechanical and index properties of Oya tuff.

8.3  Hydrothermal alteration The alteration process is due to percolating hydrothermal fluids in rock mass, and it may act on rock mass in a positive or negative way. The positive action of the alteration may heal existing rock discontinuities by rewelding through the deposition of ferro-oxides, calcite or siliceous filling material. On the other hand, the negative action of the alteration would cause the weakening of the bonding of particles of rocks and producing clayey materials similar to the chemical degradation. As a result of alteration, rocks may be transformed into kaolin or chlorite minerals. As the intact rock is one of the most important parameters influencing the mechanical response of rock masses, the negative action of hydrothermal alteration may account for the degradation of intact rock. While the weathering of rocks is mostly observed near ground surface up to a depth of 40 m and their effects disappear with depth, alteration may be observed at greater depths. Figure 8.18 shows different stages of alteration of rhyolite of Okumino.

8.4 Degradation due to surface or underground water flow Surface and underground water flow over the surface of rock surfaces or through rock mass may also cause the degradation of rock mass. Figure 8.19 shows some examples of degradation due to surface or underground water flow in the Zelve valley of the Cappadocia region in Turkey and Ishigaki island. It is observed that heavy rains and rapid surface water flow

Figure 8.18 Rhyolite of Okumino subjected to hydrothermal alteration at different stages of degradation: (1) fresh, (2) stained, (3) slight alteration, (4) moderate alteration, (5) heavy alteration, (6) decomposed

Figure 8.19 Some examples of degradation due to surface or underground water flow in Zelve valley of Cappadocia region in Turkey and Ishigaki island

Degradation of rocks  243

induces the degradation of rock mass due to both the saturation of rocks and surface erosion and the washing out of filling material of rock discontinuities in soft rocks such as tuffs. The surface flow or percolation of rock mass in limestone and evaporates may dissolve rock, resulting in the widening of pores and fractures, which may also lead to large-scale cavities.

8.5 Biodegradation As previously described, the findings have demonstrated that the degradation is ultimately formed by a combination of physical and chemical processes, and the climate or lithology are among the controlling factors affecting the rate of rock weathering. There are some new considerations for the degradation of rocks in geoscience and geoengineering (Ehrlich and Newman, 2009). This degradation is assumed to be related to bacterial activities, which may be another very important cause of degradation of rocks. In this subsection, some biodegradation studies undertaken on tuff samples in Cappadocia Region are described. (a)  Sampling locations Four tuff samples (size: approximately 1000 cm3) from the 5th floor of Derinkuyu Underground City, Zelve semiunderground (cliff) settlement and Uçhisar Fairy Chimney settlement were collected (Figure 8.20). The Derinkuyu site has been monitored by Aydan and his group (Aydan et al., 1999a, 1999b, 2007a, 2007b; Aydan and Ulusay, 2003, 2013) since 1996. Multiparameter monitoring program has been implemented in this site, and the sample collected is the underground 5th floor. Lighting at this site is due to only electricity. However, the room is about 6 m from the main ventilation shaft.

Figure 8.20 (a) Map of and (b) views of sampling locations

244  Degradation of rocks

The sample collected in the Zelve site is at the toe of semi-underground settlement where the toe erosion is severe. Multiparameter monitoring has been implemented in this area since 2004. The third sample is collected from Uçhisar where huge fairy chimneys are found. The sample was obtained from the toe of a fairy chimney suffering the heavy spalling problem. (b)  Microscopic investigations The bacteria in rocks, so-called endolithic bacteria, actively bore into the host rock by solubilizing cementing mineral grains in their attempt to gain access to nutrients and energy (Konhauser, 2007). To date, their contributions in various rock types, such as limestone, dolomite, sandstone, granite, basalt, tuff and the like, have been well documented (e.g. Büdel et al., 2004; Hoppert et al., 2004; Hall et al., 2008). In the investigation, four samples as shown in Figure 8.21 were selected, and microscopic conditions were observed by a biological microscope (Matsubara and Aydan, 2016). In the study, 1 g of rock sample was mixed with 9 g of sterile distilled water, and little mixed water samples were observed. Figure 8.22 shows the microscopic images in the rock samples. As shown in the figure, numerous microorganisms were recognized in all rock samples. Qualitatively, it is the sample from the Derinkuyu-5F that has the largest number of microorganisms in comparison to other samples. On the other hand, the number of microorganisms in the sample from the Zelve is the smallest. Although the number of microorganisms may depend on the time and period of sampling, it is understood that microorganisms would be more active at the Derinkuyu-5F. This

Figure 8.21 Views of samples obtained from Zelve valley, Fairy Chimney at Uçhisar and the fifth floor of Derinkuyu Underground City

Degradation of rocks  245

Figure 8.22 Microscopic conditions of inner rock samples (The white arrows indicate microorganisms; scale bar: 100 = 1 mm).

difference may depend on environmental conditions. It is noted that (1) the outcrops were dried almost constantly with sunlight in summer but were wet almost constantly in winter at the Zelve site; (2) the Uçhisar site is in the shade, and the outcrops were wet almost constantly; (3) the 5th floor of Derinkuyu site is always dark, and constant temperature and high humidity are continuous. Interestingly, this site is always dark, so that the chances that they

246  Degradation of rocks

are photosynthetic bacteria such as cyanobacteria are virtually nil. Therefore, they would be chemolithoautotroph or chemoheterotroph. Although the identification of the bacteria and other detailed analyses are an issue to be addressed in the future, endolithic bacteria may possibly induce the rock weathering in Cappadocia Region.

8.6  Degradation rate measurements The degradation of rock due to different causes, such as the cyclic drying–wetting, freezing– thawing, winds and rainfall, surface or underground water flow, results in erosion, slabbing, spalling and raveling (Figure 8.23). Due to these effects, tuffs, particularly those close to ground surface at the toe regions of semiunderground rock structures at valley bottoms, are prone to degradation. Furthermore, limestone and evaporates are dissolved by sea waves, winds, river flow or percolating rainwater. There are some studies on the degradation rate of soft rocks such as tuffs, mudstone, lignite (e.g. Aydan et al., 2007a, 2007b, 2008a, 2008b; Aydan and Ulusay, 2016). The studies undertaken on tuffs of the Cappadocia region by Aydan et al. (2007a, 2007b, 2008a, 2008b), Aydan and Ulusay (2013 and Kasmer et al. (2008) are described here as the preservation of antique underground cities and semiunderground settlements, as well as fairy chimneys, very much depends upon the erosion or degradation rate of tuffs. The annual erosion rate has been either computed from existing structures or measured in-situ (Figure 8.24). There are many historical structures with well-known dates of construction. If the dates of structures are known, it is possible to evaluate the average rate of degradation. The technique proposed by Aydan et al. (2007, 2008a, 2008b) adopted herein and more data are

Figure 8.23 Views of spalling and slabbing problems

Degradation of rocks  247

Figure 8.24 Some actual examples for computation of degradation rates

presented. Daily temperature changes in the Cappadocia region are very high, and freezing generally occurs between November and March. The amount of erosion was also studied by Kasmer et al. (2008) in the 1st valley in ZOAM at two selected locations between 2006 and 2011 using nails inserted in rock mass in different locations. Location 1 was the toe of a steep natural valley slope, while Location 2 was a huge tuff block at the bottom of the valley using stainless nails and a sensitive device. The measurements indicated that there is an increasing initial trend in the amount of erosion at both locations particularly in the spring season. Particularly in the winter season, the toe of the slope becomes wet due to capillarity and the stream flowing in the valley. In the spring season, the rock tends to dry. As a result of these repeated drying–wetting cycles, the amount of erosion showed an increase when compared to that in the hot summer season. The amount of erosion throughout the five-year period was between 1.07 and 6.21 mm and by assuming a homogeneous erosion, the annual average erosion is determined between 0.21 and 1.24 mm. By considering that the valley has been used for settlement for more than 1000 years and this monitoring covers a five-year period, the amount of erosion due to natural factors seems to be important. Table 8.3 gives estimated degradation rates for several locations. The erosion rate in Cappadocia generally ranges between 0.04 and 0.5 mm/yr (Aydan et al., 2007a, 2007b, 2008a, 2008b; Aydan and Ulusay, 2013). However, Kasmer et al. (2008) reported that the annual average erosion could be up to 1.6 mm in the Zelve valley as given in Table 8.3. As noted from the table, the erosion or degradation rate of tuffs with larger amounts of clay minerals is

248  Degradation of rocks Table 8.3  Degradation rate Location

Depth (mm)

Time (years)

Rate (mm a−1)

Ürgüp Avanos Derinkuyu Ihlara Açıksaray Pancarlık Göreme Özkonak Selime Zelve Ürgüp Roadcut Ürgüp-Avanos Roadcut

600–1200 280–750 60–135 300–750 150–600 150–300 75–225 300–450 150–300 150–2400 20–50 8–15

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 50 30

0.24–0.48 0.19–0.5 0.04–0.09 0.2–0.5 0.1–0.4 0.1–0.2 0.05–0.15 0.2–0.3 0.1–0.2 0.1–1.6 0.4–1.0 0.3–0.5

Figure 8.25 Views of differential erosion and weathering

much higher than that of tuffs with less clay mineral content. The degradation is particularly high at the toe of the cliffs below the semiunderground openings, roads and valleys. The resistance of the tuff layers in association with different eruption episodes of volcanoes of the region differs, and differential weathering and erosion take place. Particularly, the erosion rates are higher at the toe of rock layers, which results in overhanging configurations (Figure 8.25).

Degradation of rocks  249

8.7 Needle penetration tests for measuring degradation degree Aydan et al. (2014) proposed a method to evaluate the degree of degradation (weathering) of soft rocks utilizing the needle penetration index (NPI). Aydan et al. (2014) defined a parameter (W) for quantitative evaluation of the degradation state by the following equation:   NPI w   1−   NPI f   (8.1a) W = 100β    NPI  NPI  w w   β 1− +    NPI f  NPI f  or NPI w W (8.1b) = 1− NPI f W + β (100 −W ) where NPIW and NPIf are values of NPI for the completely decomposed and fresh conditions, respectively. The degradation state parameter (W) has a value of 0 for the fresh state and 100 for the completely decomposed state. Coefficient β in Equation (8.1) ranges between 0.2 and 0.6. A series of needle penetration index experiments were carried out on the samples shown in Figure 8.26; the results are given in Table 8.4. Aydan et al. (2006, 2008a, 2008b, 2014) also reported NPI values for fresh tuff samples at the respective places. The needle penetration index and visual weathering state results indicate that the weathering state was quite high in the Derinkuyu sample while it was light in the Zelve samples.

Figure 8.26 Relation between NPI and weathering state

250  Degradation of rocks Table 8.4  NPI values and weathering state of tuff samples Location

NPI (fresh)

NPI (degradated)

Degradation State (%)

Derinkuyu Uçhisar Zelve

33–46 14–25 20–33

1.5–5.0 2.0–3.2 7–10

85–95 65–75 30–45

8.8 Utilization of infrared imaging technique for degradation evaluation Compared to the passive-type infrared thermographic imaging technique, active-type infrared thermographic imaging technique should be also useful in evaluating the soundness of various structures for maintenance purposes (Aydan, 2019). In this respect, the utilization of artificial heat sources such as heaters, coolers, sunlight radiation, dynamic excitation, wind may be useful. Figure 8.27a shows the visible and infrared thermographic images of Nakagusuku Castle remain where castle walls were reconstructed utilizing original blocks and newly replaced blocks, which are made of Ryukyu limestone. As the original blocks are partly weathered compared with the newly replaced limestone blocks, their heat absorption characteristics under the sunlight heating are different from each other. This observational fact clearly indicates that the active infrared thermographic imaging technique could be quite useful particularly in the evaluation of the soundness of rocks as well as rock engineering structures in geoengineering for maintenance purposes. An active infrared thermographic imaging attempt was done on weathered sandstone, locally known as Niibi stone. The main purpose was to see if the weathered zone could be distinguished from the nonweathered part. Figure 8.27b shows both visible and infrared thermographic images of the partially weathered Niibi sandstone. Infrared thermographic images correspond to the images just after and 120 s after the application of the heat shock. This experiment clearly indicates that weathered and unweathered parts can be distinguished

Figure 8.27a Visible and infrared thermographic images of a reconstructed wall of Nakagusuku Castle

Degradation of rocks  251

Figure 8.27b Visible and infrared thermographic images of weathered Niibi sandstone

and that the rate of cooling, which could be determined from continuous monitoring, can yield the variation of their thermo-physical characteristics. Therefore, the active infrared thermographic imaging technique may also be another efficient tool in characterizing the weathering of rocks and its thermo-physical state.

8.9 Degradation assessment of rocks by color measurement technique Most degradation assessment procedures in rock mechanics is based on visual inspections and some index tests. When rock degrades, its color undergoes variations. Mostly the surface become whitish, yellowish or reddish due to the alteration of some ferrous minerals contained in the rock matrix. Particularly, the variation of the rock color could be a measure to assess its weathering state. For this purpose, a chroma meter developed by Minolta CR-800 was used. There are various modes of color measurements. Kano et al. (2004) reported that L-a-b mode is the most suitable mode for characterizing the degradation state of rocks. For this purpose, an orthogonal coordinate system is defined. The axis L is a measure of surface darkness. It is value +100 for white color and −100 for black color. The axis a is a measure of variation between red (+100) and green (−100), while the axis b is a measure between yellow (+100) and blue (−100). Figure 8.28 shows two views of weathered soft limestone observed outside the tomb. The weathered soft limestone gives the impression of rock of a thinly layered yellow to brown material. Figure 8.29 shows the variation of parameters of L, a, b for fresh and weathered soft limestone and hard limestone. Although the variations of hard limestone are very small, very distinct variations occur for soft limestone. Figure 8.30 shows the variation of parameters of L, a, b from the entrance to the deepest level (Jee) of the tomb. Although some fluctuations are observed, the state of rock mass throughout the tomb is remains almost the same. In other words, the weathering of rock mass in the tomb is almost negligible for 3400 years compared to that of rock subjected to the harsh weather conditions outside.

Figure 8.28 Two views of weathered soft limestone near the entrance of the tomb

Figure 8.29 Variation of parameters L, a, b for weathered and nonweathered soft and hard limestone

Figure 8.30 Variation of parameters L, a, b of rock surface from the entrance to Jee chamber of the tomb

Degradation of rocks  253

8.10 Effect of degradation process on the stability of rock structures 8.10.1  Antique rock structures (a) Observations BENDING AND TOPPLING OF BLOCKS

This problem particularly occurs at locations where the pillars left in the rock-hewn openings fail and/or when the toe of slopes is eroded by water or wind effect (Figure 8.31). As a result of changes in the stress state due to toe erosion, thermal loading due to freezing−­ thawing and property changes due to cyclic wetting and drying process, spalling and slabbing in cliffs and underground openings may occur. These phenomena result in the formation of slabs with different thicknesses on natural slopes or the walls of underground openings and consequently cause a change in geometry of openings to trigger failures (Figure 8.31)

Figure 8.31 Some partial failures, spalling, slabbing and collapses in Cappadocia region

254  Degradation of rocks

(b)  Analytical solutions The degradation process may be approximated using some functional forms. For the simplicity, we assume that the degradation is a linear function of time and is given in the following form: ΔL= a · t(8.2) where constant a is called the degradation rate. One of the main parameters influencing the long-term stability of rock structures is the creep strength characteristic of rocks. The creep strength (σcr) in terms of short-term strength (σs) can be represented in the following form (Aydan and Nawrocki, 1998). Aw(t) is assumed to be time dependent to account for the effect of erosion.   t  σcr = σ s 1− b ln   (8.3)   τ  where b, t and τ are the empirical constant, time and short-term test duration, respectively. The value of b generally ranges between 0.0186 and 0.0583 for the Cappadocian tuffs (Aydan and Ulusay, 2013; Ito et al., 2008). The tensile or uniaxial compression strength of the rock mass is estimated from the procedure suggested by Aydan et al. (2014). In this procedure, property φm, such as UCS or tensile strength of the rock mass, is obtained from the following formula by using the RMQR value of the respective property of intact rock φi, and the value of β can be taken as 6 on the basis of experimental data from construction sites in Japan (Aydan et al., 2014): ϕm RMQR = α0 − (α0 − α100 ) (8.4) ϕi RMQR + β (100 − RMQR ) STABILITY OF OPENINGS NEXT TO CLIFFS AND FAIRY CHIMNEYS

Based on the observations of the authors at several locations in Cappadocia, most of the instabilities occur as a result of the collapse of toes of openings next to cliffs (Figure 8.31). In such failures, the erosion of the toe by natural agents and decrease in long-term strength of the rock are the main causative factors. In addition, instability problems were also observed in openings in fairy chimneys (Figure 8.31). Herein, the stability of openings next to cliffs and in fairy chimneys through the utilization of creep tests is evaluated by using simplified approaches as described by Aydan et al. (2007). In the model illustrated in Figure 8.32, it is assumed that the pillar or wall at the valleyside carries half of the burden over the opening. Due to its conical shape, a fairy chimney is considered an axially symmetrical rock structure including a circular opening at its center. Although the real stress distributions in these openings are slightly different from those in these methods recommended in this study, it is considered that the approaches used will be helpful in assessing the conditions of the instabilities investigated. The time-dependent safety factor (SF) of the wall next to the cliff and in fairy chimney is written as follows: SF =

σcr (t ) Aw (t ) γH

At

(8.5)

Degradation of rocks  255

Figure 8.32 Simplified mechanical models for opening next to cliff and fairy chimney

where γ, σcr, H, At and Aw(t) are unit weight, creep strength, overburden height, total area supported and wall area. The area ratios for the continuous wall next to the cliff and in the cylindrical fairy chimney shown in Figure 8.32(a) specifically take the following forms, respectively: At w / 2 + to = * (8.6a) Aw (t ) t (t ) At

Aw (t )

=

ro2 (8.6b) r 2 (t ) − ri 2

The erosion rate in Cappadocia generally ranges between 0.04 and 0.5 mm a−1 (Aydan et al., 2007a, 2007b, 2008a, 2008b, 2008c, 2008d; Aydan and Ulusay, 2013). The reduction of supporting area (Aw) due to erosion/degradation may be counted as proposed by Aydan et al. (2006). For example, the wall thickness of the openings next to cliffs and fairy chimneys may be given in the following form: w (t ) = wo − ηt (8.7a) r (t ) = ro − ηt (8.7b) Equation 8.3 can be adopted for the creep strength versus failure time function, together with or without the degradation model to evaluate the safety of openings. Figure 8.33 shows the safety factor of the openings next to cliffs and in fairy chimneys as a function of time and overburden for strength properties of tuffs without degradation. It is clear from Figure 8.33 that an increase in overburden heightens the probability of failure of pillars in openings next to cliffs, and the collapse of fairy chimneys increases. Comparison of Figures 8.33(a) and 8.33(b) also indicates that openings next to cliffs are more likely to fail in the long term as compared with those of fairy chimneys. The observations by the authors in the Zelve, Göreme and Ihlara valleys confirm this result.

256  Degradation of rocks

Figure 8.33 Safety factor variation with depth

Figure 8.34 Effect of erosion rate and decrease in strength on pillar stability for openings next to cliffs and fairy chimneys, respectively

By considering this erosion range, the calculations carried out for an opening with a width of 10 m and an overburden height of 20 m are shown in Figure 8.34(c1). This figure suggests that the stability of openings next to cliffs is affected by erosion. Similarly, an analysis was also carried out for a fairy chimney with a base diameter of 8 m using the same erosion range and the results shown in Figure 8.34(c2) indicate that the collapse potential of fairy chimneys will increase after 1000 years.

