Practice of Constitutive Modelling for Saturated Soils [1st ed.] 9789811563065, 9789811563072

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Zhen-Yu Yin Pierre-Yves Hicher Yin-Fu Jin

Practice of Constitutive Modelling for Saturated Soils

Practice of Constitutive Modelling for Saturated Soils

Zhen-Yu Yin Pierre-Yves Hicher Yin-Fu Jin •

•

Practice of Constitutive Modelling for Saturated Soils

123

Zhen-Yu Yin Department of Civil and Environmental Engineering The Hong Kong Polytechnic University Hong Kong, China

Pierre-Yves Hicher Research Institute in Civil and Mechanical Engineering UMR CNRS GeM Ecole Centrale de Nantes Nantes, France

Yin-Fu Jin Department of Civil and Environmental Engineering The Hong Kong Polytechnic University Hong Kong, China

ISBN 978-981-15-6306-5 ISBN 978-981-15-6307-2 https://doi.org/10.1007/978-981-15-6307-2

(eBook)

Jointly published with Tongji University Press The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Tongji University Press. © Springer Nature Singapore Pte Ltd. and Tongji University Press 2020 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

Constitutive models are fundamental training for graduate students in geotechnical engineering. For a long time, however, these models have lacked a simple and easy-to-understand platform for learning, training and practice between the theory and practice of soil mechanics and geotechnical engineering. Many engineers and practitioners, and even some researchers, have reservations about this topic. There is a mismatch between research and application. Especially in recent years, constitutive models aiming at refined simulations have been continuously developed, with increasing complexity and parameters. In addition, difficulties in parameter calibration have given rise to barriers to their use in geotechnical engineering. Therefore, the establishment of a simple, easy-to-understand, easy-to-use learning, training and practice platform can serve as a bridge between theory and practice. This book, with its supporting platform software, is a welcome attempt to make constitutive models more accessible to both researchers and practitioners alike. I came to know Dr. Yin during the Mid-Autumn Festival in 2007. I participated in the annual European workshop “Alerts of Geomaterials” held at Aussois in France. At that time, Dr. Yin attended the conference as a postdoctoral researcher from University of Strathclyde. Thus, began the academic discussion and cooperation between us. Dr. Yin’s rigorous approach, deep understanding and solid numerical and analytical skills have left deep impressions on me. He has been leading a dynamic research team and has published many well cited papers. Dr. Yin also had five years of experience as an engineer before returning to academia, which prompted him to set up the platform. Professor Hicher, the second author of the book, supervised Dr. Yin’s Ph.D. thesis. Professor Hicher is a renowned expert in the study of the constitutive relations of soil. He is one of the outstanding representatives of the French School of Soil Mechanics and Geotechnical Engineering. I met with Dr. Jin, the third author of the book in the summer of 2017, when we all visited Tongji University. Dr. Jin is quite experienced in optimization-based model selection and parameter identification. Later, in the spring of 2019, Dr. Yin invited me to visit Hong Kong Polytechnic University. During that period, I had in-depth discussions with Dr. Jin

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on the application of artificial intelligence in geotechnical engineering. I am sure that this bright young man will succeed in his research career. This book begins with the theoretical fundamentals and combines geotechnical experiments to show how advanced theory can be applied in practice. This book enables readers to quickly and accurately acquire the feeling for soil properties and understand the parameter meanings and their influences on complex constitutive responses. In this way, readers can easily obtain an in-depth understanding of the deformation law and failure mechanism of soil. The main materials in the book are the accumulated knowledge and research results of authors over many years. Moreover, the authors are dedicated to the relevant software platform, which can be directly used by readers for analysis and training.

August 2019

Wei Wu Professor in Geotechnical Engineering Institut für Geotechnik Universität für Bodenkultur Vienna, Austria

Preface

With continuous development worldwide, an increasing number of large-scale infrastructure constructions involving many geotechnical engineering problems have been carried out. However, for a long time, the application of soil mechanics and geotechnical theory in engineering practice has lagged far behind the development of the theory itself. On the one hand, theoretical innovation is too complicated compared with engineering practice, and on the other hand, from students to engineers, the field is lacking a good learning, training and practice platform. We feel that advanced theories cannot be applied to the existing engineering design and construction, which makes it difficult to break through the predicaments of engineering problems and safety hazards caused by the existing conventional theoretical applications, which hinders the innovation of geotechnical engineering. This problem is a great research waste of resources. Therefore, a book that can connect theoretical innovation and engineering practice, as well as facilitate engineering reference learning is necessary. This book begins with the most basic theoretical knowledge and combines geotechnical experiments to realize how advanced theory can be applied in practice, which can help readers quickly and accurately understand the constitutive simulation of soil characteristics. In addition, this book provides source codes and a free software platform that can be directly used by readers for analysis and training. The soil constitutive theory, numerical algorithms, experimental simulations, software platforms, and so on in this book are the results of the authors’ works. At the same time, we also hope to enrich the developed platform in the future by sharing the

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latest scientific research results in the geotechnical field with readers by and promoting the application of scientific research results to achieve the purpose of geotechnical engineering innovation.

Hong Kong, China August 2019

Zhen-Yu Yin

Pierre-Yves Hicher

Yin-Fu Jin

Acknowledgments

In the process of writing this book, we received much sincere guidance and help from experts and colleagues. Here, we would like to express our sincere gratitude to Ms. Jiayuan Li, Prof. Wei Wu from Universität für Bodenkultur, Prof. Jian-Hua Yin from the Hong Kong Polytechnic University, Prof. Yang-Ping Yao from Beihang University, Prof. Shuilong Shen from Shantou University, and Prof. Hongwei Huang and Prof. Dongmei Zhang from Tongji University for their careful guidance and help during the book-writing process and also thank Dr. Hao Xiong from Shenzhen University, Dr. Junxiu Liu from Anhui Jianzhu University, Dr. Jian Li from Beijing Jiaotong University, Shuai Cao from University of Science and Technology Beijing, Dr. Chao Zhou, Dr. Wenbo Chen, Dr. Weiqiang Feng, Dr. Daoyuan Tan, Dr. Jie Yang, Dr. Pei Wang and Dr. Hanlin Wang from the Hong Kong Polytechnic University for their hard work in the process of organizing and reviewing the book. We are also grateful to Tongji University Press and Springer for performing detailed work on the publication of this book. Some of the results and publications in this book have been funded by the National Natural Science Foundation of China (41372285, 51579179) and the Research Grants Council (RGC) of Hong Kong Special Administrative Region Government (HKSARG) of China (Grant No.: 15209119, 15217220, R5037-18F), and for this funding, we would like to express our heartfelt thanks.

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Abstract

This book presents the practice of constitutive modeling for saturated soils through the development of a modeling platform for soil testing, which is one of the key components in soil mechanics and geotechnical engineering. The book discusses the fundamentals of the constitutive modeling of soils and illustrates the application of these models to simulate various laboratory tests. To aid readers in easily understanding the fundamentals and modeling of soil behaviors, this book first introduces the general stress-strain relationship of soils, then the principles and modeling approaches of various laboratory tests, followed by the ideas and formulations of the constitutive models of soils. Moving on to the application of constitutive models, this book presents the modeling platform with a practical and simple interface. The platform contains various kinds of tests and constitutive models covering clay to sand and is used for simulating most kinds of laboratory tests. The book is intended for undergraduate and graduate-level teaching in soil mechanics and geotechnical engineering, as well as in other related engineering specialties. This book is also of use to industry practitioners due to the inclusion of real-world applications, opening the door to advanced courses on modeling within the industrial engineering and operations research fields.

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Fundamentals of Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Hypothesis of a Continuous Medium for Soils . . . . . . . . . . . . . 2.2 Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Constitutive Relations of Saturated Soils: An Overview . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Compression Behavior . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Shear Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Stress–Dilatancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Small-Strain Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Induced Anisotropy and Noncoaxial Properties . . . . . . . 1.7 Time Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Temperature Dependency . . . . . . . . . . . . . . . . . . . . . . 1.9 Water Chemical Composition Effect . . . . . . . . . . . . . . . 1.10 Cyclic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Grading Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.1 Grain Breakage Effect . . . . . . . . . . . . . . . . . . . 1.11.2 Suffusion Effect . . . . . . . . . . . . . . . . . . . . . . . 1.12 Additional Mechanical Properties of Natural Soft Soils . 1.12.1 Inherent Anisotropy . . . . . . . . . . . . . . . . . . . . 1.12.2 Soil Structure and Destructuration . . . . . . . . . . 1.13 Current Difficulties in the Practice of Constitutive Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Development of the ErosLab Modeling Platform in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14.1 Introduction of the Constitutive Modeling Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14.2 Installation and Operating Environment . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2.1 2.2.2 2.2.3

Stress State of a Point in Soil . . . . . . . . . . . Mean Stress and Deviatoric Stress Tensor . . Principal Stress and Invariants of the Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . Invariants of the Deviatoric Stress Tensor . . Principal Stress Space and p-Plane . . . . . . . Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . and Strain Analyses in ErosLab . . . . . . . . . . .

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2.2.4 2.2.5 2.3 Strain 2.4 Stress References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction of Laboratory Tests for Soils 3.1 Introduction . . . . . . . . . . . . . . . . . . . 3.2 Oedometer Test . . . . . . . . . . . . . . . . 3.3 Triaxial Shear Test . . . . . . . . . . . . . . 3.4 Direct, Simple, or Ring Shear Test . . 3.5 Biaxial Shear Test . . . . . . . . . . . . . . . 3.6 True Triaxial Test . . . . . . . . . . . . . . . 3.7 Hollow Cylinder Torsional Shear Test 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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Fundamentals of Elastoplastic Theory . . . . . . . . . . . . . . . . . 4.1 Elastic Constitutive Relation . . . . . . . . . . . . . . . . . . . . . 4.1.1 Isotropic Elasticity . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Elasticity Under Undrained Conditions . . . . . . . 4.1.3 Cross-Anisotropic Elasticity . . . . . . . . . . . . . . . 4.2 Elastoplastic Constitutive Relation . . . . . . . . . . . . . . . . . 4.2.1 Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Hardening Rule . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Drücker’s Stability Hypothesis, Convexity, and Orthogonality . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Method for Solving the Plastic Problem . . . . . 4.3.1 General Explicit Solution . . . . . . . . . . . . . . . . . 4.3.2 Cutting Plane Method-Based Implicit Solution . . 4.3.3 Closest Point Projection Method-Based Implicit Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Strength Criteria for Soils . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Generalized Von Mises Criterion . . . . . . . . . . . . 4.4.2 Generalized Tresca Criterion . . . . . . . . . . . . . . . 4.4.3 Mohr–Coulomb Criterion . . . . . . . . . . . . . . . . . 4.4.4 Lade–Duncan Criterion . . . . . . . . . . . . . . . . . . . 4.4.5 Matsuoka–Nakai Criterion . . . . . . . . . . . . . . . .

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Contents

4.4.6 Generalized Nonlinear Strength Criterion . 4.4.7 Modeling Methods . . . . . . . . . . . . . . . . . 4.5 Yield Functions for Soils . . . . . . . . . . . . . . . . . . . 4.6 Potential Function for Soils . . . . . . . . . . . . . . . . . 4.7 Hardening Rule for Soils . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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Elastoplastic Modeling of Soils: From Mohr-Coulomb to SIMSAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Experimental Phenomena of the Shear Behavior of Soils 5.1.1 Shear Characteristics of Normally Consolidated Clay or Loose Sand . . . . . . . . . . . . . . . . . . . . . 5.1.2 Shear Characteristics of Overconsolidated Clay or Dense Sand . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Mohr-Coulomb Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Elastic Part . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Plastic Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Parameters of the Mohr-Coulomb Model . . . . . . 5.3 Nonlinear Mohr–Coulomb Model . . . . . . . . . . . . . . . . . 5.3.1 Enhancement 1: Nonlinear Elasticity . . . . . . . . . 5.3.2 Enhancement 2: Nonlinear Plastic Hardening . . . 5.3.3 Enhancement 3: Nonlinear Stress–Dilatancy . . . . 5.4 Critical State-Based Nonlinear Mohr-Coulomb Model . . . 5.5 Triaxial Test Simulations Using SIMSAND . . . . . . . . . . 5.5.1 Plastic Multiplier Solution . . . . . . . . . . . . . . . . . 5.5.2 Full Strain Control Simulation . . . . . . . . . . . . . . 5.5.3 Conventional Drained Triaxial Test Simulation . 5.5.4 Constant-P’ Triaxial Test Simulation . . . . . . . . . 5.5.5 Parameter Setting and Results Analysis . . . . . . . 5.5.6 Model Parameter Analysis . . . . . . . . . . . . . . . . 5.6 Some Examples of Extension of the Model . . . . . . . . . . 5.6.1 Consideration of Grading Effects . . . . . . . . . . . . 5.6.2 Consideration of the Stress Reversal Technique for Cyclic Behavior . . . . . . . . . . . . . . . . . . . . . 5.6.3 Consideration of Small-Strain Stiffness . . . . . . . 5.6.4 Consideration of Plastic Compression Behavior . 5.7 Models in the ErosLab Platform . . . . . . . . . . . . . . . . . . 5.7.1 Elastic Perfectly Plastic Model—Perfect EP . . . . 5.7.2 Nonlinear Mohr-Coulomb Model - NLMC . . . . 5.7.3 Critical State-Based Simple Sand Model—SIMSAND . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Viscoplastic Modeling of Soft Soils . . . . . . . . . . . . . . . . . . . . . . . 7.1 One-Dimensional Viscoplastic Modeling . . . . . . . . . . . . . . . 7.1.1 Strain Rate Influence . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 1D Model with Strain Rate Dependency . . . . . . . . . 7.1.3 Stress Relaxation and Coherence of Time-Dependent Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Model Extension for Structured Clays . . . . . . . . . . . 7.2 Three-Dimensional Viscoplastic Modeling . . . . . . . . . . . . . . 7.2.1 Perzyna’s Overstress Theory . . . . . . . . . . . . . . . . . . 7.2.2 Scaling Function . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Anisotropy of the Reference Surface . . . . . . . . . . . . 7.2.4 Destructuration of the Reference Surface . . . . . . . . .

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Elastoplastic Modeling of Clayey Soils: From MCC to ASCM 6.1 Experimental Phenomena of Compression . . . . . . . . . . . . 6.1.1 Isotropic Compression Behavior . . . . . . . . . . . . . 6.1.2 Anisotropic Compression Behavior . . . . . . . . . . . 6.1.3 One-Dimensional Compression Behavior . . . . . . . 6.2 Modified Cam-Clay Model . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 General Introduction to MCC . . . . . . . . . . . . . . . 6.2.2 Analyses of Special Stress Paths Under Triaxial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Parameters of the MCC Model . . . . . . . . . . . . . . 6.2.4 Limitations of the MCC Model . . . . . . . . . . . . . . 6.3 Extension from MCC to ASCM . . . . . . . . . . . . . . . . . . . 6.3.1 Enhancement 1: Nonlinear Stress–Strain Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Enhancement 2: Effect of Structure on Plasticity . 6.3.3 Enhancement 3: Effect of Structure on Elasticity . 6.4 Triaxial Test Simulations Using ASCM . . . . . . . . . . . . . . 6.4.1 Plastic Multiplier Solution . . . . . . . . . . . . . . . . . . 6.4.2 Full Strain Control Simulation . . . . . . . . . . . . . . . 6.4.3 Conventional Drained Triaxial Test Simulation . . 6.4.4 Constant-P’ Triaxial Test Simulation . . . . . . . . . . 6.4.5 Pre- and Postprocessing . . . . . . . . . . . . . . . . . . . 6.4.6 Model Parameter Analysis . . . . . . . . . . . . . . . . . 6.5 Consideration of Small-Strain Stiffness . . . . . . . . . . . . . . . 6.6 Consideration of Plastic Strain Accumulation for Cyclic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Some Elastoplastic Models for Clayey Soils in ErosLab . . 6.7.1 Modified Cam-Clay Model—MCC . . . . . . . . . . . 6.7.2 Anisotropic Structured Clay Model—ASCM . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Time Integration and Simulation of Triaxial Tests . . . . . . 7.3.1 General Explicit Algorithm . . . . . . . . . . . . . . . . 7.3.2 Simulation of Tests with Full Strain Control . . . 7.3.3 Conventional Drained Triaxial Test Simulations . 7.3.4 Creep Test Simulation . . . . . . . . . . . . . . . . . . . 7.3.5 Parameter Settings . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . 7.4 ANICREEP Model in ErosLab . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Hypoplastic Modeling of Sand Considering the Time Effect 8.1 Hypoplastic Model for the Frictional Behavior of Sand . 8.1.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Numerical Solution for Special Paths . . . . . . . . 8.2 Hypoplastic Model Considering the Critical State . . . . . 8.3 Hypoplastic Model Considering the Time Effect . . . . . . 8.3.1 Experimental Time-Dependent Behavior of Granular Materials . . . . . . . . . . . . . . . . . . . 8.3.2 Framework of Hypoplasticity with Friction and Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Acceleration-Based Formulations for Viscous Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 New Hypoplastic Model Accounting for Creep 8.3.5 Summary of Model Parameters . . . . . . . . . . . . 8.4 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Constant Strain Rate Tests . . . . . . . . . . . . . . . 8.4.2 Stepwise Strain Rate Tests . . . . . . . . . . . . . . . 8.4.3 Creep Test Under Constant Stress . . . . . . . . . . 8.5 Hypoplastic Model of Sand in ErosLab . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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293 295 296 298 298 299 302 306 308

Multiscale Modeling of Soils . . . . . . . . . . . . . . . . . 9.1 Multiscale Features of Soils . . . . . . . . . . . . . . 9.2 Fundamentals of Micromechanics . . . . . . . . . 9.2.1 Interparticle Contact Laws . . . . . . . . 9.2.2 Strain Tensor . . . . . . . . . . . . . . . . . . 9.2.3 Effective Stress Tensor . . . . . . . . . . . 9.2.4 Fabric Tensor . . . . . . . . . . . . . . . . . . 9.2.5 Averaging and Localization Operator . 9.2.6 Homogenization Integration . . . . . . . 9.3 Micromechanics-Based CH Model . . . . . . . . . 9.3.1 Interparticle Contact Law-Elasticity . . 9.3.2 Interparticle Contact Law-Plasticity . . 9.3.3 Micro–Macro Relationship . . . . . . . .

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311 311 312 312 314 315 316 317 318 321 321 322 323

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xviii

Contents

9.3.4 Stress–Strain Relationship . . . . . . . . . . . . 9.3.5 Homogenization Integration . . . . . . . . . . 9.3.6 Experimental Validation . . . . . . . . . . . . . 9.4 Micromechanical Models Based on the CH Model 9.4.1 Capillary Forces . . . . . . . . . . . . . . . . . . . 9.4.2 Chemical Forces in Grouted Sand . . . . . . 9.4.3 Surface Energy Forces . . . . . . . . . . . . . . 9.4.4 Mechanical Forces in Clayey Materials . . 9.5 MicroSoil in ErosLab . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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324 324 327 330 331 332 332 333 335 336

10 Practice of the ErosLab Modeling Platform . . . . . . . . . . . . . . . . 10.1 Stress–Strain Mixed Control Scheme . . . . . . . . . . . . . . . . . . 10.2 ErosLab Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Mixed-Language Programming . . . . . . . . . . . . . . . . 10.3 General Structure of ErosLab . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Provision of a Mechanical Calculator . . . . . . . . . . . 10.3.2 Provision of Various Types of Soil Tests . . . . . . . . . 10.3.3 Provision of a Variety of Soil Models and Support for the Extension to New Models . . . . . . . . . . . . . . 10.3.4 Provision of Different Loading Controls . . . . . . . . . 10.3.5 Provision of Comprehensive and Efficient Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.6 Provision of Visualization with Graphical Displays . 10.4 Graphical User Interface and Usage Instructions . . . . . . . . . . 10.4.1 Graphical User Interface . . . . . . . . . . . . . . . . . . . . . 10.4.2 Usage Instructions . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Case 1: Modeling Sand Behavior with the SIMSAND Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Case 2: Modeling Sand Behavior with the HYPOSAND Model . . . . . . . . . . . . . . . . . 10.5.3 Case 3: Modeling Sand Behavior with the MicroSoil Model . . . . . . . . . . . . . . . . . . . . 10.5.4 Case 4: Modeling Clay Behavior with the ASCM Model . . . . . . . . . . . . . . . . . . . . . . 10.5.5 Case 5: Modeling the Time Effects on Clay Behavior with the ANICREEP Model . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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341 341 344 344 344 344 345

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350 350 350 350 356 361

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Appendix A: Stress–Strain Mix Control Program of Modeling . . . . . . . . 379 Appendix B: Example of DLL for MCC . . . . . . . . . . . . . . . . . . . . . . . . . . 395

About the Authors

Zhen-Yu Yin has been an Associate Professor of Geotechnical Engineering at Hong Kong Polytechnic University since 2018. Dr. Yin received his B.Sc. in Civil Engineering from Zhejiang University in 1997, followed by a 5-year engineering consultancy at the Zhejiang Jiahua Architecture Design Institute. Then, he obtained his M.Sc. and Ph.D. in Geotechnical Engineering at Ecole Centrale de Nantes (France) in 2003 and 2006, respectively. Dr. Yin has been working as a postdoctoral researcher at Helsinki University of Technology (Finland), the University of Strathclyde (Glasgow, UK), Ecole Centrale de Nantes and the University of Massachusetts (Umass-Amherst, USA). In 2010, he joined Shanghai Jiao Tong University as a Special Researcher and received “Professor of Exceptional Rank of Shanghai Dong-Fang Scholar”. In 2013, he joined Ecole Centrale de Nantes as Associate Professor before moving to Hong Kong. Dr. Yin has published over 160 articles in peer reviewed international journals. Since 2012, he has been a member of the granular materials committee of the American Society of Civil Engineers. Pierre-Yves Hicher is currently Emeritus Professor at Ecole Centrale de Nantes and former Director of the Regional Institute for Research in Civil Engineering and Construction, after having served as Director of the Research Institute in Civil and Mechanical Engineering, UMR CNRS-Ecole Centrale Nantes-University of Nantes. He received his Ph.D. in material science at the Ecole Centrale de Paris in 1979 and his “doctorat d’état” in 1985 at the University Pierre and Marie Curie, Paris. He has taught and served at the Ecole Centrale de Paris, the French National Science and Technology Centre (CNRS), and the Ecole Centrale de Nantes, France. Dr. Hicher has long been committed to Sino-French cooperative research and exchange and has been employed as a “foreign expert” in the Department of Geotechnical Engineering of Tongji University. He is an internationally renowned expert in geomechanics. Among his major contributions that can be cited include the original

xix

xx

About the Authors

experimental procedures and parameter identification methods under inverse analysis techniques and the homogenization techniques for modeling the mechanical behavior of disordered granular materials. He is the author and coauthor of more than 100 papers in international journals and has published several books on his research topics. Dr. Hicher has supervised the work of 44 Ph.D. students in the fields of soil mechanics, foundation engineering, granular materials, constitutive modeling and numerical modeling. Yin-Fu Jin was born in the Shandong Province, China, in 1988. Dr. Jin received his bachelor’s degree in civil engineering from Northwest A&F University in 2011 and received his doctoral degree in geotechnical engineering from Ecole Centrale de Nantes in France in 2016. He mainly engaged in soil mechanics, parameter identification and the application of artificial intelligence in geotechnical engineering. Recently, he has published more than 20 peer reviewed papers in international journals, such as Acta Geotechnica, the International Journal for Numerical and Analytical Methods in Geomechanics, Ocean Engineering and Engineering Geology.

Symbols

a a Ad b b Caei D E e; e0 E0 ec0 ecuf ed Eh ; Ev ehc; c0 ehf; c0 emax Eu f fth G G0 Gvh

Constant of fines content effect in silty sand (SIMSAND + fr) Target inclination of yield surface related to volumetric strain (ASCM) Constant of stress-dilatancy magnitude (0.5*1.5) Constant controlling the amount of grain breakage (SIMSAND + Br) Target inclination of yield surface related to deviatoric plastic strain Intrinsic secondary compression index (remolded clay) Stiffness matrix of material Young’s modulus Void ratio and initial void ratio Referential Young’s modulus (dimensionless) Initial critical state void ratio (SIMSAND); Virgin initial critical state void ratio before breakage Fractal initial critical state void ratio due to breakage General shear strain Horizontal and vertical Young’s moduli Initial critical state void ratio of pure fine soils (fc = 0%) Initial critical state void ratio of pure coarse soils (fc = 100%) Maximum void ratio Undrained Young’s modulus Threshold fines content from coarse to fine-grained skeleton (20*35%) Fines content Shear modulus Referential shear modulus Shear modulus

xxi

xxii

I1 ; I2 ; I3 I10 ; I20 ; I30 J1 ; J2 ; J3 J10 ; J20 ; J30 K K0 kp Kw M m Mc n nd np p0 pat pb0 pc0 pexcess psteady q Rd Ra sij ux ; uy ; uz ak0 b v0 dij e1 ; e2 ; e3 ea ; er eij em ev cxy ; cyx ; cyz ; czy ; czx ; cxz /

Symbols

The first, second and third invariants of the stress tensor The first, second and third invariants of the strain tensor The first, second and third invariants of the deviatoric stress tensor The first, second and third invariants of the deviatoric strain tensor Bulk modulus Coefficient of earth pressure at rest Plastic modulus related constant in SIMSAND; Plastic modulus related parameter in ASCM Bulk modulus of water Constraint modulus in elasticity; Slope of critical state line on p'-q plane Constant of fines content effect in sandy silt Slope of critical state line in triaxial compression on p'q plane Porosity of soil; Elastic constant controlling nonlinear stiffness Phase transformation angle related constant (1) Peak friction angle related constant (1) Mean effective stress Atmosphere pressure Initial bonding adhesive stress Initial size of yield surface; Initial size of yield surface of grain breakage (SIMSAND+Br) Excess pore pressure Steady pore pressure Deviatoric stress Ratio of mean diameter of sand to silt D50 =d50 Stress relaxation coefficient Deviatoric stress tensor Displacements Initial inclination of yield surface Rate-dependency coefficient Initial bonding ratio Kronecker symbol Principle strains Axial strain and radial strain Strain tensor Mean strain Volumetric strain Engineering shear strains Friction angle

Symbols

k ki k k0 ki vu v0vh v0vv h q ra ; rr rij rm ðqÞ rn ; rh rp0 rw rx ; ry ; rz r1 ; r2 ; r3 s sxy ; syx ; syz ; szy ; szx ; sxz v x xd n nb nd w

xxiii

Swelling index of the isotropic compression test (on elnp’ plane) Intrinsic swelling index (of remolded soil, on e-lnp’ plane) Lame constant in elasticity; Compression index (in elnp’ plane); Constant controlling the nonlinearity of CSL in SIMSAND Compression index under the loge-logp′ plane Intrinsic compression index (of remolded soil, on e-lnp’ plane) Undrained Poisson’s ratio Horizontal Poisson’s ratio Vertical Poisson’s ratio Lode angle Constant controlling the movement of CSL Axial stress and radial stress Stress tensor mean stress Vertical and horizontal stresses Preconsolidation pressure Pore water pressure Normal stresses First, second and third principle stresses Reference time (Oedometer test s = 24h) (ANICREEP) Shear stresses Poisson’s ratio Absolute rotation rate of the yield surface Rotation rate of the yield surface related to the deviatoric plastic strain Constant controlling the nonlinearity of CSL (SIMSAND); Absolute rate of bond degradation Degradation rate of the interparticle cohesive bonding Constant controlling the deviatoric strain-related bond degradation rate Dilatancy angle

Chapter 1

Constitutive Relations of Saturated Soils: An Overview

Abstract In this chapter, some of the typical mechanical behaviors of saturated soils are introduced and summarized, starting with the most basic materials known as “remolded soils”: compression behavior, shear behavior, influence of intermediate principal stress, induced anisotropy, noncoaxiality, small-strain stiffness, cyclic behavior, time dependency, temperature effects, and water chemistry effects. Then, additional mechanical properties are gradually introduced with constitutive modeling methods for intact natural soils, namely, inherent anisotropy, soil structure, and destructuration. Finally, special problematic soils (i.e., organic soft soil, quick clay) are introduced with their special mechanical properties and constitutive modeling approaches. Each basic characteristic is presented with its definition or basic description, and then, the testing methods with the interpretation of test results, as well as commonly used constitutive modeling methods, are introduced.

1.1

Introduction

James [1] claimed that the nineteenth century was the “disaster area of design” because at that time, engineers rarely analyzed engineering design concepts before construction. Considering the major construction problems faced in the early twentieth century, early soil studies of soil mechanics mainly focused on preventing structural damage. In the 1970s, for many experienced engineers, soil mechanics became a mature discipline, as most of the damage mechanisms of soil had been discovered. However, numerous urbanizations and reconstructions of old urban areas involve land subsidence and its influence on adjacent buildings and underground structures. More importantly, for safety and comfort considering the widespread construction of nuclear and other important facilities, it is necessary to study the effects of even small deformations on various geotechnical structures. At the same time, the rapid development of numerical models and computational techniques also provides a powerful tool for practical engineering analysis. The basic mechanical properties of soil and its constitutive modeling method are important theoretical foundations for the implementation of geotechnical and © Springer Nature Singapore Pte Ltd. and Tongji University Press 2020 Z.-Y. Yin et al., Practice of Constitutive Modelling for Saturated Soils, https://doi.org/10.1007/978-981-15-6307-2_1

1

2

1 Constitutive Relations of Saturated Soils: An Overview

underground engineering. Since the beginning of the last century, many researchers have carried out numerous studies on the basic mechanical properties of soil to accurately understand its deformation development and failure mechanism under different stress conditions and to establish the theory of constitutive modeling. Different types of geotechnical engineering are simulated to further guide the design, construction, and maintenance of engineering structures. The key to solving the geotechnical engineering problem is to develop practical soil constitutive models under the framework of continuum mechanics.

1.2

Compression Behavior

Soil compressive behavior is characterized by a strong nonlinearity that affects the whole stress–strain relationship. The conventional test methods used to analyze soil compression behavior mainly include (1) compression tests controlling the stress ratio r′3/r′1 == constant (Fig. 1.1a) and (2) conventional one-dimensional oedometer tests e2 = e3 = 0 (Fig. 1.1b). For clay, the preconsolidation pressure (or stress) r′p0 and the compression curve before and after the yield stress are usually obtained by the above tests (Fig. 1.1c).

(a)

σ’1

Triaxial test

(c) e

κ λ

k > > e ¼ ei  j ln 0v at r0v \r0p0 > < rvi ! > r0v > > at r0v  r0p0 > : e ¼ e0  k ln r0 p0

ð1:1Þ

The bulk modulus and oedometric modulus can also be obtained as follows:
 1 þ ei 1 þ 2k 0 1 þ ei 0 K¼ rv ; Eoed ¼ rv 3 j j

ð1:2Þ

Equation (1.2) shows that the elastic behavior of the soil is nonlinear as the modulus is a function of the applied stress. The stiffness is also affected by the soil density. When the current stress is greater than the preconsolidation stress (r′v > r′p0), according to the classical elastoplastic theory, the compression deformation can be composed of two parts, elastic deformation, and plastic deformation (Fig. 1.1c): 8 > > > > >
> r0v > p > ð Þ De ¼ k  j ln > : r0p0

!

ð1:3Þ

Based on the unloading–reloading test results, it was found that the elastic deformation did not change the elastoplastic yield stress generated by the consolidation stress history (i.e., the updated preconsolidation stress, Fig. 1.1d). Therefore, the updated preconsolidation stress is only related to plastic deformation: De ¼ ðk  jÞ ln p

r0p r0p0

)

r0p

¼

r0p0




1 þ e0 p e exp kj v

 ð1:4Þ

If the preconsolidation stress is used as a hardening parameter for clay, then formula (1.4) constitutes the hardening rule for this hardening parameter. The nonlinear smooth gradient of the soil from the slope j to k during the compression process can be easily found (Fig. 1.1c enlarged to Fig. 1.2). This

4

1 Constitutive Relations of Saturated Soils: An Overview

Fig. 1.2 Schematic diagram of nonlinear compression characteristics calculation

gradual change can be achieved by setting a normal compression line (NCL) as the reference boundary line, introducing the distance R between the actual state of the soil (e, r′v) and the position on the NCL corresponding to the same value of e, establishing the nonlinear relationship of plastic strain with R, and finally achieving this nonlinear smooth process. De ¼ j

Dr0v Dr0 1 þ ðk  j Þ 0 v 0 rv rv 1 þ f ð RÞ

ð1:5Þ

In addition, another law can be obtained through the stress-controlled triaxial compression tests on remolded soil (Fig. 1.1e): (1) the yield stress decreases as the stress ratio j decreases; (2) the plastic compression curves are parallel to each other; and (3) the yield point can be obtained on the stress path plane, and an elliptical yield surface can be drawn. Thus, the one-dimensional preconsolidation stress can be extended to the size of the three-dimensional yield surface. The hardening law also naturally extends from one-dimensional to three-dimensional to describe the change in the yield surface size. In fact, the soil structure will change during natural deposition or remolding. These changes are closely related to the loading mode and loading direction. In other words, different compression stress ratios will cause differences in the compression curve, which corresponds to the induced anisotropy. If conventional experimental results with a single loading path are used as the basis for engineering design, it would cause deviations in the real soil behavior and could further lead to accidents in geotechnical engineering. Many researchers have adopted the stress-controlled triaxial apparatus to study the anisotropy of the yield surface through compression tests under paths with different stress ratios; a in Eq. (1.6) represents the inclination angle of the yield surface (Fig. 1.3); this formula summarizes the rotational hardening law of the yield surface (as an example, see Dafalias [2], Wheeler et al. [3], and so on). f ¼

ð q  p0 aÞ 2 þ p0  pc ðM 2  a2 Þp0

ð1:6Þ

1.2 Compression Behavior

q

0

(a)

CSL

5

q

α=0

IC p’ (history)

CSL

CSL

AC (history)

q

CSL

α 0

(b)

p’ CSL

α

0

(c)

CSL

p’ AC (history)

Fig. 1.3 Yield surfaces in the plane change with the stress history as the initial stress ratio increases from large to small

Notably, sand usually exhibits more reversible behavior under compression until the sand particles begin to break. At that stage, plastic deformations start to develop, and the compression curve becomes similar to that of clay, but the yield stress is related to fractural properties (mineral composition, Mohr hardness, and so on). Unlike sand, natural soft clays have a small compressive yield stress. Therefore, simple soil models such as Mohr–Coulomb and Duncan–Chang, which do not consider this specific feature, are less suitable for the calculation of engineering structures installed on clayey subsoils (such as shallow foundations and embankments). Cap models, such as the double-yield surface model [4] and the Cam–Clay model [5], are more applicable for these cases. More details can be found in Chap. 6.

1.3

Shear Behavior

The shear properties of soil refer to the contraction/dilatancy and yield strength properties or friction properties during shearing. The conventional testing method includes (1) shear tests for uneven deformation of soil samples, such as direct shear tests and simple shear tests, and (2) triaxial shear tests that lead to relatively uniform deformation of soil samples. According to different design requirements, methods such as slow shearing or fast shearing and drained or undrained test conditions are usually adopted, and different deformation and strength indexes are obtained. Triaxial shear tests are widely used due to the advantage of sample uniformity during shearing. Based on the consolidation history, the current void ratio, and the stress state, clay can be divided into normally consolidated and overconsolidated clay, while sandy soil is divided into loose and dense sand. The results of a large number of triaxial shear tests are shown in (Fig. 1.4): (a) Normally consolidated and slightly overconsolidated clays and loose sand exhibit volumetric contraction during shearing; the void ratio decreases under

6

1 Constitutive Relations of Saturated Soils: An Overview

Fig. 1.4 Schematic diagram of the mechanical properties of different overconsolidated clays: a under drained conditions, b under undrained conditions, and c effects of the critical state line

(b)

(c)

(d)

(e)

drained conditions, and the mean effective stress decreases under undrained conditions. Highly overconsolidated clay and dense sand exhibit volumetric expansion during shearing, that is, dilative characteristics (the void ratio becomes larger under drained conditions, and the mean effective stress becomes larger under undrained conditions), along with the peak stress ratio above the critical stress ratio. The stress ratios of clays with different overconsolidated ratios or sands with different densities all reach the same stress ratio line. This line is called the “critical state line”, hereinafter referred to as the “stress ratio-related critical state line”, which is related to the soil friction and determines the soil yield strength. The void ratios of a given clay at different overconsolidation ratios or a given sand at different densities also eventually reach the same “void ratio-log mean effective stress” line, which is also called the “critical state line”, hereinafter referred to as the “void ratio-related critical state line”. For clay, it is generally considered that the void ratio-related critical state line is a straight line parallel to the isotropic compression line; for sand, the void ratio-related critical state line is usually nonlinear. The relative position between the void ratio-related critical state line and the current state of the “void ratio-mean effective stress” determines the contractive or dilative behavior of the soil and the maximum volumetric strain during shearing.

