Mechanical Testing of Materials [1st ed. 2024] 303145989X, 9783031459894

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Solid Mechanics and Its Applications

Emmanuel Gdoutos Maria Konsta-Gdoutos

Mechanical Testing of Materials

Solid Mechanics and Its Applications Volume 275

Series Editors J. R. Barber, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Anders Klarbring, Mechanical Engineering, Linköping University, Linköping, Sweden

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. Springer and Professors Barber and Klarbring welcome book ideas from authors. Potential authors who wish to submit a book proposal should contact Dr. Mayra Castro, Senior Editor, Springer Heidelberg, Germany, email: [email protected] Indexed by SCOPUS, Ei Compendex, EBSCO Discovery Service, OCLC, ProQuest Summon, Google Scholar and SpringerLink.

Emmanuel Gdoutos · Maria Konsta-Gdoutos

Mechanical Testing of Materials

Emmanuel Gdoutos Academy of Athens Athens, Greece

Maria Konsta-Gdoutos University of Texas at Arlington Arlington, TX, USA

ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid Mechanics and Its Applications ISBN 978-3-031-45989-4 ISBN 978-3-031-45990-0 (eBook) https://doi.org/10.1007/978-3-031-45990-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, 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 publisher, 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 publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

This book is dedicated to our children Eleftherios and Alexandra-Kalliope

Preface

Engineering structures and machines fail of different modes of failure. The mechanical design of engineering structures and machines involves a stress and failure analysis in order to estimate how much stress and energy required to fracture can be anticipated. A postulate referred to as failure criterion is invoked to predict the event of failure itself. Sophisticated methods for determining stress distributions in loaded structure are available. Detailed theoretical analyses based on simplifying assumptions regarding material behavior and structural geometry are undertaken to obtain an accurate knowledge of the stress state. For complicated structure or load situations, experimental or numerical methods are preferable. The stresses and the failure characteristics are compared with the material’s strength and resistance to failure using a failure criterion. These material properties are determined by performing experimentally mechanical materials tests which indicate the material’s ability to resist the different modes of failure. The data obtained for the material properties can be used in specifying the suitability of materials for various applications. The various modes of materials deformation include excessive elastic, plastic, creep deformation, cracking, fatigue and environmental cracking by a hostile environment. Elastic deformation is recovered immediately upon unloading, while plastic deformation is permanent. Creep deformation increases with time. Brittle fracture is accompanied by small deformation, while ductile fracture involves considerable deformation. Creep fracture is time-dependent. Fatigue involves repeated loading and leads to the development and growth of cracks. Failure of structures due to the presence of cracks is studied by fracture mechanics. Mechanical behavior at high strain rates differs considerably from that observed at quasi-static or intermediate strain rates, and many engineering applications require characterization of mechanical behavior under dynamic conditions. Mechanical testing is a series of tests used to determine the material’s mechanical properties and suitability for its proposed application. Different types of testing are used to determine different properties. For instance, a tensile test is used to determine the tensile strength of a material, a Charpy V-notch test can qualify a material’s toughness, a Vickers hardness test can quantify a material’s toughness, a high cycle fatigue test determines the number of cycles a material can take before it fails. Some vii

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tests can provide information on several mechanical properties. For example, a tensile test determines the ultimate tensile strength, the yield stress, the modulus of elasticity and the ductility of a material. Some mechanical properties can be assessed using different types of tests. For example, toughness can be determined with a Charpy V-notch test or an Izod test, and hardness can be evaluated with Vickers, Brinell and Rockwell tests. The small differences between each testing technique allow engineers to decide which mechanical test is best suited to their requirements. The primary purpose of mechanical testing is to ensure the safety of any final products or structures. Standard test methods have been established by such national and international bodies as the International Organization for Standardization (ISO), and the American Society for Testing and Materials (ASTM). The purpose of this book is to present the most used test methods for the mechanical characterization of deformation and failure of materials. Emphasis is given on the principles of operation of the various tests, not on the detailed application. For more information on the details of the tests the various ASTM standards are provided at the end of each chapter. The book is divided into ten chapters. The first eight chapters present the test methods for the characterization of materials properties. The last two chapters are devoted to the testing of concrete and fiber composite materials due to the special experimental techniques used for the mechanical characterization of these classes of materials. Chapter 1 describes the testing procedure for the determination of the engineering and true stress-strain curves of a material in tension. It presents the yield and strain hardening behavior of a material, the hysteresis loop in plastic deformation, the Bauschinger effect, the state of stress in the neck area, the approximate simple McGregor method for determining stress-strain curves, the idealized and analytical expressions of stress-strain curves, the definitions of ductility, resilience and tensile toughness, the Poisson’s ratio, and the ASTM standards for the tensile test. Chapter 2 presents the compression, bending, torsion and multiaxial testing of materials. Even though most materials have the same elastic behavior in tension and compression their post-yield and failure characteristics in compression are quite different than in tension. Tests for the multiaxial characterization of materials include biaxial tension, biaxial tension/compression strip, tube, spherical vessel, combined tension/compression-torsion ring, combined tension/compression-torsion tube, combined tension/compression-torsion-internal pressure tube tests. Chapter 3 presents the indentation testing for the determination of mechanical and fracture properties of materials. Indentation can be categorized in macroindentation, microindentation and nanoindentation depending on the magnitude range of the applied loads. The principles of contact mechanics and the major methods used to measure the hardness, the modulus of elasticity and the critical stress intensity factor are described. Chapter 4 presents experimental methods for determining various forms of fracture toughness. They include: the critical stress intensity factor, the critical J-integral and the critical crack opening displacement. These quantities are material constants. A small introduction to fracture mechanics is also presented.

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Chapter 5 presents two different approaches for the determination of the of fatigue life of materials under repeated fluctuating loads. The first approach determines the fatigue life in terms of global measurable quantities, like stress, strain, mean stress, etc. The second approach is based on fracture mechanics. It provides a better understanding of the fatigue failure by modeling the fatigue crack initiation and propagation processes. Chapter 6 presents models for the mechanical behavior of materials for linear and nonlinear deformation with time under constant applied loads. Testing methods for creep, recovery and relaxation characterization of materials are also presented. Chapter 7 presents tests for the mechanical characterization of materials under dynamic loading conditions. Special experimental arrangements for the study of the mechanical behavior of materials at various strain rate regimes are also presented. Chapter 8 presents the following methods for the nondestructive characterization of the mechanical behavior of materials: dye penetrant, magnetic particles inspection, eddy currents, radiography, ultrasonics and acoustic emission. Chapter 9 presents the basic tests for the characterization of the mechanical behavior of concrete. They include compression, tension and bending tests. Furthermore, the basic principles of fracture mechanics applied to concrete are also presented. Finally, Chap. 10 presents test methods for the mechanical characterization of fiber reinforced composite materials under tension, compression and shear. Also methods for the determination of the interlaminar fracture toughness of laminates under modes I, II, III and mixed-mode loading are presented. Finally, the mechanical behavior and failure mechanisms of sandwich structures are presented. The book provides the basic experimental testing methods for the characterization of the mechanical behavior of materials. It is hoped that it will be used not only as a learning tool but also as an inspiration and basis on which the researcher, the engineer, the experimentalist and the student can develop their new own ideas to promote research in the study of the mechanical behavior of materials. Needless to say, that the most valuable virtues of an experimentalist are not his/her technical skills, but honesty and integrity. Experimental results should be presented as obtained from the tests, unbiased of any relations with analytical solutions, in case they exist. This most valuable virtue should be cultivated in classrooms and laboratories. We want to take this opportunity to thank our children Eleftherios and AlexandraKalliope for their love and support during the writing of the book. A great word of thanks goes to Dr. Panayiotis Danoglidis of the University of Texas at Arlington for his help in the preparation of the figures of the book in a speedy and efficient way. His hard work and dedication are greatly appreciated. A special word of thanks goes to Mrs. Mayra Castro of Springer for her kind and continuous collaboration and support and the esthetic appearance of this book. Athens, Greece Arlington, TX, USA 2023

Emmanuel Gdoutos Maria Konsta-Gdoutos

Contents

1

2

Tensile Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Specimens and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Engineering Stress–Strain Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Yield Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Strain Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Hysteresis Loop in Plastic Deformation . . . . . . . . . . . . . . . . . . . . . . 1.7 Bauschinger Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 True Stress–True Strain Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Stress State in the Neck Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Approximate McGregor Method for Determining the Stress–Strain Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Idealized Stress–Strain Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Analytical Expression of Stress–Strain Curves . . . . . . . . . . . . . . . . 1.13 Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Tensile Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Bulk Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.19 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 8 10 15 16 17 17

Compression, Bending, Torsion and Multiaxial Testing . . . . . . . . . . . 2.1 Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Stress–Strain Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Comparison of Stress–Strain Curves in Compression and Tension . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bending Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Torsion Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 35 37

19 21 23 25 26 28 29 30 31 32 33

38 40 41

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2.4

3

4

Multiaxial Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Biaxial Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Biaxial Tension/Compression Strip Test . . . . . . . . . . . . . . 2.4.4 Tube Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Spherical Vessel Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Combined Tension/Compression–Torsion Ring Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.7 Combined Tension/Compression–Torsion Tube Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.8 Combined Tension/Compression–Torsion– Internal Pressure Tube Test . . . . . . . . . . . . . . . . . . . . . . . . 2.4.9 Failure Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 ASTM Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44 44 45 46 47 47

Indentation Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Contact Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Macroindentation Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Brinell Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Meyer Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Vickers Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Rockwell Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Microindentation Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Vickers Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Knoop Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Nanoindentation Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 The Elastic Contact Method . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Nanoindentation for Measuring Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Nanoindentation for Measuring Interfacial Fracture Toughness—Conical Indenters . . . . . . . . . . . . . . 3.5.5 Nanoindentation for Measuring Interfacial Fracture Toughness—Wedge Indenters . . . . . . . . . . . . . . . 3.6 ASTM Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 64 65 65 66 67 67 69 69 70 70 70 71

Fracture Mechanics Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Critical Stress Intensity Factor Fracture Criterion . . . . . . . . . . . . . 4.1.1 The Linear Elastic Stress Field . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Strain Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Fracture Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Variation of K c with Specimen Thickness . . . . . . . . . . . . 4.1.5 Experimental Determination of K Ic . . . . . . . . . . . . . . . . . .

87 87 87 88 89 89 94

48 49 50 50 59 61

74 77 81 84 85

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4.2

J-Integral Fracture Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 J-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 J-Integral Fracture Criterion . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Experimental Determination of J-Integral . . . . . . . . . . . . 4.3 Crack Opening Displacement Fracture Criterion . . . . . . . . . . . . . . 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 COD Design Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Standard COD Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Strain Energy Density Failure Criterion . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Volume Strain Energy Density . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Basic Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Two-Dimensional Linear Elastic Crack Problems . . . . . . 4.4.5 Critical Strain Energy Density Factor S c . . . . . . . . . . . . . . 4.5 Dynamic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 ASTM Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100 100 102 104 110 110 111 114 116 116 116 116 118 119 120 122 123

Fatigue and Environment-Assisted Testing . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Fatigue Study Based on Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Stress Versus Life (S–N) Curves . . . . . . . . . . . . . . . . . . . . 5.2.3 The Effect of Mean Stress . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Multiaxial Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Variable Amplitude Loads . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fatigue Study Based on Fracture Mechanics . . . . . . . . . . . . . . . . . . 5.3.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Crack Propagation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Fatigue Life Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Variable Amplitude Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Overload Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Life Estimate Based on Summation of Crack Increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Models for Predicting Fatigue Life . . . . . . . . . . . . . . . . . . 5.4.5 Miner Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Fatigue Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Uncracked Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Cracked Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Environment-Assisted Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Time-To-Failure Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Growth Rate Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 125 127 127 128 129 133 133 134 134 136 139 140 140 140

5

141 142 145 146 146 146 147 147 147 149 149

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5.6.4 Life Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.7 ASTM Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6

Creep Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Mechanical Behavior of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Rheological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Creep Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Steady-State Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Transient Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Relaxation Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Linear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Nonlinear Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Nonlinear Stress Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Stress–Strain Relationships in Three Dimensions . . . . . . . . . . . . . 6.8 Solution of Creep Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Creep and Relaxation Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 ASTM Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 156 159 159 160 162 164 165 167 167 168 169 171 171

7

Testing at High Strain Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Tests at Different Strain Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 High-Speed Load Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Drop Weight Impact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Experimental Procedure and Instrumentation . . . . . . . . . 7.5.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Pendulum Impact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Experimental Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Energy Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.4 Charpy Impact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.5 Izod Impact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Split-Hopkinson (Kolsky) Bar Impact Test . . . . . . . . . . . . . . . . . . . 7.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Experimental Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.3 Impact Stress Waves in the Bars . . . . . . . . . . . . . . . . . . . . 7.7.4 Analysis of Experimental Data . . . . . . . . . . . . . . . . . . . . . . 7.7.5 Modifications of SHPB Arrangement for Ceramics and Soft Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.6 Split-Hopkinson Tension Bar . . . . . . . . . . . . . . . . . . . . . . . 7.7.7 Split-Hopkinson Torsion Bar . . . . . . . . . . . . . . . . . . . . . . .

173 173 174 174 175 176 176 177 177 178 179 179 179 180 181 182 182 183 183 183 185 186 189 189 190

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7.8

Taylor Impact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 Experimental Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.3 Specimen and Test Procedure . . . . . . . . . . . . . . . . . . . . . . . 7.8.4 Determination of the Yield Stress . . . . . . . . . . . . . . . . . . . 7.9 Expanding Ring Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Experimental Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.3 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Plate Impact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10.2 Experimental Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . 7.10.3 Rankine–Hugoniot Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Optical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 ASTM Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 191 192 192 193 193 194 194 196 196 196 196 197 198 199 199

Nondestructive Testing (NDT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Dye Penetrant Testing (PT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Magnetic Particle Testing (MT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Detection of Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Eddy Current Testing (ECT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Theory and Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 X-ray Diffraction Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 X-ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 Measurement of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.5 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Ultrasonic Testing (UT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Acoustic Emission Testing (AET) . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 202 202 202 203 203 203 204 204 205 205 206 206 207 207 207 207 208 209 210 210 210 210 211 212 212

8

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8.7.2 Sources of Acoustic Emission . . . . . . . . . . . . . . . . . . . . . . 8.7.3 Propagation of AE Signals . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.4 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.5 Source Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 ASTM Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212 213 214 215 216 217 225

Testing of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Compression Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Stress–Strain Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.5 Failure Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.6 Effect of Specimen Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.7 Size Effect on Strength of Cylinders . . . . . . . . . . . . . . . . . 9.2.8 Comparison of Strength of Cubes and Cylinders . . . . . . . 9.3 Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Direct Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Brazilian (Splitting) Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Flexure Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 The Ring Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Fracture Mechanics of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Why Fracture Mechanics of Concrete? . . . . . . . . . . . . . . . 9.4.3 Tensile Behavior of Concrete . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 The Fracture Process Zone . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.5 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.6 Cohesive Crack Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.7 Experimental Determination of GIc . . . . . . . . . . . . . . . . . . 9.5 Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 ASTM Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 227 228 228 228 228 229 230 230 231 231 232 232 233 233 235 237 238 238 238 239 242 243 244 245 250 251 253

10 Testing of Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Fiber Reinforced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Tension Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Compression Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 256 257 257 257 258 258 258

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Contents

10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Shear Loading Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 End Loading Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Sandwich Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.5 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Shear Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Rail Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Tensile Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.4 Iosipescu and Arcan Methods . . . . . . . . . . . . . . . . . . . . . . 10.6 Interlaminar Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Interlaminar Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.2 Mode-I Delamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.3 Mode-II Delamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.4 Mixed-Mode-I/II Delamination . . . . . . . . . . . . . . . . . . . . . 10.7.5 Mode-III Delamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Sandwich Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.2 Failure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 ASTM Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.1 ASTM Standards for Fiber Composites . . . . . . . . . . . . . . 10.9.2 ASTM Standards for Sandwich Materials . . . . . . . . . . . . Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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258 259 260 260 262 262 262 262 263 266 267 270 270 270 272 276 280 280 280 282 287 287 288 289

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

About the Authors

Emmanuel Gdoutos is Full Member of the Academy of Athens, the most prestigious academic institute in Greece in the chair of Theoretical and Experimental Mechanics (2016). He received his Diploma in Civil Engineering (1971) and Ph.D. (1973) from the National Technical University of Athens (NTUA). He served as Instructor in the Chair of Theoretical and Applied Mechanics of NTUA (1974–1977), Chair Professor (1977–2015) of Applied Mechanics of the Democritus University of Thrace (DUTH), Greece, Chairman of the Department of Civil Engineering (1987–1989) of DUTH, Chair of the Division of Natural Sciences of the Academy of Athens (2019), Visiting Professor at the universities of Toledo, Lehigh, Michigan Technological University, University of California at Santa Barbara and Davis, Northwestern University, Clark Millikan Distinguished Visiting Professor at the California Institute of Technology, Honorary Adjunct Professor of the Hebei University of Engineering of the People’s Republic of China. He is Member of the European Academy of Sciences and Arts (2001), European Academy of Sciences (2008), Academia Europaea (2008), International Academy of Engineering (2010), Fellow of the American Academy of Mechanics (2007) and the New York Academy of Sciences (2001), Foreign Member of the Russian Academy of Engineering (2009), the Bulgarian Academy of Sciences (2010), the Russian Academy of Sciences (2016) and the Ukrainian Academy of Sciences (2021). He is Honorary Member of the Italian Society of Fracture Mechanics (2004), the Polish (2009) and xix

xx

About the Authors

the Serbian (2011) Societies of Theoretical and Applied Mechanics, Doctorate Honoris Causa of the University of Nis of Serbia (2013), the Russian Academy of Sciences (2010) and DUTH (2019). He is Fellow of the American Society of Mechanical Engineers (1993), the Society for Experimental Mechanics (2004), the European Structural Integrity Society (2008), the European Society for Experimental Mechanics (2010), the International Congress on Fracture (2009) and the American Association for the Advancement of Science (2012). He is Honorary Member of the Literary Society “Parnassos”, Greece (2020). He served as President of the Society of Experimental Mechanics (2013–2014), the European Structural Integrity Society (2006–2010), the European Society for Experimental Mechanics (2004– 2007) and the International Congress on Fracture (2023– 2027). He served as President of the Hellenic Society of Linguistic Heritage (2018–2022) and the Theocaris Foundation of the Academy of Athens (2018–now). He is a recipient of many awards including: “Award of Merit” (2008), “Griffith Medal” (2010) of the European Structural Integrity Society, “Theocaris Award” (2009), “Lazan Award” (2009), “Tatnall Award” (2010) “Zandman Award” (2011) “Nemat-Nasser Award” (2023) of the Society of Experimental Mechanics, Medal and Diploma of the “International Academic Rating of Popularity Golden Fortune” (2009), “Paton Medal” of the Ukrainian Academy of Sciences (2009), “Jubilee Medal XV Year IAE” of the International Academy of Engineering (2009), “Award of Merit” (2010), “Theocaris Award” (2012) of the European Society for Experimental Mechanics, “Golden Sign” of the Russian Academy of Engineering (2011), SAGE Best Paper Award (2012), “Panetti-Ferrari Prize” of the Turin Academy of Sciences (2012), “Colonnetti Gold Medal” of the Italian Research Institute of Metrology (2012), “Blaise Pascal Medal in Engineering 2018”, of the European Academy of Sciences, “Yokobori Medal” of the International Congress of Fracture (2017), “Archon, Teacher of the Nation” of the Patriarchate of Alexandria and All Africa (2018), “Award of the Rotary Club of Mytilini” (2019), Greece, “Timoshenko Mechanics Lecture”, Louisiana State University, 2020, “Yanzhao Friendship Award” of the Hebei Province of the People’s Republic of China (2022).

About the Authors

xxi

He served as President of many national and international conferences. He published more than 130 papers in international scientific journals and more than 180 in conference proceedings. He is Author of 9 books in Greek, 6 books in English published by international publishers (Springer, Elsevier, Kluwer Academic Publishers, Martinus Nijhoff Publishers) and Editor of 24 books published by the above publishers. His book Fracture Mechanics-An Introduction, 3rd ed by Springer accompanied by “Solutions Manual”, is used as a textbook by many universities worldwide. He served Editor-in-Chief of Strain-An International Journal of Experimental Mechanics, (2007–2010), Guest Editor of special issues of international journals, Associate Editor and Member of the editorial board of international scientific journals. He is Editor of the series of SpringerBriefs in Structural Mechanics of Springer. He presented many plenary/invited lectures in conferences/ universities. His research interests include problems of the theory of elasticity, use of complex functions for the solution of singularity problems of mechanics, fracture mechanics, experimental mechanics (with emphasis in the optical methods), mechanics of composite materials, sandwich structures and nanotechnology (composite nanomaterials). In 2015 the journal Meccanica devoted a special issue in his honor. He is honorary citizen of the municipality of Lesvos (Greece). Emmanuel is married to Prof. Maria KonstaGdoutos. They have two children, a son Eleftherios, 36, who graduated from Northwestern University with BS Diplomas in Mechanical Engineering and Computer Science and received a Doctorate Degree in Aeronautics from Caltech, and a daughter Alexandra-Kalliopi, 32, who graduated from Northwestern University with a BS Diploma in Civil Engineering and earned an MBA from Kellogg School of Management. Eleftherios has currently a startup company in Los Angeles and Alexandra-Kalliopi has a consultant appointment in a major company in USA.

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About the Authors

Maria Konsta-Gdoutos is a leading scholar on concrete nanotechnology, Professor of Civil Engineering at the University of Texas at Arlington (UTA), is the cofounder and associate director of UTA’s Center for Advanced Construction Materials (CACM). She serves as Director of the US Department of Transportation University Center for Durable and Resilient Transportation Infrastructure for 2023–2028. Her research interests include the development of multifunctional smart engineered construction materials, the study of the microstructure and structural performance of advanced nanoengineered cement-based composites, exploration of the electrochemical, and surface and interface phenomena at all scales, with focus on enhancing the durability of engineered structural materials and infrastructure. She holds three US patents on nanoreinforced concrete for applications with novel functionalities. Other research interests include the design of high modulus concrete for super tall buildings; smart concrete for sensing applications in SHM; carbon-neutral concrete and novel mineralization pathways in concrete for large-scale CO2 sequestration; design of novel technologies for renewable electricity and power produced by engineering concrete’s nanostructured interfaces; and concrete’s reuse/recycling to promote circular economy. Maria is a fulbright scholar; fellow of the European Academy of Sciences and the European Academy of Arts and Sciences; member of the International Academy of Engineering; and the recipient of the A. S. Kobayashi Young Investigator Award in Experimental Science in 2012. She has published over 120 publications in peer-reviewed journals and international conference proceedings. She continues to advance the state of the art through her leadership in the American Concrete Institute, ACI 241-TG2—Nanoscale Fiber Reinforced Concrete (Chair) and ACI 236—Materials Science of Concrete; and the RILEM Technical Committee—Carbon-based Nanomaterials for Multifunctional Cementitious Matrices. She is Executive Editor-in-Chief of the Springer International Journal Frontiers in Structural and Civil Engineering. She also serves as Associate Editor of the leading peer-reviewed journal Cement and Concrete Composites by Elsevier; and the Journal of Materials in Civil Engineering by the American Society of Civil Engineers.

Chapter 1

Tensile Testing

Abstract Tensile or tension testing is a fundamental and most commonly used test for the characterization of the mechanical behavior of materials. The test consists of pulling a sample of material and measuring the load and the corresponding elongation. Main properties measured from the test include Young’s modulus, yield, ultimate and breaking strength, ductility, resilience, tensile toughness and Poisson’s ratio. The tensile properties are used in selecting materials for engineering applications and in research and development of new materials or processes. In this chapter we will present the basic characteristics of the time-independent tensile test. More specifically, we will discuss the testing procedure, the specimen characteristics, the engineering and true stress–strain curves, the yield and strain hardening behavior, the hysteresis loop in plastic deformation, the Bauschinger effect, the state of stress in the neck area, the approximate McGregor method, idealized and analytical expressions of stress–strain curves, the definitions of ductility, resilience and tensile toughness, the Poisson’s ratio, the bulk modulus, and standards for the tensile test. Finally, we will present values of material properties obtained from the tension test for some engineering metals and polymers.

1.1 Specimens and Testing In the time-independent tensile test a sample of the material called specimen is slowly pulled with axial force until it breaks. The specimen may have a circular or a rectangular cross section and its ends or shoulders are enlarged to provide extra area for gripping (Fig. 1.1). The state of stress in the gripped area is three-dimensional. It becomes uniform uniaxial tension at a distance from the grips, according to Saint Venant’s principle. The gage section of the specimen where measurement of displacements is made has a uniform cross section smaller than that of the shoulders. The gage section is long compared with its diameter or its depth (typically four times the diameter). The length of the transition region between the gage section and the shoulders should be at least as the diameter or the depth of the specimen. Detailed description of standard test specimens is given by the American Society for Testing

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_1

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Fig. 1.1 Typical cylindrical tensile specimen with reduced gage length and enlarged shoulders. To avoid end effects from the shoulders the length x of the transition region should be at least as great as the diameter d and the gage length of reduced section should be at least four times the diameter

and Materials standard ASTM E8 or other standards organizations, like the International Organization for Standardization (ISO), the Deutsche Institut für Normung (DIN) and the Japanese Industrial Standards (JIS). There are various ways of gripping the specimen at its ends. The ends may be screwed into a threaded grip, be pinned, butt ends may be used, or be held between wedges. The selection of the gripping method is to ensure that the specimen can be held at the maximum load without slippage or failure in the grip section and bending should be minimized. Tensile testing is usually carried out in a testing machine at a materials testing laboratory. Testing machines are either electromechanical or hydraulic. The main difference is the method by which the load is applied. Electromechanical machines use a variable-speed electric motor, a gear reduction system and one, two, or four screws that move the crosshead up and down. A closed-loop servosystem can be implemented to accurately control the speed of the crosshead. Hydraulic machines usually use a single-acting piston that moves the crosshead up or down. The rate of loading in manually operated machines is controlled by the orifice of a pressure-compensated needle valve, whereas in a closed-loop hydraulic servosystem the needle valve is replaced by an electrically operated servovalve for precise control.

1.2 Stress and Strain In a tensile test the specimen is mounted in a testing machine and it is subjected to tension. The tensile load P is recorded as a function of the increase of the gage length. Usually, the specimen is deformed at constant speed. If h is the distance between the fixed and moving crossheads of the universal testing machine the distance h is varied with time t so that dh = constant dt Displacements in the specimen are measured in the gage section of constant cross section over a gage length L. In case when all grip parts and the specimen ends are nearly rigid it is reasonable to assume that the crosshead motion is due to the deformation within the gage section, so that ΔL is approximately equal to Δh.

1.2 Stress and Strain

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However, in most cases measurement of ΔL is made. A plot of tensile force P versus tensile elongation ΔL is of no value since it depends on the cross-sectional area A and the gage length L. To make the force P independent of the area A and the elongation ΔL independent of the length L, we introduce the concepts of stress and strain. The engineering stress σ of a tension specimen is defined as σ =

P Ai

(1.1)

where P is the tensile load and Ai is the initial cross-sectional area. Stress is measured in N/m2 (Pa = Pascal), where N stands for Newton and m for meter. One Pascal is the stress exerted by a force of magnitude of one Newton perpendicularly upon an area of one square meter. Pascal is a very small unit, so stress is usually measured in kPa (103 Pa) or MPa (106 Pa). As the specimen is stretched the area Ai decreases. In this respect, we define the true stress σ˜ as σ˜ =

P , A

(1.2)

where A is the current (reduced) cross-sectional area (A < Ai ) that corresponds to load P. From Eqs. (1.1) and (1.2) it is deduced that σ˜ = σ

Aι . A

(1.3)

Equation (1.3) indicates that σ˜ > σ , that is, the true stress is higher than the engineering stress as the load P increases and the cross-sectional area A of the specimen decreases. Displacements in the specimen are measured in the gage section over an original length L i . The engineering strain ε is defined as ε=

L − Li ΔL L = = − 1, Li Li Li

(1.4)

where L is the current length at load P and ΔL is the change of the original length L i. Strain is a dimensionless quantity. Strains are sometimes given as percentages, where ε% = 100 ε. Strains are also expressed in millionths, called microstrain εμ , where εμ = 106 ε. Equation (1.4) can be expressed as ∫L ε= Li

dL L − Li = Li Li

(1.5)

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Note that the engineering strain is referred to the original length L i of the gage section. Unlike the engineering strain, ε, a different definition of strain called true strain (or natural or logarithmic strain) ε˜ is also used. It is associated with the instantaneous change of the specimen length while the force is acting on it and refers to the current specimen length. The length change ΔL (= L − L i = ΔL 1 + ΔL 2 + ΔL 3 + · · ·), where L is the final length and L i is the original length, is measured in small increments ΔL 1 , ΔL 2 , ΔL 3 , … at gage lengths L 1 , L 2 , L 3 , …, respectively. The total strain is L1 − Li L2 − L1 L3 − L2 + + + ··· L1 L2 L3 ∑ ΔL j ΔL 1 ΔL 2 ΔL 3 = + + + ··· = L1 L2 L3 Lj

ε˜ =

(1.6)

Note in Eq. (1.6) that the length increments ΔL 1 , ΔL 2 , ΔL 3 , … are normalized to the current lengths, L 1 , L 2 , L 3 , … The true strain ε˜ is referred to the current length, as opposed to the engineering strain which is referred to the original length of an element. Note that the engineering strain in tension that refers to the initial length is higher than the true strain that refers to the final (bigger) length. Quite the opposite happens in compression. If ΔL j ( j = 1, 2, 3, …) are assumed to be infinitesimal we obtain from Eq. (1.6) ∫L ε˜ =

L dL = ln L Li

(1.7)

Li

From Eq. (1.7) we obtain ε˜ = ln

ε3 ε4 L L i + ΔL ε2 + − + ··· = ln = ln(1 + ε) = ε − Li Li 2 3 4

(1.8)

Equation (1.8) relates the true strain ε˜ to the engineering strain ε. It indicates that for small strains the engineering and true strains are nearly equal (˜ε ∼ ε). Consider an element that takes the lengths L 0 , L 1 , L 2 , … L n . The true strains ε˜ 1 , ε˜ 2 , …, ε˜ n during the steps (L 0 , L 1 ), (L 1 , L 2 ), … (L n−1 , L n ), are ε˜ 1 = ln

L1 L2 Ln , ε˜ 2 = ln , . . . ε˜ n = ln L0 L1 L n−1

The true strain ε˜ during the step (L 0 , L n ) is ε˜ = ln

) ( L1 Ln L1 L2 Ln L2 Ln = ln = ln ... + ln + · · · + ln L0 L0 L1 L n−1 L0 L1 L n−1

(1.9)

1.3 Engineering Stress–Strain Curves

= ε˜ 1 + ε˜ 2 + · · · + ε˜ n

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(1.10)

which indicates that for a deformation consisting of several steps, the overall true strain is the sum of the true strains of the various individual steps. That is, the true strain is an additive quantity. For the engineering strain we have ε=

Ln − L0 L1 − L0 L2 − L1 L n − L n−1 /= + ··· + = ε1 + ε2 + · · · + εn L0 L0 L1 L n−1 (1.11)

that is, the engineering strain is not an additive quantity.

1.3 Engineering Stress–Strain Curves From the tension test a graph is made of the engineering stress σ versus the engineering strain ε. This graph is called engineering stress–strain curve. The stress and strain are plotted in a Cartesian coordinate system with the stress σ plotted in the ordinate axis and the strain ε plotted in the abscissa axis. Stress–strain curves vary widely for different materials. They are a fundamental characterization of the mechanical behavior of a material. Figure 1.2 presents the initial engineering stress–strain curves of three different types of material behavior. In Fig. 1.2a, b there is an initial linear part. In Fig. 1.2a the curve after the initial linear part rises continually. In Fig. 1.2b after the initial linear part there is a drop in stress, after which the stress rises again. Finally, in Fig. 1.2c there is no initial linear part and the stress rises continuously.

Fig. 1.2 Initial stress–strain curves of three different types of material behavior: a there is an initial linear part and the curve rises continually, b there is an initial linear part and a drop at the yield point and c there is no linear part

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Fig. 1.3 Stress–strain curve of many steel alloys and metals

We will now study in more detail the complete stress–strain curve of Fig. 1.2b up to the point of fracture. The curve is common in many steel alloys and metals. We can distinguish the following parts of the curve (Fig. 1.3): Part OA: It represents the initial part of the stress–strain curve. The stress σ varies linearly with strain ε and Hooke’s law applies as σ = Eε

(1.12)

where E is the Young’s modulus or modulus of elasticity. E is calculated from the slope (tangent of the angle between the line OA and the ε-axis) of the line OA. Otherwise, E can be calculated from the stresses and strains at two points on the line OA, such as M and N in Fig. 1.3 as E=

σN − σM . εN − εM

(1.13)

The point A up to which the stress varies linearly with strain is called proportional limit. Its experimental determination depends on the accuracy of the measuring device of deformations. Part AE: It follows part OA. The relation between stress and strain is not linear. The material, as in the part OA, behaves elastically, that is, upon unloading it returns to its initial shape (there is no permanent deformation, the unloading curve follows the loading curve, line OE). The point E is called elastic limit. Its experimental

1.3 Engineering Stress–Strain Curves

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determination is not accurate. It depends on the accuracy of the measuring device of deformations. A periodic loading–unloading is needed to check for permanent deformation. Part ED: It follows part AE. Part ED is usually very small. The deformations start to increase rapidly for small increases of the stress. The point D after which the resulting deformations are not recoverable upon unloading is called yield point. After the yield point the deformations are irreversible, that is, the material does not return to its initial shape. Permanent deformations occur. The experimental determination of the yield point is not accurate and depends on the accuracy of the measuring device of deformations. In most metals and alloys point D is near point E. The distinction of points D and E depends on the sensitivity of the measuring device of deformations. Part DF: It follows part ED. The strain increases rapidly with small increases of stress. The stress–strain curve rises up to a point F at which the maximum stress the material can endure occurs. Up to point F the stress–strain curve is rising. The tangent of the stress–strain curve at point F is horizontal. Beyond point F the engineering stress decreases and failure occurs. The point F is called ultimate or maximum stress point. Part FS: It follows part DF. The stress decreases with strain. Material separation takes place at point S. This point is called fracture or breaking point and the corresponding stress fracture or breaking stress. Based on the above, the following quantities are determined from the tension test. a. The modulus of elasticity E. It is the slope of the initial linear part of the stress–strain curve. b. The proportional limit (point A). Up to this point the stress varies linearly with strain. c. The elastic limit (point E). Up to this point the deformation is elastic. The material returns to its initial shape after unloading. d. The yield stress (point D). After this point the deformation is not reversible. e. The ultimate or maximum stress (point F). The engineering stress becomes maximum. The engineering stress–strain curve at this point has a horizontal tangent. The engineering stress decreases after this point. f. The fracture or breaking stress (point S). Material separation occurs. In most metals and alloys the proportional limit A, the elastic limit E and the yield point D are very close to each other. When the stress–strain curve does not have a well-defined linear part a tangent modulus, E t , is defined by the slope of the tangent to the stress–strain curve at the origin (Fig. 1.2c).

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1.4 Yield Point The yield point of the engineering stress–strain curve is a material characteristic property. It indicates the end of the elastic and the start of the plastic behavior. If the applied stress is higher than the yield stress the material is irreversibly or plastically deformed. Components or constructions cannot safely be used if the yield point is exceeded. In some engineering metals the yield point can be completely identified since the stress drops abruptly at that point (Fig. 1.4). Often, an upper (D1 ) and a lower (D2 ) yield point can be identified. The upper yield point is the highest stress value before its significant drop. Following this point the material begins to flow, the stress decreases slightly and the elongation continues to increase. The lowest tensile stress during flow corresponds to the lower yield point, D2 . This behavior where the stress–strain curve presents an upper and a lower yield point occurs mainly in mild steels. The upper yield point is defined by the tensile standard ISO 6892-1 for metals as follows: “After reaching the stress maximum, there must be a stress reduction of at least 0.5% and a subsequent flow of at least 0.05% without the tensile stress exceeding the upper yield point again”. In mild steels it can be observed that the stress remains constant after the lower yield point for an increase of the strain of the order of 4–5% (segment D2 D,2 ) until strain hardening begins (Fig. 1.5). After the lower yield point in annealed low-carbon or mild steels all of the deformation occurs within a relatively small region of the specimen. Plastic zones nucleate and propagate in the specimen. They are called Lüders bands or slip bands. Elongation of the specimen from the upper to the lower yield point and beyond occurs by

Fig. 1.4 Stress–strain curve with upper, D1 , and lower, D2 , yield points

1.4 Yield Point

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Fig. 1.5 Typical stress–strain curve of mild steels. The stress remains constant after the lower yield point (segment D2 D,2 ) until hardening begins

propagation of the Lüders bands. During the propagation of the bands the engineering stress–strain curve is flat (segment D2 D,2 of Fig. 1.5). Only after the bands propagate through the entire specimen length the deformation proceeds uniformly and the stress rises again. The stress–strain curve between the upper and lower yield points presents a noisy pattern. The Lüders bands usually start at one end of the specimen and propagate toward the other end. The front of the bands is generally at 45° to the loading direction because local plastic deformation occurs when the maximum shear stress is exceeded. Typically the bands make an angle of 50–55° with the specimen axis. The strain field within the Lüders region is strongly nonuniform. A thorough study of the strain distribution in the Lüders bands was performed by Theocaris and Koroneos [10, 11] by using the moiré method. The upper yield point is connected with the initiation of Lüders bands. In this respect, it is sensitive to testing rate, small amounts of possible bending and its values vary considerably. On the other hand, the lower yield point does not depend on such conditions and it is easily reproduced. This point is the most satisfactory means of defining the yielding. The lower yield point is considered a material property. In some materials including aluminum and cold-rolled or cold-formed materials the yield stress at which the material changes from elastic to plastic behavior is not easily detected. The stress–strain curve is continuous. In such cases an offset yield stress is arbitrarily defined. The most widely used and standardized offset strain for engineering materials is the 0.0002 (0.2%) plastic strain. The yield stress is determined from the stress–strain curve of the material by drawing a straight line parallel to the initial linear part of the stress–strain curve from the point of 0.2% strain. The point this line intersects the stress–strain curve is defined as the 0.2% offset yield point and the corresponding stress as the 0.2% offset yield stress (Fig. 1.6).

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Fig. 1.6 Definition of the 0.2% offset yield stress in a gradually increasing stress–strain curve without a drop yield point

1.5 Strain Hardening After the yield point of the engineering stress–strain curve the stress increases continually with the increase of plastic strain up to a plateau point of maximum stress. The increase of stress with increasing strain is called strain hardening, or work hardening or cold hardening. The stage of the stress–strain curve in which this occurs is called strain hardening region (part DF of the stress–strain curve of Fig. 1.3). This region starts from the yield point D and ends at the ultimate (maximum) stress point F. It is called cold-working because the plastic deformation must occur at a temperature low enough that atoms cannot rearrange themselves. It is a process of making a metal harder and stronger through plastic deformation. As we proceed from the yield point D to the point of the ultimate stress F the slope of the stress–strain curve decreases and smaller stress is needed to maintain a constant increase of strain. The hardening of the material decreases and the slope of the stress–strain curve becomes zero (the tangent of the curve is parallel to the strain axis). At this point the engineering stress reaches a maximum (Fig. 1.7). The maximum tensile strength σ u is called maximum, or ultimate tensile strength or simply tensile strength. It is the highest stress reached at any point during the tensile test. It is obtained by σu =

Pmax Ai

(1.14)

1.5 Strain Hardening

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Fig. 1.7 Engineering stress–engineering strain curve. The curve at the maximum load reaches a plateau and after that it decreases with increasing strain

where Pmax is the maximum load and Ai is the original cross-sectional area. The condition of instability at the ultimate tensile strength is dP = 0

(1.15)

d(Aσ˜ ) = σ˜ dA + Adσ˜ = 0

(1.16)

dA dσ˜ =− σ˜ A

(1.17)

or

or

where σ˜ is the true stress and A is the current cross section of the tension specimen. Assuming constancy of volume V (= AL) in plastic deformation we obtain dV = d(AL) = LdA + AdL = 0

(1.18)

or −

dL dA = A L

(1.19)

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From Eq. (1.7) we obtain: ) ( L i dL L dL = d˜ε = d ln = Li L Li L

(1.20)

From Eqs. (1.17), (1.19) and (1.20) we obtain dσ˜ = d˜ε σ˜

(1.21)

dσ˜ = σ˜ . d˜ε

(1.22)

or

Equation (1.22) indicates that the slope of the true stress–true strain curve at the ultimate tensile strength (maximum load) is equal to the true stress (Fig. 1.8). From Eq. (1.8) we obtain d˜ε =

dε . 1+ε

(1.23)

Fig. 1.8 Slope of the true stress–true strain curve at the maximum load is equal to the true stress

1.5 Strain Hardening

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Fig. 1.9 True stress–engineering strain curve. The maximum load corresponds to the point of the tangent to the curve that starts from the point − 1 of the axis of the engineering strain ε

Then, Eqs. (1.22) and (1.23) render σ˜ dσ˜ = . dε 1+ε

(1.24)

Equation (1.24) indicates that the maximum load corresponds to the point of the tangent to the true stress–engineering strain curve, (σ˜ − ε), that starts from the point − 1 of the axis of the engineering strain ε (Fig. 1.9). The condition of volume constancy in plastic deformation can be expressed as AL = Ai L i

(1.25)

where A, L are the cross-sectional area and gage length of the deformed tension specimen and Ai , L i are the corresponding quantities of the undeformed specimen. Using Eq. (1.25) we obtain from Eq. (1.8) ε˜ = ln Equation (1.26) renders

L Ai = ln(1 + ε) = ln Li A

(1.26)

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Ai =1+ε A The true stress σ˜ is obtained from Eqs. (1.3) and (1.26) as σ˜ = σ

Aι = σ (1 + ε) A

(1.27)

Equation (1.27) relates the true stress σ˜ to the engineering stress σ via the engineering strain ε under conditions of volume constancy. For a cylindrical specimen Eq. (1.26) renders (π ) 2 D Ai Di = ln ( π4 ) i2 = 2 ln ε˜ = ln A D D 4

(1.28)

where Di is the original and D is the current cross-sectional diameter. Equation (1.28) expresses the true strain in terms of the original and the current diameters of a cylindrical specimen under conditions of volume constancy. It has been observed experimentally that for specimens with orthogonal cross section the reduction of the cross-sectional area is smaller than that of cylindrical specimens. The lateral surfaces of the specimens are not maintained plane, but they are hollowed inwards (Fig. 1.10). The cross-sectional area can be approximated by the inscribed orthogonal of area, A = a, t , .

Fig. 1.10 Lateral surfaces of a specimen with orthogonal cross section are not maintained plane but they are hollowed inwards

1.6 Hysteresis Loop in Plastic Deformation

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1.6 Hysteresis Loop in Plastic Deformation Let us consider that at some point S in the plastic region of the stress–strain curve the load is removed. At first the deformation is elastic and the unloading path SK, is parallel to the linear region OA of the stress–strain curve (Fig. 1.11). When the applied load is completely removed a plastic strain of magnitude OO, remains. If the specimen is reloaded the strain varies linearly with stress up to a new proportional limit A, followed with a nonlinear elastic part up to the new yield point D, . The yield stress at point D, is higher than the original yield stress at point D. As the load increases the stress–strain curve follows the path D, S, , which is a continuation of the original path of the stress–strain curve, as if the unloading cycle has not existed. During the unloading and reloading cycle a hysteresis loop enclosed by the area (O, A, D, KK, O, ) is formed. The hysteresis loop represents the thermal energy dissipated during the unloading and reloading cycle. The right and left boundaries of the hysteresis loop are parallel to the linear part of the original stress–strain curve.

Fig. 1.11 Stress–strain curve of a material with plastic deformation. During the unloading and reloading cycle a hysteresis loop enclosed by the area (O, A, D, KK, O, ) is formed

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1.7 Bauschinger Effect Consider now that after a point in the plastic region of the stress–strain curve the applied tensile load is removed and a compressive load is applied (Fig. 1.12). In many steel and metal alloys it is observed that the yield stress under compression is lower than that in tension. On the contrary, as it was discussed previously, the yield stress under tension is higher under unloading and reloading than that of tension. This behavior is called Bauschinger effect. It may be stated as follows: if a metallic specimen is subjected to a tensile load in the plastic region its strength increases, while when it is subjected to a compressive load its strength decreases. The kinematic theory of plasticity follows the Bauschinger effect. It predicts that yielding in the reverse direction occurs when the stress change from the unloading point in the plastic region is twice the monotonic yield strength, Δσ = 2σ 0 . On the other hand, isotropic hardening predicts that yielding occurs at Δσ = 2σ , , where σ , is the highest stress reached prior to unloading in the plastic region. Fig. 1.12 Kinematic and isotropic hardening. The kinematic hardening follows the Bauschinger effect

1.9 Stress State in the Neck Area

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1.8 True Stress–True Strain Curve As we mentioned previously, the engineering stress, σ, and strain, ε, do not give a true indication of the deformation characteristics of a material because they are calculated from the dimensions of the undeformed specimen, which change continuously during the test. The engineering stress and strain approximate the state of stress and deformation in situations where there is no significant change in specimen size. When large changes occur during deformation a more realistic description of the stress–strain behavior of the material is provided by the true values of stress and strain, σ˜ and ε˜ . Ductile metals in tension become unstable at the maximum load and neck down. The load falls because the cross-sectional area of the specimen decreases rapidly. The engineering stress based on the original area decreases beyond the maximum load. Actually, the material continues to strain-harden up to fracture, and the stress required to produce further deformation actually increases. The true stress σ˜ , and the true strain, ε˜ can be calculated from the engineering stress σ and engineering strain ε from Eqs. (1.8) and (1.27), respectively. Note that Eq. (1.8) results from the definition of engineering and true strain, while Eq. (1.27) is based on the constancy of volume. A graphical plot of the true stress, σ˜ , versus true strain, ε˜ , is the true stress–true strain curve (Fig. 1.13). The stress σ˜ is plotted in the ordinate and the strain ε˜ , in the abscissa of an orthogonal system, as in the case of the engineering stress–engineering strain curve. The true stress–true strain curve increases continuously up to fracture. Note that the true stress is higher than the engineering stress at fracture (Eq. 1.27).

1.9 Stress State in the Neck Area During a tensile test of a cylindrical specimen a point is reached at which strains increase locally as a result of reduction of the cross-sectional area of the specimen. All further deformation is concentrated in a localized area and extensive extension is associated with thinning of the specimen. This phenomenon is termed necking because of the analogy of a thinned cylindrical specimen to the human neck. Necking generally begins at the maximum load of the engineering stress–strain curve, where the increase in stress due to decrease of the cross-sectional area becomes greater than the increase of the load-carrying capacity of the specimen due to strain hardening. Necking continues until fracture takes place. The stability of the material during tensile testing is dictated by the competition between strain hardening and necking of the specimen. After the onset of necking the state of stress in the specimen is not uniaxial tension. A triaxial state of stress applies in the necking area. A tensile hoop stress is developed around the circumference in the necking region. For the description of the true stress–strain curve in cylindrical specimens Bridgman’s analytical model [2] is

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Fig. 1.13 True and engineering stress–strain curves. The true stress–strain curve increases continuously up to fracture

the most widely accepted method. The equivalent (effective) true stress σ equ at the minimum cross section is given by σequ 1 ) ( , =( 2R a ) σav 1+ ln 1 + a 2R

(1.29)

where σav = P/A is the true stress, A is the instantaneous cross-sectional area corresponding to the load P, R is the radius of curvature of the neck profile and a is the radius of the specimen at the base of the neck (Fig. 1.14). Application of Eq. (1.29) requires measurement R and a. The variation of σ equ / σav versus a/R is shown in Fig. 1.14. Bridgman concluded from experimental results that the ratio a/R depends only on the reduction of the cross-sectional area. To overcome measurement of R and a he gave the following empirical relation for the calculation of the ratio a/R, as [ ( ) ]0.5 Ai a = ln − 0.1 , R A

(1.30)

where Ai is the original cross section and A is the current cross section of the cylindrical specimen.