Degradation of rocks  257 ROOF FAILURE

The analytical models are based on bending and arching theories developed by the authors to incorporate not only the gravitational load of the roof but also the effects of discontinuities, creep and degradation of rock mass (i.e. Aydan, 1989, 2008; Aydan et al., 2005a, 2005b, 2006, 2007a, 2007b; Aydan and Tokashiki, 2011). Aydan and Tokashiki (2011) recently considered also the effect of load from topsoil as well as the concentrated loads resulting from superstructures or vehicles (e.g. airplanes) in developing an analytical method based on bending theory (Figure 8.35). The limit of roof span (L) normalized by roof rock layer thickness (hr) under its deadweight and topsoil with thickness of (hs) can be obtained in the following form (Aydan and Tokashiki, 2011): σt (t ) L = β  hr (γ r hr (t ) + γ s hs )

(8.8)

where σt (t) is the tensile strength of roof layer. The value of β would take the values of 1/3, 2/3 and 2 for cantilever, simple and built-in beams. The effect of the degradation of the roof layer may be also estimated from the following relation: hr (t ) = hro − ηt (8.9) The arching theory for assessing the stability of roofs of the shallow underground openings is generally based on three-hinged beams. As discussed by Aydan (1989), the rotation of

Figure 8.35 Simplified mechanical models for estimating roof stability

258  Degradation of rocks

rock layer, in which the arch is formed, should be initiated for the arch action to take place. The failure modes of an arch are various, and they are crushing at the crown and/or abutments, vertical or horizontal sliding at abutments and sliding along an existent discontinuity within the arch. The detailed derivations for each failure mode are presented in the doctorate thesis of Aydan (1989) and also in the article by Kawamoto et al. (1991). The classical arching theory assumes that the failure mode of the arch takes place by crushing at the crown or abutments and the final form for the limit of roof span (L), normalized by rock roof layer thickness (hr) under its own weight and topsoil with thickness of (hs), as follows (i.e. Aydan and Tokashiki, 2011): σc (t ) L = β (8.10) hr (γ r hr (t ) + γ s hs ) where σc is the compressive strength of the roof layer. The value of β would take the following values 4/3 when crack length (ℓc) is zero. However, the maximum resistance of the arch would be attained when the value of β is 3/2 after some manipulations concerning the crack length in the arch. The effect of concentrated loads can be taken into account if necessary. The tensile and compressive strengths of rock mass can be obtained using the formulas suggested by Aydan et al. (2014), Aydan and Kawamoto (2000) and Tokashiki and Aydan (2010) in terms of those of intact rock and the RMQR values of rock masses (Aydan et al., 2014). When Equations (8.8) and (8.9) are used for stability assessment, the effects of creep and degradation rate of rock mass in these equations can be taken into account using the method suggested by Aydan et al. (2005a, 2006b, 2006, 2007a, 2007b, 2008a, 2008b) As shown in Figure 8.36, the stability of the roof of the largest room in the Avanos Congress Center is first analyzed using the bending theory with built-in conditions (Ulusay et al., 2013). The span of the room is 12 m with a 5 m thick solid roof and 3 m thick soil deposits at the top. The bending stress indicated that the tensile stress will exceed the tensile strength of solid rock at the top of the solid roof near the sidewalls (Figure 8.37(a)). The next computation

Figure 8.36 Plan of the Congress Center

Degradation of rocks  259

Figure 8.37 Computed results using various analytical models for the roof stability of the Congress Center

was carried out for three different conditions, namely simple-beam, built-in beam and arching (Figure 8.37(b)). The simple beam and built-in beam conditions clearly implied that some cracking should occur, and this conclusion was in accordance with our observations. The stability analyses for arching action indicated that the opening should be stable in the short term even though it may be cracked. The next computation was carried out for the long-term response with the consideration of the long-term strength of the tuffs of Cappadocia, including those of the Congress Center. The pure creep analysis with the consideration of the arching model implied that the openings should remain stable for longer than 30 years under the most unfavorable conditions (Figure 8.37(c)). However, if the degradation resulting from the cyclic freezing–thawing and wetting–drying taken into account, the stable duration becomes shorter (Figure 8.37(d)). The degradation rate is based on the observations of the tuff of Cappadocia (Aydan et al., 2008a, 2008b). The actual failure mode of semiunderground openings observed in Frig valley was analyzed by Aydan and Kumsar (2016). Figure 8.38 illustrates the physical model and idealized mechanical model of the failure mode. The tensile stresses develop above the opening as the erosion depth increases. The cracks propagate downward, and the rock body acts like dead load on the floor slab of the opening. Once the erosion depth reaches a critical depth, the whole mass topples. In other words, the final stage of the failure implies a flexural toppling

260  Degradation of rocks

Figure 8.38  Illustration of the mechanical model for assessing the stability of semi-underground openings

failure. This mode of failure of underground openings, slopes and cliffs has been previously studied in detail by Aydan and Kawamoto (1992). The maximum flexural tensile stress can be shown to be: 2 h t t  d  σ ft = 6γ r  + H 1−   (8.11)  2 d  2d  h 

where γr, h, H, t and d are unit weight of rock, thickness of the floor slab, height of unstable body, thickness of overhanging wall and erosion depth. Although the shearing failure is less likely, the maximum shear stress according to the bending theory may be given in the following form: τ max =

 t  d  3  γ r h + H    (8.12)   d  h  2 

The failure would occur whenever the tensile or shear strength of rock mass is exceeded. This approach is applied to the typical situations observed in Frig valley. The most important item is the evaluation of the tensile and cohesion of rock masses. The approach proposed by Aydan et al. (2014) is adopted in this study. For the RMQR values of rock mass given in Table 8.5, the normalized tensile strength or cohesion of rock mass would be 0.23–0.26 times that of intact rock. Furthermore, the tensile strength of rock mass would decrease with time due to sustained creep load (Aydan and Ulusay, 2013) and the long-term strength would obey Equation (8.3). First a static stress analyses is carried out, and the critical erosion depth is computed for various rock mass tensile strength ratios, as shown in Figure 8.39. Parameters assumed in

Degradation of rocks  261

Figure 8.39 Critical erosion depth as a function of slab thickness

Figure 8.40 Estimation of collapse time of semiunderground openings

the computations are also given in the figure. The results imply that the rock mass strength ratio could range between 0.4 and 1.0. Next computation is carried out to evaluate the failure time of semiunderground openings due to toe erosion and creep of rock mass. Again the rock mass strength ratio and erosion rates (Dr) were varied as a function of time. The results are shown in Figure 8.40. It is interesting to note that major collapses may start to occur after 1270 years. 8.10.2 Coupled analyses for failure in boreholes due to cyclic wetting and drying Tono mine is one of the uranium mines of Japan, and it has been the subject of extensive research regarding nuclear waste disposal in sedimentary rocks. The top formation of the mine consists of sedimentary rocks ranging from sandstone to siltstone. An exploration adit

262  Degradation of rocks

Figure 8.41 Views of yielding horizontal boreholes

was excavated by using the NATM, and this adit was therefore named the NATM adit. At an extension of this adit, a series of horizontal boreholes were excavated for measuring insitu stresses. The failure was observed in these boreholes, particularly at their crown and sidewalls (Figure 8.41). Following these observations, some new boreholes were drilled into sedimentary rocks to investigate the causes and mechanisms of failure of these nonsupported boreholes. Fine-grain sandstone starts to fracture while losing its water content, as observed in laboratory tests. The situation is similar to the reverse swelling problem. It is considered that rock shrinks as it loses its water content. This consequently induces results in shrinkage strain leading to the fracturing of rock in tension. Therefore, a coupled formulation of the problems was considered to be necessary. Since this problem was previously formulated by the first author, its summarized version is described here. (a)  Mechanical modeling The water content variation in rock may be modeled as a diffusion problem. Thus the governing equation is written as: dθ = −∇⋅ q + Q (8.13) dt where θ, q, Q and t are water content, water content flux, water content source and time, respectively. If water content migration obeys Fick’s law, the relation between flux q and water content is written in the following form: q = −k ∇θ (8.14) where k is water diffusion coefficient. If some water content is carried out by the groundwater seepage or airflow in open space, this may be taken into account through the material derivative operator in Equation (8.13). However, it would be necessary to describe or evaluate the seepage velocity or airflow.

Degradation of rocks  263

If the stress variations occur at slow rates, the equation of motion without the inertial term may be used in incremental form as given here: ∇⋅ σ = 0 (8.15) The simplest constitutive law for rock between stress and strain fields would be a linear law, in which the properties of rocks may be related to the water content in the following form: σ = D (θ ) εe (8.16) The volumetric strain variations associated with shrinkage (inversely swelling) may be related to the strain field in the following form: ε e = ε − ε s 

(8.17)

(b)  Finite element modeling The finite element form of the water content migration field takes the following form after some manipulations of Equations (8.13) and (8.14) through the usual finite element procedures:

[ M ]{θ} + [ H ]{θ} = {Q} (8.18) where T

T T [ M ] = ∫ [ N ] [ N ] dV , [ H ] = k ∫ [ B ] [ B ] dV , {Q} = ∫  N  {qn } d Γ

Similarly, the finite element form of the incremental equation of motion given by Equation (8.19) is obtained as follows:

[ K ]{U } = {F } (8.19) where

[ K ] = ∫ [ B ] [ D ][ B ] dV , {F } = ∫ [ B ] [ D ]{es } dv + ∫  N  {t}ds T

V

T

T

V

s

(c)  Analyses and discussions The first group analyses were concerned with the simulations of displacement, strain and stress field around a circular borehole in a hydrostatic stress field. Specifically, the effects of the sandstone type and borehole diameter were analyzed. Figure 8.42 shows the computed results for displacement, water content and stress fields for fine- and coarse-grain sandstones for a borehole with a diameter of 200 mm at the overburden of the adit. Since the water migration characteristics of both fine- and coarse-grain sandstones were the same, the resulting water content migration distributions with time were the same. However, displacement, strain and stress fields were entirely different for each sandstone type. Since the volumetric

264  Degradation of rocks

variation of fine-grain sandstone as a function of water content is much larger than that of coarse-grain sandstone, the shrinkage of the borehole in fine-grain sandstone is greater than that in coarse-grain sandstone. Consequently, the radial stress in the close vicinity of the borehole wall becomes tensile in fine-grain sandstone. This, in turn, implies that there would be fractures parallel to the borehole wall if the tensile strength of rock were exceeded. Furthermore, such fractures would only occur in the vicinity of boreholes in fine-grain sandstone, as observationally noted in-situ.

Figure 8.42 Variations of computed water content, displacement, stress fields for fine- and coarse-grain sandstones

Degradation of rocks  265

Figure 8.43 Variations of computed water content, displacement, stress fields of a borehole with a diameter of 100 mm for fine-grain sandstone

Next, the borehole diameter was changed from 200 mm to 100 mm, and the rock was assumed to be fine-grain sandstone. The computed results shown in Figure 8.43 indicated that the water content variation reached the steady state rapidly. However, the computed strain and stress fields are the same as those of the borehole with a diameter of 200 mm except for the magnitude of radial displacement. The final computational example was concerned with a circular borehole under two dimensional initial in-situ stress fields. It is observed that the bottom of the borehole was wet or covered with water in-situ. In order to take into account this observation in computations, the boundary conditions for the water content migration field and displacement field were assumed as illustrated in Figure 8.44. The other properties were the same as those used in axisymmetric simulations. The computed displacement field and associated yielding zone are shown in Figure 8.45.

Figure 8.44 Assumed boundary conditions in computations

Figure 8.45 Computed displacement field and yield zone

Degradation of rocks  267

As noted from the figure, the bottom of the borehole heaves, and the crown of the borehole shrinks upward. In other words, the upper part of the borehole expands outward due to water content loss. The displacement and stress fields of the surrounding rock are entirely different at the lower and upper parts of the borehole. As a result, yielding occurs only in the upper part of the borehole. This computational result is in accordance with actual observations. The yielding zone is not depleted in this computation. However, if the yielding zone were depleted in the computation region, the process would repeat itself after each depletion of the yielded zone.

References Aydan, Ö. (1989) The stabilisation of rock engineering structures by rockbolts. Doctorate Thesis, Nagoya University, Faculty of Engineering. Aydan, Ö. (2001) Modelling and analysis of fully coupled hydro-thermo-diffusion phenomena. In Proceedings of International Symposium on Clay Science for Engineering, Balkema, IS-SHIZUOKA. pp. 353–360. Aydan, Ö. (2003) The moisture migration characteristics of clay-bearing geo-materials and the variations of their physical and mechanical properties with water content. 2nd Asian Conference on Saturated Soils, UNSAT-ASIA. Aydan, Ö. (2008). New directions of rock mechanics and rock engineering: Geomechanics and Geoengineering. 5th Asian Rock Mechanics Symposium (ARMS5), Tehran, 3–21. Aydan, Ö. (2012) The inference of physico-mechanical properties of soft rocks and the evaluation of the effect of water content and weathering on their mechanical properties from needle penetration tests. ARMA 12–639, 46th US Rock Mechanics/Geomech. Symp., Paper No. 639, 10p (on CD). Aydan Ö. (2019) Infrared thermographic imaging in geoengineering and geoscience. In: LaMoreaux J. (eds) Environmental Geology, 413–438. Encyclopedia of Sustainability Science and Technology Series. Springer, New York. Aydan, Ö., Daido, M., Tano, H., Nakama, S. & Matsui, H. (2006) The failure mechanism of around horizontal boreholes excavated in sedimentary rock. 50th US Rock mechanics Symposium, Paper No. 06–130 (on CD). Aydan, Ö. & Kawamoto, T. (1992) The stability of slopes and underground openings against flexural toppling and their stabilisation. Rock Mechanics & Rock Engineering, 25(3), 143–165. Aydan, Ö. & Kawamoto, T. (2000) The assessment of mechanical properties of rock masses through RMR rock classification system. GeoEng2000, UW0926, Melbourne. Aydan, Ö. & Kumsar, H. (2016) A geoengineering evaluation of antique underground rock settlements in Frig (Phryrgian) Valley in the Afyon-Kütahya region of Turkey. EUROCK2016, Ürgüp. pp. 853–858. Aydan, Ö. & Nawrocki, P. (1998) Rate-dependent deformability and strength characteristics of rocks. International Symposium on the Geotechnics of Hard Soils-Soft Rocks, Napoli, 1, 403–411. Aydan, Ö., Sakamoto, A., Yamada, N., Sugiura, K. & Kawamoto, T. (2005a) The characteristics of soft rocks and their effects on the long term stability of abandoned room and pillar lignite mines. Post Mining 2005, Nancy. Aydan, Ö., Sakamoto, A., Yamada, N., Sugiura, K. & Kawamoto, T., (2005b): A real time monitoring system for the assessment of stability and performance of abandoned room and pillar lignite mines. Post Mining 2005, Nancy. Aydan, Ö. & Tokashiki, N. (2011) A comparative study on the applicability of analytical stability assessment methods with numerical methods for shallow natural underground openings. The 13th International Conference of the International Association for Computer Methods and Advances in Geomechanics, Melbourne, Australia. pp. 964–969. Aydan, Ö. & Ulusay, R. (2003) Geotechnical and Geoenvironmental characteristics of man-made underground structures in Cappadocia, Turkey. Engineering Geology, 69, 245–272.

268  Degradation of rocks Aydan, Ö. & Ulusay, R. (2013) Geomechanical evaluation of Derinkuyu antique underground city and its implications in geoengineering. Rock Mechanics and Rock Engineering, Vienna, 46, 738–754. Aydan, Ö. & Ulusay, R. (2016) Rock engineering evaluation of antique rock structures in Cappadocia Region of Turkey. EUROCK2016, Ürgüp. pp. 829–834. Aydan, Ö., Tano, H., Watanabe, H., Ulusay, R. & Tuncay, E. (2007a) A rock mechanics evaluation of antique and modern rock structures in Cappadocia Region of Turkey. Symposium on the Geology of Cappadocia, Nigde. pp. 13–23. Aydan, Ö., Tano, H. & Geniş, M. (2007b) Assessment of long-term stability of an abandoned room and pillar underground lignite mine. Rock Mechanics Journal of Turkey, (16), 1–22. Aydan, Ö., Ulusay, R., Tano, H. & Yüzer, E. (2008a). Studies on Derinkuyu underground city and its implications in geo-engineering. First Collaborative Symposium of Turkey-Japan Civil Engineers, 5 June, 2008, İTÜ, İstanbul, Proceedings. pp. 75–92. Aydan, Ö., Tano, H., Ulusay, R. & Jeong, G.C. (2008b). Deterioration of historical structures in Cappadocia (Turkey) and in Thebes (Egypt) in soft rocks and possible remedial measures. Proceedings of the International Symposium of Conservation Science for Cultural Heritage 2008, National Research Institute of Cultural Heritage, Korea. pp. 55–65. Aydan, Ö., Ulusay, R., Tano, H. & Yüzer, E. (2008c) Studies on Derinkuyu underground city and its implications in geo-engineering. Procs. of the First Collab. Symp. of Turkish-Japan Civil Engineers, İTÜ, İstanbul. pp. 75–92. Aydan, Ö., Tano, H., Geniş, M., Sakamoto, I. & Hamada, M. (2008d) Environmental and rock mechanics investigations for the restoration of the tomb of Amenophis III. Japan-Egypt Joint Symposium New Horizons in Geotechnical and Geoenvironmental Engineering, Tanta, Egypt. pp. 151–162. Aydan, Ö., Ulusay, R. & Tokashiki, N. (2014) A new Rock Mass Quality Rating System: Rock Mass Quality Rating (RMQR) and its application to the estimation of geomechanical characteristics of rock masses. Rock Mechanics & Rock Engineering, 47, 1255–1276. Büdel, B., Weber, B., Kühl, M., Pfanz, H., Sültemeyer, D. & Wessels, D. (2004) Reshaping of sandstone surfaces by cryptoendolithic cyanobacteria: bioalkalization causes chemical weathering in arid landscapes. Geobiology, 2, 261–268. Ehrlich, H.L. & Newman, D.K. (2009) Geomicrobiology. CRC Press, London. Hall, K, Guglielmin, M. & Strini, A. (2008) Weathering of granite in Antarctica: I. Light penetration into rock and implications for rock weathering and endolithic communities. Earth Surface Processes and Landforms, 33, 295–307. Hoppert, M., Flies, C., Pohl, W., Günzl, B. & Schneider, J. (2004) Colonization strategies of lithobiotic microorganisms on carbonate rocks. Environmental Geology, 46, 421–428. Ishijima, Y. & Fujii, Y. (1997) A study on the mechanism of slope failure at Toyohama tunnel, Feb. 10, 1996. International Journal of Rock Mechanics and Mining Sciences, 34(3–4), 87.e1–87.e12. Ito, T., Aydan, Ö., Ulusay, R. & Kasmer, Ö. (2008) Creep characteristics of tuff in the vicinity of Zelve antique settlement in Cappadocia Region of Turkey. Proceedings of the International Symposium 2008; 5th Asian Rock Mechanics Symposium, Tehran, 24–26 November 2008, Namaye Penhan. pp. 337–344. Kano, K., Doi, T., Daido, M. & Aydan, Ö. (2004) The development of electrical resistivity technique for real-time monitoring and measuring water-migration and its characteristics of soft rocks. Proceedings of 4th Asia Rock Mechanics Symposium, Kyoto. pp. 851–854. Kasmer, Ö., Ulusay, R. & Aydan, Ö. (2008) Preliminary assessments of the factors affecting the stability of antique underground openings at the Zelve Open Air Museum (Cappadocia, Turkey). 2nd European Conf. of Int. Assoc. for Engineering Geology, 15–19 September 2008, Madrid. Kawamoto, T., Aydan, Ö. & Tsuchiyama, S. (1991) A consideration on the local instability of large underground openings. Int. Conf., GEOMECHANICS’91, Hradec. pp. 33–41. Konhauser, K.O. (2007) Introduction to Geomicrobioplogy. Wiley-Blackwell, Oxford, pp. 192–234. Matsubara, H. & Aydan, Ö. (2016) The effect of biological degradation of tuffs of Cappadocia, Turkey. EUROCK2016, Ürgüp. pp. 871–876.

Degradation of rocks  269 Tokashiki, N. & Aydan, Ö., 2010. The stability assessment of overhanging Ryukyu limestone cliffs with an emphasis on the evaluation of tensile strength of rock mass. Journal of Geotechnical Engineering JSCE, 66(2), 397–406. Ulusay, R., Akagi, I., Ito, T., Seiki, T, Yüzer, E. & Aydan, Ö. (1999) Long term mechanical characteristics of Cappadocia tuff. Proceedings of the 9th International Congress on Rock Mechanics, Paris. pp. 687–690. Ulusay, R., Aydan, Ö., Geniş, M. & Tano, H. (2013) Assessment of stability conditions of an underground congress centre in soft tuffs through an integrated rock engineering methods (Cappadocia, Turkey). Rock Mechanics and Rock Engineering, 46(6), 1303–1321.

Chapter 9

Monitoring of rock engineering structures

9.1  Deformation measurements The monitoring of ground movements resulting from excavation and from creep-degradation of surrounding rock in rock engineering structures may be required for stability assessments and environmental safety. The monitoring of ground movements may be accomplished by using direct and indirect techniques. Direct techniques may involve inclinometers, extensometers of the mechanical or fiberoptic type, geodetic measurements or GPS technique. Indirect methods may involve aerial photogrammetric methods or the InSAR method. The most important aspect in monitoring is the long-term reliability and repeatability of measurements because the measurements must be carried out over a long period of time, i.e. years. When ground movements are accelerated, the measurement intervals may be required to be shortened to time units such as hours or minutes, and the combination of several techniques may be necessary. A brief outline of the major techniques is presented in the following paragraphs.

9.1.1  Direct measuring techniques (a) Inclinometers An inclinometer basically measures the deviation of inclination of each segment of the casing (Figure 9.1). The ground movements are converted to displacements on the basis of measured deviations and the location of each casing segment. The device consists of inclinometer casing, traversing probe and readout unit. The casing is permanently installed in a borehole. Important features of casing include the diameter of the casing, the coupling mechanism, groove dimensions and straightness, and the strength of the casing. The traversing inclinometer probe is the standard device for surveying the casing. Recently, some of the traversing probe is equipped with fiber-optic sensors. The traversing probe obtains a complete profile because it is drawn from the bottom to the top of the casing. The first survey establishes the initial profile of the casing. Subsequent surveys reveal changes in the profile of the casing, if movement has occurred.

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Figure 9.1 Illustration of ground movement measurement by an inclinometer

(b) Extensometers An extensometer monitors changes in the distance between one or more anchors fixed to the ground and a reference head at the borehole collar (Figure 9.2). Components of an extensometer include anchors, rods or wires with protective tubing, and a reference head. The anchors, with wires or rods attached, are installed in a borehole. Borehole extensometers have anchored benchmarks. A borehole is drilled to a depth at which the strata are stable. It is then lined with a steel casing with slip-joints to prevent crumpling as subsidence occurs. An inner pipe rests on a concrete plug at the bottom of the borehole and extends to the top. This inner pipe then transfers the stable elevation below to the surface. The wires/rods span the distance between the anchors and the reference head, which is installed at the borehole collar. Measurements are obtained at the reference head with a sensor or a micrometer, either of which measures the distance between the top (near) end of the anchor and a reference surface. A change in the distance with respect to initial measurement indicates that movement has occurred. Movement may be referenced to a borehole anchor that is installed in stable ground or to the reference head, which can be surveyed. In recent years, fiberoptic extensometers have been developed and applied in the practical monitoring of ground movements. (c)  Optical leveling technique Ground movements may be systematically measured using optical levelling devices (Figure 9.3). For this purpose, a network of levelling points is established, and measurements are performed at certain time intervals. The repeatability of the measurements obtained from

Figure 9.2 Illustration of ground movement measurement by extensometers

Figure 9.3 Illustration of ground movement measurement by levelling technique

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different surveys and different loops and stability of the monuments with respect to its local environment are the main criteria for this technique. Levelling technique can provide very precise height differences up to a few millimetres in level of accuracy. This technique is relatively flexibly implemented in areas that have usually dense housing, building and/or vegetation. The benchmarks can also be easily located. Its data processing and analysis of the measured results are also not complicated. Although it yields very accurate height differences, the levelling technique is relatively slow and time-consuming in its execution, especially when precise levelling procedures are being implemented. Its operation also depends on time, weather and also environmental conditions along the levelling routes. It should be also noted that the monitored points should generally be associated with a certain benchmark located on a stable zone outside the subsiding area. When the subsiding area is large, then connection to the stable benchmarks is another limiting constraint for implementation of the levelling technique.