Based on this phenomenon and the critical state concept, many researchers have developed various constitutive models under the framework of the elastoplasticity theory. These models can be divided into two categories according to their consideration of the void ratio-related critical state line: (a) The implicit consideration of the critical state line. The position of the void ratio-related critical state line is controlled by the intersection of the yield surface and the stress ratio of the critical state line on the “p′-q” plane (e.g.,

1.3 Shear Behavior

7

Eq. (1.5)), such as in the Cam–Clay models. Notably, the multi-surface model [6], the bounding surface model [7], and the sub-loading surface model [8] introduce the plastic strain within the yield surface so that the nonlinear stress– strain characteristics are improved in accordance with the yield strength characteristics. More details can be found in Chap. 6.

epij ¼ K

@g ; @rij


such as Cam - Clay model: g ¼ f ¼

q2 þ p0  pc M 2 p0

 ð1:7Þ

(b) The explicit consideration of the critical state line. The expression of the void ratio-related critical state line is directly introduced by establishing the relationship between the relative distance from the current “void ratio–stress” state and the critical state line, the peak friction angle, and the dilatancy angle. The double-yield-surface model (Eq. (1.8)) proposed by Yin et al. [4] simultaneously conforms to the nonlinear stress–strain and yield strength characteristics. More details can be found in Chap. 5. 8 > > > > > > > > > > > >
e > c c > > tan /l Interlocking : tan /p ¼ tan /l ; tan /pt ¼ > > e e > >
0  > > p > > Critical state line: ec ¼ ecr0  k ln : pcr0 Yield surface : fS ¼

ð1:8Þ

It should be pointed out that simple models, such as the Mohr–Coulomb and Duncan–Chang models, do not consider the void ratio-related critical state line and thus cannot describe the contraction/dilation and peak/residual yield strength characteristics of soils associated with different overconsolidated ratios or densities. Therefore, this type of model is limited to calculating the volumetric deformation or geotechnical engineering with a small change in pore water pressure (the strain is limited to before the occurrence of transformation between contraction and dilation and the occurrence of peak strength). Therefore, the rationality of the simulated results can only be guaranteed within a small-strain range. However, it is more reasonable to use the model considering the void ratio-related critical state line. The three-dimensional mechanical expression of the shear strength is one of the important properties of soils. The key to the problem lies in the influence of the intermediate principal stress on the shear strength. True triaxial tests (Fig. 1.5a), which allow control of the stresses in three independent directions, are usually used to

8

1 Constitutive Relations of Saturated Soils: An Overview Dial gauge

Piston rod Suspension

Load cell Suspension

Experimental data SMP failure criterion

Load cell

Von-Mises failure criterion

Dial gauge

Lade-Duncan failure criterion Mohr-Columb failure criterion

Specimen Ring for sealing the membrane

Transfer for horizontal displacement

Rigid platen

(a)

Drainage tube

Bi-direction bellofram cylinder

(b)

Fig. 1.5 True triaxial test: a complex true triaxial pressure chamber; b Shanghai soft clay failure surface [9]

perform loading along different stress paths on the p-plane to obtain the strength characteristics under different intermediate principal stress conditions (Fig. 1.5b). There are three main theoretical modeling methods accounting for the effects of the intermediate principal stress on soil strength: (1) the lode angle modification method or g(h) method, (2) the transformed stress tensor method, and (Fig. 1.6) (3) the micromechanical analytical method (Fig. 1.7). More details can be found in Chap. 4. Notably, the stress path within geotechnical structures is generally complicated. The strength criterion of the commonly used Mohr–Coulomb model is generally conservative. On the other hand, the Duncan–Chang model implies the generalized von Mises strength criterion, which overestimates the material strength. Therefore, it is necessary to consider the influence of the 3D stress state based on the selected constitutive model. Furthermore, with the development of computer techniques, the development and application of micromechanical methods with physical insights into soils should receive increasing attention (see details in Chap. 9).

Fig. 1.6 Yield surface of the modified Cam–Clay tij model in a the principal stress space Chowdhury and Nakai [10]; b transformed principal stress space; and c the principle of the TS method: the yield curve of the strength criterion on the p-plane Yao et al. [11]

1.4 Stress–Dilatancy

9 Macro strain-stress relationship

Δ σ ij = Cijkl Δ ε kl

Stress

micro scale

Δf jα = Δσ ij Aik nkα

Strain Localisation & Homogenization

Interparticle

N

Δu j ,i = Aik−1 ∑ Δδ αj lkα α =1

Interparticle

force

displacement

Δ fi α = Kijα Δδ αj

(a)

micro scale

(b)

Interparticle force-displacement relationship

Fig. 1.7 Schematic diagram of the micromechanical model: a statistics from representative soil elements to micro contacts; b from macro stress–strain to micro force–displacement relationships

1.4

Stress–Dilatancy

Stress–dilatancy is a physical description of the effect of the shear stress q on the volumetric strain ev. However, it is notable that dilatancy is the result of a coupling between the mean effective principal stress p’ and the generalized shear stress q in the strain-generating process. The stress–dilatancy relation is an important issue for modeling soil behavior, which can be expressed, for example, in the potential surface of an elastoplastic model. The physical manifestation of dilatancy was first discussed by Reynolds [12]. Approximately, one-half century ago, Rowe [13] and Roscoe et al. [14] introduced two different forms of stress dilatancy equations for soils, which are historical landmarks in the development of soil mechanics. Since then, these two equations have been widely used as flow rules in elastoplastic models for soils. In particular, Roscoe’s equation has become an important element of critical state soil mechanics. To date, most plasticity models are based on Roscoe’s or Rowe’s dilatancy equations, e.g., Nova [15], Jefferies [16], Manzari and Dafalias [17], Gajo and Muir Wood [18], Li et al. [19], Wan and Guo [20], Yang and Muraleetharan [21], Taiebat and Dafalias [22], and so on. Taylor’s analysis [23] of direct shear box tests on soils assumed that all the input work is dissipated by friction. This principle of energy dissipation has been extended to the conditions of triaxial tests by Roscoe et al. [14], and this principle stated that the plastic input work is equal to the energy dissipated by friction: (

p0 depv þ qdepd ¼ Mdepd for compression p0 depv þ qdepd ¼ Mdepd for extension

ð1:9Þ

The right-hand term of Eq. (1.9) represents the energy dissipated by friction, which must be positive in either the compression or extension tests. The mean effective stress p’ = (ra + 2rr)/3, the deviatoric stress q = ra − rr, the volumetric strain increment depv ¼ depa þ 2depr , the deviatoric strain increment   depd ¼ 2 depa  depr 3, and the subscripts a and r indicate the axial and radial

10

1 Constitutive Relations of Saturated Soils: An Overview

directions, respectively, in a triaxial setup. The superscript p denotes plastic components. M is the slope of the critical state line. Equation (1.9) can be rearranged into the form of the dilatancy equation:

d¼

depv depd

¼

8 q > < M  p0 for compression q > : M  for extension p0

ð1:10Þ

It should be noted that M is the stress ratio corresponding to zero dilatancy, which is also termed the slope of the phase transformation line according to Ishihara and Towhata [24] or the characteristic line according to Luong [25]. Another landmark development in the dilatancy equation is the stress dilatancy of Rowe [13], which uses the assumption that the ratio of the input energy increment to the output energy increment is a constant denoted as K. In a compression test, the input energy increment is r0a dea and the output energy increment is 2r0r der , and vice versa for an extension test. Note that in Rowe’s stress dilatancy, the total strains are introduced rather than the plastic strains. The constant K is related to the critical friction angle K = tan2(p/4 + /l/2). 8 0
 ra dev > > > ¼K 1  for compression < r0 dea r
 > r0r dev > > ¼K 1  for extension : r0a dea

ð1:11Þ

Equation (1.10) can be rearranged in terms of variables used in critical state soil mechanics, which are given as follows: 8 > >
9ðM þ gÞ > : for extension 3M þ 2Mg þ 9

ð1:12Þ

where the stress ratio η represents the ratio of deviatoric stress to mean effective stress, q/p. For both Eqs. (1.10) and (1.11), different values of M are used for compression and extension: 8 6 sin /pt > > > < 3  sin / for compression pt M¼ > 6 sin / pt > > for extension : 3 þ sin /pt where /pt = phase transformation angle.

ð1:13Þ

1.4 Stress–Dilatancy

11

Because of the discrepancy in fitting experimental tests, a constant D is often introduced in Eq. (1.10) as proposed by Nova [15], Jefferies [16], Gajo and Muir Wood [18], Li et al. [19], Yang and Muraleetharan [21]:
 depv q ¼D M  p0 depd

ð1:14Þ

The value of D is different in compression and extension. More details can be found in Chap. 5. These relations imply that the phase transformation angle and the critical state friction angle /c correspond to the same mobilized friction angle. The relations also imply a relationship between the dilatancy angle defined by the maximum of dev/de1 and the maximum friction angle /p at the peak stress, which decreases if the mean effective stress increases. For a given mean effective stress, /p increases if the initial void ratio decreases according to the following approximation: e tan /p ¼ ec tan /c ¼ constant [26]. Another frequently used alternative form of Eq. (1.14) was introduced to alter the value of M as a function of the density state, e.g., Manzari and Dafalias [17] and Wan and Guo [20] further made M dependent of the shear strain, the fabric tensor, and the orientation of bedding planes.

1.5

Small-Strain Stiffness

A large amount of measured engineering data show that a considerable part of the soil around a civil engineering structure is in a small-strain state under a normal working load. Both the back analysis of the monitoring data and the small-strain stiffness test results indicate that the actual stiffness of the soil is much larger than the stiffness values obtained from conventional experiments, such as oedometer tests or triaxial tests. In the literature Cole and Burland [27]; St John [28]; Clayton et al. [29]; Marsland and Eason [30]; Simpson [31], it has been argued that the soil strain around deep excavations and in the foundations of important buildings is basically less than 0.1%, and the maximum is less than 0.5% under working loads. Similarly, Jardine et al. [32] and Mair [33] thought that with the exception of a few plastic areas around the foundation, excavation, and tunnel, the strains in other areas are generally small, with a representative order of magnitude of 0.01–0.1%. According to Atkinson and Sallfors [34], the soil strain can be divided into three ranges: very small strain (no more than 10−6), small strain (10−6–10−3), and large strain (greater than 10−3). Figure 1.8 shows the shear strain of soil under several common geotechnical conditions and the shear strain range applicable for different test methods.

12

1 Constitutive Relations of Saturated Soils: An Overview

Retaining wall Excavation Very small Small strain strain

Tunnelling Conventional testing Large strain

Dynamic testing methods

Shear strain

Local strain gauge

Fig. 1.8 Soil strain stiffness characteristics in different strain ranges

Due to the limitation of the measurement devices, it was difficult to accurately measure the small strains in the soil before the 1980s. With the emergence of the stress path triaxial apparatus and the static and dynamic stiffness test equipment in the 1980s, studies on the small-strain stiffness characteristics gradually increased. There are many test methods for measuring the soil stiffness, which mainly include the following two categories [35]: (1) field test methods, such as the continuous surface wave method, lower hole method, and cross-hole method; and (2) laboratory test methods, such as the bending element test, resonance column test, and improved triaxial test (Fig. 1.9). At very small strains, the soil behavior is considered to be elastic. The elastic behavior is nonlinear since the modulus depends on the applied stress, which is in accordance with Hertz’s law for the assembly of identical spheres [37]. The modulus also depends on the soil density (Figs. 1.11 and 1.12). Various formulations have been proposed for the Young’s and shear moduli, which take the following general form (Table 1.1):    E0 or G0 ¼ Af ðeÞOCRk p0 pref m

ð1:15Þ

Since the elastic domain is restricted to very small strains, for which the relative displacements of the constituents are negligible, the influence of the particle deformability is particularly pronounced on the elastic properties of soils. Under these conditions, the elastic moduli measured in sands are higher than those in clays. For clays, the more rigid particles of kaolinite give higher elastic moduli than those of montmorillonite, as an example (Fig. 1.13). The small-strain stiffness characteristics can be divided into the value of very small-strain stiffness Gmax and the reduction law of small-strain stiffness with shear strain “G/Gmax-c”. For remolded soft soil, Gmax is proportional to the

1.5 Small-Strain Stiffness

13

Fig. 1.9 Illustration of laboratory testing: a bend element and shear wave velocity measure system; b Stokoe resonant column apparatus; c advanced triaxial test configuration

overconsolidation ratio and confining pressure [43, 44]. The reduction law of the small-strain stiffness is mainly related to the plasticity index of soft soils [45] (Fig. 1.10). The constitutive equations or models considering small-strain stiffness characteristics can be divided into the following three main categories: nonlinear elastic model, elastoplastic model, and block string model. Among these models, the nonlinear elastic model mainly includes the Caughey bilinear model, Ramberg– Osgood model, Hardin–Drnevich model, Jardine model, Benz model, and so on (Table 1.2). The mathematical expression for this kind of formula is simple, and the key parameters can be directly linked to the Atterberg limits of the soil based on the diagram in Fig. 1.10.

14

1 Constitutive Relations of Saturated Soils: An Overview

Fig. 1.10 Influence of plasticity index on stiffness reduction: a experimental data; b theoretical trend [36]

Fig. 1.11 Effect of soil density on the shear modulus (after Benz [36])

1000

Fig. 1.12 Effect of soil density on the elastic modulus (Biarez and Hicher [26])

Ee Ee p '0

p '0

=

Fitting Line Kaolinite C (P300) Kaolinite A Kaolinite B Kaolinite B Kaolinite C Kaolinite A

450 e

500

0

Bard (1993) 0.8

1.2

e

1.6

2.0

1.5 Small-Strain Stiffness

15

Table 1.1 Various formulations proposed for the Young’s and shear moduli (after Benz [36]) Soil tested

emin

emax

A

f(e)

m

Ref

Clean sands with Cu < 1.8

0.5

1.1

57

ð2:17eÞ2 1þe

0.4

All soils with wL < 50%

0.4

1.8

59

1 e

0.5

Undisturbed clayey soils and crushed sand Undisturbed cohesive soils

0.6

1.5

33

ð2:97eÞ2 1þe

0.5

0.6

1.5

16

ð2:97eÞ2 1þe

0.5

Iwasaki and Tatsuoka [38] Biarez and Hicher [26] Hardin and Black [39] Kim et al. [40]

14

2

0.6

Loess

1.4

4.0

ð7:32eÞ 1þe

Kokusho et al. [41]

Table 1.2 Some typical nonlinear elastic constitutive equations Name and references Caughey bilinear model [46]

Constitutive equations G0 c c\cy s¼ sy þ Gðc  cy Þ c  cy

Ramberg–Osgood model [47]

G G0

¼

Hardin–Drnevich model [44]

G G0

¼ 1 þ j1c=c j

Jardine model [32]

Eu Cu

Benz model [36] (have been applied to hardening soil model)

1 j 1 þ ajs=sy j



  c ¼ A þ B cos a log10 eCa   G ¼ G0 c +c0:7ac 0:7 Hist r

Fig. 1.13 Influence of soil nature on the elastic modulus (Biarez and Hicher [42])

16

1 Constitutive Relations of Saturated Soils: An Overview

The values of elastic coefficients are substantially affected by the loading history. In granular materials, strain hardening created by significant irrecoverable strain (>1%) increases the values of Young’s modulus and Poisson’s ratio. The soil structure is no longer isotropic, and the elastic domain and material coefficients are influenced by the geometrical anisotropy of the structure. On the other hand, in initially structured materials, a decrease in Young’s modulus was observed after a previous loading. This decrease is a function of the strain amplitude reached during the initial loading and can be explained by the degradation of the structure, and in particular by the destruction of bonds between clay particles or aggregates [48]. For these structured materials, the moduli measured on the remolded samples are much lower than those of the intact samples, which shows the importance of reducing the disturbance during boring, especially when the behavior at very small strains has to be considered. Simpson [49] showed a person pulling a string of blocks to reflect the small-strain characteristics of the soil subjected to the effect of the previous stress history, considering the different shear modulus values of the specimen under different stress paths (Fig. 1.14). The block string model has been used in many applications in different countries, such as modeling the behavior of London clay and Singapore clay. To account for the soil behavior at small strains, the bubble model [49] introduces a motion surface inside the bounding surface based on the modified Cam– Clay model. This enhancement eliminates the defect that the modified Cam–Clay model cannot reflect the hysteresis phenomenon during unloading and reloading, and thus the model response becomes closer to the actual situation when simulating unloading and reloading. Based on the bubble model, Stallerbrass [50] introduced an additional surface, called the historical surface, to reflect the small-strain characteristics of the soil by considering the influence of the recent stress history. The model is called the “three-surface model” (Fig. 1.15). Grammatikopoulou et al. [51] proposed a “modified three-surface model” based on the “original three-surface model”. The main feature of this model is the use of a hardening modulus to achieve a smooth change in stiffness with strain. Notably, this kind of model can more comprehensively and realistically reflect the change in stiffness under different

Fig. 1.14 Principle of the brick model with four bricks and stiffness decay after load reversal

1.5 Small-Strain Stiffness

17

Fig. 1.15 Three-surface model based on the bubble model

stress paths, but the parameters are more difficult to determine by conventional tests. The mathematical formula is complicated, and the calculation convergence is relatively poor, so there are still some difficulties when these models are used in engineering applications. It should be pointed out that for engineering cases with strict soil deformation control, it is necessary to consider the small-strain stiffness characteristics in the constitutive model used in the calculation analysis. More details can be found in Chap. 5.

1.6

Induced Anisotropy and Noncoaxial Properties

Soils will form a contact fabric structure during the deposition process. This contact fabric changes during loading. This change is closely related to the type and direction of loading. In other words, different loading conditions will result in different soil structure evolutions; this phenomenon is called induced anisotropy. Many researchers applied the stress-controlled triaxial apparatus to study the anisotropy of the yield surface through compression and shear tests under different stress paths and summarized a rotational hardening law of the yield surface (Table 1.3). The most representative equation is the constitutive equation proposed by Wheeler et al. [3] (Eq. 1.16, Fig. 1.3), and the verification and application of this equation to different soils has been well studied [52–57] f ¼

ð q  p0 aÞ 2 þ p0  pc ðM 2  a2 Þp0

ð1:16Þ

Furthermore, based on the contact fabric structure, the material principal axis, i.e., the anisotropic distribution, can be determined. When the stress principal axis generated by the external forces forms a different angle from the material principal axis, the stress–strain response is usually different, which is called the principal stress axis rotation effect or the noncoaxial property. With the hollow cylinder apparatus, studies on the rotational effect of the principal stress axes have

18

1 Constitutive Relations of Saturated Soils: An Overview

Table 1.3 Some typical rotational hardening rules for the yield surface References Dafalias [58] Whittle [59] Hueckel et al. [60] Newson et al. [61] Hashiguchi [62] Wheeler et al. [3] Kobayashi et al. [63] Oka et al. [64] Dafalias [65] Zhang et al. [66] Dan et al. [67]

Constitutive equations   þ ein  @f  p a_ ¼ hLi 1kj @pc p0 ðg  xaÞ h i _ konc Þ ¼ pbc kjagkkonc nj ðq  p0 agkonc Þ_epv ðag h i _ konc Þ ¼ b qp expðbeps Þ e_ ps þ r_epr ðag h i g_ 0 ¼  1  ðgs =MÞ2 p_ expðgs  gos Þf   ga _p a_ ¼ bkg  ak kga k mb  a es h       i s 3s a_ ¼ l xd 3pij0  a e_ pd  þ 4pij0  a e_ pv h iqffiffi  jg2jbs jaÞ 2 p  a_ ¼ DJ ðbs þ 2jbksga 3 Ds k    a_ ¼ Bs þ ðB0  Bs Þ exp Bt epd A_epv  a_epd  b  a_ ¼ hLihjg  aj a  a qffiffi  a_ ¼ DJ br ðbl M  aÞ 23Dps  k^g^gk nh i   vp o  a_ ¼ b ðbs þ jbs jÞ g g  2jbs ja e_ vp v  r ð2jbs jaÞ e_ s Konc

increasingly developed in recent years. This torsional shear test primarily achieves control of the principal stress axis rotation and shearing by applying torque to the specimen (Fig. 1.16a). The test results [68–72] show that the principal stress axis orientation can have different reduction degree effects on the undrained shear strength of soil (Fig. 1.16b). The theoretical methods that consider the rotation effect of the principal stress axis can be roughly divided into four categories: (1) the noncoaxial elastic–plastic method for sand proposed by Rudnicki et al. [73], which consists of adding the noncoaxial shear strain component to the plastic shear strain component. The shear

Fig. 1.16 Illustration of the hollow cylinder test and experimental results of shear strength versus principal stress rotation

1.6 Induced Anisotropy and Noncoaxial Properties

19

stress component, which induces the rotation of the principal stress axis, is the basis for introducing the effect of the principal stress axis rotation on the deformation development in the constitutive model, such as that of soft soil in Yatomi et al. [74]; (2) methods similar to the one proposed by Li et al. [75], where a joint invariant of the stress tensor and fabric tensor is introduced to describe the angle between the loading direction and the material fabric direction, the properties of the material are assigned to the fabric tensor (such as the elasticity modulus and peak strength), thereby reflecting the calculation of the invariant of the noncoaxial angle, and the principal stress axis rotation effect of soil deformation and strength is applied, such as that attempted by Yao [76] for different geotechnical materials; (3) methods that introduce a yield surface equation, including the anisotropic influence of applying the reduced Lode stress angle method of Crouch et al. [77] to correct the yield surface shape function, which is related to the principal stress axis rotation, to compute the soil deformation and strength due to the principal stress axis deviation, such as that for clay by Huang et al. [70]; and (4) micromechanical analytical methods [78, 79], in which the deformation and strength of the soil are determined by the intergranular contact law and the distribution of intergranular contacts. The distribution of intergranular contacts is directly affected by the rotation of the principal stress axes. Then, the macro stress–strain relationship influenced by the rotation of the principal stress axes can be naturally obtained by homogenizing the mechanical properties of all contact planes (Figs. 1.17, 1.18, 1.19, 1.20 and 1.21).

q

e O

A Slow loading

CSL

Rapid loading

Creep

C

Stress relaxation

Stress relaxation

Stress relaxation

B

Creep C

Creep

' v

εa

B

A C

ln σ

(a)

Rapid loading

B

Slow loading

A

Rapid loading

Slow loading O

O

p

(b)

Fig. 1.17 Illustration of time-dependent behaviors: a under 1D and b 3D conditions

Before test

Before test

Before test

Before test

Before test

Before test

Fig. 1.18 Illustration of the different definitions of particle crushing amount

Before test

Passing (%)

20

1 Constitutive Relations of Saturated Soils: An Overview dM1

q

Particle crushing

No crushing

Particle size

No crushing

Crushing No crushing

(a)

e

εv (b)

CSL

εd

Crushing

Crushing (c)

CSL

logp’

Fig. 1.19 Illustration of the stress–strain response of crushable granular materials

Fig. 1.20 Sand–silt mixture packing with different fine contents (fc: fine contents)

Fig. 1.21 Critical state lines of three sand–silt mixtures: a critical state lines on the e-logp plane, b initial critical void ratio versus fine contents, and c normalized critical state lines

1.6 Induced Anisotropy and Noncoaxial Properties

21

Among all these methods, the micromechanical analytical method has the advantage of a clear physical meaning, and the multiscale modeling method can be developed in combination with the finite element method to better explain the behaviors of various geotechnical structures. The fabric tensor-based modeling will be introduced in Chap. 5, the rotation of the yield surface will be discussed in Chap. 6, and the micromechanics-based modeling will be presented in Chap. 9.

1.7

Time Dependency

The stress–strain relationship of soft soils shows significant time dependency, such as the loading rate effect: the magnitude of the shear strength and the magnitude of the preconsolidation pressure are largely dependent on the loading rate; creep: strain increases with time while maintaining constant stress; and stress relaxation: stress decreases with time while maintaining constant strain. Therefore, the shear strength and the preconsolidation pressure obtained under the laboratory standard loading rate are used as the engineering design basis without considering the rheological characteristics of the soil, which could lead to instability of the soft soil during the construction stage or long-term settlement after construction. This behavior has been widely recognized in the practice of soft soil engineering (such as bridgehead jumping due to differential settlements of foundations on soft soil) [80]. The key parameters of the time-dependent properties can be obtained by oedometer and triaxial apparatus through different types of rheological tests. (a) Creep tests: The viscoplastic strain formula (Eq. (1.17)) [52, 81, 82, 83] can be obtained through one-dimensional consolidation tests, as well as the secondary compression index Cae. Mesri and Goldleeski [84] found that the ratio of the secondary compression index Cae to the compression index Cc is generally between 0.025 and 0.1, with the highest ratio for peat, followed by organic soil, clay, and minimal sludge. Furthermore, Singh and Mitchell [85] studied the relationship between the drained creep rate and time of soft soil, and proposed the creep rate formula (Eq. (1.18)). e_ 1 ¼ Aeaq e_ 1 ¼ Aeaq

t m i

t t m i

t

ð1:17Þ ð1:18Þ

(b) Constant strain rate tests: Different from Kim and Leroueil [86], the viscoplastic strain formula (Eq. (1.19)) and the value of the parameter b in this formula can be derived using several constant strain rate tests [55, 57, 87]; furthermore, based on the similarity of the results of constant strain rate tests between 1D

22

1 Constitutive Relations of Saturated Soils: An Overview

Table 1.4 Some typical scaling functions of elasto-viscoplastic models

Rowe and Hinchberger [91]

Formula U(F) h d i p exp N pms  1 m h d i p exp N pms  1  1 m  d N

Hinchberger and Rowe [92]

 d N

Shahrour and Meimon [93]



References Adachi and Oka [89] Fodil et al. [90]

pm psm pm psm pdm psm

1 N 1

and 3D, the 1D formula can be further extended to the 3D formula (Eq. (1.20)) [55, 57, 88]. Thus, the formula related to the loading rate influence can be used to construct the scale function in viscoplastic models, and the scale function can also directly reflect the effects of the loading rate. Table 1.4 summarizes the scaling functions of several elastic viscoplastic models.
b k  j r0z r k r0p

ð1:19Þ


b k  j pdc 1 @fd k prc ð@fd =@p0 ÞK0 @r0ij

ð1:20Þ

e_ vp z e_ vp ij

¼

e_ rv

¼

e_ rz

(c) Stress relaxation tests: The relaxation law expressing the stress change with time at a constant strain can be obtained, and the test results showed a linear relationship in double-logarithmic coordinates depending on the relaxation coefficient Ra. Under 1D conditions, the relaxation coefficient, the secondary compression index Cae, and the rate coefficient b are directly connected [94]:

Ra ¼

k  j Cae ¼ kb k

ð1:21Þ

Based on these results, a 3D constitutive model can be built, and the model rheological parameters can be obtained. Different types of constitutive models have been proposed: (1) models based on the non-stationary flow surface theory [95–98]; (2) models based on the overstress theory [54, 56, 89, 90, 92, 99]; (3) models based on the extended overstress theory [52, 55, 57, 83, 88, 100]; and (4) models based on the framework of the bounding surface theory [101–103]. It should be noted that for the extended overstress theory, due to the lack of a relationship between the preconsolidation pressure and the strain rate at very low strain rates (dev/dt < 110−8 s−1), the correctness of the assumption in the elastic

1.7 Time Dependency

23

region cannot be directly verified. The advantage of the elastic viscoplastic model established under the framework of the bounding surface is that it can describe the overconsolidation characteristics of clay, but the size of the bounding surface is difficult to determine directly. However, sandy soils do not follow this classic pattern and are referred to as non-isotach materials [104–107]. Numerous triaxial tests with constant strain rates show that the effect of the strain rate on the stress–strain relationship of sand is nonsignificant [108, 109], which means that the stress–strain relationship can be considered independent of the strain rate. However, for triaxial tests with step changes (jumps) in strain rate, the stress may exceed the unique relationship owing to the strain acceleration. The stress approaches the unique stress–strain curve upon continuation of the test by increasing the axial strain at a constant strain rate [110–112]. Modeling of the time-dependent behavior of soft soils will be presented in Chap. 7 based on the viscoplasticity theory and that of sandy soils will be presented in Chap. 8 based on the hypoplasticity theory.

1.8

Temperature Dependency

The influence of temperature change on the engineering properties of soil is a major topic in the field of geotechnical engineering. It has important practical value in the fields of thermal energy storage, geothermal resource development, nuclear waste treatment, and heating pipeline design [113, 114]. In addition, it also has important applications in geothermal construction, oil exploration, hot injections, and other production activities, as well as thermal effects in the vicinity of high-voltage cables. Temperature changes (mainly between 5 and 95 °C) can cause pore water and soil particles to expand or contract, resulting in an increase in pore water pressure, which changes the microstructure of the soil, thereby affecting the plastic properties of the soil and causing soil softening or hardening [115]. The study of the effect of temperature on soil began in the 1960s. Campanella and Mitchell [116] conducted laboratory tests to study the thermal consolidation characteristics of saturated soils, the thermal expansion of soil components, and the thermal effects on pore water pressure under undrained conditions. The existing research methods are mainly based on experiments at controlled temperatures to establish a constitutive model [117–120] considered the thermal expansion of soil particles in finite element calculations. 8 > >
> : D ¼ De þ Dvp þ bh hd _ ij ij ij ij

ð1:22Þ

24

1 Constitutive Relations of Saturated Soils: An Overview

where Dij is the total strain tensor; T is the temperature; bh is the thermal expansion coefficient of the soil particles; and h_ is the differentiation of temperature over time. In addition, Laloui et al. [121] studied the influence of strain rate and temperature on the 1D consolidation characteristics of soils. The results showed that the yield stress is affected by both the strain rate and temperature. A formula relating the vertical yield stress to the strain rate and temperature was proposed, and this formula could simulate the thermal consolidation characteristics of soils: r0y ¼ f1 ð_evp v Þf2 ðTÞ

ð1:23Þ

Yashima et al. [122] introduced the temperature to the overstress-based flow rule by introducing the influence of temperature within the scale function. The existing constitutive models that consider thermal influence are based on elasto-viscoplasticity. It is assumed that the elastic part conforms to Hooke’s law, and that the influence of temperature mainly considers the thermal expansion effect, whose influence is considered differently in the developed models [114, 123, 124, 125, 126]. At present, the temperature effect on cohesive soil under complex conditions, the variation law of pore water pressure, the quantitative analysis of the influence of structural disturbance, the law of soil strain changes under the repeated actions of hot and cold cycles, and its reversible or nonreversible mechanisms remain to be further studied. Modeling of the temperature-dependent behavior of soils is not included in this book.

1.9

Water Chemical Composition Effect

In actual geotechnical engineering, soil is often affected by groundwater with erosive ions (such as SO42−, Cl−, CO32−, etc.), domestic sewage, seawater, and other environmental factors. Under these environmental conditions, the mechanical performance of soil will be affected to various degrees [126]. In construction projects, the design is generally carried out based on the strength of undisturbed soil. However, the water–soil chemistry after the excavation or the mechanical effect caused by the chemical action of highly corrosive wastewater is less considered. With the influence of water chemistry in soft soil, the microstructure and composition of the soil are modified, which may lead to an increase in the compressibility of soil and a decrease in the shear strength of soil. In addition, the minerals contained in the soil have colloidal properties and are electrically charged; a corresponding potential difference is generated when the soil is compressed, and different binding products are formed in different pore fluids of cemented or non-cemented soils, thereby changing the physical and chemical properties of the bonds, the mineral surface, and mineral composition; then, the structure of the soil and its strength are changed.

1.9 Water Chemical Composition Effect

25

In recent years, landfills have used clay as lining to prevent seepage, while municipal solid waste (MSW) has a variety of organic or inorganic chemical components that form after landfilling. Under certain driving forces, these mixed components move underground through the impermeable layer [127]. During the migration process, the leachate will undergo complex material conversion with soil media, soil microbes, and contaminated components, which are directly related to soil type, water content, pH value, temperature, clay, and organic matter content, which changes the physical and mechanical properties of the soil. In this process, the pH value, which has an effect on the adsorption and diffusion of certain chemical ions, plays a decisive role in the transformation of the properties of organic matter in water [127]. The multiple couplings and interactions that occur between water chemistry and soil structure, which are dependent on the composition of soil, are the most important factors restricting the strength, deformation, and permeability of soil, and the influential mechanisms are complicated. Because the experimental techniques still require improvements and the theory is not sufficiently mature, there are still many tasks that need to be explored and studied. Modeling of the water chemical composition effect on the mechanical behavior of soils is not included in this book.

1.10

Cyclic Effect

Cyclic loading due to earthquakes, waves, wind, and traffic can cause an increase in the pore water pressure and the accumulation of plastic deformation in soil, which could lead to the softening of stiffness and a decrease in the strength of the soil; these changes would cause a series of problems related to engineering safety and environmental geological issues. Therefore, it is very important to study the mechanical response of soils under cyclic loading. In one-way tests, the deviatoric stress q is cycled between zero and qc. The plastic strain during the first loading is important. It is clearly less so for the following cycles even though the plastic deformations continue and accumulate with the number of cycles. Volume changes depend on the position of the stress path with respect to the contractant domain. If the average level of the cycle lies within the contractant domain, cyclic loading produces compaction; otherwise, this loading produces dilation. A two-way test consists of alternating between axisymmetric compression and extension. This test involves a sudden 90° change in the directions of the major and minor principal stresses as they interchange. Stress rotation tends to cause compaction when the stress is close to the isotropic state and therefore well inside the contractant domain. The results of the two-way cyclic tests confirm this analysis, and progressive compaction is always observed irrespective of the amplitude and the average level of the cycles. The shape of the stress–strain cycles may be altered, and large cyclic strain amplitudes will develop during loading.

26

1 Constitutive Relations of Saturated Soils: An Overview

In undrained tests, variations in the mean stress will depend on the tendency of the soil to contract or dilate. Cyclic tests generally lead to an increase in the density of soils. In undrained tests, this will translate into an increase in pore pressure and therefore a decrease in mean stress. In granular materials, the decrease could be sufficiently large in certain cases to cause complete loss of stress: this is the phenomenon known as liquefaction, which is also called cyclic mobility. Liquefaction only occurs in two-way cyclic tests. One-way tests also produce decreases in mean stress, but these decreases tend to stabilize before the occurrence of liquefaction. In the case of clay, the pore pressure increases until the stress cycle meets the line of perfect plasticity q = Mp’. At this point, there is a large increase in the strain and a general rupturing of the material. Due to the low permeability of clay, the high-frequency cyclic loading conditions are considered to be undrained, and the foundation under long-term cyclic loading is considered to be in a partially drained state. Thus, the generation and dissipation of excess pore pressure are alternately repeated. Furthermore, the axes of the principal stress change during two-way cyclic loading. Different types of cyclic loads are accompanied by different forms of stress axis rotation. The basic characteristics of cyclic loading and the test methods are summarized in Table 1.5.