1.10 Approximate McGregor Method for Determining the Stress–Strain Curve

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Fig. 1.14 Variation of the ratio of the effective stress to the axial true stress σ equ /σav versus a/R

Another equation for the calculation of the ratio σequ /σav was given by Davidenkov and Spiridonova [4] as σequ 1 =( a ). σav 1+ 4R

(1.31)

1.10 Approximate McGregor Method for Determining the Stress–Strain Curve The two-load McGregor method is used with good results for the approximate determination of the true stress–true strain curve of a material. The method is based on the use of a cylindrical tension specimen whose cross section varies radially from the ends (diameter Dmax ) to the mid-section (diameter Dmin ) (Fig. 1.15). For this specimen different stresses along its length are obtained. The diameters of the specimen are measured at various cross sections before and after the test. During the test, the maximum load, Pmax , and the fracture load, Pf , are measured. The true stresses σ˜ and true strains ε˜ at different cross sections are determined using Eqs. (1.2) and (1.28) as σ˜ =

4Pmax Di , ε˜ = 2 ln 2 πD D

where Di is the initial and D is the current diameter of the specimen.

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Fig. 1.15 True stress–true strain curve according to the McGregor approximate method using a cylindrical tension specimen whose cross-section varies radially from the ends (diameter Dmax ) to its mid-section (diameter Dmin )

Using the above equations the true stress–true strain curve up to point F of the maximum load is determined (Fig. 1.15). The stress at point F corresponds to the maximum load, Pmax and the minimum cross section. The stresses at other points of the curve before point F correspond to the maximum load and other cross sections (with greater area than the minimum cross-section), from the mid- to the end-section (Fig. 1.15). The stress at the end point G of the curve corresponds to the fracture load, Pf , and the minimum cross section. Between points F and G the stress–strain curve is taken linear. This linear behavior has been established experimentally in many steel and metal alloys. The McGregor method is simple and speedy. No measurements are needed during the test. The maximum and fracture loads, and the diameters at various cross sections of the specimen after the test are measured. Only the part of the true stress–strain curve after the yield point up to fracture is obtained. The linear elastic part of the curve is not obtained.

1.11 Idealized Stress–Strain Curves

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1.11 Idealized Stress–Strain Curves Stress–strain curves for different materials take different forms. They can be classified into the following idealized forms: a. b. c. d.

Linear elastic (Fig. 1.16a) Nonlinear elastic (Fig. 1.16b) Linear-elastic–perfectly plastic (Fig. 1.17a) Linear-elastic–linear hardening (Fig. 1.17b).

Figure 1.16a shows a linear elastic stress–strain curve. The unloading path of the curve coincides with the loading path. The stress–strain relationship is dictated by Eq. (1.12). Figure 1.16b shows a nonlinear elastic stress–strain curve. The unloading path coincides with the loading path. Figure 1.17a shows an actual and the idealized linear-elastic–perfectly plastic stress–strain curve. It is composed of a linear elastic part and a horizontal plastic part. The stress in the plastic region is constant. The unloading curve from some point in the plastic region is parallel to the linear elastic curve. The strain in the plastic region consists of an elastic part and a nonrecoverable plastic part. The stress–strain relationship is given by σ = Eε σ = σ0

(σ ( ≤ σ0 ) σ ) 0 ε ≥ ε0 = E

Fig. 1.16 Idealized stress–strain curves: a linear elastic, b nonlinear elastic

(1.32)

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1 Tensile Testing

Fig. 1.17 Idealized stress–strain curves: a linear elastic–perfectly plastic, b linear elastic–linear hardening

where σ 0 is the yield strength and E is Young’s modulus. Beyond yielding, the strain is the sum of elastic ε0 (at yield point) and plastic εp parts: ε = ε0 + εp =

( σ0 ) σ0 + εp ε ≥ E E

(1.33)

Figure 1.17b shows an actual stress–strain curve and the corresponding idealized linear elastic–linear hardening curve. This idealization is useful as an approximation of stress–strain curves that rise appreciably after yielding. The stress in the plastic region varies linearly with the plastic strain. The unloading line from some point in the plastic region is parallel to the linear elastic line. The strain in the plastic region consists of an elastic part and a nonrecoverable plastic part. The relationship between stress and strain requires two elastic constants, the modulus of elasticity E in the linear elastic region and the slope of the line following yielding, δE, where δ is the reduction factor for the slope after yielding (δ = 1 for linear elastic and δ = 0 for elastic–perfectly plastic behavior). δE is given by δE =

σ − σ0 ε − ε0

(1.34)

where σ0 and ε0 are the stress and strain at the yield point (σ0 = Eε0 ) and σ and ε are the stress and strain at a point of the straight line after yielding. The stress–strain relationship is given by σ = Eε σ = σ0 + δ E(ε − ε0 ) = (1 − δ)σ0 + δ Eε

(σ ≤ σ0 ) (σ ≥ σ0 )

(1.35)

1.12 Analytical Expression of Stress–Strain Curves

23

Solving for strain the second equation we obtain for the strain in the plastic region ε=

σ − σ0 σ0 + (σ ≥ σ0 ) E δE

(1.36)

1.12 Analytical Expression of Stress–Strain Curves Stress–strain curves of real materials generally require a more complex mathematical formulation than the two models of linear elastic–perfectly plastic and linear elastic–linear hardening behavior presented in the previous section. A power law relationship is sometimes used to relate stress and strain in the plastic region. The true stress–true strain behavior is described by σ = Eε (σ ≤ σ0 )

(1.37a)

σ = Hl εn 1 (σ ≥ σ0 )

(1.37b)

where the term n1 is called strain hardening exponent and H 1 is a strength coefficient. Both n1 and H 1 are considered material constants. The value of the strain hardening exponent lies between 0 and 1, with a value of 0 implying a perfect plastic solid and a value of 1 representing a perfectly elastic solid. Most metals have values of n1 between 0.10 and 0.50. H 1 can be found by extrapolating to ε = 1. High-strength materials have lower n1 -values than lower-strength materials. Note that ε is the total (elastic plus plastic) strain. Solving Eq. (1.37b) for strain ε we obtain ( ε=

σ H1

)1/n 1 (σ ≥ σ0 )

(1.38)

By applying both Eqs. (1.37a) and (1.37b) at the yield point (σ0 , ε0 ) and eliminating ε0 we obtain (

H1 σ0 = E E

)1/(1−n 1 )

.

(1.39)

Equation (1.39) relates the material constants σ0 , E, H 1 and n1 . From Eq. (1.37b) we obtain log σ = log H1 + n 1 log ε

(1.40)

24

1 Tensile Testing

Fig. 1.18 Stress–strain curves in linear (a) and logarithmic (b) coordinates for an elastic power– hardening relationship

which indicates a linear log σ versus log ε relationship with slope n1 in the plastic region. From Eq. (1.37a) we obtain log σ = log E + log ε

(1.41)

which indicates that in a log–log plot the linear part of the stress–strain curve has a slope of unity. Figure 1.18 presents the stress–strain curves in linear (a) and logarithmic coordinates (b) of an elastic power hardening material whose stress–strain relation is dictated by Eq. (1.37a, b). Note that in log–log plot the stress–strain curve in the linear region has a slope of unity, while in the plastic region it has a slope of n1 . An exponential relationship between stress and strain similar to the above has been proposed by Ramberg and Osgood. The relationship applies for the stress σ and strain εp in the plastic region as σ = H εpn

(1.42)

or εp =

( σ )1/n H

(1.43)

The above equation is called the Ramberg–Osgood equation. The constant n is called, as in the previous case, strain hardening exponent. Note that Eq. (1.43) applies for the plastic strain, while Eq. (1.38) of the of the previous power law applies for the total strain (elastic plus plastic). The total strain ε is equal to the sum of the elastic εe and plastic εp strains as

1.13 Ductility

25

Fig. 1.19 Stress–strain curves in linear (a) and logarithmic (b) coordinates for a Ramberg–Osgood relationship

ε = εe + εp =

( σ )1/n σ + E H

(1.44)

Equation (1.44) cannot be solved explicitly for stress σ. It provides a smooth continuous curve for all values of σ and does not exhibit a distinct yield point. This makes Eq. (1.44) amenable to analytical computations. A yield stress σ 0 can be defined from Eq. (1.42) for an offset yield strain of 0.002 as σ0 = H (0.002)n

(1.45)

The constants H and n of the Ramberg–Osgood equation can be obtained from experimental data by making a log–log plot of the stress σ versus plastic strain εp . As in the previous case, the constant H is the value of stress at εp = 1 and n is the slope of the log–log plot (Fig. 1.19). Both the power law and the Ramberg–Osgood models give equivalent results for large plastic strains. In that case, the elastic term in Eq. (1.44) is negligible and can be omitted. The values of H and n are equal to the values of H 1 and n1 .

1.13 Ductility Ductility is the ability of a material to sustain plastic deformation under tensile load before failure. We will present two measures of ductility, the engineering fracture strain, εf , and the percentage reduction in area, %RA. The engineering fracture strain, εf is defined as

26

1 Tensile Testing

εf =

Lf − Li Li

(1.46)

where L f is the length at fracture and L i is the original length. Often, it is expressed as % elongation = 100 εf

(1.47)

Measurements may be made on the broken pieces or under load. When measurements are made on broken pieces the elastic deformation is recovered and the fracture strain corresponds to the plastic component εpf of εf , given by εpf = εf −

σf , % plastic elongation = 100 εpf E

(1.48)

where σf is the engineering fracture strength. For most materials the amount of elastic deformation is so small that measurements on the broken pieces or under load give the same result, εpf = εf . The second measurement of ductility is defined as the percentage reduction in area, %RA. It is given by %RA = 100

Ai − Af Ai

(1.49)

where Af is the cross-sectional area after fracture and Ai is the original cross-sectional area. For round cross sections Eq. (1.49) takes the form %RA = 100

di2 − df2 di2

(1.50)

where d i is the original diameter and d f is the final diameter of the cross section.

1.14 Resilience Consider an applied tensile force P and let the displacement over a gage length L i be ΔL = x. The work W done by the force P in deforming the specimen to a value x = x , is ∫x , W =

Pdx 0

(1.51)

1.14 Resilience

27

This work is absorbed and stored in the material as energy. This type of energy caused by the deformation of the body is called strain energy. Let Ai be the cross-sectional area. The volume of the gage area is V = Ai L i . The strain energy U per unit volume of the material to reach strain ε, is W 1 U= = V Ai L i

∫x ,

∫x , Pdx =

0

( ) ∫ε, P x = σ dε d Ai Li

0

(1.52)

0

The strain energy per unit volume U is called strain energy density. Equation (1.52) indicates that U is equal to the area under the stress–strain curve up to the point ε, . Resilience is the ability of a material to absorb energy and to return it when it is deformed elastically. For a linear elastic material (σ = Eε) Eq. (1.52) renders U=

1 2 1 1 σε = σ = Eε2 2 2E 2

(1.53)

The resilience of a material U 0 is given by U0 =

1 2 1 σ = Eε02 2E 0 2

(1.54)

where σ 0 and ε0 are the stress and strain at the yield point of the material (which is assumed to coincide with the linear elastic limit). The resilience of a material is the area of the stress–strain curve up to the yield point (Fig. 1.20). Fig. 1.20 Resilience is the area the stress–strain curve up to the yield point

28

1 Tensile Testing

1.15 Tensile Toughness Tensile toughness U f is the ability of a material to absorb energy in the plastic range. It is given by the total area of the stress–strain curve up to fracture (Fig. 1.21). When the stress–strain curve is linear between the yield point and the ultimate strength the tensile toughness can be approximated as ( Uf = εf

) σ0 + σu , 2

(1.55)

where εf is the fracture strain. For brittle materials the stress–strain curve up to fracture may be approximated with a parabola with vertex at the origin. In that case U f is given by Uf =

2 σf εf 3

(1.56)

where σ f and εf are the fracture stress and strain. Brittle materials have low tensile toughness, while ductile material generally have high tensile toughness. Tensile toughness should not be confused with fracture toughness which is the ability of a material to resist fracture in the presence of a crack. It will be studied in Chap. 4. Fig. 1.21 Tensile toughness is given by the total area of the stress–strain curve up to fracture

1.16 Poisson’s Ratio

29

1.16 Poisson’s Ratio When a tensile specimen is loaded it elongates along the longitudinal direction and contracts laterally in both transverse directions. It has been established experimentally that the lateral contractions are constant fractions of the longitudinal extension. The ratio of the lateral strain to the axial strain is called Poisson’s ratio and is denoted by the Greek letter ν. It is expressed by ν=−

lateral strain axial strain

(1.57)

Poisson’ ratio ν is a material constant. Since lateral and axial strains have opposite signs (axial strains are positive and lateral strains are negative) ν is a positive number (this is the meaning of the negative sign in Eq. 1.57). It has a constant value in the elastic region. In the plastic region it increases from its elastic value up to a limiting value of 0.5 for deformation with constant volume (hydrostatic deformation). Values of ν for most metals and alloys are around 0.3. The variation of ν with strain in the elastic and plastic regions for an aluminum alloy is shown in Fig. 1.22 [10, 11]. For a tension specimen the strains εy , εz along the lateral directions y- and z- are given by ε y = εz = − νεx = −

ν P ν σ =− , E E A

(1.58)

where εx is the strain along the load direction x, A is the cross-sectional area and E is Young’s modulus.

Fig. 1.22 Variation of Poisson’s ratio ν with strain ε in the elastic and plastic regions for an aluminum alloy. Note that ν is constant in the elastic region and increases up to a limiting value of 0.50 in the plastic region

30

1 Tensile Testing

Equation (1.58) indicates that Young’s modulus E and Poisson’s ratio ν completely describe the linear elastic deformation of an isotropic material. When a material element is subjected to a triaxial state of stress given by the stress component σ x , σ y , σ z in an orthogonal system Oxyz the corresponding strains εx , εy , εz using the principle of superposition are )] ( 1[ σ x − ν σ y + σz E ] 1[ εy = σ y − ν(σz + σx ) E )] ( 1[ σz − ν σ x + σ y . εz = E

εx =

(1.59)

Equation (1.59) represents the so-called generalized Hooke’s law.

1.17 Bulk Modulus Consider a rectangular material element of sides a, b, c along the principal axes. The new sides of the rectangle a1 , b1 , c1 after deformation are a1 = a(1 + ε1 ), b1 = b(1 + ε2 ), c1 = c(1 + ε3 )

(1.60)

where ε1 , ε2 , ε3 are the normal strains along the sides a, b, c, respectively. The shear strains do not cause volume change, they cause only distortion. The volume of the element before V and after deformation V * is given by V = abc, V ∗ = a1 b1 c1 = a(1 + ε1 )b(1 + ε2 )c(1 + ε3 )

(1.61)

The change of volume of the element divided by the initial volume is called dilatation, D. It is given by D=

abc(1 + ε1 )(1 + ε2 )(1 + ε3 ) − abc V∗ − V = V abc

(1.62)

Deleting higher order terms of strains we obtain D = ε1 + ε2 + ε3 = J1

(1.63)

Equation (1.63) indicates that dilatation D is equal to the sum of the principal strains (first invariant of the strain tensor), and it is independent of the reference frame. Using Eq. (1.59) we obtain for D

1.18 Standards

31

D=

) 1 − 2ν 1 − 2ν ( σ x + σ y + σz (σ1 + σ2 + σ3 ) = E E

(1.64)

The average normal stress, σ h , is called the hydrostatic stress and is given by σh =

σ1 + σ2 + σ3 3

(1.65)

D=

3(1 − 2ν) σh E

(1.66)

Equation (1.64) becomes

Equation (1.66) indicates that the dilatation is proportional to the hydrostatic stress. The constant of proportionality relating the hydrostatic stress and the dilatation is called bulk modulus, K. It is defined by K =

E σh = D 3(1 − 2ν)

(1.67)

Equation (1.67) relates the bulk modulus with the elastic modulus and Poisson’s ratio. It indicates that only two of the three constants E, ν and K are independent.

1.18 Standards The tension test is the most widely used test for the characterization of the mechanical behavior of materials. The results of the test are used in engineering design and in quality control of materials to make sure that they meet established requirements. To ensure consistency of the test results standard test methods have been developed by professional societies, such as the American Society for Testing and Materials (ASTM International) in the United States, the Japanese Industrial Standards (JIS), the British Standard Institution (BSI), the International Organization for Standardization (ISO) and the European Union (EU) which are generally consistent with those of ISO. Performance of a tensile test consists of three main distinct parts: a. Specimen preparation b. Test setup and equipment c. Test. All three parts of the tensile test are standardized. The standards for tensile testing are different depending on the type of material being tested, as well as factors such as size and temperature. ASTM standards for tensile testing include:

32

1 Tensile Testing

• E8/E8M-13, “Standard Test Methods for Tension Testing of Metallic Materials” • D3039, “Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials” • A370, “Standard Test Method and Definitions for Mechanical Testing of Steel Products” • D638, “Standard Test Method for Tensile Properties of Plastics” • D828, “Standard Test Method for Tensile Properties of Paper and Paperboard Using Constant-Rate-of-Elongation Apparatus” • D7744/D7744M-20, “Standard Test Methods for Tensile Testing of High Performance Polyethylene Films” • C1557-20, “Standard Test Method for Tensile Strength and Young’s Modulus of Fibers” • D412-16 (2021), “Standard Test Methods for Vulcanized Rubber and Thermoplastic Elastomer-Tension” • D882-18, “Standard Test Method for Tensile Properties of Thin Plastic Sheeting” • D2256/D2256M-21, “Standard Test Method for Tensile Properties of Yarns by the Single-Strand Method” • F2516-22, “Standard Test Method for Tension Testing of Nickel-Titanium Superelastic Materials”. Other test standards are: • ISO 6892-1, “Metallic Materials. Tensile Testing. Method of Test at Ambient Temperature” (2009) • ISO 6892-2, “Metallic Materials. Tensile Testing. Method of Test at Elevated Temperature” (2011) • JIS Z2241, “Method of Tensile Test for Metallic Materials”.

1.19 Material Properties Mechanical properties determined from the tension test vary widely in different materials. Table 1.1 presents tensile properties of some engineering metals and Table 1.2 for some polymers at room temperature.

Further Readings

33

Table 1.1 Density, Young’s and shear moduli, ultimate stress and percentage elongation of some metals and alloys Metal/alloy

Density

Elastic moduli

Ultimate stress

ρ (kg/m3 )

E (GPa)

G (GPa)

σ u (MPa)

% Elongation

Aluminum

2710

70

27

Al–Cu alloy

2800

75

28.5

Al–Mg alloy

2725

71

26.5

330

8

Alloy steel

7900

210

83

1000

15

Brass

8500

104

39

440

8

Bronze

8800

117

45

190

10

Copper

8950

96–117

38

175

45

Iron

7850

200–206

82

300

45

Mild steel

7860

207

81

510

35

Nickel

8900

198

80

300

30

Stainless steel

7930

200

77

510

60

Titanium

4540

118

45

620

20

Ti–Al alloy

4430

110

42

860

15

Ti–Sn alloy

4600

105

40

1300

12

80

30

425

20

Table 1.2 Density, Young’s modulus, ultimate stress and percentage elongation of some nonmetals Nonmetals

Density

Young’s modulus

Ultimate stress

ρ

(kg/m3 )

% Elongation

E (GPa)

σ u (MPa)

Acetate

1220–1340

1.0–2.0

25–65

5–55

Acrylic

1185

2.7–3.5

50–70

5–8

Concrete

2400

14

4

2

Epoxy

1150

3.5–8

50–100

Nylon 66

1150

2.8–3.3

60–80

60–300

Perspex

1190

3

50

2–7

Polyester

1100–1350

2.4

40–55

650

PVC

1400–1700

2.4–4.1

48–58

2–40

Rubber

910

0.007

17

500

Further Readings 1. Bowman K (2004) Mechanical behavior of materials. Wiley 2. Bridgman PW (1944) The stress distribution at the neck of a tension specimen. Trans ASME 32:553–574 3. Courtney TH (2000) Mechanical behavior of materials. McGraw Hill 4. Davidenkov NN, Spiridonova NI (1946) Analysis of tensile stress in the neck of an elongated test specimen. Proc ASTM 46:1147–1158 5. Davis JR (ed) (2004) Tensile testing, 2nd edn. ASM International

34

1 Tensile Testing

6. Dowling NE, Kampe SL, Kral MV (2019) Mechanical behavior of materials, global edition 5th edn. Pearson 7. Hosford WF (2010) Mechanical behavior of materials, 2nd edn. Cambridge University Press 8. Meyers MA, Chawla KK (1999) Mechanical behavior of materials. Prentice Hall 9. Roesler J, Harders H, Baeker M (2007) Mechanical behaviour of engineering materials. Springer 10. Theocaris PS, Koroneos E (1963) Stress–strain and contraction ratio curves for polycrystalline steel. Phil Mag 8:1871–1893 11. Theocaris PS, Koroneos E (1964) The variation of lateral contraction ratio of low carbon steel at elevated temperatures. Proc ASTM 64:747–764

Chapter 2

Compression, Bending, Torsion and Multiaxial Testing

Abstract In this chapter we consider the compression, bending, torsion and multiaxial testing of materials. Compression is a fundamental test as it is tension. The initial portions of stress–strain curves in compression for most materials have the same characteristics as in tension. Various material properties such as modulus of elasticity, elastic and proportional limit, yield stress may be defined from the initial portion of the stress–strain curve in compression. Many materials, even though they have the same elastic behavior in tension and in compression, their post-yield and failure characteristics in compression are quite different than in tension. We present the specimen types and the stress–strain curves in compression, and we compare the material behavior in compression to that in tension. Also, we present the bending and torsion tests and a series of tests for the multiaxial characterization of materials. They include biaxial tension tests, biaxial tension/compression strip tests, tube tests, spherical vessel tests, combined tension/compression–torsion ring tests, combined tension/compression–torsion tube tests, combined tension/compression– torsion–internal pressure tube tests. Finally, we present failure envelopes under multiaxial loading for isotropic, anisotropic and uneven materials.

2.1 Compression Test 2.1.1 Specimens Compression specimens are simpler than tension specimens because they do not require special arrangements for gripping. The specimens are usually simple cylinders with length, L, to diameter, d, ratio L/d, in the range 1–3, even though values of L/d up to 10 are sometimes used. If the ratio L/d is relatively large buckling may occur. Buckling of a structural component is a form of instability accompanied by a sudden change in the shape under compression. It occurs when the compressive load reaches a critical level. After buckling any further load will cause significant and unpredictable deformations. The compression test becomes meaningless. Buckling is caused by small imperfections

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_2

35

36

2 Compression, Bending, Torsion and Multiaxial Testing

in the geometry of the specimen and its alignment with respect to the grips of the testing machine. If the ratio L/d is relatively small the test results are affected by the friction on the ends of the specimen. The lateral expansion of the material is suppressed near the ends. A cone-shaped region of dead (undeformed) material is formed at the ends (Fig. 2.1). The specimen does not expand uniformly along its height. The lateral expansion is larger near its mid-height than its ends. The specimen becomes barrel shaped. The effect of friction at the specimen ends can be reduced by lubrication. Also, it can be reduced by increasing the ratio L/d. The barreling effect in compression is opposite to the necking effect observed in tension. In tension the specimen extends by thinning leading to fracture, while in compression barreling occurs. Based on the above, a compromise should be made between the two effects that affect the shape of the compression specimen: the buckling effect and the friction effect at the specimen ends. For ductile materials a reasonable compromise is L/d = 3 and for brittle materials it is L/d = 1.5–2. Fig. 2.1 Cone-shaped region of dead (undeformed) material is formed at the ends of a compression specimen unless the ends are well lubricated. The specimen becomes barrel shaped

2.1 Compression Test

37

2.1.2 Stress–Strain Curve Compression stresses are opposite to tension, and they are indicated by negative values. In a compression test the specimen experiences two opposing forces directed toward each other upon the specimen from opposite sides. The specimen is placed in between two plates which are pushed together by a universal test machine causing the sample to flatten. A compressed specimen is usually shortened in the direction of the applied forces and expands in the direction perpendicular to the forces. When uniaxial compression stresses apply they lead to shortening and the corresponding axial strains are negative. Figure 2.2 shows a typical stress–strain curve of a metal alloy in compression and tension. The elastic curve in compression is the mirror image of the elastic curve in tension. There is an initial linear part up to the yield point (assuming that the linear, the elastic and the yield points coincide). The Young’s modulus in compression is equal to that in tension. The yield stress is the same, even a little higher, than the yield stress in tension. After the yield point with increasing load the cross-sectional area of the specimen increases. For ductile materials the specimen is barreled and fracture does not occur. In the engineering stress–strain curve the stress increases continually with increase of load and the stress–strain curve tends to become parallel to the stress axis as the specimen tends to become a thin disk. This behavior is quite opposite to that in tension in which the engineering stress reaches a plateau and then decreases up to fracture. Assuming that the volume of the specimen does not change (A0 L 0 = AL, where A0 and L 0 are the original, and A and L are the current cross-sectional area and length of the specimen), we obtain for the applied compression load P: Fig. 2.2 Typical stress–strain curve in compression and tension

38

2 Compression, Bending, Torsion and Multiaxial Testing

P = σ A0 = σ˜ A = σ˜

σ˜ A0 A0 L 0 σ˜ A0 = = L L/L 0 1+ε

(2.1)

where σ˜ is the true stress and ε (< 0) is the engineering strain. Equation (2.1) renders σ =

σ˜ . 1+ε

(2.2)

Equation (2.2) shows that the true stress, σ˜ , is smaller than the engineering stress, σ (˜ε < 0). This result is expected since the area of the specimen increases as the load increases. The engineering strain ε is related to the true strain ε˜ by Eq. (1.8) as ε = eε˜ − 1

(2.3)

where e is the base of natural logarithms (e = 2.718281828459). Note that the engineering strain ε that refers to the initial length is smaller than the true strain ε˜ that refers to the final length (in absolute value, as both strains are negative). Equations (2.2) and (2.3) relate the true and engineering stresses and strains in compression and allow the construction of the engineering stress–strain curve from the true stress–strain curve. Note that using Eqs. (2.2) and (2.3) the value of the engineering stress is obtained from the value of the true stress at a smaller strain value than the true strain that corresponds to the true stress. If we assume that the true stress–strain curves in tension and compression coincide (by removing the negative sign in the compression stresses and strains) then from the true stress–strain curve in tension we can construct the engineering stress–strain curve in compression. Figure 2.3 presents the true and engineering stress–strain curves in compression. Each point (σ, ε) on the engineering stress–strain curve is obtained from a corresponding point (σ˜ , ε˜ ) on the true stress–strain curve. The arrows connect these two points. Note that the true strain–strain curve is below the engineering stress–strain curve. This behavior is quite opposite to that in tension.

2.1.3 Comparison of Stress–Strain Curves in Compression and Tension From the above discussion the following remarks concerning the stress–strain curves in compression and tension can be made: a. The initial parts of the compression stress–strain curves of ductile metals are identical to those in tension. The true stress–true strain curves may still agree after large amounts of deformation.

2.1 Compression Test

39

Fig. 2.3 True and engineering stress–strain curves in compression. Each point (σ, ε) on the engineering stress–strain curve is obtained from the corresponding point (σ˜ , ε˜ ) on the true stress–strain curve. The arrows connect these two points

b. The modulus of elasticity, the proportional limit and the yield stress in compression take the same values as in tension. c. There is no ultimate strength behavior in compression as it is in tension. The load in compression does not reach a maximum value. The maximum load behavior in tension is related to the creation of neck and does not occur in compression. d. The engineering stress in compression increases continually as the load is increased. e. Some ductile metals and polymers never fracture in compression. The specimen deforms into a barrel shape and the compression test loses its meaning. f. Materials that are brittle in tension fracture at planes perpendicular to the specimen axis. At these planes the tension stress becomes maximum. This behavior is due to the existing cracks, pores or other flaws that grow and coalesce. Such defects have less effect in compression since compressive loads suppress and close the flaws. Thus, materials that have brittle behavior in tension have higher compressive than tensile strengths. An example is concrete in which the compressive strength is much higher than the tensile strength. g. Compressive failure is generally associated with shear, while brittle tension fracture is associated with normal stress. This results to fracture planes inclined relative to the applied load in compression and normal to the applied load in tension.

40

2 Compression, Bending, Torsion and Multiaxial Testing

2.2 Bending Test Bending tests, also called flexure tests, are commonly used for the determination of the flexure strength and the modulus of elasticity, mainly of brittle materials, like concrete, natural stone, wood, plastics, glass and ceramics. The bending test is performed in a beam of rectangular cross section loaded in three- or four-point bending (Fig. 2.4). For linear elastic behavior the normal stress σ at a distance y from the neutral axis (the centroid of the cross section) of the beam is given by σ =

My , I

(2.4)

where M is the bending moment and I is the moment of inertia of the cross section about the neutral axis of the beam. For a rectangular cross section of height 2c and width b, I is given by I =

2bc3 . 3

(2.5)

For three-point bending the concentrated load P is applied at the mid-length L of the beam, and M = PL/4 (Fig. 2.4a). From Eqs. (2.4) and (2.5) we obtain for the fracture stress σ f of the beam at the most remote point (y = c) of the mid-span section: σf =

3L Pf , 8bc2

(2.6)

where Pf is the fracture load. Equation (2.6) gives the bend strength or flexural strength of the beam. Note that for brittle materials which have higher strength in compression than in tension, σ f expresses the fracture tensile strength. Equation (2.4) indicates that the normal stress σ is not uniform over the cross section of the beam and varies linearly with the distance y from the neutral axis.

Fig. 2.4 Beam in a three-point and b four-point bending

2.3 Torsion Test

41

This type of behavior is different from that of the tension or compression test in which the stress distribution across the cross section is uniform. In the bending test yielding occurs in a thin surface layer and it progresses as the material is deformed. Thus, the load versus deflection curve in bending is not sensitive to the initiation of yielding. In this respect, the bending test is not a good test for the determination of the yield strength, as it is the tensile or compressive test. However, the bending test is particularly useful in evaluating the tensile strength of brittle materials, like glass, concrete, stone and wood which are difficult to be tested in simple uniaxial tension due to cracking at the grips. Furthermore, the bending test can be easily performed. The modulus of elasticity E may be obtained from the bending test. For a threepoint beam (Fig. 2.4a) the maximum deflection at mid-span is: v=

P L3 . 48E I

(2.7)

E can be determined from the slope (dP/dυ) of the linear part of the load, P, versus deflection, υ, curve as L3 E= 48I



dP dv



  dP L3 = . 32bc3 dv

(2.8)

Values of E determined from Eq. (2.8) are generally close to those obtained from the tension or compression test. Discrepancies may exist primarily due to the departure of the stress–strain curve from linearity, local deformations at the supports and/or points of application of the loads, significant shear deformations in short beams that are not considered in the above analysis. In case the material has different values of modulus of elasticity in tension and compression an intermediate value is obtained from the bending test. Analogous equations to those developed previously for the three-point bending test can be obtained for the four-point beam bending test (Fig. 2.4b). The difference between the two tests is that in the three-point bending test normal and shear stresses develop, while in the four-point bending test only bending stresses develop in the area between the two concentrated loads (pure bending).

2.3 Torsion Test The torsion test is used for the determination of the shear strength and shear modulus. The test is performed in round bars subjected to pure torsion and is relatively easy to perform. The state of stress and strain in the torsion test of a cylindrical bar is pure shear (Fig. 2.5). The shear stress τ at a distance ρ from the axis of a cylindrical bar for linear elastic behavior is given by

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2 Compression, Bending, Torsion and Multiaxial Testing

τ=

Tρ , J

(2.9)

where T is the applied torsion moment and J is the polar moment of inertia of the cross section of the bar with respect to its center. For a cross section of radii r 1 and r 2 (r 2 > r 1 ) J is given by   π r24 − r14 J= 2

(2.10)

Note in Eq. (2.9) that the stress τ varies linearly with the distance ρ. The shear stress becomes maximum at the most remote circular ring of the cross section. Equation (2.9) is analogous to Eq. (2.4) of bending. In both cases the stress varies linearly with the distance from the axis of the bar in torsion or the neutral axis of the beam in bending. In the shear test, as in the bending test, the stress is not uniform over the cross section. The same remarks concerning the effect of nonuniformity of stresses on the determination of material properties made in the bending test are also applied in the torsion test. The shear stress τ f at fracture is obtained from Eqs. (2.9) and (2.10) with ρ = r 2 as τf =

2Tfr2 ,  4 π r2 − r14

(2.11)

where T f is the fracture torque.

Fig. 2.5 Cylindrical bar subjected to torsion. A state of pure shear equivalent to a tensile and a compressive normal stress at 45° to the bar axis is developed

2.3 Torsion Test

43

In the torsion test a state of pure shear develops in the bar. This state is equivalent to a state of normal stresses equal to the shear stress at 45° to the bar axis (Fig. 2.5). One of the normal stresses is tensile and the other is compressive. For brittle materials which have much smaller strength in tension than in compression the fracture planes are perpendicular to the tension stress and make a 45° angle with the axis of the bar. On the other hand, the fracture planes of ductile materials which fail in shear are perpendicular to the axis of the bar (Fig. 2.5). In a torsion test the applied torque T is plotted versus the measured angle of twist θ. The shear modulus or modulus of rigidity G (= E/2(1 + ν)) is obtained from the slope of the linear part of the T versus θ curve as G=

  L dT , J dθ

(2.12)

where L is the length of the bar. For a cylindrical bar of radii r 1 and r 2 Eq. (2.12) takes the form G=

  dT 2L   4 . 4 dθ π r2 − r1

(2.13)

When G and E are measured from the torsion and the tensile tests the Poisson’s ratio ν is obtained as ν=

E −1 2G

(2.14)

The problem of nonuniform stress distribution in a torsion test is alleviated by using thin-walled tubes (Fig. 2.6). The average shear stress τ avg through the wall thickness is constant. It can be approximated by τavg =

T 2 (r − r ) 2πravg 2 1

ravg = (r1 + r2 )/2.

(2.15)

The average shear strain γ avg is obtained as γavg =

ravg θ . L

(2.16)

From Eqs. (2.15) and (2.16) the shear stress τ avg versus shear strain γ avg relationship can be obtained by measuring the applied torsion moment T and the corresponding angle of twist θ. The slope of the τ avg versus γ avg curve is the modulus of rigidity G. The shear stress–strain curve obtained by the torsion experiment is analogous to the normal stress–strain curve obtained from the tension, compression or bending experiments.

44

2 Compression, Bending, Torsion and Multiaxial Testing

Fig. 2.6 Thin-walled tube in torsion. The shear stress is approximately uniform on the cross section

The torsion test permits the direct determination of the shear modulus. It is also used for the determination of the shear strength.

2.4 Multiaxial Testing 2.4.1 Introduction So far we studied testing methods for the characterization of material behavior under uniaxial states of stress including axial tension, axial compression, bending and torsion. However, materials in structural and machine applications are generally subjected to multiaxial states of stress. In this section we present the following tests for the multiaxial characterization of materials: biaxial tension tests, biaxial tension/compression strip tests, tube tests, spherical vessel tests, combined compression–torsion ring tests, combined tension–torsion tube tests and combined tension– torsion internal pressure tube tests. In these tests the applied multiaxial loads increase proportionally until failure.

2.4 Multiaxial Testing

45

2.4.2 Biaxial Tension Test In the biaxial tension test tensile forces along two mutually perpendicular directions are applied to the specimen. A biaxial tensile machine is equipped with motor stages (high-precision positioning devices), two load cells and a gripping system. The displacement is applied to the specimen through the motor stages. For one motor stage the displacement is the same in the two directions resulting to an equibiaxial state of stress. When four motor stages are used many loading conditions can be applied. The two load cells are placed along two perpendicular directions to measure the forces applied to the specimen. The test is performed under load- or displacementscontrolled conditions. The gripping system transfers the load from the motor stages to the specimen. The geometry of the specimen is square or cruciform (Fig. 2.7).

Fig. 2.7 Cruciform specimen is loaded in the horizontal and vertical directions. Stress concentration occurs in the corners of the specimen. The stress trajectories are shown

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2 Compression, Bending, Torsion and Multiaxial Testing

2.4.3 Biaxial Tension/Compression Strip Test The test consists of subjecting a long specimen strip to a uniaxial tensile (Fig. 2.8a) or compressive (Fig. 2.8b) stress σz perpendicular to the longitudinal x-axis of the specimen. It can be assumed that the strain εx in the x direction is zero (εx = 0). From Hooke’s law εx =

1 (σx − νσz ), E

(2.17)

where ν is the Poisson’s ratio, we obtain for the stress σx along the x-axis of the specimen: σx = νσz .

(2.18)

Equation (2.18) indicates that the state of stress in the specimen is twodimensional. The applied stress σz can be tensile or compressive. Note that the

Fig. 2.8 Long specimen subjected to uniaxial tensile (a) or compressive (b) stress along its length

2.4 Multiaxial Testing

47

stress σx has the same sign as the applied stress σz . Equation (2.18) is valid for linear elastic behavior of the material. In the test the specimen is subjected to a stress σz until failure occurs. The biaxial strip test is a simple approximate method for the determination of failure of a material under a biaxial stress field. The stresses σx and σz are related by Eq. (2.18). Note that this equation does not apply along the whole length of the specimen since stress σx near the ends of the specimen is close to zero. It is worthwhile to mention that the test is restricted in biaxial stresses of the same sign which are related by Eq. (2.18).

2.4.4 Tube Test The test consists of subjecting a cylindrical vessel with closed ends to an internal pressure (Fig. 2.9). A tensile biaxial state of stress is created in the vessel walls. The longitudinal σ1 and circumferential stress σ2 are given by σ1 =

pr pr , σ2 = , t 2t

(2.19)

where p is the applied internal pressure, t is the thickness and r is the mean radius of the tube (r = (r 1 + r 2 )/2). In the test the critical value of the internal pressure at failure is monitored, and the stresses σ1 , σ2 are calculated from Eq. (2.19). Note that the test is restricted to biaxial tensile stresses one of which is half the other.

2.4.5 Spherical Vessel Test The test consists of subjecting a spherical vessel to an internal pressure (Fig. 2.10). The state of stress in the vessel is biaxial tension. The stresses σ1 , σ2 are equal and are given by σ1 = σ2 =

pr , 2t

(2.20)

where p is the internal pressure, r is the radius and t is the thickness of the sphere. In the test the critical value of the internal pressure at failure is monitored, and the two biaxial stresses are calculated from Eq. (2.20). Note that the test is restricted to equal biaxial tensile stresses.

48

2 Compression, Bending, Torsion and Multiaxial Testing

Fig. 2.9 Closed tubular vessel under internal pressure. A tension biaxial state of stress is established in the vessel

2.4.6 Combined Tension/Compression–Torsion Ring Test In the ring test the specimen is subjected to combined tension/compression and torsion (Fig. 2.11). The stresses are given by σz =

P T  , σθ = νσz , τ =  2πr 2 t π r22 − r12

(2.21)

where σ z is the normal stress along the axis of the ring, σ θ is the circumferential normal stress, τ is the shear stress due to torsion, P is the applied tensile/compressive load, T is the applied torque, r 1 , r 2 are the inner and outer radii, r = (r 1 + r 2 )/2, t is the wall thickness and ν is Poisson’s ratio. Equation (2.21) indicates that the state of stress in the ring is dictated by the stresses σ z , σ θ and τ. By varying the values of the applied load P and torque T different combinations of biaxial states of stress can be obtained.

2.4 Multiaxial Testing

49

Fig. 2.10 Spherical vessel under internal pressure. A uniform tension biaxial stress is established in the vessel

2.4.7 Combined Tension/Compression–Torsion Tube Test In the tube test specimens are subjected to combined tension/compression and torsion. The stresses σ z and τ are given by σz =

P T , τ =  , 2πr 2 t π r22 − r12

(2.22)

where P is the applied tension/compression load and T is the applied torque. By varying the values of the applied load P and torque T different combinations of biaxial states of stress can be obtained.

50

2 Compression, Bending, Torsion and Multiaxial Testing

Fig. 2.11 Ring specimen subjected to combined tension/compression and torsion

2.4.8 Combined Tension/Compression–Torsion–Internal Pressure Tube Test In the tube test specimens are subjected to combined tension/compression–torsion– internal pressure loading. The stresses σ z , σ θ and τ in the tube are given by σz =

P T pr pr  , σθ = +  2 , τ= , 2 2t t 2πr 2 t π r2 − r1

(2.23)

where P is the applied tension/compression load, p is the internal pressure and T is the applied torque. By varying the values of the applied load P, internal pressure p and torque T different combinations of biaxial states of stress can be obtained.

2.4.9 Failure Criteria 2.4.9.1

Introduction

Failure of a material by yielding or fracture in uniaxial tension or compression takes place when

2.4 Multiaxial Testing

51

σ = σY , or σ = σu

(2.24)

where σ is the applied stress, σ Y is the yield stress and σ u is the ultimate stress. When a material element is subjected to a multiaxial state of stress failure occurs when a combination of the stresses satisfies a critical condition. Postulates for determining those macroscopic stress combinations that result in failure of materials are known as failure criteria. Materials that behave in a ductile manner generally undergo yielding before they ultimately fracture. The usefulness of ductile materials is limited by yielding, while of brittle materials by fracture. Yielding criteria for ductile materials refer to yielding, while for brittle materials refer to fracture. At this point we should make clear that a material may behave in a ductile or brittle manner, depending on the temperature, rate of loading and other variables present. When we speak about ductile or brittle materials we actually mean the ductile or brittle states of materials. Although the onset of yielding is influenced by factors such as temperature, time and size effects, there is a wide range of circumstances where yielding is mainly determined by stress state itself. Under such conditions, for isotropic materials, there is an extensive evidence that yielding is a result of distortion and is mainly influenced by shear stresses. Hydrostatic stress states, however, play a minor role in the initial yielding of materials. Various failure criteria have been developed to predict failure by yielding in ductile materials and failure by fracture in brittle materials. In the following we will briefly present the maximum normal stress fracture criterion, the maximum shear stress or Tresca yield criterion, the octahedral shear stress or von Mises failure criterion and the Mohr–Coulomb fracture criterion for isotropic materials. Finally, we will present failure criteria for anisotropic (with different properties in different directions) and uneven (with different yield or fracture strengths in tension and compression) materials.

2.4.9.2

General Form of Failure Criteria

Failure criteria can be expressed in the form: f (σ1 , σ2 , σ3 ) = σc

(2.25)

where σ 1 , σ 2 , σ 3 are the principal stresses and σ c is the yield or the ultimate stress of the material obtained from a uniaxial tension or compression test. A failure criterion should be independent of the coordinate system. This condition is met by expressing the failure criterion in terms of the principal stresses σ 1 , σ 2 , σ 3 or the stress invariants J 1 , J 2 , J 3 . They are given by

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2 Compression, Bending, Torsion and Multiaxial Testing

J1 = σ1 + σ2 + σ3 = σx + σ y + σz 2 2 J2 = σ1 σ2 + σ2 σ3 + σ3 σ1 = σx σ y + σ y σz + σz σx − τx2y − τ yz − τzx

J3 = σ1 σ2 σ3 = σx σ y σz −

2 σx τ yz

−

2 σ y τzx

−

σz τx2y

(2.26)

+ 2τx y τ yz τzx ,

where σ x σ y σ z are the normal and τ xy , τ yz , τ zx are the shear stresses referred to the orthogonal system xyz. A failure criterion can be expressed in terms of stress invariants in the form F(J1 , J2 , J3 ) = σc

(2.27)

where F(J 1 , J 2 , J 3 ) is a function of the stress invariants. An effective or equivalent stress σ is usually defined as σ = f (σ1 , σ2 , σ3 ) = F(J1 , J2 , J3 ).

(2.28)

A failure criterion takes the form σ = σc

(2.29)

σ < σc

(2.30)

Failure does not occur when

If Eq. (2.25) is plotted in principal normal stress space of coordinates σ 1 , σ 2 , σ 3 the function f = f (σ 1 , σ 2 , σ 3 ) forms a surface in space called failure surface.

2.4.9.3

Maximum Normal Stress Fracture Criterion

The maximum normal stress fracture criterion states that a material element fails when the largest principal normal stress reaches the uniaxial strength of the material. The criterion is expressed by MAX(|σ1 |, |σ2 |, |σ3 |) = σu ,

(2.31)

where the tension and compression strengths of the material are assumed to be equal, σu = σut = |σuc |. The effective stress σ is σ = MAX(|σ1 |, |σ2 |, |σ3 |).

(2.32)

The failure envelope in the system of principal stresses σ 1 , σ 2 , σ 3 is a cube (Fig. 2.12a), while for plane stress (σ 3 = 0) the failure envelope is a square

2.4 Multiaxial Testing

53

Fig. 2.12 Failure envelope a on the σ 1 , σ 2 , σ 3 space, and b on the σ 1 , σ 2 plane (σ 3 = 0), according to the maximum normal stress fracture criterion

(Fig. 2.12b). For any combination of stresses that fall within the cube or the square the material is safe. The maximum principal stress fracture criterion is used for predicting the fracture of brittle materials under tension-dominated loading.

2.4.9.4

Maximum Shear Stress or Tresca Yield Criterion

The maximum shear stress or Tresca yield criterion states that yielding of a material element occurs when the maximum shear stress reaches the yield shear stress. The criterion is expressed by 

|σ1 − σ2 | |σ2 − σ3 | |σ3 − σ1 | , , MAX 2 2 2

 = τY ,

(2.33)

where τ Y is the yield stress in shear. For uniaxial tension Eq. (2.33) gives τY =

σY , 2

(2.34)

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2 Compression, Bending, Torsion and Multiaxial Testing

where σY is the yield stress in uniaxial tension. Equation (2.33) in terms of σY becomes MAX(|σ1 − σ2 |, |σ2 − σ3 |, |σ3 − σ1 |) = σY .

(2.35)

The effective stress is σ = MAX(|σ1 − σ2 |, |σ2 − σ3 |, |σ3 − σ1 |).

(2.36)

For plane stress (σ3 = 0), Eq. (2.35) becomes MAX(|σ1 − σ2 |, |σ2 |, |σ1 |) = σY .

(2.37)

The region of no yielding is bounded by the lines σ1 − σ2 = ± σY , σ2 = ± σY σ1 = ± σY .

(2.38)

The graphical representation of the yield region is shown in Fig. 2.13. Note that the first equation gives a pair of parallel lines with a slope of unity, while the other two equations give lines parallel to the coordinate axes σ1 , σ2 .