GPS LEVELING TECHNIQUE

GPS technique has also been applied to subsidence measurements in recent years (Figure 9.4). First, a network of GPS stations is established. The GPS surveys are carried out at stations using dual-frequency geodetic-type GPS receivers. There are several advantages of using GPS technique: • • • •

GPS provides the three-dimensional displacement vector with two horizontal and one vertical components so that the subsidence is obtained three-dimensionally. GPS provides the displacement vectors in a unique coordinate reference system, so that it can be used to effectively monitor subsidence in a relatively large area. GPS can yield the displacement vectors with a several millimetres in precision level, which is relatively consistent in the temporal and spatial domains, so that it can be used to detect even relatively small subsidence signals. GPS can be utilized continuously, day and night, independent of weather conditions, so that its field operation can be flexibly optimized.

However, surveys by the GPS technique may show slightly worse standard deviations due to the signal obstruction by trees and/or buildings around the station. Therefore, the obtained precision level of the survey with signal obstructions and/or multipaths may be in the order of 1–3 cm, while the surveys with good signals would be generally in the order of several millimeters. The problem may result from the destruction or alteration of observation monuments inside the urbanized areas. Furthermore, expertise in GPS data acquisition and precise data processing is required for the accurate detection of ground subsidence. (d)  Interferometric Synthetic Aperture Radar (InSAR) technique Although it is relatively new, the InSAR technique has a great potential for long-term monitoring of mine subsidence problems. Similar to aerial photogrammetry, SAR images of the same area taken at certain time intervals are necessary (Figure 9.5).

Figure 9.4 Illustration of ground movement measurement by GPS technique

Figure 9.5 Illustration of ground movement measurement by InSAR technique

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Several methods can be applied to monitor mining subsidence with the InSAR technique. Several InSAR images of the same area are used to generate an interferogram. This interferogram shows the surface topography as well as the subsidence that has occurred during the period. A digital elevation model (DEM) of the area is then used to remove the topography from the interferogram and to generate a so-called differential interferogram. This differential interferogram shows that the surface height changes occurred during the period due to the mining subsidence. Each fringe in the image represents a height change of 28 mm. From the differential interferogram, the points affected by the same surface subsidence are extracted and contoured as lines of “iso-subsidence,” which are overlaid onto a satellite image. This combination gives valuable information in the assessment of the actual and potential damage caused by subsidence. The InSAR technique yields the subsidence information on a regional scale. However, its accuracy is restricted to several centimetres. The use of the InSAR technique for studying subsidence phenomena is expected to be increasing with more radar satellites in space (e.g. ERS, Radarsat, Envisat and ALOS) and more InSAR data processing packages available (e.g. Atlantis, Gamma, Vexcel, Roi-Pac and Doris). In order to use INSAR technique for studying ground subsidence, multitemporal radar images of the area are needed, together with the InSAR processing software and hardware, as well as the expertise to process the images. Time frames for studying land subsidence will also be dictated by the passing times of radar satellites over the studied area. In the context of data processing, the relatively rapid environmental changes and relatively dynamic atmospheric conditions can also limit the potential of InSAR for the accurate detection of land subsidence. (e)  Laser scanning technique The basic principle is based on the emission of a light signal (Laser) by a transmitter and receiving the return signal by a receiver. The scanner uses different techniques for distance calculation that distinguish the type of instrument in the receiving phase. The distance is computed from the time elapsed between the emission of the laser and the reception of the return signal or phase shift based on when the computation is carried out by comparing the phases of the output and return signals. The laser scanner devices operate by rotating a pulsed laser light at high speed and measuring reflected pulses with a sensor. The scanner automatically rotates around its vertical axis, and an oscillating mirror moves the beam up and down. The scanner calculates the distance of a measured point together with its angular parameters. The measured points constitute a set of points called cloud points, which are used to quantify the geometry of the structure or surface in 3-D. As an application of this concept, the authors have tried to evaluate the performance of a tunnel in the Okinawa Prefecture. Figure 9.6 shows a digital image of the tunnel during the construction phase. This type of evaluation would provide a quick evaluation of the state of the tunnel and possible locations where some degradation of support systems may occur, and some unusual fracturing or deformed configurations of the liners resulting from large deformation or fracturing of the surrounding ground may be assessed.

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Figure 9.6 Laser scanner and a digital image of a tunnel under construction obtained from laser scanning

Furthermore, this technique could be also utilized for the maintenance and long-term deformation monitoring of rock engineering structures such as tunnels, slopes, underground powerhouses.

9.2  Acoustic emission techniques Acoustic emission (AE) signals are generated by the sudden release of elastically stored energy during rock mass fracturing caused by excavation, hydrofracturing, slippage and other engineering operations. The piezoelectric sensors receive the vibrations and transform them into electrical signals. The AE systems consist of sensors, data transmission cable, data acquisition system, and data processing unit. The data transmission may be through cable/ wires, optical fiber, or wireless. The power source may be an ordinary electric power supply or battery-operated power. The AE signals can be stored directly on the system, and they may be used to evaluate the location and type of event. However, this type of monitoring requires extensive data storage. On the other hand, if the system is based on pulse counting, the system can be quite compact. Tano et al. (2005) developed a very compact acoustic emission system consisting of AE sensor, amplifier and pulse-counting logger. The system operates using batteries (Figure 9.7). The protection of instrumentation devices during a blasting operation is extremely difficult. In previous studies, the devices were installed in larger holes at the sidewall and covered by some protection sheaths (Aydan et al., 2005a). In this study, the instruments were put in an aluminum box attached to a rock bolt head and protected by a semicircular steel cover fixed to the tunnel surface by bolts. Figure 9.8 illustrates the installation of the instruments in close proximity to the tunnel face. The instruments installed at the crown were about 1.5 m from the tunnel face while the instruments installed at the mid-height of the sidewall were about 2.5 m from the tunnel face during 1–3 September 2015 and 2 m during 5–7 October 2015 monitoring. AE counts were recorded for 1 s intervals, and the recording was started and stopped by a remotely operated switch device.

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Figure 9.7 (a) Principle of AE counting, (b) block diagram of AE counting system

Figure 9.8 Views of the instruments installed at the crown and sidewall of the tunnel

9.3  Multiparameter monitoring The mechanical properties of soft sedimentary rocks are influenced by the variation of water content. The compressive strength and elastic modulus of soft rocks generally decrease with the increase of water content, or vice versa. The effect of degradation of rock by the cycles of wetting and drying cause the flaking of rock near the surface and it falls under the effect of gravity. This process repeats itself endlessly, which subsequently causes the reduction of the supporting area of the pillars in the long term. Therefore, it is of first interest to monitor the environmental parameters such as temperature, humidity, air pressure within the abandoned mines and water content, temperature variations in surrounding rocks in association with the degradation process. It is also known that, when rocks subjected to high stresses start to fail, the stored mechanical energy in rock tends to transform itself into different forms of energy according to the energy conservation law. Some of these transformations involve the variations of electrical

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potential, magnetic field, heat release and kinetic energy in the surrounding rock mass. Recent experimental studies on various types of rocks by Aydan and his co-workers (Aydan et al., 2001, 2002, 2003), including some from abandoned mines indicated distinct variations of multiparameters previously mentioned during deformation and fracturing processes. Therefore, they may be used in the assessment of the short-term stability of abandoned mines. The real-time multiparameter measurement system involves electric potential (EP) variations, acoustic emissions (AE), temperature and water content of rocks (RT), temperature, humidity and air pressure within the abandoned lignite mines above the groundwater table and outside (Aydan et al., 2005b) (Figure 9.9). The rocks around abandoned mines above the groundwater level are much more prone to degradation due to environmental variations in time (Aydan et al., 2005a). The temperature, humidity and air pressure sensors are installed at a certain spacing from the mine entrance to monitor the spatial distribution and variations with time. Temperature and water content sensors are also installed at several depths in some selected pillars and monitored with time. Geoelectric potential monitoring devices and electrodes are set up within the abandoned mine. It is generally desirable to install the devices near geological discontinuities such as faults and fracture zones, which are much more sensitive to stress changes within the surrounding rock mass. When rock mass is damp or saturated, the electrical resistance of

Figure 9.9 Application of multiparameter system for monitoring degradation and long-term response of surrounding rock mass at an abandoned lignite mine Source: From Aydan et al. (2005b)

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ground is in the order of kiloOhms. Therefore, it would be sufficient to use devices, whose impedance is in the order of megaohms. Otherwise, it would be necessary to use devices having impedance in the order of gigaohms. Therefore, the measured electric potential variations by the devices are directly related to those of the surrounding rock mass. The amplitude and orientation of geoelectric potential variations are used to infer the likely location and magnitude of sources of instability. The AE system, limited only to counting AE events (Tano et al., 2005; Aydan et al., 2005b), has proved to be useful for the long-term monitoring of rock fracturing in abandoned lignite mines. Pulse signals, which correspond to AE waves exceeding a threshold, are discriminated through a pulsar and recorded onto a pulse counter (logger) as AE rate counts. Such a limited specification of the rate counting reduces the system cost so two AE systems can be used as one set. One of the AE systems is called an active unit, while the other one is called a dummy unit. The active unit is directly attached to the rock burst, while the dummy AE sensor is not in contact with the rock burst. If signals are counted on both systems, the count of the active unit is deleted from the measured data. This active-dummy counting system increases the reliability of the AE monitoring and checks the noise condition in field.

9.4  Applications of monitoring system 9.4.1  Amenophis III tomb (a) Instrumentation In the tomb of Amenophis III, the wall between the J-chamber and the Jd-chamber is in a critical condition concerning the overall stability of the tomb in the vicinity of J-chamber and adjacent chambers such as Je, Jc and Jd (Aydan et al., 2008). Acoustic emission monitoring units consisting of acoustic emission sensors, together with their amplifiers and loggers, were installed at the wall between J-chamber and Jd-chamber, at the pillar of Je-chamber and at pillar 3 of J-chamber (Figure 9.10). In addition, a displacement gap gauge was installed at the same wall together with amplifier and logger units. Figure 9.11 shows some views of the installation locations of acoustic emission measurement units. Figure 9.12 shows a view of the displacement gap gauge installed at the crack crossing the north wall of J-chamber adjacent to Jd-chamber.

Figure 9.10 Locations of climatic and AE sensors and displacement gap gauge

Figure 9.11 Views of installation locations of acoustic emission sensors

Figure 9.12 Views of (a) displacement gap gauge, (b) logger unit

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(b)  Temperature and humidity measurements Temperature and humidity variations from the entrance down to the Jd-chamber were monitored at an interval of 1 h (Figure 9.13). The results of measurements were downloaded on 16 February 16, 2004. Out of six instruments, we were able to download data from the four instruments installed at the entrance, F, Jd and Je chambers. Nevertheless, the results are sufficient to have a general idea of the temperature and humidity measurements for almost a period of one year throughout the tomb. Figure 9.10 shows the variations of temperature and humidity between 20 March 2003 and 16 February 2004. As noted from the figure, temperature variations of Chambers Je and Jd remain almost constant throughout the measurement period, and its value is about 28oC, except for the period of human activity in the tomb. The human activity period is associated with the cleaning of the wall art of the tomb, and it roughly starts in November and ends in the middle of May. During these periods, the temperatures fluctuate within a range of 5oC. These periods are easily noticed in the figure. The mean temperature of Chamber F is about 1oC below that of Chambers Je and Jd. Since the location of Chamber F has a higher elevation and is directly connected to the entrance, the fluctuation of the temperature of this chamber is larger. The fluctuation range is about 7oC during the human activity period, and thereafter it remains almost constant. The largest temperature variations were observed at the entrance of the tomb. The highest temperature is about 40oC in June (note that the entrance of the tomb is situated at the northern side of the west valley), and the lowest temperature is about 10oC during the period between December and February. Furthermore, the daily fluctuation is about 10–12oC during the human activity period, and it is about 5–7oC when there is no human activity in the tomb. The mean humidity in Chambers Je and Jd is about 20–22% when there is no human activiy in the tomb. However, the daily fluctuations become very large and range within 10–40%. The humidity of Chamber F is higher as it is directly connected to the entrance. The humidity variation at the entrance of the tomb ranges between 10% and 65% during the period of measurement. During summer, the humidity is lower, and it becomes larger during

Figure 9.13 Temperature and humidity variations with time

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the winter period. In other words, June is driest month of the year, while December is the most humid month of the year. (c)  Monitoring results Acoustic emissions observations were done at four locations. Since the human activities within the tomb may cause some acoustic emission events, the acoustic emission counts occurring during the daytime are not taken into account and special emphasis was given to those occurring after work hours (from 19 p.m. till 6 a.m. of the next day). If this criterion is applied to the results of measurements, there was almost no acoustic emission activity for AE instruments numbered AE1, AE2 and AE4 while AE instrument numbered AE3 showed a distinct acoustic emission activity during the period of measurements as shown in Figure 9.14(a). After a relatively calm period between April and May, acoustic emission activity started in June and continued in a linear fashion. After a calm period in August, a very sharp acoustic emission activity started at the beginning of October. Another sharp acoustic emission activity was observed at the beginning of December when the cleaning operation in the tomb started. These acoustic emission activities clearly indicate that localized cracking has been taking place in the wall between Chambers J and Jd. The data from the displacement gap gauge installed at the relatively continuous open fracture on the north wall of Chamber J is available only for the period between 15 May and 3 November 2003. Figure 9.14(b) shows the response of the displacement gap gauge during the aforementioned period. The maximum range of the displacement is within 0.05 mm. Soon after the installation of the gap gauge, the displacement of the fracture has the mode of opening. It seems that the opening displacement of the fracture resembles the temperature variations at the entrance of the tomb even though the temperature of the chamber in which the gauge is installed remains almost constant. Although it is difficult to make a quantitative statement about the effect of the outside temperature, it would be natural to expect that the deformation behavior of the tomb would be influenced by the atmospheric temperature variations and crustal straining due to the motion of the Earth. However, a sudden variation in the displacement response occurred in 11 October 2003. After a certain closure, the crack started to open up. At this moment, it is very difficult

Figure 9.14 (a) Acoustic emission response of the sensor AE3 with time, (b) displacement response of the fracture with time

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Figure 9.15 Comparison of parameters measured by in-situ monitoring systems

to make further comment. Nevertheless, the sudden changes of displacement variation should be associated with some new crack occurrence in the north wall of the main chamber. All relevant data is replotted in Figure 9.15 in order to discuss the implications of the measurement results from in-situ monitoring systems. It seems that the acoustic emission activity started at the end of May 2003 has some relation to the seismic activity along the Nile river to the north of the tomb. If the outside temperature increases, the whole hill should expand. Consequently the inward closure of the tomb should occur, which subsequently causes the opening of the fracture. On the other hand, if the outside temperature decreases, the reverse response should occur, provided that the surrounding rock behaves linear elastically. The sudden increase of AE activity and subsequent variation of displacement response simply implies that some new fracturing took place, and this further led to the opening of the fracture. In other words, there is a high possibility of further propagation of the existing fracture in the north wall of the main chamber of the tomb. 9.4.2 Application of laser surveying to an abandoned lignite mine in Mitake The laser surveying technique has been greatly improved in recent years. This method has been now applied to various rock engineering projects involving slopes, underground excavations and mining-related structures, although it was quite time-consuming to do the laser surveying at initial stages. The author utilized this technique in an abandoned mine,

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Figure 9.16 Laser surveying result of an abandoned mine at Mitake town

exploiting the room and pillar technique at Mitake town in Japan. Figure 9.16 shows one example of the laser surveying used in the abandoned mine with highly complex geometry of within 20 minutes. The processed results can be easily visualized immediately after surveying. This method may be quite useful in surveying various structures. In addition, it may be used for monitoring the response of rock engineering structures in the short and long term. 9.4.3 Application of acoustic emission measurements at underground tomb A monitoring program was undertaken to check that there was no negative effect on the nearby underground tomb of the construction of piles. The monitoring involved a multiparameter monitoring system consisting of acoustic emission (AE) sensors installed at four locations, displacement measurements between layers L2–L3, inclination at the top of the excavation and at the tomb entrance; vibration due to machinery or earthquakes was monitored as illustrated in Figure 9.17. Some of results are presented in this section. Figure 9.18 shows an example of acoustic emissions recorded at five AE sensors. The accelerometer at the ground surface was also used to check the vibrations caused by the machinery operations. In addition, major construction operations and their timing were also recorded. Most of the acoustic emissions were due to construction activities during the daytime. However, some special attention was paid to the acoustic emissions after 9:00 p.m. and before 8:00 a.m. during a typical workday. The results indicated that there were no acoustic emissions 9:00 p.m. and before 8:00 a.m and that they were due to the construction-induced vibrations during the working hours.

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Figure 9.17 Illustration of instruments

Figure 9.18 AE counts observed during 21 October–2 December 2016

9.4.4 Applications of climatic monitoring in Taru-Toge Tunnel Three locations, which were about 1.5–2.5, 31 and 76 m from the tunnel face, were chosen for temperature, humidity and air pressure variations (Figures 9.19 and 9.20). The air pressure measurements can be also used for investigating the blasting pressure wave propagation in the tunnel besides identifying the exact blasting times. Blasting operations also cause carbon dioxide (CO2) emissions. A CO2 monitoring device produced by T&D was used for this purpose and set just in front of the mobile ventilation equipment. The CO2 measurement was done during 5–7 October 2015 monitoring (Figure 9.20).

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Figure 9.19 Temperature, humidity and air pressure and CO2 devices

Figure 9.20 Measurement results

9.4.5 Applications of multiparameter monitoring system in underground powerhouse The multiparameter system fundamentally covers all measurable quantities such as displacement, acoustic emissions, electric potential or electrical resistivity, water level changes, climatic parameters such as temperature, humidity and CO2 and temperature changes of rock

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(Aydan et al., 2019). In this study, a thoroughgoing open crack at the access tunnel next to the powerhouse cavern was selected to monitor its movement (displacement), acoustic emissions (AE), together with climatic changes (temperature and humidity) (Figure 9.21). The unit is fundamentally battery operated. The locations of climatic parameters such as temperature, humidity and air pressure (THP) are measured at the ground surface (132 m elevation). Two CO2 sensors are installed at the ground surface and Underground 1F. The monitoring of CO2 is to check the air quality as well as the condition for the carbonation environment for concrete. Recently, two temperature sensors are installed in a short borehole to monitor rock temperature and air temperature around its vicinity. This measurement is expected to yield some information of cyclic thermally induced deformations of the powerhouse. Aydan et al. (2016a, 2016b) has developed a portable accelerometer, which can be used in four different modes. For strong motion observations during earthquakes, every accelerometer should have a triggering level to start and stop recording for a given time interval and sampling rate and to store in a digital format. The minimum sampling rate is 1 Hz. The device is called QV3-OAM-XXX, and it has the ability with a storage capacity of 2 GB. The power of the accelerometer can be an internal battery, external battery, solar energy or ordinary 100–240 V electricity. In the case of solar energy or ordinary electricity, the power is stored in an external battery through an adapter from a power supply such as solar panels or electric outlet. The system adopted at the powerhouse is designed to utilize the ordinary electricity (100 V) or two external batteries. Currently, the system utilizes electricity available in the powerhouse.

Figure 9.21 Views and locations of installed multiparameter monitoring system

Monitoring of rock engineering structures  289

Figure 9.22 (a) Installation locations of accelerometers, (b) views of installed accelerometers

Four accelerometers were installed at the power station site. Three accelerometers were installed in the powerhouse as illustrated in Figure 9.22. One accelerometer is at the Underground 4F, which is at the bottom of the powerhouse, two accelerometers at Underground 1F, which is the midlevel of the powerhouse. One of the accelerometers is fixed to the sidewall at the penstock side, and the other accelerometer is fixed to the middle of the end wall of the cavern. The fourth accelerometer is installed at the surface. Figure 9.22 shows some views of the installed accelerometers. The main purposes of the installation is to observe the seismic response of the cavern during earthquakes and to evaluate the ground motion amplifications as pointed out by previous pioneering researchers (Nasu, 1931; Komada and Hayashi, 1980). 9.4.6 Applications of multiparameter monitoring system at Nakagusuku Castle A multiparameter monitoring system was also initiated by the authors at Nakagusuku Castle. The system at the castle was actually installed about three years before the one installed at Katsuren Castle, which is probably the first attempt regarding masonry structures in the world. The monitoring was initiated in December 2013, and it is ongoing. During the period of measurements, some earthquakes occurred and a long-term, creep-like separation of a huge crack in Ryukyu Limestone layer extending to the Shimajiri formation layer has been taking place. Figure 9.23 shows the installation location. An earthquake with a moment magnitude of 6.5 occurred at 5:10 a.m. on 13 March 2014 (JST) in the East China Sea at a depth of 120 km on the western side of Okinawa island. Another earthquake occurred at 11:27 a.m. on the same day near Kumejima island. Although the magnitude of the earthquake was intermediate and far from the location, some permanent displacement occurred, as seen in Figure 9.24.

Figure 9.23 Views of monitoring locations and instrumentation

Figure 9.24 Monitoring results during February to March 2014

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9.5  Principles and applications of drone technology 9.5.1 Drones Drones are essentially unmanned aerial vehicles (UAV) that are equipped with high-quality cameras, which can take photos at exact intervals, and gyroscopes, an inertial measurement unit (IMU) and controllers to fly smoothly. For a drone to fly correctly, the inertial measurement unit (IMU), gyro stabilization and flight controller technology are all essential. Drones generally use three- and six-axis gyro stabilization technology to provide navigational information to the flight controller. An inertial measurement unit detects the current rate of acceleration using one or more accelerometers. A magnetometer may be used to assist the IMU on drones against orientation drift. Drones may be equipped with a number of sensors such as distance sensors (ultrasonic, laser, Lidar) or chemical sensors for digital mapping or other purposes. As Lidar, which is an acronym for laser interferometry detection and ranging, can penetrate forest canopy and are widely used for topographical mapping. Aerial photogrammetry is used in topographical mapping using digital or digitized aerial photos of area with known control points. Aerial photographs were taken from a camera mounted on the bottom of an airplane and later were digitized. These days, digital photographs are used together with a record of height and position using GPS and/or other positioning sensors. The plane flies over the area to take overlapping photographs (generally 60% overlapping) over the entire area of interest (Figure 9.25). When it is used for mapping and measuring the displacements of structures following the earthquakes, three-dimensional coordinates of the common points on pre- and post-earthquake photographs were determined. Hamada and Wakamatsu (1998) used this technique to determine the liquefaction induced displacements. This technique is now utilized together with images from drones. However, the fundamental principles remain the same.