Table 1.5 Characteristics of three typical cyclic loadings and comparisons Loading type

Seismic

Wave

Traffic load

Cyclic stress

Cyclic stress is irregular, asymmetrical Two-way cyclic Mainly 1–10 Hz; soft soil layer is generally 1–5 Hz, and mainly 1–2 Hz Short-term load

Small cyclic stress amplitude Two-way cyclic Waves are generally 0.01–0.3 Hz, mostly less than 0.2 Hz Long-term load

Small cyclic stress amplitude One-way cyclic Subway: 0.5–2.5 Hz; high-speed railway: 5– 50 Hz, even 80 Hz Long-term load

Without rotation, there will be a 90° reverse

Continuous 180° rotation and constant dynamic stress

Continuous 180° rotation, dynamic rotation stress continuously changes

One/two way Frequency range Duration time Axis Rotation of principal stress axes Stress path

τ zθ

2β

Test equipment

τ zθ

τ vh 2β

2β

σ z -σ θ

σv −σh

2

2

Cyclic simple shear, dynamic triaxial, or hollow cylindrical torsional shear

Hollow cylindrical torsional shear

σ z -σ θ 2

Hollow cylindrical torsional shear

1.10

Cyclic Effect

27

The cyclic effects on soil mainly include three characteristics: cyclic strength, cyclic cumulative deformation, and cyclic cumulative pore water pressure. The cyclic characteristics of soils are mainly affected by the following factors: (a) Cyclic strain amplitude: The larger the magnitude of the cyclic stress is, the smaller the number of cycles required for failure; an exponential relationship between the number of cycles and the cyclic stress ratio can be proposed. (b) Number of loading cycles [128–132]: The permanent strain generally increases with the increase in the number of cycles in a one-way cycle, and the cyclic strain amplitude increases in the two-way cycles. (c) Static deviatoric stress [128, 133]: When increasing the static deviatoric stress, the cyclic strength drops rapidly; the cumulative plastic strain grows rapidly; the number of cycles to failure decreases; and the cyclic pore water pressure develops rapidly. (d) Load frequency in fine soils [129, 134, 135]: For initially isotropically consolidated samples, the cyclic strength increases with the increase in frequency; at a small number of cycles, the frequency has a significant influence on the shear strain; as the number of cycles increases, the influence of frequency decreases, and there is almost no influence when the soil is close to failure; under the same cycle number, the higher the frequency is, the smaller the excess pore water pressure is. Based on test results, many researchers have suggested a relationship between the accumulative strains and the number of cycles (Table 1.6). These empirical models were obtained by fitting the laboratory or field test results. The mathematical form is simple and convenient for engineering applications. However, this formula cannot reflect the overall stress–strain relationship of soils under cyclic loading. Because of this limitation, many researchers have improved and modified models based on the elastoplasticity theory to simulate the cyclic behavior of soils. These models can be divided into the following types: models based on the Masing rule [139]; models based on generalized plasticity [140]; multi-yield surface models

Table 1.6 Equations for the permanent deformation function of the number of cycles

Reference Parr et al. [136] Monismith et al. [132] Li and Selig [131] Chai and Miura [130] Huang et al. [137, 138]

Prediction formula   log e_e_N1 ¼ log C þ f log N ep ¼AN b  m ep ¼a qd =qf N b  m  n   1 þ qqfs Nb ep ¼a qqdf ep ¼ aD m N b  m ep ¼ a qd qþf qs qd N b  c p b ep ¼ aDm d pa N

28

1 Constitutive Relations of Saturated Soils: An Overview

based on the theory of kinematic hardening plasticity [141]; models based on bounding surface theory [6, 8, 53, 142, 143]; and micromechanics-based cyclic models [144]. Note that these models require the simulation of loading cycle by cycle, which requires a high computational expense for a high number of cycles. To solve this problem, multi-time-scale techniques such as the “time homogenization method” can be combined with a given constitutive model [145]. The modeling methods on the mechanical behavior of soils under cyclic loading can be found in Chaps. 5, 7, and 8.

1.11

Grading Dependency

1.11.1 Grain Breakage Effect Clast fragmentation (also termed grain breakage) can occur during compression and shearing of granular materials, especially under high confining stresses and dynamic loadings [146–151]. Important breakage can also occur under low confining stress, particularly if the grains are fragile and present irregular shapes (e.g., Nakata et al. [152]). Grain breakage is a fundamental issue for the stability of earthen dams, embankments, and railway ballasts and for the bearing capacities of piles (e.g., Daouadji et al. [143]; Okonta [153]; Winter et al. [154]). Quantitative information about the shearing history can be retrieved from glacial sediments [155] and granular landslide deposits [156], and this information can be crucial for understanding the likelihood of the latter to exhibit further instability [157–161]. Extensive fragmentation has also been related to frictional weakening in rock avalanches, although the issue is still debated [162–166], and further dedicated experimental investigations are required. The amount of fragmentation can be quantified through changes in the grain size distribution (GSD). Different ways of measuring the evolution of fragmentation have been suggested (e.g., Marsal [167]; Hardin [168]; Einav [169]). The effect of particle crushing on various physical and mechanical properties has been investigated (e.g., Biarez and Hicher [26], McDowell et al. [170]; Bandini and Coop [171]; Zhang et al. [172]) and introduced into constitutive and numerical modeling (e.g., Ma et al. [173]; Zhou [174], Daouadji et al. [175, 176], Yin et al. [177], Einav [169]). Understanding how the breakage amount can be measured and how this measure can be explicitly related to mechanical properties is still a challenge. The three following questions have been investigated: (1) What are suitable ways of measuring the amount of grain breakage? (2) How can these measures be determined based on stress–strain histories? (3) How can these measures be related to the mechanical properties of the granular assembly? Once these three points are solved, the measured grain breakage quantity can be introduced as a variable to control the influence of particle crushing on the mechanical properties of a granular material undergoing a generic stress–strain path.

1.11

Grading Dependency

29

For the first issue, a breakage measure that varies in the range of 0–1 with changes in the GSD seems convenient. Among the proposed measures, the modified relative breakage index B*r of Einav [169] and the relative uniformity Bu, which is based on the fractality of the GSD (e.g., Coop et al. [146]) and the coefficient of uniformity Cu, appear to be good choices. Various methods to quantify the GSD change under different loading conditions were developed and validated through tests conducted under monotonic loading, i.e., the crushing surface approach [178], and the energy approach [169]. These two approaches have not been fully examined under different loading conditions, including under cyclic loading. Changes in the GSD can be correlated with the evolution of mechanical characteristics. Biarez and Hicher [26] suggested that an increase in the grading index d60/d10 due to grain ruptures produces a displacement of the critical state line toward smaller void ratios in the e-p’ plane. Several properties (stress dilatancy, peak shear strength, plastic modulus, etc.) depend on the distance of the current stress state (p’, e) from the corresponding critical state (p’, ec) of the granular material. To reach the critical state during shearing, a larger volume change will be needed in the case of significant grain breakage. This phenomenon will also result in a change in the stress–strain relationship, and in particular, this phenomenon will result in an increase in the axial strain corresponding to the maximum strength. Hence, understanding how fragmentation-induced changes in GSD affect the location of the critical state line (CSL) becomes crucial for relating particle crushing to the mechanical behavior. CSL-GSD relationships have been demonstrated in experiments on crushable and non-crushable sands [171, 179] and in discrete element simulations on crushable granular materials [180].

1.11.2 Suffusion Effect Internal erosion occurs when fine particles are plucked off by hydraulic forces and transported through the coarse matrix. The known causes are either a concentration of leak erosion, backward erosion, soil contact erosion, or suffusion [181–183]. Indeed, during rainfall infiltration, eroded fines may lead to a loosened soil structure with degraded soil strength, and these fines may cause clogging in soil, reduce soil permeability and result in the buildup of excess pore pressure and a drop in effective stress, which results in soil instability [183]. The concurrent mechanisms of internal erosion with fine clogging (or self-filtration) are broadly known as suffusion. The suffusion-induced modification of the soil microstructure may lead to deformations at the macroscopic scale, and consequently, these deformations significantly influence the mechanical behavior of the soil. Much of the damage and failures of embankment dams can be associated with internal erosion [181–188] showed that internal erosion led to nearly 46% of the damage in 128 embankment dams. In addition, sinkholes and cavities triggered by internal erosion are frequently observed within dams and dikes [189, 190]. More recently, investigations from many disaster sites indicate that one major mechanism accounting for the initiation

30

1 Constitutive Relations of Saturated Soils: An Overview

of landslides and debris flows is the mitigation of fine particles through broadly graded or gap-graded soils during rainfall infiltration [183, 191, 192, 193, 194, 195]. Indeed, during rainfall infiltration, eroded fines may lead to a loosened soil structure with degraded soil strength, and they may cause clogging in soil, reduce soil permeability, and result in the buildup of excess pore pressure and a drop in effective stress, which results in soil instability [183]. Some attempts have also been made to model the mechanical response to internal erosion by removing particles in granular materials. Wood et al. [196] modeled the mechanical consequences of internal erosion with a two-dimensional discrete element analysis and found that internal erosion changes the density state of the material from dense to loose as they compared the current soil state to the critical state line. Scholtes et al. [197] developed a three-dimensional discrete element model and an analytical micromechanical model to describe the mechanical responses induced by particle removal in granular materials. The researchers found that the mechanical behavior of the soil changes from dilative to contractive with the removal of soil particles. However, this removal was based on the size of the particles and their degree of interlocking, and the fluid phase was not considered. Hosn et al. [198] studied the macroscopic response during internal erosion and the post-internal erosion properties of granular materials by particle removal, considering a one-way coupling between the fluid and the solid phases via the discrete element and the pore-scale finite volume methods. The erosion process was simplified into the definition of detached and transportable particles by calculating the unbalanced fluid force and defining a controlling constriction size. Hicher [199] presented a numerical method to predict the mechanical behavior of granular materials subjected to particle removal and concluded that the removal of soil particles may cause diffuse failure in eroded soil masses. Recently, the suffusion effect on the mechanical behavior of granular soils was modeled by Chang and Yin [200] and Yin et al. [201, 202] by introducing the variation in fine content into the constitutive models. More specifically, the critical state line (CSL) changed with the changes in fine content in the soil. Assuming that fine particles and coarse particles are from the same origin (same mineralogical components, same particle shape, and so on), only the position of the CSL on the elogp’ plane varies with the fine content. The model has recently been applied to simulate the behavior of a dike subjected to internal erosion [203]. A brief introduction to the modeling methods of the grading-dependent behavior of granular soils due to particle crushing and suffusion can be found in Chap. 6.

1.12

1.12

Additional Mechanical Properties of Natural Soft Soils

31

Additional Mechanical Properties of Natural Soft Soils

The arrangement of soil particles and the interparticle cementation, i.e., the inherent anisotropy and the structure formed by natural soft soils during geological deposition, have a significant effect on the mechanical properties of soil. A quantitative evaluation can be conducted through a comparison between undisturbed and remolded soil samples.

1.12.1 Inherent Anisotropy The inherent anisotropy creates differences in the stress–strain relationships among soils loaded in different directions, which primarily affects the material stiffness and shear strength. The comparison between the triaxial compression and extension tests can be used to obtain the comparison between the two extreme cases. The elastic modulus in different directions and the anisotropic distribution of the failure surface on the p-plane can be obtained by true triaxial tests [204, 205]. The anisotropic distribution of shear strength under different principal stress rotation angles can also be obtained by hollow cylindrical torsional shear tests (Fig. 1.22). A sedimentary soil can usually be considered to be a transversely isotropic material. According to Graham and Houlsby [206], the elastic stress–strain relationship can be written as follows:

Fig. 1.22 Experimental results of shear strength versus principal stress orientation for intact natural clays

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1 Constitutive Relations of Saturated Soils: An Overview

3 2 1=E e_ x v 0 t =E 6 e_ y 7 6 v vv 6 7 6 t0vv =Ev 6 e_ z 7 6 6 6 e_ 7 ¼ 6 0 6 xy 7 6 4 e_ 5 4 0 yz e_ zx 0 2

t0vv =Ev 1=Eh t0vh =Eh 0 0 0

t0vv =Ev t0vh =Eh 1=Eh 0 0 0

0 0 0 1=2Gvh 0 0

0 0 0 0  1 þ t0vh =Eh 0

32 3 0 r_0 x 6 _0 7 0 7 76 r y 7 6 _0 7 0 7 76 r z 7 6 0 7 0 7 76 r_ xy 7 0 54 r_0 yz 5 1=2Gvh r_0 zx

ð1:24Þ If the ratio of the horizontal and vertical elastic moduli q = Eh/Ev is defined, Graham and Houlsby [206] derive the values of t‘vh = t‘vh/q0.5 and 2Gvh= q0.5 eV/ (1 + t‘vh). Therefore, compared to an isotropic material, an additional parameter q is required. Since the compressive modulus of an anisotropic soil varies according to the direction of loading, the deformation in each direction is different in the case of isotropic compression. The dependence of the compressive modulus of soil on the direction of loading can also be linked to the degree of compression in each direction and the variation in the corresponding dilatancy; the shear strength and the position of the critical state line will therefore be dependent on the direction of loading, which can be modeled by the maximum value of the rotational hardening of the yield surface: 
 
    p sij 3sij _ a_ ¼ l xd a 0  a e_ pd  þ  a e v p 4p0

ð1:25Þ

For some soft soils, the initial or inherent anisotropy of the shear strength is also reflected in the anisotropy of the friction angle. This aspect of the soil behavior can be modeled by applying a material fabric tensor that introduces the dependence of the friction angle /c on the orientation of the principal stress axes [78, 79].

1.12.2 Soil Structure and Destructuration In the long-term geological deposition process, more or less cementation between soil particles can be generated. If this cementation occurs early in the depositional history, a structure with a large void will be created [207]. When natural soft clay is loaded, the cementation is progressively destroyed, and the macroporous structure between the soil particles will gradually breakdown, resulting in a change in the stress–strain relationship in the soil and a sharp decrease in soil strength, which in turn causes the deformation rate to increase. This phenomenon creates a large risk of instability in soft soil engineering works or damage due to large deformations. The evaluation of the soil structure of soft soil can be directly achieved through the sensitivity test, while the

1.12

Additional Mechanical Properties of Natural Soft Soils

33

progressive evolution of the soil structure is usually obtained through conventional one-dimensional consolidation and triaxial shear tests. The results from a large number of oedometer tests show that the structure of natural soft soils enhances its yield stress, which includes the influences of both the preconsolidation pressure and the amount of cementation; however, when the external force exceeds the yield stress, the soil structure will be progressively destroyed as deformation increases [207], and the stress–strain relationship evolves to reach that corresponding to a material without cohesion. To simulate the behavior of natural soils, Hong et al. [208] proposed the field state line (FSL) in double-logarithmic coordinates to analyze the one-dimensional compression characteristics of structured clay. The expression is simple and easy to use. However, the model expressed in double-logarithmic form is less consistent with the semilogarithmic curve concept of the Cam-Clay model. Liu et al. [209] applied the disturbed state concept to analyze the difference in void ratio between remolded and undisturbed soil samples and established a theoretical calculation method that can reflect the structural damage under one-dimensional conditions. Zhu et al. [210] proposed a method that uses a simple disturbance state model to simulate the one-dimensional compression of structured soils (Fig. 1.23) [210].  p  Dep q De rz ¼ 1 þ v0 e 1 þ e0 rpi0 expð Þ when rz [ rp0 ki  j

ð1:26Þ

Figure 1.24 illustrates the behavior of intact soft clay based on a large number of oedometric test results. In three-dimensional conditions, yield surfaces similar in shape but different in size in the p’-q plane can be obtained by triaxial compression tests on remolded and undisturbed soil samples, respectively. Based on the one-dimensional intrinsic yield stress, an intrinsic yield surface can be inferred. For the undrained triaxial shear test, the stress decreases sharply with strain as the stress

Fig. 1.23 Illustration of 1D compression behaviors for intact and remolded clays

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1 Constitutive Relations of Saturated Soils: An Overview

Fig. 1.24 Illustration of the extension of destructuration from 1D to 3D

K0 oedometer test

e

Reconstituted sample

χ = f ( χ 0 , ε vp )

Natural sample K0

χ = f ( χ 0 , ε vp , ε dp ) p’

CSL

q

Current yield q surface

Undrained shear

K0 0

p’ CSL

0

εd

Intrinsic yield surface

Table 1.7 Equations of destructuration References Gens and Nova [211] Rouainia and Muir Wood [212] Kimoto and Oka [213]

Equation

   v ¼ v0 exp n epv  þ nd epd
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2 þ e0 ð1  bÞðepv Þ þ b epd v ¼ v0 exp n 1kj  qffiffiffiffiffiffiffiffi v ¼ v0 exp n epij epij

reaches the peak, which reflects the softening characteristics of soft soils with shear strain, accompanied by the degradation of the current yield surface to the intrinsic yield surface. Therefore, the structural destructuration is affected by volumetric and shear strain under three-dimensional conditions. From this viewpoint, many researchers have proposed different constitutive equations (Table 1.7). More details can be found in Chap. 6.

1.13

Current Difficulties in the Practice of Constitutive Modeling

All major projects have strict requirements for deformation amplitude, that is, the soil within a certain range around the structure is required to be in a small-strain state. Since the design parameters are generally based on conventional test results, the geotechnical engineer uses a stiffness value that is less than the actual stiffness value when predicting and analyzing the soil deformation around a civil engineering structure. On the one hand, the predicted deformation is too large, the

1.13

Current Difficulties in the Practice of Constitutive Modeling

35

Fig. 1.20 Starting interface of the MATLAB environment installation

Fig. 1.21 Completed interface of the MATLAB environment installation

underground structure support system is too conservative, and the cost is greatly increased. On the other hand, the predicted earth pressure is too small, resulting in a decrease in the safety factor of the design or even an engineering accident. Therefore, an accurate evaluation of the small-strain stiffness characteristics of the soil, usually including the two ranges of very small strain and small strain, is of great significance for accurately predicting soil deformation around the structure, which controls the engineering cost and prevents engineering disasters. For this purpose, the following sections of the book focus on the small-strain stiffness characteristics based on experiments, theories, and micromechanisms, from a computing platform to engineering applications.

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1 Constitutive Relations of Saturated Soils: An Overview

During recent decades, an increasing number of onshore and offshore large-scale infrastructure constructions have continuously taken place, resulting in many geotechnical engineering problems. However, the geotechnical engineering practice is far behind the theoretical developments in soil mechanics. On the one hand, the suggested innovations are considered to be too complex for engineering practice; on the other hand, engineers and students lack a good platform for learning, training, and practice. In recent years, more refined constitutive models have been continuously developed for more complex simulations, and at the same time, their complexity has increased, which brings more difficulties to engineers in terms of in-depth understanding, as well as difficulties in the parameter calibration, undoubtedly making it difficult to use constitutive theories in geotechnical engineering. Thus, a simple, easy-to-use, and conceptually clear simulation platform becomes necessary to bridge the gap between research and practice in geotechnical engineering.

1.14

Development of the ErosLab Modeling Platform in Practice

1.14.1 Introduction of the Constitutive Modeling Platform ErosLab is a practical and simple software for simulating laboratory tests, which currently includes three common test types and five constitutive models. ErosLab can be used for all kinds of numerical simulations of laboratory tests, and it offers a comparison between simulations and experimental data, which is helpful for selecting the best model with relevant parameters. ErosLab is one of GeoInvention’s latest software developments. GeoInvention Studio was created by Dr. Zhen-Yu YIN, who is also in charge of development. Dr. Yin-Fu JIN is in charge of the technical aspects. In the studio, there are also several software developers and senior researchers in geomechanics and geotechnics, which enables rapid responses to client demands and helps to solve practical engineering problems. The studio aims to share the latest scientific achievements in geomechanics and geotechnics, promote the application of these achievements, and thus realize scientific innovations in geotechnics.

1.14.2 Installation and Operating Environment The main program of ErosLab is an executable file that can run directly in most Windows Systems with minor requests for the operating environment. The results can be obtained directly by using the ErosLab platform. Note that the system requires a Microsoft.NET Framework 4.0 or newer version. If the version of the

1.14

Development of the ErosLab Modeling Platform in Practice

37

Microsoft.NET Framework is older than 4.0, the user can download the advanced version from https://www.microsoft.com/en-hk/download/details.aspx?id=17851. To guarantee the normal operation of ErosLab without installing the FORTRAN program, three FORTRAN environment files, “libifcoremd.dll”, “libmmd.dll”, and “msvcr100.dll”, are provided in the installation package. Note that the current version requires installation of “Intel Fortran” (installation of Visual_studio_ 2010 and Intel Visual Fortran Composer XE 2011 is recommended). The MATLAB environment is needed to plot the results. The version of the MATLAB environment adopted is “MCR_R2016b_win64_installer.exe”, which is free to download from the official website and free for use. For convenience, this program named “Matlab_env.exe” is already provided in the installation package. The starting interface of the MATLAB environment installation is shown in Fig. 1.23. By clicking “Next”, the program will automatically download and install the required files. During the installation, it is important to maintain a network connection. The completed interface is shown in Fig. 1.24.

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14. Roscoe K, Schofield A, Thurairajah A (1963) Yielding of clays in states wetter than critical. Geotechnique 13 (3):211.240 15. Nova R, Wood D (1982) A constitutive model for soil under monotonic and cyclic loading. Soil mechanics-transient and cyclic loading:343–373 16. Jefferies M (1993) Nor-Sand: a simple critical state model for sand. Geotechnique 43 (1):91.103 17. Manzari MT, Dafalias YF (1997) A critical state two-surface plasticity model for sands. Geotechnique 47(2):255–272 18. Gajo A, Wood DM (1999) Severn-Trent sand: a kinematic-hardening constitutive model: the qp formulation. Geotechnique 49(5):595–614 19. Li X-S, Dafalias YF, Wang Z-L (1999) State-dependant dilatancy in critical-state constitutive modelling of sand. Can Geotech J 36(4):599–611 20. Wan RG, Guo J (2001) Drained cyclic behavior of sand with fabric dependence. Journal of engineering mechanics 127(11):1106–1116 21. Yang Y, Muraleetharan K (2003) The middle surface concept and its application to the elasto-plastic behaviour of saturated sands. Géotechnique 53 (4):421.431 22. Taiebat M, Dafalias YF (2008) SANISAND: Simple anisotropic sand plasticity model. Int J Numer Anal Methods Geomech 32(8):915–948 23. Taylor DW (1948) Fundamentals of soil mechanics. Soil Sci 66(2):161 24. Ishihara K, Towhata I (1983) Cyclic behavior of sand during rotation of principal stress axes. In: Studies in Applied Mechanics, vol 7. Elsevier, pp 53–73 25. Luong M Stress-strain aspects of cohesionless soils under cyclic and transient loading. In: Proc., Int. Symp. on Soils under Cyclic and Transient Loading, 1980. Balkema Rotterdam, The Netherlands, pp 315–324 26. Biarez J, Hicher PY (1994) Elementary mechanics of soil behaviour: saturated remoulded soils. AA Balkema, 27. Cole K, Burland JB Observation of retaining wall movements associated with a large excavation. In: Fifth Eur Conf On Soil Proc/Sp/, 1972 28. St John H (1975) Field and Theoretical Studies of the Behaviour of Ground Around Deep Excavations in London Clay. University of Cambridge, 29. Clayton C, Edwards A, Webb M (1991) Displacements within the London Clay during construction 30. Marsland A, Eason B Measurement of Displacements in the Ground below Loaded Plates in Deep Boreholes. In: This paper was originally published in Proceeding of the British Geotechnical Society Symposium on Field Instrumentation, held May 30-June 1, 1973 at the Institution of Electrical Engineers, London., 1973 31. Simpson B, O’Riordan N, Croft D (1979) A computer model for the analysis of ground movements in London Clay. Geotechnique 29(2):149–175 32. Jardine R, Fourie A, Potts D, Burland J (1986) Studies of the influence of non-linear stress-strain characteristics in soil-structure interaction. Geotechnique 36(3):377–396 33. Mair R Unwin Memorial Lecture 1992 Developments in geotechnical engineering research: application to tunnels and deep excavations. In: Proceedings of the Institution of Civil Engineers. Civil Engineering, 1993. pp 27–41 34. Atkinson J, Sallfors G Experimental determination of stress-strain-time characteristics in laboratory and in situ tests. In, 1991. pp 915–956 35. Clayton C (2011) Stiffness at small strain: research and practice. Geotechnique 61(1):5–37 36. Benz T (2007) Small-strain stiffness of soils and its numerical consequences, vol 5. Univ, Stuttgart, Inst. f. Geotechnik 37. Hertz H (1882) Über die berührung fester elastische Körper und über die Harte. Verhandlungen des Vereins zur Beförderung des Gewerbefleisses 38. Iwasaki T, Tatsuoka F (1977) Effects of grain size and grading on dynamic shear moduli of sands. Soils Found 17(3):19–35 39. Hardin BO, Black W (1969) Closure on vibration modulus of normally consolidated clay. Journal of Soil Mechanics & Foundations Div

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184. Xu Y, Zhang L (2009) Breaching parameters for earth and rockfill dams. Journal of Geotechnical and Geoenvironmental Engineering 135(12):1957–1970 185. Zhang L, Chen Q (2006) Seepage failure mechanism of the Gouhou rockfill dam during reservoir water infiltration. Soils Found 46(5):557–568 186. Zhang L, Xu Y, Jia J (2009) Analysis of earth dam failures: A database approach. Georisk 3 (3):184–189 187. Rönnqvist H, Fannin J, Viklander P (2014) On the use of empirical methods for assessment of filters in embankment dams. Géotechnique Letters 4(4):272–282 188. Foster M, Fell R, Spannagle M (2000) The statistics of embankment dam failures and accidents. Can Geotech J 37(5):1000–1024. https://doi.org/10.1139/t00-030 189. Muir Wood D (2007) The magic of sands — The 20th Bjerrum Lecture presented in Oslo, 25 November 2005This paper represents the written version of the 20th Bjerrum Lecture. While it has been edited for the present publication, it retains the general structure of the original lecture, which was intended for a general geotechnical audience. It deliberately points out some of the still-open questions concerning soil modelling. The Bjerrum Lecture is presented in Oslo in alternate years by the Norwegian Geotechnical Society with the support of the Bjerrum Memorial Fund (Laurits Bjerrums Minnefond). Canadian Geotechnical Journal 44 (11):1329–1350. https://doi.org/10.1139/t07-060 190. Sterpi D (2003) Effects of the erosion and transport of fine particles due to seepage flow. Int J Geomech 3 (1):111.122 191. Crosta G, Prisco Cd (1999) On slope instability induced by seepage erosion. Can Geotech J 36(6):1056–1073 192. Chang D, Zhang L, Cheuk J (2014) Mechanical consequences of internal soil erosion. HKIE Transactions 21(4):198–208 193. Cui P, Guo C-x, Zhou J-w, Hao M-h, Xu F-g (2014) The mechanisms behind shallow failures in slopes comprised of landslide deposits. Eng Geol 180:34–44 194. Ke L, Takahashi A (2014) Experimental investigations on suffusion characteristics and its mechanical consequences on saturated cohesionless soil. Soils Found 54(4):713–730 195. Gao L, Zhang L, Cheung R (2018) Relationships between natural terrain landslide magnitudes and triggering rainfall based on a large landslide inventory in Hong Kong. Landslides 15(4):727–740 196. Muir Wood D, Maeda K, Nukudani E (2010) Modelling mechanical consequences of erosion. Géotechnique 60(6):447–457 197. Scholtès L, Hicher P-Y, Sibille L (2010) Multiscale approaches to describe mechanical responses induced by particle removal in granular materials. CR Mec 338(10–11):627–638 198. Hosn RA, Sibille L, Benahmed N, Chareyre B (2018) A discrete numerical model involving partial fluid-solid coupling to describe suffusion effects in soils. Comput Geotech 95:30–39 199. Hicher P-Y (2013) Modelling the impact of particle removal on granular material behaviour. Géotechnique 63(2):118 200. Chang CS, Yin ZY (2011) Micromechanical modeling for behavior of silty sand with influence of fine content. Int J Solids Struct 48(19):2655–2667. https://doi.org/10.1016/j. ijsolstr.2011.05.014 201. Yin Z-Y, Zhao J, Hicher P-Y (2014) A micromechanics-based model for sand-silt mixtures. Int J Solids Struct 51(6):1350–1363 202. Yin Z-Y, Huang H-W, Hicher P-Y (2016) Elastoplastic modeling of sand–silt mixtures. Soils Found 56(3):520–532 203. Yang J, Yin ZY, Laouafa F, Hicher PY (2019) Internal erosion in dike-on-foundation modeled by a coupled hydromechanical approach. Int J Numer Anal Methods Geomech 43 (3):663–683 204. Kirkgard M, Lade PV (1993) Anisotropic three-dimensional behavior of a normally consolidated clay. Can Geotech J 30(5):848–858 205. Callisto L, Calabresi G (1998) Mechanical behaviour of a natural soft clay. Geotechnique 48 (4):495–513

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206. Graham J, Houlsby G (1983) Anisotropic elasticity of a natural clay. Geotechnique 33 (2):165–180 207. Burland J (1990) On the compressibility and shear strength of natural clays. Geotechnique 40(3):329–378 208. Hong Z, Tsuchida T (1999) On compression characteristics of Ariake clays. Can Geotech J 36(5):807–814 209. Liu MD, Carter JP, Desai CS (2003) Modeling compression behavior of structured geomaterials. Int J Geomech 3 (2):191.204 210. Zhu Q-Y, Yin Z-Y, Wang J-H, Xia X-H (2012) One Dimension Compression Model for Natural Clays Considering Structure Disturbance. Journal of Civil, Architectural & Environmental Engineering 34(3):28–34 211. Gens A, Nova R Conceptual bases for a constitutive model for bonded soils and weak rocks. In Geotechnical Engineering of Hard Soils-Soft Rocks, Athens, Greece, Anagnostopoulos et al. (eds). Balkema: Rotterdam. 1993; 485–494 212. Rouainia M, Muir Wood D (2000) A kinematic hardening constitutive model for natural clays with loss of structure. Géotechnique 50(2):153–164 213. Kimoto S, Oka F (2005) An elasto-viscoplastic model for clay considering destructuralization and consolidation analysis of unstable behavior. Soils Found 45(2):29–42 214. Hueckel T, Baldi G (1990) Thermoplasticity of saturated clays: experimental constitutive study. Journal of geotechnical engineering 116(12):1778–1796 215. Yao Y, Hou W, Zhou A (2009) UH model: three-dimensional unified hardening model for overconsolidated clays. Geotechnique 59 (5):451.469 216. Bonelli S, Marot D On the modelling of internal soil erosion. In: The 12th international conference of international association for computer methods and advances in geomechanics (IACMAG), 2008. p 7 pages

Chapter 2

Fundamentals of Stress and Strain

Abstract The essence of soil constitutive modeling is the establishment of the stress–strain relationship for soils. In this chapter, the continuum hypothesis of soil as an assembly of grains is first introduced, then the stress and strain concepts, as well as their expressions of some important stress or strain variables used in constitutive modeling are briefly discussed, and finally, the practical stress–strain analysis module developed in the ErosLab platform is presented.

2.1

Hypothesis of a Continuous Medium for Soils

Soil is made up of particles. Therefore, the first consideration is whether the size of the sample used in the test satisfies a certain size relative to the size of the particle so that the actual force and displacement can be related to the stress and strain of the virtual continuous medium within an acceptable range [1]. Similar to continuous solid materials, the stress of the soil is still defined as the amount of force per unit area r = F/A. However, considering the size limit of the soil particles, the particles always have a certain size, so area A cannot approach zero. Therefore, the question is as follows: How much does A need to be to obtain an approximation of the virtual continuum stress relative to the size of the particle? Gourves [1] measured the resultant force applied to the plates by the cylinders, indicating that as the size of the plates is reduced, the dispersion of the resultant forces on the plates is greatly increased. To make the coefficient of variation in the average stress less than 10%, it is necessary to use a cylindrical rod with a diameter of 0.2–0.5 cm, and the diameter of the plate should be larger than 5 cm (the diameter of the cylindrical rod is 10 times or more). If the cylindrical rod is replaced by a three-dimensional particle, the coefficient of variation will be smaller. Reflected in the actual situation, the sample size in the test must be at least ten times the maximum particle size (Fig. 2.1). In addition to meeting the measured displacement relative to the particle size, which should be sufficiently long, further considerations are needed. For example, in addition to compressive strain, particle rotation or slippage between particles can © Springer Nature Singapore Pte Ltd. and Tongji University Press 2020 Z.-Y. Yin et al., Practice of Constitutive Modelling for Saturated Soils, https://doi.org/10.1007/978-981-15-6307-2_2

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Fig. 2.1 Assembly of particles to a virtual continuum

also increase strain. Now, assume that two particles touch two adjacent points on either side of the point. Normally, these two points will not always remain adjacent during motion. Gourves [1] describes complex trajectories, indicating that the virtual average points in the virtual continuum remain adjacent to each other and can be defined as strain tensors. In the case where the above stress and strain definition requirements are met, the soil can be idealized as a continuous medium, i.e., a virtual continuous medium. Continuum mechanics can be applied to special aggregate media where these particles are in continuous contact. In practical applications, it is necessary to determine the stress–strain relationship and its parameters, which are usually derived from the relationship between the force and displacement of the specimen or the field in situ test. We hope to reproduce the load path of the soil at different locations during construction and after the project. However, due to the limitations of the test instrument, the path implemented in the test is much simpler than that in an actual situation. Under the existing theoretical framework, the expression of the mechanical properties of soils requires the use of multiple sets of experimental data, which can reflect the characteristics of different stress paths. The results obtained in this way are combined to represent the constitutive relationship of soil. According to this hypothesis, the stress or strain analysis as described in many books of solid mechanics (e.g., Chen et al. [2]; Luo et al. [3]; Zhang and Ye [4]) can be adopted as the fundamentals of the constitutive modeling of soils.

2.2 Stress Analysis

2.2 2.2.1

49

Stress Analysis Stress State of a Point in Soil

By taking an infinitesimal cubic cell on an arbitrary point from a soil element and putting it in a three-dimensional coordinate system with three mutually orthogonal axes, the stress state can be shown in Fig. 2.2. There are three normal stress components (rx, ry, rz) and six shear stress components (sxy, syx, syz, szy, szx, sxz) on the six faces of the cubic cell. In soil mechanics, the normal stress is positive for compression and negative for extension. For the shear stress, on the face that is in accordance with axial directions, it is positive when contrary to the positive axial direction, while it is negative in the other direction. As shown in Fig. 2.2, both normal and shear stresses are positive. The magnitude of these nine stress components is not only related to the stress state but also to the direction of the coordinate axis, which is called the Cauchy stress tensor. 2

rx sxy sxz

3

2

rxx rxy rxz

3

2

r11 r12 r13

3

6 7 6 7 6 7 rij ¼ 4 syx ry syz 5  4 ryx ryy ryz 5  4 r21 r22 r23 5 szx szy rz rzx rzy rzz r31 r32 r33

ð2:1Þ

It can be derived from the moment equilibrium that sxy= syx, syz= szy and sxz= szx. Therefore, the stress state of a single element can be described by using six independent stress components. In constitutive model programming, the stress tensor is usually expressed as follows: rij ¼ ½ rxx

ryy

Fig. 2.2 Schematic diagram for the stress state at a point

rzz

rxy

ryz T

rxz

σz

z

τzx τxz σx

O

x

ð2:2Þ

τzy τyz τxy

σy

τ yx

y

50

2 Fundamentals of Stress and Strain

2.2.2

Mean Stress and Deviatoric Stress Tensor

If rm (or p) is defined as the average normal stress or mean stress, rm ¼

 1 rxx þ ryy þ rzz 3

ð2:3Þ

Then, the stress tensor can be transformed as follows: 2 6 rij ¼ 4

rxx  rm syx szx

sxy

sxz

3

2

rm

7 6 ryy  rm syz 5 þ 4 0 0 szy rzz  rm

0

0

3

7 rm 0 5 0 rm

ð2:4Þ

The first tensor in the equation is called the deviatoric stress tensor, while the second tensor is called the spherical stress tensor. The spherical stress tensor can be abbreviated as rm dij or pdij , where dij is the Kronecker symbol (when i ¼ j; dij ¼ 1; when i 6¼ j; dij ¼ 0). The deviatoric stress tensor can be expressed as follows: sij ¼ rij  rm dij 2 sxy rxx  rm 6 ¼4 syx ryy  rm szx

2.2.3

szy

3 2 3 2 3 sxx sxy sxz s11 s12 s13 sxz 7 6 7 6 7 syz 5¼4 syx syy syz 5¼4 s21 s22 s23 5 ð2:5Þ rzz  rm szx szy szz s31 s32 s33

Principal Stress and Invariants of the Stress Tensor

For the unit body shown in Fig. 2.2, we can go in the direction of a certain coordinate axis, just so that each surface has only normal stress and no shear stress. These three surfaces are called the principal plane of stress, and the normal stress on the principal plane is called the principal stress. Let the direction cosine on the principal plane be l, m, n, and the principal stress and the main direction can be obtained according to the vector operation: 8 8 > > < rx l þ sxy m þ sxz n ¼ rl < ðrxx rÞl þ sxym þ sxz n ¼ 0 syx l þ ry m þ syz n ¼ rm , syx l þ ryy  r m þ syz n ¼ 0 > > : : szx l þ szy m þ rz n ¼ rn szx l þ szy m þ ðrzz  rÞn ¼ 0

ð2:6Þ

The three simultaneous linear equations are homogeneous for l, m, and n. To obtain a nonzero solution, the coefficient determinant must be zero by Cramer’s rule:

2.2 Stress Analysis

51

   rxx  r sxy sxz     syz  ¼ 0 , r3  I1 r2 þ I2 r  I3 ¼ 0  syx ryy  r    szx szy rzz  r 

ð2:7Þ

The above formula is a one-dimensional cubic equation; three roots are three principal stresses r1, r2, and r3; I1, I2, and I3 are the coefficients of this one-dimensional cubic equation, whether the cubic equation is directly derived from the x–y–z coordinate system or the principal direction of stress is derived in the same way, meaning that the rotation of the coordinate axes does not change the magnitude of their I1, I2, and I3 values, so they are called the first, second, and third invariants of the stress tensor: I1 ¼ rxx þ ryy þ rzz        rxx sxy   ryy syz   rzz szx        I2 ¼  þ þ  ¼ rxx ryy þ ryy rzz þ rzz rxx  s2xy  s2yz  s2zx  syx ryy   szy rzz   sxz rxx     rxx sxy sxz      I3 ¼  syx ryy syz  ¼ rxx ryy rzz þ 2sxy syz szx  rxx s2yz  ryy s2zx  rzz s2xy    szx szy rzz 

ð2:8Þ Solving the high-order Eq. (2.7), the principal stresses r1, r2, and r3 can be obtained as follows: rffiffiffiffiffi 9 2 I1 J2 > r1 ¼ p þ q cos h > cos h r1 ¼ þ 2 > > 3 3 > 3 > > rffiffiffiffiffi  2 2p = pffiffiffi I1 J2 q cos h  r ¼ p þ ð2:9Þ r2 ¼  cos h  3 sin h or 1 3 3 > > 3 3 > >  > > r ffiffiffiffi ffi > > >   2 2p > > > pffiffiffi > > r1 ¼ p þ q cos h þ : r3 ¼ I1  J2 cos h þ 3 sin h ; 3 3 3 3 " # pffiffi 2I 3 þ 27I 9I I I 2 3I where h ¼ 13 arccos 3 2 3 J33 ð0  h  pÞ; J2 ¼ 1 3 2 ; J3 ¼ 1 273 1 2 (see details 8 > > > > > > >
> l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 > A þ B2 þ C 2 > > < B m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > A2 þ B2 þ C 2 > > > > C > > : n ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A þ B2 þ C 2

ð2:10Þ

where    sxy   sxz    ¼ sxy syz  sxz ryy  r A¼  ryy  r syz    rxx  r sxz   ¼ sxy sxz  syz ðrxx  rÞ B ¼  > sxy syz  > >   > >  rxx  r >     sxy  >   ¼ ðrxx  rÞ ryy  r  sxy 2 > C ¼ :  s  ryy  r xy 8 > > > > > > > >
> > <   1 1 2 ð2:12Þ sxx þ s2yy þ s2zz þ 2s2xy þ 2s2xz þ 2s2yz J2 ¼ sij sji ¼ > 2 2 > > : J ¼ s s s ¼ 2s s s  r s2  r s2  r s2 3

xx yy zz

xy yz xz

xx yz

yy xz

zz xy

The invariants of the deviatoric stress tensors J1, J2, and J3 are related to the invariants of the stress tensors I1, I2, and I3 through the following relations: 8 > > > >
> >   > : J3 ¼ 1 2I 2  9I1 I2 þ 27I3 27 1

ð2:13Þ

2.2 Stress Analysis

53

where the deviatoric stress q can be calculated by using the second invariant of the deviatoric stress tensor J2. q¼

pffiffiffiffiffiffiffi 3J2

ð2:14Þ

In a triaxial test, the deviatoric stress q can be simplified to q = |ra – rr|, or q = ra – rr to distinguish between compression and extension conditions. The Lode angle h can be calculated by using the invariants of the deviatoric stress tensor as follows: pffiffiffi 3 3 J3 cos 3h ¼ 2 J 32

ð2:15Þ

2

For a conventional triaxial compression test with r2 = r3, b = 0, and h = 0°; for a conventional triaxial extension test with r2 = r1, b = 1, and h = 60°; and when r2 ¼ðr1 þ r3 Þ=2, b = 0.5, and h = 30°. Note that b is the parameter of intermediate principal stress and is defined as b ¼ ðr2  r3 Þ=ðr1  r3 Þ).