2.4.9.5

Octahedral Shear Stress or von Mises Yield Criterion

The octahedral shear stress or von Mises yield criterion states that yielding occurs when the shear stress on the octahedral plane (the plane that makes equal angles with the principal axes) reaches the yield shear stress. The criterion is expressed by τh =

/ /  1 1  2 2 J1 − 3J2 = τY , (2.39) (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 = 3 3

where τ h is the octahedral shear stress. Applying Eq. (2.39) for σ1 = σY (σ2 = σ3 = 0) we obtain √ τY =

2 σY . 3

(2.40)

Then, Eq. (2.39) becomes / 1 √ (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 = σY . 2 The effective stress σ is / 1 σ = √ (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 . 2

(2.41)

(2.42)

2.4 Multiaxial Testing

55

Fig. 2.13 Failure envelope on the σ 1 , σ 2 plane (σ 3 = 0) according to the maximum shear stress or Tresca yield criterion

For plane stress Eq. (2.41) becomes σ12 − σ1 σ2 + σ12 = σY2 .

(2.43)

This is the equation of an ellipse with major axis along the line σ1 = σ2 (bisector of the angle of the axes σ1 , σ2 ) and which crosses the σ1 , σ2 axes at the points ± σY (Fig. 2.14). Note that the ellipse has the distorted hexagon of the Tresca yield criterion inscribed within it. The von Mises yield criterion is mainly used for ductile materials. Equation (2.41) indicates that a hydrostatic states of stress (σ1 = σ2 = σ3 ) has no effect on yielding.

2.4.9.6

Mohr–Coulomb Fracture Criterion

The Mohr–Coulomb fracture criterion states that fracture occurs on a given plane of the material when a critical combination of shear and normal stress that act on this plane is reached. The fracture condition can be written as |τ | = f (σ ),

(2.44)

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2 Compression, Bending, Torsion and Multiaxial Testing

Fig. 2.14 Failure envelope on the σ 1 , σ 2 plane (σ 3 = 0) according to the von Mises yield criterion. The Tresca hexagon is inscribed within the Mises ellipse

where τ and σ are the shear and normal stresses on the fracture plane and f (σ ) is a function which is determined experimentally and is considered as a material characteristic property. The simplest form of the function f (σ ) is a straight line, and the fracture criterion is expressed as |τ | + μσ = τi ,

(2.45)

where μ and τ i are material constants. This straight line is the failure envelope (Fig. 2.15). It has a slope of − μ and an intercept with the τ axis of τ i . Both τ i and μ are considered positive. From Fig. 2.15 it is shown that for compressive stresses the shear stress to cause failure is increased as the normal stress is increased. Let us plot now on the σ − |τ | plane the three Mohr’s circles that correspond to the principal stresses σ 1 , σ 2 , σ 3 (σ 1 > σ 2 > σ 3 ) (Fig. 2.15). The failure envelope touches the largest, (σ 3 , σ 1 ), of the three Mohr’s circles passing from points (σ 1 , σ 2 ), (σ 2 , σ 3 ), (σ 3 , σ 1 ) at a point (σ ' , τ ' ). This point represents the critical combination of normal and shear stresses for fracture. The orientation of the critical plane is determined from the largest circle. The angle the plane makes with the σ 1 stress is half the angle of rotation in the Mohr’s circle. From Fig. 2.15 we obtain μ = tan ϕ, ϕ = 90◦ − 2θc .

(2.46)

The Mohr–Coulomb fracture criterion is mainly used in rock and soil mechanics.

2.4 Multiaxial Testing

57

Fig. 2.15 Failure envelope on the τ –σ plane according to Mohr–Coulomb fracture criterion

2.4.9.7

Anisotropic Materials

Several empirical criteria have been developed for describing yielding or fracture of anisotropic materials (with different properties in different directions). The most widely used Hill yield criterion is a straightforward extension of the von Mises criterion. For orthotropic materials possessing symmetry about three mutually perpendicular planes the Hill criterion has the form: 2 2   2 2 F σ y − σz + G(σz − σx )2 + H σx − σ y + 2Lτ yz + 2Mτzx + 2N τx2y = 1, (2.47) where σx , σ y , σz are the normal and τx y , τ yz , τzx are the shear stresses with respect to the axes of orthotropy, and F, G, H, L, M, N are empirical constants. They are given by   1 1 1 1 F=   +  2 −  2 2 σY 2 σzY σxY y   1 1 1 1 G=   +  2 −  2 2 σY 2 σxY σ yY z   1 1 1 1 H=   +  2 −  2 2 σY 2 σY σY x

y

z

1 1 1 1 1 1 L =  2 ; M =  2 ; N =  2 , 2 τY 2 τY 2 τY yz zx xy

(2.48)

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2 Compression, Bending, Torsion and Multiaxial Testing

where σxY , σ yY , σzY are the normal yield stresses along the x, y, z axes of anisotropy Y Y and τxYy , τ yz , τzx are the shear yield stresses in the planes xy, yz, zx. The Hill criterion can also be used as a fracture criterion by replacing the various yield strengths by the corresponding ultimate strengths.

2.4.9.8

Isotropic/Anisotropic Uneven Materials

Isotropic uneven materials are isotropic materials that have different yield or fracture strengths in tension and compression. Failure depends on hydrostatic stress. A failure criterion for isotropic materials proposed by Nadai [7] is expressed by  2 1/2 R − 1 2R σt , + σ1 + σ22 + σ32 − σ1 σ2 − σ2 σ3 − σ3 σ1 (σ1 + σ2 + σ3 ) = R+1 R+1 (2.49) where R = σ c /σ t , σ t and σ c is the yield stress in tension and compression, respectively. Another yield criterion introduced by Raghava et al. [8] is expressed by  2  σ1 + σ22 + σ32 − σ1 σ2 − σ2 σ3 − σ3 σ1 + (|σc | − σt )(σ1 + σ2 + σ3 ) = |σc |σt . (2.50) Glassy polymers have higher yield strengths in compression than in tension in the range 1.20–1.33. The failure envelope for such materials is an off-center ellipse. Figure 2.16 presents the failure envelope on the axes, (σ 1 /σ t , σ 2 /σ c ) for |σc /σt | = 1.3 and biaxial yield data (σ3 = 0) for various polymers [8]. The experimental results are close to the failure envelope. A failure criterion for anisotropic uneven materials is the Tsai–Wu criterion [4]. For conditions of plane stress it takes the form f 1 σ1 + f 3 σ3 + f 11 σ12 + f 33 σ32 + 2 f 13 σ1 σ3 = 1 − k 2

(2.51)

With f1 =

1 1 − , F1t F1c

f3 =

1 1 − , F3t F3c

f 11 =

1 , F1t F1c

1 f 13 = − ( f 11 f 33 )1/2 , τ5 = k F5 , 2

f 33 =

1 F3t F3c (2.52)

where σ 1 , σ 3 and τ 5 are the normal and shear stresses referred to the principal material directions 1 and 3, F 1c and F 1t are the compressive and tensile strengths along the in-plane direction, F 3c and F 3t are the compressive and tensile strengths along the through-the-thickness direction, and F 5 = F 13 is the shear strength on the 1–3 plane.

2.5 ASTM Standards

59

Fig. 2.16 Failure envelope and experimental results on the (σ 1 /σ t ) − (σ 2 /σ c ) plane for an isotropic uneven material with |σc /σt | = 1.3 under plane stress (σ3 = 0), according to the Raghava et al. [8] failure criterion

The failure envelopes according to the Tsai–Wu criterion for an anisotropic closedcell foam under the commercial name Divinycell 250 in the σ 1 –σ 3 plane is shown in Fig. 2.16 [4]. Axis 1 is along the in-plane direction and axis 3 is along the throughthe-thickness direction of a sheet of the material. Three failure envelopes are plotted corresponding to values of k = τ 13 /F 5 = 0, 0.8 and 1, where τ 13 is the in-plane shear stress and F 5 is the shear strength. Experimental results are in close agreement with the failure envelopes of the Tsai–Wu criterion (Fig. 2.17).

2.5 ASTM Standards ASTM has published the following standards for mechanical testing of materials in compression, bending and torsion: Compression • ASTM D575 Compression Test of Rubber • ASTM D6641 Compression Testing for Polymer Matrix Composite Laminates • ASTM D695 Compression Testing for Rigid Plastics

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2 Compression, Bending, Torsion and Multiaxial Testing

Fig. 2.17 Failure envelope on the σ 1 –σ 3 plane for an anisotropic closed-cell foam under the commercial name Divinycell 250 according to the Tsai–Wu failure criterion

• ASTM D7137 Compressive Residual Strength Test Equipment for Damaged Polymer Matrix Composite Plates • ASTM D905 Wood Adhesive Bonds in Shear by Compression Loading • ASTM E9 Compression Testing of Metallic Materials at Room Temperature • ISO 14126 Compression Fiber Reinforced Plastic Composites Test Machine • ISO 1856 Flexible Cellular Polymeric Materials Compression EN • ISO 604 Compressive Plastics Testing Equipment • ISO 844 Compressive Strength of Rigid Cellular Plastics • D695-15, “Standard Test Method for Compressive Properties of Rigid Plastics” • E9-19, “Standard Test Methods of Compression Testing of Metallic Materials at Room Temperature” • F3270-F3270M-17, “Standard Practice for Compression Versus Load Properties of Gasket Materials” • D1621-16, “Standard Test Method for Compressive Properties of Rigid Cellular Plastics”

Further Readings

61

• D395-01, “Standard Test Methods for Rubber Property-Compression Set”. Bending • E290-22, “Standard Test Methods for Bend Testing of Material for Ductility” • D790, “Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials” • E190-21, “Standard Test Method for Guided Bend Test for Ductility of Welds” • C158-02(2017), “Standard Test Methods for Strength of Glass by Flexure (Determination of Modulus of Rupture)” • E855, “Standard Test Methods for Bend Testing of Metallic Flat Materials for Spring Applications Involving Static Loading” • D4032-08, “Standard Test Method for Stiffness of Fabric by the Circular Bend Procedure” • C1161, “Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature” • C1684, “Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature-Cylindrical Rod Strength” • C674, “Standard Test Methods for Flexural Properties of Ceramic Whiteware Materials”. Torsion • A938-18, “Standard Test Method for Torsion Testing of Wire” • E143-20, “Standard Test Method for Shear Modulus at Room Temperature” • D3044-16, “Standard Test Method for Shear Modulus of Wood-Based Structural Panels”.

Further Readings 1. Bowman K (2004) Mechanical behavior of materials. Wiley 2. Courtney TH (2000) Mechanical behavior of materials. McGraw Hill 3. Dowling NE, Kampe SL, Kral MV (2019) Mechanical behavior of materials, global edition 5th edn. Pearson 4. Gdoutos EE, Daniel IM, Wang KA (2002) Failure of cellular foams under multiaxial loading. Compos Part A 33:163–176 5. Hosford WF (2010) Mechanical behavior of materials, 2nd edn. Cambridge University Press 6. Meyers MA, Chawla KK (1999) Mechanical behavior of materials. Prentice Hall 7. Nadai A (1931) Plasticity. McGraw Hill 8. Raghava R, Caddell RM, Yeh GSY (1973) The microscopic yield behavior of polymers. J Mater Sci 8:225–232

Chapter 3

Indentation Testing

Abstract In this chapter we present indentation methods for measuring the modulus of elasticity and the hardness of materials at macro, micro and nanoscale levels. For macroindentation testing we present the Brinell, Meyer, Vickers and Rockwell tests; for microindentation the Vickers and Knoop tests; for nanoindentation the elastic contact method and nanoindentation tests for measuring the fracture toughness of brittle materials at small volumes and the interfacial fracture toughness of thin films on substrates using conical and wedge indenters.

3.1 Introduction The indentation test is a simple commonly used technique to measure the hardness and related mechanical properties of materials in an easy and speedy way. The method consists of touching the material of interest with another material whose properties are known. In a typical test, a hard indenter of known geometry is driven into a soft material by applying a preset load and the dimensions of the resulting imprint are measured and related to the hardness index number. The first systematic test to measure the hardness was proposed by mineralogist Mohs in 1822. It is based on the ability of one material to visibly scratch another. Materials that were able to leave a permanent scratch on another were ranked harder with diamond assigned the maximum value of 10 in the scale. At the beginning of the twentieth century indentation tests were performed by Brinell using spherical balls as indenters. Other indentation tests include the Knoop, Vickers and Rockwell tests. With the advent of nanotechnology indentation was directed down to the nanometer range. This led to the development of nanoindentation which is a combination of high-resolution recording indentation and the accompanying data analyses for the determination of mechanical properties directly from the load–displacement data without imaging the indentation. The principal goal of nanoindentation is to obtain the modulus of elasticity, hardness and fracture toughness properties of thin films and small volumes of material. The forces involved are in the milli-Newton range and the depths of penetration are in the order of nanometers.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_3

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Indentation testing can be categorized in macroindentation, microindentation and nanoindentation depending on the magnitude range of the applied load. In the following we present the principles of contact mechanics and the major methods used to measure the hardness, the modulus of elasticity and the critical stress intensity factor.

3.2 Contact Mechanics Consider an elastic sphere of radius R that indents an elastic half-plane under an applied load P (Fig. 3.1). The radius a of the circle of contact of the sphere and the half-plane is given (according to Hertz theory) by a3 =

3 PR 4 E∗

(3.1)

where 1 − ν ,2 1 1 − ν2 + = E∗ E E,

(3.2)

E and ν are the modulus of elasticity and Poisson’s ratio for the half-plane and E , and ν , are the corresponding quantities for the indenter. E * combines the modulus of elasticity and Poisson’s ratio of the half-plane and the indenter. It is often referred to as the “combined modulus” or “reduced modulus” of the system. The total displacement of the indenter δ is given by Fig. 3.1 Contact between a rigid indenter of radius R and a half-plane. The radius of the circle of contact is a and the total depth of penetration is δ. ha is the depth of the circle of contact from the specimen free surface, and hc is the distance from the bottom of the contact to the contact circle

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65

a2 = δ= R



9P 2 16E ∗2 R

1/3 (3.3)

From Eq. (3.3) we obtain P=

4 ∗ 1/2 3/2 E R δ . 3

(3.4)

Equation (3.4) expresses the load P versus displacement δ relation of the indenter. Note that P varies with δ in a power law with exponent 3/2.

3.3 Macroindentation Testing In macroindentation tests the applied load P is in the range 2 N < P < 30 kN. The major tests are the Brinell, Meyer, Vickers and Rockwell tests. The tests determine the resistance of the material to penetration of a nondeformable indenter in the form of a ball, pyramid or cone. We briefly present these tests.

3.3.1 Brinell Test The material is indented by a hard spherical ball of diameter D through a fixed load P for a certain period of time (Fig. 3.2). The load is then removed and the chordal diameter d of the impression is measured with an optical microscope. The Brinell hardness number (BHN) is calculated as the load divided by the actual area Ac of the curved surface of the impression BHN =

P 2P   = √ Ac π D D − D2 − d 2

(3.5)

The load P is expressed in kilograms force. If Newton is used for the load the BHN must be divided by 9.81. The Brinell test has been standardized by the American Society for Testing and Materials (ASTM) and by the International Organization for Standardization (ISO). The load is applied for 10–30 s. The diameter d is taken as the mean value of two diameters of the impression at right angles. In a typical test the diameter of the ball d = 10 mm and the applied load P = 3000 kgf (~ 29.4 kN). Smaller loads of P = 1500 kgf (~ 14.9 kN) and P = 500 kgf (~ 4.9 kN) are used for softer materials. Tests on small parts use balls of diameter of D = 1 mm and load of P = 1 kgf (~ 9.8 N).

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Fig. 3.2 Brinell indentation test

3.3.2 Meyer Test The Meyer test is based on the same principle as the Brinell test (Fig. 3.2). The test was originally defined for spherical indenters but can be applied to any indenter shape. The Meyer hardness number (MHN) is expressed by the load divided by the projected contact area of the impression as MHN =

P 4P = Ap π d2

(3.6)

From experiments Meyer deduced the following relation which is known as Meyer’s law P = kd n

(3.7)

where k is a proportionality constant. The exponent n is known as the Meyer index. It was found that it is independent of the diameter D of the ball. Its value is between 2 and 2.5. The effective strain εeff of the indentation imposed by a spherical tip, as it was established experimentally, can be approximated by

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67

εeff = 0.2

d D

(3.8)

Equation (3.8) can be used to create the indentation stress–strain curve of a material by measuring the impression diameters for different applied loads. The yield stress σ y for materials such as copper and steel can be approximated as σ y = MHN/ 2.8.

3.3.3 Vickers Test The indenter in the Vickers test is a diamond in the form of a square-based pyramid. Its opposite sides meet at the apex at an angle of 136°, the edges at 148° and the faces at 68° (Fig. 3.3). The Vickers hardness number (HV) is calculated from the ratio of the applied load P in kilograms force (kgf) and the actual surface area of the impression Ac in mm2 . Ac is given by Ac =

d2 d2  136◦  = , 1.8544 2 sin 2

(3.9)

where d (mm) is the length of the diagonal measured from corner to corner on the residual impression of the specimen surface. The Vickers hardness number is calculated by HV =

P 1.8544P = Ac d2

(3.10)

where P is measured in kgf and d in mm. The applied loads vary in the range of 1–120 kgf with standard values of 5, 10, 20, 30, 50, 100, and 120 kgf. The time of application of the load is 10–15 s. The size of the impression is measured with a microscope with a tolerance of ∓ 1/1000 mm. The Vickers contact area is related to the penetration depth t (d = 7t) by Ac = 24.5 t 2 if the elastic recovery of the material is not important. This allows calculation of VHN from measurement of the penetration depth t.

3.3.4 Rockwell Test The previous three methods of indentation (Brinell, Meyer, Vickers) require measurement of the diameter of the indentation by an optical microscope. In the Rockwell test the depth of penetration of an indenter under load into the sample is measured. The hardness is determined from the applied load on a spherical indenter and the depth of penetration. The test involves the application of a minor load of P0 = 10 kgf

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Fig. 3.3 Vickers indentation test

(~ 98.1 N) followed by a major load P1 (Fig. 3.4). The minor load establishes the zero position. It is used to eliminate errors in measuring the penetration depth. The major load is applied and removed, while the minor load is still maintained. The Rockwell hardness (HR) is calculated by HR = N − 500 h,

Fig. 3.4 Rockwell indentation test

(3.11)

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69

Table 3.1 Main Rockwell scales Scale

Name

Indenter

A

HRA

120° diamond spheroconical

Load (kgf)

B

HRB

1/16-in.-diameter (1.588 mm) steel sphere

100

C

HRC

120° diamond spheroconical

150

D

HRD

120° diamond spheroconical

100

E

HRE

1/8-in.-diameter (3.175 mm) steel sphere

100

F

HRF

1/16-in.-diameter (1.588 mm) steel sphere

60

G

HRG

1/16-in.-diameter (1.588 mm) steel sphere

150

60

where N is a scale factor and h measured in mm is the difference of the penetration depths of the major and minor loads. The value of N depends on the used indenter. It is 100 for spheroconical indenters and 130 for a ball. Equation (3.11) shows that the penetration depth and the hardness are inversely proportional. Loads of 60, 100 and 150 kgf and ball diameters of 1/2, 1/4, 1/8, 1/16 in. are used, as described in the standards ISO 6508-1 and ASTM E18 for metallic materials and ISO 2039-2 for plastics. The main Rockwell scales are established by letters as: A, B, C, D, E, F, G, H, K, L, M, P, R, S and V. The most used scales are shown in Table 3.1. The main advantage of the Rockwell test is that the hardness values are displayed directly, thus avoiding calculations involved in the other methods.

3.4 Microindentation Testing In microindentation tests the applied load P is P < 10 N and the depth of penetration h is > 0.2 µm. The main tests are the Vickers and Knoop tests. As in the macroindentation tests, the material resistance to the penetration of a diamond indenter with a shape of a pyramid under a given load and within a specific time period is determined.

3.4.1 Vickers Test It is similar to the macroindentation test, with the only difference that the applied load is smaller (< 1 kgf (~ 9.81 N)). The test is described by ISO 6507 and ASTM E384.

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3.4.2 Knoop Test The indenter is a rhombic-based pyramidal diamond that produces an elongated indent. The angles from the opposite faces of the indenter are 172.5° for the long edge and 130° for the short edge. The indenter produces a rhombic-shaped indentation with approximate ratio of the long and short diagonals of 7–1. The Knoop hardness number (KHN) is determined from the projected area Ap as KHN =

P =  Ap d 2 cot

2P 172.5◦ 2

tan

130◦ 2

 = 14.24

P , d2

(3.12)

where d is the length of the longest diagonal in mm and P is the indentation load in kgf. The load is maintained for 10–15 s and then the indenter is removed leaving an elongated impression. A high-magnification microscope is used to measure the impression size. The applied load is in the range 10–1000 g (98 mN–9.8 N). A high-magnification microscope is needed to measure the indent size. The test is used particularly for very brittle materials or thin sheets.

3.5 Nanoindentation Testing 3.5.1 Introduction Study of the mechanical properties of materials at the nanoscale range has received much attention in the last years due to the development of nanostructured materials and the application of nanometer-thick films in engineering and electronic components. Nanoindentation is the combination of high-resolution recording indentation and the accompanying data analyses for the determination of mechanical properties directly from the load–displacement data without imaging the indentation. A major difference between macro and microindentation on one hand and nanoindentation on the other is that in nanoindentation the load and indentation depth is continuously monitored during the test. Nanoindentation employs ultra-low load indentation instruments (sensors and actuators) of high-resolution capable of continuously monitoring the loads and displacements on an indenter as it is driven and withdrawn from a material. Loads of the order of a nano-Newton (10−9 N) and displacements of the order of an Angstrom (10−10 m) can be accurately measured. Mechanical properties of materials in small dimensions can be very different from those of bulk materials having the same composition. From a nanoindentation test the elastic modulus, the hardness and the fracture toughness of a material can be measured. These mechanical properties characterize the three fundamental modes of deformation of solids, elasticity, plasticity and fracture, respectively.

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71

In the following we will first present a data analysis method for determining Young’s modulus and hardness based on the elastic contact method. We will then present nanoindentation testing for measuring the fracture toughness of brittle materials at small volumes.

3.5.2 The Elastic Contact Method A schematic representation, of a typical load, P, versus indenter displacement, h, data for an indentation experiment is shown in Fig. 3.5. For the analysis of the load–displacement curve we make the following assumptions [30]: i. Deformation upon unloading is purely elastic. ii. The compliances of the specimen and the indenter tip can be combined as springs in series. iii. The contact can be modeled using the analytical model developed by Sneddon for the indentation of an elastic half-space by a punch that can be described by an axisymmetric solid of revolution. A cross section of an indentation is shown in Fig. 3.6. During loading the total displacement h is written as h = hc + hs

(3.13)

where h is the vertical distance along which contact is made (called contact depth), hs is the vertical displacement of the surface at the perimeter of the contact and hc Fig. 3.5 Schematic representation of a typical load versus indenter displacement data for an indentation experiment

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Fig. 3.6 Cross section of an indentation

is the penetration depth of the indenter under load. When the indenter is withdrawn the final depth of the residual hardness impression under load is hf . The determination of Young’s modulus E ∗ defined in Eq. (3.2) is based on Hertz contact equation according to which ∗

E =

√

π S √ 2 A

(3.14)

where A is the contact area and S is the stiffness of the unloading curve (S = dP/dh) (Fig. 3.5). According to Oliver and Pharr [30], the unloading data for stiffness measurement are fitted into equation P = B(h − h f )m ,

(3.15)

where P is the load, (h − hf ) is the elastic displacement and B and m are material constants. The quantities B, m and hf are determined by a least squares fitting procedure of the unloading curve. For the analysis it is assumed that the geometry of the indenter is described by an area function A = A(h) which relates the cross-sectional area of the indenter to the distance from its tip. The contact area at maximum load is given by A = A(h c ).

(3.16)

The contact depth at maximum load, hc , that is, the depth along the indenter axis to which the indenter is in contact with the specimen, is determined by h c = h max − ε(h max − h i ),

(3.17)

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73

where hmax is the maximum depth and hi is the intercept depth, that is, the intercept of the tangent to the unloading load–displacement curve at maximum load with the depth axis. The constant ε is a function of the shape of the indenter tip. It takes the value 1 for a flat punch, the value of 0.7268 for a cone indenter and the value of 0.75 for a spherical or paraboloidal indenter. The quantities hmax and hi are determined from the load–displacement curve (Fig. 3.5). The area function A(hc ) depends on the shape of the indenter. For a Berkovich indenter (a three-sided pyramid with angles between the axis of symmetry and a face of 35.3°) it takes the form A(h c ) = 24.54 h 2c .

(3.18)

The hardness H c is defined by Hc =

Pmax Ac

(3.19)

where Pmax is the peak indentation load and Ac is the contact area under maximum load. This definition of hardness is different from that used in an imaging indentation test. In the latter case the area is the residual area measured after the indenter is removed, while in the nanoindentation test the area is the contact area under maximum load. This distinction is important for materials with large elastic recovery, for example rubber. A conventional hardness test with zero residual area would give infinite hardness, while a nanoindentation test would give a finite hardness. Equation (3.18) gives the area function A(hc ) for an ideal Berkovich indenter. However, real tips are never ideally sharp and generally, are characterized by a radius of curvature at the tip. In such cases the function A(hc ) must be determined. Methods for determining A(hc ) include the transmission electron microscope (TEM) replica method in which replicas of indentation are made and their areas are measured in TEM, the scanning force microscope (SFM) method in which the indenter tip is measured with a sharper SFM tip of known shape and a method based on Eq. (3.14) applied to a number of materials with known Young’s modulus. The above analysis suggests the following procedure for the determination of Young’s modulus and hardness in nanoindentation from Eqs. (3.14) and (3.19): i. Use Eq. (3.15) to fit the unloading data. ii. Find hc from Eq. (3.17) using the value of depth at maximum load, hmax , the slope of the fit at Pmax to obtain hi and the appropriate value of ε. iii. Use the area function A(hc ) of the indenter to find the contact area at maximum load from the contact depth hc . From the value of E * determined from Eq. (3.14) the elastic modulus, E, of the material is determined from Eq. (3.2) when the elastic modulus E , of the indenter is known.

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Nanoindentation testing is used for measuring the elastic modulus of a thin film on a substrate. The major difficulty encountered in the test is to avoid unintentional probing of the properties of the substrate. In order to achieve this the maximum depth of penetration in a test is no more than 10% of the thickness of the film. The substrate influences measurements for the determination of the elastic modulus. A few equations have been proposed for this reason. King [26] proposed the following formula for the combined modulus of the film, substrate and indenter, E eff √  1 − νs2 −αt/√ A 1 − νi2 1 1 − νf2  1 − e−αt/ A + = e + , E eff Ef Es Ei

(3.20)

where the subscripts f, s and i refer to the film, substrate and indenter, respectively, t is the film thickness, α is an empirical constant evaluated from a series of experimental results on films of known properties and thicknesses, A is the area of indentation and ν is Poisson’s ratio. Jung et al. [25] proposed a simple empirical power law as  E eff = E s +

Ef Es

L ,

(3.21)

where L=

1 1 + A(h/t)c

(3.22)

The constants A and c are found by calibration on specimens of known elastic modulus, and h is the penetration depth.

3.5.3 Nanoindentation for Measuring Fracture Toughness Nanoindentation testing is perhaps the most widely used method to measure fracture toughness of brittle materials, like glass and ceramics. The method was developed by Lawn et al. [27]. During elastic/plastic contact two types of cracks may form: those cracks which form on symmetry median planes containing the load axis and those which form on planes parallel to the specimen surface. The pertinent cracks used to determine fracture toughness are the first of these, the median/radial cracks. These cracks emanate from the edge of the contact impression, are oriented normal to the specimen surface on median planes coincident with the impression diagonals and have a half-penny configuration (Fig. 3.7). It was observed that most of the crack development occurs not on loading, but on unloading the indenter. Thus, the main driving force for the formation of these cracks is the irreversible component of the contact stress. After unloading the indenter, characteristic radial traces are left on the specimen

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75

Fig. 3.7 Median/radial cracks emanating from the edge of the contact impression oriented normal to a specimen surface

surface (Fig. 3.8). They provide the necessary information for the evaluation of fracture toughness. The critical stress intensity factor, K c , is given by Lawn et al. [27] 

E Kc = α H

1/2

P c3/2

(3.23)

where E H P c α

Young’s modulus Hardness Peak indentation load Characteristic crack length (crack length plus half diagonal impression length) Empirical dimensionless constant which depends on the geometry of the indenter (α = 0.016 for a Vickers pyramidal indenter, α = 0.040 for a cube-corner indenter).

Equation (3.23) provides a simple formula for the determination of fracture toughness of brittle materials by measuring the length of the radial cracks emanating from the impression corners. Note that for the determination of K c both E and H

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Fig. 3.8 Characteristic radial traces of the median/radial cracks left on the specimen surface after unloading

are needed. They can be determined from the analysis of the nanoindentation data. Thus, from one test all three quantities E, H and K c can be determined. For an accurate determination of the crack length the test surface must be prepared to an optical finish. The method applies to those materials which produce a well-defined radial/median crack system. In the application of the method it should be mentioned that there are well-defined loads called cracking thresholds, which depend on the material and the type of indenter below which cracks do not develop. For most ceramic materials the cracking thresholds for Vickers and Berkovich indenters are about 250 mN or more. The crack lengths produced at these loads are relatively large. The cracking thresholds can be reduced by using indenters with smaller tip angles, for example, the cube-corner indenter has an angle of 35.3° between the axis of symmetry and a face, as compared to 65.3° for the Berkovich indenter. Using cube-corner indenters thresholds less than 10 mN can be achieved, while for Vickers indenters threshold are of 1 N or greater. Table 3.2 [33] shows values of Young’s modulus, E, hardness, H, and fracture toughness, K c , for various materials.

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Table 3.2 Properties of materials used in indentation cracking measurement of fracture toughness √ Material Cube-corner Vickers threshold E a (GPa) H a (GPa) K c (MPa m) threshold (mN) (mN) Soda-lime glass

0.5–1.5

250–500

76.1

6.1

0.70b

Fused quartz

0.5–1.5

1000–1500

69.3

8.3

0.58b

Pyrex glass

1.5–4.4

500–1000

60.5

6.3

0.63b

Silicon (100)

0.5–1.5

20–50

185.6

11.5

0.7c

Silicon (111)

0.5–1.5

50–100

205.8

11.2

0.7c

Germanium (111)

0.5–1.5

< 10

133.6

10.1

0.5c

Sapphire (111) 4.4–13.3

50–100

433.1

25.9

2.2c

Spinel (100)

4.4–13.3

100–150

286.2

18.4

1.2c

Silicon nitride (NC 132)

40–120

1000

319.9

21.6

4.7c

Silicon carbide 4.4–13.3 (SA)

100–150

454.7

30.8

2.9c

Silicon carbide Grain pushout (ST)

500–1000

427

21.8

4.1c

a b c

Nanoindentation measurements with Berkovich indenter 3-pt bend chevron notch method From material data sheet or literature

3.5.4 Nanoindentation for Measuring Interfacial Fracture Toughness—Conical Indenters (a) Introduction As it was previously discussed, determination of the mechanical properties of thin films is one of the most popular applications of nanoindentation. An important property of thin coatings on substrates is the interfacial fracture toughness. It dictates the separation of the coating from the substrate. Interfacial fracture toughness can be determined using nanoindentation. For this reason a coated surface is loaded with an indenter until a critical load is reached to initiate an interface crack. The load is then increased causing the interface crack to propagate in a stable fashion. The interfacial fracture toughness is calculated from the indenter load and the length of the debond crack. Calculation of the interfacial fracture toughness will be performed in this section using a conical indenter (plane stress). In the next section the calculation will be performed using a wedge indenter (plane strain). (b) Compressed films Consider a circular delamination of radius a at the interface between a uniformly prestressed thin film and a substrate (Fig. 3.9). The interface crack is parallel to

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the free surface and does not disturb the stress field in the film. Hence a stress concentration at the edge of the crack is not induced. At a critical applied compressive stress the film buckles away from the substrate. The resulting separation generates large tensile stresses at the perimeter of the interface crack and may cause crack growth. The strain energy release rate, G, associated with the propagation of the crack across a buckled delaminated area was calculated by Evans and Hutchinson [24] as   (1 − ν)(1 − α)t σ02 − σc2 , G= E

(3.24)

where σ0 E, ν σc t

Biaxial compression applied to the film Young’s modulus and Poisson’s ratio of film material Critical buckling stress for a clamped circular plate Film thickness

Fig. 3.9 Circular delamination at the interface between a uniformly prestressed thin film and a substrate: a unconstrained film, b constrained film prior to buckling and c buckled film

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and α=

1 1 + 1.207(1 + ν)

(3.25)

(α = 0.383 for ν = 1/3). The stress σ c is given by σc =

t2 kE  2 12 1 − ν 2 a 

(3.26)

where k = 14.68. (c) Indented stress-free films Consider a circular delamination of radius a at the interface between a thin film and a substrate. When the film is indented at the center of the delamination, in the absence of plastic pile-up and when the plastic deformation due to indentation is confined to the film, the film is subjected to dilatation governed by the indentation volume, V 0 (Fig. 3.10). When the interface contains a crack of radius a and the film is considered as a clamped plate of radius a the displacement Δ0 imposed at the plate edge to offset the dilatation, is given by Marshall and Evans [29] Δ0 ≈

V0 , 2π at

(3.27)

where t is the film thickness. The corresponding biaxial compressive stress induced in the unbuckled film is then σ0 =

E V0 EΔ0 = 2π (1 − ν)ta 2 (1 − ν)a

(3.28)

The strain energy release rate for crack growth is given by Evans and Hutchinson [24]

Fig. 3.10 Indentation of volume V 0 and plastic zone configuration used to calculate the residual strain energy and the expansion Δ0

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 1 − ν 2 tσ02 G= 2E

(3.29)

Note that contrary to the previous case of the uniformly compressed film, a strain energy release rate exists in the unbuckled film. When the indentation-induced stress σ 0 becomes equal to the critical stress σ c given by Eq. (3.26) the film buckles. From Eqs. (3.26) and (3.28) with σ 0 = σ c we obtain the critical indentation volume for buckling of the film: V0c =

π kt 3 6(1 + ν)

(3.30)

If buckling occurs the strain energy release rate for crack growth is given by Marshall and Evans [29]  (1 − ν)t σ02 (1 + ν) − 2(1 − α)(σ0 − σc )2 , G= 2E

(3.31)

where the buckling stress σ c (σ 0 ≥ σ c ) is given by Eq. (3.26). (d) Prestressed indented films Consider a thin film subjected to a prestress σ 0 and indented by a sharp diamond tip (Fig. 3.11a). The material deforms to an indentation volume V 0 . Due to indentation an interfacial crack nucleates and propagates. If the indenter is driven deep enough, so that the crack reaches its critical buckling length on each side of the indenter, the film double buckles (Fig. 3.11b). Otherwise, single-buckling might occur when the indenter is removed (Fig. 3.11c). The strain energy release rate is given by Marshall and Evans [29]  (1 − ν)t σI2 (1 + ν) + 2(1 − α)σ02 − 2(1 − α)(σI − σc )2 G= 2E

(3.32)

where σ I Indentation stress calculated from Eq. (3.28) σ 0 Prestress in the film σ c Buckling stress calculated from Eq. (3.26). Note from Eq. (3.32) that when the film does not buckle (α = 1) the energy release rate depends only on the indentation stress dictated by the indented volume and not on the prestress.

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Fig. 3.11 Thin film on a substrate indented by a sharp diamond indenter: a no buckling, b doublebuckling and c single-buckling after removed of indenter tip. Figure refers to both conical and wedge indenters

3.5.5 Nanoindentation for Measuring Interfacial Fracture Toughness—Wedge Indenters A microwedge indentation test was proposed by De Boer and Gerberich [23] for the evaluation of interfacial fracture toughness of thin films on structures. In the test a symmetric diamond wedge is indented uniformly on a thin film line of finite width to cause an interfacial crack to nucleate and propagate (Fig. 3.11). The indenter plastically deforms an indentation volume in the film. Conditions of plane strain dominate during indentation. Three different film configurations may occur. If the indentation depth is small a short interfacial crack forms and the film does not buckle (Fig. 3.11a). When the film is purely adhered a large interfacial crack forms, so that the film may buckle when the indenter is removed (Fig. 3.11c). This configuration is referred to as single-buckling. When the indentation depth is deep, symmetric buckling may occur on each side of the indenter tip during indentation (Fig. 3.11b). This configuration is referred to

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as double-buckling. Following removal of the indenter single-buckling may occur. Otherwise the film remains double-buckled. The strain energy release rate, G, for the above three film configurations was calculated by De Boer and Gerberich [23] as: i. No buckling during indentation G=

E , V02 2b2 ha 2

(3.33)

where E, V0 b h a

E/(1 − ν 2 ) Half of the indentation volume Width of thin film Thickness of thin film Half-length of interfacial crack.

The 1/a2 dependence in Eq. (3.33) is much stronger than the 1/a4 dependence in the axisymmetric case (Eqs. 3.28 and 3.29). This means that cracks will travel much further for the same indentation volume. This occurs because of the greater driving force of the wedge indenter tip relative to an axisymmetric indenter. ii. Double-buckling during indentation The interfacial strain energy release rate is given by De Boer and Gerberich [23]

   db 2  σ σ02 h σcdb −3 c G= 4 , 2E , σ0 σ0

(3.34)

where   E , V0 π2 E, h 2 db , σc = σ0 = abh 3 a

(3.35)

σcdb = Critical stress for double-buckling. Once crack is long enough it is constrained at its center and the Euler buckling condition is met. The threshold volume for double-buckling is Vthdb =

π2 b 3 h . 3 a

(3.36)

iii. Single-buckling after indentation Consider the case in which double-buckling is not reached during indentation. The interfacial crack is no longer constrained at the center. The buckling stress reduces

3.5 Nanoindentation Testing

83

as the crack length is doubled. The Euler buckling stress σcsb,i for a beam of length 2a is σ0sb,i

    π2 E, h 2 d2 = 1− 2 12 a h

(3.37)

where d is the depth of penetration into the film. The interfacial strain energy release rate is calculated as G=−

   3σcsb,i σcsb,i σ02 h 1 − 1 − 2E , σ0 σ0

(3.38)

where σ0 ≥ σcsb,i . If 1 ≤ σ0 /σcsb,i ≤ 3, then G ≥ 0. The film may single-buckle after removing the indenter tip, but crack extension will likely not occur. If σ/σcsb,i > 3 then G < 0, that is buckling can take place, but crack extension does not occur. A similar wedge indentation test has been developed by Vlassak et al. [35] to measure the interfacial fracture toughness of strongly adhering brittle films on ductile substrates. When a wedge is driven through a thin film coating and into the substrate plastic deformation of the substrate forces the coating to be displaced away from the wedge. This increases the stress in the coating and therefore the driving force for delamination of the coating. In this test the indenter penetrates into the substrate, so that the coating does not affect significantly the substrate displacement field. Delamination of the film from the substrate is caused by the plastic deformation of the substrate. The strain energy release rate is given by Vlassak et al. [35]   1 − ν 2 σx2x h , G= 2E

(3.39)

where σ xx is the stress in the film perpendicular to the wedge line. It is given by σx x = σr − ν

E W 2 tan β 1 − ν 2 πa 2

(3.40)

where σr W β a

Residual stress in the film Half width of the wedge indentation Inclination of the face of the wedge to the surface of the film Interfacial crack length.

The wedge delamination test has the advantage over the axisymmetric indentation test that due to prevailing plane strain conditions no tensile hoop stresses develop in the film and, therefore, no radial cracks. Using the wedge delamination test it is possible to apply much larger energy release rates to the interface between the film and the substrate. This makes the technique suitable for the study of brittle coatings that exhibit good adhesion to their substrates. An experimental advantage

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of the wedge test is that plane surface of the wedge may be used as a mirror allowing in situ optical measurements of crack growth. A problem with the wedge test is the alignment. It is difficult to align the wedge perpendicular to the plane of the film and wedges may not be perfectly symmetric. Misalignment causes asymmetric crack growth on both sides of the wedge. The wedge delamination test was applied for an interface of diamond films on titanium substrates [35]. The interfacial strain energy release rate was estimated at 51 ± 11 J/m2 . This value is much larger than what one would expect for the atomistic work of fracture for the titanium-diamond interface because a large portion of the energy required for delamination of the diamond film is spent in plastic deformation of the substrate.

3.6 ASTM Standards ASTM has published the following standards for indentation testing: • • • • • • • • • • • • •

E18-22, “Standard Test Methods for Rockwell Hardness of Metallic Materials” E2546-15, “Standard Practice for Instrumented Indentation Testing” C730, “Standard Test for Knoop Indentation Hardness of Glass” D1474, “Standard Test Methods for Indentation Hardness of Organic Coatings” A833-17, “Standard Test Method for Indentation Hardness of Metallic Materials by Comparison Hardness Testers” E140-12B (2019) e1, “Standard Hardness Conversion Tables for Metals Relationship Among Brinell Hardness, Vickers Hardness, Rockwell Hardness, Superficial Hardness, Knoop Hardness, Scleroscope Hardness, and Leeb Hardness” E384-17, “Standard Test Method for Microindentation Hardness of Materials” E92-17, “Standard Test Methods for Vickers Hardness and Knoop Hardness of Metallic Materials” D2240-15, “Standard Test Method for Rubber Property-Durometer Hardness” D1415-18, “Standard Test Method for Rubber Property-International Hardness” C661-15, “Standard Test Method for Indentation Hardness of Elastomeric-Type Sealants by Means of a Durometer” E103-17, “Standard Practice for Rapid Indentation Hardness Testing of Metallic Materials” E1015, “Standard Test Method for Brinell Hardness of Metallic Materials”.

Further Readings

85

Further Readings Books 1. Alta K (2011) Effects of varying humidity in polymers by nanoindentation: investigation of the effects of varying humidity in additive manufactured by depth sensing indentation. LAP LAMBERT Academic Publishing 2. Antunes J (2010) On depth sensing indentation of materials. VDM Verlag 3. Argatov I, Mishuris G (2018) Indentation testing of biological materials. Springer 4. Bahr DF, Morris DJ (2008) Nanoindentation: localized probes of mechanical behavior of materials. In: Sharpe WN (ed) Handbook of experimental solid mechanics. Springer, pp 389–407 5. Bhattacharyya A (2018) Substrate effect and nanoindentation failure. LAP LAMBERT Academic Publishing 6. Bourhis EL, Morris DJ, Oyen ML, Schwaiger R, Staedler T (eds) (2008) Fundamentals of nanoindentation and nanotribology IV: volume 1049 (MRS proceedings). Cambridge University Press 7. Cagliero R (2016) Instrumented indentation test: in the macro hardness range. LAP LAMBERT Academic Publishing 8. Chen L (2015) Micro-nanoindentation in materials science. NY Research Press 9. Dey A, Mukhopadhyay AK (2018) Nanoindentation of natural materials: hierarchical and functionally graded microstructures. CRC Press 10. Fischer-Cripps AC (2011) Nanoindentation, 3rd edn. Springer 11. Gdoutos EE (2020) Fracture mechanics, 3rd edn. Springer, pp 371–385 12. Handadi UP, Udupa KR (2017) Indentation creep studies on stainless steel welds and solder alloys. Scholars’ Press 13. Mohanty P, Behera A (2020) Nanoindentation study of NiTi thin film shape memory alloys: varying annealing temperature. LAP LAMBERT Academic Publishing 14. Murthy CSN (2021) Rock indentation: experiments and analyses. CRC Press 15. Navamathavan R, Nirmala R (2011) Mechanical properties of some III–V and II–VI semiconductor alloys: a micro and nanoindentation approaches. LAP LAMBERT Academic Publishing 16. Oyen ML (2019) Handbook of nanoindentation: with biological applications. Jenny Stanford Publishing 17. Solomah AG (ed) (2004) Indentation techniques in ceramic materials characterization: theory and practice. Wiley 18. Tiwari A, Natarajian S (2017) Applied nanoindentation of advanced materials. Wiley 19. Tsui T, Pharr M (2018) Advanced nanoindentation of materials. MDPI AG 20. Tsui T, Volinsky A (2019) Small scale deformation using advanced nanoindentation techniques. MDPI AG 21. Wang H, Zhu L, Xu B (2018) Residual stresses and nanoindentation testing of films and coatings. Springer

Articles 22. Broitman E (2017) Indentation hardness measurements at macro-, micro-, and nanoscale: a critical review. Tribol Lett 65:1–18 23. De Boer MP, Gerberich WW (1996) Microwedge indentation of the thin film fine line—I. Mechanics. Acta Mater 44:3169–3175 24. Evans AG, Hutchinson JW (1984) On the mechanics of delamination and spalling in compressed films. Int J Solids Struct 20:455–466

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25. Jung Y-G, Lawn BR, Martyniuk M, Huang H, Hu XZ (2004) Evaluation of elastic modulus and hardness of thin films by nanoindentation. J Mater Res 19:1–5 26. King RB (1987) Elastic analysis of some punched problems for a layered medium. Int J Solids Struct 23:1657–1664 27. Lawn BR, Evans AG, Marshall DB (1980) Elastic/plastic indentation damage in ceramics: the median/radial crack system. J Am Ceram Soc 63:574–581 28. Li X, Bhushan B (2002) A review of nanoindentation continuous stiffness measurement technique and its applications. Mater Charact 48:11–36 29. Marshall DB, Evans AG (1984) Measurement of adherence of residually stressed thin films by indentation. I. Mechanics of interface delamination. J Appl Phys 56:2632–2638 30. Oliver WC, Pharr GM (1992) An improved technique for determining hardness anelastic modulus using load and displacement sensing indentation experiments. J Mater Res 7:1564– 1583 31. Oliver WC, Pharr GM (2004) Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology. J Mater Res 19(1):3– 20 32. Pathak S, Kalidindi SR (2015) Spherical nanoindentation stress–strain curves. Mater Sci Eng 91:1–36 33. Pharr GM (1998) Measurement of mechanical properties of ultra-low load indentation. Mater Sci Eng A 253:151–159 34. Prasanna HU, Udupa KR (2016) Indentation creep studies to evaluate the mechanical properties of stainless steel welds. Aust J Mech Eng 14:39–43 35. Vlassak JJ, Drory MD, Nix WD (1997) A simple technique for measuring the adhesion of brittle films to ductile substrates with application to diamond-coated titanium. J Mater Res 12:1900–1910 36. Wen W, Becker AA, Sun W (2017) Determination of material properties of thin films and coatings using indentation tests: a review. J Mater Sci 52:12553–12573

Chapter 4

Fracture Mechanics Testing

Abstract Fracture mechanics constitutes a powerful method for the determination of the load-carrying capacity of structures and machine components in the presence of cracks. This approach is in contrast to traditional failure criteria (maximum stress/ strain, Tresca, von Mises, Coulomb–Mohr, etc.) which ignore the presence of defects. Since structures and machine components cannot be constructed without defects, on the grounds of practicality, fracture mechanics is used to determine either the safe operating load for a prescribed crack size or the safe crack size for a prescribed operating load. Design by fracture mechanics necessitates a parameter known as fracture toughness which characterizes the resistance of a material to crack extension. Fracture toughness is a material property and it should be size independent. It expresses the ability of a material to resist fracture in the presence of cracks. Fracture toughness is analogous to the yield or ultimate stress used in design by the conventional failure criteria. In this chapter we consider the following fracture mechanics failure criteria: the stress intensity factor criterion, the J-integral criterion, the crack opening displacement criterion and the strain energy density criterion, and present the experimental procedure for the determination of fracture toughness for each criterion. Furthermore, we discuss the stress intensity factor failure criterion for dynamic fracture.