Figure 9.25 Illustration of concept of aerial photogrammetry

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9.5.2  Applications to slope and cliffs (a) Slope stability problems and landslide caused by the 2016 Kumamoto earthquakes Kumamoto Prefecture suffered by two successive earthquakes occurring on 14 and 16 April 2016 (Aydan et al., 2018a). These two earthquakes were associated with well-known faults in the region. While the first earthquake on 14 April had a moment magnitude of 6.1–6.2 (Mj 6.5), the strong motions at Mashiki town were more than 1500 cm s−2. The second earthquake with a moment magnitude of 7.0 (Mj 7.3) occurred on April 16, resulting in surface ruptures due to faulting and induced strong motions over a large area. The second earthquake was particularly destructive and caused widespread damage to rock engineering structures, including the built environment. The causes of the damage were high ground motions and permanent straining, which is one of the well-known characteristics of intraplate earthquakes associated with surface faulting. The drone was used the first time to estimate the geometry of the landslide body and the volume of landslide body. Figure 9.26 shows the digital image of the landslide, while Figure 9.27 compares the longitudinal profiles of the landslide area before and after the earthquake. As noted from the figures, it is very easy to evaluate the geometry of the slope failures for evaluating the landslide body.

Figure 9.26 Digital images obtained from drones using the aerial photogrammetry technique

Figure 9.27 Comparison of longitudinal profiles before and after landslide due to the Kumamoto earthquake

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(b)  Application of drone surveying to cliffs and steep slopes The investigation of the possibility of the failures of slopes and cliffs or the back-analysis of the failed cliffs and slopes requires the exact geometry of the topography. Figure 9.28 shows the applications in the shore of the Gushikawa in Itoman City in the south of Okinawa island and the shore at the southern part of the Miyako island. As noted from the figures, it is quite easy to evaluate digitally the geometry of slopes and cliffs. In particular, the evaluation of cliffs is quite cumbersome due to the overhanging rock mass with toe erosion. 9.5.3  Applications to sinkholes The evaluation of the geometry of sinkholes is an extremely difficult and dangerous task due to the unstable configuration and unseen cracks. A sinkhole recently occurred in a Ryukyu limestone quarry in Kumejima island during the quarrying. The excavator fell into the sinkhole together with its operator. Luckily no one was hurt. The evaluation of the size and geometry of the sinkhole was necessary. The drone utilizing the aerial photogrammetry technique was applied at this site, and the results are shown in Figures 9.29 and 9.30. It is interesting to note that the overhanging part of the sinkholes can be accurately evaluated.

Figure 9.28 (a) Digital image of the cliffs in the vicinity of Gushikawa Castle remains in Itoman City, (b) digital image of the cliffs in the southern shore of Miyako island

Figure 9.29 Evaluation of the sinkhole geometry

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Figure 9.30 Selection of cross sections and tracing

9.5.4  Applications to tsunami boulders There are many tsunami boulders in the major islands of the Ryukyu archipelago (Aydan and Tokashiki, 2019). The largest tsunami boulder is probably the one in Shimoji island near Shimoji Airport. The quantification of the geometry and position of these boulders are of great importance in assessing the past major earthquake and tsunami events in a given region. Both drones based on the aerial photogrammetry technique and the laser scanning technique were used to evaluate the tsunami boulders in Okinawa island and Shimoji island. (a)  Kasakanja tsunami boulder in Okinawa island A drone based on the aerial photogrammetry technique was utilized to evaluate the geometry and the position of the tsunami boulder at Kasakanja of Okinawa island (Aydan 2018; Aydan and Tokashiki, 2019). Figure 9.31 shows a view of the drone in operation near the tsunami boulder in Kasakanja. Figure 9.32 shows the topography of the investigated area together with projections on a chosen cross section and in plan. The skill of the operator is also important when the investigations are carried out in areas where overhanging cliffs exist. As noted from Figure 9.32, the geometry of the overhanging cliffs can also be accurately evaluated. (b)  Tsunami boulder in Shimoji island Shimoji Airport has a 4 km long runway near this tsunami boulder. As drones could not fly near the airports due to restrictions, which are automatically imposed on drones, the laser scanning technique was used. Figure 9.33(a) shows the actual tsunami boulder, and Figure 9.33b shows the laser-scanned image of the boulder from the same angle. Although the laser scanning technique can evaluate the geometry of the tsunami boulder, it is somewhat affected by the existence tress and bushes. In other words, the existence of trees and bushes disturb the digital data needed for a proper evaluation of the geometry of the tsunami boulders.

Figure 9.31 View of the drone operation near the tsunami boulder at Kasakanja

Figure 9.32 Processed digital topography of the Kasakanja tsunami boulder and its close vicinity

Figure 9.33 (a) View of the tsunami boulder, (b) digital laser scanned image of the tsunami boulder in Shimoji island

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9.5.5  Applications to masonry castles Many historical masonry structures are in the Okinawa Prefecture, Japan. The northeast corner of the Katsuren Castle collapsed during the 2010 earthquake off Okinawa island (Figure 9.34). Therefore, there is a great concern about the long-term performance and stability of masonry structures during earthquakes in the Okinawa Prefecture. In Katsuren Castle and Nakagusuku Castle, some long-term monitoring and strong motion observations are implemented (Figures 9.35–9.37). The drone-based aerial photogrammetry technique was used to observe the current state of Katsuren Castle and Nakagusuku Castle with a particular attention to locations where continuous measurements were undertaken (Aydan et al., 2016b). These measurements are going to be repeated and compared with those from continuous monitoring results. The repetitions of the measurements using the aerial photogrammetry technique are expected to provide the overall behavior of the castles in the long term three-dimensionally. These types of drone monitoring are also among the first to utilize the drone technology in the world.

Figure 9.34 Collapse and damage to the retaining walls of Katsuren Castle during the 2010 off-Okinawa island earthquake Source: After Aydan et al. (2016b, 2018b)

Figure 9.35 3-D digital image of Katsuren Castle

Figure 9.36 3-D digital image of Katsuren Castle at its northeast corner, where continuous monitoring is implemented

Figure 9.37 3-D digitized image of Nakagusuku Castle

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9.6  Applications to maintenance monitoring Japan has established regulations to carry out compulsory checks on the long-term performance of infrastructures every five years. For this purpose, the authorities or public and private companies and establishments owning the structures have been implementing various techniques to evaluate the state of the structures every five years. Needless to say, such evaluations should be such that they are independent of the techniques employed. The techniques vary from very simple procedures to the very sophisticated. In this respect, the utilization of the drone-based and/or laser scanning techniques could be of great use. As an application of this concept, the authors have tried to evaluate the performance of a tunnel in Okinawa Prefecture. Figure 9.38 shows a digital image of the tunnel during the construction phase. As the tunnels have concrete liners at the final stage of construction, it would be quite practical to evaluate the configuration of the tunnel in a 3-D digital form and check its geometrical changes every five years. This type of evaluation would provide a quick evaluation of the state of the tunnel and possible locations where some degradation of support systems may occur, and some unusual fracturing or deformed configurations of the liners resulting from large deformations or fracturing of the surrounding ground may be assessed. The concept described in the previous structures could be also utilized for the maintenance and long-term deformation monitoring of rock engineering structures such as tunnels, slopes and underground power houses.

9.7  Monitoring faulting-induced deformations The permanent deformation of ground may be induced due to earthquake faulting, or creeping faults may also be monitored by the utilization of the drone-based and/or laser scanning techniques in a similar fashion to those described for previous structures (Aydan 2012; Aydan 2017). Figure 9.39 shows an application of an application of the drone-based aerial

Figure 9.38 Digital images of tunnels obtained from laser scanners

Figure 9.39 (a) View of the evaluation of ground deformations induced by earthquake faulting at a site in the 2016 Kumamoto faulting, (b) measured subsidence

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photogrammetry technique at a site during the Kumamoto earthquake. It is expected that the utilization of the drone-based and/or laser scanning techniques would be quite useful for the evaluations of the deformation of the ground, as well as structures induced by earthquake faulting or permanent movements resulting from ground liquefaction or other causes would be quite effective in years to come (Aydan et al., 2018a; Aydan 2016). These achievements may also lead to better evaluations of the effects of earthquake faulting on structures, as well as that of permanent ground movements.

References Aydan, Ö. (2012) Ground motions and deformations associated with earthquake faulting and their effects on the safety of engineering structures. In: Meyers, R. (ed.) Encyclopedia of Sustainability Science and Technology. Springer, New York, 3233–3253. Aydan, Ö. (2016) Issues on Rock Dynamics and Future Directions. Keynote. ARMS2016, Bali, 20p, on USB. Aydan, Ö. (2019) Some thoughts on the risk of natural disasters in Ryukyu Archipelago. International Journal of Environmental Science and Development, 9(10), 282–289. Aydan, Ö. & Tokashiki, N. (2019) Tsunami Boulders and Their Implications on the Mega Earthquake Potential along Ryukyu Archipelago, Japan. Bulletin of Engineering Geology and Environment. DOI: 10.1007/s10064-09-1378-3. Aydan, Ö., Minato, T. & Fukue, M. (2001) An experimental study on the electrical potential of geomaterials during deformation and its implications in Geomechanics. 38th US Rock Mechanics Symposium, Washington, Vol. 2, 1199–1206. Aydan, Ö., Ito, T., Akagi, T., Watanabe, H. & Tano, H. (2002). An experimental study on the electrical potential of geomaterials during fracturing and sliding. Korea-Japan Joint Symposium on Rock Engineering, Seoul, Korea, July, 211–218. Aydan, Ö., Tokashiki, N., Ito, T., Akagi, T., Ulusay, R. & Bilgin, H.A. (2003) An experimental study on the electrical potential of non-piezoelectric geomaterials during fracturing and sliding. 9th ISRM Congress, South Africa, 73–78. Aydan, Ö., Daido, M., Tano, H., Tokashiki, N. & Ohkubo, K. (2005a) A real-time multi-parameter monitoring system for assessing the stability of tunnels during excavation. ITA Conference, Istanbul. pp. 1253–1259. Aydan, Ö., Sakamoto, A., Yamada, N., Sugiura, K. & Kawamoto, T. (2005b) A real time monitoring system for the assessment of stability and performance of abandoned room and pillar lignite mines. Post Mining 2005, Nancy. Aydan, Ö., Tano, H., Ulusay, R. & Jeong, G.C. (2008) Deterioration of historical structures in Cappadocia (Turkey) and in Thebes (Egypt) in soft rocks and possible remedial measures. 2008 International Symposium on Conservation Science for Cultural Heritage, Seoul. pp. 37–41. Aydan, Ö., Tano, H., Imazu, M., Ideura, H. & Soya, M. (2016a) The dynamic response of the Taru-Toge tunnel during blasting. ITA WTC 2016 Congress and 42st General Assembly, San Francisco, USA. Aydan, Ö., Tokashiki, N. & Tomiyama, J. (2016b) Development and application of multi-parameter monitoring system for historical masonry structures. 44th Japan Rock Mechanics Symposium. Tokyo, pp. 56–61. Aydan, Ö., Tomiyama, J., Matsubara, H., Tokashiki, N. & Iwata, N. (2018a) Damage to rock engineering structures induced by the 2016 Kumamoto earthquakes. The 3rd Int. Symp on Rock Dynamics, RocDyn3, Trondheim, 6p, on CD. Aydan, Ö., Tokashiki, N., Tomiyama, J., Morita, T., Kashiwayanagi, M., Tobase, T. & Nishimoto, Y. (2019). A study on the dynamic and multi-parameter responses of Yanbaru Underground Powerhouse. Proceedings of 2019 Rock Dynamics Summit in Okinawa, 7–11 May 2019, Okinawa, Japan, ISRM (Editors: Aydan, Ö., Ito, T., Seiki T., Kamemura, K., Iwata, N.), pp. 414–419.

300  Monitoring of rock engineering structures Aydan, Ö., Nasiry, N.Z., Ohta, Y. & Ulusay, R. (2018b) Effects of earthquake faulting on civil engineering structures. Journal of Earthquake and Tsunami, 12(4), 1841007 (25 pages). Hamada, M. & Wakamatsu, K. (1998) A study on ground displacement caused by soil liquefaction. Geotechnical Journal JSCE, 596(III-43), 99–208. Tano, H., Abe, T. & Aydan, Ö. (2005) The development of an in-situ AE monitoring system and its application to rock engineering with particular emphasis on tunneling. ITA Conference, Istanbul. pp. 1245–1252.

Chapter 10

Earthquake science and earthquake engineering

10.1 Introduction Earthquakes are known to be one of the natural disasters resulting in huge losses of human lives as well as of properties experienced in the 1999 Kocaeli, Düzce, Chi-chi, and 1995 Kobe earthquakes. It is well-known that ground motion characteristics, deformation and surface breaks of earthquakes depend on the causative faults. While many large earthquakes occur along the subduction zones, which are far from the land, and their effects appear as severe shaking, the large in-land earthquakes may occur just beneath or near urban and industrial zones as observed in the recent great earthquakes. Earthquakes are due to the temporary instability of Earth’s crust resulting from stress state changes. While the accumulation of stress takes a long time, from seconds to thousands of years, which is called the stick phase, the stress release occurs in a few seconds to 500–600 s, and it is called the slip phase. This chapter addresses the scientific and engineering aspects of earthquakes.

10.2  Earthquake occurrence mechanics 10.2.1 Uniaxial compression experiments in relation to earthquakes Tests on instrumented samples of various rocks, such as Ryukyu limestone, tuff, granite, porphryte, andesite, sandstone and the like, were performed (e.g. Aydan et al., 2007, 2011; Ohta, 2011; Ohta and Aydan, 2004, 2010). Two examples (Fuji-TV No.1 and Mitake Sandstone MS2). Fuji-TV No.1 is a prismatic granite sample (100 × 100 × 200 mm). The acceleration responses start to develop when the applied stress exceeds the peak strength and it attains the largest value just before the residual state is achieved, as seen in Figure 10.1(a). This pattern was observed in all experiments. Another important aspect is that the acceleration of the upper plate is much larger than that of the lower plate. This is also a common feature in all experiments. In other words, the amplitude of accelerations of the mobile part of the loading system is higher than that of the stationary part. Mitake Sandstone MS2 sample (height: 93 mm; diameter: 45 mm) is a soft rock, and an accelerometer was attached to the sample at the mid-height. Figure 10.1(b) shows the axial stress and acceleration response as a function of time. The failure of this rock sample is ductile, and the maximum acceleration is much less than that for the granite sample. Nevertheless, the maximum acceleration occurs just before the residual state, which is very similar to that observed during the fracturing of hard brittle granite sample.

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Figure 10.1 Acceleration and axial responses of a granite sample denoted: (a) Fuji-TV No.1, (b) MitakeGifu sandstone (MS-2)

Fundamentally, the observed acceleration responses during the fracturing of various rocks are similar to one another except their absolute values. The most striking feature is the chaotic acceleration response during the initiation and propagation of the macroscopic fracture of the sample. This chaotic response is very remarkable for the radial acceleration component in particular, and probably this phase is associated with the small fragment detachments before the final burst of rock samples. The small fragments result from splitting cracks aligned along the direction of loading before they coalesce into a large shear band. Furthermore, the audible sounds of fracturing are emitted from the rock during this phase. As reported by Aydan (2003a) and his coworkers (Aydan et al., 2007, 2011), work done (according to the definition in continuum mechanics) on tested samples and maximum acceleration (Aydan, 2003a) increases proportionally to the maximum acceleration, and it is always higher on the mobile part compared with that of the stationary side of the loading system. This result should probably have very important implications in many disciplines of geoscience and earthquake engineering for inferring and understanding ground motions.

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10.2.2 Stick-slip phenomenon for simple mechanical explanation of earthquakes and some experiments Brace and Byerlee (1910) and Byerlee (1970) were first to suggest the stick-slip phenomenon as a possible explanation of mechanics of earthquakes. Nevertheless, the stick-slip phenomenon is well-known in the field of tribology (e.g. Bowden and Leben, 1939, Jaeger and Cook, 1979). 10.2.2.1  Simple theory of stick-slip phenomenon In this model, the basal plate is assumed to be moving with a constant velocity vm, and the overriding block is assumed to be elastically supported by the surrounding medium, as illustrated in Figure 10.2. The basic concept of modeling assumes that the relative motion between the basal plate and overriding block is divergent and follows the formulation by Bowden and Leben (1939). Let as assume that the motion of the plate can be modeled as a stick-slip phenomenon. The governing equation of the motion of the overriding block may be written. During the stick phase, the following holds: x = vs , Fs = k ⋅ x (10.1) where vs is belt velocity, and k is the stiffness of the system. The initiation of slip is given as (Figure 10.3): Fy = µs N (10.2)

Figure 10.2 Mechanical modeling of stick-slip phenomenon

Figure 10.3 Frictional forces during a stick-slip cycle

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where μs is static friction coefficient, N is normal force. For the block shown in Figure 10.4, it is equal to block weight W, and it is related to the mass m and gravitational acceleration g through mg. During the slip phase, the force equilibrium yields: −kx + µkW = m

d 2 x (10.3) dt 2

where μk is dynamic friction angle. The solution of the preceding equation can be obtained as: x = A1 cos Ωt + A2 sin Ωt + µk

W  k

(10.4)

If initial conditions (t = ts, x = xs and ẋ = vs) are introduced in Equation (10.4), the integration constants are obtained as follows: v W W (µs − µk ) cos Ω(t − ts ) + s sin Ω(t − ts ) + µk k Ω k W x = − (µs − µk )Ω sin Ω(t − ts ) + vs cos Ω(t − ts ) (10.5) k W x = − (µs − µk )Ω2 cos Ω(t − ts ) − vs Ω sin Ω(t − ts ) k x=

W . k At t = tt, velocity becomes equal to belt velocity, which is given as ẋ = vs. This yields the slip period as: where Ω = k m and xs = µs

tt =

 (µ − µk )W Ω  2   + ts (10.6) π − tan −1  s   k ⋅ vs Ω 

where xs= vs .ts. The rise time, which is the slip period, is given by: tr = tt − ts (10.7) Rise time can be specifically obtained from Equations (10.10) and (10.5) as: tr =

 (µ − µk )W Ω  2   (10.8) π − tan −1  s   k ⋅ vs Ω 

If belt velocity can be omitted, that is, vs ≈ 0, the rise time reduces (tp) to the following form: tr = π

m (10.9) k

The amount of slip is obtained as: xr = xt − xs = 2

W (µs − µk ) (10.10) k

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The force drop during slip is given by: Fd = 2(µs − µk )W 

(10.11)

It should be noted that this formulation does not consider the damping associated with slip velocity. If the damping resistance is linear, the governing equation (10.4) will take the following form: −kx − η x + µkW = m

d2x  dt 2

(10.12)

10.2.2.2  Device of stick-slip tests Figure 10.4 shows a view of the experimental device. The experimental device consists of an endless conveyor belt and a fixed frame. The inclination of the conveyor belt can be varied so that tangential and normal forces can be easily imposed on the sample as desired. To study the actual frictional resistance of the interfaces of the rock blocks, the lower block is stuck to a rubber belt while the upper block is attached to the fixed frame through a spring as illustrated in Figure 10.4(a). Some experiments were conducted using the rock samples of granite with planes having different surface morphologies. The base blocks were 200–400 mm long, 100–200 mm wide and 40–100 mm thick. The upper block was 100–200 mm long, 100 mm wide and 50–100 mm high. When the upper block moves together on the base block at a constant velocity (stick phase), the spring is stretched at a constant velocity. The shear force increases to some critical value, and then a sudden slip occurs with an associated spring force drop. Because the instability sliding of the upper block occurs periodically, the upper block slips violently over the base block. Normal loads can also be easily increased in experiments. To measure the frictional force acting on the upper block, the load cell (KYOWA LURA-200NSA1) is installed between the spring and fixed frame. During experiments, the displacement of the block is measured through a laser displacement transducer produced by KEYENCE and a contact type displacement transducer with a measuring range of 70 mm,

Figure 10.4 Stick-slip experimental setup

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while the acceleration responses parallel and perpendicular to the belt movement are measured by a three-component accelerometer (TOKYO SOKKI) attached to the upper block. The measured displacement, acceleration and force are recorded onto laptop computers. 10.2.2.3  Stick-slip experiment Many stick-slip experiments were performed on various natural rock blocks as well as on other types of blocks made of foam, plastic, wood and aluminum (e.g. Aydan, 2003a; Aydan, 2019; Aydan et al., 2019; Ohta and Aydan, 2010). Here experimental results on discontinuities in granite are quoted as an example. Three different combinations of the surface roughness conditions of granite blocks were investigated while keeping the system stiffness, upper block weight and base velocity constant. These combinations are rough-to-rough (tension joint), rough-to-smooth (saw-cut), smooth-to-smooth interfaces. The measured response of a discontinuity with a rough-to-rough combination is shown in Figure 10.5. As noted from the figure, the velocity of the upper block starts to change before the slippage.

Figure 10.5 The response of a discontinuity surface during a stick-slip experiment

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Figures 10.6 and 10.7 show a series of stick-slip experiments on discontinuities in granite and Ryukyu limestone, together with interpretations of peak and residual friction angles. The peak (static) friction angle can be evaluated from the T/N response while the residual (kinetic) friction angle is obtained from the theoretical relation (10.11). Some tilting experiments were carried out on the same discontinuity planes (Aydan et al., 2019). The peak (static) friction angles for both the discontinuity plane obtained from tilting tests and stickslip experiments were very close to each other. The residual or kinetic friction angles for a rough discontinuity plane of granite are also very close to each other. Similarly, the residual (kinetic) friction angles of saw-cut discontinuity plane of Ryukyu limestone obtained from stick-slip experiments are very close to those obtained from tilting experiments. Nevertheless, the kinetic or residual friction angle is generally lower than those obtained from the tilting experiments. As expected from the theoretical formulas derived in the previous subsection, the phenomenon would be periodic. If the peak and residual friction angles are the

Figure 10.6 Stick-slip response of rough discontinuity plane of granite

Figure 10.7 Stick-slip response of saw-cut plane of Ryukyu limestone

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same, the slip would be continuous with a given velocity, which may be the fundamental explanation of fault-creep observed in some segments of the North Anatolian Fault in Turkey and the San Andreas Fault in the United States.