2.2.5

Principal Stress Space and p-Plane

If the three principal stresses r1, r2, and r3 are used as the coordinate axes, the composed three-dimensional space is called the principal stress space. In this space, the stress state of r1 = r2 = r3 = rm is under the condition of isotropic spherical stress. Its trajectory is a straight line passing through the origin and having the same angle with each coordinate axis, called the space diagonal, also called the isocline (Fig. 2.3). A plane perpendicular to the diagonal of the space is called the deviatoric plane, or the p-plane (in traditional plasticity theory, only the deviatoric plane passing through the origin is called the p-plane), which can be expressed as follows: σ3

σ3

P Q

σ2

O

σ2

O

(π plane)

(a)

σ1

(π plane)

Fig. 2.3 p-plane and stress projection diagram

(b)

σ1

54

2 Fundamentals of Stress and Strain

r1 þ r2 þ r3 ¼ constant

ð2:16Þ

! An arbitrary stress state P (r1, r2, r3) can be expressed as a vector OP . This vector can be expressed as the sum of the projection of the spatial diagonal direction

!

! OQ and the projection of the p-plane QP , which is called the normal stress component rp on the p-plane and is called the shear stress component sp on the p-plane. According to Fig. 2.3, we have the following expression: pffiffiffi   pffiffiffi pffiffiffi 3 1 1 1  ! ðr1 þ r2 þ r3 Þ ¼ 3rm ¼ 3p rp ¼ OQ  ¼ r1 pffiffiffi þ r2 pffiffiffi þ r3 pffiffiffi ¼ 3 3 3 3 ð2:17Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi 1 r21 þ r22 þ r23  pffiffiffi ðr1 þ r2 þ r3 Þ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 ¼ pffiffiffi ðr1  r2 Þ þ ðr2  r3 Þ þ ðr3  r1 Þ 3 rffiffiffi pffiffiffiffiffiffiffi 2 ¼ 2J2 ¼ q 3

         !2  !2  !2  ! s2p ¼ QP  ¼  OP   OQ  ) sp ¼  QP  ¼

ð2:18Þ For the conventional triaxial condition r2 = r3, we have p ¼ ðr1 þ 2r3 Þ=3; q ¼ r1  r3 . Looking at the p-plane from top to bottom against the axis, three major axes with an angle of 120° appear between Or”1, Or”2, and Or”3 in the plane of the ridge (see Fig. 2.4), which are projections of the three perpendicular axes of the principal stress in the principal stress space. It is known that the direction cosine of the space pffiffiffiffiffiffiffiffi

! diagonal On in Fig. 2.3 is cos a ¼ 1=3, so the direction cosine between the pffiffiffiffiffiffiffiffi p-plane and the principal stress axis in the figure is cos b ¼ 2=3, and the relationship between the coordinate axis on the p-plane and the principal stress space pffiffiffiffiffiffiffiffi coordinate axis can be obtained as “r00i ¼ri cos b ¼ 2=3ri ”. If the Cartesian coordinate system O’xy is taken on the p-plane, see Fig. 2.4, the projection of the stress on the p-plane (r”1, r”2, r”3) on the x- and y-axes is calculated as follows:

2.2 Stress Analysis

55

Fig. 2.4 Representation of stress in the p-plane

(

8  1  00 > 00 00 rffiffiffi > < x ¼ 2r1  r2  r3 x ¼ r001  r002 þ r3 cos 60 2 2 pffiffiffi 00  00  ; with ri ¼ ri 00  ) >   3 3 00 y ¼ r2  r3 cos 30 > : y¼ r2  r003 2 8 1 > > < x ¼ pffiffiffi ð2r1  r2  r3 Þ 6 ) 1 > > : y ¼ pffiffiffi ðr2  r3 Þ 2 ð2:19Þ 

 00

If the polar coordinates r and h are taken on the p-plane, the projection of any point P(r1, r2, r3) in the principal stress space on the p-plane is P’(r”1, r”2, r”3), and the vector diameter r and the stress Lode angle h of P’ on the p-plane are, respectively, calculated as follows: 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 > 2 þ y2 ¼ pffiffiffi > r ¼ x ðr1  r2 Þ2 þ ðr2  r3 Þ2 þ ðr3  r1 Þ2 ¼sm > > < 3 pffiffiffi 3 p 2r1  r2  r3 > > cos h ¼ ¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > r : 6 ðr  r Þ2 þ ðr  r Þ2 þ ðr  r Þ2 1 2 2 3 3 1

ð2:20Þ

Defining the intermediate principal stress parameter b ¼ ðr2  r3 Þ=ðr1  r3 Þ, the Lode angle can also be expressed as follows:

56

2 Fundamentals of Stress and Strain 2 2 rr11 r x 1 r3 þ b cos h ¼ ¼ pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 r1 r2 2 þ ð bÞ 2 þ 1 r1 r3

ð2:21Þ

Applying the invariants of the deviatoric stress tensor, the Lode angle h can also be written as Eq. (2.15).

2.3

Strain Analysis

Strain is defined as “deformation of a solid due to stress”. For example, two typical strains can be easily understood: (1) normal strain is defined as the elongation or contraction of a line segment and (2) shear strain is defined as the change in the angle between two line segments that were originally perpendicular. Note that strain is a dimensionless unit since it is the ratio of two lengths. However, it is also common practice to state strain as the ratio of two length units. Under small deformation conditions, the strain state at a given point can be described by the strain tensor: 2 ex

6 6 6 1 eij ¼ 6 cyx 6 2 4 1 c 2 zx

1 3 c 2 3 2 3 2 xz 7 exx exy exz e11 e12 e13 7 1 7 6 7 6 7 cyz 7  4 eyx eyy eyz 5  4 e21 e22 e23 5 ey 7 2 5 ezx ezy ezz e31 e32 e33 1 czy ez 2

1 c 2 xy

ð2:22Þ

where c is the engineering shear strain. The strain tensor can be divided into the deviatoric strain tensor eij and the spherical strain tensor as follows: 2

ex  em

6 6 6 1 eij ¼ 6 cyx 6 2 4 1 c 2 zx

1 c 2 xy ey  em 1 c 2 zy

1 3 c 2 3 2 xz 7 em 0 0 7 1 7 6 7 cyz 7 þ 4 0 em 0 5 7 2 5 0 0 em ez  em

  where the mean strain em is defined as em ¼ ex þ ey þ ez 3.

ð2:23Þ

2.3 Strain Analysis

57

Similar to the stress tensor, the invariants of the strain tensor are calculated as follows: 8 > > > >
> c c c  c 2 c 2 c 2 > > xy yz yz xy zx zx : I0 ¼ e e e þ 2 ey ez  ex x y z 3 2 2 2 2 2 2

ð2:24Þ

The invariants of the deviatoric strain tensor are calculated as follows: 8 > > > >
> c c c  c 2 c 2 c 2 > > xy yz yz xy zx :  ex J30 ¼ ðexx  em Þðeyy  em Þðezz  em Þ þ 2 ey zx ez 2 2 2 2 2 2

ð2:25Þ The general shear strain ed is defined as follows: ed ¼

rffiffiffi pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2pffiffiffiffiffiffiffiffiffi ðe1  e2 Þ2 þ ðe2  e3 Þ2 þ ðe3  e1 Þ2 eij eij ¼ 3 3

ð2:26Þ

For a triaxial test (e2 ¼ e3 ), the general shear strain ed can be reduced to the following expression: 2 ed ¼ ðe1  e3 Þ 3

ð2:27Þ

The volumetric strain ev is defined as follows (under the small deformation assumption): ev ¼

DV ¼ ð1 þ e1 Þð1 þ e2 Þð1 þ e3 Þ  1  e1 þ e2 þ e3 V

ð2:28Þ

Using three principal strains “e1 , e1 , and e3 ” as the coordinate axes, a three-dimensional strain space can be formed. A point in this space can describe the

! strain state at a point in the soil. On the isosceles of the main strain space On with e1 ¼ e2 ¼ e3 ¼ em , the plane perpendicular to this line is called the strain p-plane, and its equation is defined as follows: pffiffiffi e1 þ e2 þ e3 ¼ 3r

ð2:29Þ

58

2 Fundamentals of Stress and Strain

The normal strain component in the strain p-plane ep is expressed as follows: pffiffiffi e p ¼ 3e m

ð2:30Þ

The shear strain component in the strain p-plane cp is expressed as follows: cp ¼

2.4

2 3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiqffiffiffiffiffi ðe1  e2 Þ2 þ ðe2  e3 Þ2 þ ðe3  e1 Þ2 ¼2 2 J20

ð2:31Þ

Stress and Strain Analyses in ErosLab

According to the previously presented equations, the stress and strain analyses in ErosLab are presented in Figs. 2.5 and 2.6, respectively. If a stress tensor or a strain tensor is given, the corresponding principal stress or strain and deviatoric stress or strain can be obtained by clicking the button “Calculate”. Furthermore, all the results can be cleared by clicking the button “Clear”.

Fig. 2.5 Stress analysis in ErosLab

References

59

Fig. 2.6 Strain analysis in ErosLab

References 1. Biarez J, Hicher PY (1994) Elementary mechanics of soil behaviour: saturated remoulded soils. AA Balkema 2. Chen H, Yu T, Wang X, Liu Z (2005) Elasticity and plasticity. China Architecture & Building Press 3. Luo T, Yao Y, Hou W (2010) Constitutive relationship of soil. Beijing: China Communications Press 4. Zhang F, Ye GL (2007) Computational Soil Mechanics. Beijing: China Communications Press

Chapter 3

Introduction of Laboratory Tests for Soils

Abstract This chapter summarizes six common laboratory tests, including the oedometer test, triaxial test, simple shear test, biaxial test, true triaxial test, and hollow cylinder torsional shear test, and their associated stress and strain paths for developing constitutive models and calibration are also introduced. Additionally, the operation for simulating each test in the ErosLab platform is introduced for in-depth understanding and practice.

3.1

Introduction

The stress path is the trajectory of the change in stress state. As mentioned earlier, a stress state can be represented by a point in the stress plane or stress space. Then, a continuous change in the stress state can be represented by a line or curve in the stress plane or stress space, which is called the stress path. The stress path can be divided into the total stress path and the effective stress path depending on whether the stress space is expressed by the total stress rij or the effective stress r‘ij (where the effective stress r0ij ¼ rij  udij is specially defined in soil mechanics, different from solid mechanics, and u is the pressure of the water filling the voids between the grains for saturated media). To describe the stress path more intuitively, the three-dimensional problem is often two-dimensionalized. For example, the stress pffiffiffi path is plotted in a coordinate system, such as p’−q, r‘1−r‘3, r‘1− 2r‘3, s−r. The strain path is also similar. To develop constitutive models, the mechanical responses of some simple stress paths usually need to be clarified through several common laboratory tests. Six test types are available in the platform: the oedometer test, triaxial test, simple shear test, biaxial test, true triaxial test, and hollow cylinder apparatus (HCA) test, as shown in Fig. 3.1. The schematic diagram picture of the selected test below will display according to the user’s selection to further clarify the selected test type.

© Springer Nature Singapore Pte Ltd. and Tongji University Press 2020 Z.-Y. Yin et al., Practice of Constitutive Modelling for Saturated Soils, https://doi.org/10.1007/978-981-15-6307-2_3

61

62

3 Introduction of Laboratory Tests for Soils

Fig. 3.1 Six test types available in ErosLab

3.2

Oedometer Test

Oedometer tests are performed by applying different loads to a soil sample and measuring the deformation response [1]. The results from these tests are used to predict how a soil in the field will deform in response to a change in effective stress (first suggested by Terzaghi). Oedometer tests are designed to simulate the one-dimensional deformation and drainage conditions that soils experience in the field (Fig. 3.2). The soil sample in an oedometer test is typically a circular disk with a diameter-to-height ratio of approximately 3:1. The sample is held in a rigid confining ring, which prevents lateral displacement of the soil sample but allows the sample to swell or compress vertically in response to changes in the vertically applied load. Known vertical stresses are applied to the top and bottom faces of the sample, typically using free weights and a lever arm. The applied vertical stress varies, and the change in the sample thickness is measured. For samples that are saturated with water, porous stones are placed on the top and bottom of the sample to allow drainage in the vertical direction, and the entire sample is submerged in water to prevent drying. Saturated soil samples exhibit the phenomenon of consolidation, whereby the soil’s volume gradually changes and presents a delayed response to the change in the applied vertical stress. This typically takes minutes or hours to complete in an oedometer test, and the change in sample thickness with time is recorded, providing measurements of the coefficient

3.2 Oedometer Test

63

Fig. 3.2 Oedometer testing system (GDS) and soil sample

of consolidation and the permeability of the soil. Note that for sandy soils, the dissipation of excess pore pressure due to applied load is very fast, and only compression properties can be measured in this case. Additionally, for the loading system, an oedometer test can be conducted through stress control (deadweight) or strain control (displacement). In the platform, the oedometer test is simulated as a one-dimensional compression test without considering the coupled consolidation theory, where the lateral deformation is constrained to zero and only vertical deformation is allowed (e2 ¼ e3 ¼ 0 & e1 [ 0), as shown in Fig. 3.3 (note: here, the subscript “v” represents vertical). The lateral stress keeps changing necessarily during the loading process because of the restriction of lateral deformation. Therefore, it is convenient that the test can be controlled by pure strain loading (e2 ¼ e3 ¼ 0 & de1 [ 0) or by strain and stress mixed loading (e2 ¼ e3 ¼ 0 & dr01 [ 0). The oedometer curve corresponds to the relation between e1 and r1 or the relation between r1 and the void ratio e (De = ev/1 + e) (ev = e1).

Fig. 3.3 Schematic diagram of an oedometer test for a stress and strain conditions, and b the typical result

64

3 Introduction of Laboratory Tests for Soils

Fig. 3.4 Loading conditions of the oedometer test in ErosLab

From the oedometer test, one can measure the soil parameters, which mainly include the compression index Cc and swelling index Cs. The interface of the oedometer test in ErosLab is shown in Fig. 3.4. Both stress control and strain control are optional, and the specific loading values must be entered. For the selected control, the values of loading and the duration time are needed for each loading stage. If the value of time is negative, the loading process will be stopped. For stress-controlled loading, the default loading is 25–50–100– 200–400–50–400–800–1600 kPa, and the duration of each loading process is 24 h. For the strain control loading, the default loading is 0.05–0.1–0.15–0.145–0.2–0.3– 0.35–0.4–0.45, and the duration of each loading process is also 24 h. Users can change the default loading value and duration according to their requirements. Note that the effect of duration can be considered only when the time-dependent constitutive model is selected, e.g., ANICREEP in this platform.

3.3

Triaxial Shear Test

A triaxial test is a common method used to measure the mechanical properties of many deformable solids, especially soil (e.g., sand, clay), rock, and other granular materials or powders. In a triaxial test, stress is applied to a soil sample in a way that results in stresses along one axis being different from the stresses in the

3.3 Triaxial Shear Test

65

perpendicular direction [2–4]. This differential stress is typically created by placing the sample between two parallel platens that apply stress in one (usually vertical) direction, and fluid pressure is applied to the specimen to achieve identical stresses in the perpendicular direction (usually the horizontal direction). The application of different compressive stresses in the test apparatus causes shear stress to develop in the sample; the loads can be increased, and deflections can be monitored until failure of the sample (Fig. 3.5). During the test, the surrounding fluid is pressurized, and the stress on the platens is increased until the material in the cylinder fails and forms sliding regions within itself, which is known as shear bands. The geometry of the shearing in a triaxial test typically causes the sample to become shorter while bulging out along the sides. During the test, the pore pressures of the water in the sample may be measured using Bishop’s pore pressure apparatus. Note that for developing constitutive models, only homogenous stress and strain fields from triaxial tests are considered, which requires that the sample be homogenously deformed as much as possible during loadings. From the triaxial test data, it is possible to extract the fundamental soil parameters about the sample, including its angle of shearing resistance, apparent cohesion, and dilatancy angle. Then, these parameters are used in computer models to predict how the soil will behave in a larger scale engineering application. An example would be to predict the stability of the soil on a slope, whether the slope will collapse or whether the soil will support the shear stresses of the slope and remain in place. Triaxial tests are used along with other tests to make such engineering predictions. During shearing, the soil will typically have a net gain or loss of volume. If it had originally been in a dense state, then the soil typically gains volume, which is a characteristic known as Reynolds dilatancy. If the soil had originally been in a very loose state, then contraction may occur before the shearing begins or in conjunction with the shearing. Occasionally, testing of cohesive soil samples is done with no confining pressure in an unconfined compression test, which is considered to be a special case in a triaxial test. This type of test requires a much simpler and less expensive apparatus and sample preparation, although the applicability is limited to samples in which the sides will not crumble when exposed, and the confining stress being lower than the in situ stress may give overly conservative results. Only consolidated drained and undrained triaxial tests are available in the developed ErosLab platform. For the conventional consolidated drained triaxial compression test, the soil sample is first consolidated to a given confining pressure, and then the axial load is increased up to the point of sample failure (dr0a ¼ dr01 [ 0 or dea ¼ de1 [ 0) while maintaining a constant confining pressure (dr0r ¼ dr02 ¼ dr03 ¼ 0). The slope of this loading path on the p′−q plane is dq=dp0 ¼ 3, which is noted as the conventional triaxial compression path (CTC). Another approach to conducting this test is reducing the axial load until the sample reaches failure (dr0a ¼ dr03 \0 or dea ¼ de3 \0) while maintaining a constant

Fig. 3.5 Triaxial testing system (GDS) and soil sample

GDS

Cell Pressure Controller

Impervious membrane

O-ring

Water

Perspex cell

Load frame

Load cell 50 kN FPISTON

Alumina cement

Open valves

Pore pressure Volume change

Porous stone

Displacement transducer

Specimen (LHC/LRC)

66 3 Introduction of Laboratory Tests for Soils

3.3 Triaxial Shear Test

67

confining pressure (dr0r ¼ dr01 ¼ dr02 ¼ 0). The slope of this loading path is dq=dp0 ¼ 3, which is the conventional triaxial extension path. The above stress schematic diagrams are shown in Fig. 3.6a. In the conventional consolidated undrained triaxial compression test (Fig. 3.6b), the axial stress is progressively increased while the total confining stress is kept constant (drr ¼ 0). Thus, the slope of the loading path on the p-q plane is still 3 (dq=dp ¼ 3). During the test, the pore water is prevented from leaving or entering the specimen, which leads to a change in the pore pressure. On the p′-q plane, the horizontal distance between the total stress path and the effective stress path is the excess pore water pressure. The excess pore water pressure is always positive for the normally consolidated soil during the loading process; thus, the effective stress path is to the left of the total stress path on the p′-q plane. For overconsolidated soil, the excess water pore pressure is negative during the postloading process. Therefore, the effective stress path is to the right of the total stress path. Under the conventional confining pressure, both the soil particle and the water are considered to be incompressible, which leads to the fact that in undrained conditions, the volumetric strain remains constant (dev ¼ 0 ) dea ¼ 2der ). In ErosLab, all undrained simulations (except for the creep simulation using the ANICREEP model) are performed by keeping the volumetric strain constant while increasing (compression) or decreasing (extension) the axial strain. In addition, ErosLab also sets the p’-q full stress control and ev-ed full strain control under drainage conditions, which can be selected in the panel of “multistage loading”. The interface of the loading condition for triaxial tests is shown in Fig. 3.7. The consolidation process before the shear loading stage is optional. If the consolidation process is selected, the confining pressures r‘r and r‘a should be filled in (r‘r = ‘22 = r‘33). If not, the confining pressures r‘r and r‘a would be equal to the initial stress state. The shear loading is divided into two types of controls: (a) strain control and (b) stress control. The triaxial test is simulated by choosing either mode or giving a proper load value. Furthermore, if the ANICREEP model is selected, the

Fig. 3.6 Schematic diagram of the triaxial test for the a drained test, b undrained test, and c typical results

68

3 Introduction of Laboratory Tests for Soils

Fig. 3.7 Interface of the loading condition for triaxial tests

3.3 Triaxial Shear Test

69

Fig. 3.8 Settings for cyclic loading in triaxial tests

time effect will be controlled by “loading time”. In addition, for convenience, a multistage setting is provided: select the button “Multistage” and click the button “Settings” to complete the multistage simulation. Apart from monotonic loading, cyclic loading is also available for triaxial tests, as shown in Fig. 3.8. The user can set the number of cycles, loading mode, and amplitude of loading. Then, the load is generated as a sinusoidal function for the calculations.

3.4

Direct, Simple, or Ring Shear Test

A direct shear test is a laboratory or field test used by geotechnical engineers to measure the shear strength properties of soil or rock material or of discontinuities in soil or rock masses [5, 6]. The test is performed on three or four specimens from a relatively undisturbed soil sample. A specimen is placed in a shear box that has two stacked rings to hold the sample; the contact between the two rings is at approximately the mid-height of the sample. A vertical confining stress is applied to the

70

3 Introduction of Laboratory Tests for Soils

specimen, and the upper ring is pulled laterally until the sample fails or through a specified strain. The load applied and the strain induced are recorded at frequent intervals to determine a stress–strain curve for each confining stress. Several specimens are tested at varying confining stresses to determine the shear strength parameters, the soil cohesion c, and the angle of internal friction, commonly known as the friction angle /. The results of the tests on each specimen are plotted on a graph with the peak (or residual) stress on the y-axis and the confining stress on the x-axis. The y-intercept of the curve that fits the test results is the cohesion, and the slope of the line or curve is the friction angle. Direct shear tests can be performed under several conditions. The sample is normally saturated before the test is run but can be run with the in situ moisture content. The rate of strain can be varied to create a test of undrained or drained conditions, depending on whether the strain is applied slowly enough for water in the sample to prevent pore water pressure buildup in clayey soils. A direct shear test machine is required to perform the test (Fig. 3.9). The test using the direct shear machine determines the consolidated drained shear strength of a soil material in direct shear. The main advantage of the torsional ring shear apparatus is that it shears the specimen continuously in one direction for any magnitude of displacement. This allows soil particles to be oriented parallel to the direction of shear and a residual strength condition to develop. Therefore, the shortcomings raised for the direct shear box may be overcome by using the ring shear apparatus (Fig. 3.9). The ring shear specimen is annular with a smaller inside diameter (e.g., 7 cm) and a larger outside diameter (e.g., 10 cm). Drainage is provided by annular bronze porous stones secured to the bottom of the specimen container and the loading platen. The specimen container confines the specimen radially (e.g., 0.5 cm deep). A simple shear apparatus was developed because of the shortcomings of the direct or ring shear test (ASTM [7]) (Fig. 3.9). In these shear tests, different shearing conditions are applied to soil samples. For the direct shear test, shearing occurs at a predetermined center of the specimen, which may not be the weakest plane of the soil, while for indirect simple shear, the entire specimen is distorted without the formation of a single shearing surface. For the ring shear test, there are a number of limitations, including potentially nonuniform stress and strain distributions, potential soil extrusion, and wall friction. However, in simple shear, the shear sliding surface is more representative of the tested soils. The mode of shearing established in the simple shear device is similar to that which occurs around the shaft of a pile. In general, when the soil subjected to shear stress reaches its maximum strength, the soil will slide along a surface, which leads to failure. Such phenomena can be studied through a simple shear test, equivalent to the direct shear and ring shear

3.4 Direct, Simple, or Ring Shear Test

71

Loading cap

Porous plate Top perforated plate

Normal load

Carriage

Load from drive unit

Loading yoke

Reaction from load ring Split box

Water

Machine Lower perforated bed plate Base plate

Porous plate

Linear low friction bearing Typical setup for a direct shear test

Normal Stress

Rotation Axis

Inner Confining Ring Normal Stress

Shear Stress Soil Sample Outer confining Ring

Top Cap

Fixed Platen

Vertical Stress

Brass Rings

Shear Stress Base Pedestal

Fig. 3.9 Testing system and soil sample for a direct shear, b ring shear, and c simple shear

(Fig. 3.10). In the simple shear test, the shear strain c is defined as the ratio of the horizontal displacement to sample height. Under the vertical strain loading, the shear stress s, vertical stress and vertical displacement can be obtained from a simple shear test. Under these three kinds of shear testing conditions, the stress– strain tensors can be reduced as follows:

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3 Introduction of Laboratory Tests for Soils

Fig. 3.10 Three typical shear tests: a simple shear, b direct shear, c ring shear, and d typical result

2

r0n 4 rij ¼ sxy 0

sxy r0n 0

 3 2 en 0 cxy 2  0 5; eij ¼ 4 cxy 2 en r0n 0 0

3 0 05 en

ð3  1Þ

where cxy is the shear strain c, en is the vertical strain, sxy is the shear s, and r‘n is the vertical effective stress. There are two ways to conduct this simple shear test: (1) maintain a constant vertical load, which is the case in the drained simple shear test, and (2) maintain the volume of the sample constant, which can be regarded as the undrained simple shear test. In ErosLab, the simple shear test is controlled by rn and c, as shown in Fig. 3.11. First, the sample is K0-consolidated under a vertical stress rn, and then, a tangential strain c is applied at the bottom of the sample after consolidation. In this version, multistage loading is also available. The input time of a loading stage is less than zero, and the loading will stop up to the previous stage.

Fig. 3.11 Loading conditions of the simple shear test

3.5 Biaxial Shear Test

3.5

73

Biaxial Shear Test

Most field cases, such as soil behavior under strip footings, behavior of earthen dams, and stability analysis of slopes and levees, are better represented if the soil is tested under plane-strain (PS) conditions, rather than under other conditions (e.g., triaxial). Higher peak stress values followed by severe softening have been reported in the literature for specimens tested under PS loading compared to triaxial specimens, and the difference between the two cases increases as the specimen density and confining pressure increase. Furthermore, there is a fundamental difference between the failure modes of specimens tested under PS loading compared to triaxial conditions (Fig. 3.12), where the failure of a PS specimen is characterized by distinct shear bands, whereas triaxial specimens develop more complex deformation patterns. In recent decades, biaxial tests have been used to study the stress–strain–strength behaviors of soil under plane-strain conditions. For biaxial tests, the displacement in one direction (usually the horizontal direction) is constrained to zero, and the sample is constrained in the perpendicular direction by applying a horizontal confining pressure (rh). Then, the sample is loaded through the application of vertical loading by controlling either the vertical strain ev or stress rv). The schematic diagram of the biaxial test is shown in Fig. 3.13 (note: here, the subscript “v” represents vertical). In the ErosLab platform, the biaxial test can be achieved in two stages: (1) first, the sample is isotropically compression to the confining pressure r‘3; and (2) then, the confining pressure r‘3 is kept constant, and the axial strain ea is applied, as shown in Fig. 3.14. Furthermore, if the ANICREEP model is selected, the time effect will be controlled by the “loading time”.

Fig. 3.12 Plane-strain shear testing system and soil sample

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3 Introduction of Laboratory Tests for Soils

Fig. 3.13 Schematic diagram of the biaxial test for a stress and strain conditions and b a typical result

Fig. 3.14 Interface of loading for the biaxial test

3.6

True Triaxial Test

True triaxial testing systems have been developed to enable independent control of the stress in three perpendicular directions [8] (Fig. 3.15), which allows for the investigation of stress paths not capable of being generated in axisymmetric triaxial test machines. The test cell is cubical, and there are six separate plates applying pressure to the specimen, with LVDTs reading the movement of each plate. Pressure in the third direction can also be applied using hydrostatic pressure in the test chamber, which requires only four plates for the stress application in the two

3.6 True Triaxial Test

75

Fig. 3.15 True triaxial testing system and soil sample

other directions. The apparatus is significantly more complex than that used for axisymmetric triaxial tests and is therefore less commonly used. Thus, the true triaxial apparatus can be adopted to study the stress–strain– strength behaviors of soil in 3D conditions, as shown in Fig. 3.16. The word “true” is meant to distinguish it from the traditional triaxial apparatus, by which only the axial stress (or strain) and the horizontal confining stress can be controlled independently. Since the stresses in all three directions can be controlled independently, the true triaxial test can be used to apply more complicated stress paths. A common stress path is to carry out a series of drained tests with different constant intermediate principal stress factors b (b = (r‘2-r‘3)/(r‘1-r‘3)) or Lode angles (h) while maintaining a constant mean stress. p′ and b are known as input values. That is, the sample is loaded to failure by increasing the major principal stress along with the intermediate and minor principal stresses, which are calculated according to the following equations:

30

0 330

1.0 60

1

300

0.5 0.0 90

Mises Lade SMP

270 MC

3 2

(a)

0.5 240

120 1.0

(b)

150

210 180

Fig. 3.16 Schematic diagram of the true triaxial test for a stress and strain conditions and b a typical result

76

3 Introduction of Laboratory Tests for Soils

Fig. 3.17 Interface of loading for the true triaxial test in the ErosLab platform

8 8 r01 þ r02 þ r03 3ð1  bÞp0 þ ð2b  1Þr01 > > 0 > > p ¼ < < r02 ¼ 3 2b 0 0 ) r > > 3p0  ð1 þ bÞr01 2  r3 > > 0 : : b¼ 0 r3 ¼ r1  r03 2b 8 ð2b  1Þ 0 > 0 > dr1 < dr2 ¼ 2b ) >  ð 1 þ bÞ 0 > : dr03 ¼ dr1 2b

ð3:2Þ

Furthermore, another test can be conducted in which the radius soct in the p-plane is kept constant and the stress path is a circle (usually p′ is also kept constant). 8 2b > 0 0 > ffi 2p0 > r1 ¼ p þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2  bþ1 > 3 b > > < 2b  1 r02 ¼ p0 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p0 > 3 b2  b þ 1 > > > > bþ1 > > ffi 2p0 : r03 ¼ p0  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 b2  b þ 1

ð3:3Þ

3.6 True Triaxial Test

77

In the ErosLab platform, the true triaxial test is controlled by two parameters p0 and b, as shown in Fig. 3.17. The loading is totally stress-controlled; therefore, the drainage condition cannot be chosen. First, the sample is isotropically compressed to a given confining pressure, and then, the loading along the directions of the three principal stresses is calculated according to Eq. (3) and then applied. The values of the three principal stresses can be adjusted by changing the value of b. During the whole process of loading, p’ is kept constant. Furthermore, if the ANICREEP model is selected, the time effect will be controlled by the “loading time”.

3.7

Hollow Cylinder Torsional Shear Test

The hollow cylinder apparatus (HCA) is a versatile testing device that is widely employed to investigate the constitutive behavior of soils under generalized stress conditions, including principal stress rotation, anisotropy, and noncoaxiality [9]. The soil hollow cylinder test was developed to allow for independent control of all principal stresses observed in in situ soil, including shear stress, which is generally ignored in standard soil triaxial tests and true triaxial tests. The hollow cylinder test is commonly used to determine the dynamic properties of soil, such as liquefaction potential, dynamic shear strength, and damping, and can also be used to perform standard static triaxial tests. The hollow cylinder test is performed by applying three different stresses to a hollow, cylindrical soil specimen. These three stresses can each be individually controlled, as in the true triaxial test. However, in the hollow cylinder test, the stresses are applied outside the specimen (confining stress) and inside the specimen, and a torsional stress is applied at the top of the specimen. By controlling each of these three stresses during the test, the effects of different stress paths on the shear strength of the soil can be determined, including paths with rotation of the principal stress directions. The hollow cylinder test system has a few drawbacks. This test system is much more complicated than a typical triaxial system and requires much more control to use. Therefore, it is used most often for research. In addition, specimen preparation requires much more time and care (Fig. 3.18), as a hollow specimen is much more difficult to create than a solid specimen used in a triaxial test. The hollow cylinder soil specimen can be subjected to the internal pressure pi, the external pressure po, the axial load pv, and the rotational displacement or torque T. Thus, the hollow cylinder apparatus can impose four stresses rr, rh, rz, and szh, which makes it possible to change the values of the three principal stresses independently and impose the rotation of the principal stress directions, as shown in

78

3 Introduction of Laboratory Tests for Soils

pv

po

T

1

pi 3

z z r= 2

Normalized undrained shear strength

Fig. 3.18 Hollow cylinder torsional shear testing system and soil sample

1

Shanghai clay

0.8 San-francisco clay

0.6 0.4 Hangzhou clay

0.2 0 0

(a)

(b)

30 60 90 Rotation angle of pricipal stress axis /

Fig. 3.19 Schematic diagram of the HCA test for a stress and strain conditions and b a typical result

Fig. 3.19. Therefore, the hollow cylinder torsional shear test is an effective way to study the influence of the principal stress rotation on the stress–strain relationship, as well as the anisotropy of soil. When the principal stresses do not rotate, the apparatus can also be used for conducting true triaxial tests along different stress paths.