4.1 Critical Stress Intensity Factor Fracture Criterion 4.1.1 The Linear Elastic Stress Field The singular linear elastic stress field for opening-mode loading (loads are perpendicular to the crack plane) at a point P(r, θ ) in the vicinity of the crack tip (Fig. 4.1) is given by the stresses σ x , σ y and τ xy as [19]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_4

87

88

4 Fracture Mechanics Testing

Fig. 4.1 Crack of length 2a in an infinite plate subjected to a uniform stress σ at infinity

 θ KI θ 1 − sin sin σx = √ cos 2 2 2πr  θ KI θ σy = √ 1 + sin sin cos 2 2 2πr θ 3θ KI θ τx y = √ cos sin cos , 2 2 2 2πr

 3θ 2  3θ 2

(4.1)

where K I is the opening-mode stress intensity factor. It expresses the strength of the singular elastic stress field. K I depends on the applied loads, the crack length and the geometrical configuration of the cracked plate.

4.1.2 Strain Energy Release Rate The strain energy release rate GI expresses the energy released per unit area of crack extension. It is related to the stress intensity factor K I by Gdoutos [19] GI =

K I2 , E

(4.2)

4.1 Critical Stress Intensity Factor Fracture Criterion

89

for generalized plane stress, and by   1 − ν 2 K I2 GI = , E

(4.3)

for plane strain.

4.1.3 Fracture Criterion When the plastic deformation at the crack tip is small we can assume that crack growth occurs when the strain energy release rate G becomes equal to the energy required to create a unit area of material surface, R. The condition of crack growth can be expressed as G I = G c = R,

(4.4)

where Gc is the critical strain energy release rate. Equation (4.4) is usually expressed in terms of the opening-mode stress intensity factor K I . Let us introduce a new material constant K c by / Kc =

ER , β

(4.5)

where β = 1 for plane stress and β = 1 − ν 2 for plane strain. From Eqs. (4.2), (4.3) and (4.5) we obtain KI = Kc

(4.6)

Equation (4.6) expresses the critical stress intensity factor fracture criterion for crack growth. The left-hand side of the equation, K I , depends on the applied load, the crack length and the geometrical configuration of the cracked body. The righthand side of the equation, K c , is a material parameter that can be determined experimentally.

4.1.4 Variation of Kc with Specimen Thickness Laboratory experiments indicate that Gc or K c varies with the specimen thickness B (Fig. 4.2). Three distinct regions in the K c versus B curve can be distinguished. They correspond to “very thin”, “very thick” and “intermediate range thickness” specimens. Study of the load–displacement response and the appearance of the fracture

90

4 Fracture Mechanics Testing

Fig. 4.2 Critical fracture toughness Gc (or K c2 ) versus plate thickness B

surfaces of the specimen are helpful in understanding the fracture mechanisms in each of these three regions. The fractures are classified as square or slant according to whether the fracture surface is normal to or forms a 45° angle with the direction of the applied tensile load. We now analyze the state of affairs in the three regions of Fig. 4.2. In region I, that corresponds to thin specimens, the critical fracture toughness Gc (which is proportional to K c2 ) increases almost linearly with B up to a maximum value at a critical thickness Bm . The load–displacement response is linear, and the fracture surface is completely slant (Fig. 4.3a). In this case, plane stress predominates in the specimen and yielding occurs on planes through thickness at 45° to the specimen flat surfaces. Under such conditions, the crack extends in an antiplane shear mode. In region III, corresponding to thick specimens, the load–displacement response is linear, and the state of stress is predominantly plane strain, except for a thin layer at the free surfaces where plane stress dominates (Fig. 4.3c). The fracture surface is almost completely square with very small slant parts at the free surfaces. A triaxial state of stress is produced in most parts of the specimen, which reduces the ductility of the material, and fracture takes place at the lowest value of the critical strain energy release rate Gc . For increasing thickness beyond a critical minimum value, Bc , plane strain conditions dominate and the fracture toughness remains the same. The critical value of stress intensity factor in region III for plane strain conditions is denoted by K Ic and is independent of the specimen thickness. K Ic is the so-called fracture toughness and represents an important material property. The larger the value of K Ic , the larger the resistance of the material to crack propagation. Experimental

4.1 Critical Stress Intensity Factor Fracture Criterion

91

Fig. 4.3 Load–displacement response for a plane stress, b transitional behavior and c plane strain

determination of K Ic takes place according to the ASTM specification described in the next section. Region II corresponds to intermediate values of specimen thickness. The fracture behavior is neither predominantly plane stress nor predominantly plane strain. The central and edge regions of the specimen are under plane strain and plane stress conditions, respectively. They are of comparable size. The fracture toughness in this region changes between the minimum plane strain and the maximum plane stress value. In the load–displacement curve (Fig. 4.3b) the crack extends mainly from the

92

4 Fracture Mechanics Testing

Fig. 4.4 Thumbnail crack growth with square and slant fracture

center of the specimen thickness, while the edge regions are plastically deformed. The crack grows in a “thumbnail” shape (Fig. 4.4) under constant or decreasing load while the overall displacement is increased. This behavior is known as “pop-in” (Fig. 4.3b). After crack growth at pop-in, the stiffness (slope) of the load–displacement curve decreases, since it corresponds to a longer crack. A simplified model for the explanation of the decrease of fracture toughness with the increase of the depth of square fracture was proposed by Krafft et al. [27]. Figure 4.5 shows the part of square fracture of length (1 − S)B and the two slant fracture parts of length BS/2, where S is a nondimensional coefficient that indicates the percentage of the square and slant lengths of the specimen thickness. The slant fracture surface makes an angle 45° with the crack plane. If dW f /dA is the work consumed to produce a unit area of flat fracture (dA = Bda) and (dW p /dV ) is the work for plastic deformation per unit volume, the work done for an advance of crack length by da is  dW =

   dWp B 2 S 2 dWf da (1 − S)Bda + dA dA 2

(4.7)

The strain energy release rate G (which is the rate of the energy released per unit area of crack extension) is calculated as dW = G= dA



   dWp B S 2 dWf . (1 − S) + dA dA 2

(4.8)

By fitting the experimental data to this equation and assuming that the slant fracture has a thickness of 2 mm, Krafft et al. [27] obtained for the critical strain energy release rate Gc the expression G c = 20(1 − S) + 200S 2 .

(4.9)

4.1 Critical Stress Intensity Factor Fracture Criterion

93

Fig. 4.5 Calculation of crack growth resistance according to the Kraft et al. [1] model

Equation (4.9) relates Gc to S. The variation of Gc and 100(1 − S) with the specimen thickness B for the experiments of Krafft et al. is shown in Fig. 4.6. Irwin [25] suggested the following semi-empirical equation which relates the critical stress intensity factor K c for a plate of thickness B to the critical stress intensity factor K Ic (plane strain fracture toughness) for plates of large thickness / K c = K Ic 1 +

  1.4 K Ic 4 , B 2 σY

(4.10)

where σ Y is the yield stress. Equation (4.10) indicates that K c depends on the specimen thickness B. For large values of B, K c ≈ K Ic .

94

4 Fracture Mechanics Testing

Fig. 4.6 Crack growth resistance Gc and percentage square fracture 100(1 − S) versus plate thickness B according to experiments by Kraft et al. [1]

4.1.5 Experimental Determination of KIc For the experimental determination of the plane strain fracture toughness K Ic (the critical value of stress intensity factor for plates of large thickness) special requirements must be fulfilled in order to obtain reproducible values under conditions of maximum constraint at the crack tip. The size of the plastic zone at the crack tip must be very small relative to the specimen thickness and the K I -dominant region. The procedure for measuring K Ic has been standardized by the American Society for Testing and Materials (ASTM) [2] to meet these requirements in small specimens that can easily be tested in the laboratory. In this section we present the salient points of the ASTM standard test method for the experimental determination K Ic . (a) Test specimens The specimens used to measure K Ic must be designed to ensure that the size of the plastic zone is very small relative to the specimen thickness, and plane strain conditions dominate at the crack tip. According to the ASTM standard, the specimen thickness B, the crack length a and the specimen width W, must be fifty times greater than the radius r c of the plane strain plastic zone at fracture, that is B, a, W ≥ 50rc

(4.11)

The radius r c for plane strain according to the Irwin model is given by Gdoutos [19]

4.1 Critical Stress Intensity Factor Fracture Criterion

1 rc = 6π



K Ic σY

95

2 .

(4.12)

From (4.11) and (4.12) we obtain   K Ic 2 B, a, W ≥ 2.5 . σY

(4.13)

Many precracked test specimens are described in the ASTM Specification E39981. They include the three-point bend specimen; the compact tension specimen; the arc-shaped specimen and the disk-shaped compact specimen. The geometrical configurations of the most widely used three-point bend specimen and compact tension specimen are shown in Figs. 4.7 and 4.8. Several formulas have been proposed for the calculation of stress intensity factor K I for the standard specimens. According to ASTM the following expressions are used: 

  a 1/2 a a a a2 1.99 − 2.15 − 3.93 + 2.7 2 1− 3 PS W W W W W KI = ,  3/2  3/2 a a BW 1− 2 1+2 W W (4.14) for the bend specimen, and

KI =

P BW 1/2

  a 3  a 4

 a 2 a a 2+ + 14.72 − 5.6 0.886 + 4.64 − 13.32 W W W W W  a 3/2 1− W

(4.15)

Fig. 4.7 Three-point bend specimen according to ASTM standards

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4 Fracture Mechanics Testing

Fig. 4.8 Compact tension specimen according to ASTM standards

for the compact tension specimen. The quantities a, W and B are shown in Figs. 4.7 and 4.8. S is the distance between the points of support of the beam in Fig. 4.7. Equation (4.14) is accurate to within 0.5%, over the entire range of a/W (a/W < 1), while Eq. (4.15) is accurate to within 0.5% for 0.2 < a/W < 1. (b) Precrack The precrack introduced in the specimen must simulate the ideal plane crack with zero root radius. The effect of the notch radius ρ on the critical value of the stress intensity factor K c is shown in Fig. 4.9. K c decreases with decreasing ρ until a limiting radius ρ c is obtained. Below ρ c , K c is approximately constant. This shows that a notch with radius smaller than ρ c can simulate the theoretical crack. The crack front must be normal to the specimen free surfaces, and the material around the crack should experience little plastic deformation or damage. To meet these requirements, a special technique is used for the construction of the precrack in the specimen. A chevron starter notch (Fig. 4.10) of length 0.45W is first machined in the specimen. The notch is then extended by fatigue at a length 0.05W beyond the notch root. The advantage of the chevron notch is that it forces crack initiation in the center, so that a straight crack front is obtained. The crack length a used in the calculation of stress intensity factor is the average of the crack lengths measured at the center of the crack front and midway between the center and the end of the crack front on each surface of the specimen (a = (a1 + a2 + a3 )/3). The surface crack length should not differ from the average length by more than 10%.

4.1 Critical Stress Intensity Factor Fracture Criterion

Fig. 4.9 Effect of notch radius ρ on the critical stress intensity factor K c

Fig. 4.10 Chevron starter notch

97

98

4 Fracture Mechanics Testing

To ensure that the material around the crack front does not experience large plastic deformation or damage, and that the fatigue crack is sharp, the fatigue loading should satisfy some requirements. The maximum stress intensity factor to which the specimen is subjected during fatigue must not exceed 60% of K Ic and the last 2.5% √ of such that K /E < 0.002. in (= the crack length should be loaded at a maximum K I I √ 0.32 × 10−3 m). (c) Experimental procedure The precracked standard specimen is loaded by special fixtures recommended by ASTM. The load and the relative displacement of two points located symmetrically on opposite sides of the crack plane are recorded simultaneously during the experiment. The specimen is loaded to produce a rate of increase of stress intensity, K I , within the range 0.55–2.75 MPam1/2 /s. A test record consisting of an autographic plot of the output of the load-sensing transducers versus the output of the displacement gage is obtained. A combination of load-sensing transducer and autographic recorder is selected so that the maximum load can be determined from the test record with an accuracy of 1%. The specimen is tested until it can sustain no further increase of load. (d) Interpretation of test record and calculation of K Ic For perfectly elastic behavior until fracture, the load–displacement curve should be a straight line. Most structural materials, however, present elastoplastic behavior which, combined with some stable crack growth before fracture, leads to nonlinear load–displacement diagrams. The principal types of the load–displacement curve observed in experiments are shown in Fig. 4.11. Type I corresponds to nonlinear response, type III to purely linear response and type II reflects the phenomenon of pop-in. For the determination of a valid K Ic , a conditional value K Q is first obtained. This involves a geometrical construction on the test record, consisting of drawing a secant line OP through the origin with slope equal to 0.95 of the slope of the tangent to the initial linear part of the record. The load P5 corresponds to the intersection of the secant line with the test record. The load PQ is then determined as follows: if the load at every point on the record which precedes P5 is lower than PQ then PQ = P5 (type I); if there is a maximum load preceding P5 which is larger than P5 then PQ is equal to this load (types II and III). The test is not valid if Pmax /PQ is greater than 1.10, where Pmax is the maximum load the specimen was able to sustain. In the geometrical construction, the 5% secant offset line represents the change in compliance due to crack growth equal to 2% of the initial length. After determining PQ , we calculate K Q using Eq. (4.14) for the bend specimen or Eq. (4.15) for the compact tension specimen. When K Q satisfies Inequality (4.13), then K Q is equal to K Ic and the test is a valid K Ic test. When Inequality (4.13) is not satisfied, it is necessary to use a larger specimen. Its dimensions can be estimated on the basis of K Q . Values of the critical stress intensity factor K Ic and the ultimate stress σ u for some common metals and alloys are given in Table 4.1.

4.1 Critical Stress Intensity Factor Fracture Criterion

99

Fig. 4.11 Determination of PQ for three types of load–displacement response according to ASTM standards Table 4.1 Values of critical stress intensity factor K Ic and 0.2% offset yield stress σ Y at room temperature for various alloys Material

σY

K Ic MPa

√ m

ksi

√ m

MPa

ksi

300 maraging steel

1669

242

93.4

85

350 maraging steel

2241

325

38.5

35

D6AC steel

1496

217

66.0

60

AISI 4340 steel

1827

265

47.3

43

A533B reactor steel

345

50

197.8

180

Carbon steel

241

35

219.8

200

Al 2014–T4

448

65

28.6

26

Al 2024–T3

393

57

34.1

31

Al 7075–T651

545

79

29.7

27

Al 7079–T651

469

68

33.0

30

Ti 6AJ–4 V

1103

160

38.5

35

Ti 6AJ–6 V–2Sn

1083

157

37.4

34

945

137

70.3

64

Ti 4Al–4Mo–2Sn–0.5Si

100

4 Fracture Mechanics Testing

4.2 J-Integral Fracture Criterion 4.2.1 J-Integral J-integral for a two-dimensional plane elastic body is a line integral defined by Gdoutos [19] (Fig. 4.12) ∫ J⎡ =

ωdy − Tk ⎡

∂u k ds (k = x, y) ∂x

(4.16)

where ⎡ is a closed contour bounding a region of a two-dimensional space. Equation (4.16) is referred to an orthogonal system Oxy, uk is the displacement vector, ds is the arch length along ⎡, Tk is the traction vector on ⎡ and ω is the strain energy density function defined as Fig. 4.12 Closed contour ⎡ and paths ⎡ 1 and ⎡ 2 between two points O1 and O2 in a continuum

4.2 J-Integral Fracture Criterion

101

∫εi j ω=

σi j dεi j (i, j ) = (x, y).

(4.17)

0

For a closed contour ⎡, J = 0. The values of J-integral along two paths ⎡ 1 , ⎡ 2 connecting any two points O1 , O2 within the region R are equal (since J is zero for the closed path O1 ⎡ 1 O2 ⎡ 2 O1 ) (Fig. 4.12) ∫ Jl =

∫ [. . .] = J2 =

⎡1

[. . .]

(4.18)

⎡2

Consider now a notch or a crack with flat surfaces parallel to the x-axis with an arbitrary root radius (Fig. 4.13). The region bounded by the closed contour AB⎡ 1 CD⎡ 2 A is free of singularities (does not contain the crack tip). The J-integral along AB⎡ 1 CD⎡ 2 A is zero J AB⎡1 C D⎡2 A = J AB + J B⎡1 C + JC D + J D⎡2 A = 0.

(4.19)

When the flat surfaces AB and CD of the notch (dy = 0) are traction free (T k = 0), we obtain from Eq. (4.16) JC D = J AB = 0. Equation (4.19) becomes J B⎡1 C = J A⎡2 D ,

(4.20)

when the contour A⎡2 D is described in a counter clockwise sense. Equation (4.20) indicates that J-integral calculated along a path Γ in a counterclockwise sense starting from an arbitrary point on the flat part of the lower notch surface and ending at an arbitrary point on the flat part of the upper notch surface is path independent. Note that path independence of J-integral is based on the assumption that the notch surfaces are traction free and parallel to the x-axis. The integration path may be taken close or far away from the crack tip, and can be chosen in such a way to simplify the calculation of J-integral. The potential energy ⊓ = ⊓(a) of a body (of unit thickness) is given by Gdoutos [19] ∫ ∫ (4.21) ⊓ (a) = ωdA − Tk u k ds A

A

J-integral is related to the rate of decrease of potential energy with respect to the crack length by Gdoutos [19]

102

4 Fracture Mechanics Testing

Fig. 4.13 a Path ⎡ starting from the lower and ending up to the upper face of a crack in a two-dimensional body. The flat notch surfaces are parallel to the x-axis. b Paths ⎡ 1 and ⎡ 2 around a crack tip

J =−

d⊓ da

(4.22)

Equation (4.22) indicates that J-integral is equal to the strain energy release rate G [19] J =G

4.2.2 J-Integral Fracture Criterion J-integral has the following properties:

(4.23)

4.2 J-Integral Fracture Criterion

(i) (ii) (iii) (iv) (v)

103

It is path independent for linear or nonlinear elastic material response It is equal to − d⊓/da for linear or nonlinear elastic material response It is equal to G It can easily be determined experimentally It can be related to the crack-tip opening displacement δ by a simple relation of the form J = Mσ Y δ (for the Dugdale model M = 1 [19]).

Because of these properties J has been proposed as a fracture criterion. For opening-mode loading, the criterion takes the form J = JIc ,

(4.24)

where J Ic is the critical value of J. J Ic is a material property for a given thickness under specified environmental conditions. For plane strain conditions, the critical value of J, J Ic , is related to the plane strain critical stress intensity factor K Ic by Gdoutos [19] JIc =

1 − ν2 2 K Ic . E

(4.25)

The above properties of J-integral are derived under elastic material response. Attempts have been made to extend the realm of applicability of J-integral fracture criterion to ductile fracture where extensive plastic deformation and possibly stable crack growth precede crack growth. Strictly speaking, the presence of plastic enclaves nullifies the path independence of J-integral. For any closed path surrounding the crack tip and taken within the plastic zone the necessary requirements for path independence are not satisfied. The stress is not uniquely determined by the strain, and the stress–strain constitutive equations relate strain increments to stresses and stress increments. In an effort to establish path independence for J-integral the deformation theory of plasticity is invoked. This theory is a nonlinear elasticity theory, and no unloading is permitted. The results of the deformation theory of plasticity coincide with the results of the flow (incremental) theory of plasticity under proportional loading (the stress components change in fixed proportion to one another). No unloading is permitted at any point of the plastic zone. Although, strictly speaking, the condition of proportional loading is not satisfied in practice, it is argued that in a number of stationary problems, under a single monotonically applied load, the loading condition is close to proportionality. Finite element solutions supported this proposition. J is used today as a fracture criterion in situations of appreciable plastic deformation. J-dominance conditions were formulated for such circumstances. For the special cases of the bend and center cracked specimen they take the form [19] bσY > 25, J for the bend specimen, and

(4.26)

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bσY > 175, J

(4.27)

for the center cracked specimen, where b is the uncracked ligament.

4.2.3 Experimental Determination of J-Integral For the experimental determination of J-integral and its critical value J Ic we will present the following methods: (a) the multiple-specimen method, (b) the onespecimen method and (c) the standard test method according to ASTM specifications. Before presenting these methods, we will consider some general equations which relate the J-integral to the load–displacement curves under “fixed-grips” (prescribed displacement) or “dead-load” (prescribed load) conditions. (a) General equations The experimental determination of J follows from Eq. (4.22), according to which J is equal to the rate of decrease of potential energy (defined from Eq. 4.21) with respect to the crack length. Experiments are usually performed under “fixed-grips” (prescribed displacement) or “dead-load” (prescribed load) conditions. In the load– displacement diagram the potential energy is equal to the area included between the load–displacement curve and the displacement axis or the load axis, for fixed-grips or dead-load conditions, respectively (shaded areas of Fig. 4.14a, b). Observe that the potential energy is positive for fixed-grips and negative for dead-load conditions. Consider in Fig. 4.15a, b the load–displacement (P − u) curves corresponding to crack lengths a and (a + Δa) for “fixed-grips” or “dead-load” conditions. The area included between the two curves represents the value of JΔa. We obtain for crack growth for “fixed-grips”

Fig. 4.14 Potential energy shown as shaded area for a “fixed-grips” and b “dead-load” conditions

4.2 J-Integral Fracture Criterion

105

Fig. 4.15 Load–displacement curves for crack lengths a and a + Δa for a “fixed-grips” and b “dead-load” conditions

   ∫u 0  ∂P ∂⊓ J =− =− du ∂a u ∂a u

(4.28)

  ∫P0   ∂u ∂⊓ J =− =− dP ∂a P ∂a P

(4.29)

0

and for “dead-load”

0

Equations (4.21), (4.28) and (4.29) form the basis for the experimental determination of J. (b) Multiple-specimen method The method is based on Eq. (4.22). A number of identically loaded specimens with neighboring crack lengths is used (Fig. 4.16a). The procedure is as follows: (i) Load–displacement (P − u) records, under fixed-grips, are obtained for several precracked specimens, with different crack lengths (Fig. 4.16b). For given values of displacement u, the area underneath the load–displacement record, which is equal to the potential energy ⊓ of the body at that displacement is calculated. (ii) ⊓ is plotted versus crack length for the previously selected displacements (Fig. 4.16c). (iii) The negative slopes of the ⊓ − a curves are determined and plotted versus displacement for different crack lengths (Fig. 4.16d). Thus, the J − u curves are obtained for different crack lengths.

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Fig. 4.16 Multiple-specimen method for calculating the J-integral

The critical value J Ic of J is determined from the displacement at the onset of crack extension. Since J Ic is a material constant, the values of J Ic obtained from different crack lengths should be the same. The multiple-specimen method presents the disadvantage that several specimens are required to obtain the J versus displacement u relation. Furthermore, accuracy problems enter in the numerical differentiation of the ⊓ − a curves. A technique for determining J from a single test becomes attractive and is described next. (c) Single-specimen method

4.2 J-Integral Fracture Criterion

107

Fig. 4.17 a A deeply cracked bend specimen and b bending moment M versus angle θ of relative rotation of the specimen end sections

Three types of deeply cracked specimens are used for the experimental determination of J: the bend specimen (Fig. 4.17), the compact specimen (Fig. 4.18) and the threepoint bend specimen. For the bend specimen (Fig. 4.17) we have [19] 2 J= b

∫θ Mdθ

(4.30)

0

where b is the crack ligament length, M is the applied bending moment and θ is the angle of relative rotation of the end sections of the specimen. Equation (4.30) indicates that the critical value of J, J Ic , can be obtained from the area under the M versus θ curve up to the point of crack extension. For the compact specimen (Fig. 4.18) we have [19] 2 1+β J= b 1 + β2

∫δP PdδP + 0

 ∫P  2 β 1 − 2β − β 2 δP dP,  2 b 1 + β2 0

(4.31)

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Fig. 4.18 Deeply cracked compact specimen at plastic collapse

where δP is the plastic contribution to the load-point displacement, and β is given by β=2

  a 2 b

+

1 a + b 2

1/2

 −2

1 a + b 2

 (4.32)

For a/W > 0.5 the total displacement δ instead of δP can be used in Eq. (4.31). For deeply cracked specimens β = 0, and Eq. (4.31) becomes 2 J= b

∫δ Pdδ.

(4.33)

0

Equation (4.33) is similar to Eq. (4.30) of the bend specimen. For the deeply cracked three-point bend specimen J is given by Eq. (4.30). (d) ASTM standard test method ASTM issued a standard test method [47] for the determination of the plane strain value of J at initiation of crack growth for metallic materials, J Ic . The recommended

4.2 J-Integral Fracture Criterion

109

specimens are the three-point bend and the compact specimen with deep initial cracks. The specimens are loaded to special fixtures. The applied loads and the load-point displacements are simultaneously recorded during the test. For a valid J Ic value, the crack ligament b and the specimen thickness B in both the three-point bend and the compact specimen must be greater than 25J Ic /σ Y (Eq. 4.26). For the three-point bend specimen the initial crack length must be at least 0.5W, but not greater than 0.75W, where W is the specimen width. The overall specimen length is 4.5W, and the specimen thickness is 0.5W. For the compact specimen (Fig. 4.18) the specimen thickness is 0.5W and the condition 0.5W < a < 0.75W should be satisfied. For the determination of the value of J Ic at the onset of slow stable crack growth the following procedure is followed: The value of J-integral for the bend specimen (Eq. (4.30)) or for the compact specimen (4.31) is calculated. Equation (4.31) is approximated by 2 1+β J= b 1 + β2

∫δ Pdδ,

(4.34)

0

where β is given from Eq. (4.32). J is plotted against the crack growth length, using at least four data points within specified limits of crack growth (Fig. 4.19). A power law expression J = C1 (Δa)C2

(4.35)

is fitted to the experimental data. The point at which this curve intersects the line originating at Δa = 0.2 mm and parallel to the blunting line (J = 2σ Y Δa) is the critical value of J, J Ic . The blunting line approximates the apparent crack advance due to crack-tip blunting when there is no slow stable crack tearing. We choose this line because we assume that, before tearing, the crack advance is equal to one half of the crack-tip opening displacement (Δa = 0.5 δ). Two additional offset lines parallel to the blunting line and starting from the points Δa = 0.15 and 1.5 mm of the crack extension axis are drawn. For a valid test all experimental data should fall inside the area enclosed by these two parallel offset lines and the line J = J max = b0 σ Y /15. Data outside this area (shown with open circles in Fig. 4.19) are not valid. Only data inside the above area (shown with bold circles in Fig. 4.19) are used for the determination of the regression curve of Eq. (4.35). The value of J Ic can be used to obtain K Ic in situations where large specimens are required for a valid K Ic test (Eq. 4.25), as   K Ic2 = JIc E/ 1 − ν 2

(4.36)

The minimum specimen thickness for a valid J Ic test is much smaller than that for a valid K Ic test (Eq. 4.13).

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Fig. 4.19 Determination of J Ic according to ASTM standards

4.3 Crack Opening Displacement Fracture Criterion 4.3.1 Introduction Wells [51] and Cottrell [12] introduced independently the concept of the critical crack opening displacement fracture criterion for the study of crack initiation in situations where significant plastic deformation precedes fracture. Under such conditions, they argued that the stresses near the crack tip reach the critical values and, therefore, fracture is controlled by the amount of plastic strain. Crack extension takes place by void growth and coalescence with the original crack tip, a mechanism for which the crack-tip plastic strain is responsible. A measure of the amount of crack-tip plastic strain is the separation of the crack faces or crack opening displacement (COD), especially very close to the crack tip. Crack extension begins when the crack opening displacement reaches a critical value, which is characteristic of the material at a given temperature, plate thickness, strain rate and environmental conditions. The critical crack opening displacement fracture criterion is expressed by δ = δc

(4.37)

4.3 Crack Opening Displacement Fracture Criterion

111

where δ is the crack opening displacement and δ c is its critical value. It is assumed that δ c is a material constant independent of specimen configuration and crack length. This assumption has been confirmed experimentally. In order to obtain an analytical expression for Eq. (4.37) in terms of applied load, crack length, specimen geometry and other fracture parameters the Irwin or the Dugdale models are invoked [19]. In both models δ is taken as the separation of the faces of an effective crack at the tip of the physical crack. According to the Irwin model for plane stress [19], δ c is given by δc =

4 K Ic2 π EσY

(4.38)

where K Ic is the critical stress intensity factor, σY is the yield stress and E is the modulus of elasticity. For the Dugdale model valid for conditions of plane stress δ c for small values of σ /σ Y is given by Gdoutos [19] δc =

K Ic2 EσY

(4.39)

Equations (4.2), (4.38) and (4.39) yield π σY δc 4

(4.40)

G Ic = σY δc ,

(4.41)

G Ic = for the Irwin model, and

for the Dugdale model. Equations (4.40) and (4.41) express the critical strain energy release rate GIc in terms of the critical crack opening displacement δ c . They show that under conditions of small-scale yielding, the stress intensity factor, the strain energy release rate and the crack opening displacement criterion are equivalent.

4.3.2 COD Design Curve The objective of the COD design curve is to establish a relationship between the crack opening displacement and the applied load and crack length. In this way, when the critical crack opening displacement is known the maximum permissible stress for a given crack size or the maximum allowable crack size for a given applied load in a structure can be determined. The COD design curve is constructed using an analytical model. Burdekin and Stone [8] used the Dugdale model to obtain the

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Fig. 4.20 Points P at equal distances y from a crack of length 2a

following equation for the overall strain ε (ε = u/2y) of two equidistant points P from the crack (Fig. 4.20): ⎤ ⎡ / ⎤ ⎡ / 2 + n2 2 + n2 1 ε k k 2⎣ ⎦ + (1 − ν) cot−1 = + ν cos−1 k ⎦, 2n coth−1 ⎣ εY π n 1 − k2 1 − k2 (4.42) with n=

  πσ a σY , εY = , k = cos y 2σY E

(4.43)

where a is half crack length, y is the distance of point P from the crack, σY is the yield stress of the material in tension and σ is the applied stress. We define the dimensionless crack opening displacement Φ by Φ=

δ , 2π εY a

(4.44)

where δ is determined from the Dugdale model. If we eliminate the stress σ from Eqs. (4.42)–(4.44) we obtain the design curves shown in Fig. 4.21, which present the variation of Φ versus ε/εY for various values of a/y. Figure 4.21 enables the

4.3 Crack Opening Displacement Fracture Criterion

113

Fig. 4.21 Design curves according to crack opening displacement criterion

determination of the maximum allowable overall strain ε in a cracked body when the critical crack opening displacement δ c and the crack length a are known. Experimental data that relate δ c and the maximum strain at fracture fall into a single scatter band of Fig. 4.21 for a wide range of values a/y. This indicates that the design curve based on the Dugdale model is far from reality. An empirical equation was obtained by Dawes [13] to describe the experimental data of Fig. 4.21. It has the form ⎧  2 ε ⎪ ⎪ ⎪ ⎨ εY , Φ= ⎪ ⎪ ⎪ ⎩ ε − 0.25, εY

ε < 0.5 εY ε > 0.5 εY

.

(4.45)

Dawes [14] argued that for small cracks (a/W < 0.1, W is the width of the plate) and for applied stresses below the yield stress we may assume ε σ = εY σY

(4.46)

From Eqs. (4.45) and (4.46) the maximum allowable crack length amax is obtained as

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amax

⎧ δc EσY σ ⎪ ⎨ , < 0.5 2 2π σ σY = . δc E σ ⎪ ⎩ , 0.5 < 0. ∂θ 2

(4.55)

According to hypothesis (2) crack growth occurs when S(θc ) = Sc ,

(4.56)

where S c is the critical strain energy density factor which is a material constant. For a crack in a two-dimensional stress field governed by the opening-mode K I and sliding-mode K II stress intensity factors the stresses σ x , σ x , τ xy are given by Gdoutos [19]     θ 3θ K II θ 3θ KI θ θ 1 − sin sin −√ 2 + cos cos σx = √ cos sin 2 2 2 2 2 2 2πr 2πr   3θ θ 3θ θ K II KI θ θ σy = √ 1 + sin sin +√ cos sin cos cos 2 2 2 2 2 2 2πr 2πr   θ 3θ K II 3θ θ KI θ θ +√ 1 − sin sin . τx y = √ cos sin cos cos 2 2 2 2 2 2 2πr 2πr (4.57) Introducing these values of stresses into Eq. (4.53) we obtain the following quadratic equation for the strain energy density factor S from Eq. (4.54). S = a11 kI2 + 2a12 kI kII + a22 kII2 ,

(4.58)

where 16μa11 = (1 + cos θ )(κ − cos θ ) 16μa12 = sin θ [2 cos θ − (κ − 1)] 16μa22 = (κ + 1)(1 − cos θ ) + (1 + cos θ )(3 cos θ − 1),

(4.59)

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√ with k j = K j / π , j = I, II. From Eqs. (4.55), (4.58) and (4.59) we obtain [2 cos θ − (κ − 1)] sin θ kI2 + 2[2 cos 2θ − (κ − 1) cos θ ]kI kII + [(κ − 1 − 6 cos θ ) sin θ ]kII2 = 0, [2 cos 2θ − (κ − 1) cos θ ]kI2 + 2[(κ − 1) sin θ − 4 sin 2θ ]kI kII + [(κ − 1) cos θ − 6 cos 2θ ]kII2 > 0

(4.60a)

(4.60b)

Relations (4.60a) and (4.60b) can be used for the determination of the crack extension angle θ c when the stress intensity factors k I and k II are known. Introducing the value of the angle θ c into Eqs. (4.58) and (4.59) we obtain the value of the strain energy density factor S(θ c ). Then, the critical value of the applied load at the onset of crack extension is determined from Eq. (4.56).

4.4.4 Two-Dimensional Linear Elastic Crack Problems Consider a central crack of length 2a in a large plate subjected to a uniform uniaxial stress σ that makes an angle β with the stress σ (Fig. 4.24). The stress intensity factors k I and k II are given by Gdoutos [19] kI = σ a 1/2 sin2 β, kII = σ a 1/2 sin β cos β

(4.61)

Introducing the values of k I and k II into Eq. (4.60a) we obtain the following equation for the calculation of critical angle θ c of crack extension (κ − 1) sin(θc − 2β) − 2 sin[2(θc − β)] − sin 2θc = 0.

(4.62)

The value of the strain energy density factor at the critical angle θ c , S(θ c ), is then obtained from Eq. (4.58) which takes the form   S = σ 2 a a11 sin2 β + 2a12 sin β cos β + a22 cos2 β sin2 β,

(4.63)

where the coefficients aij (i, j = 1, 2) are calculated from Eq. (4.59) with θ = θ c . The fracture envelope in the k I -k II plane obtained from Eqs. (4.56), (4.58), (4.59) and (4.60) is shown in Fig. 4.24.

4.4 Strain Energy Density Failure Criterion

119

Fig. 4.24 Mixed-mode fracture criterion for cracks under tension

4.4.5 Critical Strain Energy Density Factor Sc For a crack of length 2a in a large plate subjected to a uniform uniaxial stress σ perpendicular to the crack plane we have [19] kI = σ a 1/2 , kII = 0.

(4.64)

Introducing these values of k I and k II into Eq. (4.62) we obtain [2 cos θ − (κ − 1)] sin θ = 0

(4.65)

Inequality (4.60b) takes the form 2 cos 2θ − (κ − 1) cos θ > 0.

(4.66)

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4 Fracture Mechanics Testing

From Eq. (4.65) we obtain θc = 0 or, θc = arc cos[(κ − 1)/2]. The angle θ c = 0 satisfies Inequality (4.66), while the angle θ c = arc cos[(κ − 1)/2] does not (1 ≤ κ ≤ 3). This indicates the obvious result that the crack extends in its own plane. S min is then obtained from Eq. (4.58) as Smin =

(κ − 1)σ 2 a . 8μ

(4.67)

For the general case of mode-I loading dictated by the stress intensity factor K I we obtain from Eq. (4.56) Sc =

(1 + ν)(1 − 2ν)K I2 . 2π E

(4.68)

Equation (4.68) relates the critical strain energy density factor S c to the critical stress intensity factor K Ic . Sc is a material constant and can be determined experimentally.

4.5 Dynamic Problems Dynamic crack propagation occurs when a crack moves rapidly under slowly varying applied loads or when a stationary crack is subjected to rapidly varying loads, like impact or impulsive loads. Problems of interest include initiation of rapid crack growth, crack speed, crack branching and crack arrest. Dynamic crack growth is dictated by Gdoutos [19] K (t) = K ID (V ),

(4.69)

where K(t) is the dynamic stress intensity factor and K ID (V) is the critical dynamic stress intensity factor, where V = da/dt is the crack speed. K ID (V) is a material parameter that depends on crack speed. K(t) is determined from the solution of the elastodynamic crack problem and is a function of loading, crack length and geometrical configuration of the cracked body and time. Figure 4.25 shows a typical form of the curve K ID = K ID (V ) for Homalite 100. Note that K ID is nearly independent of the crack speed at low crack speeds and increases as the crack speed increases. Crack arrest may take place in some cases of crack propagation. When energy is constantly supplied to the crack-tip region the crack continues to propagate. This is the case of a crack in a tensile stress field. On the other hand, crack growth under

4.5 Dynamic Problems

121

Fig. 4.25 Dynamic fracture toughness versus crack speed for Homalite 100

constant displacements leads to crack arrest, since the energy supplied to the cracktip region progressively decreases with time. When the distance between the energy source and the crack tip increases with time the capability of a system to arrest a crack increases. This occurs, for example, in the splitting of long cantilever beam specimens. The crack arrest condition can be put in the form K (t) = K IA = min[K ID (V )]

(4.70)

where K(t) is the dynamic stress intensity factor and K IA is the critical dynamic stress intensity factor at crack arrest. It is the minimum value of K ID . The critical dynamic stress intensity factor K ID can be measured by using several types of specimens including the double cantilever beam, the single edge-notched and the wedge-loaded specimen. The wedge-loaded specimen presents many advantages over the other types of specimens and is mainly used. Besides the dynamic fracture toughness K Id , which depends on crack speed, the critical value K Id of stress intensity factor for crack initiation under a rapidly applied load is of interest in practical applications. K Id depends on the loading rate and temperature and is considered to be a material parameter. K Id is determined experimentally by using the three-point bend specimen. The specimen is loaded by a

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4 Fracture Mechanics Testing

falling weight and K Id is determined by static analysis. K Id decreases with increasing loading rate, below the transition temperature, and increases with load rate above the transition temperature. Crack velocity in dynamic crack problems is measured by using a series of conducting wires placed at certain intervals along the crack path and perpendicular to the direction of crack propagation. As the crack propagates the wires break and the corresponding times are obtained from the trace on an oscilloscope. Highspeed photography is perhaps the most widely used method of recording rapid crack propagation. The multiple-spark Cranz–Schardin camera, which is capable of operating at rates of up to 106 frames per second is widely employed. Other high-speed cameras are also used. Stress intensity factors during dynamic crack growth are determined by various methods of experimental mechanics, such as, photoelasticity, caustics, moiré, interferometry, holography and digital image correlation [20].

4.6 ASTM Standards ASTM has published the following standards for fracture mechanics testing: • E399-81, “Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials” • E1820-18 “Standard Test Method for Measurement of Fracture Toughness” • E740/E740M, “Standard Practice for Fracture Testing with Surface-Crack Tension Specimens” • E561-20, “Standard Test Method for KR Curve Determination” • E1290, “Standard Test Method for Crack-Tip Opening Displacement (CTOD) Fracture Toughness Measurement” • D5045-14, “Standard Test Methods for Plane-Strain Fracture Toughness and Strain Energy Release Rate of Plastic Materials” • D6068-10, “Standard Test Method for Determining J-R Curves of Plastic Materials” • C1421-18, “Standard Test Methods for Determination of Fracture Toughness of Advanced Ceramics at Ambient Temperature” • E813-87, “Standard Test Method for J Ic , a Measure of Fracture Toughness” • A514-81 “Standard Specification for High-Yield-Strength, Quenched and Tempered Alloy Steel Plate, Suitable for Welding”.

Further Readings

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Further Readings 1. Anderson TL (2017) Fracture mechanics: fundamentals and applications, 4th edn. CRC Press 2. ASTM STP 514-81. Standard specification for high-yield-strength, quenched and tempered alloy steel plate, suitable for welding 3. Begley JA, Landes JD (1972) The J-integral as a fracture criterion. In: Fracture toughness, ASTM STP 514, American Society for Testing and Materials, Philadelphia, pp 1–39 4. Broek D (1982) Elementary engineering fracture mechanics. Kluwer Academic Publishers 5. Broek D (1988) The practical use of fracture mechanics. Kluwer Academic Publishers 6. BS 5762 (1979) Methods for crack opening displacement (COD) testing. British Standards Institution, London 7. Bucci RJ, Paris PC, Landes JD, Rice JR (1972) J-integral estimation procedures. In: Fracture toughness, ASTM STP 514. American Society for Testing and Materials, Philadelphia, pp 40–69 8. Burdekin FM, Stone DEW (1966) The crack opening displacement approach to fracture mechanics in yielding materials. J Strain Anal 1:145–153 9. Carpinteri A (2021) Fracture and complexity. Springer 10. Campbell J (2022) Fracture mechanics: fundamentals and applications. Murphy & Moore Publishing 11. Coates C, Sooklal V (2022) Modern applied fracture mechanics. CRC Press 12. Cottrell AH (1961) Theoretical aspects of radiation damage and brittle fracture in steel pressure vessels. Iron Steel Inst Spec Rep 69:281–296 13. Dawes MG (1974) Fracture control in high yield strength weldments. Weld J Res Suppl 53:369S–379S 14. Dawes MG (1980) The COD design curve. In: Larsson LH (ed) Advances in elastic-plastic fracture mechanics. Applied Science Publishers, pp 279–300 15. Dharan CKH, Kang BS, Finnie I (2016) Finnie’s notes on fracture mechanics: fundamental and practical lessons. Springer 16. Gdoutos EE (1984) Problems of mixed mode crack propagation. Martinus Nijhoff Publishers 17. Gdoutos EE (1990) Fracture mechanics criteria and applications. Kluwer Academic Publishers 18. Gdoutos EE, Pilakoutas K, Rodopoulos CA (eds) (2000) Failure analysis of industrial composite materials. McGraw Hill 19. Gdoutos EE (2020) Fracture mechanics, 3rd edn. Springer 20. Gdoutos EE (2022) Experimental mechanics. Springer 21. Gross D, Seelig T (2017) Fracture mechanics: with introduction to micro mechanics, 2nd edn. Springer 22. Hertzberg RW, Vinci RP, Hertzberg JL (2020) Deformation and fracture mechanics of engineering materials, 6th edn. Wiley 23. Hutchinson JW (1983) Fundamentals of the phenomenological theory of nonlinear fracture mechanics. J Appl Mech Trans ASME 50:1042–1051 24. Hutchinson JW, Paris PC (1979) Stability analysis of J-controlled crack growth. In: Elasticplastic fracture, ASTM STP 668. American Society for Testing and Materials, Philadelphia, pp 37–64 25. Irwin GR (1948) Fracture dynamics. Fracture of metals. American Society for Metals, Cleveland, USA, pp 147–166 26. Janssen M, Zuidema J, Wanhill R (2002) Fracture mechanics, 2nd edn. Spon Press 27. Krafft JM, Sullivan AM, Boyle RW (1961) Effect of dimensions on fast fracture instability of notched sheets. In: Proceedings of crack propagation symposium, vol 1. College of Aeronautics, Cranfield (England), pp 8–28 28. Kumar R (2009) Elements of fracture mechanics. Tata McGraw Hill Education Private Limited 29. Kuna M (2013) Finite elements in fracture mechanics. Springer 30. Liebowitz H (1969–1972) Fracture mechanics—an advanced treatise, vols 1–7. Academic Press

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31. Maiti SK (2015) Fracture mechanics: fundamentals and applications. Cambridge University Press 32. McMeeking RM (1977) Finite deformation analysis of crack opening in elastic-plastic materials and implications for fracture. J Mech Phys Solids 25:357–381 33. McMeeking RM, Parks DM (1979) On criteria for J-dominance of crack tip fields in large scale yielding. In: Elastic-plastic fracture, ASTM STP 668. American Society for Testing and Materials, Philadelphia, pp 175–194 34. Merkle JG, Corten HT (1974) A J-integral analysis for the compact specimen, considering axial force as well as bending effects. J Press Vessel Technol 96:286–292 35. Orowan E (1948) Fracture and strength of solids. In: Reports on progress in physics XII, pp 185–232 36. Paris PC, Tada H, Zahoor A, Ernst H (1979) The theory of instability of the tearing mode of elastic-plastic crack growth. In: Elastic-plastic fracture, ASTM STP 668. American Society for Testing and Materials, Philadelphia, pp 5–36, 251–265 37. Perez N (2016) Fracture mechanics, 2nd edn. Springer 38. Popelar CH, Kanninen MF (1985) Advanced fracture mechanics. Oxford University Press 39. Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech Trans ASME 35:379–386 40. Rice JR, Paris PC, Merkle JG (1973) Some further results of J-integral analysis and estimates. In: Progress in flaw growth and fracture toughness testing, ASTM STP 536. American Society for Testing and Materials, Philadelphia, pp 213–245 41. Ritchie RO, Liu D (2021) Introduction to fracture mechanics. Elsevier 42. Saxena A (2019) Advanced fracture mechanics and structural integrity. CRC Press 43. Shukla A (2004) Practical fracture mechanics in design. CRC Press 44. Standard Practice for R-Curve-Determination (1981) ASTM annual book of standards, Part I0, E561-81. American Society for Testing and Materials, Philadelphia, pp 680–699 45. Shih CF, German MD (1981) Requirements for a one parameter characterization of crack tip fields by the HRR singularity. Int J Fract 17:27–43 46. Sih GC (1977–1981) Mechanics of fracture, vols 1–7. Martinus Nijhoff Publishers 47. Standard test method for J Ic , a measure of fracture toughness (1987) In: ASTM annual book of standards, Part 10, E813-87. American Society for Testing and Materials, Philadelphia, pp 968–990 48. Sun C-T, Jin Z (2011) Fracture mechanics. Academic Press 49. Unger D (2011) Analytical fracture mechanics. Dover Publications 50. Wei RP (2014) Fracture mechanics: integration of mechanics, materials science and chemistry. Cambridge University Press 51. Wells AA (1961) Unstable crack propagation in metals: cleavage and fracture. In: Proceedings of the crack propagation symposium, vol 1. College of Aeronautics, Cranfield, pp 210–230 52. Zehnder AT (2012) Fracture mechanics. Springer

Chapter 5

Fatigue and Environment-Assisted Testing

Abstract Fatigue is the process of damage and failure of materials and structures due to cycling loading. It was demonstrated that under repeated loading materials fail at stresses well below the ultimate strength, and microscopic damage accumulates until cracks or other forms of macroscopic damage develop. Failure due to fatigue loading is called “fatigue failure”. The number of loading cycles leading to failure of structural or machine components is called fatigue life. The main objective of fatigue analysis is to determine the fatigue life for a repeated fluctuating load of constant or variable amplitude. In this chapter we will consider two major approaches for the study of fatigue failure. The stress-based approach which is based on nominal stresses and does not account on the mechanisms of fatigue failure, and the fracture mechanics approach which considers the micro and macromechanisms of fatigue failure and uses the principles of fracture mechanics for initiation and propagation of cracks. This approach provides a better understanding of fatigue failure by analyzing the initiation, propagation and instability processes of fatigue cracks. For both approaches we present the mechanical tests performed on specimens subjected to cyclic loading. The objective of the tests is to generate fatigue life and crack growth data and demonstrate the safety of a material or structure. The chapter concludes with a study of the environment-assisted fracture.