10.3  Causes of earthquakes The main cause of earthquakes is the stress changes in the Earth’s crust and its temporary mechanical instability. Let’s consider the average stress changes on a fault plane in view of results of stick-slip experiments. The stress state and geometrical parameters of the faults can be illustrated as shown in Figure 10.8. Vectors n, s and b are normal, sliding and neutral vectors with respect to the fault plane, respectively. Neutral vector b is perpendicular to the plane defined by normal and sliding vectors n, s. Parameters p, d and i stand for dip (plunge), dip direction and sliding direction on the fault plane. When the value of i is 0–180, it corresponds to the faults with a normal component. On the other hand, if the value of i is 180–360, it will correspond to faults with reverse components (Figure 10.9).

Figure 10.8 Illustration of notation for a fault plane with directions of principal stresses, slip, normal and neutral vectors

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Figure 10.9 Illustration of the definition of striation angle on a fault surface

The normal and sliding vectors in terms of their components can be specifically written: n = {nx

ny

nz }, s = {sx

sy

sz }(10.13)

where nx = sin p sin d , n y = sin p cos d , nz = cos p , sx = − cos p sin d sin i + cos d cos i , s y = − cos p cos d sin i − sin d cos i , sz = cos p sin i. As the neutral vector b is perpendicular to the plane of normal and sliding vectors, it can be mathematically expressed as given here: b = s ×n (10.14) The traction vector t acting in the fault plane can be related to stress tensor σ and the normal n of the fault plane through the Cauchy equation (e.g. Mase, 1970; Eringen, 1980): t = σ ⋅n The stress tensor can be written in matrix form as given here: σ xx  σ = σ xy σ  xz

σ xy σ yy σ yz

σ xz  σ yz  (10.15) σ zz 

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Furthermore, the following relations can be written for the normal, shear and neutral vectors: σ N = n ⋅ t or σN = n ⋅ σ ⋅ n(10.16a) σ S = s ⋅ t or σ S = s ⋅ σ ⋅ n(10.16b) σ B = b ⋅ t or σ B = b ⋅ σ ⋅ n (10.16c) Aydan (1995) has both theoretically and numerically shown that the vertical component σ zz of the stress tensor can be taken as a quantity obtained by the multiplication of depth h and unit weight γ of rock, as given here, by taking into account the sphericity of the Earth and gravitational acceleration: σ zz = γ h (10.17) Let us introduce a normalized stress obtained by dividing the component of the stress tensor by its vertical component:  N xx  N =  N xy N  xz

N xy N yy N yz

N xz  N yz  (10.18) N zz 

where N zz is equal to 1. Let us consider a coordinate system ox′y′z′ whose axes are aligned with the principal stress components as shown in Figure 10.8: σ I  ′ σ =0  0 

0 σ II 0

0   0  (10.19)  σ III 

Similarly, the normalized principal stress tensor by the vertical stress can be written as:  NI  N′ =  0  0 

0 N II 0

0   0  (10.20)  N III 

The shearing of rock takes place along the direction of the maximum shear stress according to the least work principle of the mechanics. Therefore, the shear stress on the plane of normal and sliding vectors must be nil, implying that the direction of the neutral vector must coincide with that the intermediate principal stress. Thus this can be mathematically expressed as: σ B ≡ σ II (10.21) and b ≡ s II and b ≡ {bx

by

bz }; s II = {l2

m2

n2 } (10.22)

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In light of experimental facts on rocks, the following relations may be written among normal, sliding and neutral vectors and the maximum and minimum principal stresses: φ φ s ⋅ s I = cos( 45 − ) ve s ⋅ s III = cos(135 − ) (10.23a) 2 2 φ φ n ⋅ s I = cos( 45 + ) ve n ⋅ s III = cos(45 − ) (10.23b) 2 2 and (10.23c) s ⋅ s = 0 b ⋅ sI = 0 III Therefore the direction vectors sI and sII of the maximum and minimum principal stresses can be easily obtained from the preceding relations as:

 l1   sx   s I = m1  =  nx     n1   bx

s III

 l3   sx   = m3  =  nx     n3   bx

sy ny by

sy ny by

sz  nz  bz 

−1

sz  nz  bz 

ϕ   cos( 45 − ) 2     ϕ  cos( 45 + ) (10.24a)  2    0    

−1

  cos(135 − ϕ )  2    ϕ  cos( 45 − )  (10.24b)  2    0    

When the Mohr-Coulomb yield criterion is used, the value of intermediate principal stress becomes indeterminate, although its value is bounded by the maximum and minimum principal stresses. Aydan (2000a) established a relation through the use of the Mohr-Coulomb criterion and Drucker-Prager yield criterion for frictional condition, which is commonly used in the numerical analysis of structures in geomaterials, and he derived the following inequality relation to obtain the value of intermediate principal stress: β2 −

(1 + 6α 2 )(q + 1) (1− 3α 2 )(q 2 + 1) − q (1 + 6α 2 ) β+ = 0 (10.25) (1− 3α 2 ) (1− 3α 2 )

where β=

σ σ

II (10.26) III

The inequality relation yields two roots, and one of the roots is chosen so that the intermediate stress would have a value between the maximum and minimum principal stresses. When the peak friction angle is utilized, the stress state would correspond to

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Figure 10.10 Normalized stress states obtained for peak and residual friction angles for the 1984 Nagano Prefecture Seibu earthquake

Table 10.1 Stress and orientation changes for 1984 Nagano Prefecture Seibu earthquake Normal Stress ∆σn / σv

Shear Stress ∆τ d / σv

∆θ

Orientation

0.14

0.114534

2.11

the stress state at the time slip. On the other hand, if the residual friction is utilized, it should correspond to the stress state at the equilibrium following the termination of the earthquake. An example of this concept has been applied to the 1984 Nagano Prefecture Seibu earthquake with a moment magnitude of 6.2. The normalized stress states by the vertical stress for peak (30 degrees) and residual friction (26 degrees) angles are obtained and shown in lowerhemisphere stere-net projections in Figure 10.10. The computed normalized stress changes are given in Table 10.1. The average shear stress change on the fault plane would be about 14.88 MPa for a linear distribution of vertical stress.

10.4  Earthquake-induced waves It is known that earthquakes cause fundamentally two types of earthquakes waves (Figure 10.11). The first type of waves, called body waves, are the P-wave and S-wave. P-waves, or primary waves, pass through all materials. S-waves, or secondary waves, arrive at the observation point after the P-wave. The second-type waves are called surface waves, and

Earthquake science and engineering  313

Figure 10.11 Illustration of wave types

Figure 10.12 Seismogram of the 1939 Erzincan earthquake at Harvard University Source: From Ketin (1973)

they are further subdivided into Rayleigh and Love waves. Figure 10.12 shows a record of 1939 Erzincan earthquake taken at Harvard University. It is known that shear waves are not transmitted through materials in the liquid phase. As the outer core of the Earth is in liquid phase, no shear waves are observed beyond 143 degrees from the earthquake focus. Surface waves (Rayleigh and Love) are observed near the Earth surface, and they disappear as the depth increases. The equation of motion can be written in the following form: ∂σij ∂x j

+ bi = ρ

∂ 2 ui (10.27a) ∂t 2

314  Earthquake science and engineering

or specifically: ∂σ11 ∂σ12 ∂σ13 ∂ 2u + + + b1 = ρ 21 (10.27b) ∂x1 ∂x2 ∂x3 ∂t ∂σ12 ∂σ 22 ∂σ 23 ∂ 2u + + + b2 = ρ 22 (10.27c) ∂x1 ∂x2 ∂x3 ∂t ∂σ13 ∂σ 23 ∂σ33 ∂ 2u + + + b3 = ρ 23 (10.27d) ∂x1 ∂x2 ∂x3 ∂t Normal strain components are related to components of the displacement vector if the infinitesimal strain approach is adopted: ∂u j  1  ∂u  (10.28a) εij =  i + 2  ∂x j ∂xi  or specifically: ε11 =

∂u ∂u ∂u1 , ε22 = 2 , ε33 = 3 (10.28b) ∂x3 ∂x2 ∂x1

Engineering shear strains are related to the components: γ ij = 2εij with i ≠ j (10.28c) or specifically: γ 23 =

∂u ∂u ∂u2 ∂u3 ∂u ∂u , γ12 = 1 + 2 , γ13 = 1 + 3 (10.28d) + ∂x3 ∂x1 ∂x3 ∂x2 ∂x2 ∂x1

Rotational strains are defined as: ∂u  ∂u  1  ∂u 1  ∂u ∂u  1  ∂u ω1 =  3 − 2 , ω2 =  1 − 3 , ω3 =  2 − 1 (10.28e) 2  ∂x1 ∂x2  2  ∂x3 ∂x1  2  ∂x2 ∂x3  The constitute law between stress and strain can be expressed if material is an isotropic elastic body as: σij = λδij εkk + 2µεij ; εkk = ε11 + ε22 + ε33;(10.29a) or specifically: λ λ 0 0 0   ε11  σ11  λ + 2µ      λ + 2µ λ 0 0 0   ε22  σ 22   λ   σ   λ λ λ + 2µ 0 0 0   ε33   33  =     σ12   0 0 0 µ 0 0   γ12       0 0 0 µ 0  γ 23  σ 23   0      0 0 0 0 µ  γ13  σ13   0

(10.29b)

Earthquake science and engineering  315

where λ and µ are Lamé coefficients, specifically given in the following form: λ=

E Eυ and µ = (10.30) 2 (1 + υ ) (1 + υ ) (1− 2υ )

Let us introduce the following: ∆=

∂u1 ∂u2 ∂u3 + + (10.31a) ∂x1 ∂x2 ∂x3

∇2 = ∇⋅∇ =

∂2 ∂2 ∂2  + + 2 2 ∂x2 ∂x32 ∂x1

(10.31b)

Equation (10.31a) corresponds to volumetric strain, while Equation (10.31b) is called Laplacian operator. Inserting constitutive law given by Equation (10.29), together with relations between strain and displacement components given by Equation (10.28), into the equation of motion and differentiating Equations (10.28b), (10.28c) and (10.28d) with respect to x1, x2 and x3, respectively, yields for each respective directions provided that elastic coefficients, density and body forces are constant as follows:

(λ + µ)

∂ 2∆ ∂  ∂ 2 u1  2 ∂u1  (10.32a) + ∇ µ ρ =  ∂x1 ∂x1  ∂t 2  ∂x12

(λ + µ)

∂u ∂ 2∆ ∂  ∂ 2 u2   (10.32b) + µ∇ 2 2 = ρ  2 ∂x2 ∂x2  ∂t 2  ∂x2

(λ + µ)

∂ 2∆ ∂  ∂ 2 u3  2 ∂u3  (10.32c) + ∇ = µ ρ  ∂x3 ∂x3  ∂t 2  ∂x32

Summing up Equations (10.32a), (10.32b) and (10.32c) results in the following equation:

(λ + µ)∇2 ∆ + µ∇2∆ = ρ

∂ 2 εv ∂ 2∆ ∂ 2∆ 2 2 2 2 or or V ∇ ε = (10.33) V ∇ ∆ = p v p ∂t 2 ∂t 2 ∂t 2

where λ + 2µ =

Vp =

E (1− υ ) Eυ E + = (10.34a) (1 + υ )(1− 2υ ) 1 + υ (1 + υ )(1− 2υ ) E (1− υ )

ρ (1 + υ )(1− 2υ )



10.34b)

εv = ∆ (10.34c) Equation (10.33) is known as the governing equation of P-wave propagation in solids. As noted from this equation, P-wave propagation is directly related to volumetric straining. During the propagation of P-wave, solids will undergo dilatational and compressive volumetric straining.

316  Earthquake science and engineering

Similarly inserting constitutive law given by Equation (10.33), together with relations between strain and displacement components given by Equation (10.28), into the equation of motion takes the following form specifically for each respective direction:

(λ + µ)

∂ 2u ∂∆ + µ∇2 u1 + b1 = ρ 21 (10.35a) ∂x1 ∂t

(λ + µ)

∂ 2u ∂∆ + µ∇2 u2 + b2 = ρ 22  ∂x2 ∂t

(λ + µ)

∂ 2u ∂∆ + µ∇2 u3 + b3 = ρ 23 (10.35c) ∂x3 ∂t

(10.35b)

Differentiating Equations (10.35b) and (10.35c) with respect to x3 and x2 yields the following, provided that elastic coefficients, density and body forces are constant:

(λ + µ)

∂u ∂ 2∆ ∂ 2  ∂u  + µ∇2 2 = ρ 2  2  (10.36a) ∂x3∂x2 ∂x3 ∂t  ∂x3 

(λ + µ)

∂u ∂ 2∆ ∂ 2  ∂u  + µ∇2 3 = ρ 2  3  (10.36b) ∂x2∂x3 ∂x2 ∂t  ∂x2 

Subtracting Equations (10.36a) from (10. 36b) results in:  ∂u  ∂ 2∆ ∂u  ∂u  ∂ 2  ∂u ∂ 2 ∆   + µ∇2  3 + 2  = ρ 2  3 − 2  (10.37) −  ∂x  ∂x2∂x3 ∂x3∂x2  ∂x3  ∂t  ∂x2 ∂x3  2

(λ + µ)

Using the rotational strain definition given by Equation (10.28e) and dividing Equation (10.37) gives: ∂2ω µ 2 ∇ ω1 = 21 (10.38a) ρ ∂t Using the same procedure for other directions together with rotation strain components given by Equation (10.28e), one can easily derive the following: ∂ 2 ω2 µ 2  ∇ ω2 = ρ ∂t 2

(10.38b)

∂ 2 ω3 µ 2 (10.38c) ∇ ω3 = ρ ∂t 2 Equation (10.38) is the governing equation of Rayleigh waves. The coefficient in Equation (10.38) is interpreted as the propagation velocity of rotational waves: Vs =

E µ or Vs = (10.39) 2 ρ 1 ( + υ) ρ

Earthquake science and engineering  317

If ∆ is 0 and body force is negligible, one easily gets the following expressions from Equation (10.36): ∂ 2u µ 2 ∂ 2u ∇ u1 = 21 or Vs 2 ∇2 u1 = 21 (10.40a) ρ ∂t ∂t ∂ 2u ∂ 2u µ 2 2 ∇ u2 = 22 or Vs ∇ u2 = 22 (10.40b) ∂t ρ ∂t ∂ 2u µ 2 ∂ 2u ∇ u3 = 23 or Vs ∇2 u2 = 22 (10.40c) ρ ∂t ∂t Equation (10.40) is the fundamental equation of distortion (shear) waves known as S-waves. It should be noted that the propagation velocity of S-waves is the same as that of Rayleigh waves.

10.5  Inference of faulting mechanism of earthquakes The striations and internal structure of these faults are evidence of what type of stress state caused them, and they may also indicate what type of earthquake they produced. The methodology for the inference of the possible stress state and focal plane solutions of earthquakes from the faults require data on dip, dip direction and striation orientation (Aydan, 2000a; Aydan and Kim, 2002). Figure 10.9 shows an illustration of how striation angle is measured. Figure 10.13 shows the focal mechanism of earthquakes estimated from the fault striations obtained for the 1891 Kiso-Beya earthquake in Japan and the 1999 Kocaeli earthquake in Turkey. The focal mechanism of the earthquakes may also be inferred from the P-waves and S-waves induced by earthquakes. Figure 10.14 illustrates the concept of obtaining the focal mechanism of a vertical strike-slip and associated wave responses at the observation points around the focus of the earthquake. Figure 10.15(b) shows the focal mechanism solution for the motion of the fault plane shown in Figure 10.15(a), and the seismograms for this solution would look like those shown in Figure 10.15.

Figure 10.13 Inferred faulting mechanism for some earthquakes from fault striations

318  Earthquake science and engineering

Figure 10.14 Illustration of fundamental concept to obtain focal mechanism solutions from seismic waves

Figure 10.15 (a) Fault motion, (b) inferred focal mechanism

The focal plane solutions used in geoscience for inferring the faulting mechanism of earthquakes are derived by assuming that the pure-shear condition holds. As a result of this assumption, one of the principal stresses is always compressive, while the other one is tensile in focal plane solutions. This condition may also imply that the friction angle of the fault is assumed to be nil. Therefore, the principal stresses are inclined at an angle of 45 degrees with respect to the normal of slip direction. This condition is used to determine p-axis and t-axis in focal plane solutions. Each focal plane solution involves the fault plane on which the sliding takes place and the auxiliary plane (Figure 10.16). The normal of the auxiliary plane corresponds to the slip vector, and it is orthogonal to the neutral plane on which p-axis and t-axis exist.

Earthquake science and engineering  319

Figure 10.16 Illustration of fault plane, auxiliary fault, slip vector on a lower hemisphere stereo net

10.6  Characteristics of earthquake faults The fault is geologically defined as a discontinuity in geological medium along which a relative displacement takes place. Faults are broadly classified into three big groups, namely normal faults, thrust faults and strike-slip faults, as seen in Figures 10.17 and 10.18. A fault is geologically defined as active if a relative movement took place in a period of less than 2 million years. It is well-known that a fault zone may involve various kinds of fractures as illustrated in Figure 10.19(a), and it is a zone having a finite volume (Aydan et al., 1997; Ulusay et al., 2002). In other words, it is not a single plane. Furthermore, the faults may have a negative or positive flower structure as a result of their transtensional or transpressional nature and the reduction of vertical stress near the Earth surface (Aydan et al., 1999). For example, even a fault having a narrow thickness at depth may cause broad rupture zones and numerous fractures on the ground surface during earthquakes (Figure 10.19(b)). Furthermore, the movements of a fault zone may be diluted if a thick alluvial deposit is found on the top of the fault (e.g. the 1992 Erzincan earthquake (Hamada and Aydan, 1992)). The appearance of ground breaks is closely related to geological structure, the characteristics of sedimentary deposits, their geometry, the magnitude of earthquakes and fault movements.

Figure 10.17 Fault types Source: From Aydan (2003b, 2012)

Figure 10.18 Some examples of faulting

Figure 10.19 (a) Fractures in a shear zone or fault, (b) negative and positive flower structures due to transtension or transpression faulting and zoning Sources: (a) From Aydan et al. (1997), (b) modified from Aydan et al. (1997, 1999), Aydan (2003b)

Earthquake science and engineering  321

10.7  Characterization of earthquakes from fault ruptures Turkey is one of the well-known earthquake-prone countries in the world, and most of her large earthquakes involve ground surface rupturing. The data from the past and present earthquakes of Turkey, as well as those of other countries, may be quite useful to establish and/or to revise empirical relations among the characteristics of earthquakes accompanying ground surface rupturing. The data compiled by the author come from the Turkish earthquake database (TEDBAS) developed by the author (Aydan, 1997) and additional inputs from recent earthquakes (Ambraseys, 1988; Hamada and Aydan, 1992; Aydan et al. 1991; Aydan and Kumsar, 1997; Aydan et al., 1998, 1999, 2000a, 2003b, 2005a, 2005b, 2010; Ergin et al., 1960; Soysal et al., 1981; Eyidogan et al., 1991; Gencoglu et al., 1990; Ohta, 2010). The data for other countries is compiled by Wells and Coppersmith (1994), Matsuda (1975) and Sato (1989). The data on the source properties of earthquakes is gathered from the well-known seismological institutes such as the U.S. Geological Survey (USGS), Harvard, ERI of Tokyo University and Swiss Seismological Institute. The number of a data set varies depending upon the studied empirical relations. For example, the number of data for the relation between Mw and Ms is 2010. The following items are chosen as the characteristics of earthquakes: 1 2

3 4 5 6 7 8

Magnitude (moment and surface wave magnitudes, Mw, Ms) Length of earthquake fault (L), which denotes the length of the source fault or that estimated by the ground surface trace observed in the field or aftershock distribution if the surface rupture is hindered by the thick sedimentary deposits (i.e. 1992 Erzincan, 1998 Adana-Ceyhan earthquake) Depth of earthquake hypocenter (D) Rupture area (S), which denotes the ruptured area of the earthquake fault inferred from aftershock distribution or the multiplication of surface rupture length produced by the earthquake by its hypocenter depth with the assumption of a rectangular source area Net slip of the earthquake fault (Umax), which denotes the maximum slip along the slip direction (Whitten and Brooks, 1972) Maximum ground acceleration and velocity (amax, vmax) (hypocenter distance is mostly in the range of 15–25 km) Rupture mode–striation orientation Ratio of vertical maximum acceleration to the horizontal maximum acceleration (RVAHA)

It should be noted that the minimum value of Mw is assumed to be 0 in all-empirical relations presented hereafter. 10.7.1 Relation between surface wave magnitude and moment magnitude Aydan (Aydan, 1997; Aydan et al., 1996) selected surface wave magnitude Ms in developing his empirical relations for Turkish earthquakes since a lot of data based on surface wave magnitude Ms is available, and the magnitude of earthquakes did not exceed 8 so far. It is pointed out that surface wave magnitude Ms becomes unreliable if it exceeds the value of 8 (i.e. Fowler, 1990). Furthermore, it is becoming more popular to use the moment magnitude Mw in place of surface wave magnitude Ms since, recently, many seismological institutes release moment magnitude data rather than surface wave magnitude data. Nevertheless, the moment magnitude data determined by various institutes for the same earthquake is not always the same. Furthermore, the moment magnitude data must be assigned to previous earthquakes

322  Earthquake science and engineering

before the development of moment magnitude concept. Kanamori (1983) suggested that the surface wave and moment magnitudes of earthquakes can be taken as equal to each other within the range of 5–7.6 Aydan (Aydan, 1997; Aydan et al., 1996) proposed the following relation between surface and moment magnitudes of earthquakes for Turkish earthquakes: M w = 1.044 M s or M s = 0.958M w (10.41) However, Ulusay et al. (2004) recently suggested the following formula for their data set on Turkish earthquakes: M w = 0.6798M s + 2.0402 (10.42) Figure 10.20 compares the data set of Turkish earthquakes compiled by the author, including all recent data on Turkish earthquakes and worldwide data, which is fitted to the following empirical relation: M w = 1.2 M s e−0.028 M s (10.43) As noted from the figure, all data sets generally support the suggestion of Kanamori (1983) for relating surface wave magnitude to moment magnitude for the magnitude range of 4–8. Therefore, it can be safe to adopt the previous empirical relations proposed by Aydan (1997, 2001) based on surface wave magnitude for the seismic characteristics of Turkish earthquakes, together with the replacement of surface wave magnitude with moment magnitude. However, the constants of functions will have to be recalculated if the independent variable is chosen as moment magnitude.