3.7 Hollow Cylinder Torsional Shear Test

8 8 DH pv po ro2  pi ri2 > > 0 > > ez ¼ > > > rz ¼ pr 2  r 2  þ r 2  r 2 > H > > > > i o i o > > > > Dr  Dr > > i > > po r o þ pi r i > > e ¼ o > > r0r ¼ < < r r  r i o ro þ ri and po r o  pi r i > > e ¼  Dro þ Dri 0 > > h r ¼ > > h > > r þ ri > > ro  ri > > > >  3o  > > > > 3T 2Dh ro  ri3 > > > > > >    szh ¼  3 : : czh ¼ 2p ro  ri3 3H ro2  ri2

79

ð3:4Þ

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0  e  e 2 c 2 > > r0z þ r0h rz  r0h 2 > e þ e > z z h h 0 > 2 > > þ r1 ¼ þ szh þ þ zh e1 ¼ > > > > 2 2 > > 2 2 2 > > > > > > 0 0 > > > ¼ r r e < 2 ¼ er < 2 r ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   and e  e 2 c 2 ez þ eh r0 þ r0h r0z  r0h 2 > > z h > > >  e3 ¼ þ zh  r03 ¼ z þ s2zh > > > > > 2 2 2 2 2 > > > > > > > czh > > 2s > zh > > tan 2ae ¼ : : tan 2ar ¼ 0 ez  eh rz  r0h

ð3:5Þ 8 > > > > > > > > > >
0 0 0  r0 2 4 þ s 2 > r  r þ r 2 þ r > z r z h h zh > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi br ¼ > >  2 . > > : 2 r0z  r0h 4 þ s2zh 8 ev ¼ ez þ er þ eh > > > pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > < 2 ðe1  e2 Þ2 þ ðe2  e3 Þ2 þ ðe3  e1 Þ2 ed ¼ 3 > > > ðe  e3 Þ > > : be ¼ 2 ; where e1 ; e2 ; e3 using Eq:ð3:5Þ ðe1  e3 Þ

ð3:6Þ

Therefore, utilizing the hollow cylinder torsional shear test, the following studies can be conducted: (a) the anisotropy of soil, (b) the effects of the principal stress rotation, and (c) the effects of intermediate principal stress. In the ErosLab platform, the loading of the HCA test is controlled by four variables: p0, q, a, and b. During the whole process of loading, p0 is kept constant. The loading is controlled by the total stress; therefore, the drainage condition cannot be chosen. The values of loading can be calculated by Eq. (3.7). In the program, the

80

3 Introduction of Laboratory Tests for Soils

Fig. 3.20 Interface of loading for the HCA test in the ErosLab platform

sample is first isotropically compressed to a given confining pressure p0, and then, the loading is applied by changing the values of q, a, and b, as shown in Fig. 3.20. 8 8 r0 þ r03 r01  r03 > > 2  b r0z ¼ 1 þ cosð2aÞ > > 0 > > r1 ¼ p þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q > 2 2 > > > 2 > > 3 b  bþ1 > > > > > > r0r ¼ r02 < < 2b  1 0 r2 ¼ p þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q þ r01 þ r03 r01  r03 > > 0 3 b2  b þ 1 > >  cosð2aÞ r ¼ > > h > > 2 2 > > > > b þ 1 > > > > : r3 ¼ p0  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q > r0  r03 > : 3 b2  b þ 1 szh ¼ 1 sinð2aÞ 2 8 1  2b 1 > > r0z ¼ p0 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q cosð2aÞ > > 2 2 > 6 b  bþ1 2 b  bþ1 > > > > > 2b  1 > > r0r ¼ p0 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q > < 3 b2  b þ 1 ) > 1  2b 1 > > r0h ¼ p0 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q cosð2aÞ > > 2 2 > 6 b  bþ1 2 b  bþ1 > > > > > 1 > > : szh ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q sinð2aÞ 2 b2  b þ 1

ð3:7Þ

3.8 Conclusion

3.8

81

Conclusion

Six typical soil laboratory tests, i.e., the oedometer test, triaxial test, simple shear test, biaxial test, true triaxial test, and hollow cylinder torsional shear test, were presented with their stress and strain conditions and representative stress paths. The operations for simulating each test in the ErosLab platform were also introduced for in-depth understanding and practice.

References 1. Soil ACD-O, Rock (2004) Standard test methods for one-dimensional consolidation properties of soils using incremental loading. ASTM International 2. ASTM (2011) Method for consolidated drained triaxial compression test for soils, vol 4 3. Soil ACD-o, Rock (2011) Standard test method for consolidated undrained triaxial compression test for cohesive soils. ASTM International 4. Bs BS (1990) 2,” Methods of Test for Soils for Civil Engineering Purposes. General Requirements and Sample Preparation, British Standard Institution, London 5. Bjerrum L, Landva A (1966) Direct simple-shear tests on a Norwegian quick clay. Geotechnique 16(1):1–20 6. ASTM D- (2011) Standard test method for direct shear test of soils under consolidated drained conditions. D3080/D3080M 7. ASTM (2008) Standard test method for torsional ring shear test to determine drained fully softened shear strength and nonlinear strength envelope of cohesive soils (using normally consolidated specimen) for slopes with no preexisting shear surfaces 8. Reddy KR, Saxena SK, Budiman JS (1992) Development of a true triaxial testing apparatus. Geotech Test J 15(2):89–105 9. Logeswaran P, Sivathayalan S (2013) A new hollow cylinder torsional shear device for stress/ strain path controlled loading. Geotech Test J 37(1):24–35

Chapter 4

Fundamentals of Elastoplastic Theory

Abstract This chapter presents the classic constitutive modeling method known as the conventional elastoplastic theory with different components, such as the elastic stress–strain relationship, the elastoplastic relationship, which includes the yield surface, flow rule, and hardening rule, and Drucker’s stability hypothesis, convexity, and orthogonality. Some typical numerical methods for solving plastic problems are also presented, such as the general explicit solution, the cutting plane method-based implicit solution, and the closest point projection method-based implicit solution.

4.1 4.1.1

Elastic Constitutive Relation Isotropic Elasticity

Due to the nonlinearity of the stress–strain behavior of soils, the elastic constitutive relation is normally expressed in incremental form using the generalized form of Hooke’s law: 1þt 0 t drij  dr0kk dij E E

ð4:1Þ

E tE deij þ de dij 1þt ð1 þ tÞð1  2tÞ kk

ð4:2Þ

deij ¼ or dr0ij ¼

where two parameters are needed: Young’s modulus E and Poisson’s ratio t. To calculate the stress–strain relationship, we need to define a stiffness matrix for material D. In most finite
element codes, the engineering shear strain (cxy ¼ exy þ eyx ¼ @ux =@y þ @uy @x) is used. Then, the elastic stiffness matrix with the stress–strain relationship in incremental form can be expressed as follows: © Springer Nature Singapore Pte Ltd. and Tongji University Press 2020 Z.-Y. Yin et al., Practice of Constitutive Modelling for Saturated Soils, https://doi.org/10.1007/978-981-15-6307-2_4

83

84

4

2

3 2 dr0xx 1t 6 dr0yy 7 6 t 6 0 7 6 6 drzz 7 6 t E 6 0 7¼ 6 6 drxy 7 ð1  2tÞð1 þ tÞ 6 0 6 0 7 6 4 dryz 5 4 0 0 drzx 0

t 1t t 0 0 0

Fundamentals of Elastoplastic Theory

3 32 dexx t 0 0 0 6 7 t 0 0 0 7 76 deyy 7 6 dezz 7 1t 0 0 0 7 7 76 6 7 0 0:5  t 0 0 7 76 dcxy 7 0 0 0:5  t 0 54 dcyz 5 dczx 0 0 0 0:5  t

ð4:3Þ which can also be written in the inverse with an elastic flexibility matrix: 2 3 dexx 1 6 t 6 deyy 7 6 7 6 6 dezz 7 1 6 t 7¼ 6 6 6 dcxy 7 E 6 0 6 7 6 4 0 4 dcyz 5 dczx 0 2

t 1 t 0 0 0

t t 1 0 0 0

32 0 3 drxx 0 0 0 76 dr0yy 7 0 0 0 76 0 7 76 drzz 7 0 0 0 76 0 7 76 drxy 7 2ð 1 þ t Þ 0 0 76 0 7 54 dryz 5 0 2ð1 þ tÞ 0 dr0zx 0 0 2ð 1 þ t Þ

ð4:4Þ

Different elastic constants (E, G, K, t, k, M) are related to each other. If we know two of these constants, we can calculate the others, as summarized in Table 4.1.

Table 4.1 Summary of elastic constants Shear modulus G

Yang’s modulus E

Constraint modulus M

Bulk modulus K

Lame coefficient k

Poisson’s ratio t

G, E

G

E

Gð4G3Þ 3GE

GE 9G3E

GðE2GÞ 3GE

E2G 2G

G, M

G

M

M  43 G

M  2G

M2G 2ðMGÞ

G, K

G

Gð3M4GÞ MG 9GK 3K þ G

K þ 43 G

K

K  2G 3

3K2G 2ð3K þ GÞ

G, k

G

Gð3k þ 2GÞ kþG

k þ 2G

kþ

k

k 2ðk þ GÞ

G, t

G

2Gð1 þ tÞ

2Gð1tÞ 12t

2Gð1 þ tÞ 3ð12tÞ

2Gt 12t

t

E, K

3KE 9KE

E

K

E 2ð1 þ tÞ

E

E 3ð12tÞ

K ð9K3EÞ 9KE tE ð1 þ tÞð12tÞ

3KE 6K

E, t

K ð9K þ 3E Þ 9KE Eð1tÞ ð1 þ tÞð12tÞ

K, k

3ðKkÞ 2 3ðMK Þ 4

9K ðKkÞ 3Kk 9K ðMK Þ 3K þ M

3K  2k

K

k

k 3Kk

M

K

3KM 2

3K=M1 3K=M þ 1

3K ð12tÞ 2ð1 þ tÞ

2K ð1  2tÞ

3K ð1tÞ 1þt

K

3Kt 1þt

t

K, M K, t

2G 3

t

4.1 Elastic Constitutive Relation

85

Different from metal materials or other geomaterials (i.e., concrete, rock), the elastic modulus of soils is nonlinear depending on many factors, such as stress level and density. According to experimental observations, for clays, we can directly adopt the swelling index of the isotropic compression test (j = −De/Dlnp′) as the input parameter to calculate Young’s modulus [1]. Note that the swelling index from the oedometer test is slightly different but can be accepted to determine the value of the bulk modulus. K¼

1 þ e0 0 p ; E ¼ 3K ð1  2tÞ j

ð4:5Þ

For sand, the shear modulus is usually adopted as the input parameter to calculate Young’s modulus. When the isotropic compression curve is available, the shear modulus can be directly measured as an input parameter [1, 2] extended from Hardin and Black [3]:   ð2:97  eÞ2 p0 n1 G ¼ G0  pat ; E ¼ 2Gð1 þ tÞ ð 1 þ eÞ pat

ð4:6Þ

where e is the void ratio, pat is the atmospheric pressure (pat = 101.325 kPa), G0 is the reference shear modulus, and n1 is the parameter controlling the nonlinearity of the modulus with the applied mean effective stress. In the case of a lack of measurement of the shear modulus, it is suggested to use the bulk modulus as an input parameter from the isotropic compression test (which is easy to perform in the laboratory). Then, a typical value of Poisson’s ratio m = 0.25 can be adopted to complete the input setting for elasticity. Note that nonlinear elasticity is usually adopted in constitutive modeling of soils.

4.1.2

Elasticity Under Undrained Conditions

Pore pressure includes steady pore pressure psteady and excess pore pressure pexcess: rw ¼ psteady þ pexcess

ð4:7Þ

Steady pore pressure is data generated according to the depth of water. Thus, the differential of steady pore pressure by time is zero. Then, we have the following equation: r_ w ¼ p_ excess Then, the following matrix can be obtained by Hooke’s law:

ð4:8Þ

86

4

3 2 e_ exx 1 6 e_ eyy 7 6 t0 6 e 7 6 0 6 e_ zz 7 6 6 e 7 ¼ 1 6 t 6 c_ xy 7 E 0 6 0 6 e 7 6 4 c_ yz 5 4 0 e c_ zx 0 2

t0 1 t0 0 0 0

t0 t0 1 0 0 0

0 0 0 2 þ 2t0 0 0

Fundamentals of Elastoplastic Theory

0 0 0 0 2 þ 2t0 0

32 0 3 r_ xx 0 6 r_ 0yy 7 0 7 76 0 7 7 6 0 7 76 r_ zz 7 6 r_ xy 7 0 7 76 7 0 54 r_ yz 5 2 þ 2t0 r_ zx

ð4:9Þ

Substituting Eq. (4.9) with the relationship between effective stress and total stress (r_ 0ij ¼ r_ ij  r_ w dij ), we obtain the following matrix: 3 2 e_ exx 1 6 e_ eyy 7 6 t0 6 e 7 6 0 6 e_ zz 7 6 6 e 7 ¼ 1 6 t 6 c_ xy 7 E 0 6 0 6 e 7 6 4 c_ yz 5 4 0 e c_ zx 0 2

t0 1 t0 0 0 0

t0 t0 1 0 0 0

0 0 0 2 þ 2t0 0 0

0 0 0 0 2 þ 2t0 0

32 3 r_ xx  r_ w 0 7 6 0 7 76 r_ yy  r_ w 7 7 6 0 76 r_ zz  r_ w 7 7 ð4:10Þ 7 6 0 7 76 r_ xy 7 5 4 0 r_ yz 5 0 2 þ 2t r_ zx

Considering the slight compressibility of water, the pore pressure rate can be expressed as follows: r_ w ¼

 Kw  e e_ xx þ e_ eyy þ e_ ezz n2

ð4:11Þ

where Kw is the bulk modulus of water and n2 is the porosity of the soil. Then, Hooke’s law can be expressed by using the total stress increment with the undrained Young’s modulus Eu and the undrained Poisson’s ratio tu: K0 ¼ 3 2 e_ exx 1 6 e_ eyy 7 6 tu 6 e 7 6 6 e_ zz 7 6 6 e 7 ¼ 1 6 tu 6 c_ xy 7 Eu 6 0 6 e 7 6 4 c_ yz 5 4 0 c_ ezx 0 2

tu 1 tu 0 0 0

1 þ e0 0 0 p ; E ¼ 3K 0 ð1  2t0 Þ j tu tu 1 0 0 0

0 0 0 2 þ 2tu 0 0

0 0 0 0 2 þ 2tu 0

32 3 r_ xx 0 7 6 0 7 76 r_ yy 7 7 6 0 76 r_ zz 7 7 7 6 0 7 76 r_ xy 7 5 4 0 r_ yz 5 2 þ 2tu r_ zx

ð4:12Þ

ð4:13Þ

where Eu ¼2Gð1 þ tu Þ ¼ l¼

t0 þ l ð 1 þ t0 Þ 1 þ 2lð1 þ t0 Þ

1 Kw 0 E0 K ¼ 3n2 K 0 3ð1  2t0 Þ

ð4:14Þ ð4:15Þ

4.1 Elastic Constitutive Relation

87

Thus, according to previous equations, the consideration of undrained behavior results in the parameters of effective stress G and t being replaced by the undrained constants Eu and tu, respectively. Note that this part is only used in the software when we use the ANICREEP model to simulate the undrained creep test. The full incompressibility of water is represented by tu = 0.5. However, this will result in the singularity of the stiffness matrix. In fact, the water is slightly compressible with a very high bulk modulus value (Kw  nK 0 ). To avoid this computational problem, tu = 0.495 is adopted. Then, for the undrained soil behavior, the bulk modulus of water is automatically added to the stiffness matrix, which is expressed as follows: Kw 3ð t u  t 0 Þ 0:495  t0 0 K 0 ¼ 300  ¼ K [ 30K 0 0 n2 ð1  2tu Þð1 þ t Þ 1 þ t0

ð4:16Þ

For the undrained elastic constitutive law, more information can be found in the material manual of PLAXIS [4].

4.1.3

Cross-Anisotropic Elasticity

During natural sedimentation, soil exhibits a significant cross-anisotropy in elastic stiffness, friction angle, and even critical state line. In this software, we consider the cross-anisotropic elasticity of Graham and Housley [5], which is expressed as follows: 2 6 6 6 6 6 4

de11 de22 de33 de12 de23 de13

3

2

1=Ev 0 t 7 6 vv =Ev 7 6 6 7 6 t0vv =Ev 7¼6 0 7 6 5 4 0 0

t0vv =Ev 1=Eh t0vh =Eh 0 0 0

t0vv =Ev t0vh =Eh 1=Eh 0 0 0

0 0 0 1=2Gvh 0 0

0 0 0 0  1 þ t0vh =Eh 0

32 0 6 0 7 76 7 0 76 6 6 0 7 76 0 54 1=2Gvh

dr011 dr022 dr033 dr012 dr023 dr013

3 7 7 7 7 7 7 5

ð4:17Þ pffiffiffiffiffi where Eh ¼ n3 Ev ; t0vh ¼ n3 t0vv with Ev and Eh representing the vertical and horizontal Young’s moduli, respectively; t0vv and t0vh are the vertical and horizontal Poisson’s ratios, respectively; and Gvh is the shear modulus. For convenient utilization, according to Yin et al. [6] the modification of the elastic modulus increment was obtained as follows based on the stress-controlled isotropic compression test:  dev ¼ de11 þ de22 þ de33 ¼

1

4t0vv

 2 2t0vv dp0 þ  pffiffiffiffiffi n3 n3 Ev

ð4:18Þ

88

4

Fundamentals of Elastoplastic Theory

According to K = dp’/dev, the vertical Young’s modulus Ev can be calculated by the following equation: Ev ¼

    2 2t0vv 2 2t0vv 1 þ e0 0 0 p 1  4t0vv þ  pffiffiffiffi þ  K ¼ 1  4t ffi pffiffiffiffiffi vv n3 n3 n3 n3 j

ð4:19Þ

Then, the shear modulus becomes the following expression: pffiffiffiffiffi n3 E v Gvh ¼  pffiffiffiffiffi  2 1 þ n3 t0vv

ð4:20Þ

Thus, for cross-anisotropic elasticity, we need the following three input parameters: Ev, t’vv, and n3. Compared to the isotropic elasticity, one extra parameter n3 is added for the cross-anisotropic elasticity. K or j can be obtained from the curve of the isotropic compression test. Since the elastic stiffness is normally a function of the applied normal stress, which implies that the parameter n3 is 
n n3 ¼ Eh =Ev ¼ r0h r0v 1 ¼ K0n1  ð1  sin /c Þn1 , considering that the natural deposition of soils suffers from this K0 condition, the default value of n3 can thus be set as n3 ¼ ð1  sin /c Þn1 if we know the friction angle of the soil.

4.2

Elastoplastic Constitutive Relation

The plastic constitutive relation has three key contents: the yield surface equation related to the yield condition, the flow rule with the plastic potential function related to the stress dilatancy during shearing, and the hardening law of the yield surface. This section presents the classical elastoplastic theory. In the classical elastoplastic theory, the total strain increment is the sum of the elastic strain increment and the plastic strain increment: deij ¼ deeij þ depij

ð4:21Þ

Then, the purpose of the plastic constitutive relationship is mainly to solve the   0 p plastic strain part to update all variables (e.g., dr ¼D de  de ). This aspect can be found in many books on solid mechanics or plasticity (e.g., Chen et al. [7]; Luo et al. [8]; Zhang and Ye [9]).

4.2.1

Yield Condition

When subjected to loading, a material evolves from an elastic state to a plastic state as the load increases. This process is called yielding. At that point, the condition

4.2 Elastoplastic Constitutive Relation

89

that must be satisfied by the stress or strain is called the yield condition. Therefore, the yield condition is a criterion for judging whether the material is in an elastic phase or in a plastic phase. In a uniaxial stress state, the elastic limit of a material is defined by two yield stress points. In a more general stress state, the elastic limit becomes a curve, a surface, or a hypersurface in the stress space. The mathematical expression of the elastic limit is determined as follows:   f r0ij ¼ 0

ð4:22Þ

Equation (4.22) is called the yield criterion. The particular form of the function f is material-dependent and contains several material constants. The function f is called the yield function, and the surface f = 0 is called the yield surface. During the hardening phase, the size, shape, and position of the yield surface may change. Therefore, for the sake of clarity, the yield surface and the yield function of the initial state are called the initial yield surface and the initial yield function, respectively. Accordingly, the surface and function of the corresponding hardening phase are called the subsequent yield surface and the subsequent yield function, respectively. It should be noted that the word “loading” can be used instead of “yielding”, such as replacing the yield surface with the loading surface. For isotropic materials, the direction of the principal stress does not play a role, and therefore, the three principal stress values r1, r2, r3 are sufficient to determine the unique stress state. The yield criterion can be expressed as follows:     f r01 ; r02 ; r03 ¼ 0 or f I10 ; J2 ; J3 ¼ 0

ð4:23Þ

where, I 1, J2, and J3 are the first invariant of the stress tensor r′ij and the second and third invariants of the deviatoric stress tensor sij, respectively. The yield criterion can be determined experimentally. For anisotropic materials, the material properties are different in different directions; thus, the direction of the principal stress plays a decisive role, and the yield criterion of an anisotropic material must take the general form of Eq. (4.22). The yield surface in the stress space determines the boundary of the current elastic domain. If a stress point is inside this domain, it is called an elastic state, and the strain variation is reversible, depending on only the elastic properties of the material; on the other hand, if the stress state is on the yield surface, it corresponds to a plastic state, producing elastic, or elastoplastic strains according to the direction of loading. Mathematically, the elastic state and the plastic state are defined as follows: elastic state when f < 0; plastic state when f = 0. For hardening materials, if the stress state tends to move out of the yield surface, a loading process can be obtained, and elastoplastic deformation can be observed; additional plastic strain will be produced, and the current yield (or loading) surface configuration will also change so that the stress state is always maintained on the

90

4

Fundamentals of Elastoplastic Theory

subsequent loading surface. If the stress state tends to move into the yield surface, it is the unloading process under which only elastic deformations occur and the yield surface remains the same. Another possibility is that the stress point moves along the current yield surface. This process is called neutral transposition, and the deformation associated with this process is elastic. The mathematical expressions that distinguish these phenomena are called the loading criterion and can be expressed by the following formula: 8 > > > > > > > > < > > > > > > > > :

f ¼ 0 and f ¼ 0 and

@f dr0 [ 0 ) Loading @r0ij ij

@f dr0 ¼0 ) Neutral loading @r0ij ij

f ¼ 0 and

ð4:24Þ

@f dr0 \0 ) Unloading @r0ij ij

In general, . the f-function form is defined such that the direction of the gradient vector @f @r0ij ¼ nfij always follows the normal direction of the yield surface f = 0. Therefore, these loading criteria can be briefly illustrated in Fig. 4.1. For a perfect plastic material, elastoplastic deformations can develop as the stress point stays on the yield surface. However, the stress state does not always cause plastic deformation and may be classified as a neutral load, so the loading criteria for this material are given by the following expressions: 8 @f > > dr0 ¼0 ) Loading or neutral loading > f ¼ 0 and < @r0ij ij > > > :

@f f ¼ 0 and dr0 \0 ) Unloading @r0ij ij

ð4:25Þ

It should be noted here that the loading and neutral loading processes cannot be distinguished by the above criteria. Different forms of expression have been proposed, and strain increments can be used instead of stress increments to make judgments: 8 @f > > f ¼ 0 and > 0 Dijkl dekl [ 0 ) Loading > > @r ij > > > < @f f ¼ 0 and Dijkl dekl ¼0 ) Neutral loading ð4:26Þ @r0ij > > > > > > @f > > f ¼ 0 and Dijkl dekl \0 ) Unloading : @r0ij Here, Dijkl is the elastic stiffness tensor. This form is more general and more applicable to perfectly plastic materials. Especially in a finite element analysis, the

4.2 Elastoplastic Constitutive Relation

91

stress increment needs to be calculated from the given or known strain increments. This calculation requires determining the form of deformation that is occurring. The criteria used in Eqs. (4.25) and (4.26) are not convenient because they require knowledge of the stress increment. Equation (4.26) permits us to solve this problem in a very straightforward way. In constitutive models for soils, several yield functions have been proposed and adopted, for example, (1) the Mohr–Coulomb criterion, (2) the Lade–Duncan criterion, and (3) the spatially mobilized plane (SMP) criterion. In addition, many functions of the yield surface have been proposed in recent decades, such as those of Cam-Clay and its extended models [10–14]. More details can be found in Sect. 4 of this chapter.

4.2.2

Flow Rule

Plastic strain is generated during the loading process. To describe the stress–strain relationship in the case of elastoplastic deformation, the direction and magnitude of the plastic strain increment vector depij must be defined for a given state of stress rij and a given stress increment: (1) the ratio of the components; and (2) the components correspond to the size of the stress increment drij . For the direction of the plastic strain increment, the common understanding is as follows: what kind of evolution does the material experience when plastic deformation occurs? What is the relationship between the strain increment and the normal direction of the yield surface? This rule is similar to the flow of an ideal fluid and is therefore called the flow rule. The magnitude of the plastic strain increment will be determined later by the consistency conditions. The concept of the plastic potential energy function g is described below in a manner similar to the ideal fluid flow problem. We define the flow rule as follows: depij ¼ dk

@g @r0ij

ð4:27Þ

where dk is a nonnegative scalar called a plastic multiplier. The gradient vector @g=@r0ij specifies the direction of the plastic strain increment vector depij , that is, the potential energy plane g = 0 is in the normal direction to the current stress point. For this reason, the flow rule is also called the orthogonal condition. On the other hand, the length or magnitude of the plastic strain increment vector is determined by dk. If g = f, the flow rule is said to be associated; if f 6¼ g, the flow rule is non-associated. The models for cohesive soil often adopt the associated form, whereas a non-associative form is usually chosen for sandy soils to better reproduce the amplitude of dilation in dense materials, which cannot be well controlled by the application of the associated flow rule.

92

4.2.3

4

Fundamentals of Elastoplastic Theory

Hardening Rule

During the loading process, the yield surface constantly changes its shape so that the stress state remains on the yield surface. However, there are countless yield surfaces that can satisfy this condition, and thus, how the loading surface develops is not a simple question. In fact, this is one of the main problems in the theory of plastic hardening. The rule governing the development of the loading surface is called the hardening law. Several rules have been proposed, and the material response will be very different after initial yielding, depending on the particular hardening rule used. To state the hardening properties, it is necessary to (i) record the history of plastic loading and (ii) describe the relationship between hardening and plastic loading history. For the former, the enhancement or growth function k has been introduced, and for the latter, a monotonous growth scalar called the hardening parameter j* has been introduced. The hardening function of j* is a function of hardening parameters, and its functional form is material-dependent. Two assumptions are made when recording the plastic history: one assumption is that the hardening depends on the plastic work Wp, called the processing hardening hypothesis. The other hypothesis, called the strain hardening hypothesis, assumes that the hardening is related to the plastic strain ep. The materials that meet one of these two assumptions are called processing hardening materials and strain hardening materials, respectively. The two hardening parameters W p and ep are denoted as j*. From a practical point of view, it is easier to use ep than Wp; thus, ep is used more frequently than Wp in elastoplastic models. Here, a brief introduction to the two most basic hardening rules is presented.

(a)

σ

(b) Loading

Neutral loading n'ijdσij=0 n'ij

σ1 Unloading

O

σ2

Loading n'ijdσij>0 Unloading n'ijdσijk1

Initial yield surface f0=k B A

σ2

O

D

C

(1) Isotropic hardening The isotropic hardening rule assumes that the yield surface is uniformly expanding during loading without distortion and movement, as shown in Fig. 4.2. Therefore, the mathematical expression of the yield surface can be written as follows:     f r0ij ; j ¼ f0 r0ij  kðj Þ ¼ 0

ð4:28Þ

where k is an increasing function of j*, which controls the size of the yield surface. (2) Kinematic hardening The kinematic hardening rule assumes that during the plastic deformation process, the loading surface moves in the stress space as a rigid body without rotation, so the size, shape, and direction of the initial yield surface remain unchanged. Initially, as suggested by Prager, this hardening rule provides a simple way to consider the Bauschinger effect, as illustrated in Fig. 4.3 (https://en.wikipedia.org/wiki/ Bauschinger_effect). The yield surface with kinematic hardening is generally expressed as follows:     f r0ij ; aij ¼ f0 r0ij  aij  k ¼ 0

ð4:29Þ

where k is a constant fixing the size of the yield surface (same as k in isotropic hardening but kept constant herein); aij is called the reverse stress, which provides the coordinates of the center of the yield surface. The reverse stress varies during the plastic loading process with the kinematic hardening rule to account for the ij ¼ rij  aij is often used for convenience. hardening response. r

94

4

Fundamentals of Elastoplastic Theory

σ1

Fig. 4.3 Subsequent yield surface for kinematic hardening

Loading and unloading paths

B A C

OI(αij)

σ2

O Subsequent yield surface Initial yield surface

f0(σij-αij)=k

f0(σij)=k

4.2.4

Drücker’s Stability Hypothesis, Convexity, and Orthogonality

A material that satisfies the Drücker stability hypothesis is called a stabilizing material: during the application of a complete cycle of an additional force on the material, the net work done by the external force, including the unloading stage, is nonnegative. Considering a material unit that is subjected to a uniform stress, rij , rij is inside the yield surface (Fig. 4.4). Supposing an external force increases the stress along path ABC, AB is inside the yield surface, and point B is only on the yield surface. This stress continues to move outward, causing the yield surface to expand until it reaches point C, and then, the external force is removed and the stress state rij returns along the elastic path CDA. Therefore, the work done by the external force during this cycle is determined as follows: Z DW ¼

 ABCDA

Z  r0ij  rij deij ¼

 ABCDA

Z  r0ij  rij deeij þ

 BC

 r0ij  rij depij ð4:30Þ

Since the elastic deformation is reversible and independent of the path, all elastic energy is recoverable. After the loading cycle is completed, only the net work done by the plastic work as an external force is retained.

4.2 Elastoplastic Constitutive Relation

95 σij-σ*ij

σ σ σ*

σij

BC

B C

D

(σ-σ*) dε p>0 O

A

dε

C

σ*ij

A*

A

dε p

B

p

O

ε

(a)

(b)

Fig. 4.4 Cycle stability: a stress state in the yield plane (point A); b stress path ABC generated by external force

Z DW ¼

 BC

 r0ij  rij depij

ð4:31Þ

Equation (4.31) indicates that DW  0. Considering that path BC is arbitrary, we obtain the mathematical expression of the loop stability condition as follows: 

 r0ij  rij depij  0

ð4:32Þ

If the plastic strain coordinates are superimposed on the stress coordinates, as shown in Fig. 4.4, then Eq. (4.32) can be geometrically represented as the scalar  product of the stress increment vector r0ij  rij

and the strain increment vector

depij .

A positive scalar product indicates that the angle between the two vectors is an acute angle. Therefore, the stability hypothesis derives the following conclusions for processing hardening materials [15]: (1) convexity, where the initial yield surface and all subsequent loading surfaces must be convex; and (2) orthogonality, where the plastic strain increment vector must be orthogonal   . tothe yield or loading surface f r0ij ; epij ; k ¼ 0 at a smooth point: depij ¼ dk @f @r0ij , where the point is

at a corner and between adjacent normals. Note that the convexity is a postulate but not a theorem. As noted by Yu [16], although Drücker’s postulate only covers certain types of real stress–strain behavior for engineering materials, it does provide a unique way of unifying a whole set of features for the plastic stress–strain relations. It must be stressed that while Drücker’s postulate implies that the material must obey Hill’s maximum plastic work (Eq. (4.32)), the reverse is not true, which is because Drücker’s stability postulate also requires a nondecreasing hardening rate (dr0ij depij  0).

96

4.3 4.3.1

4

Fundamentals of Elastoplastic Theory

Numerical Method for Solving the Plastic Problem General Explicit Solution

For isotropic hardening materials, the yield surface equation can be expressed as follows:   f r0ij ; j ¼ 0

ð4:33Þ

The consistency condition states that the stress state must remain on the yield surface during loading:   @f @f df r0ij ; j ¼ 0 dr0ij þ  dj ¼ 0 @rij @j

ð4:34Þ

Assuming that the hardening parameter is a function of plastic strain, the consistency condition can be written in the following form: @f @f @j p 0 dr þ de ¼ 0 @r0ij ij @j @epij ij

ð4:35Þ

The general process for deriving the full stress–strain relationship of a plastic hardened material is given below: (1) Dividing the total tress increment and elastic strain increment into elastic and plastic parts according to Hooke’s law and elastic stiffness matrix, we obtain the following equations:

deij = deeij þ depij

ð4:36Þ

  dr0ij ¼Dijkl deekl ¼ Dijkl dekl  depkl

ð4:37Þ

(2) The general form of the non-associated plastic flow rule is used to represent the following equation: ! depkl

  @g 0 @g ¼ dk 0 drij ¼ Dijkl dekl  dk @rkl @rkl

ð4:38Þ

(3) By substituting the above formula into the consistency condition, we obtain the following expression:

4.3 Numerical Method for Solving the Plastic Problem

97

  @f @f @j p @f @g @f @j @g 0 dr þ de ¼ D de  dk dk ¼0 þ ijkl kl p ij ij @r0ij @j @eij @r0ij @r0kl @j @epij @rij h iT

 @f Dijkl dekl @r0ij ) dk ¼ h iT h i @f @g @f @j @g Dijkl @r  @j 0  @ep @r @r0 ij

kl

ij

ij

ð4:39Þ Then, Hooke’s law is used again:

)

Dep ijkl

  @g drij ¼ Dijkl dekl  dk ¼ Dep ijkl dekl @rkl   h iT @g @g @f Dijkl dekl  dk @r D ijkl @rkl @rij Dijkl kl ¼ ¼ Dijkl  h iT dekl @f @g @f @j @g @rij Dijkl @rkl  @j @ep @r

ð4:40Þ

ij

According to this ratio, there are 6  6 independent variables. Therefore, Dep can be turned into a 6  6 matrix. We can make the following transformation: instead of the subscript ij, a is used, and instead of the subscript kl, b is used; the following relationship exists between a, b, and ij, kl: 8 a > > > >1 > > > > 2 > > > > 3 > > < 2 4 > > > > 5 > > > > > 3 > > > > 5 > : 6

i 1 1 1 2 2 2 3 3 3

9 j> >b > 1> > >1 > 2 2> > > > 3 3> > > = 1 2 4 2> > > 5 3> > > > > 3 1> > > > 5 2> > ; 3 6

k 1 1 1 2 2 2 3 3 3

9 l> > > 1> > > > 2> > > > 3> > > = 1 2> > > 3> > > > > 1> > > > 2> > ; 3

ð4:41Þ

According to this formula, the expression of each component in the matrix Dep can be expressed by the following equation: 2

Dep ijkl

= Dab

D11 6 D21 6 =4 ::: D61

D12 D22 ::: D62

::: ::: ::: :::

2

Dep 1111 ep 6 D16 6 D1211 6 Dep D26 7 7 = 6 1311 ep ::: 5 6 2211 6 Dep 4 D66 D2311 Dep 3311 3

Dep 1112 Dep 1212 Dep 1312 Dep 2212 Dep 2312 Dep 3312

Dep 1113 Dep 1213 Dep 1313 Dep 2213 Dep 2313 Dep 3313

Dep 1122 Dep 1222 Dep 1322 Dep 2222 Dep 2322 Dep 3322

Dep 1123 Dep 1223 Dep 1323 Dep 2223 Dep 2323 Dep 3323

3 Dep 1133 7 Dep 1233 7 7 Dep 1333 7 7 Dep 2233 7 ep 5 D2333 Dep 3333

ð4:42Þ

98

4

Fundamentals of Elastoplastic Theory

The plasticity theory described above is limited to the case of a smooth yield surface and does not include the multi-segment yielding line that appears, for example, in the Mohr–Coulomb model. Koiter et al. [17] extended the plasticity theory to this yield surface condition to handle the flow function vertices that include two or more plastic potential functions: e_ pij ¼ k1

@g1 @g2 þ k2 0 þ    r0ij rij

ð4:43Þ

Similarly, several quasi-independent yield functions (f1, f2,…) are used to determine the size of the multiplier (k1, k2..).

4.3.2

Cutting Plane Method-Based Implicit Solution

For complex models, it is difficult to derive the second gradients, and the advantage of the cutting plane method is that we only need to perform the first gradient derivation [18]. In the numerical integration process, it is necessary to judge whether the current strain increment de is caused by the elastic stress increment or elastoplastic stress increment. Under the stress point ri, for the strain increment de, we first assume that it is elastic strain, based on which the updated stress point r‘i+1 = r‘i + dr‘ can be obtained. Relative to the current yield surface f, as shown in Fig. 4.5, there are three possibilities for r‘i+1: (1) in the elastic region (f < 0); (2) on the current yield surface (f = 0); or (3) located outside the current yield surface (f > 0). For the updated stress point r‘i+1, if f 0, then the strain increment de is equal to elastic strain dee , and the following stress update formula is directly valid: ri þ 1 ¼ ri þ dre ¼ ri þ Ddee

ð4:44Þ

If f [ 0, the plastic strain dep generated during the stress update process needs to be calculated. This process can be implemented using a cutting plane algorithm combined with return mapping.