5.1 Introduction The word fatigue comes from the Latin verb “fatigare” which means “to tire out” or “to exhaust”. In engineering terminology it means damage or failure of materials under cyclic loads. It was first realized in the middle of the nineteenth century that engineering components and structures often fail when subjected to repeated fluctuating loads whose magnitude is well below the critical load under monotonic loading. Early investigations were primarily concerned with axle and bridge failures which occurred at cyclic load levels less than half their corresponding monotonic load magnitudes.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_5

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Study of fatigue failure takes place along two different approaches. The first considers the fatigue life in terms of global measurable quantities, like stress, strain, mean stress, etc. It does not account for the details of the failure mode nor for the existence and growth of initial imperfections in the material. The results of tests performed on small laboratory specimens subjected to repeated sinusoidally fluctuating loads are interpreted in diagrams expressing the stress amplitude, S, versus the number of cycles to failure, N, known as S–N curves. A number of empirical relationships for estimation of fatigue life, derived from curve fitting of test data, have been proposed in the literature. The S–N curve method and other available procedures based on gross specimen quantities lead to an inaccurate prediction of the fatigue life of engineering components due to the large scatter of experimental results, as they are influenced by specimen size and geometry, material and the nature of the fluctuating load. Furthermore, the physical phenomena and mechanisms governing the fatigue process are completely ignored. The second approach of fatigue failure is based on the initiation and propagation of fatigue cracks and uses the principles of fracture mechanics. It provides a better understanding of fatigue failure by modeling the fatigue crack initiation and propagation processes. Usually, crack initiation is analyzed at the microscopic level, while the continuum mechanics approach is used for the study of crack propagation. The necessity for addressing the initiation and propagation processes separately arises from the inability of the current theory to bridge the gap between material damage that occurs at microscopic and macroscopic levels. It is generally accepted that, when a structure is subjected to repeated loads, energy is accumulated in the neighborhood of voids and microscopic defects which grow and coalesce, forming microscopic cracks. Eventually larger macroscopic cracks are formed. A macrocrack may be defined as one that is large enough to permit the application of the principles of continuum mechanics of homogeneous media. A macrocrack is usually referred to as a fatigue crack. The number of cycles required to initiate a fatigue crack in a structural or machine component is the fatigue crack initiation life N i . Following the initiation of a fatigue crack, slow stable crack propagation begins, until the crack reaches a critical size corresponding to the onset of global instability leading to catastrophic failure. The fatigue life of an engineering component may be considered to be composed of three stages: the initiation or stage I; the propagation or stage II; and the fracture or stage III. In the last stage the crack growth rate increases rapidly as global instability is approached. The number of cycles required to propagate a fatigue crack until it reaches its critical size is the fatigue crack propagation life Np. Depending on the material, the amplitude of the fluctuating load and environmental conditions, the fatigue crack initiation life may be a small or a substantial part of the total fatigue life. In this chapter we will study the stress-based and the fracture mechanics approaches of fatigue failure. At the end of the chapter, we will briefly present the environment-assisted fracture.

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5.2 Fatigue Study Based on Stress 5.2.1 Basic Definitions Most tests and practical applications of fatigue involve constant amplitude cycling between a minimum σ min and a maximum σ max stress level. This is called constant amplitude fatigue (Fig. 5.1). The following quantities are relevant: The stress range Δσ is defined as Δσ = σmax − σmin σmax > σmin , Δσ > 0.

(5.1)

Tension is considered positive and compression negative. The stress range Δσ is always positive. σ max and σ min can be either positive or negative. The stress amplitude σ a is half the stress range defined as σa =

σmax − σmin Δσ = 2 2

(5.2)

The stress amplitude σ a is always positive. The mean stress σ m is the average of the maximum and minimum stress values. It is defined as σm =

σmax + σmin 2

(5.3)

Fig. 5.1 Constant amplitude fatigue involves constant amplitude cycling between a maximum σ max and a minimum σ min stress levels. a Completely reversed cycling, σ m = 0, b Nonzero mean stress σ m , c Zero-to-tension cycling, σ min = 0

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The stress ratio R and the amplitude ratio A are defined as R=

σmin , σmax

A=

σa σm

(5.4)

Between the above five quantities Δσ, σ a , σ m , R and A the following relations apply: σmax = σm + σa , σmin = σm − σa , σa =

σmax Δσ σmax = (1 − R), σm = (1 + R), 2 2 2 R=

1− A , 1+ A

A=

1− R . 1+ R

(5.5a) (5.5b) (5.5c)

A cycling loading is specified by two independent values of the above five quantities. Some combinations include σ max and σ m ; σ m and σ a ; σ max and R; Δσ and R; σ max and σ min ; σ a and A. When the mean stress is zero (σ m = 0, R = − 1) the cycling stress can be specified by the amplitude σ a , which is numerically equal to the maximum stress σ max . This situation is called completely reversed cycling (Fig. 5.1a). The case of nonzero mean stress σ m is shown in Fig. 5.1b. The case of σ min = 0, or R = 0 (Fig. 5.1c) refers to zero-to-tension cycling. The stress σ = σ (t) in a sinusoidal cycling load varies with time t according to the following equation: σ (t) = σm + σa sin

2π t , T

(5.6)

where T is the period of the load (the time required for one load cycle).

5.2.2 Stress Versus Life (S–N) Curves From fatigue tests performed on material specimens or engineering components the number of stress cycles to failure N f at different stress amplitudes σ a is obtained. The resulting stress-life curve of N f versus σ a is called S–N curve (Fig. 5.2). The stress amplitude σ a is usually plotted in the ordinate and the number of cycles, N f , is plotted in the abscissa. The stress to failure decreases as the number of cycles increases. Usually, N f changes rapidly with the stress amplitude σ a , and on a linear plot it cannot be read accurately. For this reason the number of cycles N f in a S–N curve is plotted on a logarithmic scale. In some cases a logarithmic scale is also used for the stress amplitude σ a .

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Fig. 5.2 Stress amplitude, S a versus number of cycles to failure, N f

Phenomenological equations have be obtained to approximate the S–N curves. When σ a varies linearly with the logarithm N f a mathematical representation of the S–N curve is σa = C + D log N f ,

(5.7)

where C and D are material constants. When σ a varies linearly with N f on a log–log plot the equation of the S–N curve is σa = AN Bf

(5.8)

where A and B are material constants. In some materials there is a limiting stress value below which fatigue failure does not occur. In such cases the S–N curve approaches asymptotically to that stress (Fig. 5.3). Such limiting stress is called fatigue limit or endurance limit. It is a material property.

5.2.3 The Effect of Mean Stress The mean stress σ m has a strong effect on the S–N curve. Generally speaking, for tensile mean stresses, the fatigue life decreases as the mean stress increases for a given stress amplitude σ a . Tensile mean stresses give shorter fatigue lives than zero mean stresses. Quite opposite occurs for compressive mean stresses. This is illustrated in

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Fig. 5.3 S–N curve approaching asymptotically to fatigue or endurance limit

Fig. 5.4 for tensile mean stresses. Note that as the mean stress is increased the S–N curve is lowered. Another form of representation of the effect of the mean stress on the S–N curves is to plot the variation of the stress amplitude σ a versus the mean stress σ m for constant values of the fatigue life N f (Fig. 5.5). The intersections of the constant-life

Fig. 5.4 S–N curves for various tensile mean stresses. As the mean stress is increased the S–N curve is lowered

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131

curves N f = constant with the line σ m = 0 represent the values of the stress amplitude σ ar for the particular case of zero mean stress. A normalized plot of the graph σ a / σ ar versus σ m is shown in Fig. 5.6. The curve passes from the point (0, 1) on the σ a / σ ar axis (for σ m = 0 σ a = σ ar ) and from the point (σ u , 0), on the σ m axis, where σ u is the ultimate stress (for σ a = 0 the mean stress σ m equals the ultimate stress σ u ). Assuming a straight line between the points (0, 1) on the σ a /σ ar axis and the point (σ u , 0) on the σ m axis, the equation of this line is σa σm + =1 σar σu

(5.9)

Equation (5.9) is called the Goodman–Smith equation. It quantifies the interaction of mean, σm , and alternating, σa , stresses on the fatigue life of a material. It is used for the calculation of alternating stress cycles a material will withstand when subjected to an alternating stress σa with mean stress σm when the function σar = F(σar ) for σm = 0 is known. Other equations have been proposed to fit the experimental data. An early equation is the Gerber parabola given by σa + σar



σm σu

2 = 1, σm ≥ 0

(5.10)

Equation (5.10) applies to tensile mean stresses.

Fig. 5.5 Stress amplitude σ a versus mean stress σ m for constant values of the fatigue life N f

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Fig. 5.6 Normalized stress amplitude σ a /σ ar versus σ m . The curve passes from point (0, 1) on the σ a /σ ar axis and from point (σ u , 0), on the σ m axis

Equations (5.9) and (5.10) relate the stress amplitude σ a for any type of fatigue load to the stress amplitude σ ar for the specific case of fatigue load with σ m = 0. Therefore, they can be used to obtain the S–N curve for any type of loading when the S–N curve for σ m = 0 is known. Let the fatigue curve for σ m = 0 is given by Eq. (5.8) as σar = AN Bf

(5.11)

From Eq. (5.9) we obtain σar =

σu σa σu − σm

(5.12)

Substituting this value of σ ar into Eq. (5.11) we obtain the S–N curve for a stress amplitude σ a  σa =

 σu − σm A N Bf σu

(5.13)

Equation (5.13) provides the σ a versus N f relationship for a stress amplitude σ a with mean stress σm for an ultimate stress σu .

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133

5.2.4 Multiaxial Stresses We consider the simple case of elastic applied cyclic principal stresses of amplitudes σ 1a , σ 2a , σ 3a having the same frequency and in-phase or 180° out-of-phase with one another. Under these circumstances we can compute an effective stress amplitude σ a based on the octahedral shear yield criterion as / 1 σ a = √ (σ1a − σ2a )2 + (σ2a − σ3a )2 + (σ3a − σ1a )2 . 2

(5.14)

The fatigue life N f is calculated by introducing this value of σ a into Eq. (5.7) or (5.8) for uniaxial load. The effective mean stress σ m is calculated from the mean stresses σ 1m , σ 2m , σ 3m in the three principal directions as σ m = (σ1m + σ2m + σ3m )/3

(5.15)

Equations (5.9) to (5.13) for uniaxial loads can be generalized for multiaxial stresses using the effective stress amplitude and the effective mean stress.

5.2.5 Variable Amplitude Loads Consider a variable amplitude loading that consists of a stress amplitude σ a1 applied for N 1 cycles, a stress amplitude σ a2 applied for N 2 cycles, and so on, up to a stress amplitude σ an applied for N n cycles (Fig. 5.7 for n = 3). Let the number of cycles to failure for the stress amplitude σ a1 is N f 1 , for the stress amplitude σ a2 is N f 2 , and so on, up to the stress amplitude σ an for which is N fn . When the stress amplitudes σ a1 , σ a2 , …, σ an are applied simultaneously the fatigue failure is expressed by the Palmgren–Miner rule (it has been proposed by A. Palmgren in 1924 and was popularized by M. A. Miner in 1945) as ∑ Nj N1 N2 Nn + + ··· + = = 1. Nf1 Nf2 Nfn Nfj

(5.16)

When the number of repetitions of the above history of loading (stress amplitude σ a1 applied for N 1 cycles, a stress amplitude σ a2 applied for N 2 cycles, and so on) to failure is Bf Eq. (5.16) becomes ∑  Nj Bf =1 N f j one rep

(5.17)

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Fig. 5.7 Life prediction according to the Palmgren–Miner rule for a completely reversed variable amplitude loading

5.3 Fatigue Study Based on Fracture Mechanics 5.3.1 General Considerations Fatigue failure is better understood using the principles of fracture mechanics. The growth of a dominant crack under fatigue loading is studied. The fatigue life of materials and engineering components may be considered to be composed of three stages: the initiation or stage I; the propagation or stage II; and the fracture or stage III. Stage II represents a large portion of the fatigue life. Accurate prediction of this stage is of utmost importance for the determination the fatigue life. The main question of fatigue crack propagation may be stated as: Determine the number of cycles N c required for a crack to grow from a certain initial size a0 to the maximum permissible size ac , and the form of this increase a = a(N), where the crack length a corresponds to N loading cycles. Figure 5.8 presents a plot of a versus N. ai is the crack length that is big enough for fracture mechanics to apply, but too small for detection, while a1 is the nondestructive inspection detection limit. The crack first grows slowly until the useful life of the component is reached. The crack then begins to propagate very rapidly, reaching a length af at which catastrophic failure begins. Fatigue crack propagation data are obtained from precracked specimens subjected to fluctuating loads, and the change in crack length is recorded as a function of loading cycles. The crack length is plotted against the number of loading cycles for different load amplitudes. The stress intensity factor is used as a correlation parameter in analyzing the fatigue crack propagation results. The experimental results are usually plotted in a log(ΔK) versus log (da/dN) diagram, where ΔK is the amplitude of the stress intensity factor corresponding to the fluctuating load and da/dN is the crack propagation rate. The load is usually sinusoidal with constant amplitude and frequency (Fig. 5.9). Two of the four parameters K max , K min ; ΔK = K max − K min

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Fig. 5.8 Crack size, a, versus number of cycles, N, for constant amplitude loading

or R = K min /K max are needed to define the stress intensity factor variation during a loading cycle. A typical plot of the characteristic sigmoidal shape of a log(ΔK) − log(da/dN) fatigue crack growth rate curve is shown in Fig. 5.10. Three regions can be distinguished. In region I, da/dN diminishes rapidly with decreasing ΔK to a vanishingly small level, and for some materials there is a threshold value of the stress intensity

Fig. 5.9 Sinusoidal load with constant amplitude and frequency

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Fig. 5.10 Typical form of a fatigue crack growth rate curve. log (da/dN) is plotted versus log (ΔK)

factor amplitude ΔK th . For ΔK < ΔK th no crack propagation takes place. In region II there is a linear log(ΔK) − log(da/dN) relation. Finally, in region III the crack growth rate curve rises and the maximum stress intensity factor K max in the fatigue cycle becomes equal to the critical stress intensity factor K c , leading to catastrophic failure. Experimental results indicate that the fatigue crack growth rate curve depends on the ratio, R, and is shifted toward higher da/dN values as R increases.

5.3.2 Crack Propagation Laws A number of different quantitative continuum mechanics models of fatigue crack propagation have been proposed in the literature. All these models lead to relations based mainly on experimental data correlations. They relate da/dN to such variables

5.3 Fatigue Study Based on Fracture Mechanics

137

as the external load, the crack length, the geometry and the material properties. Representative examples of such relations will be analyzed in this section. One of the earlier mathematical models of fatigue crack propagation was proposed by Head [19]. He considered an infinite plate with a central crack of length 2a subjected to a sinusoidally applied stress ± σ. Modeling the material elements ahead of the crack tip as rigid plastic work-hardening tensile bars and the remaining elements as elastic bars, he arrived at the relation da = C1 σ 3 a 3/2 , dN

(5.18)

where C 1 is a constant which depends on the mechanical properties of the material and is determined experimentally. Equation (5.18) can be written in terms of stress intensity factor K I as da = CK I3 dN

(5.19)

One of the most widely used fatigue crack propagation laws was proposed by Paris and Erdogan [27] and is usually referred to in the literature as the “Paris law”. It has the form da = C(ΔK )m dN

(5.20)

where ΔK = K max − K min , and K max and K min refer to the maximum and minimum values of the stress intensity factor in a load cycle. Constants C and m are determined empirically from a log(ΔK) − log(da/dN) plot. The value of m is usually taken equal to 4, resulting in the so-called 4th power law, while the coefficient C is assumed to be a material constant. Equation (5.20) represents a linear relationship between log(ΔK) and log(da/dN) and is used to describe the fatigue crack propagation behavior in region II of the curve of Fig. 5.10. Fatigue crack propagation data are well predicted from Eq. (5.20) for specific geometrical configurations and loading conditions. The effect of mean stress, loading and specimen geometry is included in the constant C. Despite many shortcomings, Eq. (5.20) has been widely used to predict the fatigue crack propagation life of engineering components. Equation (5.20) does not account for the crack growth characteristics at low and high levels of ΔK. At high ΔK values, as K max approaches the critical level K c , an increase in crack growth rate is observed. For this case (region III of Fig. 5.10) Forman et al. [16] proposed the equation C(ΔK )n da = , dN (1 − R)K c − ΔK

(5.21)

where R = K min /K max and C and n are material constants. Equation (5.21) arises from a modification of Eq. (5.20) by the term (1 − R)K c – ΔK in the denominator;

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this term decreases with increasing load ratio R and decreasing fracture toughness K c , both of which give rise to increasing crack growth rates at a given ΔK level. Note that for K max = K c , corresponding to instability, the denominator of Eq. (5.21) becomes zero, and Eq. (5.21) predicts an unbounded value of da/dN. For low values of ΔK (region I of Fig. 5.10) Donahue et al. [8] have suggested the relation da = K (ΔK − ΔK th )m , dN

(5.22)

where ΔK th denotes the threshold value of ΔK. Klesnil and Lucas [22] gave the following value of ΔK th ΔK th = (1 − R)γ ΔK th(0) ,

(5.23)

where ΔK th(0) is the threshold value at R = 0 and γ is a material parameter. A generalized fatigue crack propagation law that can describe the sigmoidal response exhibited by the data of Fig. 5.10 has been suggested by Erdogan and Ratwani [14]. It has the form C(1 + β)m (ΔK − ΔK th )n da = , dN K c − (1 + β)ΔK

(5.24)

where C, m, n are empirical material constants, and β=

K max + K min . K max − K min

(5.25)

The factor (1 + β)m has been introduced to account for the effect of mean stress level on fatigue crack propagation, while the factor [K c − (1 + β)ΔK] considers the experimental data at high stress levels. Finally, the factor (ΔK − ΔK th )n accounts for the experimental data at low stress levels and the existence of a threshold value ΔK th of ΔK at which no crack propagation occurs. By proper choice of constants, Eq. (5.24) can fit the experimental data over a range of 10−8 to 10−2 in/cycle. Attempts have been made to apply the J-integral to elastic–plastic fatigue crack propagation. A relation in complete analogy to the Paris law has been suggested [9, 10]. It takes the form da = C(ΔJ )m dN

(5.26)

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139

5.3.3 Fatigue Life Calculations When a structural component is subjected to fatigue loading, a dominant crack reaches a critical size under the peak load during the last cycle. This leads to catastrophic failure. The basic objective of fatigue crack propagation analysis is to determine the relation a = a(N), where a is the crack size and N is the number of cycles (Fig. 5.8). In this way, the fatigue crack propagation life N p is obtained. When the applied loads and the stress intensity factor are known, application of one of the foregoing fatigue laws enables a calculation of the fatigue crack propagation life. As an example, we consider a plane fatigue crack of initial length 2a0 in a plate subjected to a uniform stress σ perpendicular to the crack plane. The stress intensity factor K is given by √ K = f (a)σ πa,

(5.27)

where 2a is the crack length, σ is the applied stress and f (a) is a geometry dependent function. Integrating Eq. (5.20) of Paris law, we obtain a N − N0 = a0

da , C(ΔK )m

(5.28)

where N 0 is the number of load cycles corresponding to the initial half crack length a0 . Introducing the value of ΔK from Eq. (5.27), into Eq. (5.28) we obtain a N − N0 = a0

da

√ m . C f (a)Δσ πa

(5.29)

Assuming that the function f (a) is equal to its initial value f (a0 ) we obtain / ΔK = ΔK 0

√ a ΔK 0 = f (a0 )Δσ π a0 . a0

(5.30)

Then, we obtain from Eq. (5.29)  a m/2−1  2a0 0 1− for m /= 2. N − N0 = (m − 2)C(ΔK 0 )m a

(5.31)

Unstable crack propagation occurs when √ K max = f (a)σmax πa = K I C ,

(5.32)

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where K IC is the critical plane strain stress intensity factor. From Eq. (5.32) the critical crack length ac at instability is obtained. Then Eq. (5.31) for a = ac gives the fatigue crack propagation life N p = N c − N 0 , where N c denotes the number of cycles at instability. Usually f (a) varies with the crack length a and integration of Eq. (5.29) cannot be performed directly, but only through numerical analysis.

5.4 Variable Amplitude Loading 5.4.1 Introduction Fatigue crack propagation discussed so far concerns with constant amplitude load fluctuation. Although this type of loading occurs frequently in practice, the majority of engineering structures are subjected to complex fluctuating loading. Unlike the case of constant cyclic load where ΔK increases gradually with increasing crack length, abrupt changes in ΔK take place in variable amplitude fatigue due to changes in applied load. Load interaction effects take place which greatly influence the fatigue crack propagation behavior.

5.4.2 Overload Effect It was first recognized empirically in the early 1960s that the application of a tensile overload in a constant amplitude cyclic load leads to crack retardation following the overload; that is, the crack growth rate is smaller than it would have been under constant amplitude loading. This effect is shown schematically in Fig. 5.11. The amount of crack retardation is dramatically decreased when a tensile overload follows a constant amplitude cyclic load. Slower than normal crack growth continues beyond the region affected by the overload. The tensile overload may increase significantly the fatigue life. This effect of increasing the load cycles to failure due to tensile overload is called crack growth retardation. An explanation of the crack retardation phenomenon may be obtained by examining the behavior of the plastic zone ahead of the crack tip. The overload has left a large plastic zone behind. The elastic material surrounding this plastic zone after unloading acts like a clamp on this zone causing compressive residual stresses. As the crack propagates into the plastic zone, the residual compressive stresses tend to close the crack. Hence the crack will propagate at a decreasing rate into the zone of residual stresses. When these stresses are overcome and the crack is opened again, subsequent fluctuating loading causes crack growth.

5.4 Variable Amplitude Loading

141

Fig. 5.11 Typical form of crack length versus number of cycles curve for constant amplitude loading and constant amplitude plus overloading

5.4.3 Life Estimate Based on Summation of Crack Increments A simple approach to estimate fatigue life is to assume that growth by a given cycle is not affected by the prior history. The crack growth increment Δa in each cycle is determined by the da/dN versus ΔK relationship, and the increments Δa are summed up. Consider a crack of length aj and an increment of crack length Δaj. The new crack length aj+1 for the next cycle is  a j+1 = a j + Δa j = a j +

da dN

 j

(5.33)

The crack length aN after N cycles is aN = a j +

 N  ∑ da j=1

dN

j.

(5.34)

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Each da/dN is calculated from the da/dN versus ΔK relationship for that particular cycle, where ΔK is calculated from the current crack length aj and the applied stress Δσ . The summation is continued until the critical stress intensity factor calculated from Eq. (5.32) is reached. The summation procedure can also be applied for constant amplitude loading, as an alternative to numerical integration. Values of ΔN other than unity can be accommodated by this approach. Values of ΔN need to be sufficiently small that da/dN does not change much during an interval of N cycles, so that its value at the beginning of the interval is representative of the entire interval.

5.4.4 Models for Predicting Fatigue Life Due to the crack retardation phenomenon, the determination of fatigue life under a variable amplitude loading by simply summing the fatigue lives of the various constant amplitude loads of the loading history presented previously leads to conservative predictions. To accommodate the crack retardation phenomenon various methods have been proposed. We will present the root mean square, the crack retardation and the crack closure model. The root mean square model proposed by Barsom [1] applies to variable amplitude narrow-band random loading spectra. It is assumed that the average fatigue crack growth rate under a variable amplitude random loading fluctuation is approximately equal to the rate of fatigue crack growth under constant amplitude cyclic load; this load is equal to the root mean square of the variable amplitude loading. The fatigue crack propagation laws presented in Sect. 5.3.2 can be equally applied for a variable amplitude random loading when ΔK is replaced by ΔK rms given by /∑ ΔK rms =

(ΔK ι )2 n i ∑ , ni

(5.35)

where ni is the number of loading amplitudes with a stress intensity factor range of ΔK i . The crack retardation model proposed by Wheeler [34] assumes that, after a peak load, there is a load interaction effect when the crack-tip plastic zones for the subsequent loads are smaller than the plastic zone due to the peak load. Consider that at a crack length a0 an overload stress σ 0 creates a crack-tip plastic zone of length cpo which is given by c po =

1 σ02 a0 , A σY2

(5.36)

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143

where A = 1, or 3 for plane strain or plane stress conditions, respectively. When the crack has propagated to a length ai a stress σ i will produce a plastic zone of length cpi given by

c pi =

1 σi2 ai . A σY2

(5.37)

The plastic zone due to the stress σ i is included inside the plastic zone due to the overload. A retardation factor ϕ is introduced by ϕ=

c m pi

λ

,

ai + c pi < a0 + c p0 ,

Fig. 5.12 Crack retardation model proposed by Wheeler [34]

(5.38)

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where λ = a0 +c p0 −ai and m is an empirical parameter. The crack growth increment for ai + cpi < a0 + cp0 is given by 

da dN



 R

 da =ϕ , dN

(5.39)

where da/dN is the constant amplitude crack growth rate corresponding to the stress intensity factor range ΔK i of load cycle i. When ai + cpi > a0 + cp0 the crack has propagated through the overload plastic zone and the retardation factor ϕ = 1. The crack closure model introduced by Elber [12, 13] is based on the observation that the faces of fatigue cracks subjected to zero-tension loading close during unloading, and compressive residual stresses act on the crack faces at zero load. The crack closes at a tensile rather than zero or compressive load. An effective stress intensity factor range is defined by (ΔK )eff = K max − K op ,

(5.40)

where K op corresponds to the point at which the crack is fully open. Then, using the Paris crack propagation law we obtain da = C(U ΔK )m , dN

(5.41)

where U=

K max − K op , ΔK = K max − K min . K max − K min

(5.42)

A number of empirical relations have been proposed for the determination of U. Elber [12, 13] suggested the form U = 0.5 + 0.4R,

(5.43)

where R=

K min , K max

for − 0.1 ≤ R ≤ 0.7.

(5.44)

Schijve [28] proposed the relation U = 0.55 + 0.33R + 0.12R 2 ,

(5.45)

which extends the previous equation to negative R ratios in the range − 1.0 ≤ R ≤ 0.54. Equations (5.44) and (5.45) were obtained for a 2024-T3 aluminum. De Koning [7] suggested a method to determine K op .

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145

5.4.5 Miner Rule Miner rule concerns with crack growth under variable stress amplitudes. Consider a large plate with a crack of length 2a0 subjected to a series of stress amplitudes Δσ i (i = 1,2,…,n) normal to the crack plane. Assume that the crack length at instability 2af is the same for all stress amplitudes Δσ i , and fatigue crack growth is governed by Paris law of Eq. (5.20) with m = 2 as da = C(ΔK )2 . dN

(5.46)

The number of cycles N 1 required to grow the crack from its initial length 2a0 to a length 2a1 is calculated by integrating Eq. (5.46) as 1 Nl = πC(Δσ1 )2

a1 a0

  1 da a1 . = ln 2 a a0 πC(Δσ1 )

(5.47)

In a similar manner, we obtain for the number of cycles (N 1 )f required to grow the crack from its initial length 2a0 to its final length 2af at instability (Nl ) f =

  af 1 . ln a0 π C(Δσ1 )2

(5.48)

From Eqs. (5.47) and (5.48) we obtain     af N1 a1 / ln . = ln a0 a0 (N1 ) f

(5.49)

In a similar manner we obtain     af N2 a2 / ln , = ln a1 a0 (N2 ) f

(5.50)

    af Ni ai / ln . = ln ai−1 a0 (Ni ) f

(5.51)

and

From Eqs. (5.49) to (5.51) we obtain N1 N2 Ni Nn + + ··· + + ... + (N1 ) f (N2 ) f (Ni ) f (Nn ) f            af a2 ai an a1 + ln + ln + · · · + ln / ln = ln a0 a1 ai−1 an−1 a0

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5 Fatigue and Environment-Assisted Testing

 = ln

a1 a0



a2 a1



 ...

ai ai−1



 ...

an an−1

   af / ln = 1, a0

(5.52)

or n ∑ Ni =1 (N i) f i=1

(5.53)

Equation (5.53) expresses the Miner rule for fatigue crack growth under variable stress amplitudes. It is analogous to Palmgren–Miner rule expressed by Eq. (5.16).

5.5 Fatigue Testing 5.5.1 Introduction Fatigue testing is the mechanical testing performed on a range of components from specimens to full size structures subjected to a cyclic loading. The objective of the tests is to generate fatigue life and crack growth data and demonstrate the safety of a structure. Fatigue tests on specimens are usually performed using servohydraulic test machines which apply variable amplitude cyclic loads. Test data are used to create stress/strain or crack growth versus number of cycles curves. Fatigue tests of full size structures are performed on special test rigs built to apply loads through a series of hydraulic or electric actuators, which reproduce the loads experienced by the structure. Instrumentation such as load cells, strain gauges and displacement gauges are installed on the structure to ensure the correct loading has been applied. Fatigue tests are addressed by several ASTM standards. In the following we briefly present fatigue testing of uncracked and cracked specimens.

5.5.2 Uncracked Specimens The specimens may range from tiny samples tested in a chamber of a scanning electron microscope to full structures, like automobiles, aircrafts, etc. The specimen is cycled through ranges of stress amplitudes and the number of cycles to failure is recorded. The results are reported in the form of S–N curves. There are two main types of loading: rotating bending tests and direct stress tests. Rotating bending tests are the simplest and oldest types of fatigue testing. A known bending moment is applied to a rotating round specimen in a cantilever or two- or four-point bending situation. In the test any point of the specimen is subjected to

5.6 Environment-Assisted Fracture

147

a sinusoidally varying stress as it rotates form the tension side to the compression side of the beam, completing one cycle each time the specimen rotates 360°. In the four-point loading the bending moment is constant over the entire section of the specimen. In the cantilever loading machine the specimen has a narrow waist, so that the maximum bending stress occurs at the smallest diameter, or has a tapered cross section such that the maximum bending stresses are constant at all cross sections. Direct stress tests are conducted using closed-loop servohydraulic test machines which are capable of applying large variable amplitude cyclic loads. Constant amplitude testing can also be applied by simpler oscillating machines. Testing of coupons can be carried out inside environmental chambers where the temperature, humidity and environment that may affect the rate of crack growth can be controlled. For fatigue tests the value of the maximum number of cycles to failure N for a host of stress amplitudes S is determined. Based on these values the stress amplitude versus the number of cycles to failure S–N curve is obtained. Tests are performed for different values of R = σmin /σmax . The frequency of fatigue tests is usually in the range of 10 to 100 Hz. At a range of 100 Hz test of 107 cycles takes 28 h, a test of 108 cycles takes 12 days and a test of 109 cycles almost 4 months. Long test times place a limit on the range of lives that can be studied.

5.5.3 Cracked Specimens In these tests the crack growth rate da/dN versus the stress intensity factor range ΔK = K max − K min during a loading cycle is obtained for different values of the ratio R = K min /K max . A variety of test specimens are used. The most common ones are compact tension, center-cracked tension and single-edge-notch-tension specimen. A precrack is introduced in the specimen. This is accomplished by first machining a sharp notch and then applying a low level cycling load to grow the notch. The specimen is fatigued between a maximum and a minimum stress intensity factor for various R values. The crack growth length is monitored during the test. Crack length versus number of cycles curves are obtained for various ΔK values. In this way the da/dN–ΔK curves are obtained.

5.6 Environment-Assisted Fracture 5.6.1 Introduction It has long been recognized that failure of engineering components subjected to an aggressive environment may occur under applied stresses well below the strength of the material. Failure under such conditions involves an interaction of complex

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chemical, mechanical and metallurgical processes. Environment-assisted cracking refers to a brittle fracture of a typical ductile material where the environment’s corrosive effect is the actual causing agent. The basic subcritical crack growth mechanisms include stress corrosion cracking, hydrogen embrittlement and liquid metal embrittlement. Stress corrosion cracking is the propagation of cracks in a material within a corrosive environment, potentially leading to catastrophic failure of a component/ structure, as the cracking appears brittle. This form of corrosion can occur as either intergranular stress corrosion cracking or as transgranular stress corrosion cracking: Intergranular stress corrosion cracking is where the fracture (crack) forms along the grain boundaries of a material. Transgranular stress corrosion cracking is where the fracture (crack) forms through the grains of a material (and not along the boundaries). There are three main factors that work in combination to affect and cause the stress corrosion cracking of a material. These include the material, the environment and the tensile stress. Different materials are more/less susceptible to stress corrosion cracking than others. The environment that the material is operating within can contain chemical species which cause stress corrosion cracking to occur. A material can experience stress from either residual stress or the direct application of stress. In the case of stress corrosion cracking, crack propagation is caused by mostly static stress. Hydrogen embrittlement is a reduction in the ductility of a metal due to absorbed hydrogen. Hydrogen atoms are small and can permeate solid metals. Once absorbed, hydrogen lowers the stress required for cracks in the metal to initiate and propagate, resulting in embrittlement. Hydrogen embrittlement occurs most notably in steels, as well as in iron, nickel, titanium, cobalt, and their alloys. Copper, aluminum and stainless steels are less susceptible to hydrogen embrittlement. Hydrogen embrittlement requires the presence of both atomic (“diffusible”) hydrogen and a mechanicalstress to induce crack growth. Liquid metal embrittlement is a phenomenon of practical importance, where certain ductile metals experience drastic loss in tensile ductility or undergo brittle fracture when exposed to specific liquid metals. Generally, a tensile stress is needed to induce embrittlement. Cracking can occur catastrophically and very high crack growth rates have been measured. The experimental methods for the evaluation of the stress corrosion susceptibility of a material under given environmental conditions fall into two categories: the timeto-failure tests and the growth rate tests. Both kinds of tests are performed on fatigue precracked specimens. The most widely used specimens are the cantilever beam subjected to constant load, and the wedge-loaded subjected to constant displacement. The stress intensity factors for the specimens are calculated by appropriate calibration formulas.

5.6 Environment-Assisted Fracture

149

Fig. 5.13 Initial stress intensity factor, K Ii , versus time to failure, t, for environment-assisted fracture

5.6.2 Time-To-Failure Tests In the time-to-failure tests the specimens are loaded to various initial stress intensity factor levels K Ii and the time required to failure is recorded. The test results are represented in a K Ii , versus time t diagram, a representative form of which is shown in Fig. 5.13. Observe that, as K Ii decreases, the time to failure increases. The maximum value of K Ii is equal to K Ic or K c , where K Ic is the plane strain fracture toughness and K c is the fracture toughness at thicknesses smaller than the critical thickness for which plane strain conditions apply. A threshold stress intensity factor K ISCC is obtained, below which there is no crack growth. It is generally accepted that K ISCC is a unique property of the material-environment system. The time required for failure can be divided into the incubation time (the time interval during which the initial crack does not grow) and the time of subcritical crack growth. The incubation time depends on the material, environment and K Ii , while the time of subcritical crack growth depends on the type of load, the specimen geometry and the kinetics of crack growth caused by the interaction of material and environment.

5.6.3 Growth Rate Tests In the crack growth rate method for the study of stress corrosion cracking, the rate of crack growth per unit time, da/dt, is measured as a function of the instantaneous

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5 Fatigue and Environment-Assisted Testing

Fig. 5.14 Logarithm of subcritical crack growth rate, log (da/dN), versus stress intensity factor, K I , for environment-assisted fracture

stress intensity factor, K I . Figure 5.14 shows a typical form of the curve log (da/dt) − K I . This can be divided into three regions. In regions I and III the rate of crack growth, da/dt, depends strongly on the stress intensity factor, K I , while in region II da/dt is almost independent of K I . This behavior in region II indicates that crack growth is not of a mechanical nature, but it is caused by chemical, metallurgical and other processes occurring at the crack tip. Note that in region I the threshold stress intensity factor corresponds to K ISCC .

5.6.4 Life Estimate Crack growth can be estimated using a relationship between the crack growth rate da/dt and the stress intensity factor K I of the form da = C K Im dt

(5.54)

where C and m are material constants that depend on the particular environment.

5.6 Environment-Assisted Fracture

151

Equation (5.54) is analogous to Eq. (5.20) of Paris law. Integrating this equation the time for a crack to grow from an initial length to a final length can be obtained. As an example, consider a semi-circular surface crack of radius a0 in a semiinfinite body that grows under stress corrosion cracking according to √ da = C K Im , K I = 0.66σ πa. dt

(5.55)

It is assumed that the crack remains semi-circular during growth and the applied stress increases linearly with time (σ = σ 0 t, where σ 0 is a constant). Fracture takes place at a critical stress intensity factor K IC . We have from Eq. (5.55)  √ √ m da = C(0.66σ πa)m = C 0.66σ0 t π a . dt

(5.56)

By separation of variables Eq. (5.56) becomes  √ m a −m/2 da = C 0.66σ0 π t m dt,

(5.57)

and integrating both parts we obtain

 √ m 1 1 1−m/2 C 0.66σ0 π tcm+1 , ac1−m/2 − a0 = 1 − m/2 m+1

(5.58)

where t c is the critical time to fracture. From Eq. (5.58) we obtain for t c ⎡

⎛



2(m + 1) ⎢ ⎜ tc = ⎣  √ m ⎝a0 C 0.66σ0 π (m − 2)

1−

m 2

 1−

− ac

m  ⎞⎤1/(m+1) 2 ⎟⎥ . ⎠⎦

(5.59)

The critical crack length ac is obtained from √ K I C = 0.66σc πac ,

(5.60)

as  ac =

KIc √ 0.66 π σc

2 .

(5.61)

Introducing the value of ac into Eq. (5.59) we obtain for the critical time t c to fracture

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5 Fatigue and Environment-Assisted Testing

⎡

⎛



2(m + 1) ⎢ ⎜ tc = ⎣  √ m ⎝a0 C 0.66σ0 π (m − 2)

1−

m 2

 −

KIc √ 0.66 π σc

(2−m)

⎞⎤1/(m+1) ⎟⎥ ⎠⎦

. (5.62)

Equation (5.62) relates the critical time to fracture t c and the critical stress σ c at fracture.

5.7 ASTM Standards ASTM Standards for Fatigue Testing of Uncracked Specimens • ASTM E466-18, Standard Practice for Conducting Force-Controlled Constant Amplitude Axial Fatigue Tests of Metallic Materials • E2207-15-21 Standard Practice for Strain-Controlled Axial–Torsional Fatigue Testing with Thin-Walled Tubular Specimens • E606/E606M-21 Standard Test Method for Strain-Controlled Fatigue Testing • E468-18 Standard Practice for Presentation of Constant Amplitude Fatigue Test Results for Metallic Materials • E2368-10(2017) Standard Practice for Strain-Controlled Thermomechanical Fatigue Testing • E2714-13(2020) Standard Test Method for Creep-Fatigue Testing • E2789-10(2021) Standard Guide for Fretting Fatigue Testing • E2948-22 Standard Test Method for Conducting Rotating Bending Fatigue Tests of Solid Round Fine Wire. ASTM Standards for Fatigue Testing of Cracked Specimens • E647-15, “Standard Test Method for Measurement of Fatigue Crack Growth Rates” • E606/E606M-19e1, “Standard Test Method for Strain-Controlled Fatigue Testing” • E2789-10, “Standard Guide for Fretting Fatigue Testing” • E466-21, “Standard Practice for Conducting Force-Controlled Constant Amplitude Axial Fatigue Tests of Metallic Materials” • E2368-10(2017), “Standard Practice for Strain-Controlled Thermomechanical Fatigue Testing” • E2760-19e1, “Standard Test Method for Creep-Fatigue Crack Growth Testing” • F382-17, “Standard Specification and Test Method for Metallic Bone Plates” • F1801-20, “Standard Practice for Corrosion Fatigue Testing of Metallic Implant Materials” • E739-10, “Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S–N) and Strain-Life (ε-N) Fatigue Data”

Further Readings

153

• E1049-85(2017), “Standard Practice for Cycle Counting in Fatigue Analysis” • D3479/D3479M-19, “Standard Test Method for Tension-Tension Fatigue of Polymer Matrix Composite Materials”. ASTM Standards for Environment-Assisted Fracture • E1681-03, “Standard Test Method for Determining a Threshold Stress Intensity Factor for Environment-Assisted Cracking of Metallic Materials” • G129-21, “Standard Practice for Slow Strain Rate Testing to Evaluate the Susceptibility of Metallic Materials to Environmentally Assisted Cracking” • G139-05, “Standard Test Method for Determining Stress-Corrosion Cracking Resistance of Heat-Treatable Aluminum Alloy Products Using Breaking Load Method”.

Further Readings 1. Barsom JM (1973) Fatigue-crack growth under variable-amplitude loading in ASTM A514B steel. Progress in flaw growth and fracture toughness testing, ASTM STP 536. American Society for Testing and Materials, Philadelphia, pp 147–167 2. Bathias C, Pineau A (2011) Fatigue of materials and structures: application to damage and design. Wiley 3. Beevers DJ (1980) The Measurement of crack length and shape during fracture and fatigue. Chamelon Press 4. Beevers CJ (ed) (1982) Advances in crack length measurement. Chameleon Press 5. Cavaliere P (2021) Fatigue and fracture of nanostructured materials. Springer 6. Cullen WH, Landgraf RW, Kaisand LR, Underwood JH (eds) (1985) Automated test methods for fracture and fatigue crack growth. ASTM STP 877 7. De Koning AU (1981) A simple crack closure model for prediction of fatigue crack growth rates under variable-amplitude loading. Fatigue mechanics-thirteenth conference. ASTM STP American Society for Testing and Materials, Philadelphia, pp 63–85 8. Donahue RJ, Clark HM, Atanmo P, Kumble R, McEvily AJ (1972) Crack opening displacement and the rate of fatigue crack growth. Int J Fract Mech 8:209–219 9. Dowling NE, Begley JA (1976) Fatigue crack growth during gross plasticity and the J integral. Mechanics of crack growth, ASTM STP 590. American Society for Testing and Materials, Philadelphia, pp 82–103 10. Dowling NE (1977) Crack growth during low-cycle fatigue of smooth axial specimens. Cyclic stress-strain and plastic deformation aspects of fatigue crack growth, ASTM STP 637. American Society for Testing and Materials, Philadelphia, pp 97–121 11. Dowling NE, Kampe SL, Kral MV (2019) Mechanical behavior of materials, Global Edition 5th edn. Pearson 12. Elber W (1970) Fatigue crack closure under cyclic tension. Eng Fract Mech 2:37–45 13. Elber W (1976) Equivalent constant-amplitude concept for crack growth under spectrum loading. Fatigue crack growth under spectrum loads, ASTM STP 595. American Society for Testing and Materials, Philadelphia, pp 236–250 14. Erdogan F, Ratwani M (1970) Fatigue and fracture of cylindrical shells containing a circumferential crack. Int J Fract Mech 6:379–392 15. Fatigue Test Methodology (1981) AGARD Lecture Series No. 118 16. Forman RG, Kearney VE, Engle RM (1967) Numerical analysis of crack propagation in cyclicloaded structures. J Basic Eng Trans ASME 89:459–464

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17. Gdoutos EE (2022) Fracture mechanics, 3rd edn. Springer, Heidelberg 18. Hans-Jürgen C (ed) (2018) Fatigue of materials at very high numbers of loading cycles: experimental techniques, mechanisms, modeling and fatigue life assessment. Springer Spektrum 19. Head AK (1953) The growth of fatigue crack. Phil Mag 44:924–938 20. Heinrich G, Kipscholl R, Stoˇcek R (eds) (2021) Fatigue crack growth in rubber materials. Springer, Heidelberg 21. Hudak SJ, Bucci RJ (eds) (1981) Fatigue crack growth measurement and data analysis. ASTM STP 738 22. Klesnil M, Lucas P (1972) Effect of stress cycle asymmetry on fatigue crack growth. Mater Sci Eng 9:231–240 23. Lesiuk G, Duda S, Correia JAFO, Jesus AMP (eds) (2022) Fatigue and fracture of materials and structures: contributions from ICMFM XX and KKMP2021. Springer, Heidelberg 24. Manson SS (2005) Fatigue and durability of structural materials. ASM International 25. Marsh KJ, Smith RA, Ritchie RO (eds) (1991) Fatigue crack measurement: techniques and applications. Engineering Materials Advisory Services (EMAS) 26. Marsh KJ (ed) (1988) Full-scale fatigue testing of components and structures. Butterworth 27. Paris P, Erdogan F (1963) A critical analysis of crack propagation laws. J Basic Eng Trans ASME 85:528–534 28. Schijve J (1981) Some formulas for the crack opening stress level. Eng Fract Mech 14:461–465 29. Schijve J (2009) Fatigue of structures and materials, 2nd edn. Springer, Heidelberg 30. Skibicki D (2014) Phenomena and computational models of non-proportional fatigue of materials. Springer, Heidelberg 31. Stephens RI, Fatemi A, Stephens RR, Fucks HO (2001) Metal fatigue in engineering, 2nd edn. Wiley, Hoboken 32. Suresh S (1998) Fatigue of materials, 2nd edn. Cambridge University Press 33. Swanson SR (ed) (1974) Handbook of fatigue testing. ASTM STP 566, American Society for Testing and Materials, Philadelphia 34. Wheeler OE (1972) Spectrum loading and crack growth. J Basic Eng Trans ASME 94:181–186

Chapter 6

Creep Testing

Abstract Creep is the slow, continuous, inelastic, permanent deformation of materials that increases with time under the action of constant stress. Generally occurs at high temperature, but can also happen at room or very low temperatures, albeit much slower. Creep results in large mechanical stress when loads are applied for long time. The rate of creep deformation is a function of the material’s properties, exposure time, temperature and applied load. In this chapter we first present rheological models for the understanding of the three major types of material deformation including elastic, plastic and creep deformation. Based on these models we give constitutive equations for steady-state and transient creep. Also we study relaxation behavior, which is the decrease of stress when a material is held at constant strain. Linear and nonlinear creep and relaxation are analyzed. Finally, we present mechanical tests for the creep, recovery and relaxation characterization of materials.