Figure 10.20 Relation between moment magnitude and surface wave magnitude

Earthquake science and engineering  323

10.7.2  Relation between MMI intensity and moment magnitude The magnitude of historical earthquakes is mainly inferred from the Modified Mercalli Seismic (MMI) intensity. Aydan et al. (1996) and Aydan (1997) proposed the following empirical relation between MMI intensity and surface wave magnitude Ms (see also Aydan, 2001): I o = 1.317 M s (10.44) In this study, the following relation between moment magnitude and MMI intensity is proposed with the consideration of recent large earthquakes in Turkey and worldwide: I o = 1.32 M w (10.45) Yarar et al. (1980) and Gürpınar et al. (1979) also studied the relation between MKS intensity and earthquake magnitude for Turkish earthquakes. Their proposed relations were essentially similar to those given by Equations (10.44) and (10.45) except for the constant with a minus sign. However, the magnitude in their formula is local magnitude. Furthermore, the coefficient for magnitude is about 1.10 times the one given in Equations (10.44) and (10.45). Also, Kudo (1983) studied the relation between the MKS intensity and the Intensity Scale of the Japan Meteorological Agency. As the maximum values of intensity were different from each other, it can be roughly said that two intensity values of MKS would be designated as one intensity value in the Intensity Scale of the Japan Meteorological Agency. However, the intensity scales of 4, 5 and 6 of the Japan Meteorological Agency have been recently revised and subdivided into weak and strong intensity levels. These days the broad intensity scale of the Japan Meteorological Agency discussed by Kudo (1983) is no longer used in Japan. 10.7.3 Relation between moment magnitude and rupture length, area and net slip of fault It is natural to expect that the rupture sense of faulting of earthquakes greatly influences their seismic characteristics. In this study, the faulting sense is considered in empirical relations between moment magnitude and rupture length (L), rupture area (A) and net slip (Umax) of fault given here as an extension of the previous works of Aydan et al. (1996) and Aydan (1997, 2001): L, S or U max = A ⋅ M w e M w / B (10.46) The functional form of the empirical relations is the same, while their constants A and B differ depending upon the faulting sense, which are given in Table 10.2. Kudo (1983) and Table 10.2 Values of constants for Equation (10.46) for each fault parameter  

Normal faulting Strike-slip faulting Thrust faulting

Rupture Length L (km)

Rupture Area S (km2)

Maximum Displacement Umax (cm)

A

B

A

B

A

B

0.0014525 0.0014525 0.0014525

1.21 1.25 1.19

0.003 0.001 0.0032

1.5 1.7 1.5

0.0003 0.00035 0.0014

1.6 1.6 1.4

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Ambraseys (1988) proposed similar empirical relations between rupture length and earthquake magnitude for Turkish earthquakes. Most of them were caused by strike-slip faulting. Although their relations are good fits to the data set they used, they have limited applicability for the range of data set used in this study. Figure 10.21 shows the plot of data for several parameters previously listed together with empirical functions given by Equation (10.46) in Aydan (1997) and Wells and Coppersmith (1994). The horizontal axis of the plots is the moment magnitude of earthquakes. As seen from the figure, the data is somewhat scattered. Nevertheless, the proposed function, together with constants for each seismic parameter, is the best fit to observational data. The standard deviations of fitted equations to observational data were obtained. Nevertheless, they will not be presented in this article for the purpose of clarity. Furthermore, the relation proposed by Aydan (1997) without considering faulting sense is still valid as noted from Figure 10.21.

Figure 10.21 Comparison of proposed relations with observed data and other empirical relations

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Figure 10.22 Comparison of proposed relation with observed data and other empirical relations

Another important source parameter is the duration of fault rupture. The duration is longer if the rupture propagation is unilateral. However, it is shortened if bilateral rupture propagation takes place. Figure 10.22 shows a compilation of data on Turkish earthquakes and worldwide including the most recent events such as the 2004 and 2005 Sumatra earthquakes and the 2005 Pakistan earthquake. The functional form of the empirical equation has the form of Equation (10.46), and it is shown in Figure 10.22 together with empirical relations proposed by Dobry et al. (1978). The proposed empirical relation holds for the data of Turkish earthquakes as well as worldwide data.

10.8  Strong motions and permanent deformation 10.8.1 Observations on strong motions and permanent deformations 10.8.1.1  Observations on maximum ground accelerations It is observationally known that the ground motions induced by earthquakes could be much higher in the hanging wall block or mobile side of the causative fault as observed in the recent earthquakes such as the 1999 Kocaeli earthquake (strike-slip faulting), the 1999 Chi-chi earthquake (thrust faulting), the 2004 Chuetsu earthquake (blind thrust faulting) and the 2000 Shizuoka earthquake and l’Aquila earthquake (normal faulting) (Ohta, 2011; Chang et al., 2004; Somerville et al., 1997; Tsai and Huang, 2000; Aydan et al., 2009; Abrahamson and Somerville, 1991; Aydan, 2003b; Aydan et al., 2007) as seen in Figures 10.23–10.26. Figure 10.26 illustrates the effect of the hanging wall effect on the attenuation of maximum ground accelerations observed in the 1999 Chi-chi earthquake (Taiwan), 1999 Düzce earthquake (Turkey) and 2001 Geiyo earthquake (Japan) with different faulting mechanisms (Ohta and Aydan, 2010).

Figure 10.23 Footwall and hanging wall effects on the maximum ground accelerations (thrust faulting)

Figure 10.24 Mobile and stationary block effects on the maximum ground accelerations observed in 2000 Tottori Seibu earthquake (strike-slip faulting)

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Figure 10.25 Footwall and hanging wall effects on the maximum ground accelerations (normal faulting)

Figure 10.26 Attenuation of maximum ground accelerations for some earthquakes

Figure 10.27 shows the records of accelerations at the ground surface and at bedrock 260 m below at the Ichinoseki strong motion station (IWTH25) of the KiK-NET (2008) strong motion network of Japan measured during the 2008 Iwate-Miyagi earthquake. The strong motion station was located on the hanging wall side of the fault, and it was very close to the surface rupture. As noted from the figure, the ground acceleration of the UD component was amplified 5.67 times that at the bedrock, and the acceleration records are not symmetric with respect to time axis. This record is also the highest strong motion recorded in the world so far.

328  Earthquake science and engineering

Figure 10.27 Acceleration records at ground surface and bedrock at Ichinoseki strong motion station IWTH25 of KiK-NET in Iwate-Miyagi earthquake

10.8.1.2  Permanent ground deformation The recent global positioning system (GPS) also showed that permanent deformations of the ground surface occur after each earthquake (Figures 10.28 and 10.29). The permanent ground deformation may result from different causes such as faulting, slope failure, liquefaction and plastic deformation induced by ground shaking (Aydan et al., 2010). These type of ground deformations have a limited effect on small structures as long as the surface breaks do not pass beneath those structures. However, such deformations may cause tremendous forces on long and/or large structures such as rock engineering structures. The ground deformation may induce large tensile or compression forces, as well as bending stresses in structures depending upon the character of permanent ground deformations. Blind faults and folding processes may also induce some peculiar ground deformations and associated folding of soft overlaying sedimentary layers. Such deformations caused tremendous damage on tunnels during the 2004 Chuetsu earthquake, although no distinct rupturing took place.

Figure 10.28 Permanent ground deformations and associated straining induced by the 1999 Kocaeli earthquake Source: Reilinger et al. (2000)

Figure 10.29 Ground deformation induced by the great East Japan earthquake Source: Measured using GPS by GSI (2011)

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10.8.1.3  Strong motion estimations (A)  EMPIRICAL APPROACH

There are many empirical attenuation relations for estimating ground motions in the literature (i.e. Joyner and Boore, 1981; Campbell, 1981; Ambraseys, 1988; Aydan et al., 1991). Including so-called next (?) generation attenuation (NGA) relations, all these equations are essentially spherical or cylindrical attenuation relations, and they cannot take into account the directivity effects. As shown in the beginning of this section, ground motions such as maximum ground acceleration (Amax) and maximum ground velocity (Vmax) have strong directivity effects in relation to fault orientation. Furthermore, these relations are generally far below the maximum ground acceleration, and they are incapable of obtaining the maximum ground acceleration (AMAX) or the preferred term, “peak ground acceleration (PGA).” Aydan (2012) proposed an attenuation relation by combining their previous proposals (Aydan et al., 1997; Aydan, 1997, 2001, 2007; Aydan and Ohta, 2010, 2011) with the consideration of the inclination and length of the earthquake fault using the following functional form (Figure 10.30): α max = F1 (Vs ) * F2 ( R, θ, ϕ, L* ) * F3 ( M ) (10.47) where Vs, θ, φ, L* and M are the shear velocity of ground and the angle of the location from the strike and dip of the fault (measured anticlockwise with the consideration of the mobile side of the fault) and earthquake magnitude. The following specific forms of functions in Equation (10.47) were put forward as: F1 (Vs ) = Ae−vs / B (10.48a) (

)

− R 1− D sin θ + E sin 2 θ (1+ F cos ϕ )/ L*

F2 ( R, θ, ϕ, L* ) = e

(10.48b)

F3 ( M ) = e M / G −1 (10.48c)

Figure 10.30 Illustration of geometrical fault parameters (R, θ, φ)

Earthquake science and engineering  331

The same form is also used for estimating the maximum ground velocity (Vmax). L* (in km) is a parameter related to half of the fault length. And it is related to the moment magnitude in the following form: L* = a + becM w (10.49) The specific values of constants of Equations (10.48)–(10.49) for this earthquake are given in Tables 10.3–10.5. The most important parameter in this approach is the estimation of magnitude of the potential earthquake. If a very reliable database exists for a given region, one may estimate the magnitude of the most likely earthquake from such as database. Another approach may be the estimation from the characteristics (length, area, maximum relative slip) of active faults. Matsuda (1975), Sato (1989), Wells and Coppersmith (1994) and Aydan (1997) proposed empirical relations. Aydan (2007, 2012) recently established several relations between moment magnitude and rupture length (L), rupture area (A) and net slip (Umax) of fault given here and checked their validity with available data as well as the data from the most recent event of the 2011 Great East Japan earthquake: L, S orU max = A ⋅ M w e M w / B (10.50) The functional form of the empirical relations is the same, while their constants A and B differ depending upon the faulting sense, which are given in Table 10.4. If striation or sense of deformation of the potential active fault is known, it is also possible to infer its focal mechanism. Such a method is proposed by Aydan (2000a) and compared with the focal mechanism solutions inferred from fault striations or sense of deformation with those from telemetric wave solutions. Table 10.3 Values of constants in Equation (10.48) for interplate earthquakes  

A

B(m s−1)

D

E

F

A max Vmax

2.8 0.4

1000 1000

0.5 0.5

1.5 1.5

0.5 0.5

 

G(Mw) 1.05 1.05

Table 10.4 Values of constants in Equation (10.48) for intraplate earthquakes  

A

B(m/s)

D

E

F

G(Mw)

A max Vmax

2.8 0.4

1000 1000

0.5 0.5

1.5 1.5

0.5 0.5

1.16 1.16

Table 10.5 Values of constants in Equation (10.49) for earthquakes Faulting Type

a

b

c

Normal faulting Strike-slip faulting Thrust faulting

30 20 20

0.002 0.002 0.002

1.35 1.40 1.27

332  Earthquake science and engineering Table 10.5 Values of constants for Equation (10.48) for each fault parameter Fault Type

Parameter

L (km)

S (km2)

Umax (cm)

Normal Faulting

A B A B A B

0.0014525 1.21 0.0014525 1.19 0.0014525 1.25

0.003 1.5 0.001 1.7 0.0032 1.5

0.0003 1.6 0.00035 1.6 0.0014 1.4

Strike-slip Faulting Thrust Faulting

L is rupture length, S is rupture area, Umax is maximum displacement.

Figure 10.31 Comparison of estimated attenuation of maximum ground acceleration and ground velocity with observations for the 2011 Great East Japan earthquake

Figure 10.32 Comparison of estimated attenuation of maximum ground acceleration and ground velocity with observations for the 1999 Kocaeli earthquake

The attenuation relation given by (Equation 10.48) was used to evaluate the maximum ground acceleration and ground velocity of the 2011 Great East Japan earthquake (GEJE) and 1999 Kocaeli earthquake and compared with actual observation data in Figures 10.31– 10.32. The same equation is used to evaluate the areal distribution of maximum ground

Earthquake science and engineering  333

Figure 10.33 Comparison of estimated contours of maximum ground acceleration and ground velocity with observations for the 1999 Kocaeli earthquake

Figure 10.34 Comparison of single and double source models for maximum ground acceleration for the 2008 Wenchuan earthquake

acceleration and velocity for Kocaeli earthquake and compared with observational data in Figure 10.33. For large earthquakes, the use of Equation (10.48) and the estimations based on the segmentation of faults may be more appropriate. Figure 10.34 shows the single- and double-source models for the 2008 Wenchuan earthquake. (B)  GREEN-FUNCTION-BASED EMPIRICAL WAVE FORM ESTIMATION

The empirical Green’s function method was initially introduced by Hartzell (1978). Follow-up methods proposed by Hadley and Helmberger (1980) and Irikura (1983) are modifications of Hartzell’s method of summing empirical Green’s functions. In the empirical Green’s function approach, rupture propagation and radiation pattern were specified deterministically, and the source propagation and radiation effects were included empirically by assuming that the motions observed from aftershocks contained this information (Somerville et al., 1991). A semiempirical Green’s function summation technique has been used by Wald et al. (1998), Cohee et al. (1991) and Somerville et al. (1991), which allows the

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Figure 10.35 Illustration of the fundamental concept of the empirical Green’s function method Source: From Hutchings and Viegas (2012)

gross aspects of the source rupture process to be treated deterministically using a kinematic model based on first motion studies, teleseismic modeling and distribution of aftershocks. Gross aspects of wave propagation are modeled using theoretical Green’s functions calculated with generalized rays (Figure 10.35). The empirical Green’s function method can be used only for a region where small events (i.e., aftershocks or foreshocks) of the target event are available. Ikeda et al. (2016) recently performed an analysis of strong motion induced by the 2014 Nagano-ken Hokubu earthquake using the empirical Green’s function method. The earthquake fault was assumed to be 9.16 km long and 7.2 km wide with a dip angle of 50 degrees. The stress drop was about 12.10. MPa, and the rupture time was 2.7 km s−1 with a rise time of 0.16 s. Figure 10.36 shows the observed and simulated responses of acceleration, velocity and displacement for the north-south direction for the Hakuba strong motion station of K-NET. As seen from the figure, the simulated strong motions are close to the observations. (C)  NUMERICAL APPROACHES

There are several numerical techniques, which are known to be finite difference method (FDM), finite element method (FEM), boundary element method (BEM). The FDM is the earliest numerical model, while FEM and BEM have become available after the 1960s and 1970s, respectively. Therefore, the first application of the numerical methods for strong

Earthquake science and engineering  335

Figure 10.36 Comparison of simulations with observations for the north-south component of Hakuba record

motion estimation is related to the FDM. When this method is applied for strong motion estimation, one needs to solve Equation (10.27) together with appropriate constitutive laws for the medium and the assumption of a rupture plane. In particular, the geometrical definition of the rupture plane and its rupture velocity would be also the key parameters of the simulations. Furthermore, both the FDM and FEM consider the finite size domain; the prevention of reflections of waves from the boundaries would be necessary. This issue is generally dealt with by the introduction of the Lysmer-type viscous boundaries into the numerical model. Both the FDM and FEM would evaluate the wave propagation without any assumption on how waves generated at the source is transferred to any point of particular interest, which is a major issue in Green’s function–based strong motion simulations. While the FEM can easily handle the irregular boundaries such as the surface of the model

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with its topography as a free boundary, the FDM has a severe restriction dealing with such boundaries with irregular geometry. Nevertheless, some procedures dealing with irregular surface topography have been proposed (i.e. Hestholm, 1999; Gravers, 1996)). For irregular surface topography, FDM and FEM are also combined (i.e. Ducellier and Aochi, 2012). In addition, there are also some proposals to combine FDM or FEM with BEM in order to deal with newly developing ruptures. (i) Finite difference method (FDM)  The constitutive law for the medium adjacent to the rupture plane is generally assumed to be visco-elastic (i.e. Graves 1996). The most difficult aspect is the simulation of the fault plane associated with rupture process in the FDM schemes. The most conventional technique is to assume that the fault plane coincides with the grid planes. Forced displacement field is introduced at the domain where two points occupy the same space initially and can move relative to each other after the rupture. Many schemes also explore the incorporation of the finite element method or boundary integral model to simulate the fault plane, and the rest of the domain is discretized using the FDM. Figure 10.37 shows a simulation of strong motion induced by the 1995 Kobe earthquake using the FDM by Pitarka et al. (1998). (ii) Finite element method (FEM)  Toki and Miura (1985) utilized Goodman-type joint elements in 2-D-FEM to simulate both the rupture process and the ground motions (Figure 10.38). Fukushima et al. (2010) utilized this method to simulate ground motions caused by the 2000 Tottori earthquake (Figure 10.39). Later Mizumoto et al. (2004) extended the same method to 3-D. Iwata et al. (2016) recently investigated the strong motions induced by the 2014 NaganoHokubu earthquake. The model is based on 3-D FEM version. Figure 10.40(a) shows the fault parameters, and Figure 10.40(b) shows the 3-D mesh of the earthquake fault and its vicinity. Figure 10.41(a) shows the time histories of surface acceleration at distances of 1 km and 2 km from the surface rupture in the 3-D-FEM model. Rupture time is about 7–8 s. The maximum acceleration is higher in the east-side (hanging wall) than that in west-side (footwall), which is close to the general trend observed in strong motion records. Nevertheless, the computed acceleration was less than the measured accelerations. Figure 10.41(b) shows the time histories of surface displacement at distances of 1 km and 2 km from the surface rupture. The east side of the fault moves upward with respect to the footwall together with movement to the north, and the vertical displacement of the east side is larger than that of the footwall, and the computed results are close to the observations. However, it is necessary to utilize finer meshes for better simulations of ground accelerations, which requires use of the supercomputers. (iii) GPS method  The recent global positioning system (GPS) also showed that permanent deformations of the ground surface occur after each earthquake (Figure 10.28–10.29). The permanent ground deformation may result from different causes such as faulting, slope failure, liquefaction and plastic deformation induced by ground shaking (Aydan et al., 2010). These types of ground deformations have a limited effect on small structures as long as the surface breaks do not pass beneath those structures. However, such deformations may cause tremendous forces on long and/or large structures. The ground deformation may induce large tensile or compression forces, as well as bending stresses in structures depending upon the character of permanent ground deformations. As an example, the ground deformations

Figure 10.37 Fault normal ground velocity propagation induced by the 1995 Kobe earthquake Source: From Pitarka et al. (1998)

Figure 10.38 Representation of joint elements for faults and its constitutive law Source: From Fukushima et al. (2010)

Figure 10.39 Comparison of computed and observed maximum ground acceleration for 2000 Tottori earthquake Source: From Fukushima et al. (2010)

Earthquake science and engineering  339

Figure 10.40 Fault model and 3-D FEM mesh

Figure 10.41 Computed acceleration (a) and displacement (b) responses

reported by Reilinger et al. (2000) are shown in Figure 10.42, which were caused by a strikeslip fault during the 1999 Kocaeli earthquake in Turkey. Blind faults and folding processes may also induce some peculiar ground deformations and associated folding of soft overlaying sedimentary layers. Such deformations caused tremendous damage on tunnels during the 2004 Chuetsu earthquake although no distinct rupturing took place. (iv) InSAR method  Interferometric synthetic aperture radar, abbreviated InSAR or IfSAR, is a radar technique used in geodesy and remote sensing. This geodetic method uses two or more synthetic aperture radar (SAR) images to generate maps of surface deformation or digital elevation, using differences in the phase of the waves returning to the satellite or aircraft.