Yield surface f

Fig. 4.5 Schematic plot for the loading path

σi+1 σi

(1)

(2)

σi+1

(3)

σi+1

4.3 Numerical Method for Solving the Plastic Problem

99

Yield surface f n = 2 i +1

σ in+=10

Yield surface f i +n1=1 Yield surface f i +n1= 0

σ in+=11 σ in+=12

σi Evolution of yield surface

Fig. 4.6 Iterative calculation of plasticity modification of the cutting plane method

It is assumed that the strain increment de is complete elastic deformation. When Eq. (4.45) is satisfied, the soil enters the elastoplastic state from the elastic state:     f r0i ; #i \0 and f r0i þ dre ; #i [ 0

ð4:45Þ

where # is the yield surface hardening parameter. At this time, the stress point r‘i+1 after the trial calculation is outside the yield surface, so the plasticity correction is entered. As shown in Fig. 4.6, in the first n¼1 plastic modification, the stress point rn¼0 i þ 1 is reduced to ri þ 1 , and as the hardening n¼1 n¼0 parameter #n¼0 i þ 1 is updated as #i þ 1 , the yield surface fi þ 1 is also developed outward n¼1 as fi þ 1 to obtain a plastic strain increment. If the stress point at this time rn¼1 i þ 1 still , the plasticity correction needs to be does not coincide with the yield surface fin¼1 þ1 p n¼2 performed again to obtain the plastic strain increment ðde Þ , and then, the hardening parameter is updated, and the yield surface is expanded again until the corrected stress point coincides with the yield surface. The updated stress is n P ri þ 1 ¼ rniþ 1 , and the plastic strain increment is dep ¼ ðdep Þi . Finally, the stress i¼1

state point coincides with the yield surface (f = 0) here, which means that f ðr; #Þ\tolerance with the tolerance is generally small, e.g., 10−7. The cutting plane algorithm is based on the stress surface obtained by the elastic stress trial (the shape is the same as the yield surface) f ðri þ dre Þ as the start point and is gradually pulled back (or the stress relaxation process) during the plastic modification process. In each plastic correction calculation, compared with the previous iteration result, as the plastic strain increment increases, the elastic strain increment will decrease correspondingly to ensure that the total strain increment is unchanged. Therefore, in the iterative process, we can establish the following equation:

100

4

Fundamentals of Elastoplastic Theory

dep ¼ dee

ð4:46Þ

Through further derivation, we can obtain the amount of stress relaxation and hardening development: @g @r0

ð4:47Þ

@# p @# @g de ¼ p dk 0 p @e @e @r

ð4:48Þ

dr0 ¼ Ddep ¼ Ddk d# ¼

For the first iteration correction, the yield equation f is first-order expanded with the Taylor formula to obtain an approximate equation: n¼0 fin¼1 þ 1 ¼ fi þ 1 þ

@f @f d# dr0 þ 0 @r @#

ð4:49Þ

Substituting formulas (4.47)-(4.48) into formula (4.49) and obeying the target value f = 0, we can obtain the following expression: dk ¼

@f  @r 0

fin¼0 þ1

@g D @r 0 þ

@f @# @g @# @ep @r0

ð4:50Þ

 n¼0 n¼0   0  e where fin¼0 þ 1 ¼ f ri þ 1 ; #i þ 1 ¼ f ri þ dr ; #i . Then, from formulas (4.47)–(4.50), the stress and hardening parameter update formula can be obtained after the plastic modification: @g @r0 @# @g ¼ #n¼1 dk iþ1 þ @ep @r0

n¼0 rn¼1 i þ 1 ¼ ri þ 1  Ddk

#n¼1 iþ1

ð4:51Þ

At this point, the first plastic correction is completed. If the convergence condition f ðr0 ; #Þ\tolerance is not satisfied at this time, the plastic correction calculation is continued until convergence according to formulas (4.46)–(4.51). Figure 4.7 shows the plasticity correction process of the cutting plane algorithm for one incremental step of strain.

4.3.3

Closest Point Projection Method-Based Implicit Solution

The closest point projection method (CPPM) solves the plastic multiplier equation in conjunction with Newton’s iterative technique [19]. Consequently, a consistent tangential modulus needs to be derived to obtain the quadric convergence speed.

4.3 Numerical Method for Solving the Plastic Problem

101

Fig. 4.7 Flowchart for the cutting plane algorithm for plastic modification

In CPPM, we need to derive the second gradients’ derivation. Since the yield criterion is plastic displacement dependent, the residue can be expressed as follows: ðkÞ

ðkÞ

ðkÞ

Rn þ 1 ¼ ðDeÞn þ 1 þ ðDkÞn þ 1 bkn þ 1

ð4:52Þ

. where the first-order derivatives of the potential function b ¼ @g @r0ij are defined. ðkÞ

If the yield point satisfies the condition fn þ 1 TOL1 and the remainder satisfies the ðkÞ condition Rn þ 1 TOL2 , it will not be necessary to calculate the plastic corrector. Otherwise, the consistent tangential modulus is computed as follows: " ðkÞ Cn þ 1

¼

ðkÞ I þ ðDkÞn þ 1

@b @r0

ðkÞ # nþ1

Den þ 1

ð4:53Þ

where Den þ 1 is the elastic matrix, as shown in Eq. (4.3), and the second derivatives of the potential function @b=@r0 can be derived for a given potential surface. The consistency parameter increment can be obtained as follows: 

Dk 2

ðkÞ nþ1

ðkÞ

¼

ðkÞ

ðkÞ

ðkÞ

fn þ 1  an þ 1 Cn þ 1 Rn þ 1 ðkÞ

ðkÞ

ðkÞ

ð4:54Þ

an þ 1 Cn þ 1 bn þ 1

. where the first-order derivatives of the yield function a ¼ @f @r0ij are defined. The stress increments can be obtained as follows:

102

4

Fundamentals of Elastoplastic Theory

 ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðDrÞn þ 1 ¼ Cn þ 1 Rn þ 1  D2 k n þ 1 Cn þ 1 an þ 1

ð4:55Þ

Then, the plastic multiplier can be updated by the following expression:  ðkÞ ðk þ 1Þ ðkÞ ðDkÞn þ 1 ¼ ðDkÞn þ 1 þ D2 k n þ 1

ð4:56Þ

The incremental plastic strains can be calculated as follows:  ðkÞ ðkÞ ðk þ 1Þ ðDep Þn þ 1 ¼  D1 n þ 1 ðDr0 Þn þ 1

ð4:57Þ

Finally, the stresses can be updated by the following equation: ðk þ 1Þ

ðkÞ

ðkÞ

ðr0 Þn þ 1 ¼ ðr0 Þn þ 1 þ ðDr0 Þn þ 1

ð4:58Þ

The flowchart for implementing the elastoplastic model by CPPM is presented in Table 4.2.

4.4

Strength Criteria for Soils

It has been shown that soil has an elastic limit under both compression and shear loadings. The failure of the material manifests itself in the rapid development or continuous development of the deformation or cumulatively develops to the extent that it is considered destructive. Soil is an assembly of soil particles that are more prone to relative slip damage than soil particles themselves. Usually, the strength of the soil refers to the shear strength rather than compression, particularly for sandy soils. The adhesion between continuous soil particles is almost nonexistent, so the tensile strength of the soil is very low, and it is generally considered that it cannot withstand tensile stress. In the isotropic stress (hydrostatic pressure) and the confined compression (e.g., consolidation test), the void of the soil can only be reduced, becoming more compact, and no sliding shear failure occurs. Soil is a kind of friction material. The shear strength is related to the normal stress. The greater the normal stress is, the greater the shear stress that can be withstood. The hardness of the soil is that the shear strength increases with increasing normal stress. This is a significantly different feature from metallic materials. The strength criterion of soil is the formula under the condition that the soil will be destroyed if its stress  state  is satisfied. The mathematical description of the strength criterion is F rij ; kf ¼ 0, where kf is the soil characteristic parameter. Depending on the stresses used, the strength criteria for soil can have many different      representations, such as F r01 ; r02 ; r03 ; kf ¼ 0; F I10 ; J2 ; h3 ; kf ¼ 0; F r0n ; sn ; h; kf ¼ 0. When graphically represented, the failure envelope determined by the strength

4.4 Strength Criteria for Soils

103

Table 4.2 Close point projection method (CPPM) for the elastoplastic model

( )

1) Initialize: k = 0 , ε

p

( 0)

= ( ε p ) , ( Δε p ) = 0 , ( Δλ )n +1 = 0 . n n+1 (k )

(k )

n +1

2) Compute the yield condition and evaluate hardening law residuals:

(

( Δσ′ )n+1 = ( D)n+1 ( Δε )n+1 − ( Δ ε p )n+1 (k )

(k )

(k )

( ( σ′ )

f n(+k1) = f Rn( k+1) = − ( Δε p )

(k ) n +1

(k )

) , ( σ′ )

, ( ε )n +1 (k )

n +1

(k ) n +1

= ( σ′ )n + ( Δ σ′ )n+1

)

+ ( Δλ )n +1 b nk +1 (k )

(k ) IF: f n(+k1) ≤ FTOL and Rn +1 ≤ RTOL , THEN: EXIT.

3) Compute the consistent tangent moduli: Cn−1+1 = ( D−1 ) (k )

(k ) n +1

+ ( Δλ )n +1 b (nk+)1 (k )

4) Obtain the increment of the consistency parameter:

(Δ λ ) 2

(k ) n +1

=

f n(+k1) − a(nk+)1C(nk+)1 Rn( k+1) a (nk+)1C(nk+)1b (nk+)1

5) Calculate the force increments:

( Δσ )n+1 = −C(nk+)1Rn( k+1) − ( Δ 2λ )n+1 C(nk+)1a(nk+)1 (k )

(k )

6) Obtain the incremental plastic strains and internal variables:

( Δλ )n+1

( k +1)

= ( Δλ )n +1 + ( Δ 2 λ ) (k )

(k ) n +1

7) Incremental plastic displacements:

( Δε )

p ( k +1) n +1

= − (D −1 )

(k ) n +1

( Δ σ′ )n+1 (k )

8) Update the forces at the interparticle contact:

( σ′)n+1

( k +1)

Set k

k + 1 and GO TO step 2.

= (σ ′ )n +1 + ( Δσ′ )n +1 (k )

(k )

(k )

104

4

Fundamentals of Elastoplastic Theory

criterion can be plotted in the three-dimensional principal stress space. For ease of analysis, the failure envelope is often plotted on both the p-plane and the space meridian plane.

4.4.1

Generalized Von Mises Criterion

The von Mises criterion expresses that when the deviatoric stress J2 reaches a certain limit, the material is destroyed. This criterion is expressed as follows: F ð J2 Þ ¼ J2  k 2 ¼ 0

ð4:59Þ

Or in terms of principal stresses,  0 2  2  2   r  r02 þ r02  r03 þ r03  r01  k2 ¼ 0 F r01 ; r02 ; r03 ¼ 1 6

ð4:60Þ

In the principal stress space, this criterion is represented by a cylinder with the space diagonal set as the axis. The intercept with the p-plane is a circle. The von Mises criterion is independent of the hydrostatic pressure. In this aspect, it does not reflect the properties of soils that are pressure-dependent materials. Drücker and Prager added the hydrostatic pressure effect within the von Mises criterion to suggest the generalized von Mises criterion, which is expressed as follows:   pffiffiffiffiffi F J2 ; I10 ¼ J2  aI10  k ¼ 0

ð4:61Þ

In the principal stress space, this criterion is represented by a conical surface with the space diagonal set as the axis. The above equation can be expressed in the p’-q plane as follows: pffiffiffi pffiffiffi Fðp; qÞ ¼ q  3 3ap0  3k ¼ 0

ð4:62Þ

It can be seen from Eqs. (4.61) and (4.62) that as the mean stress increases, the maximum shear stress also increases, which is in agreement with the behavior of the pressure-dependent materials.

4.4.2

Generalized Tresca Criterion

The Tresca criterion expressed the fact that the maximum shear stress smax cannot exceed a certain limit:

4.4 Strength Criteria for Soils

105

 1 0 r1  r03 ¼ k 2

ð4:63Þ

The Tresca criterion can also be expressed in terms of q and h: Fðq; hÞ ¼

 pffiffiffi p pffiffiffiffiffi 2qsin h þ  6k ¼ 0 3

ð4:64Þ

When r01  r02  r03 , 0 h 60 . In the p-plane, the failure envelope is a regular hexagon. In the principal stress space, it is a regular hexagonal cylinder with the space diagonal set as the axis. Considering the hydrostatic pressure effect, the Tresca criterion can also be modified to obtain the generalized Tresca criteria:  1 0 r  r03  k  aI10 ¼ 0 2 1

ð4:65Þ

 pffiffiffi pffiffiffi p pffiffiffi 2q sin h þ  6k  3 6ap0 ¼ 0 3

ð4:66Þ

or Fðp0 ; q; hÞ ¼

In the principal stress space, this is a regular hexagonal cone with the space diagonal set as the axis.

4.4.3

Mohr–Coulomb Criterion

The Mohr–Coulomb strength criterion states that when the stress ratio ðs=rÞmax cannot exceed a certain limit, the equation can be obtained as follows: s r

max

¼ const

ð4:67Þ

Or can be expressed as follows:  1  1 0 r1  r03  r01 þ r03 sin /  c cos / ¼ 0 2 2

ð4:68Þ

where c is the cohesion force and u is the internal friction angle. The Mohr–Coulomb strength criterion can be expressed in terms of p’, q, h: Fðp0 ; q; hÞ ¼ 6p0 sin/ þ q

h   i pffiffiffi pffiffiffi cosh  3sinh sin/  3cosh þ 3sinh þ 6c cos /

¼0 ð4:69Þ

106

4.4.4

4

Fundamentals of Elastoplastic Theory

Lade–Duncan Criterion

The intermediate principal stress has a significant influence on the shear strength of soils, as investigated through true triaxial tests. The strength criterion proposed by Lade–Duncan [20] based on the true triaxial test results on sand is determined as follows:   I 031 F I10 ; I20 ¼ 0  k1 ¼ 0 I2

ð4:70Þ

The Lade–Duncan criterion can be expressed by p’, q, h: F ðp0 ; q; hÞ ¼ 2q3 cos 3h  9p0 q2 þ 729

4.4.5



 1 1 0  p ¼0 27 k1

ð4:71Þ

Matsuoka–Nakai Criterion

The SMP criterion proposed by Matsuoka and Nakai [21] also considers the influence of the intermediate principal stress and establishes the strength of the soil under three-dimensional stress conditions. The failure criterion considers that the ratio of shear stress to normal stress is reached. The failure surface under this three-dimensional stress condition becomes the SMP space sliding surface. The SMP criterion can be expressed as follows: I10 I20 ¼ constant I30

ð4:72Þ

  I0 I0 F I10 ; I20 ; I30 ¼ 1 0 2  8 tan2 /  9 ¼ 0 I3

ð4:73Þ

or

The criteria can also be expressed in terms of p’, q, h: Fðp0 ; q; hÞ ¼



  2    16 2 2 3 tan2 /  1 q 3 8 2 q tan / þ tan þ / þ 2 8 tan2 / ¼ 0 27 3 ðtan2 / þ 1Þ12 p0 3 p0

ð4:74Þ The SMP criterion given by the above expressions is applicable to non-cohesive granular materials (c = 0); in the case of c 6¼ 0, the criterion can be transformed by the coordinate translation method.

4.4 Strength Criteria for Soils

107

Although the Lade–Duncan and Matsuoka–Nakai criteria have relatively similar shapes in the p-plane, some differences can be pointed out: – Matsuoka–Nakai criterion, such as the Mohr–Coulomb criterion, corresponds to the same friction angles in triaxial compression and extension; and – The Lade–Duncan criterion corresponds to higher friction angle values in extension than in compression.

4.4.6

Generalized Nonlinear Strength Criterion

Yao et al. [22] proposed a generalized nonlinear strength criterion based on the SMP criterion and the generalized von Mises criterion. Let the von Mises criterion coincide with the SMP criterion under triaxial compression, and then, the expression of the generalized shear stress qc is determined as follows: qc ¼ aqM þ ð1  aÞqS

ð4:75Þ

where a is the material parameter reflecting the strength ratio of compression to extension in the p-plane, (i) a = 1 corresponds to the generalized von Mises criterion, where the failure line in the p-plane is a circle; (ii) a = 0 corresponds to the SMP criterion, and the failure line in the p-plane is a smoothly curved triangle; and (iii) 0 < a < 1 corresponds to a smooth curve located between the generalized von Mises criterion and the SMP criterion; therefore, the strength characteristics of various materials can be described. qM is the generalized shear stress corresponding to the von Mises criterion, which can be expressed as follows: qM ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I 01  3I20

ð4:76Þ

qS is the generalized shear stress corresponding to the SMP criterion and can be expressed as follows: 2I10 qS ¼ qffiffiffiffiffiffiffiffiffiffiffiffi I 0 I 0 I 0 3 I 01I 029I30  15 1 2

ð4:77Þ

3

The material parameter a can be determined from the slopes of the critical state lines Mc and Me in the p-q plane under compression and extension conditions. a¼

Me Mc þ 3ðMe  Mc Þ 3ð3 þ sin /e Þðsin /e  sin /c Þ or a¼ Me2 2 sin2 /e ð3  sin /c Þ

ð4:78Þ

108

4.4.7

4

Fundamentals of Elastoplastic Theory

Modeling Methods

Two general methods for modifying the strength in the stress space are introduced herein. These two methods have been widely adopted in phenomenological models. (1) g(h)-method This method mainly works for soil models that consider the slope of the critical state line in the p′-q plane as the main parameter. This method considers the influence of the Lode angle on the yield strength by using M = Mcg(h), namely, the g(h)-method (see Table 4.3). For example, the widely adopted modification proposed by Sheng et al. [23] is expressed as follows:

2c4 M ¼ Mc 1 þ c4 þ ð1  c4 Þ sin 3h

14

ð4:79Þ

where c = Me/Mc is the ratio of the slope of the critical state line in extension and compression. For a friction angle independent of the Lode angle, the ratio c can be derived as c ¼ ð3  sin /c Þ=ð3 þ sin /c Þ ¼ 3=ð3 þ Mc Þ. In this case, the strength criterion is close to the Matsuoka–Nakai criterion. The expression of the Duncan–Lade strength criterion can also be obtained by giving the value of c in Eq. (4.79). Here, we present the derivation process: (1) According to Duncan–Lade’s criterion I 01 =I 01 I30  I30 = k1 and under the triaxial 2 condition,I10 ¼ r0a þ 2r0r and I30 ¼ r0a r0r , we can obtain the following expression: 3



r0a þ 2r0r 2 r0a r0r

3

3 ¼ k1

ð4:80Þ

(2) According to the slope of the critical state line in extension and compression,
we can derive the stress ratio r0a r0r at the critical state or on the failure line as a function of Mc or Me:

Table 4.3 Some of the typical Lode angle modification methods for yield strength criteria

References

Equation g(h)

Arygris [24]

2m ð1 þ mÞð1mÞ sinð3hÞ  pffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 8 þ sin2 u0 sin u0

Satake [25] Sheng et al. [23] Zheng [26]

h

4

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

2 þ sin u0 cos w

i14

2m4 1 þ m4 þ ð1m4 Þ sin 3h

2m ð1 þ mÞð1mÞ sinð3hÞ þ a1 cos2 ð3hÞ

4.4 Strength Criteria for Soils

109

8 q r0a  r0r r0a 3 þ 2Mc > > > M ¼ ¼ 3 ) ¼ c < p0 r0a þ 2r0r r0r 3  Mc 0 0 0 > q r  r r 3  2Me >M ¼ ¼ 3 r a > ) a0 ¼ : e 0 0 0 p ra þ 2rr rr 3 þ Me

ð4:81Þ

   3  3  0 02  2 (3) Thus, in the expression r0a þ 2r0r = r0a þ 2r0r ra r r  r0a r0r ¼ k1 , the right term representing the material strength can be expressed by Mc, which is equal to that expressed by Me. Therefore, we have the following equation: 

32Me 3 þ Me þ 2 32Me 3 þ Me

3

 ¼

3 þ 2Mc 3Mc þ 2 3 þ 2Mc 3Mc

3 ð4:82Þ

(4) By setting a ¼ ð3 þ 2Mc Þ=ð3  Mc Þ and b ¼ ð3  2Me Þ=ð3 þ Me Þ and substituting them into the above equation, b can be solved as a function of a: b¼

ða2 þ 6aÞ þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ða2 þ 12a þ 36Þ þ 32a 2a

ð4:83Þ

(5) b ¼ ð3  2Me Þ=ð3 þ Me Þ implies (can be transformed to) Me ¼ ð3  3bÞ=ð2 þ bÞ, and by substituting the above b into this equation, Me is a function of a and Mc can be obtained as follows:

Me ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3  3b 3a2 þ 24a  3 a2 ða2 þ 12a þ 36Þ þ 32a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2þb a2  2a þ a2 ða2 þ 12a þ 36Þ þ 32a

ð4:84Þ

(6) Therefore, if we know Mc, c can be obtained for the Duncan–Lade strength criterion by the following equation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3a2 þ 24a  3 a2 ða2 þ 12a þ 36Þ þ 32a Me  c¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mc Mc a2  2a þ a2 ða2 þ 12a þ 36Þ þ 32a

ð4:85Þ

where a ¼ ð3 þ 2Mc Þ=ð3  Mc Þ (2) Transformed stress method Nakai and Matsuoka [27] derived the transformed stress tensor tij based on the spatial sliding surface (SMP) under a three-dimensional stress state and then developed the constitutive model using tij instead of the stress space rij. The equivalent relationship between tij and rij is determined as follows:

110

4

Fundamentals of Elastoplastic Theory

tij ¼ aik rkj rffiffiffiffi  1 I3 aij ¼ frik þ I2 dik g I1 rkj þ I3 dkj I2 rffiffiffiffiffiffiffiffi I3 ði ¼ 1; 2; 3Þ ai ¼ I2 ri

ð4:86Þ

The generalized nonlinear strength criterion can scientifically and reasonably describe the nonlinear effects of friction, cohesion, hydrostatic pressure and Lode angle on strength of geomaterials. In order to overcome the difficulty of combining the three-dimensional yield or strength criterion with the constitutive model, Yao [28] innovatively derived the transformed stress tensor by transforming the strength failure surface in three-dimensional principal stress space into the conical surface in the transformed stress space and transformed the strength yield surface considering the influence of the intermediate principal stress on the expanded Mises criterion in the transformed stress space, as shown in Fig. 4.8. It provides a general way for the three-dimensional strength problem that exists in the constitutive modelling. The equivalent relationship between the transformed stress tensor rij and the Cauchy stress tensor rij is determined as follows: rij ¼ p0 dij þ

 q 0 rij  p0 dij q

ð4:87Þ

The expression of the strength criterion has been obtained for two different typical strength criteria: 2I10 ~q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðSMP criteriaÞ 0 0 
0 0 ffi I1 I2  9I30  1 3 I1 I2  I30 8
1 or ec–e > 0 for dense packing, and ec/e = 1 or ec–e = 0 when the initial state lies on the CSL in the e–p’ plane. In agreement with the experimental results obtained by Biarez and Hicher

5 Elastoplastic Modeling of Soils …

140

= 0.5

= 0.01

= 0.71

= 0.1

ec0 = 0.68 = 0.1, = 0.1

= 0.022

=1

ec0 = 1.33 = 1, = 0.12

ec0 = 0.937 = 0.022

ec0 = 0.937 = 0.5, = 0.1

ec0 = 0.937 = 0.71

(a)

(b)

(c)

Fig. 5.16 Influence of material parameters on the linearity of the critical state line in a double logarithmic plot

[3], the density state effect on the stress–strain-strength behavior was implemented into the model by the following relation: /p ¼ arctan

h e np c

e


nd i e tan /c ; /pt ¼ arctan tan /c ec

ð5:37Þ

where np and nd are parameters controlling the effect of particle interlocking; these parameters can be taken as 1 according to Biarez & Hicher [3]. The expression of the mobilized peak friction angle /p in Eq. (5.37) shows that in a loose structure, /p is initially smaller than the critical friction angle /c. When the stress ratio overcomes the phase transformation state, a dense structure provides a higher degree of interlocking. Thus, /p is greater than /c. When the loading stress reaches the peak stress, a dense structure will dilate, and the degree of interlocking will decrease. As a consequence, the peak friction angle will be reduced, which results in a strain-softening phenomenon. The mobilized phase transformation angle /pt in Eq. (5.37) implies that in a loose structure with e > ec, /pt is larger than /c, which leads to a continuously contractive behavior; in a dense structure with e < ec, /pt is smaller than /c, which imposes a dense structure that is initially contractive and then dilative during deviatoric loading. For both loose and dense structures, when the stress state reaches the critical state line, the void ratio e becomes equal to the critical void ratio ec, and volume changes will no longer occur. Thus, the constitutive equations guarantee that the stress and void ratio simultaneously reach the critical state in the p’-q-e space. Note that Eq. (5.37) gives a similar density effect as in the model suggested by Manzari and Dafalias [11] but in a different form. Using the above-mobilized peak friction angle /p and the mobilized phase transformation angle /pt and accounting for the Lode angle dependency, we can express the peak strength ratio Mp and the phase transformation stress ratio Mpt by the following relations:

5.4 Critical State-Based Nonlinear Mohr-Coulomb Model

Mp ¼

8
10

2

.3

=1

120

80

40 20 0.6 0 10-3

10-2

BARD(1993) 10-1

1

10

γd

4

0.54

3

0.68

2

0.90

1

1.35

0

2.70

Aggregate La Noubleau Debernardy

σ'v (MPa)

Fig. 6.3 e-logp’ curves of isotropic compression tests for soils with different liquid limits

expressed by a constant slope Cc between “w = wL and r0vc ¼ 7 kPa” and “w = wP and r0vc ¼ 1 MPa” (see Fig. 6.3). According to the statistics of a large number of clay compression tests, the correlation formula Cc= 0.009(wL-13) can be used to obtain the compression index of the soil.

6.1.2

Anisotropic Compression Behavior

The behavior under anisotropic compression at a constant stress ratio r03 =r01 ¼ k 6¼ 1 or stress path g ¼ q0 =p0 ¼ 3ð1  k Þ=ð1 þ 2kÞ is similar to that observed under isotropic compression, and the normal consolidation characteristics can also be represented by a straight line with a slope on the plane e  log p0 (Fig. 6.4). A special case is the oedometric consolidation test path ðe2 ¼ e3 ¼ 0Þ, under which a normally consolidated soil exhibits characteristics close to those of the constant stress ratio or stress path r03 =r01 ¼ K0  1  sin /. This reproduces the sedimentary stress path and compression process of fine-grained sedimentary soil.

6.1.3

One-Dimensional Compression Behavior

The strain conditions of a one-dimensional compression test are e2 ¼ e3 ¼ 0. The stress on the soil sample during compression is related to the vertical strain or void ratio (Fig. 6.5). The relationship can be divided into two phases: the first phase corresponds to a loading that reaches the preconsolidation pressure, and the second

6 Elastoplastic Modeling of Clayey Soils …

182 e e (p', q/p' )

0.0 (Isotropic compression line) 0.4 0.8 =η=q'/p' 1.0 1.15

λ=0.062 η=M =1.25 Critical state line p' (MPa)

Fig. 6.4 e-logp’ curves of anisotropic compression tests under different constant stress ratios

σ'vc

0

σ'v

σ'h

400

600

σ' kPa

e e (p', q/p' )

1 0

0.0 (Isotropic compression line) 0.4 0.8 =η=q'/p' 1.0

2 0 ε1 %

1.15

e

λ=0.062

1.3 1.1 0.9

η=M =1.25 Critical state line

σ'vc 10

100

1000

σ'v ( kPa)

p' (MPa)

Fig. 6.5 Schematic plot of the oedometer test and its representative results

phase corresponds to a loading that exceeds the preconsolidation pressure. The deformation in the first stage is reversible, while in the second stage, only a small part of the strain can be recovered. The preconsolidation pressure r0vc corresponds to the elastic limit or yield surface in the stress space when the soil is subjected to the consolidation tensor r
0 c . (1) Compression behavior of normally consolidated soils If the lateral stress can be measured in oedometer tests, i.e., r02 ¼ r03 ¼ r0h , the following relation is obtained: r0h r02 r03 ¼ ¼ ¼ K0 r0v r01 r01

ð6:4Þ

6.1 Experimental Phenomena of Compression

183

where K0 = 1  sin /0pp is a constant in the normally consolidated state. Thus, the corresponding stress ratio q=p0 is also a constant:     q 1  K0 ¼ 3 p0 1 þ 2K0

ð6:5Þ

Therefore, the confining compression of the normally consolidated soil can be expressed by the intersection of the constant surface in the e-q-p’ space with the yield surface of the normally consolidated soil. This plane passes through the void ratio axis. If p0 is changed to log p0 , the line becomes a straight line. Its projection on the plane e  log p0 is parallel to the isotropic compression line and the critical state line with slope Cc . (2) Compression behavior of overconsolidated soils As shown in Fig. 6.6, when the value of r0V during unloading decreases from tensor r
0 c (point A) until point E, the deformation decrease is small and relatively linear. The slope of this line in the semi-logarithmic space is Cs ¼ De=D log r0v or j ¼ De=D ln r0v . We can define the area enclosed by the line r
0 c (point A) and point E in the stress space and the boundary between the two points and the origin as the quasi-elastic area, as shown in the shaded part of Fig. 6.6. If the unloading continues, the reversibility will diminish, and plastic deformations will develop. In the r0v  r0h or q  p0 stress space, the unloading stress path passes through the isotropic state r01 ¼ r02 ¼ r03 . Subsequently, the sample is subjected to a horizontal load r02 ¼ r03 ¼ r0h that is greater than the vertical load r0v . It is likely that the point P’ on the curve reaches the ideal plastic state:   r0v r01 p /0 þ ¼ ¼ tan 4 r0h r03 2

ð6:6Þ

As shown in Fig. 6.6, for clay, the value of this stress ratio is 1/2.29. Cracking often occurs during plastic unloading, and the same mechanism is observed in situ. If the overconsolidation ratio is extremely high, cracks in the depth range of 10 m can be observed in a clayey layer. If a sedimentary rock is regarded as a material with a very high consolidation ratio, it can be understood under these conditions that the applied vertical stress is relatively small compared to the horizontal stress, so the crack is perpendicular to the vertical stress, and horizontal cracks can occur within the rock.

6 Elastoplastic Modeling of Clayey Soils …

184

log σ'vc (MPa) E

0.40 σ' v

Cc=0.21

0.20 Cs=0.03

0.10

Clay

0.04 0.02 P'

e 0.90

0.80

0.70

0.60 0.50 0.8

0.01 σ'v (MPa)

Consolidation tensor A

0.6 E 0.4

E

0.2

P'

e 0.90

0.80 0.70

0.60

æp fö s1 =tan 2 çç - ÷÷÷ σ' (MPa) çè 4 2 ø h s3 0.0 p' 0.50 0.0 0.2 0.4 0.6 0.8

0.4 A 0.2 E

e 0.90

0.80

0.0 0.70 P' 0.60 0.50 P'

0.2

0.4

0.6

0.8

Fig. 6.6 Results of oedometer tests on clay

6.2 6.2.1

Modified Cam-Clay Model General Introduction to MCC

The Cam-Clay model and later on the modified Cam-Clay (MCC) model were developed by researchers at the University of Cambridge according to the mechanical behavior of remolded clay [2]. Since then, these models have been widely adopted in geotechnical analyses for clayey soils. The principle of the MCC is illustrated in Fig. 6.7. The basic constitutive equations are summarized in

6.2 Modified Cam-Clay Model

185

Cone with constant q/p'

Modified Cam-Clay model

q M Von-mises model

f=g pc

(a)

p’

(b)

Fig. 6.7 Principle of the modified Cam-Clay model: a on the p’-q plane and b in the r01  r02  r03 space

Table 6.1 Basic constitutive equations of MCC

Components

Constitutive equations

Elasticity

1 deeij ¼ 2G dsij þ

Yield surface

f ¼ Mq 2 þ p02  p0 pc

Potential surface Hardening rule

g¼f

j 0 3ð1 þ e0 Þ dp dij

2

dpc ¼ pc


1 þ e0  kj

depv

Table 6.1. Model parameters and their definitions are summarized in Table 6.2. Note that in the MCC model, the adopted strength criterion is the generalized Von-Mises criterion.

6.2.2

Analyses of Special Stress Paths Under Triaxial Conditions

According to elastoplastic theory, the consistency equation can be expressed as follows: df ¼

@f 0 @f @f dq þ dp þ dpc ¼ 0 @p0 @q @pc

Then, the plastic multiplier can be obtained in a general form:

ð6:7Þ

6 Elastoplastic Modeling of Clayey Soils …

186

Table 6.2 Model parameters and definitions of MCC Parameters

Definitions

e0 t j k Mc pc0

Initial void ratio Poisson’s ratio Swelling index Compression index Slope of the critical state line on the p´-q plane Initial size of the yield surface

    @f @g @f @g @f @pc @g 3G de K de  dk  dk dk 0 ¼ 0 þ þ v d @p0 @p0 @q @q @pc @epv @p ) dk ¼

@f @p0

K

@f @p0 @g @p0

@f @q 3Gded @f @g @f @pc @g @q 3G @q  @pc @epv @p0

Kdev þ þ

ð6:8Þ

where 8 q2 @f @g 0 0 > > @p0 ¼ @p0 ¼ 2p  pc ¼ p  M 2 p0 > > > < @f ¼ @g ¼ 2q2 @q

@q

M

@f 0 > > @pc ¼ p >  >

 > : @ppc ¼ 1 þ e0 pc ¼ 1 þ e0 q22 0 þ p0 kj kj M p @e

ð6:9Þ

v

For specific stress paths, the plastic multiplier can be simplified. (1) Compression under constant stress ratio The loading condition under a constant stress ratio can be written as dq/dp’ = q/p’ = η. In the elastic region, the ratio between the incremental volumetric strain and the incremental deviatoric strain is expressed as follows: 

de dp0 3G 9ð1  2tÞ 1 depv ¼ dp0 =K ¼ ) v¼ p ded ¼ dq=ð3GÞ 2ð1 þ tÞ g ded dq K

ð6:10Þ

In the plastic region, the ratio between the incremental volumetric strain and the incremental deviatoric strain is expressed as follows: 8   q2 0 < dep ¼ dk @g ¼ dkð2p0  p Þ ¼ dk p0  q2 depv p  M 2 p0 M 2 p02  q2 c v M 2 p0 @p0 ) p¼ ¼ 2q : dep ¼ dk @g ¼ dk 2q2 ded 2qp0 M2 d

@q

M

M 2  g2 ¼ 2g ð6:11Þ

6.2 Modified Cam-Clay Model

187

According to this equation, for normally consolidated or slightly overconsolidated soils, the condition M > η is usually fulfilled, and the volumetric change is positive (compression positive in soil mechanics). Thus, the soil is contractive. If the soil is heavily overconsolidated (e.g., OCR > 4), the condition M < η is imposed in the plastic domain. Then, the volumetric change is negative. Thus, the soil is dilative. Note that the modified Cam-Clay model assumes a value for K0 according to its stress–dilatancy rule: depv depd

¼ M 2gg 2

9 =

2

Oedometer:

dev ded

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 þ 4Mc2  3 3  gK0 ) K0 ¼ ) g ¼ K0 3; 2 3 þ 2gK0 ¼2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9  9 þ 4M 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 2 9 þ 4Mc2

ð6:12Þ

where the elastic strain is assumed to be negligible compared to the plastic strain. Compared with Jacky’s assumption, Mc ¼ 6 sin /c =ð3  sin /c Þ , sin /c ¼ 3Mc =ð6 þ Mc Þ K0 ¼ 1  sin /c ¼ ð6  2Mc Þ=ð6 þ Mc Þ gK0 ¼ 3Mc =ð6  Mc Þ The comparison can be plotted in Fig. 6.8. Fig. 6.8 Comparison of different compression paths under a constant stress ratio

ð6:13Þ

6 Elastoplastic Modeling of Clayey Soils …

188

(2) Triaxial test under constant confining stress The conventional drained triaxial test is a stress–strain mixed control test, with the radial effective stress constant (drr = 0) and the axial strain dea increasing, which implies that the stress path on the p′ − q plane has a slope of 3, i.e., dq = 3dp′, so the increment dp′ can be directly replaced by dq/3. Therefore, the plastic multiplier dk can be simplified by the following formula:     @f @g @f @g @f @pc @g 3G ded  dk G ded  dk ¼0 þ þ p dk 0 @p @q @q @q @pc @ev @p0   @f @f @p0 þ 3 @q Gded   ) dk ¼ @f @f @g @f @pc @g @p0 þ 3 @q G @q  @pc @epv @p0   2 p0  Mq2 p0 þ 3 M2q2 Gded    ¼
þ e0   q 2 2 2 0 0 q p0  Mq2 p0 þ 3 M2q2 G M2q2 þ p0 1kj þ p p 2 0 2 0 M p M p   q2 2 2 0 M M p  p0 þ 6q Gded    ¼
þ e q2 2 2 0 2 0 M 2 p0  qp0 þ 6q G2q þ p0 1kj M 2 p0  qp0 p0 þ M p

ð6:14Þ

  @g dq ¼ 3G ded  dk @q

  2 M 2 M 2 p0  qp0 þ 6q Gded 2q    ¼ 3Gded  3G 
1 þ e0 q2 q2 q2 M 2 2 0 0 2 0 2 0 M p  p0 þ 6q G2q þ p kj p0 þ M p M p  p0
þe  4 0 p0 1kj ð M  g4 Þ @q
þ e0  ) ¼ 3G @ed ðM 4  g4 Þ 2ðM 2  g2 þ 6gÞGg þ p0 1kj ð6:15Þ

When η approaches zero, the slope is equal to 3G, which is the same as in elasticity. (3) Triaxial test under constant mean stress The constant-p′ triaxial test can also be considered as a stress–strain mixed control test, with the mean effective stress p′ composed of the radial and axial effective stress being constant (dra + 2drr = 0) and the axial strain dea increasing, which implies that the stress path in the p′ − q plane is a straight line parallel to the q-axis, i.e., dp′ = 0. Therefore, the plastic multiplier dk can be simplified as follows:

6.2 Modified Cam-Clay Model

189

  @f @g @f @pc @g 3G ded  dk dk 0 ¼ 0 þ @q @q @pc @epv @p @f @q 3Gded ) dk ¼ @f @g @f @pc @g @q 3G @q  @pc @epv @p0 3 M2q2 Gded  ¼
þ e0   q 2 0 3 M2q2 G M2q2 þ p0 1kj þ p p0 2 0 M p

¼

2

 Mq2 p0



ð6:16Þ

M 2 6qGded  
þ e0 q2 2 2 0 6qG2q þ p0 1kj M 2 p0  qp0 p0 þ M p

  @g dq ¼ 3G ded  dk @q M 2 6qGded 2q   2
1 þ e0 q2 2 q 6qG2q þ p0 kj p0 þ M 2 p0 M 2 p0  p0 M 2 3 2 @q 12q G  5 ) ¼ 3G41 
1 þ e0 q2 q2 @ed 2 0 2 0 2 0 12q G þ p kj p0 þ M p M p  p0
 1 þ e p0 kj0 ðM 4  g4 Þ
þ e0  ¼ 3G 2 ðM 4  g4 Þ 12g G þ p0 1kj ¼ 3Gded  3G

ð6:17Þ

When η approaches zero, the slope is equal to 3G, which is the same as in elasticity. (4) Undrained triaxial test The undrained triaxial test is a special case of a full strain control test with the incremental volumetric strain dev = 0 and the incremental deviatoric strain ded fixed at a given value. The plastic multiplier dk must be calculated to obtain the plastic strains and then update the stresses:     @f @g @f @g @f @pc @g 3G de K de  dk  dk dk 0 ¼ 0 þ þ v d @p0 @p0 @q @q @pc @epv @p ) dk ¼

@f @p0

K

@g @p0

þ

@f @q 3Gded @f @g @f @pc @g @q 3G @q  @pc @epv @p0 3 M2q2 Gded

¼ 2  
þ e0  q2 2 2 0 p0  Mq2 p0 p0  Mq2 p0 K þ 3 M2q2 G M2q2 þ p0 1kj M 2 p0 þ p M 2 6qGded ¼   
þ e0 q2 2 2 2 q 2 0 M 2 p0  qp0 M 2 p0  p0 þ 12q2 G þ p0 1kj p0 þ M p

ð6:18Þ

6 Elastoplastic Modeling of Clayey Soils …

190

  @g dq ¼ 3G ded  dk @q M 2 6qGded 2q 2   2
 2 M q þ e0 2 0 M 2 p0  qp0 þ 12q2 G þ p0 1kj M 2 p0  qp0 p0 þ M p
þ e0  4 2 ðM 2  g2 Þ þ p0 1kj ð M  g4 Þ @q ) ¼ 3G
 þ e0 @ed ðM 2  g2 Þ2 þ 12g2 G þ p0 1kj ðM 4  g4 Þ ¼ 3Gded  3G 

 2 2

ð6:19Þ When η approaches zero, the slope is equal to 3G, which is the same as in elasticity. 