6.1 Mechanical Behavior of Materials Deformation of materials can be classed into three major types: elastic, plastic and creep. They are closely related to the physical mechanisms that occur in the materials. In the elastic deformation there is stretching, but not breaking, of the atomic bonds. The material returns to its initial position it occupied before loading after the loads are removed. The other two types of deformation, plastic and creep, are inelastic, that is, there remains permanent deformation after the loads are removed. Atoms change their relative position, and slip of crystal planes or sliding of chain molecules takes place. Plastic deformation is independent of time. Creep deformation depends on time. For the solution of engineering problems relationships between stress, strain and time, called constitutive equations, are needed. They are different for the elastic, plastic and creep types of deformation. For the creep deformation the time should be included. In the next section we will present mechanical devices which are helpful for the understanding of the three types of material behavior.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_6

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6 Creep Testing

6.2 Rheological Models Mechanical devices are used for modeling the elastic, plastic and creep deformation. We now present these models for each type of deformation. Elastic deformation: It is simulated by an elastic spring (Fig. 6.1a). When a force P is applied to the spring the deformation x is proportional to the force. It is given by x=

P , k

(6.1)

where k is the spring constant in N/m. When the force is removed the deformation is recovered. The displacement versus time (x–t) and the force versus displacement (P–x) response of the spring to an input force versus time (P–t) is shown in Fig. 6.1. Note that the x–t response (Fig. 6.1c) follows exactly the P–t response (Fig. 6.1b). The (P–x) response (Fig. 6.1d) is linear, and the displacement returns to zero when the force is removed. Plastic deformation: It is simulated by a block of mass m on a frictional plane (Fig. 6.2a). The block is pulled by a force P. Assume that the static and kinetic coefficients of friction μ are equal. When the force P is smaller than P0 = μmg (P < P0 ), where g is the acceleration of gravity, the block does not move (x = 0). For a constant force P ' > P0 the block moves with an acceleration a=

P ' − P0 . m

(6.2)

When the force is maintained for a time t the block has moved by a distance x = at 2 /2. After the force is removed at time t the distance x remains. The deformation x = x p represents the permanent deformation in plasticity. The displacement versus time (x–t) and the force versus displacement (P–x) response of the block to an input force versus time (P–t) for P ' < P0 and for

Fig. 6.1 Displacement versus time (x–t) (c) and force versus displacement (P–x) (d) response of a spring (a) to an input force versus time (P–t) (b)

6.2 Rheological Models

157

Fig. 6.2 Displacement versus time (x–t) (c) and force versus displacement (P–x) (d) response of a block of mass m on a frictional plane (a) to an input force versus time (P–t) (b)

P ' > P0 is shown in Fig. 6.2. Note that for P ' < P0 the block does not move (x = 0). For P ' > P0 and for constant applied force the displacement increases according to x = at 2 /2 (part between points (0, 1) and (2, 3) of the curve). Upon removal of the force the displacement is kept constant (part between points (2, 3) and (4) of the curve). It represents the constant plastic deformation. The force versus displacement curve (Fig. 6.2d) shows that there is a permanent displacement x p which remains upon removal of the applied force. The block of mass m on a frictional plane corresponds to rigid plastic behavior. Creep deformation: It is simulated by a linear dashpot also known as a damper (Fig. 6.3a). A dashpot is constructed by placing a piston in a cylinder filled with a viscous fluid. It resists force by viscous friction. When a force is applied oil leaks past the piston, allowing the piston to move. The force is proportional to the velocity, but acts in the opposite direction. For a constant applied force P ' the velocity x˙ = dx/dt is constant. It is given by x˙ =

P' dx = , dt c

(6.3)

where c is the dashpot constant. Equation (6.3) is analogous to Eq. (6.1) where the displacement x in Eq. (6.1) is replaced by the displacement velocity x˙ and the modulus of elasticity E by the dashpot constant c. The displacement versus time (x − t) and the force versus displacement (P − x) response of the dashpot to an input force versus time (P − t) is shown in Fig. 6.3. In the x-t graph the slope of the curve between points (0, 1) and (2, 3) is constant, equal to P ' /c according to Eq. (6.3). After the removal of the load from point 3 to point 4 a permanent deformation remains. This is shown in the (P − x) graph which indicates the permanent deformation xsc on the x-axis at point (3, 4). The spring constant k can be related to the modulus of elasticity E. Consider a bar of length L and constant cross-sectional area A loaded by a uniaxial force P. We

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6 Creep Testing

Fig. 6.3 Displacement versus time (x–t) (c) and the force versus displacement (P–x) (d) response of the dashpot (a) to an input force versus time (P–t) (b)

have ε=

ΔL x σ P kx = = = = L L E AE AE

(6.4)

kL . A

(6.5)

from which we obtain E=

Equation (6.5) relates the modulus of elasticity E with the spring constant k. Note that in the above rheological models the force, P, simulates the stress, σ , and the displacement, x, the strain, ε. In creep studies a material constant η is defined, analogous to the dashpot constant c, as η=

σ , ε˙

(6.6)

where ε˙ = dε/dt is the strain rate. The constant η is called coefficient of tensile viscosity. We have η=

P PL cL σ = = = ε˙ A˙ε A x˙ A

(6.7)

Equation (6.7) relates η and c. It is analogous to Eq. (6.5) that relates E and k.

6.3 Creep Deformation

159

6.3 Creep Deformation The three rheological models discussed previously, the spring for elastic deformation, the block on a frictional plane for plastic deformation and the dashpot for creep deformation, can be combined to better simulate the mechanical behavior of materials. We consider the creep deformation under constant applied stress and study the increase of the resulting strain with time. In order to incorporate the elastic deformation, we introduce two rheological models, one comprised of a dashpot and a spring in series (Fig. 6.4a), and a second comprised of a spring and dashpot in parallel with this assembly connected with a spring in series (Fig. 6.5a). The first model predicts the steady-state creep and the second model the transient creep.

6.3.1 Steady-State Creep The steady-state creep is simulated by a dashpot and a spring in series (Fig. 6.4a). Consider the strain response to a constant applied stress, σ ' , versus time, t, (σ –t)

Fig. 6.4 Steady-state creep is simulated by a dashpot and a spring in series (a). Strain, ε, versus time, t, (c) and stress, σ, versus strain ε, (d) response to a constant applied stress σ ' versus time (σ –t) history (a)

Fig. 6.5 Transient creep is simulated by a spring and dashpot in parallel connected to a spring in series (a). Strain, ε, versus time, t, (c) and stress, σ, versus strain, ε, (d) response to a constant applied stress σ ' versus time (σ –t) history (b)

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6 Creep Testing

history 01234 (Fig. 6.4b). Let E 1 be the modulus of elasticity of the spring and η1 the coefficient of viscosity of the dashpot. The total strain ε of the spring-dashpot system is obtained by adding the elastic strain, εe , of the spring and the creep strain, εc , of the dashpot, as: ε = εe + εc =

σ' + εc E1

(6.8)

The creep strain εc is obtained by integrating Eq. (6.7) with respect to time t as εc =

σ' t. η1

(6.9)

Introducing the value of εc into Eq. (6.8) we obtain for the total strain ε ε=

σ' σ' + t. E1 η1

(6.10)

For a constant applied stress σ ' , the first part of Eq. (6.10) is constant, while the second part varies linearly with time t. Consider a stress versus time response (Fig. 6.4b). The strain ε versus time t response (Fig. 6.4c) consists of a vertical line 01 corresponding to the vertical line 01 of the stress versus time graph. For t = 0, ε = σ ' /E 1 . This is the elastic strain for an applied stress σ ' (point 1 of the ε–t graph). For a constant stress σ ' between points 1 and 2 (Fig. 6.4b) the strain ε, according to Eq. (6.10), varies linearly with time t along the line 12 of Fig. 6.4c. When stress σ ' is removed at point 2 the elastic strain disappears (part 23 of Fig. 6.4c, length 23 is equal to length 01) and the remaining strain is the creep strain. The creep strain εsc accumulated for a constant stress σ ' during loading 1–2 remains as a permanent strain. The stress–strain behavior is shown in Fig. 6.4d. The strain ε varies linearly with stress during the interval 01 and is constant during the interval 12. Upon unloading at point 2 the strain decreases linearly with stress up to the point 3 (Fig. 6.4d). The creep strain εsc accumulated during interval 1–2 remains as a permanent strain (strain between points 0 and (3, 4) (Fig. 6.4d).

6.3.2 Transient Creep The transient creep is simulated by a spring (with modulus of elasticity E 2 ) and dashpot (with coefficient of tensile viscosity η2 ) in a parallel combination with this assembly connected with a spring (with modulus of elasticity E 1 ) in series (Fig. 6.5a). For an applied constant stress σ ' during interval 1–2 the resulting strain ε is the sum of the elastic strain, εe , in the spring and the creep strain, εc , in the spring-dashpot parallel combination (Fig. 6.5b). To analyze the creep strain we note that the stress

6.3 Creep Deformation

161

σ ' in the (η2 , E 2 ) stage is the sum of the elastic stress in the spring and the creep stress in the dashpot. We have σ ' = E 2 εc + η2 ε˙ c .

(6.11)

From this equation we obtain for ε˙ c ε˙ c =

σ ' − E 2 εc dεc = . dt η2

(6.12)

This equation can be written as E 2 εc dεc σ' + = . dt η2 η2

(6.13)

This is a first-order linear differential equation that can be solved by multiplying both sides of the equation by e(E2 t/η2 ) . We obtain

e(E2 t/η2 )

dεc E 2 εc σ' = e(E2 t/η2 ) . + e(E2 t/η2 ) dt η2 η2

(6.14)

Equation (6.14) can be written in the form

d ( (E2 t/η2 ) ) σ' e ec = e(E2 t/η2 ) dt η2

(6.15)

By integration between 0 and t we obtain

εc =

) σ' ( 1 − e−(E2 t/η2 ) E2

(6.16)

This is the transient creep strain. Adding the elastic strain εe we obtain for the total strain ε

) σ' σ' ( ε = εe + εc = + 1 − e−(E2 t/η2 ) E1 E2 ) ( 1 1 1 ( −(E2 t/η2 ) ) ' e =σ + − E1 E2 E2

(6.17)

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6 Creep Testing

Equation (6.17) gives the strain as a function of time for a constant applied stress. It is the constitutive creep equation that relates strain, stress and time. Note that strain ε varies linearly with stress σ ' . For t = 0 Eq. (6.17) gives ε = σ ' /E 1 which is the elastic strain (part 0–1 of the graph ε–t of Fig. 6.5c). The strain rate ε˙ is obtained by differentiating Eq. (6.17) with respect to time t as: ε˙ =

σ ' −(E2 t/η2 ) e η2

(6.18)

Equation (6.18) indicates that the strain rate decreases with time. When the stress is removed at point 2 of graph ε–t (Fig. 6.5b) the strain reduces by the amount 2–3 which is equal to the elastic strain εc = σ ' /E 1 . Following unloading, the creep strain εc given by the second part of Eq. (6.17) approaches to the limit εc = σ ' /E 2 (for t → ∞). The stress is being transferred from the dashpot to the spring as time passes, until all stress is resisted by the spring at infinite time. After the removal of the stress the strain varies according to the curve 3–4 of Fig. 6.5c. The curve 3–4 represents the recovery response of creep. The stress–strain response is shown in Fig. 6.5d. Note that the elastic strain develops first as the stress increases up to a constant value. Under constant stress the creep strain increases. Upon unloading, the elastic strain is relieved and the creep strain diminishes as time passes.

6.3.3 Relaxation Behavior Relaxation is the decrease of stress when a material is held at constant strain. We will study relaxation by using the model of a dashpot and spring in series (Fig. 6.6α). The total applied strain ε' is the sum of the elastic, εe , and creep, εc , strain as: ε' = εe + εc

(6.19)

Fig. 6.6 A dashpot and spring in series for the study of relaxation under constant strain. The step in strain (b) causes stress-time behavior as in (c) and stress–strain behavior as in (d)

6.3 Creep Deformation

163

The stress σ = σ (t) is related to the elastic strain εe by σ = E 1 εe

(6.20)

The rate of creep strain ε˙ c is related to the stress by ε˙ c =

dεc σ = dt η1

(6.21)

Since the applied strain ε' is constant we obtain by differentiating Eq. (6.19) with respect to time ε˙ e + ε˙ c = 0

(6.22)

By differentiating Eq. (6.20) with respect to time we obtain σ˙ =

dσ = E 1 ε˙ e dt

(6.23)

Substituting the values of ε˙ c and ε˙ e from Eqs. (6.21) and (6.23) into Eq. (6.22) we obtain σ 1 dσ + =0 E 1 dt η1

(6.24)

By separating the variables Eq. (6.24) becomes E1 dσ = − dt σ η1

(6.25)

Integrating Eq. (6.25) we obtain ln σ = −

E1 t +C η1

(6.26)

where C is a constant of integration. C can be obtained by noting that at t = 0 the creep strain εc is zero (all of the applied strain is absorbed by the spring, since the dashpot requires a finite time to respond), so that σ = E 1 ε' (from Eqs. (6.19) and (6.20)). For t = 0 Eq. (6.26) gives C = ln σ = ln E 1 ε'

(6.27)

Substituting this value of C into Eq. (6.26) we have σ = σ (t) = E 1 ε' e−(E1 t/η1 )

(6.28)

164

6 Creep Testing

Equation (6.28) represents the variation of stress with time for a constant applied strain. It is the relaxation constitutive equation. It indicates that the stress decreases, relaxes, with time (Fig. 6.6c). Note that the stress is linearly related to strain. The relaxation phenomenon is observed under constant strain, whereas creep is observed under constant stress. Both creep and relaxation are of the same nature, and they are indicative of the rheological behavior of materials. Materials that exhibit creep will also exhibit relaxation. When the applied strain is removed after a period of relaxation the stress becomes compressive (Fig. 6.6d). For zero strain the stress relaxes further and approaches to zero asymptotically. The stress–strain response of the material is shown in Fig. 6.6d.

6.4 Linear Viscoelasticity In the three rheological models presented above, the steady-state creep, the transient creep and the relaxation the relationship between stress and strain is linear. The linearity condition is valid for any combination of the rheological models. Note that in creep the stress is held constant and the change of strain with time is studied, while in relaxation the strain is held constant and the change of stress with time is studied. The study of problems with linear stress–strain relations that include the time dependency based on the above simple rheological models is termed linear viscoelasticity. The strain–time equations for constant applied stress for the above-presented rheological models can be combined to provide a better simulation of the timedependent material behavior. By combining the models of steady-state and transient creep we obtain the model of Fig. 6.7. Due to the linear stress–strain relationship for a given time for both models the strain in the combined model is obtained from Eqs. (6.10) and (6.17) as:

ε = εe + εsc + εtc =

Fig. 6.7 Strain versus time behavior for a viscoelastic model obtained by combining the models of steady state and transient creep

) σ σt σ ( 1 − e−(E2 t/η2 ) + + E1 η1 E2

(6.29)

6.5 Nonlinear Creep

165

The first term of the above equation corresponds to the instantaneous linear strain εe in spring E 1 , the second term to the steady-state creep strain εsc in dashpot η1 and the third term to the transient creep strain εtc in the (E 2 , η2 ) parallel combination. Strain rates are obtained by differentiating Eq. (6.29) with respect to time, as

σ σ ( −(E2 t/η2 ) ) ε˙ = ε˙ sc + ε˙ tc = + e η1 η2

(6.30)

Note from Eq. (6.30) that the steady-state creep strain (first term of Eq. (6.30)) has a constant rate and the transient creep strain (second term of Eq. (6.30)) has a decreasing rate. The transient creep strain approaches the limiting value of εtc = σ/E2 (Eq. (6.29) for t → ∞). Equations (6.10), (6.17) and (6.29) show that stress–strain curves for a given time are straight lines. These curves are called isochronous curves. For the transient creep model (Fig. 6.5) we obtain from Eq. (6.17) for large t ε = εe + εc =

σ' σ' σ ' (E 1 + E 2 ) σ' + = = E1 E2 E1 E2 Ee

(6.31)

where Ee =

E1 E2 (E 1 + E 2 )

(6.32)

Equation (6.31) indicates that for the transient creep model for large values of t the dashpot has no effect and the stiffness is the combination of the stiffnesses of springs E 1 and E 2 in series given by Eq. (6.32) (Fig. 6.8a). For small values of t only the spring E 1 deforms and the slope of the stress–strain curve is E 1 (Eq. (6.17)) (Fig. 6.8a). For the combined steady-state and transient creep models of Fig. 6.7 for small values of t the slope of the stress–strain curve is E 1 , while for large values of t the slope is zero (Eq. (6.29)) (Fig. 6.8b).

6.5 Nonlinear Creep Creep behavior for many materials at elevated temperatures cannot by described by the equations of linear viscoelasticity based on the rheological models presented above. In this case, the creep deformation is better described for nonlinear viscoelastic equations. Equations involving powers of stress σ have been proposed. Modifying Eq. (6.29) the stress–strain-time behavior can be described as ( ) ε = εi + Bσ m t + Dσ α 1 − e−βt

(6.33)

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6 Creep Testing

Fig. 6.8 Linear stress–strain curves for small and large values of time for two linear viscoelastic models

where B, m, D, α and β are empirical constants obtained from creep experistrain εi can include both elastic and plastic components (ments. The instantaneous ) εi = εe + ε p . For the special case the plastic strain is given by ε p = B1 σ n with 1/n = α = m Equation (6.33) can be put in the form ε=

[ ( )] σ + B1 + B2 t + B3 1 − e−βt σ m E

(6.34)

Stress–strain-time relationships in the form of the Ramberg–Osgood equation are also used to describe the viscoelastic behavior of materials. Such a form is given by σ + ε= E

(

σ Hc

)1/ηc (6.35)

where n c is an empirical constant and Hc includes a time dependency. Equation (6.35) can be obtained from Eq. (6.34) with nc =

1 m

[ ( )]−1/m Hc = B1 + B2 t + B3 1 − e−βt

(6.36)

6.7 Stress–Strain Relationships in Three Dimensions

167

6.6 Nonlinear Stress Relaxation Consider the case of relaxation of Sect. 6.3.3, Fig. 6.6. The applied strain ε is kept constant and the variation of stress σ with time t is studied. The creep strain rate is not linearly related to strain, but follows the equation ε˙ c = Bσ m

(6.37)

This equation is a generalization of the linear relaxation model of Eq. (6.21) with σ 1 = ε˙ c Bσ m−1

(6.38)

1 dσ + Bσ m = 0 E dt

(6.39)

η1 = Then, from Eq. (6.22) we obtain

By separation of variables and integration we obtain {t

1 dt = − BE

o

{σ σi

dσ σm

(6.40)

Integrating this equation and noting that at the beginning of relaxation, t = 0 the initial stress is σi = Eε' . we obtain for the variation of stress σ = σ (t) with time t σi σ =[ ]1/(m−1) t B E (m − 1)σim−1 + 1

(m /= 1)

σ = σi e−B Et (m = 1)

(6.41a) (6.41b)

Equation (6.41b) is equivalent to Eq. (6.28) of the linear model with m = 1, η1 = 1/B.

6.7 Stress–Strain Relationships in Three Dimensions The strain rate-stress relationships for creep in three dimensions can result from the assumption that the material is incompressible, that is ε x + ε y + εz = 0 from which it follows for the strain rates

(6.42)

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6 Creep Testing

ε˙ x + ε˙ y + ε˙ z = 0

(6.43)

For a uniaxial stress σ x (σ y = σ z = 0) the resulting strain rate is: ε˙ x =

σx η

(6.44)

where η is the tensile viscosity. We obtain from Eq. (6.43) with ε˙ y = ε˙ z ε˙ y = ε˙ z = −0.5˙εx = −0.5

σx η

(6.45)

Using Eq. (6.45) and the principle of superposition we obtain the following strain rate-stress equations for a triaxial stress field of stresses σx , σ y , σz ( )] 1[ σx − 0.5 σ y + σz η ] 1[ ε˙ y = σ y − 0.5(σz + σx ) η )] ( 1[ ε˙ z = σz − 0.5 σx + σ y η

ε˙ x =

(a) (b) (c)

(6.46)

The shear strain rates are obtained from the shear stresses using the shear viscosity coefficient ητ . Since ητ = η/3, we have γ˙x y =

3 3 3 τx y , γ˙yz = τ yz , γ˙zx = τzx η η η

(6.47)

Equations (6.46) and (6.47) are analogous to Hooke’s stress–strain law, except that they involve strain rates. Poisson’s ratio ν is replaced by 0.5 and shear modulus of rigidity by η/3.

6.8 Solution of Creep Problems The previous linear isochronous stress–strain equations between stress and strain for constant time suggest that creep problems can be solved within the frame of linear viscoelasticity as linear elasticity problems. For nonlinear isochronous stress–strain relationships a nonlinear stress analysis can be performed in the same manner as for the deformation theory of plasticity.

6.9 Creep and Relaxation Testing

169

6.9 Creep and Relaxation Testing Creep testing involves the application of a prolonged constant tensile or compressive stress to a material specimen at constant temperature. The force may be applied by a dead weight and a lever system. Universal testing machines are usually used. During testing, the material’s deformation is recorded at specific time intervals. Maintaining a constant temperature during a creep test is critical due to the possible thermal expansion or shrinkage of the material. Tests are performed at various stresses and temperatures. The duration of the creep test can range from less than a minute to several years. Creep data are presented on graphs of strain versus time (Fig. 6.9). There are three different stages in a creep test. a. Primary or transient stage. Following an initial elastic and perhaps plastic strain a gradual accumulation of primary creep strain takes place. The creep rate ε˙ = dε/dt which is the slope of the strain versus time plot begins by rising quickly and then slows down and decreases. b. Secondary of steady stage. The creep rate remains fairly constant. c. Tertiary stage. The creep rate is much steeper than it is in the secondary stage, and it increases in an unstable manner as rupture failure approaches. The deformation becomes localized by the formation of a neck as in a tension specimen or voids may form inside the material. If failure occurs the time to failure is recorded. In the creep tests the following quantities enter: the constant stress, σ, the strain, ε, and the time to rupture t r . The tests are performed at different temperatures. The results of a creep test are usually presented in the form of strain versus time curves for different stress levels (Fig. 6.10). Other forms of curves include plot of stress σ versus strain rate ε˙ at different temperatures, stress σ versus time to rupture t r at different temperatures, stress σ versus time to rupture t r at different strains. Figure 6.11a Fig. 6.9 Strain, ε, versus time, t, behavior during creep under constant stress. The three stages of creep (primary, secondary, tertiary) are shown

170

6 Creep Testing

shows strain ε versus time t curves for four values of applied stress σ. From these curves the isochronous stress-strain curves can be plotted (Fig. 6.11b). For the determination of the stress relaxation behavior of a material the specimen is deformed at a given amount and the decrease of stress is recorded over prolonged period of exposure at constant temperature. From such experiments stress versus time curves are plotted at constant strains.

Fig. 6.10 Creep curves of strain, ε, versus time, t, for different stress levels

Fig. 6.11 a Strain ε versus time t curves for four values of applied stress σ. b Isochronous stress– strain curves

Further Readings

171

6.10 ASTM Standards ASTM standards for creep testing include: • E139-11(2018), “Standard Test Methods for Conducting Creep, Creep-Rupture, and Stress-Rupture Tests of Metallic Materials” • D2294-96(2016), “Standard Test Method for Creep Properties of Adhesives in Shear by Tension Loading (Metal-to-Metal)” • D2990-17, “Standard Test Methods for Tensile, Compressive, and Flexural Creep and Creep-Rupture of Plastics” • E2760-19e1 “Standard Test Method for Creep-Fatigue Crack Growth Testing” • E1457-19e1 “Standard Test Method for Measurement of Creep Crack Growth Times in Metals” • C512-02 “Standard Test Method for Creep of Concrete in Compression” • D6992-03(2009) Standard Test Method for Accelerated Tensile Creep and CreepRupture of Geosynthetic Materials Based on Time–Temperature Superposition Using the Stepped Isothermal Method • C1291, “Standard Test for Elevated Temperature Tensile Creep Strain, Rate, and Time-to-Failure for Monolithic Advanced Ceramics” • E328-21, “Standard Test Methods for Stress Relaxation for Materials and Structures” • D6147-97(2020), “Standard Test Method for Vulcanized Rubber and Thermoplastic Elastomer-Determination of Force Decay (Stress Relaxation) in Compression” • D6746-15(2020), “Standard Test Method for Determination of Green Strength and Stress Relaxation of Raw Rubber or Unvulcanized Compounds”

Further Readings 1. Dowling NE, Kampe SL, Kral MV (2019) Mechanical behavior of materials. Global Edition 5th edn. Pearson 2. Gooch DJ, How IM (eds) (1986) Techniques for multiaxial creep testing. Elsevier Applied Science 3. Kraus H (1980) Creep analysis. Wiley, Hoboken 4. McKeen LW (2009) The effect of creep and other time related factors on plastics and elastomers (Plastics Design Library). William Andrew 5. Penny RK, Marriott DL (2012) Design for Creep, 2nd edn. Springer Science + Business Media, BV 6. Rees D (2023) Handbook on mechanics of inelastic solids: vol 1: plasticity, creep and viscous deformation, vol 2: Finite and cyclic deformation: Structural applications. World Scientific Publishing Company

Chapter 7

Testing at High Strain Rates

Abstract In this chapter we present methods of material testing at high strain rates. Mechanical behavior of materials at high strain rates differs considerably from that at quasi-static rates. Special testing equipment, methods and experimental arrangements are needed. We will present the following tests: high-speed load frames for low rates, drop weight impact and pendulum impact including the Charpy and Izod tests for intermediate rates, the split-Hopkinson (Kolsky) bar impact for compression, tension and torsion, the Taylor impact and the expanding ring for high rates, and the plate impact for very high rates. In all tests we will provide the testing equipment, experimental procedure and instrumentation, data analysis and discussion. The chapter concludes with the ASTM standards for testing at high strain rates.

7.1 Introduction Materials in engineering applications are subjected to high strain rates. They include drop of personal items, vehicle collision, sports impact, armor penetration by a bullet, earthquakes, blast loading, structural impacts, ballistics, metalworking, etc. Mechanical behavior at high strain rates differs considerably from that observed at quasistatic strain rates. Experimental results, for example, indicate that the yield stress of many metals increases with the rate of loading. The characterization of mechanical properties of materials at high strain rates is of great importance. In impact problems the range of strain rates differs widely. During the impact of an asteroid on the Earth strain rates of the order of 108 s−1 may be developed. On the other hand, for impacts related to defense ballistics peak value of the order of 105 s−1 to 106 s−1 develops. Damage usually occurs at lower strain rates because they sustain for longer times, and it takes time for many damage mechanisms to develop. During impact stress waves or shock waves propagate inside the impacted bodies, inelastic deformations may develop and the impacted bodies may be excited leading to structural dynamics and vibration problems. A fundamental difference between a high strain test and a quasi-static test is that inertia and wave propagation effects become more pronounced at high strain rates. At

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_7

173

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7 Testing at High Strain Rates

medium strain rates the measurement of load is the first to be affected by propagation of stress waves. For further increase of the strain rate uniform deformation on the specimen becomes more critical. At very high strain rates shock wave propagation occurs. Measurements at high strain rates cannot be performed by conventional testing machines. For this reason special methods and experimental arrangements are used. They will be presented in the following sections of this chapter.

7.2 Strain Rate The concept of strain rate is very important in problems of dynamic loading. The strain rate, ε˙ , is defined as ε˙ =

dε , dt

(7.1)

where ε is the engineering or the true strain and t is the time. The unit of strain rate is s−1 (second−1 ). For a compression test the negative sign of the strain and strain rate is omitted. When the strain rate is constant during a test, it is given by ε ε˙ = . t

(7.2)

For an engineering strain ε we have from Eq. (7.1) ε˙ =

1 dL V dε = = , dt L 0 dt L0

(7.3)

where L 0 is the original and L is the current length of the specimen and V is the speed at which the specimen is being deformed. A testing machine operating at a constant crosshead speed (V is constant) provides a constant strain rate.

7.3 Tests at Different Strain Rates Characterization of the mechanical behavior of materials at high strain rates is important for many engineering applications. Conventional servohydraulic testing machines are generally used for testing at quasi-static strain rates of 1 s−1 and with special design it is possible to attain greater strain rates up to 102 s−1 . For higher strain rates other special methods have been developed. We will present the following methods with the corresponding strain rate range: testing machines for ε˙ = 10–1 ~ 102 s−1 , drop towers for ε˙ = 1 ~ 103 s−1 ; the split-Hopkinson (Kolsky) bar for ε˙ = 102 ~ 104 s−1 ; the Taylor rod for ε˙ = 102 ~ 104 s−1 ; and the plate impact methods

7.4 High-Speed Load Frames

175

Table 7.1a Experimental methods for high strain rate testing Applicable strain rates, s−1

Testing technique

Compression tests < 0.1

Conventional load frames

0.1–100

Special servohydraulic frames

0.1–500

Cam plastometer and drop test

200–104

Hopkinson (Kolsky) bar in compression

103 –105

Taylor impact test

Tension tests < 0.1

Conventional load frames

0.1–100

Special servohydraulic frames

100–103

Hopkinson (Kolsky) bar in tension

104

Expanding ring

> 105

Flyer plate

Shear and multiaxial tests < 0.1

Conventional shear tests

0.1–100

Special servohydraulic frames

10–103

Torsional impact

100–104

Hopkinson (Kolsky) bar in torsion

103 –104

Double-notch shear and punch

104 –107

Pressure-shear plate Impact

for ε˙ = 106 ~ 108 s−1 . The various methods with the corresponding strain rates they can achieve are summarized in Tables 7.1a and 7.1b. In dynamic testing the time to produce 1% strain (characteristic time) is a useful guide for the test duration. From Eq. (7.2) we obtain for the characteristic time, t, for the strain rate ranges: t = 10–2 ~ 10−5 s for ε˙ = 1 ~ 103 s−1 ; t = 10–4 ~ 10−6 s for ε˙ = 102 ~ 104 s−1 ; and t = 10–8 ~ 10−10 s for ε˙ = 106 ~ 108 s−1 .

7.4 High-Speed Load Frames At high strain rates inertia and wave propagation effects become more pronounced than in quasi-static tests. The measurement of load is the first to be affected by stress wave propagation. At even higher strain rates, uniform deformation within the specimen becomes an issue. At very high strain rates shock wave propagation becomes critical. The speed capability of pneumatic and hydraulic machines is influenced by several factors including the load the ram is attempting to apply and the distance traveled. In a long-stroke machine a given speed is attained only after a significant amount of time. Depending on the specimen considerable strain occurs before final

176

7 Testing at High Strain Rates

Table 7.1b Strain rate regimes and associated instruments and experimental conditions 0

10-8

10-6

10-4

10-2

102

100

104

106

Creep

Quasi-static

Intermediate strain rate

High strain rates

Constant load or stress machine

Servohydraulic and screw machines

Special servohydraulic machines

Hopkinson techniques

Light gas gun or explosively driven plate impact

Strain versus time or creep rate recorded

Constant strain rate tests

Uniaxial stress and torsion tests

Uniaxial strain and shear tests

Constant strain rate tests

Very high strain rates

Inertia forces neglected

Inertia forces important

Isothermal

Adiabatic/quasi-isothermal Uniaxial and shear stress

Strain rate (s-1)

Shock loading

Dynamic considerations in testing

Uniaxial strain and simple shear

Increasing stress levels

velocity is obtained. The ability to control the speed depends on the response capability of the servo-controlled machine (“servo” means “feedback control”) which works in a closed system. Load cell ringing at intermediate strain rates can mask the desired measurements. The load-cell response time must be small compared to the duration of the test. Servohydraulic testing machines for strain rates up to ε˙ = 102 s−1 are available from many companies, such as MTS and Instron. Full description of the various machine models and their specifications are provided and will not be presented here.

7.5 Drop Weight Impact Test 7.5.1 Introduction The drop weight impact test (DWIT) or drop tower compression test is the most widely used method to simulate medium strain rates impact loading. As the name implies the method uses a falling weight to provide an impact compressive load to the specimen. The method can provide high loads at medium strain rates, which cannot be obtained from servohydraulic testing machines. The strain rates produced are in the range of ε˙ = 1 ~ 103 s−1 . The method is useful for the study of damages under impact loading and is widely applied to the automotive, aerospace, defense and packaging industries. It is characterized by its simplicity in implementation and its good repeatability.

7.5 Drop Weight Impact Test

177

Fig. 7.1 Drop tower impact test system

7.5.2 Experimental Procedure and Instrumentation A drop weight impact system consists of a mechanical system and a data acquisition system. The mechanical system consists of a drop tower with a massive foundation, a dynamic compression fixture and stop blocks (Fig. 7.1). The test technique has the capability to generate high loads at medium strain rates, which cannot be readily obtained by servohydraulic load frames. The data acquisition system senses the force and the displacement of the drop weight. Two transducer systems are used. The first system employs an accelerometer transducer embedded in the falling weight to measure the acceleration versus time, and a velocity transducer to measure the initial impact velocity. The second system records the force versus time history of the drop weight.

7.5.3 Data Analysis The dynamic energy E d of the drop weight of mass m from a height h is E d = mgh

(7.4)

where g is the acceleration of gravity (g = 9.81 m/c2 ). The kinetic energy E k of the moving drop weight is Ek =

1 mV 2 , 2

(7.5)

178

7 Testing at High Strain Rates

where V is the velocity when the drop rigid body just contacts the specimen. Since the dynamic energy is equal to the kinetic energy of the rigid body, we obtain for the velocity of the rigid body at the impact point of the specimen V =

√

2gh.

(7.6)

Equation (7.6) indicates that the velocity of rigid body can be adjusted by changing the drop height h. Due to the limitation of the height, very high velocities cannot be achieved. For example to achieve a velocity of 30 m/s a frame with a height 45 m is required. To accelerate the drop weight to a higher velocity a spring system may be installed between the frame and the drop weight. From the accelerometer transducer the displacement s (t) can be calculated as ¨ s(t) = V0 t +

a(t)dt,

(7.7)

where a (t) is the acceleration, V 0 is the initial velocity and t is the time. The velocity V (t) can be calculated as { V (t) = V0 +

a(t)dt.

(7.8)

The impact energy E (t) is calculated as { E(t) =

F(t)s(t)dt.

where F(t) is the force. The force can be calculated from the accelerometer transducer as F(t) = m a(t).

(7.9)

7.5.4 Discussion The DWIT is neither a constant-displacement rate nor a constant-loading rate test. The rate and form of the load depend on the compliances of the specimen and test system, the impact velocity and the energy of the drop weight. The test velocities and, therefore, the strain rates are limited by the response time of the instrumentation and the inertia loading of the system. In general, it is better to drop a larger weight from a lower height than the converse. The lower velocity will reduce the inertia loading and generate comparable maximum loads.

7.6 Pendulum Impact Test

179

The DWIT has been standardized by ASTM. Standards E436 and E604 are used to determine the fracture ductility of metals in order to establish the temperature range over which ferritic steels undergo a fracture mode transition from ductile to brittle. E208 provides guidelines for conducting the DWIT on welded specimens in order to determine the nil-ductility transition temperature of ferritic steels. Specimen requirements have not been developed for the DWIT. Cylindrical specimens of lengths two times their diameter have been used. As with all compression tests, the end constraint of the specimen due to friction is a primary concern. Hardened steel and modulus-matched inserts have been used between the platens and the specimen to minimize the constraint.

7.6 Pendulum Impact Test 7.6.1 Introduction The pendulum impact test is a standardized high strain rate test used to measure the impact energy absorbed by a material during fracture. A notched specimen is broken by a swinging pendulum. The notch provides a point of stress concentration within the specimen and improves the reproducibility of the results. The energy absorbed during fracture is measured from the loss of the potential energy of a pendulum through breaking the specimen. Absorbed energy is a measure of the material’s notch toughness. The test is used to measure the toughness (sometimes referred to as notch toughness) or the impact strength of a material. Toughness measures the ability of a material to absorb energy. Strain rates in the range of ε˙ = 1 ~ 103 s−1 can be achieved in the test depending upon the design or size of the pendulum impact machine. Two versions of the test are available, the Charpy and the Izod tests. In both tests the instrumentation is the same, only the type of the specimen differs. The test is easy to prepare and conduct, and results can be obtained quickly and cheaply. It is widely used in industry.

7.6.2 Experimental Arrangement The experimental arrangement consists of a base frame with a pendulum and a specimen support fixture (Fig. 7.2). The pendulum is raised to a specific height and then released. The pendulum swings down through an axis located on the base frame hitting the notched specimen and breaking it. The specimen is supported in the fixture. Commercial pendulum impact test systems are available. The test conditions are governed by: the dimensions of the specimen, the height and the mass of the pendulum at the start position and the curvature of the tip of the notch.

180

7 Testing at High Strain Rates

Fig. 7.2 A pendulum-type test machine for Charpy or Izod impact tests

7.6.3 Energy Considerations The initial height of the pendulum and the height the pendulum reaches after breaking the specimen are used to measure the kinetic energy the pendulum hits the specimen. The potential energies of the specimen at an initial height h1 and at a height h2 after breaking the specimen are E 1 = mg h 1 , E 2 = mg h 2

(7.10)

where m is the mass of the pendulum. The energy loss E k = E 1 − E 2 is equal to the kinetic energy absorbed by the specimen. We have E k = El − E 2 = mg(h l − h 2 ). This energy is absorbed by fracturing the specimen.

(7.11)

7.6 Pendulum Impact Test

181

7.6.4 Charpy Impact Test In the Charpy impact test a notched three-point bend specimen is used. To carry out the test the specimen is supported at its two ends on an anvil and struck on the opposite face to the notch by a pendulum (Fig. 7.3). The specimen is fractured and the pendulum swings through. From the height of the swing the amount of energy absorbed in fracturing the specimen at the notch is calculated from Eq. (7.11). The results of the test are expressed in energy lost per unit of thickness at the notch (Jm−1 ) called impact energy or notch toughness. Alternatively, the results of the test are reported as energy loss per unit of cross-sectional area of the notch (Jm−2 ) as: R = E k /A,

(7.12)

where A is the cross-sectional area of the specimen where the notch is located. The standard methods for “Notched Bar Impact Testing of Metallic Materials” can be found in ASTM E23 and ISO 148, where all aspects of the test and equipment are described in detail. According to ASTM A370 (Standard Test Method and Definitions for Mechanical Testing of Steel Products), the standard specimen size for Charpy impact testing is 10 mm × 10 mm × 55 mm. Subsize specimen sizes are: 10 mm × 7.5 mm × 55 mm, 10 mm × 6.7 mm × 55 mm, 10 mm × 5 mm × 55 mm, 10 mm × 3.3 mm × 55 mm, 10 mm × 2.5 mm × 55 mm. According to ISO 148, standard specimen sizes are 10 mm × 10 mm × 55 mm. Subsize specimens are: 10 mm × 7.5 mm × 55 mm, 10 mm × 5 mm × 55 mm and 10 mm × 2.5 mm × 55 mm.

Fig. 7.3 Specimen and loading configuration for Charpy V-notch test

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7 Testing at High Strain Rates

Fig. 7.4 Specimen and loading configuration for Izod test

7.6.5 Izod Impact Test In the Izod impact test the specimen is a cantilever beam (Fig. 7.4), as opposed to the three-point bend specimen of the Charpy test. The V-notch is positioned at the fixed end of the specimen facing the striker. The specimen is stroked by the pendulum and fractures at its notch. The impact energy Ek is calculated from Eq. (7.11). The results of the test are also reported by the impact strength E k /L, where L is the net section length ahead of the notch. The ASTM International standard for Izod impact testing of plastics is ASTM D256. In Europe, ISO 180 methods are used. The dimensions of a standard specimen for ASTM D256 are 63.5 × 12.7 × 3.2 mm. The most common specimen thickness is 3.2 mm, but the width can vary between 3.0 and 12.7 mm.

7.6.6 Discussion Pendulum impact tests provide a rough estimate of the notch toughness of materials. Results depend on the size of the specimen and the geometry of the notch. More sophisticated methods for measuring the notch toughness are based on fracture mechanics. However, pendulum impact tests, despite their shortcomings, remain popular as simple, economical and quick tests for the characterization and comparison of materials with respect to their notch toughness.

7.7 Split-Hopkinson (Kolsky) Bar Impact Test

183

7.7 Split-Hopkinson (Kolsky) Bar Impact Test 7.7.1 Introduction The split-Hopkinson pressure bar (SHPB) impact test, also called Kolsky bar impact test is a widely used experimental method based on impact stress wave measurement for the determination of material properties at high strain rates in the range of ε˙ = 50 ~ 104 s−1 . It constitutes a simple and effective means for the study of the dynamic properties of materials. It is considered as one of the major breakthroughs in the investigation of the material response at high strain rates. The method started from John Hopkinson and his son Bertram Hopkinson in 1914 who proposed a technique to measure the shape of an impact stress pulse in a long elastic bar. Kolsky in 1949 extended the Hopkinson bar method to measure stress–strain response of materials under impact loading. The pressure bar technique of Kolsky was similar to that developed by Davis in 1948, except that Kolsky used two bars and the specimen was sandwiched in between. This allows deformation of a specimen of ductile material at high strain rates, while maintaining a uniform uniaxial state of stress within the specimen. The maximum strain rate in a Kolsky bar varies inversely with the length of the specimen. It is limited by the elastic limit of the two bars that are used to transmit the stress pulse to the specimen. With the Kolsky bar it is possible to develop the uniaxial stress–strain behavior of materials at a variety of strain rates. This allows the development of constitutive relations that express the uniaxial stress as a function of strain, strain rate and temperature. The SHPB impact test was first developed for compression loading and then extended to tension, torsion or combinations of them. In the following we will first present the method for compressive loading, and then, for the other forms of loading.

7.7.2 Experimental Arrangement The experimental arrangement of the SHPB consists of a mechanical, a data acquisition and recording system (Fig. 7.5). Mechanical system: It consists of a loading device and the bar components. The dynamic loading is created by launching a striker that impacts the incident bar. Gas guns usually provide a controllable and repeatable impact on the incident bar. The striker is launched by the release of compressed air and accelerates in a long gas barrel until it impacts the incident bar at constant speed. The striking velocity just before impact is measured optically. The speed of the striker is controlled by the pressure of the compressed gas in the tank and/or the depth of the striker inside the gun barrel. The bar components of the mechanical system consist of two long bars, the incident/input bar and the transmission/output bar. An optional extension bar and a

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7 Testing at High Strain Rates

Fig. 7.5 Experimental arrangement of the split-Hopkinson (Kolsky) bar impact test

momentum trap device may be added to the system. All bars are made of the same material, have the same diameter and are aligned along a single axis. The specimen is sandwiched between the incident and the transmission bars (Fig. 7.5). The bars are constructed from a high-strength structural metal, AISI-SAE 4340 steel, maraging steel, or a nickel alloy such as Inconel. The stress waves in the bars are measured by strain gages. The stress in the bars should not exceed the yield limit. Thus, the yield strength of the pressure bar material determines the maximum stress of the specimen. The length and diameter of the pressure bars should be chosen to meet a number of criteria for test validity as well as the maximum strain and strain rate desired in the specimen. The length of the bars must ensure one-dimensional wave propagation for a given pulse length. For most engineering materials this requires the length to diameter ratio of the bars to be approximately 10. Taking into consideration that the incident and reflected waves should be separated for data reduction the length to diameter ratio of the bars should exceed ~ 20. The maximum strain rate also influences the diameter of the bars because the higher strain rate tests require the smallest diameter pressure bars. The amount of the total strain in the specimen affects the bar length. The magnitude of this strain is related to the length of the incident wave. The pressure bars must be at least twice as long as the incident wave if the incident and reflected waves are to be recorded without interference. Depending on the specimen size, for strains greater than 30% the bars need to have a length to diameter ratio of 100 or more. The bars must be straight, free to move without binding and mounted to ensure optimal axial alignment. The bars should not being over constrained because this clamping violates the boundary conditions for one-dimensional wave propagation in an infinite cylindrical solid. The bars are mounted to a common rigid base to provide a rigid and straight mounting platform.

7.7 Split-Hopkinson (Kolsky) Bar Impact Test

185

Acquisition and recording system: Strains in the bars of the Kolsky arrangement are measured with strain gages. Two strain gages are usually attached on the surface of the bars across a bar diameter. A Wheatstone bridge is used to condition the signals from the strain gages. The voltage output from the bridges is generally of the order of milli-Volts, so an amplifier is necessary to record the low-amplitude voltage with an oscilloscope or a computer board. The amplifier and the oscilloscope should have a high frequency response to record a signal of duration usually shorter than a milli-second.

7.7.3 Impact Stress Waves in the Bars The stress wave in the Kolsky bar experiment is generated by the impact of the striker on the incident bar. A position-time (x − t) diagram of the stress wave propagation in the bars is shown in Fig. 7.6. When the striker impacts the incident bar two compressive stress waves are generated in the striker and incident bars. The stress wave propagates along the incident bar up to the interface between the incident bar and the specimen as a compressive wave. At the interface between the incident bar and the specimen part of the wave is reflected back into the incident bar as a tensile wave, while the rest transmits as a compressive wave into the specimen. This wave is reflected back and forth inside the specimen due to the mismatch of the wave impedance between the specimen and the bars. The successive reflections build up the compressive stress level inside the specimen. Following the passage of the compressive stress wave through the specimen it propagates in the transmission bar. Due to the small length of the specimen, the stress wave propagation in the specimen is usually ignored by assuming equilibrium in the specimen.

Fig. 7.6 A position-time (x − t) diagram of the stress wave propagation in the bars of the Kolsky experiment

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7 Testing at High Strain Rates

In this way, three pulses are generated, an incident compressive pulse, a reflected tensile pulse and a transmitted compressive pulse. These pulses are recorded by the data acquisition system. The impact of the striker generates a compression wave in the striker, which is reflected back at the free end of the striker as a tensile wave. This tension wave transmits in the incident bar as an unloading wave. Part of this wave is reflected back and the rest transmits into the transmission bar at the bar/specimen interface, while the specimen is unloaded.

7.7.4 Analysis of Experimental Data Determination of the stress–strain behavior of materials using the SHPB test is based on the principles of one-dimensional elastic wave propagation within the pressure loading bars. The displacements or stresses generated can be deduced by measuring the elastic wave at any point as it propagates along the bar. The one-dimensional wave equation is 1 ∂ 2u ∂ 2u = , ∂2x cb2 ∂ 2 t

(7.13)

where u is the displacement at position x, cb is the longitudinal wave speed, c = √ E b /ρ, E b is the modulus of elasticity, ρ is the density and t is the time. The solution of this equation can be written as u = f (x − ct) + g(x + ct),

(7.14)

where f and g are functions that describe the incident and reflected waves. Applying this equation for the incident bar of the SHPB test we have: u i = f (x − ct), u r = g(x + ct),

(7.15)

where the indices i and r stand for the incident and the reflected waves. The strain ε is given by ε=

∂u . ∂x

(7.16)

By differentiating Eq. (7.14) with respect to x we obtain for the strain ε in the incident bar ε = εi + εr = f ' (x − ct) + g ' (x + ct) = f ' + g ' ,

(7.17)

7.7 Split-Hopkinson (Kolsky) Bar Impact Test

187

where the strains εi and εr refer to the incident and reflected waves in the incident bar, and the prime symbol ´ refers to differentiation with respect to x. By differentiating Eq. (7.14) with respect to time t we obtain for the velocity at the incident bar ∂ f (x − ct) ∂(x − ct) ∂g(x + ct) ∂(x + ct) ∂u = + ∂t ∂x ( ∂t ) ∂x ∂t = −c f ' + cg ' = c − f ' + g ' = c(−εi + εr ).

u˙ =

(7.18)

Similarly, by differentiating Eq. (7.14) with respect to time t we obtain for the velocity at the transmission bar (there is only incident and no reflected wave in the transmission bar) u˙ = −c f ' = −cεt ,

(7.19)

where the strain εt refers to the transmission bar. The stress σ in the incident bar is given by ( ) σi = E b (εi + εr ) = E b f ' + g '

(7.20)

and in the transmission bar by σt = E b εt = E f '

(7.21)

where E b is Young’s modulus of the incident and transmission bars. Let us now consider an expanded view of the incident bar/specimen/transmission bar region (Fig. 7.7) and use the indices 1 and 2 for the incident bar/specimen interface and specimen/transmission bar interface, respectively. Equations (7.18) and (7.19) apply all over the incident and transmission bars, and, therefore, they apply at interfaces 1 and 2, respectively. The strain rate ε˙ in the specimen is: ε˙ =

u˙ 2 − u˙ 1 , ls

(7.22)

where ls is the instantaneous length of the specimen. Introducing the values of u˙ 1 and u˙ 2 from Eqs. (7.18) and (7.19) for the incident and transmission bars, respectively, we obtain for ε˙ ε˙ =

c (−εi + εr + εt ). ls

(7.23)

If the specimen deforms uniformly the strain at the incident bar-specimen interface should be equal to the strain at the specimen transmission bar interface, that is εi + εr = εt .