340  Earthquake science and engineering

Figure 10.42 Computed ground straining from GPS measurements Source: From Aydan et al. (2010)

The technique can potentially measure centimeter-scale changes in deformation over spans of days to years. It has applications for geophysical monitoring of natural hazards, for example earthquakes, volcanoes and landslides, and in structural engineering, in particular the monitoring of subsidence and structural stability. Figure 10.43 shows an application of the InSAR to estimate ground deformations induced by the 1999 Kocaeli earthquake. (v) EPS method  Ohta and Aydan (2007) and Aydan and Ohta (2011) have recently showed that the permanent ground deformations may be obtained from the integration of acceleration records. The erratic pattern screening (EPS) method proposed by Ohta and Aydan (2007) and Aydan and Ohta (2011) can be used to obtain the permanent ground displacement with the consideration of features associated with strong motion recording. The duration of shaking should be naturally related to the rupture time tr. Depending upon the arrival time difference of S-wave and P-wave, the shaking duration would be a sum of rupture duration and S-P arrival time difference ∆tsp. If the ground exhibits a plastic response due to yielding or ground liquefaction, the duration of shaking would be elongated. If coseismic crustal deformations are to be obtained, the integration duration should be restricted to the rupture duration with the consideration of the S-P arrival time difference. The existence of plastic deformation can be assessed by comparing the effective shaking duration and the sum of rupture duration tr and S-P arrival time difference ∆tsp. If the effective shaking duration is longer than the sum of rupture duration tr and S-P waves arrival time difference ∆tsp, the integration can be carried out for both durations, and the difference can be interpreted as the

Earthquake science and engineering  341

Figure 10.43 Interferogram produced using ERS-2 data from 13 August and 17 September 1999 for the 1999 Kocaeli earthquake Source: NASA/JPL-Caltech

plastic ground deformation. The most critical issue is the information of rupture duration. The data on rupture duration is generally available for earthquakes with a moment magnitude greater than 5.10. worldwide. The effective shaking duration may be obtained from the acceleration records using the procedure proposed by Housner (1965). When such data is not available, the empirical relation proposed by Aydan (2007) may be used: tr = 0.005M w exp (1.25M w ) (10.51) S-P arrival time difference ∆tsp can also be easily evaluated from the acceleration record. They divided an acceleration record into three sections and applied filters in Section 1 and Section 3, and the integration is directly carried in Section 2 without any filtering. The times to differentiate sections are t1 and t2. Time t1 is associated with the arrival of the P-wave,

342  Earthquake science and engineering

while time t2 is related to the arrival time of P-wave, rupture duration and S-P waves’ arrival time difference for the crustal deformations as given here: t2 = t1 + tr + ∆tsp (10.52) Any deformation after time t2 must be associated with deformations related to the local plastic behavior of the ground at the instrument location. We show one example for defining times t1 and t2 on a record taken at HDKH07 strong motion station in the 2003 Tokachi-oki earthquake, Japan (Figure 10.44). The estimated rupture time for this earthquake is about 40 s (Yamanaka and Kikuchi, 2003) with about 18 s S-P waves’ arrival time difference (∆tsp). Another important issue is how to select filter values in Sections 1 and 2. This is somewhat a subjective issue, and it depends upon the sensitivity of the accelerometers. The filter value ε1 is generally small, and this stage is associated with the pretrigger value of instruments. Our experience with the selection of ε1 for K-NET and KiK-NET accelerometer records implies that its value be less than ±2 gals. As for the value of ε2, higher values must be assigned. Again, our experiences with the records of K-NET and KiK-NET accelerometers imply that its value should be ±6 gals. The threshold values in the acceleration records of Turkey and Italy are much less than those from the highly sensitive accelerometers of networks in Japan.

Figure 10.44 Definition of sections in the EPS method

Earthquake science and engineering  343

This method is applied to the results of laboratory faulting and shaking table tests, in which shaking was simultaneously recorded using both accelerometers and laser displacement transducers. Furthermore, the method was applied to the strong motion records of several large earthquakes with measurements of ground movements by GPS as seen in Figure 10.45. The comparison of computed responses with actual recordings was almost the same, implying that the proposed method can be used to obtain actual recoverable as well as permanent ground motions from acceleration recordings. Figure 10.46 shows the application of the EPS method to the strong motions records of the 2009 L’Aquila earthquake to estimate

Figure 10.45 Comparison of the permanent ground deformation by the EPS method with measured GPS recordings Source: From Ohta and Aydan (2007)

Figure 10.46 Estimated permanent ground displacements by EPS method Source: From Aydan et al. (2009)

344  Earthquake science and engineering

the coseismic permanent ground displacements. These results are very consistent with the GPS observations. However, it should be noted that the permanent ground deformations recorded by the GPS does not necessarily correspond to those of the crustal deformation. Surface deformations may involve crustal deformation as well as those resulting from the plastic deformation of ground due to ground shaking. The records at ground surface and 210 m below the ground surface taken at IWTH25 during the 2008 Iwate-Miyagi earthquake clearly indicated the importance of this fact in the evaluation of GPS measurements (KiKNET, 2008). (vi) Okada’s method  Okada (1992) proposed closed-form solutions for dislocation in a half-space isotropic medium (Figures 10.47 and 10.48). Closed-form analytical solutions are presented in a unified manner for the internal displacements and strains due to shear and tensile faults in a half space for both point and finite rectangular sources. These expressions evaluate deformations in an infinite medium, and a term related to surface deformation is obtained through multiplying by the depth of observation point. Stein (2003) utilized the solutions of Okada’s method in his software to compute permanent ground deformation and associated stress changes. This method is also used to forecast earthquakes with the introduction of superposing displacement field and associated stress changes together with the use of the Mohr-Coulomb criterion (King et al). His method has been upgraded by his research group and applied to various earthquakes in recent years. Figure 10.49 shows an example of computations by Toda et al. (2002) for earthquake activity in the Izu islands.

Figure 10.47 Geometrical illustration of assumed fault and relative displacements

Figure 10.48 Illustration of ground deformation associated with faulting

Figure 10.49 Relations between static stress changes and seismicity Source: From Toda et al. (2002)

346  Earthquake science and engineering

Figure 10.50  3D-FEM mesh for strong motion simulation of 2011 Great East Japan earthquake Source: From Romano et al. (2014)

(vii) Numerical methods  FDM, FEM, BEM or combined FDM, FEM and BEM produce permanent ground deformation as a natural output of computations provided that the rupture process is well simulated. Figures 10.50 and 10.51 show an example of such a computation for 2011 Great East Japan earthquake (or Tohoku earthquake). Although, the displacement response could be more easily simulated, acceleration responses simulation requires fine meshes, which undoubtedly require the use of supercomputers. It should be also noted that FEM models could easily simulate both permanent displacements in addition to strong ground motions.

10.9 Effects of surface ruptures induced by earthquakes on rock engineering structures In this section, typical examples of damage to various structures induced by the fault breaks observed in recent large earthquakes since 1995 are presented, and details can be found in the quoted references (e.g. Aydan et al., 2011; Ohta et al., 2014).

Earthquake science and engineering  347

Figure 10.51 3-D-FEM simulations of ground motions associated with the 2011 Great East Japan earthquake Source: From Romano et al. (2014)

Figure 10.52 (a) Collapse of the overpass, (b) collapse of Pefong Bridge (Note the uplifted ground on RHS.)

10.9.1  Bridges and viaducts Along the damaged section of the TEM motorway previously mentioned, there were several overpass bridges. Among them, a four-span overpass bridge at Arifiye junction collapsed as a result of faulting (Figure 10.52(a)). The fault rupture passed between the northern abutment and the adjacent pier. The overpass was designed as a simply supported structure according to the modified AASHTO standards, and girders had elastometric bearings. However, the girders were connected to one another through prestressed cables. The angle between

348  Earthquake science and engineering

the motorway and the strike of the earthquake fault was approximately 15 degrees, while the angle between the axis of the overpass bridge and the strike of the fault was 65 degrees. The measurements of the relative displacement in the vicinity of the fault range between 330 and 450 cm. Therefore an average value of 390 cm could be assumed for the relative displacement between the pier and the abutment of the bridge. The Pefong bridge collapsed due to thrust faulting in the 1999 Chi-Chi earthquake, which passed between the piers near its southern abutment, as seen in Figure 10.52(b). 10.9.2 Dams The Shihkang dam, which is a concrete gravity dam with a height of 25 m, was ruptured by thrust-type faulting during the 1999 Chi-Chi earthquake (Figure 10.53). The relative displacement between the uplifted part of the dam was more than 980 cm. Liyutan rockfill dam with a height of 90 m and a crest width of 210 m, which was on the overhanging block of Chelongpu fault, was not damaged, even though the acceleration records at this dam showed that the acceleration was amplified 4.5 times that at the base of the dam (105 gals). The deformation zone of faulting during the 2008 Wenchuan earthquake caused some damage at the Zipingpu dam with concrete facing. 10.9.3 Tunnels The past experience on the performance of tunnels through active fault zones during earthquakes indicates that the damage is restricted to certain locations. Portals and the locations where the tunnel crosses the fault may be damaged, as occurred in the 2004 Chuetsu, 2005 Kashmir and 2008 Wenchuan earthquakes (Figure 10.54). A section nearby Elmalık portal of Bolu Tunnel collapsed (Figure 10.55). This section of the tunnel was excavated under very heavy squeezing conditions. The well-known examples of damage to tunnels at locations where the fault rupture crossed the tunnel are mainly observed in Japan. The Tanna fault ruptured during the 1930 Kita-Izu earthquake, causing damage to a railway tunnel; the relative displacement was about 100 cm. The 1978 Izu-Oshima Kinkai earthquake induced damage to the Inatori railway tunnel. Similar type of damage to the tunnels of Shinkansen and subway lines through the Rokko mountains with a small amount of relative displacement due to motions of the Rokko, Egeyama and Koyo faults was also observed. During the

Figure 10.53 Failure of Shihkang dam due to thrust faulting

Earthquake science and engineering  349

Figure 10.54 Examples of damaged portals of tunnels

Figure 10.55 Collapse of Bolu Tunnel during the 1999 Düzce earthquake

1999 Chi-Chi earthquake, the portal of the water intake tunnels was ruptured for a distance of 10 m as a result of thrust faulting. Except for this section, the tunnel was undamaged for its entire length. Jiujiaya Tunnel is a 2282 m long double-lane tunnel that is 226.6 km away from the earthquake epicenter and that is about 3–5 km away from the earthquake fault of the Wenchuan earthquake. The tunnel face was 983 m from the south portal at the time of the earthquake. The concrete lining follows the tunnel face at a distance of approximately 30 m. Thirty workers were working at the tunnel face, and one worker was killed by the flying pieces of rock bolts, shotcrete and bearing plates caused by the intense deformation of the tunnel face during the earthquake. The concrete lining was ruptured and fallen down at several sections (Figure 10.56). However, the effect of the unreinforced lining rupturing was quite large and intense in the vicinity of the tunnel face. The rupturing of the concrete lining generally occurred at the crown sections, although there was rupturing along the shoulders of the tunnel at several places. Furthermore, the invert was uplifted due to buckling at the middle sections.

350  Earthquake science and engineering

Figure 10.56 Earthquake damage at Jiujiaya Tunnel due to permanent deformations

The Kumamoto earthquake on 16 April 2016 caused heavy damage to several tunnels in the vicinity of Tateno and Minami-Aso villages. Damage to the Tawarayama Roadway Tunnel and Aso Railway Tunnel and Minami-Aso Tunnel was publicized. The damage to Tawarayama Tunnel occurred at two locations (Figure 10.57). The first damage occurred approximately 50–60 m from the west portal of the tunnel, and the concrete lining was displaced by about 30 cm almost perpendicularly to the tunnel axis. The heaviest damage occurred for a length of 10 m about 1600 m from the west portal and about 460 m from the east portal. The angle between the relative movement and tunnel axis was about 20–30 degrees. At this location, the nonreinforced concrete lining collapsed for a length of about 5 m. Although the tunnel is located about 2 km away from the main fault, the tunnel was damaged by secondary faults associated with the transtension nature of the earthquake fault. 10.9.4  Landslides and rockfalls The recent large earthquakes caused mega-scale slope failures and rockfalls particularly along the surface ruptures on the hanging wall side of the fault. The slope failure induced in Beichuan town during the 2008 Wenchuan earthquake is of great interest. In association

Earthquake science and engineering  351

Figure 10.57 Views of damage and their locations at Tawarayama Tunnel

with the sliding motion of the earthquake fault, northwest- or southeast-facing slopes failed during this earthquake. There were two large-scale slope failures (landslides) in Beichuan town, which destroyed numerous buildings and facilities. The northwest-facing landslide (Jingjiashan) involved mainly limestone while the southeast-facing landslide (Wangjiaya) involved phyllite (mudstone, according to some) rock unit (Figure 10.58). Limestone layers dipped toward the valley-side with an inclination of about 30 degrees. Furthermore, there are several faults dipping parallel to the failure surface within the rock mass. The angles of the lower and upper parts of the failed slope are 10.0 degrees and 30–35 degrees, respectively. The existence of several faults dipping parallel to the slope with an inclination of about 60–65 degrees creates a stepped failure surface. The SE facing slope (Wangjiaya landslide) may involve a slippage along the steeply dipping bedding plane (fault plane?) and shearing through the layered rock mass. In other words, it may be classified as a combined sliding and shearing sliding (Aydan et al., 1992). The angles of the lower and upper parts of the failed slope are 40–45 degrees and 30–35 degrees, respectively. The layers dip at an angle of 40 degrees toward the valley, and the shearing plane is inclined at an angle of 20 degrees.

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Figure 10.58 Views of landslides in Beichuan

10.10 Response of Horonobe underground research laboratory during the 20 June 2018 Soya region earthquake and 6 September 2018 Iburi earthquake 10.10.1  Characteristics of Soya region earthquake The Soya region earthquake occurred on 20 June 2018 at 5:28 a.m. The moment magnitude of the earthquake was 4.0 according to F-NET of NIED. The focal mechanism of the earthquake was estimated to be thrust fault (Figure 10.59a), which is a consistent mechanism in view of the tectonics of the Soya region. Figure 10.59b. shows the inferred stress state for the earthquake. The maximum horizontal stress acts in almost in an east-west (EW) direction. 10.10.2  Characteristics of Iburi earthquake Another major earthquake with a moment magnitude of 6.6 (Mj 6.7) occurred on 6 September 2018 at 3:08 in Iburi Region of Hokkaido island, which is about 216 km away from Horonobe. The focal mechanism of this earthquake was due to the blind steeply dipping thrust fault. The earthquake was felt in Horonobe as recorded by the Kik-Net network. However, the maximum ground acceleration was 3.3 gals.

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Figure 10.59 (a) Redrawn focal mechanism obtained by F-NET, (b) inferred stress state for the focal mechanism obtained by F-NET by Aydan’s method (2000a)

Figure 10.60 Seismic records of the 20 June 2018 Soya region earthquake (20 June 2018, M = 4.8) observed in Horonobe URL (maximum acceleration is 8.0, 5.8 and 2.0 gals for N-S, E-W and Z directions.)

10.10.3  Acceleration records at Horonobe URL Accelerometers are set at the ground surface, GL.-250 m and GL.-350 m galleries in Horonobe URL. Figure 10.60 shows the seismic records of the 20 June 2018 Soya Region earthquake observed at GL.-350 m gallery and shaft bottoms (Figure 10.61). Table 10.6

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Figure 10.61 Locations of strong motion observation stations

Table 10.6  Maximum acceleration and amplification Locations

NS (gals)

EW (gals)

UD (gals)

Surface (+10.0 m) 250 m level 350 m level West shaft (310.5 m) Ventilator shaft (380 m)

8.0 8.72 10.4 7.79 0.87

5.8 10.04 8.710. 5.410. 0.10.4

2.0 3.8 3.45 2.74 0.94

Source: Kik-NET and K-NET data.

compares the maximum acceleration at each strong motion station installed at various depths. The ground motions are amplified toward the ground surface. The data even in the same level is scattered, which may imply some local effects such as the geological conditions and the geometry of the opening where the devices are installed. The National Research Institute for Earth Science and Disaster Prevention of Japan (NIED) has been operating the Kik-Net and K-Net strong motions networks. There is a strong motion station of the Kik-Net in Horonobe town that recorded the accelerations at the ground surface and at a depth of 100 m from the ground surface (-70 m). Figure 10.62 shows the acceleration records taken at the ground surface and at the base (100 m below the

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Figure 10.62 Acceleration records taken at the ground surface and at the base

ground). Table 10.7 gives the maximum ground accelerations and their amplifications. Theoretically, the amplification is expected to be greater than 2 for elastic ground (Nasu, 1931). The comparison indicates that the amplification is more than 3 times. Compared to data from the Kik-NET, the measurements at the Horonobe URL are somewhat scattered.

356  Earthquake science and engineering Table 10.7  Maximum acceleration and amplification  

NS (gals)

EW (gals)

UD (gals)

Surface (+30 m) Base (−70 m) Amplification

3.4 10.7 3.6

2.5 11.8 4.72

1.3 4.2 3.23

10.10.4  Fourier and acceleration response spectra analyses The Fourier and accelerations response spectra analyses have been carried out for each strong motion stations. We report some of them here. (a)  Fourier spectra analyses The Fourier spectra analyses of acceleration records measured by the Kik-NET (RMIH01) at the ground surface and base are shown in Figure 10.63. As noted from the figure, the dominant frequency ranges between 4 Hz and 8 Hz, and the Fourier Spectra characteristics do not change with depth, although the amplitude of the ground surface is at least 3 times that at the base. Figure 10.64 shows the FFT of records taken at the ground surface and at a depth of 380 m at the shaft bottom in Horonobe URL. The FFT amplitude of the shaft bottom records are

Figure 10.63 FFT of records at the ground surface and base

Figure 10.64 FFT of records at the ground surface and base

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almost the same as that of the ground surface. The frequency characteristics are also quite similar, resembling those of the Kik-Net records. Figure 10.65 shows the FFT spectra of the record taken at No.9 observation station during the 2018 Iburi earthquake. Except for the UD component, the other components, except amplitude, are quite similar to those of the Soya region earthquake shown in Figure 10.64. The normalized amplitude may be useful for comparison purposes. (b)  Acceleration response spectra analyses A series of acceleration response analyses are carried out. Figure 10.66 shows the acceleration response spectra for RHIM01 and Horonobe URL No. 1 strong ground motion stations. The amplitude and frequency characteristics are somewhat different. The ground conditions at the RHIM01 may be softer than those at the Horonobe URL site.

Figure 10.65 FFT of records at the ground surface for the records due to 2018 Iburi earthquake of 6 September 2018

Figure 10.66 Comparison of acceleration response spectra for RHIM01 and Horonobe URL site

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10.10.5  Modal analyses A series of 3-D finite element modal analyses were carried out for four conditions: no shafts, single shaft, double shafts and triple shafts. The software used was 3-D MIDAS-FEA. Table 10.8 gives the material properties used in numerical analyses, while Table 10.9 compares the eigenvalues for four different conditions; Figure 10.67 shows the displacement response for Mode 1. Table 10.8  Material properties Material

UW (kN m−3)

E (GPa)

Poisson’s ratio

Rock mass Concrete

26.5 23.5

0.600 11.042

0.37 0.20

Table 10.9  Eigenvalues for Mode 1  

No Shaft

Single

Double

Triple

Mode 1(s) Mode 2(s) Mode 3(s)

1.763 1.645 1.564

1.752 1.635 1.554

1.203 1.889 1.117

1.199 1.172 1.111

Figure 10.67 Displacement response for Mode 1

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10.11 Global positioning method for earthquake prediction As stated previously, if the stress state and the yielding characteristics of the earth’s crust are known at a given time, one may be able to predict earthquakes with the help of some mechanical, numerical and instrumental tools. The GPS method may be used to monitor the deformation of the Earth’s crust continuously with time. From these measurements, one may compute the strain rates and probably the stress rates. The stress rates derived from the GPS displacement rates can be effectively used to locate the areas with high seismic risk, as proposed by Aydan et al. (2000b). Thus, daily variations of derived strain–stress rates from dense continuously operating GPS networks in Japan and the United States may provide high-quality data to understand the behavior of the Earth’s crust preceding earthquakes. 10.11.1  Theoretical background First we describe a brief outline of the GPS method proposed by Aydan (2000b, 2004, 2008). The crustal strain rate components can be related to the displacement rates at an observation point (x, y, z) through the geometrical relations (i.e. Eringen, 1980) as given here: ε xx =

∂u ∂w ∂u ∂v ∂w ∂v ∂u ∂w ∂v + (10.53) ; ε yy = , ε zz = , γ xy = + , γ yz = + , γ zx = ∂x ∂x ∂z ∂y ∂y ∂x ∂y ∂y ∂z

 v and w are displacement rates in the direction of x, y and z, respectively. ε xx, ε yy and where u, ε zz are strain rates normal to the x, y and z planes and γ xy, γ yz , γ zx are engineering shear strain rates. The GPS measurements can only provide the displacement rates on the Earth’s surface (x (EW) and y (NS) directions), and it does not give any information on displacement rates in the z-direction (radial direction). Therefore, it is impossible to compute normal and shear strain rate components in the vertical (radial) direction near the Earth’s surface. The strain rate components in the plane tangential to the Earth’s surface would be ε xx, ε yy and γ xy . Additional strain rate components γ yz and γ zx, which would be interpreted as tilting strain rate in this article, are defined by neglecting some components in order to make the utilization of the third component of displacement rates measured by GPS as follows: γ zx =

∂w ∂w , γ zy = (10.54) ∂x ∂y

Let us assume that the GPS stations are rearranged so that a mesh is constituted similar to the ones used in the finite element method. It is possible to use different elements as illustrated in Figure 10.68. Using the interpolation technique used in the finite element method, the displacement in a typical element may be given in the following form for any chosen order of interpolation function:

{u } = [ N ]{U } (10.55) where {u } , [ N ] and {U } are the displacement rate vector of a given point in the element, shape function and nodal displacement vector, respectively. The order of shape function [N] can be chosen depending upon the density of observation points. The use of linear interpolation functions has been already presented elsewhere (Aydan, 2000b, 2003b). From Equations (10.53),

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Figure 10.68 Finite elements for GPS method

(10.54) and (10.55), one can easily show that the following relation holds among the components of the strain rate tensor of a given element and displacement rates at nodal points:

{ε } = [ B ]{U } (10.56) Using the strain rate tensor determined from the Equation 10.56, the stress rate tensor can be computed with the use of a constitutive law such as Hooke’s law for elastic materials, Newton’s law for viscous materials or Kelvin’s law for visco-elastic materials (Aydan and Nawrocki, 1998). For simplicity, Hooke’s law is chosen and is written in the following form: σ xx  λ + 2µ 0   ε xx  λ      σ  =  λ λ + 2µ 0   ε yy  (10.57) yy      σ xy   0 0 µ γ xy  where λ and µ are Lamé’s constants, which are generally assumed to be λ = µ =30 GPa (Fowler, 1990). It should be noted that the stress and strain rates in Equation (10.57) are for the plane tangential to the Earth’s surface. From the computed strain rate and stress rates, principal strain and stress rates and their orientations may be easily computed as an eigenvalue problem. To identify the locations of earthquakes, one has to compare the stress state in the Earth’s crust at a given time with the yield criterion of the crust. The stress state is the sum of the stress at the start of GPS measurement and the increment from GPS-derived stress rate given as: t

{σ } = {σ }0 + ∫ {σ } dt (10.58) T0

If the previous stress {σ}0 is not known, a comparison for the identification of the location of the earthquake cannot be made. The previous stress state of the Earth’s crust is generally

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unknown. Therefore, Aydan et al. (2000a) proposed the use of maximum shear stress rate, mean stress rate and disturbing stress for identifying the potential locations of earthquakes. The maximum shear stress rate, mean stress rate and disturbing stress rate are defined here: τ max =

σ + σ 3 σ 1 − σ 3 , σ m = 1 , τ d = τ max + βσ m(10.59) 2 2

where β may be regarded as a friction coefficient. It should be noted that one (vertical) of the principal stress rates is neglected in the preceding equation since it cannot be determined from GPS measurements. The definition of disturbing stress rate is analogous to the wellknown Mohr-Coulomb yield criterion in geomechanics and geoengineering. The concentration locations of these quantities may be interpreted as the likely locations of the earthquakes as they imply the increase in disturbing stress (Figure 10.69). If the mean stress has a tensile character and its value increases, it simply implies the reduction of resistance of the crust. Figure 10.70 shows the flowchart for the implementation of the procedure just described. The computation programs are written in FORTRAN and True BASIC programming codes.