@g dp ¼ K dev  dk 0 @p 0



  M 2 6qGded q2 2 0 ¼ K     M p  0
þ e0 q2 2 2 2 p 2 0 M 2 p0  qp0 þ 12q2 G þ p0 1kj M 2 p0  qp0 p0 þ M p ¼ K )

0

ðM 2 

g2 Þ2


2  M 2 6gGded M  g2
þ e0  þ 12g2 G þ p0 1kj ðM 4  g4 Þ

@p ðM 2  g2 ÞM 2 6gG ¼ K
þ e0  @ed ðM 2  g2 Þ2 þ 12g2 G þ p0 1kj ðM 4  g4 Þ

ð6:20Þ When η is increased from zero, the slope of the stress path in the p’, q plane has a negative value, indicating a decrease in p’ for NC clay (M > η), whereas a positive value indicates an increase of p’ for OC clay (M < η) starting from the end of the elastic part.

6.2.3

Parameters of the MCC Model

The model parameters (Table 6.2) can be divided into two groups: the parameters controlling the elastic behavior: j and t; and the parameters controlling the plastic behavior: k, Mc, and pc0. For ease of understanding, the conventional triaxial drained compression test (in which the radial stress is constant and the axial stress increases to yield) is taken as an example to display the physical meaning of the model parameters and how to calibrate them.

6.2 Modified Cam-Clay Model

191

(1) From the q − ea curve For overconsolidated soils, the initial stress–strain curve is located within the elastic region (with an initial mean effective stress of p’0); thus, @q 1 þ e0 0 @q 3ð1  2tÞ 1 þ e0 0 p0 or p0 ¼ E ¼ 2ð1  2tÞ ¼ 3G ¼ @ea @ed 2ð 1 þ t Þ j j

ð6:21Þ

For normally consolidated soil, the stress–strain curve is immediately located within the plastic domain. According to the above derivation, G degrades with the level of stress ratio η. However, when η has a very small value close to zero, @q=@ed ¼ 3G. (2) From the ev − ea curve For overconsolidated soils, the following equation is available according to the elasticity theory: @ev ¼ 1  2t @ea

ð6:22Þ

In the plastic domain, according to the stress–dilatancy equation implied by the potential function of MCC, we have the following expression: 

ev ¼ ea þ 2er ) ed ¼ 2ðea  er Þ=3



dep dep ea ¼ ed þ ev =3 ) vp ¼ p v p er ¼ ev =3  ed =2 dea ded þ dev =3 3ð M 2  g2 Þ ¼ 6g þ M 2  g2

ð6:23Þ

If the dilation angle defined in the Mohr-Coulomb model is adopted ðdepv =depa ¼ 2 sin w=ð1  sin wÞÞ, then we have the following expression: 2 sin w 3ð M 2  g 2 Þ ¼ 1  sin w 6g þ M 2  g2

ð6:24Þ

The dilation angle can be obtained for a given OCR, implying a stress ratio η when plastic strains start to develop, i.e, corresponding to the initial slope of the plastic volumetric change, as follows:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  M 2  OCR2 þ 36OCR  36  M  OCR OCR ) g ¼ )w 6   3M 2  3g2 ¼ sin1 M 2  g2  12g M

ð6:25Þ

or the corresponding stress ratio η (or OCR) can be obtained for a given dilation angle as follows:

6 Elastoplastic Modeling of Clayey Soils …

192

w)g¼ ¼

6 sin w 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi ð5M 2 þ 36Þ sin w2 þ M 2 ð9  18 sin wÞ ð3  sin wÞ

3ð M þ g Þ M 2 ð 3  gÞ 2

2

) OCR ð6:26Þ

Remark To find the stress state on the yield surface for a conventional drained test on OC clay (


 pc q ¼ 3 p0  OCR q2 M2

02

0

þ p  p pc ¼ 0

) OCR ¼

)

3ðM 2 þ g2 Þ M 2 ð 3  gÞ

 8 1e-7) % for plasticity, explicit solution dfdq = 2.0*(q-(p + pb)*alpha)/(dM^2-alpha^2); dgdq = dfdq; dfdp = -2.0*(q-(p + pb)*alpha)*alpha/(dM^2-alpha^2) + 2.0*p + pb(1.0 + chi)*pci; dgdp = dfdp; dgdpMac = dfdp; if(dgdpMac < 0.0) dgdpMac = 0.0; end dfdpci = -1.0*(p + pb)*(1.0 + chi); dpcidepsv = ((1.0 + e0) * pci) / (lambda - kappa); dhh1 = dfdpci * dpcidepsv * dgdp; dfdpb = -2.0*(q-(p + pb)*alpha)*alpha/(dM^2-alpha^2) + p(1.0 + chi)*pci; dpbdepsd = -1.0 * dksi_pb * pb; dhh2 = dfdpb * dpbdepsd * dgdq; dfdchi = -1.0*(p + pb)*pci; dchidepsv = -1.0 * dksi * chi; dchidepsd = -1.0 * dksi * dksi_d*chi; dhh3 = dfdchi * (dchidepsv * abs(dgdp) + dchidepsd * dgdq); dfdalpha = -2.0*(q-(p + pb)*alpha)*(p + pb)/(dM^2-alpha^2); dfdalpha = dfdalpha + 2.0*(q-(p + pb)*alpha)^2*alpha/ (dM^2-alpha^2)^2; dalphadepsv = dw * (al_v * q / (p + pb) - alpha); dalphadepsd = dw * dw_d * (al_d * q / (p + pb) - alpha); dhh4 = dfdalpha * (dalphadepsv * dgdpMac + dalphadepsd * dgdq); dhh = dhh1 + dhh2 + dhh3 + dhh4; AA = dfdp*K*depsv + dfdq*3.0*G*depsd; BB = dfdp*K*dgdp + dfdq*3.0*G*dgdq; dlambda = AA/(BB-dhh); depspv = dlambda*dgdp; depspd = dlambda*dgdq; depspvMac = depspv; if(depspvMac < 0.0) depspvMac = 0.0; end p = p0 + K*(depsv-depspv); q = q0 + 3*G*(depsd-depspd); % update all state variables for output

203

6 Elastoplastic Modeling of Clayey Soils …

204

dpci = ((1.0 + e0) * pci * depspv) / (lambda - kappa); dchi = -1.0 * chi * dksi * (depspvMac + dksi_d * depspd); dpb = -1.0 * pb * dksi_pb * depspd; dalpha = dw*(al_v*q/(p + pb)-alpha)*depspvMac; dalpha = dalpha + dw*dw_d*(al_d*q/(p + pb)-alpha)*depspd; pci = pci + dpci; chi = chi + dchi; pb = pb + dpb; alpha = alpha + dalpha; pspd = epspd0 + depspd; epspv = epspv0 + depspv; e = e0-(1 + ei)*depsv;

Since the program is full strain control, it can be used to simulate a test under a constant strain ratio, the so-called R-test 00 der ¼ Rde00a : 

dev 3ð1 þ 2RÞ dev ¼ dea þ 2der ! der ¼ Rdea ¼ ded ¼ 2ðdea  der Þ=3 2ð 1  R Þ ded

ð6:58Þ

From this formula, it is known that R = 1 corresponds to isotropic compression, R = 0 corresponds to one-dimensional compression, and R = 0.5 corresponds to an undrained test. In general, the results of the R-test can be analyzed to see the relationship between the stress combination 00 ra þ Rr00r and ea. Interested readers can copy this code directly for training or analysis.

6.4.3

Conventional Drained Triaxial Test Simulation

The conventional drained triaxial test is a stress–strain mixed control, that is, the radial effective stress is constant (drr = 0) and the axial strain dea is increased, implying that the stress path on the p’-q plane has a slope of 3, i.e., dq = 3dp’; thus, the increment dp’ can be directly replaced by dq/3. Therefore, the plastic multiplier dk can be simplified by the following formula:     @f @g @f @g @f @pci @g @f @pb @g þ 3G de þ G de  dk  dk dk 0 þ dk d d @p0 @q @q @q @pci @epv @p @pb @epd @q     @f @v @g @v @g @f @a @g @a @g þ þ þ ¼0 dk dk dk dk þ @v @epd @q @epv @p0 @a @epd @q @epv @p0   @f @f @p0 þ 3 @q Gded h    ) dk ¼ @f @g @f @g @f @pci @g @f @pb @g @f @v @g @v @g @f @a @g @p0 G @q þ @q 3G @q  @pci @ep @p0 þ @pb @ep @q þ @v @ep @q þ @ep @p0 þ @a @ep @q þ v

d

d

v

d

@a @g @epv @p0

i

ð6:59Þ

6.4 Triaxial Test Simulations Using ASCM

205

The pseudocode for an incremental step is outlined as follows: Known : p00 ; q0 ; e0 ; epd0 ; epv0 ; ded 1 þ e0 0 Calculate : K ¼ ½p þ pb0 ð1  Rb Þ þ ðpc0  pci0 Þð1  Rc Þ j Calculate stresses of elastic trial : q ¼ q0 þ 3Gded ; p0 ¼ p00 þ Gded Get previous plastic strains : epv ¼ epv0 ; epd ¼ epd0 Calculate the yield function : f ¼ If f [ 0 then

½q  ðp0 þ pb Þa2 þ ðp0 þ pb Þ½p0  ð1 þ vÞpci  ¼ 0 M 2  a2

Calculate partial derivatives: @f 2½q  ðp0 þ pb Þa ¼ @q M 2  a2 @f 2½q  ðp0 þ pb Þaa ¼ þ 2p0  ð1 þ vÞpci þ pb @p0 M 2  a2 @f 2½q  ðp0 þ pb Þaa ¼ þ p0  ð1 þ vÞpci @pb M 2  a2 @pb ¼ pb nb @epd @f 2½q  ðp0 þ pb Þaðp0 þ pb Þ 2½q  ðp0 þ pb Þa2 a ¼ þ @a M 2  a2 ðM 2  a2 Þ2 @f ¼ ðp0 þ pb Þpci @v @v ¼ vn @epd @v ¼ vnnd @epv @f ¼ ðp0 þ pb Þð1 þ vÞ @pci  Calculate : dk ¼

@f @p0

G @g @q þ

@f @g @q 3G @q



h

@f @pci @g @pci @epv @p0

 @f þ 3 @q Gded  @f @pb @g @f @v @g þ @p þ @v þ p @ep @q b @e @q @f @p0

d

d

@v @g @epv @p0



þ

@f @a



@a @g @epd @q

@g @g ; depd ¼ dk @p0 @q Update plastic strain : epv ¼ epv0 þ depv ; epd ¼ epd0 þ depd

 Update stress : q ¼ q0 þ 3G ded  depd ; p0 ¼ p00 þ G ded  depd

  G
Calculate the volumetric strain : K dev  depv ¼ G ded  depd ) dev ¼ ded  depd þ depv K Update void ratio : e ¼ e0  ð1 þ ei Þdev Calculate plastic strain : depv ¼ dk

End if Variables are updated and then used in the next step: p00 ¼p0 ; q0 ¼q; e0 ¼e; epd0 ¼epd ; epv0 ¼epv

þ

@a @g @epv @p0

i

206

6 Elastoplastic Modeling of Clayey Soils …

The MATLAB code for this incremental step is outlined as follows:

Psyn = p0 + pb + chi*pci0; K = abs((1.0 + e0) * Psyn / kappa); G = 3.0*K*(1.0-2.0*pois)/(2.0*(1.0 + pois)); pci0_sub = ((q0-(p0 + pb)*alpha)^2/(dM*dM-alpha*alpha)/(p0 + pb) + p0)/(1. + chi); q = q0 + 3.0*G*depsd; p = p0 + G*depsd; depsv = G*depsd/K; f = ((q-(p + pb)*alpha)^2)/(dM^2-alpha^2) + (p + pb)*(p-(1.0 + chi) *pci0_sub); if (f > 1e-7) % for plasticity, explicit solution dfdq = 2.0*(q-(p + pb)*alpha)/(dM^2-alpha^2); dgdq = dfdq; dfdp = -2.0*(q-(p + pb)*alpha)*alpha/(dM^2-alpha^2) + 2.0*p + pb(1.0 + chi)*pci; dgdp = dfdp; dgdpMac = dfdp; if(dgdpMac < 0.0) dgdpMac = 0.0; end dfdpci = -1.0*(p + pb)*(1.0 + chi); dpcidepsv = ((1.0 + e0) * pci) / (lambda - kappa); dhh1 = dfdpci * dpcidepsv * dgdp; dfdpb = -2.0*(q-(p + pb)*alpha)*alpha/(dM^2-alpha^2) + p(1.0 + chi)*pci; dpbdepsd = -1.0 * dksi_pb * pb; dhh2 = dfdpb * dpbdepsd * dgdq; dfdchi = -1.0*(p + pb)*pci; dchidepsv = -1.0 * dksi * chi; dchidepsd = -1.0 * dksi * dksi_d*chi; dhh3 = dfdchi * (dchidepsv * abs(dgdp) + dchidepsd * dgdq); dfdalpha = -2.0*(q-(p + pb)*alpha)*(p + pb)/(dM^2-alpha^2); dfdalpha = dfdalpha + 2.0*(q-(p + pb)*alpha)^2*alpha/ (dM^2-alpha^2)^2;

6.4 Triaxial Test Simulations Using ASCM

207

dalphadepsv = dw * (al_v * q / (p + pb) - alpha); dalphadepsd = dw * dw_d * (al_d * q / (p + pb) - alpha); dhh4 = dfdalpha * (dalphadepsv * dgdpMac + dalphadepsd * dgdq); dhh = dhh1 + dhh2 + dhh3 + dhh4; pci_sub = ((q-(p + pb)*alpha)^2/(dM*dM-alpha*alpha)/(p + pb) + p)/(1. + chi); dR = (pci*(1. + chi) + pb)/(pci_sub*(1. + chi) + pb); % for bounding surface if(dR < 1.0) % for bounding surface dR = 1.0; end dkp_b = dhh*dR*dR; % for bounding surface dhh_m = dkp_b -para_kp*(1 + e0)/(lambda-kappa)*(dR*p)^3* (1.-1./dR); AA = (dfdp + 3.0*dfdq)*G*depsd; % dfdp*K*depsv + dfdq*3.0*G*depsd; BB = dfdp*G*dgdq + dfdq*3.0*G*dgdq; dlambda = AA/(BB-dhh_m); depspv = dlambda*dgdp; depspd = dlambda*dgdq; depspvMac = depspv; if(depspvMac < 0.0), depspvMac = 0.0; end p = p0 + G*(depsd-depspd); q = q0 + 3.0*G*(depsd-depspd); dpci = ((1.0 + e0) * pci * depspv) / (lambda - kappa); dchi = -1.0 * chi * dksi * (depspvMac + dksi_d * depspd); dpb = -1.0 * pb * dksi_pb * depspd; dalpha = dw*(al_v*q/(p + pb)-alpha)*depspvMac + dw*dw_d*(al_d*q/ (p + pb)-alpha)*depspd; depsv = G/K*(depsd-depspd) + depspv; % Give back updated state variables for next step pci = pci + dpci; chi = chi + dchi; pb = pb + dpb; alpha = alpha + dalpha; epspd = epspd0 + depspd;

6 Elastoplastic Modeling of Clayey Soils …

208

epspv = epspv0 + depspv; e = e0-(1 + ei)*depsv;

Interested readers can copy this code directly for training or analysis.

6.4.4

Constant-P’ Triaxial Test Simulation

The constant-p’ triaxial shear test can also be considered as a stress–strain mixed control test, that is, the mean effective stress p’ composed of the radial and axial effective stresses is constant (dra + 2drr = 0) and the axial strain dea is increased, implying that the stress path on the p’-q plane is a straight line parallel to the q-axis, i.e., dp’ = 0. Therefore, the plastic multiplier dk can be simplified to the following formula:   @f @g @f @g 3G ded  dk ¼0 þ p dk @q @q @ed @q ) dk ¼

ð6:60Þ

@f @q 3Gded @f @g @f @g @q 3G @q  @epd @q

The pseudocode for an incremental step is outlined as follows:     @f @g @f @g @f @pci @g @f @pb @g 3G de þ þ K de  dk  dk þ dk v d p dk 0 0 0 @p @p @q @q @pci @ev @p @pb @epd @q     @f @v @g @v @g @f @a @g @a @g þ þ þ dk dk dk dk þ @v @epd @q @epv @p0 @a @epd @q @epv @p0 ) dk ¼

@f @g @q 3G @q



h

@f @q 3Gded @f @pci @g @pci @epv @p0

þ

@f @pb @g @pb @epd @q

þ

@f @v



@v @g @epd @q

þ

@v @g @epv @p0



þ

@f @a



@a @g @epd @q

þ

@a @g @epv @p0

i

ð6:61Þ

6.4 Triaxial Test Simulations Using ASCM

209

Known : p00 ; q0 ; e0 ; epd0 ; epv0 ; ded 1 þ e0 0 Calculate : K ¼ ½p þ pb0 ð1  Rb Þ þ ðpc0  pci0 Þð1  Rc Þ j Calculate stresses of elastic trial : q ¼ q0 þ 3Gded ; p0 ¼ p00 Calculate : Get previous plastic strains : epv ¼ epv0 ; epd ¼ epd0 Calculate the yield function : f ¼ If f [ 0 then

½q  ðp0 þ pb Þa2 þ ðp0 þ pb Þ½p0  ð1 þ vÞpci  ¼ 0 M 2  a2

Calculate partial derivatives: @f 2½q  ðp0 þ pb Þa ¼ @q M 2  a2 @f 2½q  ðp0 þ pb Þaa ¼ þ 2p0  ð1 þ vÞpci þ pb @p0 M 2  a2 @f 2½q  ðp0 þ pb Þaa ¼ þ p0  ð1 þ vÞpci @pb M 2  a2 @pb ¼ pb nb @epd @f 2½q  ðp0 þ pb Þaðp0 þ pb Þ 2½q  ðp0 þ pb Þa2 a þ ¼ @a M 2  a2 ðM 2  a2 Þ2 @f ¼ ðp0 þ pb Þpci @v @v ¼ vn @epd @v ¼ vnnd @epv @f ¼ ðp0 þ pb Þð1 þ vÞ @pci Calculate : dk ¼

@f @g @q 3G @q



h

@f @pci @g @pci @epv @p0

@f @q 3Gded

þ

@f @pb @g @pb @epd @q

þ

@f @v



@v @g @epd @q

þ

@v @g @epv @p0



þ

@f @a



@a @g @epd @q

@g @g ; depd ¼ dk @p0 @q Update plastic strain : epv ¼ epv0 þ depv ; epd ¼ epd0 þ depd
 Update stress : q ¼ q0 þ 3G ded  depd ; p0 ¼ p00

Calculate plastic strain : depv ¼ dk

Calculate the volumetric strain : dev ¼ depv Update void ratio : e ¼ e0  ð1 þ ei Þdev End if Variables are updated and then used in the next step: p00 ¼ p0 ; q0 ¼ q; e0 ¼ e; epd0 ¼ epd ; epv0 ¼ epv

The MATLAB code for this incremental step is outlined as follows:

þ

@a @g @epv @p0

i

210

6 Elastoplastic Modeling of Clayey Soils …

Psyn = p0 + pb + chi*pci0; K = abs((1.0 + e0) * Psyn / kappa); G = 3.0*K*(1.0-2.0*pois)/(2.0*(1.0 + pois)); pci0_sub = ((q0-(p0 + pb)*alpha)^2/(dM*dM-alpha*alpha)/(p0 + pb) + p0)/(1. + chi); p = p0; q = q0 + 3.0*G*depsd; f = ((q-(p + pb)*alpha)^2)/(dM^2-alpha^2) + (p + pb)*(p-(1.0 + chi) *pci0_sub); if (f > 1e-7) % for plasticity, explicit solution dfdq = 2.0*(q-(p + pb)*alpha)/(dM^2-alpha^2); dgdq = dfdq; dfdp = -2.0*(q-(p + pb)*alpha)*alpha/(dM^2-alpha^2) + 2.0*p + pb(1.0 + chi)*pci; dgdp = dfdp; dgdpMac = dfdp; if(dgdpMac < 0.0) dgdpMac = 0.0; end dfdpci = -1.0*(p + pb)*(1.0 + chi); dpcidepsv = ((1.0 + e0) * pci) / (lambda - kappa); dhh1 = dfdpci * dpcidepsv * dgdp; dfdpb = -2.0*(q-(p + pb)*alpha)*alpha/(dM^2-alpha^2) + p-(1.0 + chi) *pci; dpbdepsd = -1.0 * dksi_pb * pb; dhh2 = dfdpb * dpbdepsd * dgdq; dfdchi = -1.0*(p + pb)*pci; dchidepsv = -1.0 * dksi * chi; dchidepsd = -1.0 * dksi * dksi_d*chi; dhh3 = dfdchi * (dchidepsv * abs(dgdp) + dchidepsd * dgdq); dfdalpha = -2.0*(q-(p + pb)*alpha)*(p + pb)/(dM^2-alpha^2); dfdalpha = dfdalpha + 2.0*(q-(p + pb)*alpha)^2*alpha/(dM^2-alpha^2)^2; dalphadepsv = dw * (al_v * q / (p + pb) - alpha); dalphadepsd = dw * dw_d * (al_d * q / (p + pb) - alpha); dhh4 = dfdalpha * (dalphadepsv * dgdpMac + dalphadepsd * dgdq); dhh = dhh1 + dhh2 + dhh3 + dhh4; pci_sub = ((q-(p + pb)*alpha)^2/(dM*dM-alpha*alpha)/(p + pb) + p)/ (1. + chi); dR = (pci*(1. + chi) + pb)/(pci_sub*(1. + chi) + pb); if(dR < 1.0) dR = 1.0; end dkp_b = dhh*dR*dR; % for bounding surface

6.4 Triaxial Test Simulations Using ASCM

211

dhh_m = dkp_b -para_kp*(1 + e0)/(lambda-kappa)*(dR*p)^3*(1.-1./dR); AA = dfdq*3.0*G*depsd; BB = dfdq*3.0*G*dgdq; dlambda = AA/(BB-dhh_m); % for bounding surface depspv = dlambda*dgdp; depspd = dlambda*dgdq; depsv = depspv; depspvMac = depspv; if(depspvMac < 0.0) depspvMac = 0.0; end p = p0; q = q0 + 3.0*G*(depsd-depspd); dpci = ((1.0 + e0) * pci * depspv) / (lambda - kappa); dchi = -1.0 * chi * dksi * (depspvMac + dksi_d * depspd); dpb = -1.0 * pb * dksi_pb * depspd; dalpha = dw*(al_v*q/(p + pb)-alpha)*depspvMac + dw*dw_d*(al_d*q/ (p + pb)-alpha)*depspd; depsv = depspv; epsv = epsv0 + depsv; epsd = epsd0 + depsd; % Give back updated state variables for next step pci = pci + dpci; chi = chi + dchi; pb = pb + dpb; alpha = alpha + dalpha; epspd = epspd0 + depspd; epspv = epspv0 + depspv; e = e0-(1 + ei)*depsv;

Interested readers can copy this code directly for training or analysis.

6.4.5

Pre- and Postprocessing

A complete calculation program requires pre- and postprocessing sections. The preprocessing section is mainly used to process the parameter input, state variables initialization, calculation parameter setting, and so on. Therefore, the following definitions are provided:

212

6 Elastoplastic Modeling of Clayey Soils …

%% Parameters pois = 0.25; kappa = 0.036; %! slope of the loading-reloading curve in e-ln(p) lambda = 0.17; %! slope of the post-yield compression curve in e-ln(p) e0 = 1.0; %! initial void ratio dM = 1.04; %! slope of the critical state line on the q-p plane pc0 = 400; chi0 = 5.0; dksi_d = 0.32; pb0 = 18.5; dksi_pb = 2.0; dksi = 11.0; para_kp = 500; %% Initialization p0 = 100; %kPa q0 = 0; %kPa epsd0 = 0; %update epsv0 = 0; %update epspd0 = 0; %update epspv0 = 0; %update pci0 = pc0/(1.0 + chi0); eta_K0 = 3.0*dM/(6.0-dM); alpha = eta_K0 - (dM^2-eta_K0^2)/3.0; dw_d = 3.0*(4.0*dM^2-4.0*eta_K0^2-3.0*eta_K0)/(8.0* (eta_K0^2 + 2.0*eta_K0-dM^2)); dw = (1.0 + e0)/(lambda -kappa)*log((10.0*dM^2-2.0*alpha*dw_d)/ (dM^2-2.0*alpha*dw_d)); pci = pci0; %update chi = chi0; %update pb0 = pb0; %update alpha = alpha; %update pc0 = pc0; %update ei = e0; %update

6.4 Triaxial Test Simulations Using ASCM

6.4.6

213

Model Parameter Analysis

Simulations of three typical tests are given for a set of model parameters: (1) MCC model parameters: t = 0.25, j = 0.036, k = 0.17, M = 1.04, and e0 = 1.0; (2) parameters related to anisotropy a = 0.4, x = 41.75, and xd = 0.56; and (3) parameters related to soil structure destructuration v = 5.0, n = 11, nd = 0.32, npb = 2.0, pb0 = 18.5, and, kp = 500. All simulations are shown in Figs. 6.13, 6.26. Figure 6.13 shows the simulated results of the undrained triaxial test with different OCRs. Figure 6.14 shows the simulated result of the one-dimensional compression test accounting for structural destructuration. Figure 6.15 shows the simulations of the conventional drained triaxial tests with different OCRs. Figure 6.16 shows the simulations of constant p’ tests with different OCRs. Figures 6.17, 6.18, 6.19, 6.20, 6.21, 6.22, 6.23, 6.24, 6.25 and 6.26 show the parametric study of ASCM with respect to different parameters. All simulation results appear reasonable regarding the experimental phenomena on different soils: for slightly overconsolidated soil, pc0 = 100 kPa, p0 = 75 kPa, and q0 = 0 kPa; for overconsolidated soil, pc0 = 300 kPa, p0 = 75 kPa, and q0 = 0 kPa; and for the simulations of oedometer tests, some initial stresses are set as pc0 = 100 kPa, p0 = 10 kPa, and q0 = 0 kPa.

Fig. 6.13 Undrained triaxial test simulations

214

6 Elastoplastic Modeling of Clayey Soils …

Fig. 6.14 An oedometer test simulation

Fig. 6.15 Conventional drained triaxial test simulations

6.4 Triaxial Test Simulations Using ASCM

Fig. 6.16 Constant p’ test simulations

Fig. 6.17 Parametric study of undrained triaxial tests (pb = 0, 10.0, 18.5)

215

216

6 Elastoplastic Modeling of Clayey Soils …

Fig. 6.18 Parametric study of undrained triaxial tests (pb = 18.5, npb = 0, 2, 10)

Fig. 6.19 Parametric study of undrained triaxial tests (v = 0, 5, 10)

6.4 Triaxial Test Simulations Using ASCM

Fig. 6.20 Parametric study of undrained triaxial tests (v = 5, n = 0, 11, 20)

Fig. 6.21 Undrained triaxial test simulations (a = 0, 0.2,0.4)

217

218

6 Elastoplastic Modeling of Clayey Soils …

Fig. 6.22 Parametric study of an oedometer test (p0 = 10 kPa, pb = 0, 10.0, 18.5)

Fig. 6.23 Parametric study of an oedometer test (pb = 18.5, npb = 0, 2, 10)

6.4 Triaxial Test Simulations Using ASCM

Fig. 6.24 Parametric study of oedometer tests (v = 0, 5, 10)

Fig. 6.25 Parametric study of oedometer tests (v = 5, n = 0, 11, 20)

219

6 Elastoplastic Modeling of Clayey Soils …

220

Fig. 6.26 Oedometer test simulations (a = 0, 0.2, 0.4)

6.5

Consideration of Small-Strain Stiffness

The degradation of stiffness with increasing strain has been expressed by various relationships [16], but all of these expressions ignore the effect of volumetric strain on stiffness degradation. In fact, isotropic and K0-compression tests with unloading–reloading loops on different clays indicated an increase in j (corresponding to the degradation of the bulk modulus) due to the volumetric strain (shown in Fig. 6.27, Reconstituted Bothkennar clay by Allman and Atkinson [17], and Kaolin

1

1.2

0.8 e

e

1.4

1

0.6

0.8

0.4 1

(a)

10

100 p' / kPa

1000

1

(b)

10

100 p' / kPa

Fig. 6.27 Isotropic compression curve: a Kaolinite and b Bothkennar clay

1000

10000

6.5 Consideration of Small-Strain Stiffness

221

clay by Hattab and Hicher [18]). Considering a constant Poisson’s ratio, the effect of the volumetric strain on the shear modulus should also be considered. According to Hardin and Drnevich [19], the shear modulus G was expressed as follows: G¼

G0     1 þ cc  a

ð6:62Þ

Using the threshold shear strain ca ¼c0:7 , Santos and Correia [43] proposed the following modified Hardin–Drnevich relationship: G¼

G0   1 þ a cc

ð6:63Þ

0:7

where the constant a = 3/7 is obtained by Santos and Correia [43] through curve fitting from many test results. According to Benz [39], the secant modulus given in Eq. (6.63) should be converted to a tangent modulus for the convenience of numerical application: G¼

G0 1 þ a cc

2

ð6:64Þ

0:7

Taking cref ¼ c0:7 =a, Eq. (6.64) can be simplified as follows: G¼


G0 1 þ c=cref

2

ð6:65Þ

where the shear strain c can be replaced by the deviatoric strain ed for the extension, and cref ¼ 7e70 =3 is defined with e70 as the deviatoric strain level corresponding to 70% of the maximum stiffness. For simplicity, by adopting the shear modulus at 0.1% of strain G0.1% as the input parameter, the initial shear modulus G0 can be obtained as follows:
2 G0 ¼ 1 þ 0:001=cref G0:1%

ð6:66Þ

For clayey soils, G0.1 % can be calculated using the swelling index j as follows: G0:1% ¼

ð1  2tÞ ð1 þ e0 Þ 0 p 2ð 1 þ t Þ j

ð6:67Þ

6 Elastoplastic Modeling of Clayey Soils …

222

Thus, the shear modulus G can be expressed as follows:


2 1 þ 0:001=cref ð1  2tÞ ð1 þ e0 Þ 0 p G¼
2 2ð 1 þ t Þ j 1 þ ed =cref

ð6:68Þ

To consider the effect of the volumetric strain for describing this stiffness degradation, a new relationship was proposed by modifying the swelling index j (detailed in the following remark):   2  3ð1  2tÞð1 þ e0 Þ 3 eeq  pat 1 þ Prev   js ¼ 2ð 1 þ t Þ 7 e70 Gref 0

ð6:69Þ

with Gref 0

 2  3ð1  2tÞð1 þ e0 Þ 3 0:001 pat 1 þ Prev  ¼ 2ð 1 þ t Þ 7 e70  j

ð6:70Þ

where js is the apparent swelling index under small to large strain levels, Prev is the controlling factor of stiffness with Prev = 1 when loading and Prev = 2 when qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi unloading; eeq ¼ ðev Þ2 þ ðed Þ2 is the average equivalent strain, and thus,e70 is the equivalent strain level corresponding to 70% of the maximum stiffness, pat is the atmospheric pressure pat= 101.325 kPa, and Gref0 is the initial reference secant stiffness. After the apparent swelling index js has been obtained, the elastic modulus Ev in Eq. (6.52) can be calculated by replacing j with js. Therefore, using the above equations, j is still used as the input parameter, and only one additional parameter e70 is required. The value of e70 can be related to the plasticity index (PI) for clayey soils according to Benz’s database [16]. Note that for capturing cyclic loading behavior, the proposed small-strain stiffness relationship can be achieved by the stress reversal technique proposed by Yin et al. [20] with the definition of eeq as follows: eeq

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   
2 2 R R R eij  eij : eij  eij þ ev  ev ¼ 3

ð6:71Þ

where eRij is the deviatoric strain tensor at the reverse point and eRv is the volumetric strain at the reverse point. Then, the model can be applied to more general cases with complex loading conditions. Remark Derivation of the initial reference shear modulus and apparent swelling index Based on Eq. (6.17), the shear modulus at 0.1% of strain and that at any strain level is defined as follows:

6.5 Consideration of Small-Strain Stiffness

8 < G0:1% ¼ :G ¼

223 G0

ð1 þ 0:001=cref Þ

2

ð6:72Þ

G0 2 ð1 þ c=cref Þ

According to the elastic relationships, the corresponding bulk modulus at 0.1% strain and that at any strain level can be obtained as follows: 8 2 ð 1 þ tÞ 2ð1 þ tÞ > < K0:1% ¼ 3ð12tÞ G0:1% ¼ 3ð12tÞ 2ð1 þ tÞ 2ð1 þ tÞ > : K ¼ 3ð12tÞ G ¼ 3ð12tÞ

G0

ð1 þ 0:001=cref Þ

2

G0 2 ð1 þ c=cref Þ

ð6:73Þ

Assuming that the ordinary swelling index j is adopted to compute K0.1 %, a modified swelling index js corresponding to any strain level is adopted to compute K: (

K0:1% ¼ ð1 þj e0 Þ p0 K ¼ ð1 þjse0 Þ p0

ð6:74Þ

Combining Eqs. (A2) and (A3) and noting a reference value pat for replacing p’, the initial reference shear modulus Gref0 for replacing G0 can be defined as follows: (

2 Þð1 þ e0 Þ
1 þ 0:001=cref pjat Gref 0 ¼ 3ð12t 2ð1 þ tÞ 2 Þð1 þ e0 Þ
Gref 0 ¼ 3ð12t 1 þ c=cref pjats 2ð1 þ tÞ

ð6:75Þ

Adopting cref ¼ 7e70 =3 and adding a controlling factor of stiffness Prev (with Prev = 1 when loading and Prev = 2 when unloading), Eq. (A4) can be expressed as follows: 8  2  > < Gref 0 ¼ 3ð12tÞð1 þ e0 Þ 1 þ Prev 3 0:001 pat 7 e70 j 2ð1 þ tÞ    2  eeq  pat  > 3 ð 12t Þ ð 1 þ e Þ 0 3 : Gref 0 ¼ 1 þ Prev 7 e70  js 2ð1 þ tÞ

ð6:76Þ

From the 2nd expression in Eq. (A5), the modified swelling index js corresponding to any average equivalent strain level can be obtained as follows:  2  3ð1  2tÞð1 þ e0 Þ 3 eeq  pat 1 þ Prev   js ¼ 2ð 1 þ t Þ 7 e70 Gref 0

ð6:77Þ

This equation suggests that the apparent swelling index increases (in correspondence to the decrease in the shear modulus and bulk modulus) with the increase in the average equivalent strain.