(7.24)

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7 Testing at High Strain Rates

Fig. 7.7 Expanded view of the incident bar/specimen/transmission bar region of the splitHopkinson (Kolsky) bar impact test

Using Eq. (7.24) we obtain for the strain rate in the specimen ε˙ =

2c (εi − εt ). ls

(7.25)

The strain ε in the specimen is {t ε= 0

2c ε˙ dt = ls

{t (εi − εt )dt

(7.26)

0

Equation (7.26) is used for the determination of the strain in the specimen by measuring the strains εi and εt in the incident and transmission bars. These strain are measured by strain gages, usually installed at the mid-lengths of the bars. The load F on the specimen is equal to the average of the two loads F 1 and F 2 at the incident bar/specimen interface and the specimen/transmission bar interface. From Eqs. (7.20), (7.21) and (7.24) we have F=

σi Ab + σt Ab Ab E b F1 + F2 = = (εi + εr + εt ) = Ab E b εt 2 2 2

(7.27)

where Ab is the cross-sectional area of the bars. The stress σ in the specimen is σ =

F Ab E b Ab E b = εt (εi + εr + εt ) = As 2 As As

(7.28)

where As is the cross-sectional area of the specimen. Equations (7.25), (2.26) and (7.28) are the basic equations of the SHPB test method. They are used for the determination of the stress–strain relationship of a material by measuring the strains εi and εt in the incident and transmission bars. The stress σ and strain ε are calculated from Eqs. (7.28) and (7.26), respectively, at a strain rate ε˙ given by Eq. (7.25).

7.7 Split-Hopkinson (Kolsky) Bar Impact Test

189

7.7.5 Modifications of SHPB Arrangement for Ceramics and Soft Materials The conventional arrangement of SHPB test used to obtain the stress–strain curves for metals at high strain rates should be modified for some engineering materials which present different mechanical behavior of metals. We will examine two types of such materials, ceramics and soft materials. Ceramics: Ceramic materials are hard, extremely brittle and do not develop plastic deformation. They are likely to be damaged at the bar ends, fail at small strains in the elastic domain and develop defects which cause premature failure. To address these issues special precautions should be taken. To prevent damage at the bar ends special inserts of harder ceramics are introduced at the ends of the bars. It is very important to ensure that equilibration occurs in the specimen before failure develops. Equilibration may be considered to occur when at least five reverberations of the elastic wave occur in the specimen. The specimen design plays a crucial role. Soft Materials: Soft materials exhibit low modulus, flow stress and mechanical impedance. This results in small transmitted stresses which are difficult to be measured with accuracy. Furthermore, equilibration of the stress in the specimen can take a very long time. According to Eq. (7.28) for the same stress level the magnitude of the transmitted strain signal εt can be increased by reducing the elastic modulus and/or decreasing the cross-sectional area Ab of the transmission bar. This suggests replacing the steel bars in the SHPB arrangement with aluminum bars. When aluminum bars are not sufficient polymeric bars are employed to further enhance the transmitted signal. However, the use of polymeric bars introduces further problems, because polymers are viscoelastic materials and cause attenuation and dispersion of the stress pulses. The viscoelastic behavior of the bars needs to be taken into consideration resulting in complicated stress–strain relationships. Concerning the use of transmission bars with small cross-sectional area the most recommended practice is the use of hollow transmission bars. This results in an amplification of the strain signal. At the specimen/transmission bar interface an aluminum bar is fitted into the transmission bar to support the specimen.

7.7.6 Split-Hopkinson Tension Bar The principles of the split-Hopkinson bar in tension are similar to those in compression. The primary differences are: (a) the methods of generating a tensile loading pulse and (b) the method of attaching the specimen to the incident and transmission bars. There are two basic approaches to generate a tensile wave in the input bar. In the first approach a tubular striker is fired at a flange attached at the end of the incident bar

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7 Testing at High Strain Rates

Fig. 7.8 Experimental arrangement for the split-Hopkinson (Kolsky) tension bar impact test. A tubular projectile is fired at a flange attached at the end of the incident bar

Fig. 7.9 Experimental arrangement for the split-Hopkinson (Kolsky) tension bar impact test. A large tensile strain is stored within a section of the incident bar and then it is suddenly released

(Fig. 7.8). This generates a compression wave that is reflected from the free surface of the flange as a tension stress wave in the incident bar. In the second approach a large tensile strain is stored within a section of the incident bar and then it is suddenly released to generate a tensile pulse in the incident bar (Fig. 7.9). This pulse loads the specimen in tension. The incident bar is restrained using a friction clamp, and a tensile force is applied using a hydraulic actuator system. The tension specimen must be firmly connected to the incident and transmission bar ends. The joints between the specimen and the bars may be threaded, clamped or bonded. The gripping of the specimens requires some care, because spurious wave reflections from the grips create difficulties in interpreting the experiments.

7.7.7 Split-Hopkinson Torsion Bar As with tension testing, there exist a variety of methods for specimen attachment and loading when subjecting materials to torsion on a SHPB. One way of applying loading, called the stored-torque method, involves clamping the midsection of the incident bar while a torque is applied to the free end. A large amount of torsional strain energy is stored in the incident bar within the section to the left side of the clamp. The clamp is suddenly released, and the stored torsional strain energy generates the torsional stress pulse that applies the torsional load on the specimen. An experimental

7.8 Taylor Impact Test

191

Fig. 7.10 Experimental arrangement for the split-Hopkinson (Kolsky) torsion bar impact test. A large amount of torsional strain energy is stored in the incident bar and is suddenly released

arrangement is shown in Fig. 7.10. It is similar to the experimental arrangement of the tension test of Fig. 7.9. Another loading technique known as explosive-loading uses explosive charges on the free end of the incident bar to create the incident wave. This method has the advantage of having a very small rise time as compared to the stored-torque method. The equations of the split-Hopkinson torsion bar test can be developed in the same way as for the compression bar presented in Sect. 7.7.4. The longitudinal displacement u is replaced by the angular displacement θ, the longitudinal strain ε by the shear strain γ and the longitudinal stress σ by the shear stress τ. The strain gages are installed at ± 45° to the axis of the bar to record the principal strains. The shear strain is then obtained by stress transformation.

7.8 Taylor Impact Test 7.8.1 Introduction The test was invented by Sir Geoffrey I. Taylor in 1948. It is a simple test for measuring the dynamic compressive yield stress of a material in the strain rate range of ε˙ = 102 ~ 105 s−1 . A long cylindrical specimen of the material to be tested is fired against a rigid solid anvil. From the deformed shape of the specimen the dynamic stress is determined with an accuracy of ± 10%.

7.8.2 Experimental Arrangement The experimental arrangement consists of a mechanical and a measurement system (Fig. 7.11). The mechanical system includes a gas gun and a fixed rigid solid anvil. The measurement part consists of laser beams and detectors placed in the path of the specimen for measuring the specimen velocity. High-speed photography is used to

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7 Testing at High Strain Rates

Fig. 7.11 A sketch of typical Taylor impact

acquire the real-time deformation of the specimen. The impact velocity is controlled by the gas pressure in the gas gun.

7.8.3 Specimen and Test Procedure A long cylindrical specimen is used. The aspect ratio (length to diameter) of the specimen is an important factor because after the test an undeformed part of the specimen is needed to perform the data analysis. When the aspect ratio of the specimen is small there may not be an undeformed part. A standard aspect ratio of the specimen has not been established. The specimen is fired perpendicularly against the rigid solid anvil, and the deformed and undeformed lengths of the specimen are measured. From these lengths and the impact velocity the dynamic yield stress is calculated. The impact velocity is controlled from the gas pressure in the gas gun.

7.8.4 Determination of the Yield Stress The yield stress σ y is calculated by (Fig. 7.12): σy =

) ( 1 ρU02 l0 − H ( ), l0 2 l0 − lf ln H

(7.29)

7.9 Expanding Ring Test

193

Fig. 7.12 Taylor cylinder impact test. a Original cylinder, b during deformation and c after deformation

where U0 l0 lf H ρ

impact velocity initial specimen length final specimen length undeformed specimen length density.

Equation (7.29) was derived by performing a one-dimensional analysis and assuming a rigid-perfectly plastic material behavior. Note that the yield stress is calculated from the geometry of the deformed cylindrical specimen. The Taylor test applies for ductile materials. Brittle materials disintegrate after impact and, therefore, this technique cannot be applied.

7.9 Expanding Ring Test 7.9.1 Introduction The expanding ring test is a simple test for determining the tensile stress–strain behavior of materials at large strains and at high strain rates over 104 s−1 . The test involves the sudden radial acceleration of a ring due to the detonation of an explosive charge or electromagnetic loading. The test is of special value because the tensile dynamic behavior of materials is evaluated at strain rates higher than those of the SHPB method.

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7 Testing at High Strain Rates

Fig. 7.13 Schematic of ring test

7.9.2 Experimental Arrangement The test is conducted by placing a thin ring of test material in a state of uniform radial expansion and then measuring its subsequent displacement–time history (Fig. 7.13). The ring is usually propelled by a high explosive driving system. The ring is mounted on a cylinder which has a hollow core for explosive detonation. When the explosive detonates a shock wave in the outward direction enters the ring. The displacement– time history of the ring is measured using high-speed photography techniques.

7.9.3 Mathematical Analysis The analysis applies in a thin ring. The wall thickness should be less than one-tenth the ring diameter. For the stress analysis consider a small segment of the circular ring of angle dθ, radius r, thickness h and unit length along the perpendicular direction (Fig. 7.14). Newton’s second law in the radial direction is Fr = mar ,

(7.30)

where F r is the force in the radial direction, m is the mass of the segment and ar (= d2 r/dt 2 ) is the acceleration in the radial direction. We have: Fr = 2F sin

dθ , m = ρ(r dθ h), 2

(7.31)

7.9 Expanding Ring Test

195

Fig. 7.14 Ring section

where F is the force applied at the ends of the ring cross section and ρ is the density of the material of the ring. Equation (7.30) becomes 2F sin

d2 r dθ = ρ r dθ h 2 . 2 dt

(7.32)

For F = σ h, where σ is the tensile stress, and for small angles sin (d θ/2) ≈ d θ/2 Eq. (7.32) becomes σ h dθ = ρ r dθ h

d2 r dt 2

(7.33)

or σ =ρr

d2 r . dt 2

(7.34)

Equation (7.34) is used for the determination of the stress σ when the deceleration history of the radius of the ring is known. The true strain εr of the ring in the radial direction is ε = ln

r r0

(7.35)

where r0 is the original ring radius. Differentiating Eq. (7.35) with respect to time t we obtain ε˙ =

1 dr r˙ dε = = . dt r dt r

(7.36)

Equation (7.36) is used for the determination of the strain rate when the rate of change of the radius of the ring with respect to time is known.

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7 Testing at High Strain Rates

Equations (7.34) and (7.35) are used for the determination of stress and strain, while Eq. (7.36) is used for the determination of the strain rate.

7.9.4 Discussion A great advantage of the expanding ring test is that the material is subjected to a state of dynamic uniaxial stress without the wave propagation complications that accompany other high strain rate tests. Furthermore, the maximum strain rate attained in the test is higher than any other tension test involving large plastic strains. The strain rate dε/dt computed from Eq. (7.36) is not usually constant during the test. Usually, it is greatest at the start of ring deceleration when strain is smallest and falls to zero at the end of the test. This constitutes a limitation of the test. The ring test is an expensive test, and only a few laboratories are capable of performing it. However, if the determination of the stress–strain curve is of no primary interest this test can be easily performed for the determination of the strain to failure under dynamic loading. Determination of stresses needs double differentiation of the displacement which generally is a divergent process. If stresses are not calculated the accurate determination of radial displacement versus time is not critical.

7.10 Plate Impact Test 7.10.1 Introduction The plate impact test is performed by launching down a flat flyer plate by a gun barrel or an explosive against a second stationary target plate (Fig. 7.15) with a velocity ranging from a few tens of meters per second up to several kilometers per second. Due to the high velocity impact extremely high rate deformation is induced in the impacting materials. A three-dimensional stress state is induced in the flyer plate. The test is convenient for material testing at extremely high three-dimensional stress states. Strain rates induced by the test are in the range of ε˙ = 106 ~ 108 and are the highest strain rates that can be achieved in the laboratory. From the test the Rankine–Hugoniot curve is obtained. It expresses the relationship between stress and particle velocity of the material.

7.10.2 Experimental Arrangement A schematic of the simplest form of experimental arrangement is shown in Fig. 7.15. It consists of a test system and a measurement system. The test system includes a

7.10 Plate Impact Test

197

Fig. 7.15 Experimental arrangement for the plate impact test

gas gun, a holder (sabot), a flyer plate and a target plate. The flyer plate impacts the target plate at normal incidence, resulting to the so-called normal plate impact test. At an oblique plate impact the test is described as the pressure-shear plate impact test. The normal plate impact test is the most used. The flyer and target plates need to be aligned in the test. The measurement system includes stress gages, particle velocity sensors and an acquisition system. Piezoelectric stress gages are usually used. One or more of the stress gages are placed within the target plate. Velocity pins are positioned in the path of the flyer in order to measure initial impact velocity. An interferometer is used to measure the rear surface particle velocity of the target plate. Just after impact normal uniaxial strain compressive waves will propagate in both the flyer and the target plates. Both plates are expected to deform inelastically.

7.10.3 Rankine–Hugoniot Curve From the test the Rankine–Hugoniot curve is constructed. This curve presents the dynamic response of materials under intensive impulse loading, such as shock wave loading. The curve expresses the relation between the inner pressure p (stress) and the particle velocity u of the material. Let us assume that the flyer has an initial impact velocity u0 . The initial state of the target plate is p = 0, u = 0 (point 1 of Fig. 7.16b). The initial state of the flyer plate is p = 0, u = u0 (point 2 of Fig. 7.16b). After the impact of two plates shock waves are produced and propagate in opposite directions. One of the waves travels into the flyer plate and the other into the target plate (Fig. 7.16a). At the contact interface between the two plates the pressure p and the particle velocity u are equal.

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7 Testing at High Strain Rates

Fig. 7.16 Impact process and Rankine–Hugoniot curves

Fig. 7.17 Rankine–Hugoniot curves for symmetric impact

This stage is obtained from the intersection of the two curves p-u of the flyer and target plate (Fig. 7.16b). When both the flyer and the target plates are of the same material the Rankine– Hugoniot p-u curve should be symmetric (Fig. 7.17). The point of the intersection of the two curves has final particle velocity equal to u0 /2. This point can be obtained by measuring the final (maximum) pressure in the target plate and the initial impact velocity of the flyer plate.

7.11 Optical Methods Optical methods of experimental mechanics, including photoelasticity, caustics, moiré, digital image correlation and other [4] in conjunction with high-speed photography have extensively been used for the study of dynamic problems [11, 12]. These methods are not intrusive and provide full field stresses, strains and displacements. They are convenient for the study of dynamic problems.

Further Readings

199

7.12 ASTM Standards ASTM standards for testing at high strain rates include: • E604-18, “Standard Test Method for Dynamic Tear Testing of Metallic Material” • D5023-15, “Standard Test Method for Plastics: Dynamic Mechanical Properties: In Flexure (Three-Point Bending)” • D5992-96(2011), “Standard Guide for Dynamic Testing of Vulcanized Rubber and Rubber-Like Materials Using Vibratory Methods” • D4092, “Standard Terminology for Plastics: Dynamic Mechanical Properties” • D5279-21, “Standard Test Method for Plastics: Dynamic Mechanical Properties: In Torsion” • D4945-00, “Standard Test Method for High-Strain Dynamic Testing of Piles” • E23-18, “Standard Test Methods for Notched Bar Impact Testing of Metallic Materials” • E2298-18, “Standard Test Method for Instrumented Impact Testing of Metallic Materials” • D950, “Standard Test Method for Impact Strength of Adhesive Bonds” • D5420-21, “Standard Test Method for Impact Resistance of Flat, Rigid Plastic Specimen by Means of a Striker Impacted by a Falling Weight (Gardner Impact)” • D256-10(2018), “Standard Test Methods for Determining the Izod Pendulum Impact Resistance of Plastics”

Further Readings 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Achenbach JD (1973) Wave propagation in elastic solids. Elsevier Freund LB (1998) Dynamic fracture mechanics. Cambridge University Press Gdoutos EE (2020) Fracture mechanics, 3rd edn. Springer Gdoutos EE (2022) Experimental mechanics. Springer Kuhn H, Medlin D (2000) ASM handbook, mechanical testing and evaluation, vol 8. ASM International Meyers MA (1994) Dynamic behavior of materials. Wiley Ramesh KT (2008) High rates and impact experiments. In: Sharpe WN (ed) Handbook on experimental solid mechanics. Springer, pp 929–959 Sih GC (ed) (1977) Mechanics of fracture, vol 4, Elastodynamic crack problems. Noordhoff International Publishing Rao CL, Narayanamurthy V, Simba KRY (2016) Applied impact mechanics. Wiley Ravi-Chandar K (2005) Dynamic fracture. Wiley Shukla A (ed) (2006) Dynamic fracture mechanics. World Scientific Publishing Co Pte Ltd Shukla A, Dally JW (2014) Experimental solid mechanics, 2nd edn. College House Enterprise, LLC, pp 513–55 Yu X, Chen L, Fang Q, Jiang X, Zhou, Y (2018) A review of the torsional split Hopkinson bar. Advances in Civil Engineering Article ID 2719741 https://doi.org/10.1155/2018/2719741

Chapter 8

Nondestructive Testing (NDT)

Abstract Nondestructive testing (NDT) methods include a wide group of analysis techniques to evaluate the properties of a material, component or system without causing damage. They are important for the assessment of fabrication quality and for the detection of early damage through alteration in microstructures leading to premature failure. They allow the investigator to carry out examinations without invading the integrity of the engineering component under investigation. In this chapter we will present the following NDT methods: dye penetrant, magnetic particles inspection, eddy currents, radiography, ultrasonics and acoustic emission.

8.1 Introduction Nondestructive testing (NDT) refers to the science and technology of noninvasive methods of testing, evaluation and characterization of materials, components or systems. It provides techniques to detect and characterize flaws in materials and structures and plays an important role in the prevention of failure. The American Heritage Dictionary defines “nondestructive” as “Of, relating to, or being a process that does not result in damage to the material under investigation or testing”. The terms nondestructive examination (NDE), nondestructive inspection (NDI) and nondestructive evaluation (NDE) are also commonly used to describe this technology. NDT is important for the in-service inspection of load-bearing structures whose failure could have catastrophic consequences. In most of NDT methods some form of energy, electromagnetic, acoustic, is sent through the material and response of the material is analyzed by sensors. Sensor developments led to increased sensitivity and reliability of NDT methods. The concept of fracture tolerance in fracture mechanics put new challenges to NDT methods. Structures are safe as long as the existing cracks do not surpass a critical size. In this respect, it became possible to accept structures containing defects under the condition that the sizes of those defects are smaller than a critical size. Components with known defects below the critical size could continue in service as long as the defects cannot grow to a critical size. Fracture mechanics puts a new challenge to

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_8

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NDT methods. Detection of defects is not enough. The location, sizing and orientation of the defects should be determined. Our ability to use fracture mechanics in design is largely due to the reliability of the NDT methods. At the production or service inspection stage, parts containing flaws larger than those determined according to fracture mechanics design standards must be rejected or replaced. There are many NDT methods. In this chapter, we will briefly present the following major methods: dye penetrant, magnetic particles inspection, eddy currents, radiography, ultrasonics and acoustic emission.

8.2 Dye Penetrant Testing (PT) 8.2.1 Principle The basic principle the dye penetrant testing is based on the capillarity effect, according to which in tubes having very small diameters liquids rise or fall relative to the level of the surrounding liquid depending on the relative strengths of the adhesive (between molecules of different types) and cohesive (between molecules of the same type) forces. The capillary forces are very strong, and, if a penetrant test is performed on a specimen in an overhead position, the penetrant would be drawn into the opening, against the force of gravity. The capillary force is greater than the gravity force, and therefore, discontinuities can be detected even though they may be in an overhead specimen. The test involves the application of a colored or fluorescent dye onto a cleaned surface of the component. After allowing sufficient time for penetration, the excess penetrant is washed off and the surface is dusted with a post-penetrant material (developer) such as chalk. The developer acts as a blotter and helps to draw penetrant out of the flaw. The defects show up under ultraviolet or white light, depending on the type of dye used (fluorescent or nonfluorescent (visible)).

8.2.2 Application Application of the method involves the following steps: Precleaning. The surface of the material to be tested is properly cleaned to remove any contaminants such as dirt, oil, water and oxides that restrict the entry of the penetrant into surface openings. Application and removal of penetrant. Penetrants are classified as visible or fluorescent. Visible penetrants use a color contrast (usually red) dye, while fluorescent penetrants use a dye which fluoresces under dark light. Fluorescent penetrants are more sensitive than visible dye penetrants. The penetrant is applied to the surface of the body under study for a “dwell time” (5–30 min) to soak into the flaws. The

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dwell time depends on the type of penetrant, the material tested and the size of the flaws (smaller flaws require longer time). The excess penetrant is removed from the surface. Application of developer. After the removal of the excess surface penetrant a developer is applied. The function of the developer is to draw penetrant out of the defect onto the surface. A development time from 7 to 30 min usually applies. A visible indication known as bleed-out is formed on the surface and indicates the location, orientation and possible types of defects. Inspection. Visible light is used for inspection for visible dye penetrant and ultraviolet radiation for fluorescent penetrants. The inspection time depends on the penetrant and developer used.

8.2.3 Discussion Dye penetrant is a simple and highly sensitive nondestructive testing method used widely for the detection of surface discontinuities in nonporous solid materials. It is the most commonly used surface NDT method because it can be applied to any magnetic or nonmagnetic material. It is well-suited for detection of all types of defects. The reliability of the method depends on the surface preparation of the component. Advantage of the method include speed of the test, low cost and sensitivity. Proper cleaning of the surface is required. The main disadvantage of the method is that it applies to only discontinuities open to the surface of the test piece and it is difficult to apply on rough and porous surfaces. The method can detect small cracks.

8.3 Magnetic Particle Testing (MT) 8.3.1 Introduction The method of magnetic particle testing (MT) is used for detecting discontinuities that are at or near the surface of ferromagnetic (strongly attracted to a magnet and can easily be magnetized, like iron, nickel and cobalt) materials. Magnetic flux in test objects is produced either using DC storage batteries or alternating current. If an electric current flows through a conductor, a magnetic flux will be produced that flows around the conductor.

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8.3.2 Detection of Discontinuities Detection of discontinuities in MT is based on the distortion in the internal field of a magnetized material. When a material is homogeneous the flux lines will distribute themselves evenly through the material. However, when there are discontinuities in the material they produce a distortion to an induced magnetic field. In MT the body under examination is magnetized directly when the electric current is passed through it, or indirectly, when no electric current is passed, but a magnetic field is applied from an external source. The lines of magnetic flux are perpendicular to the direction of the electric current, which may be alternating or direct. The presence of flaws causes the magnetic flux to leak since air cannot support a magnetic field, as metals do. Measuring this distortion provides information on the existing defects. For detecting the distortion of the magnetic field the surface under inspection is coated with detection media. A variety of different sizes and shapes are used. Elongated particles will rotate to align with the flux lines, whereas rounded particles have greater mobility to move to the areas of flux leakage. The smaller particles accumulate at small discontinuities, whereas larger particles have the ability to bridge across larger discontinuities. Particles may have dry or liquid form. The particles can be coated to provide a greater contract with the test surface. The coatings are color contrast (available in several colors) of fluorescent liquids that contain magnetic particles in suspension. After the test the part is demagnetized.

8.3.3 Discussion MT has the added advantage over the dye penetrant testing in detecting subsurface defects. The test results are instantaneous, and no developing or processing times are involved. The method can be applied in situ by using permanent magnets and the indications formed by the particles represent closely the shape and type of the discontinuity. MT equipment is much less expensive than other NDT equipment. Any size and shape of component can be inspected, and part surface preparation is less critical than the penetrant testing. Disadvantages on the method include that it applies only to ferromagnetic materials, and the detection of discontinuities is limited to those at or near the surface. Some MP techniques may cause damage to the part. Demagnetization may be required before, between and after inspections. The inspection should be carried out with magnetic field perpendicular to the plane of the flaw. More than one evaluation is needed to cover all orientations of the flaws. For thick parts large currents are required. MT is easy to apply, speedy and economical.

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8.4 Eddy Current Testing (ECT) 8.4.1 Theory and Principle When a specimen is brought near an eddy current coil producing an alternating flux field electrons in the specimen circulate in a swirling pattern. These are the eddy currents induced in the specimen. Because their flow patterns resemble swirling eddies or whirlpools in a river, they are called eddy currents. They have the following properties: a. Eddy currents flow in closed concentric circular loops within conductors parallel to the turns of a bobbin-type coil and perpendicular to the axis of the coil’s flux field (Fig. 8.1). b. A discontinuity in the specimen is least detectable when its longest dimension is parallel to the eddy current flow paths and most detectable when it is perpendicular to the paths. c. The flow of eddy currents in the specimen alternates clockwise and counterclockwise depending on the alternating flux field. The frequency of alteration of eddy currents depends on the frequency of alteration of the flux field. d. Eddy current exhibits the skin effect; that is, current density is maximum at the material surface and decreases exponentially with depth. Standard depth of penetration, δ, defined as the depth at which eddy current density has decreased to 1/e (e = 2.71828) is given by

Fig. 8.1 Bobbin-type coil’s flux and eddy currents

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/ δ(mm) = 50

ρ , f μr

(8.1)

where ρ is the resistivity, f the frequency and μr the relative permeability. e. The extent of a coil’s flux field varies with coil diameter. Effective eddy current penetration is limited by the diameter of the coil. If the coil is too small the current density at a particular depth will be less than that of Eq. (8.1).

8.4.2 Application A coil carrying alternating current is placed near the specimen. It induces eddy currents in the surface of the specimen. The eddy currents create a magnetic field that affects the coil; its impedance changes when a defect is present. By measuring this change we can find information about the defect. Eddy current hardware includes AC generator, coil circuit and processing/display circuitry. The AC generator provides the voltage that drives the coil. The coil circuits may range from a single specific coil, a limited range of specific coils or with any coil configuration available. In a ferritic conductor, the penetration depth is smaller than 1 mm at most frequencies, while in nonmagnetic conductors it may be several millimeters. The sensitivity of the method is high for defects near the surface but decreases with increasing depth. Problems in the method arise from the difficulty of relating the defect size to the change in impedance, and the influence of a number of other factors on the impedance. These include the relative position of the coil and the conductor; the presence of structural variations; material inhomogeneities. Measurement of defect size is made by comparing its effect to that observed from a standard defect.

8.4.3 Discussion The eddy current method is sensitive to many variables including material conductivity and thickness, size of discontinuities, spacing between test coil and specimen, permeability variations. The response of the method to these variables is additive, and it may be difficult to resolve it into its separate components. This constitutes a major advantage and simultaneously a limitation of the method. Advantages of ECT method include: ability to detect surface defects of 0.5 mm in length, defects in multilayer structures, defects through nonconducting surface coatings of more than 5 mm thickness, little precleaning requirement, can be automated. Test results are usually instantaneous. The method is safe; there is no danger from radiation. No material preparation and cleanup are required. Disadvantages of ECI method are: susceptibility to magnetic permeability changes, the method applies to conductive materials and has limited penetration

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(to fraction of an inch in most materials), does not detect defects parallel to the surface, and is not suitable for large areas and complex geometries.

8.5 X-ray Diffraction Testing 8.5.1 Introduction X-ray diffraction (XRT) is a nondestructive testing method for measuring strains and detecting subsurface defects. When X-rays are pointed at an object at an incident angle, they slightly penetrate the material and interact with the atoms. The scattered rays provide a pattern of the atoms inside the material. From the analysis of the intensity of the scattered pattern the atomic and molecular structure of materials can be determined. Defects absorb less X-rays than the surrounding material, and therefore, they can be detected. In the following we will briefly present the nature of X-rays, their diffraction with atoms/molecules inside materials and their use in measuring strains.

8.5.2 X-rays X-rays are a high-energy electromagnetic radiation of very short wavelength capable to pass through many materials opaque to light. Most X-rays have wavelength ranging from 10 × 10–12 to 10 × 10–9 m, and corresponding frequencies ranging from 3 × 1015 to 3 × 1018 Hz and energies ranging from 124 keV to 124 eV. In the electromagnetic range X-rays have wavelengths shorter than those of the UV rays and longer than those of the gamma rays. X-rays were discovered by W.C. Roentgen in 1895. The name of X-rays comes from their unknown character at the time of their discovery after the algebraic symbol x used to denote an unknown quantity.

8.5.3 X-ray Diffraction It was proposed by Max von Laue in 1912 that a crystalline solid in which the atoms are arranged in regular patterns with spacing between neighboring atoms of the order of 10–10 m might serve as a three-dimensional diffraction grating for X-rays. A beam of X-rays is scattered, that is, absorbed and re-emitted by the atoms of the crystalline body. The scattered waves might interfere in the same way as a diffraction grating. Xrays offer greater resolution than visible light because they have shorter wavelengths. They are very effective in studying the microscopic world of atoms and molecules.

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X-ray diffraction can be used to study the structure of crystals and is the basis of measuring strains by X-rays.

8.5.4 Measurement of Strain Consider a crystal in which the atoms are arranged in a cubical fashion. Let the distance between neighboring atoms is d (Fig. 8.2). A beam of X-rays is incident on the crystal at an angle ϕ with the surface. The two rays shown in Fig. 8.2 are reflected from two subsequent planes of atoms. The optical path length difference between the two rays is: 2d sinϕ. Constructive interference occurs when 2d sinϕ is a multiple of wavelengths λ, that is mλ = 2d sin ϕ,

(8.2)

where m is an integer (m = 0, 1, 2,…). Equation (8.2) is called the Bragg equation. It can be used for the determination of the distance d between adjacent atoms when the angle of incidence ϕ and the wavelength λ are known. Equation (8.2) is the basis for strain measurement by X-ray diffraction. From Eq. (8.2) the strain ε is calculated as ε=

Fig. 8.2 X-ray diffraction by a crystal for strain measuring

d − d0 sin ϕ0 −1, = d0 sin ϕ

(8.3)

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where d and ϕ are the distance between two adjacent atoms and the angle of incidence with respect to the surface of the crystal of the stressed body, and d 0 and ϕ 0 are the corresponding quantities for the unstressed body. Equation (8.3) refers to strain at a given direction. By using six different directions defined by the angle of incidence ϕ we can determine the normal strain in six different directions. From the equations of strain transformation we can then determine all six components of the strain tensor. From the strains the stresses can be determined by using the appropriate constitutive equations.

8.5.5 Instrumentation X-rays are generated when electrons accelerated by a high voltage in a vacuum tube strike a metal surface inside the tube. The vacuum tube consists of a cathode and an anode of a high melting heavy metal. Electrons are emitted from the cathode and accelerated to the anode. Voltages in the order of 30–150 kV are typically used. X-rays have a wave-particle duality. The energy E and the frequency f of the wave nature of X-rays are related by the equation c E = hf = h , λ

(8.4)

where h is Planck’s constant (h = 6.626 × 10–34 J s), c is the speed of light (c = 3 × 108 m/s), f is the frequency and λ is the wavelength. Equation (8.4) can be put in the form E=

12.4 , λ

(8.5)

where λ is in Angstroms and E is in keV. In applications X-rays are produced by diffractometers which are laboratory-based or portable. A diffractometer is an X-ray stress analysis equipment consisting of an X-ray source and has the capability of measuring diffraction angles. Synchrotron facilities provide high-energy X-ray with fluxes orders of magnitude higher than Xray tubes and offer excellent capabilities for strain measurements. They can penetrate depths over 1 mm in most materials. X-ray strain measurements based on laboratory equipment are restricted to small penetration depths typically a few tens of microns into materials yielding strain near the surface of the body. For deeper depths destructive layer removal or more high-energy X-rays are used.

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8.5.6 Discussion XRT has extensively been used in the measurement of residual stresses. Actually, the strain in the crystal lattice is measured and the residual stress related to the strain is calculated using a constitutive law. Macroscopic and microscopic residual stresses can be determined. Macroscopic strains extend over distances that are large relative to the grain size of the material, while microscopic stresses are associated with strains within the crystal lattice that traverse distances on the order of or less than the dimensions of the crystals. XRT is widely used in welds and castings. It provides an extremely accurate method for the measurement of strain and characterization of discontinuities. XRT has been applied to many other product forms spanning a wide range of industries including power generation, aerospace, petrochemical, medicine, law enforcement and security, food, objects of art or historic value.

8.6 Ultrasonic Testing (UT) 8.6.1 Introduction Ultrasonic testing (UT) uses high frequency sound waves well above the range of human hearing that travel in the object to be tested to detect internal flaws or to characterize materials. In most applications the frequencies of the waves range from 0.1 to 15 MHz. UT can be used to detect flaws and discontinuities on steel and other metals and alloys. It can also be used on concrete, composites and wood, however, with less resolution. Ultrasound is used in medicine to create images of soft tissue structures, such as the gall bladder, liver, heart, kidney, female reproductive organs and even of babies still in the womb. It cannot be used to image bones because they are too dense to penetrate.

8.6.2 Operation A UT inspection system consists of several functional units, such as the pulser/ receiver, transducer and display devices. A pulser/receiver is an electronic device that can produce high voltage electrical pulses. The ultrasound transducer is driven by the pulser to generate ultrasonic waves which propagate through the material. It is separated from the test object by a couplant (such as oil) or by water. The sound waves follow the laws of optic waves in reflection, refraction, etc. During their travel in the material they are reflected at interfaces. They are picked-up by the receiver and an

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analysis of the signal gives information about discontinuities, flaws, microstructures, etc. There are two methods of receiving the ultrasound waves: reflection (or pulseecho) and attenuation (or through transmission). In the reflection method the sound is reflected back to the device from an interface, such as the back wall of the object or from an imperfection. The transducer sends and receives the waves. The reflected wave signal is transformed into an electrical signal by the transducer and is displayed on a screen. Signal travel time can be directly related to the distance that the signal traveled. From the signal, information about the reflector location, size, orientation and other features can be gained. In the attenuation method the transmitter sends the ultrasound and a separate receiver detects the amount that has reached it. When discontinuities exist between the transmitter and the receiver the amount of transmitted sound is reduced. From this amount the presence of discontinuities is revealed. Automated systems typically consist of an immersion tank, scanning system and recording system for a printout of the scan. The sound is transmitted from the transducer to the material and reflected from the surface back through the liquid. This creates a strong signal that can be used to detect very small flaws. The resultant Cscan provides a plan or top view of the component. The scan information is collected by a computer for evaluation and archiving. Quantitative theories have been developed to describe the interaction of the interrogating fields with flaws. Measurement procedures initially developed for metals have been extended to engineered materials such as composites, where anisotropy and inhomogeneity have become important issues. The rapid advances in digitization and computing capabilities have totally changed the faces of many instruments and the type of algorithms that are used in processing the resulting data. High-resolution imaging systems and multiple measurement modalities for characterizing a flaw have emerged.

8.6.3 Discussion Advantages of UT include: inspection can be accomplished from one surface, sensitivity of the method to both surface and subsurface discontinuities, detection of small discontinuities, examination of thick parts, accuracy in determining reflector position and estimating size and shape of discontinuity, minimal part preparation, instantaneous results, thickness measurement in addition to discontinuity measurements. Limitations of the method include: accessibility of surface to transmit ultrasound, requirement of a coupling medium for transfer of energy sound into the material, defects parallel to the sound beam may go undetected, very small, rough or materials of irregular shape are difficult to inspect, discontinuities that are similar to or smaller than the material’s grain structure may not be detected, coarse grain material like cast iron are difficult to inspect due to low sound transmission and high signal noise,

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subjectivity of the interpretation of the echoes by the operator, high level of skill and training required.

8.7 Acoustic Emission Testing (AET) 8.7.1 Introduction When a solid body is subjected to an external stimulus (loads, change of temperature, etc.) irreversible changes in its internal structure may occur resulting in sudden release of elastic energy in the form of stress waves. These waves propagate in the body and result in small transient surface displacements which are related to the internal damage of the body. The displacements are recorded by sensors and provide information about the damage in the body. Acoustic emission testing (AET) is a nondestructive method for the detection, recording and analysis of the acoustic emission (AE) signals using specialized equipment with the objective to obtain valuable information regarding the damage in the body. AET is of great importance in structural health monitoring (to detect, locate and characterize damage), quality control, process monitoring and other fields. AET has been used as early as 6,500 BC by potters to listen for audible sounds during the cooling of their ceramics, signifying structural failure. Acoustic emission in this form is to the ears what visual inspection is to the eyes. Sources of AE vary from natural events like earthquakes to the initiation and growth of cracks, slip, dislocation movements, cavitation processes and phase transformations in metals. In composite materials matrix cracking, fiber breakage and debonding contribute to AEs. They have also been recorded in polymers, wood and concrete, among other materials. AE differs from other nondestructive testing methods in two regards: First, energy is not supplied to the test object. AE listens the energy released by the body while in operation. In this respect AE is passive method, while the other NDT methods are for the most part active. The second difference is that AET deals with dynamic processes in the body. Only active damages (for example crack growth) are detected. Defects go undetected if they cannot cause an acoustic event.

8.7.2 Sources of Acoustic Emission In acoustic emission (AE) elastic waves are generated by the rapid change in the state of stress in some region of the material caused by the application of an external stimulus. The stress change must be rapid enough to transmit some energy to the surrounding material. On the macroscopic scale AE includes earthquakes and thunder, while on the microscopic scale the fracture of crystallites and martensitic

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phase transformations. Acoustic energy is released when materials break and new surfaces are formed, as in the case of crack initiation. Detection of emission from growing cracks has been the most common goal in many applications of acoustic emission technology. High amplitude signals are generated from growing cracks. For AE the so-called Kaiser effect applies, i.e., that no signals were generated by a sample upon the second loading until the previous maximum load was exceeded.

8.7.3 Propagation of AE Signals Following the generation of the elastic wave, it travels throughout the material and can be detected at considerable distances from its point of origin. The energy of the wave as it travels from its source to the point of detection is attenuated. In most structures attenuation is due to geometric spreading, scattering at boundaries and absorption. A simulated AE source is used for attenuation measurements. The most widely used simulated AE source is the breaking of a pencil lead pressed against a structural member (Hsu-Nielsen pencil lead test). The breaks are used to generate sound waves enabling the characterization of acoustic wave speed in complex structures. The test consists of breaking a 0.5 mm diameter pencil lead approximately 3 mm from its tip by pressing it against the surface of the piece. This generates an intense acoustic signal, quite similar to a natural AE source that the sensors detect as a strong burst. The purpose of this test is twofold. First, it ensures that the transducers are in good acoustic contact with the part being monitored. Generally, the lead breaks should register amplitudes of at least 80 dB for a reference voltage of 1 mV and a total system gain of 80 dB. Second, it checks the accuracy of the source location setup. This last purpose involves indirectly determining the actual value of the acoustic wave speed for the object being monitored. Acoustic waves follow the general rules of wave propagation. The wave velocity V is determined from the characteristics of the material. It is given by / V =

Ci , ρ

(8.6)

where ρ is the density of the material and Ci is the elastic constant for that type of wave. Acoustic waves can propagate in a solid in the form of longitudinal, shear and Raleigh waves. The velocity of Rayleigh waves is slightly lower than the shear velocity. For plates of thickness on the order of a few acoustic wave lengths or less Lamb waves can occur.

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8.7.4 Testing Sensors generate an electrical signal when they are stimulated by an acoustic wave. Most of the sensors in NDT are piezoelectric. Piezoelectric (from the Greek word piezein means to squeeze) materials generate an electric voltage when they are squeezed (deformed). AE elastic waves propagate in all directions and hit the piezoelectric sensor that is attached to the surface of the body (Fig. 8.3). They contain information about the internal behavior of the material and the geometry of the body. The wave motion at the surface of the body on picometer to nanometer scale is converted into an electric signal by the piezoelectric sensor. Signals can be detected at frequencies under 1 kHz. Typically, most of the released energy is in the range of 1 kHz to 1 MHz. The sensitivity of the sensor is key issue for the detection of AE signals and source location. For testing small components only one sensor may be sufficient. Typically, multiple sensors are used. Different sensors pick up different signal characteristics for the same acoustic emission event. A pattern of interlocking triangles or rectangles is used to set up sensors. A fluid couplant is used to bond the sensor to the surface and help the sensor to obtain stronger signal. The sensors are connected to an amplifier, and additional electronic equipment is used to filter and isolate the sound. The AET system records any AE along with the exact time it occurred. Data related to emission count, signal length, peak amplitude, emission length and other parameters are recorded. After the test is completed the results are analyzed. By measuring the arrival time of an AE signal to each sensor the location of the defect can be determined by knowing the velocity of the wave in the material and the difference in arrival times among the sensors. Following the location of the defects additional testing can be performed by other NDT methods for measuring the size and inclination of the defect.

Fig. 8.3 Schematic of the acoustic emission process

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8.7.5 Source Location The primary information carried by the acoustic wave is the time of arrival and the elastic energy detected at each sensor on the structure. The excitation of a sensor indicates that something happened in the specimen at a specific time, while the amplitude indicates the level of the disturbance. The traditional methods of AE source location are usually based on the time difference of arrival, wave velocity and distances between sensors. The basic idea in source location is to cover the surface with a network of sensors. If the arrival times of an AE signal at several sensors are determined, then knowing the acoustic velocity it is possible to determine the location of the source of that signal. The material is assumed to be isotropic so that there is one acoustic speed. Two of the most widely used location techniques are the linear location and the planar location. The linear location technique is suitable for rod-like media whose lengths are much larger than their widths. The principle of the technique is illustrated in Fig. 8.4. Consider an AE source and two sensors S1 and S2 at distances L 1 and L 2 from the source. Let the times of arrival of the signal from the source to sensors 1 and 2 be T 1 and T 2 . The difference of arrival times at the two sensors ∆T 2-1 is ∆T2−1 = T2 − T1 =

L2 L1 2x − = , V V V

(8.7)

where V is the speed of the sound wave and x is the distance of the AE source from the midpoint between the two sensors. Equation (8.7) indicates that when the source is at the midpoint the wave arrives at sensors S1 and S2 simultaneously and the time of arrival difference is zero (∆T 2-1

Fig. 8.4 Schematic of the linear location technique

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Fig. 8.5 Principle of the planar location technique

= 0). If the source is beyond one of the sensors ∆T 2-1 has a constant value L/V, and the source location cannot be determined. Thus, a source between two sensors can be identified, but not a source beyond a sensor. The principle of the planar location technique is shown in Fig. 8.5. Three sensors are located around the source. The AE signal is produced at the initial moment of T 0 and spreads to any sensor at moment T i . As the wave propagates from the source, it reaches first the nearest sensors S1, then the further sensors S2 and S3 . The first hit on sensor S1 occurs at time R1 /V after the source event, the second at time r 2 /V and the third at time r 3 /V. From the difference between the hit times and using special algorithms the location and the time of the source event are calculated. Techniques and algorithms based on more sensors have been developed. The principle of the 2-D planar location technique can be extended to 3-D.

8.7.6 Discussion AET can be used for early detection and real-time monitoring of defects. Major advantages of the method include: early detection of small defects, no need to shutdown the unit that can be inspected while in operation, detection of only active defects which may impose an immediate threat to the structure, real-time evaluation, remote scanning and reduced cost. AET is limited to only qualitative, not quantitative, results. It can detect the existence of a defect, but cannot determine its size and orientation. This requires other test methods. It can identify only active defects which can be an advantage, but in

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some cases identification of stagnant defects is needed. Loud environments present challenges in the accuracy of the results and filtering of the noise is required. Finally, AET requires an experienced and skilled operator.