Figure 10.69 Illustration of stress rates in the space of mean and shear stresses

Figure 10.70 Flowchart of computation codes

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The computed results can be visualized through the embedded graphical programs in the codes or other visualization software. 10.11.2 Applications The earthquake prediction involves three fundamental parameters: location, time and magnitude. If any of these parameters cannot be predicted, the prediction cannot be claimed to be true. In this section, the applications of the GPS technique to some specific countries are described in this section in view of these three fundamental parameters of earthquake prediction. The earthquakes plotted in this section are from the catalogue of the U.S. Geological Survey (USGS). 10.11.2.1  Prediction of earthquake epicenters (A) TURKEY

As Aydan (2000b) have shown previously, the recent earthquakes in Turkey fall within the maximum shear stress concentration regions. Similarly, close correlations exist between mean stress rate and disturbing stress rate concentrations and epicenters of the earthquakes. Therefore, the concentrations of maximum shear stress rate and disturbing stress rate may serve as indicators for identifying the location of potential earthquakes. The high mean stress rate of tensile character may also be used to identify likely earthquakes due to normal faults (Aydan, 2000b). Figure 10.71 shows the contours of the disturbing stress rate, together with the epicenters of the earthquakes with a magnitude greater than 4 occurring during 1995 and 1999 using the GPS data reported by Reilinger et al. (1997). In particular, the epicenters of 1999 Kocaeli, 1999 Düzce-Bolu, 2000 Orta-Çankırı and 2000 Honaz-Denizli earthquakes coincide with the regions of concentration of these stress rates. Therefore, the GPS method implies that it is possible to locate the earthquakes. (B) TAIWAN

Yu et al. (1997) reported the annual crustal displacement rate (velocity) as shown in Figure 10.72(a). Figure 10.72(b) shows the disturbing stress rate contours with earthquakes greater than 6 until 2000. As noted from the figure, earthquake activity is very high in areas where stress concentrations occur. The M7.4 1999 Chi-Chi earthquake occurred in one such area, as indicated in Figure 10.72(b). (C) JAPAN

Japan has the most extensive network of continuous GPS, called GEONET. An evaluation of GPS measurements by this network for the 2003 is shown in Figure 10.73. As noted from this figure, high stress concentrations occur along the eastern shore of Japan compared to those along the west shore. The seismic activities along the east coasts of Hokkaido island in 2003 (M8.3 Tokachi earthquake), Honshu island in 2004 and 2011 (M7.6 Tokaido-oki earthquake, M9.0 East Japan Mega Earthquake), the Suruga Bay earthquake in 2008 in the area of the anticipated Tokai earthquake coincide with the largest concentrations of disturbing and maximum shear stress rates.

Figure 10.71 Computed disturbing stress rate and earthquakes

Figure 10.72 Displacement rate vectors and disturbing stress rate contours with earthquakes

Figure 10.73 Comparison of disturbing stress contours of 2003 with earthquake activity

Earthquake science and engineering  365 (D) INDONESIA

Indonesia has suffered many large earthquakes along Sumatra island and Java island since 2004. Figure 10.74(a) shows the seismic activity and locations of major earthquakes. Figure 10.74(b) shows the contours of disturbing stress rates obtained from GPS stations in the region bounded by latitudes 15S–15N and longitudes 90E–140E (Aydan, 2008). As noted from the figures, stress rate concentrations are clearly observed in the Moluccas area (Banda

Figure 10.74 Distribution of contours of the disturbing stress rate and seismic activity

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Sea). Concentrations in the vicinity of Sunda strait and west of Sumatra island are worth noticing. However, it should be noted that the GPS stations in the west of Sumatra island are sparse. Therefore it is expected that the actual concentrations may be larger than those seen in Figure 10.74(b). Nevertheless, it is of great interest that the stress rate concentrations are closely associated with the regional seismicity. 10.11.3  Prediction of time of occurrence Aydan (2003c, 2004) also showed that the time of occurrence of earthquakes in terms of weeks may be possible using the GPS measurements recorded during the 2003 MiyagiHokubu earthquake (Figures 10.75 and 10.76). Parameter MRI shown in Figure 10.76 is defined as: MRI =

M ×100 (10.60) R

where M and R are the magnitude and hypocenter distance of an earthquake, respectively. The MRI is a measure of the effect of earthquakes at a given point. As noted from Figure 10.76, the stress rate components of the Yamoto-Rifu-Oshika element indicated that remarkable stress variations started in October 2002. However, the strain rate components of the elements of Yamoto-Oshika-Onagawa, Yamoto-Onagawa-Wakuya and YamotoWakuya-Miyagi-Taiwa started to change remarkably at the beginning of May 2003 about one month before the M7.0 Kinkazan earthquake that occurred on 26 May 2003. The

Figure 10.75 GPS stations and configuration of GPS mesh Source: From Aydan (2004)

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Figure 10.76 Time series of disturbing stress rates of GPS elements Source: From Aydan (2004)

high rate of variations continued after the M7.0 earthquake and resulted in the 26 July 2003 Miyagi-Hokubu earthquakes. Variations before the earthquake resembles those observed in creep tests. As the variations of disturbing stress rates were greater than those of the mean and maximum shear stress rates, Aydan (2004) concluded that the disturbing stress rate may be a good indicator of regional stress variations and precursors of following earthquakes. Therefore, the time of the earthquake may be obtained from the GPS measurements. 10.11.4  Prediction of magnitude The prediction of the magnitude of the earthquake is still difficult. Nevertheless, the area of stress rate concentrations with a chosen value may be used to determine the magnitude. As a result, the fundamental parameters of the earthquake prediction, i.e. location, time and magnitude, may be determined from the evaluation of GPS measurements. However, there are still some technical problems associated with GPS observations and artificial disturbances as pointed out by Aydan (2000c,2004).

10.12 Application to Multi-Parameter Monitoring System (MPMS) to earthquakes in Denizli basin Some experience has been gained during the past earthquakes in Denizli and its close regions, as well as in the Aydın-Germencik region. 10.12.1  June 2003 Buldan (Denizli) earthquakes There was earthquake activity in Buldan and its surroundings in Denizli on 23 July 2003 (Figure 10.77). An earthquake with a magnitude of 5.2 occurred at 07:56 a.m. on 23 July

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Figure 10.77 Acoustic emission (AE) count variations at Tekkehamam station associated with July 2003 Buldan earthquakes

2003. There were aftershocks with magnitudes of 4.1 on the following days. On 26 July 2003, another earthquake with a magnitude of 5.10 happened near Buldan at 11:21 a.m. local time. This earthquake caused no loss of life but caused some damage to kerpiç (made of soil) and stonewall houses. The Tekkehamam Puf-Puf count is defined as the acoustic emission count (AE) in a unit time of gas pressure of thermal spring mud bubbles. There is an important increase of the Puf-Puf numbers 2 days before the M5.2 earthquake in Buldan. The activity of the Puf-Puf count continued as the earthquake activity occurred. After 3 days, an earthquake hit with a magnitude of 5.6. When the aftershocks eased to less than magnitude 4.0, the Puf-Puf count decreased to low levels (Figure 10.77). This shows that there is a relation between the earthquake activity of the region and Puf-Puf count changes. The electric potential data variations of Honaz, Tekkehamam and Çukurbağ stations look like the change of tidal wave height. During the measurements, some unwanted artificial sudden voltage changes are due to human touch. At Çukurbağ station, there are electric potential changes, which are thought to be related to the M5.2 and M5.10. earthquakes (Figure 10.78). At Tekkehamam station, there are sudden changes. The changes in the east-west direction of Honaz station and the north-south direction at Çukurbağ station started at the same time 2 days before the magnitude 5.2. When the magnitude 5.6 earthquake occurred, the Honaz station’s north-south direction value was down to minimum voltage value, and then the electric potential started to increase (Figure 10.78). This type of changes was also observed by 2003) in the laboratory uniaxial tests. The changes at the Honaz and the Çukurbağ stations point out the possible changes of the regional stress.

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Figure 10.78 Electric potential variations at Honaz, Çukurbağ and Tekkehamam stations associated with July 2003 Buldan earthquakes

10.12.2  Sarayköy (Denizli) earthquake on 20 August 2005 There was an earthquake activity in Denizli basin with magnitudes between 2.0 and 3.8. There is a remarkable increase of AE counts from 10.0 to 700 at Tekkehamam station between 7 July and 27 July 2005. In this period of time, the magnitudes of earthquake increased from 3.0 to 3.9. AE counts went down to 25 after the earthquake activity stopped on 28 July 2005. AE counts started to climb and reached 420 on 21 August 2005. After the increase of AE counts, secondary earthquake activity in the Denizli basin started, and magnitudes between 2.2 and 3.9 earthquakes occurred (Figure 10.79). Electric potential measurements at Tekkehamam station show that there is a remarkable increase in the east-west direction measurements. It increases from a negative value to positive with increasing earthquake activity. When earthquake activity increased to the maximum magnitude of 3.9, the east-west direction electric potential value also reached its maximum value (Figure 10.80). There is an important decrease in the temperature of the thermal spring at Karahayıt Bibiana station. Temperature decreased from 49oC to 46oC before the earthquake activity started. The temperature of Karahayıt station dropped to 29.5 oC at 23:10 on 19 August 2005. After two days, the main shock of 3.9 occurred (Figure 10.81). 10.12.3  Denizli earthquake on 5 June 2006 There was other earthquake activity in the Denizli region between May and July 2006. Although the earthquake activity was weak with magnitudes between 2.3 and 3.5, the AE counts were between 25 and 200. There was a sudden increase of AE count from 80 to 700

Figure 10.79 Relation between acoustic emission counts and earthquake activity near Tekkehamam Puf-Puf station in Denizli basin

Figure 10.80 Relationship between electric potential changes and earthquake activity at Tekkehamam station between July and September 2005

Figure 10.81 Change of thermal water temperatures of Karahayıt and Yenice stations and relations between the earthquake activity in Denizli basin

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on 30 May 2006. After this increase, earthquake activity in the Denizli basin increased, and on 5 June 2006 at 04:20, an earthquake with a magnitude of 4.1 occurred (Figures 10.81 and 10.82). There were some changes of electric potential values of the Tekkehamam (VR) station. There was a remarkable increase of the north-south and east-west directions’ electric potential measurements before the earthquake activity in the Denizli region. When electric potential values reached a maximum value, earthquake activity in the Denizli basin increased, and a 4.1 magnitude main shock occurred on 5 June 2009 (Figure 10.83).

Figure 10.82 Relationship between AE count of Tekkehamam and earthquake activity in Denizli basin between May and July 2006

Figure 10.83 Relationship between electric potential of Tekkehamam station and earthquake activity in Denizli basin between May and July 2006

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10.12.4  Denizli earthquake on 27 December 2008 A seismic activity in Denizli basin was observed between November 2008 and February 2009. The peak value of the thermal spring temperature increased from 79 oC to 82oC. When the maximum value of thermal spring was reached, seismic activity in the Denizli region also increased, and a 3.9 magnitude earthquake occurred (Figure 10.84). Electric potential value changes in the north-south direction also indicate that there is a sudden increase of electric potential values from 0.03 mV to 0.16 mV before the earthquake activity increased in the Denizli region (Figure 10.85). Bubbling count values of the thermal spring at Tekkehamam station show that, Puf-Puf count values (with 2 min interval) increased to 140, 4 days before the intensive seismic activity. After that, the Puf-Puf count values dropped to 110 and then climbed to 130. Within this period, earthquake magnitudes increased to 3.9 and the number of earthquakes also increased (Figure 10.86).

Figure 10.84 Relation between earthquake activity and temperature change of thermal spring at Tekkemamam station

Figure 10.85 Relation between earthquake activity and electric potential change at Tekkemamam station

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Figure 10.86 Relation between earthquake activity and AE numbers of thermal spring at Tekkemamam station

Figure 10.87 Relation between earthquake activity and temperature change of thermal spring at Germencik station

10.12.5  Germencik earthquakes on 27 December 2008 Three parameters, namely, electric potential AE counts and temperature, are measured at Germencik (Aydın) geothermal field: However, the AE station did not work properly due to device error. The temperature change of the thermal spring shows that the increasing values of the spring are associated with the increasing seismic activity of the region. When the temperature increased to its maximum value of 59oC, the magnitude of the earthquake reached 4.0, and the number of earthquakes also increased (Figure 10.87). The electric potential

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Figure 10.88 Relation between earthquake activity and electric potential change at Germencik station

values of the east-west and north-south directions at Germencik station show that there is a sudden increase from zero to 0.24 mV after the decrease to 0.05 mV. Later an intensive earthquake activity started, and the magnitude of earthquakes reached 4.0. Earthquakes with higher magnitudes occurred after the decrease of electric potential values (Figure 10.88).

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Index

action 56, 86, 258; arch 56, 258; chemical 86 airflow 88, 262 analytical solution 68, 100, 254, 344 atmospheric 2, 229–230, 276, 283 axisymmetric 23, 26, 53, 63, 265 bar 20, 179, 222–226 Barla, G. 376 Barton, N. 71, 76, 78–81, 88, 90, 100–101 behavior 1–2, 21, 23, 28, 30, 39, 41, 47–48, 51, 71, 94, 103, 105, 124, 129, 132, 146, 230, 283, 296, 342, 359; elastic 39; elasto-viscoplastic 86; linear 47; long-term 86; material 48, 51; non-linear 132; plastic 342; slip 124; squeezing 94 Bieniawski, Z.T. 69, 71, 75, 88, 90–91, 93, 98–101, 106, 153 blasting 1, 50–51, 75, 80, 155–168, 170, 172–181, 183–184, 186, 188–191, 193–196, 198–203, 218, 220–221, 227, 277, 286, 299 body force 315–317 borehole 84, 94, 181, 240, 261–265, 267, 271, 272, 288 Brown, E.T. 49, 90–93, 96, 99–101 cavern 51, 69, 277 cavity 51, 69, 277; circular 51, 69; underground 277 chemical reaction 230 concrete lining 53, 65, 88, 349–350 condition 1, 5, 6, 13, 18, 42, 51, 55, 197, 265–266, 304; boundary 1, 5, 6, 13, 42, 55, 265–266; initial 18, 51, 197, 304 Cook, N.G.W. 47, 69, 303, 377 corrosion 68, 100, 222–223, 226 creep 254–255, 367; strength 54, 255; test 254–255, 367

dam 37–38, 101, 348; arch 37; foundation 37–38; gravity 37–38, 348; Kurobe 101; Malpasset 37 damping coefficient 191 degradation 2, 54, 82, 84, 229–269, 271, 276, 278–279, 298 discontinuity 21, 37–39, 73, 75, 81–82, 84, 109, 113, 115–117, 131–133, 134, 152, 307; orientation 73, 109; plane 21, 37–39, 131–133, 134, 307; set 81–82, 84, 113, 115–117, 152; spacing 75, 81, 82, 84 displacement transducer 112, 115, 120, 128, 131, 136, 237, 305, 343 drilling 50, 84, 176, 178, 198 earthquake 68, 131, 301–366; 2003 Buldan 367–369; 2003 Miyagi-Hokubu 68, 366–367, 375; 2005 Kashmir 348; 2008 Wenchuan 131, 333, 348, 350; 2008 Iwate-Miyagi 327, 344; 2009 L’Aquila 343, 376; 2011 Great East Japan 331, 332, 341, 346–347 effect 20, 99; arch formation 99; dowel 20 Egger, P. 103, 110, 114, 153 electrical resistivity 237, 238, 268, 287 equation of motion 17, 263, 313, 315–316 Eringen, A.C. 51, 69, 309, 359, 376 fault 336–337, 339, 343–352, 362, 374–378 forepole 68 foundation 23–24, 37–38; bridge 23–24; dam 37–38; design 37 friction angle 9, 14, 17, 21–22, 39, 65–66, 73, 76, 78, 81, 89, 91, 93, 95, 97, 103, 105, 114, 120–121, 128, 135–136, 304, 307, 311, 312, 318 geoengineering 61, 92, 243, 250, 267–268, 361 Goodier, J.N. 26, 43, 45

382 Index Hoek, E. 49, 89–93, 96, 99, 101, 155 Ikeda, K. 26, 45, 186, 201, 334, 377 impact wave 223 Jaeger, J.C. 47, 69, 303, 377 Kumsar, H. 8, 19, 45, 68, 128, 152–153, 188, 201, 259, 267, 321 Ladanyi, B. 92, 101 law 47, 262, 360; Fick 62; Hooke 7, 360; Kelvin 360; Newton 360 lining 53, 65, 88–89, 198, 349–350 measurement 28, 170, 185, 187, 203–207, 206, 208, 209, 211, 213, 215–217, 221–227, 233; ground stress 28; ın-situ 215; ın-situ stress 185; vibration 203–207, 209, 211, 213, 215–217, 221–227; wave velocity 170, 187, 204, 206, 208, 211, 233 method 5, 7, 10, 13–14, 19, 21, 26, 35–39, 45, 53, 57, 65, 69, 334, 336, 359, 378; discrete finite element 13–14, 35–38, 45, 65, 69; finite difference 334, 336; finite element 5, 7, 13–14, 26, 35–38, 45, 53, 57, 65, 69, 334, 336, 359, 378; limiting equilibrium 10, 19, 21; secant 39 mine 58, 60–61, 68–69, 211, 278–279, 284–285; abandoned 61, 68, 211, 278–279, 284–285; room and pillar 58, 60, 69 Mitake 58–59, 61, 211, 239, 284–285, 301–302 modulus 23, 26, 42, 44, 56–57, 66, 81, 84–85, 89–90, 92–94, 98, 101–102, 233; deformation 81, 84–85, 89–90, 93–94, 98, 101–102, 233; elastic 23, 26, 42, 44, 56–57, 66, 90, 92; shear 42 mudstone 161, 169–170, 211, 246, 351 nondestructive testing 221, 233 nuclear waste disposal 261 numerical analysis 59, 63, 68, 100, 226, 311 overburden 57, 211, 255–256, 263 Oya tuff 236, 237, 241 Poisson’s ratio 5, 7, 26, 44, 56–57, 66, 81, 91–94, 358 prismatic block 145–147

rock 98, 100, 169, 261–262, 267, 278; igneous 98; sedimentary 100, 169, 261–262, 267, 278 rockanchor 222, 226 rockbolt 45, 68, 99–100, 267 rock classifications 26, 71, 75, 77, 81–82, 84–86, 88–100, 201, 268; DENKEN (CRIEP) 26, 86, 89; NEXCO 71; Q-value system 76, 89, 98; rock mass quality rating (RMQR) 71, 81–82, 84–86; rock mass rating (RMR) 71, 75; rock quality designation (RQD) 71, 76, 77, 88–100, 201, 268 rock reinforcement 45, 99, 152, 227 rock support 45, 81, 86, 152, 227 Ryukyu limestone 69, 94, 96, 101–102, 120, 134, 136, 142, 145, 147–149, 151, 213, 227, 250, 269, 289, 293, 301, 307 sandstone 149–152, 157, 161, 169–170, 176, 211, 237–239, 244, 250, 251, 261–268, 301–302 shape function 359 shear 42–44, 66, 96, 107, 260, 310, 312, 314, 359, 361–362, 367; modulus 42; strain 47, 314, 359; stress 43–44, 66, 96, 107, 260, 310, 312, 361–362, 367 sliding plane 128 softening 76, 78 strength 5, 7–8, 26, 30, 35, 39, 50, 71, 81, 89, 91, 93–95, 97–99, 102–103, 233, 254, 257–260, 264, 269; long-term 254, 259, 260; tensile 5, 7–8, 26, 30, 35, 39, 50, 81, 93–95, 102–103, 233, 254, 257–258, 260, 264, 269; uniaxial compressive 71, 89, 91, 93–95, 97–99, 101, 103, 186–187, 233, 240–241 Terzaghi, K. 71, 101 test 21, 96, 113, 120, 121, 128, 240, 305, 307, 376; compression 240; direct shear 96, 113; stick-slip 305, 376; tilting 21, 120, 121, 128, 307 timoshenko 26, 43, 45 toppling 37, 45, 100, 109–110, 117, 131, 152, 259, 267; columnar 109; failure 37, 100, 109–110, 117, 152; flexural 45, 100, 110, 117, 131, 152, 259, 267 tunnel 28, 50, 52, 53, 180, 187, 194–195, 201, 227; anchorage 28; Bolu 348, 349; circular 50, 52, 53; Tarutoge 180, 187, 194–195, 201, 227

Index 383 Ulusay, R. 9–11, 17–19, 45–46, 89, 92, 100, 102, 153, 188, 201, 227, 231, 234, 241, 243, 246–247, 254–260, 267–269, 299–300, 319, 322, 375, 378 Vardar, M. 92, 102

yield criterion 17–18, 26, 49, 90, 96, 311, 360–361; Aydan 49, 90; Bingham type 17–18; Drucker-Prager 49, 311; Hoek-Brown 49, 90, 96, 101; Mohr-Coulomb 12, 49, 90, 96, 311, 360–361 yield zone 184, 266