6 Elastoplastic Modeling of Clayey Soils …

224

6.6

Consideration of Plastic Strain Accumulation for Cyclic Effect

To account for the accumulation of plastic strains during cyclic loading, the parameter kp in Eq. (6.34) can be modified as (Ahayan [21]), kp ¼

kp0 1þd

ð6:78Þ

where the new variable d is introduced to take into account the deformation features of cyclically loaded soils, such as the ratcheting and the shakedown effects. The increment of such a new variable takes the following expression:     d_ ¼ ½ad  av ð1 þ d Þe_ pij 

ð6:79Þ

Thus the accumulation of both the volumetric   and deviatoric components of the   plastic strain is considered, via the variable e_ pij , defined as the norm of the plastic strain tensor. The new model parameters ad and av take only positive or null values. When av is selected equal to zero, the state parameter d is increasing with the plastic strain, leading to a decrease of the plastic modulus, which produces the ratcheting effect. When selecting a positive value of av and ad = 0, the shakedown effect is predicted due to the accumulation of the volumetric strain. When av and ad have nonzero values, a transition behavior from ratcheting to shakedown is predicted. Accordingly, the proposed equation can predict ratcheting and shakedown effects as well as the transition behavior from shakedown to ratcheting.

6.7 6.7.1

Some Elastoplastic Models for Clayey Soils in ErosLab Modified Cam-Clay Model—MCC

Thus, when the modified Cam-Clay model is used, the relationship between the preconsolidation pressure from the oedometer test and the initial size of the yield surface can be established as follows: (Fig. 6.28) 9 2 " # fK0 ¼ 0 ) pc0 ¼ Mq2 p0 þ p0 = 3ð1  K0 Þ2 ð1 þ 2K0 Þ ) pc0 ¼ þ rp0 ð6:80Þ q ¼ ð1  K0 Þrp0 ; 3 ð1 þ 2K0 ÞM 2 p0 ¼ ð1 þ 2K0 Þrp0 =3

6.7 Some Elastoplastic Models for Clayey Soils in ErosLab

225

Fig. 6.28 Model parameters of MCC in ErosLab

6.7.2

Anisotropic Structured Clay Model—ASCM

An anisotropic structured clay model was developed under the framework of the Modified Cam-Clay model and considering the behavior of intact clays due to its structure (Yang et al. [16]). The model can be used to predict the mechanical behavior of soft-structured clay, stiff clay, and artificially reinforced clay. The principle of the ASCM model is illustrated in Fig. 6.29. The basic constitutive equations are summarized in Table 6.3. The model parameters and their definitions are summarized in Table 6.4 and Fig. 6.30. Fig. 6.29 Principle of ASCM

6 Elastoplastic Modeling of Clayey Soils …

226 Table 6.3 Basic equations of ASCM Components

Constitutive equations

Elasticity

t 0 t 0 deeij ¼ 1 þ E drij  E drkk dij E ¼ 3Kð1  2tÞ K ¼ 1 þjie0 ½p0 þ pb0 ð1  Rb Þ þ ðpc0  pci0 Þð1  Rc Þ

Yield surface Potential surface Hardening rule

f ¼ 32

Bounding surface rule

½sij ðp0 þ pb Þaij :½sij ðp0 þ pb Þaij  þ ðp0 þ pb Þðp0  pc Þ ðM 2 32aij :aij Þ

g¼f dpci ¼ pci 1kiþje0i depv

 qffiffiffiffiffiffiffiffi pc ¼ ppci0ci pc0 ð1  Rc Þ þ pci Rc with Rc ¼ 1  exp nc epij epij
 pb ¼ pb0 ð1  Rb Þ with Rb ¼ 1  exp nb epd h  i
 as daij ¼ x p0 þijpb  aij depv  aij0 ð1  Ra Þxd depd with Ra ¼ 1  exp na epd
ij ¼ brij r

  þ e0 Kp ¼ K p þ kp 1kj ðbp0 Þ3 1  b1

Table 6.4 Parameters of ASCM Parameters

Definitions

e0 t ji ki Mc pc0 ak0 a b x xd kp

Initial void ratio Poisson’s ratio Intrinsic swelling index (of remolded soil) Intrinsic compression index (of remolded soil) Slope of the critical state line on the p′ – q plane Initial size of the yield surface Initial inclination of the yield surface Target inclination of the yield surface related to the volumetric strain Target inclination of the yield surface related to the deviatoric plastic strain Absolute rotation rate of the yield surface Rotation rate of the yield surface related to the deviatoric plastic strain Plastic modulus-related parameter in the bounding surface Initial bonding ratio (pci0 = pc0/(1+0)) Degradation rate of the bonding ratio related to the plastic volumetric strain Degradation rate of the bonding ratio related to the plastic deviatoric strain Initial interparticle bonding Degradation rate of interparticle bonding

0

n nd pb0 nb

6.7 Some Elastoplastic Models for Clayey Soils in ErosLab

227

Fig. 6.30 Parameters of ASCM in ErosLab

Note that the anisotropy-related parameters are directly calculated by using Mc: a ¼ 0:75; aK0 ¼ gK0 

Mc2  g2K0 3

b¼0 with

ð6:81Þ gK0 ¼

3Mc 6  Mc

 3ðagK0  aK0 Þ 3 Mc2  g2K0  3ð1  aÞgK0  ¼
2 xd ¼ 2ðbgK0  aK0 Þ 2 gK0  Mc2 þ 3ð1  bÞgK0 x¼

1 þ e0 10Mc2  2aK0 xd ln ð k i  ji Þ Mc2  2aK0 xd

ð6:82Þ ð6:83Þ ð6:84Þ

References 1. Biarez J, Hicher P-Y (1994) Elementary mechanics of soil behaviour: saturated remoulded soils. AA Balkema 2. Roscoe KH, Burland J (1968) On the generalized stress-strain behaviour of wet clay. Paper presented at the Engineering Plasticity, Cambridge, UK 3. Wheeler SJ, Näätänen A, Karstunen M, Lojander M (2003) An anisotropic elastoplastic model for soft clays. Can Geotech J 40(2):403–418 4. Manzari MT, Nour MA (1997) On implicit integration of bounding surface plasticity models. Comput Struct 63(3):385–395. https://doi.org/10.1016/S0045-7949(96)00373-2 5. Dafalias YF (1986) Bounding surface plasticity. I: Mathematical foundation and hypoplasticity. Journal of engineering mechanics 112 (9):966–987 6. Dafalias YF (1986) An anisotropic critical state soil plasticity model. Mech Res Commun 13 (6):341–347

228

6 Elastoplastic Modeling of Clayey Soils …

7. Yang J, Yin Z-Y, Liu X-F, Gao F-P (2020) Numerical analysis for the role of soil properties to the load transfer in clay foundation due to the traffic load of the metro tunnel. Transp Geotech 23:100336 8. Karstunen M, Koskinen M (2008) Plastic anisotropy of soft reconstituted clays. Can Geotech J 45(3):314–328 9. Kobayashl I, Soga K, IlZuka A, Ohta H (2003) Numerical interpretation of a shape of yield surface obtained from stress probe tests. Soils Found 43(3):95–103. https://doi.org/10.3208/ sandf.43.3_95 10. Yin ZY, Chang CS (2009) Microstructural modelling of stress-dependent behaviour of clay. Int J Solids Struct 46(6):1373–1388 11. Chang CS, Yin ZY (2010) Micromechanical Modeling for Inherent Anisotropy in Granular Materials. Journal of Engineering Mechanics-Asce 136(7):830–839. https://doi.org/10.1061/ (asce)em.1943-7889.0000125 12. Yin ZY, Karstunen M (2011) Modelling strain-rate-dependency of natural soft clays combined with anisotropy and destructuration. Acta Mech Solida Sin 24(3):216–230 13. Yin ZY, Chang CS, Hicher PY (2010) Micromechanical modelling for effect of inherent anisotropy on cyclic behaviour of sand. Int J Solids Struct 47(14–15):1933–1951. https://doi. org/10.1016/j.ijsolstr.2010.03.028 14. Yin Z, Hicher PY (2013) Micromechanics-based model for cement-treated clays. Theoretical and Applied Mechanics Letters 3(2):021006 15. Graham J, Houlsby G (1983) Anisotropic elasticity of a natural clay. Geotechnique 33 (2):165–180 16. Benz T (2007) Small-strain stiffness of soils and its numerical consequences, vol 5. Univ, Stuttgart, Inst. f. Geotechnik 17. Allman M, Atkinson J (1992) Mechanical properties of reconstituted Bothkennar soil. Géotechnique 42(2):289–301 18. Hattab M, Hicher P-Y (2004) Dilating behaviour of overconsolidated clay. Soils Found 44 (4):27–40 19. Hardin BO, Drnevich VP (1972) Shear modulus and damping in soils: Measurement and parameter effects (Terzaghi Leture). Journal of the Soil Mechanics and Foundations Division 98(6):603–624 20. Yin Z-Y, Xu Q, Hicher P-Y (2013) A simple critical-state-based double-yield-surface model for clay behavior under complex loading. Acta Geotech 8(5):509–523 21. Ahayan S (2019) A constitutive Model for natural Clays: From Laboratory Testing to Modelling of Offshore Monopiles. PhD dissertation, École centrale de Nantes and Université de Liège

Chapter 7

Viscoplastic Modeling of Soft Soils

Abstract First, a one-dimensional elastic viscoplastic model is developed based on the experimental results of strain rate dependency in oedometer tests. Using this model, the coherence of strain rate dependency, creep, and stress relaxation is established. Then, the model is extended for structured clays by accounting for the effect of interparticle bonds. A three-dimensional viscoplastic model called ANICREEP is presented under the framework of Perzyna’s overstress theory, and the choice of the scaling function is discussed. The developed ANICREEP model also accounts for anisotropy and destructuration. To simulate the different tests, a general explicit algorithm is presented. The development of a MATLAB code for simulating the triaxial tests is also presented. Finally, the ANICREEP model in the ErosLab Platform is presented for practical exercises.

7.1 7.1.1

One-Dimensional Viscoplastic Modeling Strain Rate Influence

The 1D controlled strain rate test (CRS test) is conducted by controlling the rate of vertical displacement and measuring the stresses and strains of the specimen. Then, the stress–strain and preconsolidation stress–strain rate can be obtained, which is a basis of developing time-dependent constitutive models. Based on the published data of CRS tests, we have investigated (1) the strain rate dependence of the preconsolidation stress, (2) the normalized compression curves, and (3) the evaluation of different equations expressing the influence of the strain rate on the preconsolidation pressure. Several 1D CRS tests [1–7] etc. have shown that a larger loading rate can result in a larger preconsolidation pressure r′p. Leroueil et al. [1] gathered test results showing the strain rate dependency of different clays and pointed out that the isotach line approach [8] can describe the correlation between the preconsolidation pressure and strain rate. Based on CRS tests on 17 different clays [1–3, 9–11] etc., we plotted the relationship between the preconsolidation pressure and strain rate in © Springer Nature Singapore Pte Ltd. and Tongji University Press 2020 Z.-Y. Yin et al., Practice of Constitutive Modelling for Saturated Soils, https://doi.org/10.1007/978-981-15-6307-2_7

229

230

7 Viscoplastic Modeling of Soft Soils

Normalized preconsolidation pressure ˄σp/σv0˅

5

4

3

0.002%/h

27%/h

2

1

0 0.001

0.01

0.1

1

10

Berthierville clay St-Cesaire clay Gloucester clay Varennes clay Joliette clay Ste-Catherine clay Mascouche clay St-Alban clay Fort Lennox clay Louiseville clay Batiscan clay Wenzhou clay Tungchung clay Backebol clay Bothkennar clay St-Herblain clay Xiaoshan clay 100

Volumetric Strain rate (%/h)

Fig. 7.1 Preconsolidation pressure versus applied strain rate

Fig. 7.1, which shows that the preconsolidation pressure r′p is proportional to the log of the applied strain rate from 0.002%/h to 27%/h. To date, test results under very low strain rates (100%/h) have not been available. Thus, it is difficult to determine the strain rate dependence of the preconsolidation pressure for very low/high strain rates. The possible limiting factors of the low strain rate tests are as follows: (1) the time consumption of the tests (i.e., 417 days are required to reach ev = 10% under a strain rate of 0.001%/h); (2) the accuracy/difficulty of testing due to limitations of the equipment; and (3) chemically induced interparticle bonds during long test durations. On the other hand, high strain rate CRS tests are also influenced by various factors: (1) rapid loading induces excess pore pressure gradients, resulting in non-homogeneous effective stress fields; (2) some energy dissipation phenomena are not considered by the effective stress theory (e.g., acoustic/thermal energy); and (3) some technological reasons, e.g., the sensors cannot record the changes in pore pressure under high speed conditions. Consequently, the investigation of the rate dependency of soft clay at very low/high strain rates remains challenging.

7.1.2

1D Model with Strain Rate Dependency

Based on the conventional elastoplasticity theory, the one-dimensional total strain rate contains two components: the elastic strain rate and the viscoplastic strain rate: e_ ¼ e_ e þ e_ vp

ð7:1Þ

7.1 One-Dimensional Viscoplastic Modeling

231

where e_ denotes the one-dimensional strain rate and the superscripts e and vp denote the elastic and viscoplastic components, respectively. The elastic strain rate is expressed as follows: e_ e ¼

j r_ 0 1 þ e0 r0

ð7:2Þ

where j is the slope of the swelling line on the e-lnr′ plane; e0 is the initial void ratio; and r′ is the current effective stress. For the viscoplastic strain rate, we present a one-dimensional strain rate dependency-based model [5], which has the same bases but different formulations than the expressions developed by Kim and Leroueil [12] or the expressions of the creep-based models developed by Kutter and Sathialingram [13], Vermeer et al. [14], Stolle et al. [15] and Yin et al. [16]. For the strain rate effect on the preconsolidation stress, a linear relationship can be obtained on the log(r′p0) − log(de/ dt) plane. Then, if an arbitrarily applied strain rate is selected as a reference, the following expression can be obtained: e_ ¼ e_ r

r0p0 r0rp0

!b ð7:3Þ

where the preconsolidation stress r′p0 corresponds to the applied strain rate e_ ; the reference preconsolidation stress r0rp0 corresponds to the reference strain rate e_ r (see Fig. 7.2); and b is a material constant corresponding to the slope of this linear relationship (see Fig. 7.2). Similar to the plastic strain defined in conventional elastoplasticity, the viscoplastic strain can be derived from the e-ln(r′) curve at a constant strain rate. In Fig. 7.2, based on the compression line at a given e_ v with loading from r0 to r0 þ dr0 during dt, the viscoplastic strain rate is obtained as follows: e_ vp ¼

k  j r_0 1 þ e0 r 0

ð7:4Þ

where k is the compression index measured from the e-ln(r′) curve. Thus, the ratio between the elastic strain rate (Eq. (7.2)) and the viscoplastic strain rate (Eq. (7.4)) can be derived as follows: j e_ e ¼ e_ vp k  j Then, the total strain rate can be written as follows:

ð7:5Þ

232

7 Viscoplastic Modeling of Soft Soils

Fig. 7.2 Principle of the strain rate-based model: a 1D compression curves for strain rate e_ v and reference strain rate e_ rv , strain rate versus preconsolidation pressure for different plastic strain levels, and c elastic and plastic strain increments due to a stress increment during 1D compression at constant strain rate e_ v

e_ ¼

k vp e_ kj

ð7:6Þ

Substituting Eq. (7.6) into Eq. (7.3), the viscoplastic strain rate in the stress state of the initial preconsolidation pressure can be written as follows: e_

vp

k  j r0p0 ¼ e_ k r0rp0

!b

r

ð7:7Þ

Along the compression line corresponding to the strain rate e_ (see Fig. 7.2a), the current stress r′ (equal to the current preconsolidation stress r′p) at a given viscoplastic strain evp can be derived as follows: 0

r ¼

r0p

¼

r0p0

  1 þ e0 vp e exp kj

ð7:8Þ

Similarly, along the compression line corresponding to the reference strain rate e_ r (see Fig. 7.2a), the current reference stress r0rp at the same viscoplastic strain level can be derived as follows:

7.1 One-Dimensional Viscoplastic Modeling

r0rp

¼

r0rp0

233



1 þ e0 vp e exp kj

 ð7:9Þ

By substituting Eqs. (7.8)–(7.9) to Eq. (7.7), the proposed strain rate-based model is given as follows: e_

vp

k  j r0 ¼ e_ k r0rp

!b ð7:10Þ

r

Equation (7.10) implies that for a given state of stress (constant r′), the viscoplastic strain rate decreases with time since r0rp increases with time due to the hardening rule controlled by Eq. (7.9). Four model parameters (j, k, b, e_ r ) with two initial state variables (e0, r0rp0 ) are needed for the model, and all of these parameters can be determined easily from the 1D CRS (constant rate of strain) test. Meanwhile, Kutter and Sathialingram [13], Vermeer et al. [14] and Stolle et al. [15, 17] have also proposed a one-dimensional formulation based on the creep phenomenon: e_ vp

Cae r0 ¼ ð1 þ e0 Þs r0rp

!kj C ae

ð7:11Þ

where s is the reference time, which denotes the constant period in which each load step is maintained as constant, which is precisely one day (24 h) in conventional oedometer testing for simplicity. Comparing Eqs. (7.10) and (7.11), we can obtain the following relationship: e_ r

kj Cae kj ¼ and b ¼ k Cae ð1 þ e0 Þs

ð7:12Þ

Thus, the model parameters b and e_ r can also be obtained easily if a conventional oedometer test is available with a measured Cae and s = 24 h. Note that s relates to hypotheses A and B. Different from Perzyna’s overstress approach [18], the proposed modeling method uses the reference yield stress instead of the static yield stress. Therefore, the viscoplastic strain occurs even when the current stress is smaller than the reference yield stress, while for the overstress approach, only elastic strain occurs when the current stress is smaller than the static yield stress.

234

7 Viscoplastic Modeling of Soft Soils

7.1.3

Stress Relaxation and Coherence of Time-Dependent Behavior

According to the rate dependency-based 1D elasto-viscoplastic model, the total strain rate is expressed as follows: !b 0 Cs r_ 0v e vp r Cc  Cs rv þ e_ v ð7:13Þ e_ v ¼ e_ v þ e_ v ¼ ð1 þ e0 Þ ln 10 r0v Cc r0rp When the soil is under stress relaxation conditions, the total strain rate is zero. Thus, Eq. (7.13) can be rewritten as follows: Cs r_ 0v ð1 þ e0 Þ ln 10 r0v

0

¼

1b

Cc  Cs @ r0  v A e_ rv 0 Þ ln 10 vp Cc r0rpi exp ð1 þCceC e v s

ð7:14Þ

where r0rpi is the initial value of the current reference stress corresponding to the start of stress relaxation. During stress relaxation, the viscoplastic strain is equal to the negative value of the elastic strain: Z evp v ¼

Cs r_ 0v dt ð1 þ e0 Þ ln 10 r0v

ð7:15Þ

R r_ 0 Since rv0 dt ¼ ln r0v  ln r0vi (where r0vi is the initial vertical stress at the start of v stress relaxation), by substituting Eqs. (7.15) and (7.14) can be further transformed as follows: 0 Cs r_ 0v ð1 þ e0 Þ ln 10 r0v

¼  e_ rv

Cc  Cs B B Cc @

1b C C  0 C CCs A r0v

r0rpi

rv r0vi

c

ð7:16Þ

s

In general, form, the equation can be written as follows:  m r_ 0v ¼ A r0v

ð7:17Þ

with 0 1b ln 10  ð 1 þ e Þ  ð C  C Þ 1 0 c s @ A and m ¼ Cc b þ 1 A ¼  e_ rv Cs 0r 0 Cc  Cs Cc  Cs rpi  rvi Cc Cs ð7:18Þ

7.1 One-Dimensional Viscoplastic Modeling

235

where A and m have both constant values during stress relaxation. Solving this first-order differential equation with the initial condition r0vi for t = 0, the analytical solution for stress relaxation is determined as follows: 1  1m r0v ¼ Að1  mÞt þ r01m vi

ð7:19Þ

Substituting the constants A and m into Eq. (7.19), the evolution of the vertical stress is expressed as follows: 0

0

ln 10  ð1 þ e0 Þ  ðCc  Cs Þ @ 1 B r0v ¼ @ e_ rv Cs 0r Cc  Cs rpi  rvi 0 Cc Cs

1b

 A 

1CcCCb s c  Cc b Cc b 0 Cc Cs C t þ rvi A Cc  Cs

ð7:20Þ This equation also indicates the decrease in ln(r′v) with ln(t); therefore, a positive value of r′v is maintained for an infinite time since a logarithmic scale is used. According to the experimental results, a linear relationship between the vertical stress and the time in the plot ln(r′v)—ln(t) during stress relaxation can be generally assumed. The slope of ln(r′v) versus ln(t) is then defined as the stress relaxation coefficient Ra: Ra ¼ 

D ln r0v D ln t

ð7:21Þ

Based on the analytical solution of a stress relaxation test (Eq. (7.19)), Ra can be derived as follows: in the stage of stress relaxation, after a certain time, r01m vi becomes small compared to Að1  mÞt: Hence, the differential of ln(r′v) with ln (t) can be expressed as follows: @ ln r0v 1 Cc  Cs ¼ ¼ 1m @ ln t Cc b

ð7:22Þ

Comparing Eq. (7.21) with Eq. (7.22), the stress relaxation coefficient Ra can be expressed by the compression parameters Cc and Cs and the rate dependency coefficient b: Ra ¼

Cc  Cs Cc b

or

b¼

Cc  Cs Cc Ra

ð7:23Þ

Furthermore, Kutter and Sathialingam [13] presented the quantity (Cc – Cs)/Cae, and Yin et al. [10, 19] related it to the rate dependency coefficient b (Leoni et al. [20]):

236

7 Viscoplastic Modeling of Soft Soils

b¼

Cc  Cs Cae

or

Cae ¼

Cc  Cs b

ð7:24Þ

Hence, by substituting b in Eq. (7.24) into Eq. (7.23), Ra can also be expressed by Cae or the inverse: Ra ¼

Cae Cc

or

Cae ¼ Ra  Cc

ð7:25Þ

Mesri and Castro [21] showed that Cae is related to Cc and, more precisely, that the ratio Cae/Cc is constant for a given soil. Moreover, they summarized the range of values for a number of clays published in the literature and found that Cae/Cc is in the remarkable range of 0.02-0.1. For a majority of inorganic soft clays, Cae/Cc is 0.04 ± 0.01, and for highly organic plastic clays, Cae/Cc is 0.05 ± 0.01. Based on Eq. (7.25), it is interesting to point out that this classification is also related to the stress relaxation coefficient. Consequently, previous studies on the properties of Cae/Cc can be used for Ra. Overall, based on the derived expressions between Ra, b, and Cae, a unique relationship among three time-dependent-related parameters was obtained, which suggests that once Ra is measured, b and Cae are equally obtained. This relationship will be helpful for the case when b and Cae cannot be easily measured. Since the time-dependency related parameters Ra, b, and Cae can each be obtained from the other two, consequently, the analytical solution of the 1D stress relaxation (Eq. (7.20)) using b can also be directly expressed by Ra. Thus, substituting Eq. (7.23) into Eq. (7.20), the stress relaxation can be expressed as follows by using Ra as a key parameter: 0

0

1CCc C s R

1 B ln 10  ð1 þ e0 Þ  ðCc  Cs Þ @ A r0v ¼ @e_ rv Cs 0r Cc  Cs rpi  rvi 0 Cc Cs

c a

1Ra 1 1C t þ rvi 0 Ra A Ra ð7:26Þ

Then, Eq. (7.26) can be used to simulate the stress relaxation behavior of soft soils. The uniqueness of the strain rate dependency, creep, and stress relaxation was validated by the experimental results of reconstituted illite and Berthierville clay, as shown in Figs. 7.3, 7.4 and 7.5.

7.1.4

Model Extension for Structured Clays

For natural soft clay, interparticle bonds are usually formed during deposition, and these bonds are referred to as the soil structure. These bonds represent the

7.1 One-Dimensional Viscoplastic Modeling 500

100 90 80 70 60

Experiment by β by ψ by Rα

400

σ'P0 (kPa)

300

σ'P0 (kPa)

237

200

Experiment by β by ψ by Rα

50 40 30

Reconstituted Illite

100 1E-9

(a)

1E-7

Berthierville clay

1E-5

1E-3

dεv/dt (s)

(b)

20 1E-10

1E-8

1E-6

1E-4

dεv/dt (s)

Fig. 7.3 Comparison of the influence of experimental and derived b on the relationship between preconsolidation pressure and strain rate. a Reconstituted illite; b Berthierville clay

0.05 Berthierville clay

0.006 Reconstituted Illite

ψ

ψ /(1+e0)

0.04 0.004

0.03 0.02

0.002

0.01 0.000

(a)

A Experment Experiment

byBβ

byCRα

0.00

(b)

A Experment Experiment

byBβ

byCRα

Fig. 7.4 Comparison of w measured from experiment and derived by b and Ra: a Reconstituted illite; b Berthierville clay

interparticle cementation of the soil resulting in higher compression yield stresses, and thus, they are key factors influencing the mechanical characteristics of soils. The amount of these bonds will decrease progressively during straining. By experimentally comparing the compression curves on the e − log(r′) plane (void ratio versus vertical stress on the logarithmic scale) for intact and reconstituted samples, Leroueil et al. [22], Burland [23], Smith et al. [24], Karstunen and Yin [25], Yin et al. [26], and others concluded that the differences observed were caused by bond degradation. This physical phenomenon is called destructuration by Leroueil [22], and we will retain this denomination in our text. Based on the destructuration effects on the compression behavior during loading, a modeling approach for creep degradation of structured clay was developed by Yin and Wang [5].

238

7 Viscoplastic Modeling of Soft Soils 600

Reconstituted Illite

500 450

Experiment byβ byψ byRα

Berthierville clay

150

R3

100

σ 'v (kPa)

σ 'v (kPa)

550

200

Experiment byβ byψ byRα

R2 R1

50

400 350

(a)

1

10

100

Time (min)

1000

30

10000

(b)

1

10

100

1000

10000

Time (min)

Fig. 7.5 Comparison of experimental and predicted results of stress relaxation in three ways: a Reconstituted illite; b Berthierville clay

(1) Constitutive equations During CRS and conventional oedometer tests on soft structured clays, the shape of the post-yield compression curve (or the apparent compression index k) is significantly influenced by the debonding process during straining, as shown in Fig. 7.6a–c (Vanttila clay by Yin et al. [19], Shanghai clay by the authors, and Wenzhou clay by Zeng [27]). Figure 7.2d shows the schematic plot of the stress– strain curve at a constant strain rate for soft structured clays. At a given viscoplastic strain level, the bond degradation results in the current stress r′p reaching point D instead of point E (assuming no destructuration). Corresponding to this current stress r′p at evp, we define an intrinsic stress r′pi, which is the stress for a reconstituted sample at the same strain rate. The intrinsic line for a reconstituted sample is expressed as follows: r0pi ¼ r0pi0 exp



 1 þ e0 vp 1 þ e0 vp e or in rate form r_0 pi ¼ r0pi e_ ki  j ki  j

ð7:27Þ

where ki is the slope of the intrinsic normal compression line on the e-ln(r′) plane for a reconstituted sample as defined by Burland [23] and r0pi0 is the intrinsic preconsolidation pressure (see Fig. 7.6d).  A structure indicator defined by v ¼ r0 r0i  1 (similar to Gens and Nova [28]) is proposed, since a definition based on the stress ratio can guarantee that the compression line of an intact sample will approach the line of a reconstituted sample when the stress ratio decreases in correspondence to destructuration. The current stress r′ (equal to r′p due to viscoplastic strain hardening) during straining can be expressed as follows:

7.1 One-Dimensional Viscoplastic Modeling

(a)

(b)

Shanghai clay

χ0 = 2.05

1.2

239

κ = 0.025

Wenzhou clay

χ0 = 14.5

2.2

σ'p0,24 = 95 kPa

κ = 0.027

σ'p0,24 = 96 kPa

1

1.8

λ0 = 0.68

0.8

e

e

λ0 = 0.289

1.4

λi = 0.135

0.6 0.4

10

(c) 4

λi = 0.204

1

Conventional test: 24h Intact Reconstituted

Conventional test: 24h Intact Reconstituted

0.6

100 σ 'v (kPa)

10

(d)

A

λ0 = 4.1

3

σ'pi

σ'pi0

σ'p0,24 = 29 kPa

κ = 0.057

100

1

e

C

Intact Reconstituted 10

σ ' (kPa)

Δεe Δεvp E

D

Reconstituted sample

1 Conventional test: 24h

0.1

ln(σ')

σ'p0 σ'p

2

0

10000

κ/(1+e0) B

λi = 0.31

(e)

1000

σ ' (kPa)

Vanttila clay

χ0 = 77.0

1

1000

Intact sample

χ = σ'p/σ'pi-1

λi/(1+e0)

εv

1000

1

(f) χ0 = 77 Vanttila clay Wenzhou clay χ0 = 14.5 Shanghai clay χ0 = 2.05 vp Theory χ =χ0 exp[-ρ·ε ]

χ/ χ0

1

ρ = 6.5

0.5

ρ = 12.5 ρ = 13.5

0

0

0.1

0.2

ε vp

0.3

0.4

Fig. 7.6 a–c One-dimensional compression curves for intact and reconstituted samples of structured clays, d schematic plot for the compression behavior of soft structured clay at a constant strain rate, and e evolution of the structure indicator v versus viscoplastic strain

240

7 Viscoplastic Modeling of Soft Soils

r0p ¼ ð1 þ vÞr0pi

ð7:28Þ

. Initially, the structure indicator v ¼ v0 ¼ r0p0 r0pi0  1 can be obtained. When the strain increases, some of the bonds are broken and v decreases from its initial value v0 and ultimately moves toward zero when the bonds are completely destroyed, as shown in Fig. 7.6a–d. To clarify the viscoplastic strain effect on the destructuration, an oedometer test on the highly sensitive Vantilla clay with an incremental loading period of 100 days, which was conducted by Yin et al. [19], was adopted. In this test, there is a loading increment from 14.1 to 28.1 kPa smaller than state B in Fig. 7.6d (with rp0 = 29 kPa). Thus, the sample has more than 99 days of pure creep deformation at this loading stage. Then, based on the void ratio change under the next loading stage that lasts 1 day, a local compression index between 28.1 and 56.2 kPa was measured with k = 0.539 (see the star symbol in Fig. 7.6e). Compared to the sample with less creep duration (1 day for a conventional test in this figure), k is much smaller, which indicates that the pure creep deformation has a significant influence on the interparticle bond degradation (saying k decreases to the value of reconstituted clay during destructuration). Furthermore, in Fig. 7.6e, for the same stress level, k is very different due to different histories of creep deformation, which means that the structural degradation is better correlated by the viscoplastic strain instead of the stress. Moreover, the test was continued for a period of 100 days for this loading step of 56.2 kPa and for the following loading steps. Based on all these results, k was measured and plotted in Fig. 7.6e. The k values measured from 1 day conventional tests on intact clay and reconstituted clay were also plotted in this figure for comparison. Since these tests have the same stress levels but different creep times, it can be concluded that the intact clay with more creep deformation has more destructuration and earlier convergence with the reconstituted clay behavior. Therefore, the viscoplastic strain can be reasonably adopted for describing bond degradation. According to the schematic plot in Fig. 7.6d, the structure indicator functions of the viscoplastic strains during one-dimensional compression were measured and plotted in Fig. 7.6f for different clays (Fig. 7.6a–c). Based on these results, the following relationship between the structure indicator v and the magnitude of the viscoplastic strain was proposed: v ¼ v0 eqe or in rate form v_ ¼ vq_evp vp

ð7:29Þ

where the soil constant q controls the rate of destructuration (q = 13.5 for Vanttila clay, q = 12.5 for Wenzhou clay, and q = 6.5 for Shanghai clay in Fig. 7.6f). Substituting Eqs. (7.27) and (7.29) into Eq. (7.28), r′p can be derived as follows:

7.1 One-Dimensional Viscoplastic Modeling

r0p

    0 1 þ e0 vp qevp rpi0 exp e ¼ 1 þ v0 e ki  j

241

ð7:30Þ

Considering the elastic strain, the stress–strain curve for a given strain rate can be obtained. Two additional parameters v0 and q are required compared to the model for reconstituted clay, and they can be measured from the oedometer tests (see Fig. 7.6). In the model, the reference preconsolidation pressure can also be determined by adopting Eq. (7.30):    1 þ e0 vp vp  r0rp ¼ 1 þ v0 eqe r0rpi0 exp e ki  j

ð7:31Þ

Then, the effect of destructuration on the compression behavior has been introduced into the model, and the apparent value of Cae can be measured from the creep curves. Considering that creep degradation is indirectly modeled using the stress-based structure indicator, this model is termed “creep-implicit” in the following sections. (2) Coupled consolidation analysis The one-dimensional models were implemented in the commercial finite element code PLAXIS. In the following numerical simulations on the laboratory tests, hydromechanical coupling analyses were performed. Biot’s theory was adopted as the basis for the fully coupled one-dimensional consolidation analysis. The basic finite element scheme for the proposed models is similar to that presented by Oka et al. [29]. For the consolidation analysis, the permeability coefficient k plays an important role. Based on the experimental results and the suggestions by Berry and Poskitt [30], the evolution of k can be expressed as follows: k ¼ k0 10ðee0 Þ=ck

ð7:32Þ

where the initial permeability coefficient k0 corresponds to e0; the permeability index ck can be easily measured from the oedometer test results by plotting the relation e-logk. Using the above equations and a standard discretization method, the finite difference equations can be established for modeling the consolidation process in an oedometer test. Details of coupled consolidation and creep analysis can be found in Kim and Leroueil [12], Stolle et al. [15, 17], etc. and are not repeated here. (3) Parameter determination and validation The proposed model for soft structured clay contains a number of soil parameters that can be determined as follows: (a) Parameters related to compressibility: the initial void ratio (e0), the intrinsic compression index (ki), the swelling index (j), and the reference

242

7 Viscoplastic Modeling of Soft Soils

 preconsolidation pressure

 r0rp0 . Note that ki can be measured from the

compression curve of reconstituted samples, and if tests on reconstituted samples are not available, this value can be estimated from oedometer tests on intact samples at a high strain level (Yin et al. 2011). (b) Parameters related to the time effect: the intrinsic creep coefficient (Caei) and the reference time (s). Caei can be obtained from the e-lnt curves of a reconstituted sample or can be estimated from intact clay when the evolution of Cae is stable at high stress levels. s is a reference time, which is usually taken as one day if the conventional oedometer test is adopted. (c) Parameters related to consolidation. The initial permeability (k0) and the permeability coefficient (ck). The two parameters can be measured from either the conventional oedometer test or CRS test on intact samples. (d) Initial structure indicator (v0) and destructuration rate (q). The value of v0 can be measured from oedometer tests on intact samples together with reconstituted samples. If tests on reconstituted samples are not available, v0 can be estimated from the stress–strain curve of an intact sample assuming an intrinsic compression line (with slope ki) passing through points at high strain levels. Notably, for this key parameter v0, intact samples with high quality (very slightly disturbed, such as block samples) are required. The parameter q can be obtained by the following equation derived from Eq. (7.30): 8 2 39