8.8 ASTM Standards ASTM has published the following standards for nondestructive testing of materials and structures: Acoustic Emission Method E1106-12(2021), “Standard Test Method for Primary Calibration of Acoustic Emission Sensors” E2076/E2076M-15, “Standard Practice for Examination of Fiberglass Reinforced Plastic Fan Blades Using Acoustic Emission” E2191/E2191M-16, “Standard Practice for Examination of Gas-Filled FilamentWound Composite Pressure Vessels Using Acoustic Emission” E2478-11(2016), “Standard Practice for Determining Damage-Based Design Stress for Glass Fiber Reinforced Plastic (GFRP) Materials Using Acoustic Emission” E650/E650M-17, “Standard Guide for Mounting Piezoelectric Acoustic Emission Sensors” E1139/E1139M-17, “Standard Practice for Continuous Monitoring of Acoustic Emission from Metal Pressure Boundaries” E751/E751M-17, “Standard Practice for Acoustic Emission Monitoring During Resistance Spot-Welding” E1888/E1888M-17, “Standard Practice for Acoustic Emission Examination of Pressurized Containers Made of Fiberglass Reinforced Plastic with Balsa Wood Cores” E1495/E1495M-17, “Standard Guide for Acousto-Ultrasonic Assessment of Composites, Laminates, and Bonded Joints” E1211/E1211M-17, “Standard Practice for Leak Detection and Location Using Surface-Mounted Acoustic Emission Sensors” E1930/E1930M-17, “Standard Practice for Examination of Liquid-Filled Atmospheric and Low-Pressure Metal Storage Tanks Using Acoustic Emission” E1932-12(2017), “Standard Guide for Acoustic Emission Examination of Small Parts” E2863-17, “Standard Practice for Acoustic Emission Examination of Welded Steel Sphere Pressure Vessels Using Thermal Pressurization” E3100-17, “Standard Guide for Acoustic Emission Examination of Concrete Structures” E1067/E1067M-18, “Standard Practice for Acoustic Emission Examination of Fiberglass Reinforced Plastic Resin (FRP) Tanks/Vessels”

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E1736-15, “Standard Practice for Acousto-Ultrasonic Assessment of FilamentWound Pressure Vessels” E2907/E2907M-13(2019), “Standard Practice for Examination of Paper Machine Rolls Using Acoustic Emission from Crack Face Rubbing” E2983-14(2019), “Standard Guide for Application of Acoustic Emission for Structural Health Monitoring” E569/E569M-20, “Standard Practice for Acoustic Emission Monitoring of Structures During Controlled Stimulation” E750-15(2020), “Standard Practice for Characterizing Acoustic Emission Instrumentation” E1118/E1118M-16(2020), “Standard Practice for Acoustic Emission Examination of Reinforced Thermosetting Resin Pipe (RTRP)” E1419/E1419M-15a(2020), “Standard Practice for Examination of Seamless, Gas-Filled, Pressure Vessels Using Acoustic Emission” E2075/E2075M-15(2020), “Standard Practice for Verifying the Consistency of AE-Sensor Response Using an Acrylic Rod” E2661/E2661M-20e1, “Standard Practice for Acoustic Emission Examination of Plate-like and Flat Panel Composite Structures Used in Aerospace Applications” E1781/E1781M-20, “Standard Practice for Secondary Calibration of Acoustic Emission Sensors” E976-15(2021), “Standard Guide for Determining the Reproducibility of Acoustic Emission Sensor Response” E2374-16(2021), “Standard Guide for Acoustic Emission System Performance Verification” E749/E749M-17(2021), “Standard Practice for Acoustic Emission Monitoring During Continuous Welding” E2984/E2984M-21, “Standard Practice for Acoustic Emission Examination of High Pressure, Low Carbon, Forged Piping using Controlled Hydrostatic Pressurization” E2598/E2598M-21, “Standard Practice for Acoustic Emission Examination of Cast Iron Yankee and Steam Heated Paper Dryers”. Digital Imaging and Communication in Nondestructive Evaluation (DICONDE) E2699-20, “Standard Practice for Digital Imaging and Communication in Nondestructive Evaluation (DICONDE) for Digital Radiographic (DR) Test Methods” E3267-21, “Standard Guide for Building Information Models and Archiving for Digital Imaging and Communication in Nondestructive Evaluation (DICONDE)” E2767-21, “Standard Practice for Digital Imaging and Communication in Nondestructive Evaluation (DICONDE) for X-ray Computed Tomography (CT) Test Methods” E3147-18, “Standard Practice for Evaluating DICONDE Interoperability of Nondestructive Testing and Inspection Systems” E2663-14(2018), “Standard Practice for Digital Imaging and Communication in Nondestructive Evaluation (DICONDE) for Ultrasonic Test Methods”

8.8 ASTM Standards

219

E1475-13(2018), “Standard Guide for Data Fields for Computerized Transfer of Digital Radiological Examination Data” E3169-18, “Standard Guide for Digital Imaging and Communication in Nondestructive Evaluation (DICONDE)” E2738-18, “Standard Practice for Digital Imaging and Communication in Nondestructive Evaluation (DICONDE) for Computed Radiography (CR) Test Methods” E2934-22, “Standard Practice for Digital Imaging and Communication in Nondestructive Evaluation (DICONDE) for Eddy Current (EC) Test Methods” E2339-21, “Standard Practice for Digital Imaging and Communication in Nondestructive Evaluation (DICONDE)”. Editorial Review E1316-22a, “Standard Terminology for Nondestructive Examinations”. Electromagnetic Method E2905/E2905M-20, “Standard Practice for Examination of Mill and Kiln Girth Gear Teeth—Electromagnetic Methods” E426-16(2021), “Standard Practice for Electromagnetic (Eddy Current) Examination of Seamless and Welded Tubular Products, Titanium, Austenitic Stainless Steel and Similar Alloys” E1312-18, “Standard Practice for Electromagnetic (Eddy Current) Examination of Ferromagnetic Cylindrical Bar Product Above the Curie Temperature” E243-18, “Standard Practice for Electromagnetic (Eddy Current) Examination of Copper and Copper-Alloy Tubes” E1004-17, “Standard Test Method for Determining Electrical Conductivity Using the Electromagnetic (Eddy Current) Method” E2928/E2928M-17, “Standard Practice for Examination of Drill string Threads Using the Alternating Current Field Measurement Technique” E566-19, “Standard Practice for Electromagnetic (Eddy Current/Magnetic Induction) Sorting of Ferrous Metals” E309-16, “Standard Practice for Eddy Current Examination of Steel Tubular Products Using Magnetic Saturation” E376-19, “Standard Practice for Measuring Coating Thickness by Magnetic-Field or Eddy Current (Electromagnetic) Testing Methods” E1629-12(2020), “Standard Practice for Determining the Impedance of Absolute Eddy-Current Probes” E690-15(2020), “Standard Practice for In Situ Electromagnetic (Eddy Current) Examination of Nonmagnetic Heat Exchanger Tubes” E570-20, “Standard Practice for Flux Leakage Examination of Ferromagnetic Steel Tubular Products” E571-19, “Standard Practice for Electromagnetic (Eddy-Current) Examination of Nickel and Nickel Alloy Tubular Products” E1606-20, “Standard Practice for Electromagnetic (Eddy Current) Examination of Copper and Aluminum Redraw Rod for Electrical Purposes”

220

8 Nondestructive Testing (NDT)

E703-20, “Standard Practice for Electromagnetic (Eddy Current) Sorting of Nonferrous Metals” E1571-21, “Standard Practice for Electromagnetic Examination of Ferromagnetic Steel Wire Rope” E2261/E2261M-17(2021), “Standard Practice for Examination of Welds Using the Alternating Current Field Measurement Technique” E3052-21, “Standard Practice for Examination of Carbon Steel Welds Using An Eddy Current Array” E2096/E2096M-22, “Standard Practice for In Situ Examination of Ferromagnetic Heat-Exchanger Tubes Using Remote Field Testing” E2884-22, “Standard Guide for Eddy Current Testing of Electrically Conducting Materials Using Conformable Sensor Arrays” E2338-22, “Standard Practice for Characterization of Coatings Using Conformable Eddy Current Sensors without Coating Reference Standards” E215-22, “Standard Practice for Standardizing Equipment and Electromagnetic Examination of Seamless Aluminum-Alloy Tube”. Liquid Penetrant and Magnetic Particle Methods E2297-15, “Standard Guide for Use of UV-A and Visible Light Sources and Meters used in the Liquid Penetrant and Magnetic Particle Methods” E709-21, “Standard Guide for Magnetic Particle Testing” E1219-21, “Standard Practice for Fluorescent Liquid Penetrant Testing Using the Solvent-Removable Process” E1208-21, “Standard Practice for Fluorescent Liquid Penetrant Testing Using the Lipophilic Post-Emulsification Process” E1220-21, “Standard Practice for Visible Penetrant Testing Using SolventRemovable Process” E1418-21, “Standard Practice for Visible Penetrant Testing Using the WaterWashable Process” E1210-21, “Standard Practice for Fluorescent Liquid Penetrant Testing Using the Hydrophilic Post-Emulsification Process” E3022-18, “Standard Practice for Measurement of Emission Characteristics and Requirements for LED UV-A Lamps Used in Fluorescent Penetrant and Magnetic Particle Testing” E1209-18, “Standard Practice for Fluorescent Liquid Penetrant Testing Using the Water-Washable Process” E1135-19, “Standard Test Method for Comparing the Brightness of Fluorescent Penetrants” E165/E165M-18, “Standard Practice for Liquid Penetrant Testing for General Industry” E433-71(2018), “Standard Reference Photographs for Liquid Penetrant Inspection” E125-63(2018), “Standard Reference Photographs for Magnetic Particle Indications on Ferrous Castings” E1417/E1417M-21e1, “Standard Practice for Liquid Penetrant Testing”

8.8 ASTM Standards

221

E3024/E3024M-22a, “Standard Practice for Magnetic Particle Testing for General Industry” E1444/E1444M-22a, “Standard Practice for Magnetic Particle Testing for Aerospace’. Radiology (Neutron) Method E803-20, “Standard Test Method for Determining the L/D Ratio of Neutron Radiography Beams” E748-19, “Standard Guide for Thermal Neutron Radiography of Materials” E545-19, “Standard Test Method for Determining Image Quality in Direct Thermal Neutron Radiographic Examination” E2003-20, “Standard Practice for Fabrication of the Neutron Radiographic Beam Purity Indicators” E2971-16(2020), “Standard Test Method for Determination of Effective Boron10 Areal Density in Aluminum Neutron Absorbers using Neutron Attenuation Measurements” E2861-16(2020), “Standard Test Method for Measurement of Beam Divergence and Alignment in Neutron Radiologic Beams” E2023-21, “Standard Practice for Fabrication of Neutron Radiographic Sensitivity Indicators”. Radiology (X and Gamma) Method E1165-20, “Standard Test Method for Measurement of Focal Spots of Industrial X-Ray Tubes by Pinhole Imaging” E1453-20, “Standard Guide for Storage of Magnetic Tape Media that Contains Analog or Digital Radioscopic Data” E999-20, “Standard Guide for Controlling the Quality of Industrial Radiographic Film Processing” E1114-20, “Standard Test Method for Determining the Size of Iridium-192, Cobalt-60, and Selenium-75 Industrial Radiographic Sources” E1161-21, “Standard Practice for Radiographic Examination of Semiconductors and Electronic Components” E1030/E1030M-21, “Standard Practice for Radiographic Examination of Metallic Castings” E2662-15, “Standard Practice for Radiographic Examination of Flat Panel Composites and Sandwich Core Materials Used in Aerospace Applications” E1411-16, “Standard Practice for Qualification of Radioscopic Systems” E1647-16, “Standard Practice for Determining Contrast Sensitivity in Radiology” E1255-16, “Standard Practice for Radioscopy” E2007-10(2016), “Standard Guide for Computed Radiography” E1734-16a, “Standard Practice for Radioscopic Examination of Castings” E1416-16a, “Standard Practice for Radioscopic Examination of Weldments” E1000-16, “Standard Guide for Radioscopy” E1815-18, “Standard Test Method for Classification of Film Systems for Industrial Radiography”

222

8 Nondestructive Testing (NDT)

E2903-18, “Standard Test Method for Measurement of the Effective Focal Spot Size of Mini and Micro Focus X-ray Tubes” E1742/E1742M-18, “Standard Practice for Radiographic Examination” E1025-18, “Standard Practice for Design, Manufacture, and Material Grouping Classification of Hole-Type Image Quality Indicators (IQI) Used for Radiography” E746-18, “Standard Practice for Determining Relative Image Quality Response of Industrial Radiographic Imaging Systems” E94/E94M-17, “Standard Guide for Radiographic Examination Using Industrial Radiographic Film” E1254-13(2018), “Standard Guide for Storage of Radiographs and Unexposed Industrial Radiographic Films” E2446-16, “Standard Practice for Manufacturing Characterization of Computed Radiography Systems” E1032-19, “Standard Practice for Radiographic Examination of Weldments Using Industrial X-Ray Film” E1441-19, “Standard Guide for Computed Tomography (CT)” E1570-19, “Standard Practice for Fan Beam Computed Tomographic (CT) Examination” E2737-10(2018), “Standard Practice for Digital Detector Array Performance Evaluation and Long-Term Stability” E2033-17, “Standard Practice for Radiographic Examination Using Computed Radiography (Photostimulable Luminescence Method)” E747-18, “Standard Practice for Design, Manufacture and Material Grouping Classification of Wire Image Quality Indicators (IQI) Used for Radiology” E2698-18e1, “Standard Practice for Radiographic Examination Using Digital Detector Arrays” E1935-97(2019), “Standard Test Method for Calibrating and Measuring CT Density” E1735-19, “Standard Practice for Determining Relative Image Quality Response of Industrial Radiographic Imaging Systems from 4 to 25 meV” E2736-17, “Standard Guide for Digital Detector Array Radiography” E1817-08(2014), “Standard Practice for Controlling Quality of Radiological Examination by Using Representative Quality Indicators (RQIs)” E1672-12(2020), “Standard Guide for Computed Tomography (CT) System Selection” E2445/E2445M-20, “Standard Practice for Performance Evaluation and LongTerm Stability of Computed Radiography Systems’ E1390-21, “Standard Specification for Illuminators Used for Viewing Industrial Radiographs” E1079-21, “Standard Practice for Calibration of Transmission Densitometers” E801-21, “Standard Practice for Controlling Quality of Radiographic Examination of Electronic Devices” E2597/E2597M-22, “Standard Practice for Manufacturing Characterization of Digital Detector Arrays”

8.8 ASTM Standards

223

E3327/E3327M-21, “Standard Guide for the Qualification and Control of the Assisted Defect Recognition of Digital Radiographic Test Data” E2002-22, “Standard Practice for Determining Image Unsharpness and Basic Spatial Resolution in Radiography and Radioscopy” E2104-22, “Standard Practice for Radiographic Examination of Advanced Aero and Turbine Materials and Components” E1695-20e1, “Standard Test Method for Measurement of Computed Tomography (CT) System Performance” E1931-16(2022), “Standard Guide for Non-computed X-Ray Compton Scatter Tomography” E1814-14(2022), “Standard Practice for Computed Tomographic (CT) Examination of Castings”. Ultrasonic Method E494-20, “Standard Practice for Measuring Ultrasonic Velocity in Materials by Comparative Pulse-Echo Method” E273-20, “Standard Practice for Ultrasonic Testing of the Weld Zone of Welded Pipe and Tubing” E2479-16(2021), “Standard Practice for Measuring the Ultrasonic Velocity in Polyethylene Tank Walls Using Lateral Longitudinal (LCR) Waves” E127-20, “Standard Practice for Fabrication and Control of Flat Bottomed Hole Ultrasonic Standard Reference Blocks” E2534-20, “Standard Practice for Targeted Defect Detection Using Process Compensated Resonance Testing Via Swept Sine Input for Metallic and NonMetallic Parts” E114-20, “Standard Practice for Ultrasonic Pulse-Echo Straight-Beam Contact Testing” E317-21, “Standard Practice for Evaluating Performance Characteristics of Ultrasonic Pulse-Echo Testing Instruments and Systems without the Use of Electronic Measurement Instruments” E1324-21, “Standard Guide for Measuring Some Electronic Characteristics of Ultrasonic Testing Instruments” E3081-21, “Standard Practice for Outlier Screening Using Process Compensated Resonance Testing via Swept Sine Input for Metallic and Non-Metallic Parts” E797/E797M-21, “Standard Practice for Measuring Thickness by Manual Ultrasonic Pulse-Echo Contact Method” E2904-17, “Standard Practice for Characterization and Verification of Phased Array Probes” E2580-17, “Standard Practice for Ultrasonic Testing of Flat Panel Composites and Sandwich Core Materials Used in Aerospace Applications” C1331-18, “Standard Practice for Measuring Ultrasonic Velocity in Advanced Ceramics with Broadband Pulse-Echo Cross-Correlation Method” C1332-18, “Standard Practice for Measurement of Ultrasonic Attenuation Coefficients of Advanced Ceramics by Pulse-Echo Contact Technique” E2192-13(2018), “Standard Guide for Planar Flaw Height Sizing by Ultrasonics”

224

8 Nondestructive Testing (NDT)

E2375-16, “Standard Practice for Ultrasonic Testing of Wrought Products” E1816-18, “Standard Practice for Measuring thickness by Pulse-Echo Electromagnetic Acoustic Transducer (EMAT) Methods” E1901-18, “Standard Guide for Detection and Evaluation of Discontinuities by Contact Pulse-Echo Straight-Beam Ultrasonic Methods” E2223-13(2018)e1, “Standard Practice for Examination of Seamless, Gas-Filled, Steel Pressure Vessels Using Angle Beam Ultrasonics” E3167/E3167M-18, “Standard Practice for Conventional Pulse-Echo Ultrasonic Testing of Polyethylene Electrofusion Joints” E3170/E3170M-18, “Standard Practice for Phased Array Ultrasonic Testing of Polyethylene Electrofusion Joints” E2001-18, “Standard Guide for Resonant Ultrasound Spectroscopy for Defect Detection in Both Metallic and Non-metallic Parts” E2491-13(2018), “Standard Guide for Evaluating Performance Characteristics of Phased-Array Ultrasonic Testing Instruments and Systems” E3213-19, “Standard Practice for Part-to-Itself Examination Using Process Compensated Resonance Testing Via Swept Sine Input for Metallic and NonMetallic Parts” E2985/E2985M-14(2019), “Standard Practice for Determination of Metal Purity Based on Elastic Constant Measurements Derived from Resonant Ultrasound Spectroscopy” E164-19, “Standard Practice for Contact Ultrasonic Testing of Weldments” E1962-19, “Standard Practice for Ultrasonic Surface Testing Using Electromagnetic Acoustic Transducer (EMAT) Techniques” E2373/E2373M-19, “Standard Practice for Use of the Ultrasonic Time of Flight Diffraction (TOFD) Technique” E1065/E1065M-20, “Standard Practice for Evaluating Characteristics of Ultrasonic Search Units E587-15(2020), “Standard Practice for Ultrasonic Angle-Beam Contact Testing” E664/E664M-15(2020)e1 Standard Practice for the Measurement of the Apparent Attenuation of Longitudinal Ultrasonic Waves by Immersion Method” E2700-20, “Standard Practice for Contact Ultrasonic Testing of Welds Using Phased Arrays” B594-19e1 Standard Practice for Ultrasonic Inspection of Aluminum-Alloy Wrought Products” E1001-21, “Standard Practice for Detection and Evaluation of Discontinuities by the Immersed Pulse-Echo Ultrasonic Method Using Longitudinal Waves” E1961-16(2021), “Standard Practice for Mechanized Ultrasonic Testing of Girth Welds Using Zonal Discrimination with Focused Search Units” E213-22, “Standard Practice for Ultrasonic Testing of Metal Pipe and Tubing” E3044/E3044M-22, “Standard Practice for Ultrasonic Testing of Polyethylene Butt Fusion Joints” E1774-17(2022), “Standard Guide for Electromagnetic Acoustic Transducers (EMATs)”.

Further Readings

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Further Readings 1. Almer JD, Winholtz RA (2008) X-ray stress analysis. In: Sharpe WN (ed) Handbook of experimental solid mechanics. Springer, Heidelberg, pp 801–820 2. Balayssac J-P, Garnier V (eds) (2017) Non-destructive testing and evaluation of civil engineering structures. ISTE Press–Elsevier 3. Barkanov EN, Dumitrescu A, Parinov IA (eds) (2018) Non-destructive testing and repair of pipelines. Springer, Heidelberg 4. Beattie AG (2013) Acoustic emission non-destructive testing of structures using source location techniques. Sandia Report, SAN2013-7779 5. Blitz J, Simpson G (1995) Ultrasonic methods for non-destructive testing. Springer, Heidelberg 6. Blitz J (2012) Electrical and magnetic methods of non-destructive testing. In: Czichos H (ed) Handbook of technical diagnostics, 2nd edn. Springer, Heidelberg 7. Breysse D, Balayssac J-P (2021) Non-destructive in situ strength assessment of concrete. Springer, Heidelebrg 8. Cavalcanti WL, Brune K, Noeske M, Tserpes K, Ostachowicz W, Schlag M (eds) (2021) Adhesive bonding of aircraft composite structures: non destructive testing and quality assurance concepts. Springer, Heidelberg 9. Halmshaw R (1991) Non-destructive testing, 2nd edn. Butterworth-Heinemann, Oxford 10. Halmshaw R (2012) Industrial radiology: theory and practice, 2nd edn. Springer, Heidelberg 11. Hellier CJ (2001) Handbook of nondestructive evaluation. McGraw-Hill, New York 12. Huang S, Wang S (2016) New technologies in electromagnetic non-destructive testing. Springer, Heidelberg 13. Doherty JE (1993) Nondestructive evaluation. In: Kobayashi AS (ed) Handbook of experimental mechanics, 2nd edn. Society for Experimental Mechanics, pp 527–555 14. Ida N, Meyendorf N (eds) (2018) Handbook of advanced non-destructive evaluation. Springer, Heidelberg 15. Kleinert W (2016) Defect sizing using non-destructive ultrasonic testing. Springer, Heidelberg 16. Mix PE (2005) Introduction to nondestructive testing: a training guide. Wiley, Hoboken 17. Murashov V (2017) Non-destructive testing and evaluation designs by the acoustic methods: quality–reliability–safety. LAP Lambert Academic Publishing 18. Ohtsu M (2020) Acoustic emission and related non-destructive evaluation techniques in the fracture mechanics of concrete: fundamentals and applications. Woodhead Publishing 19. Papaelias M, Garcia Marquez FP, Karyotakis A (2019) Non-destructive testing and condition monitoring techniques for renewable energy industrial assets. Elsevier, Amsterdam 20. Prasad J, Nair CCK (2011) Non-destructive testing and evaluation of materials, 2nd edn. McGraw-Hill, New York 21. Raj B, Subramanian CV, Jayakumar T (2000) Non-destructive testing of welds. Woodhead Publishing 22. Raj B, Jayakumar T, Thavasimuthu T (2007) Practical non-destructive testing, 3rd edn. Alpha Science 23. Rao BPC (2006) Practical non-destructive testing. Alpha Science Int’l Ltd 24. Summerscales J (1990) Non-destructive testing of fibre-reinforced plastic composites. Elsevier, Amsterdam 25. Wong BS (2014) Non-destructive testing—theory, practice and industrial applications. LAP Lambert Academic Publishing 26. Zoughi R (2012) Microwave non-destructive testing and evaluation principles. Springer, Heidelberg

Chapter 9

Testing of Concrete

Abstract Concrete is a heterogeneous composite material full of cracks, which play a vital role in its mechanical behavior and failure. A distinct characteristic of concrete is that it presents a softening behavior, as opposed to steels which present a hardening behavior. In concrete the crack-tip region is accompanied by a large fracture process zone in which microfailure mechanisms including matrix microcracking, debonding of cement-matrix interface, crack deviation and branching take place. All these mechanisms contribute to the energy of fracture. Concrete structures exhibit the so-called size effect according to which the failure load of a structure depends on its size. The size effect can only be explained with fracture mechanics concepts, while classical theories, such as elastic or plastic limit analysis cannot take into consideration this effect. In this chapter we briefly present the basic tests for the characterization of the mechanical behavior of concrete. They include compression, tension and bending tests. Furthermore, we present the basic principles of fracture mechanics applied to concrete, test methods for the determination of the critical strain energy release rate and an analysis of the size effect of concrete structures.

9.1 Introduction Concrete is the most important and widely used building material. It is made of cement, water, fine and coarse aggregates (such as sand and gravel), and sometimes admixtures. Each of these materials has a contributing function in concrete. The role of cement is to bond large aggregates and sand to form a durable matrix. Water serves a crucial role in the curing process of concrete. Curing is a chemical reaction (specifically a hydration reaction) which consumes water molecules to produce crystals and minerals that strengthen concrete matrix. Aggregates comprise more than half of the total composition of concrete and mostly contribute to its mechanical properties. Admixtures may be added to enhance the properties of concrete. Strength is the most important technical property of concrete. It appears to be a good index for other properties. Strength test is simple and can easily be performed.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_9

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Concrete is primarily employed in most structural applications to resist compressive loads. The compression strength is used as a measure of the overall quality of concrete. However, tensile strength is of vital importance despite its low magnitude. Concrete is not used to resist tensile loads. Generally speaking, concrete fracture is essentially a tensile failure, regardless of whether the failure is caused by compression, or other combined types of load. Concrete is full of cracks, which exist even before the application of loads. Failure of concrete is a result of initiation, coalescence and growth of cracks. Cracks play a vital role in the characterization of concrete within the frame of fracture mechanics and the explanation of the so-called size effect, according to which the failure load of a structure depends on its size.

9.2 Compression Test 9.2.1 Introduction Compressive strength is one of the most important engineering properties of concrete, not only because concrete is employed primarily to resist compressive loads, but also because it is used as a measure of its overall quality. In this section we present the basic characteristics of the compression test of concrete.

9.2.2 Compression Stress The compressive stress–strain curve of concrete is a graph of stress versus strain. The stress is obtained by submitting a specimen of constant cross section to a uniformly distributed progressively increasing axial compression load until failure occurs. The stress σ is given by σ =

P , A

(9.1)

where P is the applied compressive load and A is the cross-sectional area. The compressive load is provided by a testing machine. The strain ε corresponding to stress σ is measured by strain gages or other strain measuring devices.

9.2.3 Specimens The type, size and preparation of specimens used for the determination of the compressive strength are standardized. This is due to the fact that the compression

9.2 Compression Test

229

test is affected by many factors including the type and size of specimen, the curing, the end surfaces, the rate of application of the load and the rigidity of the testing machine. There are two types of specimens: the 150 mm or 200 mm cubes, or the cylinders with a length equal to twice the diameter. The standard cylinder size is 150 × 300 mm, if the coarse aggregate does not exceed 50 mm nominal size. Smaller more economical 100 × 200 mm cylinders are also used for many purposes. The cube specimens are mainly used in European countries and the cylinders in the USA and Canada. The length to diameter ratio of the cylindrical specimens is important, because if this ratio is too high it is likely that the material will fail under buckling.

9.2.4 Stress–Strain Curve A typical stress–strain curve is shown in Fig. 9.1a. The curve up to the point of maximum stress can be considered to consist of the following four parts (Fig. 9.1b), which are related to the existence and growth of cracks in concrete: Part I (below 30% of the ultimate stress): Even before loading cracks exist along the interface of coarse aggregates and matrix. The cracks are due to the different elastic moduli of aggregate sand matrix. The stress–strain curve is linear elastic.

Fig. 9.1 a Stress–strain curve of concrete in uniaxial compression. b Failure mechanisms including interfacial and matrix cracks depending on the load level for four regions of the stress–strain curve corresponding to 30%, 50%, 75% and between 80 and 100% of the ultimate stress

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Part II (between 30 and 50% of ultimate stress): The interfacial cracks propagate along the aggregates-matrix interface. The stress–strain curve begins to deviate from linearity. Part III (between 50 and 80% of the ultimate stress): The interfacial cracks and other cracks appear in the matrix. The stress–strain curve deviates appreciably from linearity. Part IV (between 80% and ultimate stress): Macrocracks appear and grow in the matrix for small increases of stress. The stress–strain curve reaches at a peak at the ultimate stress. After reaching the peak stress the stress–strain curve presents a descending (softening) branch. In order to obtain the softening part a deformation or strain-controlled testing machine must be used. In conventional testing machines, where the test is performed under control of loading rate, a sudden failure of the specimen occurs as soon as the maximum load level is reached. The machine gives small increments of load to the specimen and when the incremental load goes over the maximum level, the specimen fractures suddenly.

9.2.5 Failure Mechanisms When concrete is subjected to compression tensile strains are generated in the lateral direction (perpendicular to the loading direction) due to Poisson effect. Since the tensile strength of concrete is much lower than the compressive strength, cracks due to the tensile strains will first appear. The failure planes in uniaxial compression are parallel to the direction of the applied load (Fig. 9.2a). Concrete fails in tension under uniaxial compression. Failure planes for other loading combinations, biaxial compression (Fig. 9.2b), compression and tension (Fig. 9.2c) and biaxial tension (Fig. 9.2d) are shown.

9.2.6 Effect of Specimen Ends Friction forces are developed between the end surfaces of the specimen and the steel plates of the testing machine. A combined shear and compression state of stress is developed leading to an increase of the failure stress. The magnitude of the shear stress decreases as we recede from the specimen ends. Thus, concrete is stronger near its ends. In the specimen there is an undamaged cone or pyramid of height approximately equal to 0.87d, where d is the lateral dimension of the specimen. Specimens with lengths less than 2 × 0.87d = 1.73d show higher strength than those with a greater length (as the standard cylinder specimen of length twice the diameter).

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Fig. 9.2 Failure planes for uniaxial compression (a), biaxial compression (b), compression and tension (c) and biaxial tension (d)

9.2.7 Size Effect on Strength of Cylinders The effect of height to diameter ratio h/d of cylindrical specimens on the strength of low- and medium-strength concrete is shown in Fig. 9.3. For values of h/d < 1.5 the strength increases rapidly which is due to the restraining effect of the friction at the ends of the specimen. The decrease of strength is not much for h/d > 2, whereas in the region 1.5 < h/d < 2 the deviation from the standard specimen of h/d = 2 is only 5%. This indicates that a slight departure of h/d = 2 does not have much effect on the strength of concrete. For high-strength concrete the strength is less influenced by the specimen geometry. The end friction effect decreases for homogeneous concretes.

9.2.8 Comparison of Strength of Cubes and Cylinders The restraining effect of the platens of the testing machine extends over the entire length of a cube, whereas it leaves a central part unaffected in cylindrical specimens. Thus, it is expected that the strength of cube is higher than the strength of cylinder. A coefficient of 0.8 is usually used (the strength of cylindrical specimens is 0.8 of the strength of cubic specimens), even though there is no simple relation for the

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Fig. 9.3 Strength of concrete versus ratio of height to diameter h/d of cylindrical specimens

strength of the two specimen types. The ratio increase with the increase of strength and is nearly 1 for high-strength concretes. On the question of which specimen type is better the general consensus is that the cylinder is better than the cube. The end effect is less in cylinders, which contributes to more uniformity of the results.

9.3 Tension Test 9.3.1 Introduction Although concrete is used to withstand compressive forces, significant tensile stresses may develop in structures associated with multiaxial states of stress, Furthermore, tensile stresses play a fundamental role in the fracture mechanism of concrete. Generally, any kind of fracture occurs through cracking caused by tensile stresses. Concrete is much weaker in tension than it is in compression. The strength of concrete in tension is about 10% of its strength in compression. This difference can be attributed to the fact that cracks in concrete need less tensile than compressive stresses to propagate. The shape of the stress–strain curve in tension, generally speaking, is similar to that in compression. Besides the great difference in the tensile and compressive strength of concrete, it is difficult to obtain the descending part of the stress–strain curve in tension due to the rapid propagation of cracks under tension. In the following, we will present a few test methods for the determination of the tensile strength of concrete. They include: the direct tension test, the Brazilian (splitting) test, the flexural test and the ring test.

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9.3.2 Direct Tension Test The test is similar to the tensile test of metals and other materials studied in Chap. 1. A specimen is subjected to a uniformly distributed increasing axial tension load in a testing machine until it breaks into two parts. The stress is determined by Eq. (9.1), where P is the tensile load. The strain corresponding to a stress value is measured by strain gages or other strain measuring devices. The failure plane is the plane of maximum principal strain, which is perpendicular to the applied load. Gripping of the specimens is made using bonded end plates. Difficulties are introduced due to the clamping stresses and misalignment. Stress concentrations develop resulting in the breaking of the specimens near their ends. Dog bone specimens with reduced cross section of the central portion are often used. The specimens have a shoulder at each end and a gage section in between. The shoulders are wider than the gage section which causes a stress concentration to occur in the middle. This stress concentration ensures that the specimen will fail in the gage area away from the ends due to maximum tensile loading in that area. Direct tension tests present many problems. In this respect, other methods have been developed for the indirect measurement of the tensile strength of concrete.

9.3.3 Brazilian (Splitting) Test The test consists of subjecting a cylindrical specimen to compressive load applied along two diametrically opposite generatrices (Fig. 9.4). The load is increased until failure occurs by splitting the specimen along the vertical diameter. The normal compressive stress perpendicular to the horizontal diameter (along the direction of the applied load), σ c , and the normal tensile stress perpendicular to the vertical diameter (perpendicular to the direction of applied load), σ t , in an element located along the vertical diameter of the cylinder are given by σt =

[ ] D2 2P 2P , σc = −1 , πLD π L D r (D − r )

(9.2)

where P = compressive concentrated applied load on the cylinder D = diameter of the cylinder L = length of the cylinder r, (D − r) = distances of the element from the two loads, respectively. The distribution of stresses σ t and σ c along the vertical axis of the cylinder is shown in Fig. 9.5. Note that the tensile stress σ t is constant along the vertical diameter of the cylinder. The compressive stress σ c takes its minimum value (equal to 6P/ (π LD)) at the center of the cylinder and becomes infinite at the points of application

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Fig. 9.4 Brazilian (splitting) test specimen

of the load. Equation (9.2) indicates that a biaxial state of stress exists along the vertical diameter of the cylinder, with the compressive stresses higher more than three times the (constant) tensile stresses. Under the load high compressive stresses are induced. To reduce these stresses narrow strips of a packing material are placed between the cylinder and the platens of the testing machine. The real distribution of the tensile stress σ t along the vertical diameter is shown in Fig. 9.6. Note that the stresses are constant along most parts of the vertical diameter of the specimen and increase rapidly near the ends. The specimen fails due to tensile stresses. The tensile strength f sp is calculated by f sp =

2Pmax , πLD

(9.3)

where Pmax is the maximum load applied. Determination of the tensile strength from Eq. (9.3) is approximate. The state of stress along the vertical diameter of the cylinder is not plane stress, but in most parts of the specimen it is plane strain. Furthermore, there is a compressive stress along the vertical diameter much higher than the horizontal tensile stress. Thus, a three-dimensional state of stress is developed along the vertical diameter. The result of the stress triaxiality is to overestimate the direct tensile strength of concrete (as it is obtained from the direct tension test) by about 10–15%. The Brazilian test is the most popular test for the determination of the tensile strength of concrete because it is simple, and it gives relatively good results.

9.3 Tension Test

235

Fig. 9.5 Distribution of stresses σ t and σ c along the vertical axis of a cylindrical specimen subjected to diametral compression

Fig. 9.6 Distribution of tensile stress σ t along the vertical diameter of a cylindrical specimen subjected to diametral compression. In order to reduce the compressive stresses in the neighborhood of the applied load narrow strips of a packing material are placed between the cylinder and the platens of the testing machine

9.3.4 Flexure Test A rectangular beam specimen is subjected to bending. The flexural strength f fl is calculated from the Bernoulli bending theory formula as

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Md , I 2

f fl =

(9.4)

where M = bending moment to cause failure I = moment of inertia of the beam cross section d = depth of the beam. The flexural strength f fl is the critical stress at the most remote point of the cross section of the beam. The moment of inertia I of a beam with a rectangular cross section is given by bd 3 , 12

I =

(9.5)

where b is the width of the beam. The bending moment M for a three-point bend specimen of length L subjected to a concentrated load P at the mid-span is given by PL , 4

(9.6)

1.5P L . bd 2

(9.7)

M= and Eq. (9.4) becomes f fl =

For a four-point bend specimen (Fig. 9.7) of length L subjected to two equal concentrated forces of P/2 Eq. (9.4) with M = PL/6 becomes f fl =

PL . bd 2

(9.8)

Equation (9.8) provides the flexural strength provided that fracture occurs within the central one-third length of the beam. The flexural strength of concrete calculated by Eqs. (9.7) or (9.8) is always greater than the actual strength calculated from the tension test. The main reason for the higher strength is that in the bending test the strength is calculated from the failure of the most stressed fiber of the specimen, whereas in the tension test the entire volume of the specimen is subjected to the maximum uniform stress, and, therefore, the probability of a weak element is high. In the bending specimen the stresses change rapidly from a maximum to a zero value, while the stresses in the tension specimen are uniform. The cracks are blocked in the bending specimen from the less stressed material nearer to the neutral axis, whereas the cracks expand from its entire cross section in the tension specimen.

9.3 Tension Test

237

Fig. 9.7 A three-point bend specimen

9.3.5 The Ring Test A ring specimen is subjected to an internal pressure. The tensile strength f r is determined from the ultimate pressure from the following equation fr =

pD , 2t

(9.9)

where p = ultimate hydrostatic pressure D = average diameter of the ring t = wall thickness of the ring. Equation (9.9) applies when the radius of the ring is high enough compared to the wall thickness (higher than 10). The ring test is a simple test for the determination of the tensile strength of concrete. A disadvantage of the test is that for large aggregate sizes large rings are required. Calculation of the tensile strength from Eq. (9.9) is based on the validity of Hooke’s law.

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9 Testing of Concrete

9.4 Fracture Mechanics of Concrete 9.4.1 Introduction The applicability of linear elastic fracture mechanics (LEFM) to concrete was explored in the 1950s, but with unfavorable results. The values of critical strain energy release rate, GIc, obtained from notched concrete specimens were specimen size-dependent, and therefore, they could not be used as a characteristic material property. The fundamental research in fracture mechanics of concrete has been performed in the 1980s. It was realized that cementitious materials like pastes, mortars and concretes as well as many other materials (rock, particulate composites, grouted soils, bone, paper, wood, etc.) require a different kind of fracture mechanics than metals. In both metal and concrete structures nonlinear zones of small (small-scale yielding approximation treated by LEFM) or large sizes (ductile fracture) develop at the crack tip. However, in ductile/brittle metals the material in the nonlinear zones undergoes hardening or perfect plasticity, whereas in concrete the material undergoes softening damage. Materials that undergo softening damage are called quasi-brittle, because even though no appreciable plastic deformation takes place, the size of the nonlinear region is large enough and needs to be taken into account, whereas in brittle materials the size of the nonlinear region is negligible and LEFM applies. The foundations of the application of fracture mechanics to concrete were laid down by the pioneering work of Hillerborg [13–15] who introduced the fictitious crack model of concrete, in an analogous way to the Dugdale–Barenblatt model of metals. After that, the development of the field was explosive and the theory appears to be matured for applicability to design. In this chapter the basic principles of fracture mechanics of cementitious materials with emphasis on concrete will be presented.

9.4.2 Why Fracture Mechanics of Concrete? Concrete is a quasi-brittle material that develops cracks under service loads and exposure to regular environmental conditions. Concrete structures are full of cracks. Failure of concrete structures involves the initiation and growth of large cracking zones up to a critical point of instability at the maximum load. Even though cracks play an important role, concrete structures have been successfully designed and built without any use of fracture mechanics. This may be attributed to the difficulties involved in the application of fracture mechanics to concrete. However, intensive research efforts have progressed to the point that the theory is ripe to be used for the design of both reinforced and unreinforced concrete structures. A few reasons for the application of fracture mechanics to concrete structures and the resulting benefits are:

9.4 Fracture Mechanics of Concrete

239

a. The fracture process involves creation of new surfaces in the material. It is generally accepted that material separation is better described by energy principles than by stress or strain. The energy necessary to create new material surfaces is a fundamental quantity characteristic of the material. Fracture mechanics uses mostly energy concepts than stress or strain. Therefore, use of fracture mechanics will give a fundamental basis for the understanding of fracture phenomena in all materials, including concrete. b. High-strength concretes with compressive strength in excess of 100 N/mm2 present extremely brittle behavior. The use of such concretes in structures needs new design principles to accommodate crack growth relevant to brittle materials. Thus, it is necessary to develop test methods to quantify the degree of brittleness (or fracture toughness) of concrete. The area under the complete tensile stress– strain curve of the material cannot be used to quantity fracture toughness, since it is specimen size and geometry dependent. Brittleness or fracture toughness is commonly quantified by using principles of fracture mechanics. c. Design of concrete structures will benefit significantly from fracture mechanics. It will make it possible to achieve more uniform safety factors which will improve economy and reliability. This is apparent in structures like dams, nuclear reactor vessels or containments which due to their large size behave in a rather brittle manner and the consequences of a potential failure are enormous. d. The experimentally observed size effect of structures can be adequately explained by fracture mechanics. According to the size effect the ultimate stress for geometrically similar structures of different sizes depends on the size of the structure. Classical theories such as elastic analysis with allowable stress, plastic limit analysis and other theories (viscoelasticity, viscoplasticity, etc.) cannot explain the size effect. The size effect which is ignored in current codes is significant in design.

9.4.3 Tensile Behavior of Concrete In plain or reinforced concrete design little attention is paid to the tensile behavior of concrete. Its tensile strength is small and it is usually ignored, even though its tensile ductility is large. This prevented the efficient use of concrete for many years. The behavior of concrete under tensile loads plays a key role in the analysis of fracture since the material in the nonlinear region ahead of the crack is under tension. In the previous section we studied the behavior of concrete under tensile load up to the ultimate load. In this section we will present the stress-strain and stressdisplacement behavior of concrete after the ultimate load. Consider a concrete specimen of uniform cross section subjected to tension. It has been observed experimentally that when the maximum load is reached the weakest cross section of the specimen cannot carry more load, and damage is concentrated on a small volume of material adjacent to the weakest cross section. This is a microcracked material zone or fracture zone. Thus after the maximum load is reached additional

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9 Testing of Concrete

Fig. 9.8 Stress localization in a tension specimen

deformation takes place in the fracture zone, while the material outside the fracture zone unloads elastically. As the fracture zone increases the load decreases. A concrete tension specimen of initial length L is shown in Fig. 9.8. The elongation of the specimen, ∆L, after the maximum load is ∆L = ε0 L + w,

(9.10)

where ε0 is the strain (uniform) in the material outside the fracture zone and w is the width of the fracture zone. The average strain εm is εm =

∆L w = ε0 + . L L

(9.11)

Equation (9.11) suggests that after the maximum stress the average strain depends on the specimen length, L. The stress-strain curve is not a material property. This behavior is known as the strain localization effect. The behavior of concrete in tension should be described by two curves: the stress-strain curve up to the maximum stress and the stress-displacement curve after the maximum stress. It has been shown experimentally that the stress-displacement curve after the maximum stress does not depend on the size of the specimen and is characteristic of the material. Figure 9.9 shows a stress-displacement curve of concrete in tension. It consists of three regions: In region I which extends up to about 60% of the maximum stress

9.4 Fracture Mechanics of Concrete

241

the curve is linear elastic. In region II which extends up to the maximum stress the deformation becomes nonlinear because of the development of cracks. Up to the maximum load, as the load increases the strain remains uniformly distributed along the specimen. At the maximum stress the damage becomes localized and a fracture zone starts to form. From this point the displacement increases under a decreasing stress. The displacement measured from the elastic unloading line passing from the point of maximum load becomes independent of the gage length. The material outside the fracture zone recovers elastically, while the deformation is concentrated in the fracture zone up to a critical value. The stress-displacement relation for the fracture zone is obtained by subtracting the elastic displacement from the total displacement. The stress-displacement curve up to the critical displacement is characteristic of the material. In conclusion, the tensile behavior of concrete is characterized by two relations: the stress-strain relation for the undamaged material outside the fracture zone and the stress-displacement relation for the damaged material in the fracture zone. These two curves for a linear stress-strain and stress-displacement relation are shown in Fig. 9.10. The softening stress-displacement curve is difficult to be obtained experimentally due to unstable fracture, secondary bending and cracking.

Fig. 9.9 An idealized stress-displacement curve of concrete in tension

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9 Testing of Concrete

Fig. 9.10 Tensile behavior of cementitious materials. a Linear stress–strain relation of the material outside the damage zone and b linear stress-displacement relation of the material in the damage zone

9.4.4 The Fracture Process Zone Concrete is a heterogeneous material with complex microstructure. It can be modeled at various scale levels including the nano-, micro-, meso- and macrolevels. A simple way of modeling concrete is to consider it as a two-phase particulate composite. The matrix is the mortar, and the reinforcement is the coarse aggregates. As we noticed previously, defects play a vital role on the mechanical behavior of concrete. Microcracks are usually present, even before loading, at regions of high material porosity near the interfaces between the coarse aggregates and the mortar. They are caused by shrinking of the mortar during drying out of the concrete. Cracks are also present in the mortar matrix. Under an applied load both types of cracks start to increase and new cracks are formed. The interface cracks extend inside the mortar and are connected with the mortar cracks. When a sufficient number of microcracks coalesce a macrocrack is formed. Figure 9.11a shows a macrocrack (continuous traction-free crack) with its surrounding zone of concrete. The damage zone ahead of the traction-free crack is referred to as the fracture process zone (FPZ) and plays a vital role in the analysis of growth of the crack. Within the FPZ many microfailure mechanisms including matrix microcracking, debonding of cement-matrix interface, crack deviation and branching take place. All these mechanisms contribute to the energy of fracture. In the FPZ, Young’s modulus is smaller than that of the undamaged material and stress relaxation takes place. The closure stress in the FPZ associated with localized damage takes a maximum value at the tip of the FPZ and decreases to a zero value at the tip of the macrocrack (Fig. 9.11b). Experimental methods including optical scanning electron microscopy, moiré interferometry, dye penetrants, acoustic emission among others, and methodologies using compliance measurements and multicutting techniques have been applied to

9.4 Fracture Mechanics of Concrete

243

Fig. 9.11 a Fracture process zone (FPZ) ahead of a macrocracking concrete and b closure stresses in the FPZ

detect the shape and size of the FPZ. The FPZ depends on the geometry and size of the structure and the type of material. For cement paste the FPZ length is of the order of a millimeter, for mortar is about 30 mm, for normal concrete or coarse-grained rock is up to 500 mm, for dam concrete with extra large aggregates is around 3 m, for a grouted soil mass is around 10 m and in a mountain with jointed rock values of 50 m may be typical. In concrete the length of a fully developed FPZ is about 1.8 ch , where ch is the characteristic length. Typical values of ch for concrete with aggregate sizes of 8 to 32 mm are 250 to 800 mm. On the other hand, the length of the FPZ in a fine-grained silicon oxide ceramic is of the order of 0.1 mm and in a silicon wafer of the order of 10–100 nm.

9.4.5 Fracture Mechanics The proper fracture mechanics theory to be applied for a crack growth problem depends on the relative size of the FPZ, , with respect to the smallest critical dimension, D, of the structure, under consideration. Approximately, we may define that linear elastic fracture mechanics applies for D/ > 100, while nonlinear quasibrittle fracture mechanics for 5 < D/ < 100. For D/ < 5 nonlocal damage models, particle models or lattice models are applied. This indicates the importance of the size of the FPZ in the applicability of the proper fracture mechanics theory. However, there is another factor that differentiates the application of fracture mechanics to concrete from metals. In ductile or brittle metals the deformation in the

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9 Testing of Concrete

FPZ is dictated by hardening plasticity or perfect yielding, whereas in cementitious materials the material undergoes softening damage.

9.4.6 Cohesive Crack Models The nonlinear and dissipative phenomena that take place in the FPZ can be described by the cohesive or fictitious crack model. A fictitious crack equal to the original stressfree crack plus the length of the FPZ with cohesive forces in the FPZ is considered. The cohesive crack model for concrete is similar to the Dugdale model presented for metals. It is developed under the following assumptions: i. The FPZ localizes into a very narrow (line) band ahead of the crack tip. ii. The effect of the inelastic deformation in the FPZ is modeled by introducing a fictitious (equivalent) crack of length equal to the length of the true crack and the length of the FPZ. iii. The constitutive modeling in the FPZ is the stress-displacement relation of the material in tension. iv. The material outside the fictitious crack is elastic. Figure 9.12 shows a crack, the FPZ, the fictitious crack and the stress distribution in the FPZ and in the elastic material ahead of the fictitious crack tip. Note that the stress at the critical displacement δ c is zero, while the stress at the fictitious crack tip is equal to the tensile strength f t of the material (this stress is not exactly equal to the tensile strength as it is influenced by the stress normal to the crack front). A simple estimate of the length d c of the FPZ for a parabolic stress distribution of degree n along the FPZ is [12]: ( ) n + 1 KI 2 dc = . π ft

(9.12)

Note that n for concrete takes values between 7 and 14 and the length of the FPZ in concrete is many times the length of the plastic zone in metals. This differentiates the fracture behavior between ductile and quasi-brittle materials, like concrete. The size of the FPZ in quasi-brittle materials can be one order of magnitude larger than that of ductile materials of the same strength and toughness. Equation (9.12) suggests that d c can be put in the form dc = η

EG =η f t2

(9.13)

ch

since K I2 = E G, where η is dimensionless constant and characteristic length, given by

ch

is the so-called

9.4 Fracture Mechanics of Concrete

245

Fig. 9.12 Stress distribution ahead of the crack tip according to the cohesive crack model

ch

=

EG . f t2

(9.14)

Reported values of ch fall in the range 0.15–0.40 m and, therefore, the length of a fully developed fracture process zone takes values in the range 0.3–2 m.

9.4.7 Experimental Determination of GIc (a) Introduction The fracture energy, GIc , which is the specific work of fracture necessary for developing and fracturing the FPZ, can be obtained from the area of the stress-displacement curve of a uniaxial tension test (Fig. 9.10b). Although this is the most direct way of determining GIc the tensile test is not easy to perform because of stability problems. For this reason many test methods using precracked specimens have been developed for the experimental determination of GIc . In the following we will present briefly the (early) LEFM method, the compliance methods, the Jenq–Shah method and the RILEM method. (b) LEFM method Early researchers in the 1960s concerned with fracture mechanics of concrete used a methodology similar to the experimental determination of critical stress intensity

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9 Testing of Concrete

factor K Ic , in metals. Three-point beam specimens with notches have been the most popular. The specimens were subjected to a progressively increasing load and the load versus deflection or load versus crack mouth opening displacement response was recorded. The value of K Ic is determined from the peak load or the load at the intersection with a secant of slope 95% of the initial slope and the initial notch length. The values of K Ic obtained by this procedure varied with beam size or notch depth. This was attributed to the arrest of the growing crack by the aggregate particles and the fact that an initial notch, not a true crack, was used. Thus, the method based on LEFM cannot be used for the determination of K Ic . (c) Compliance methods The idea behind the compliance methods is to introduce an effective, not the initial, crack length. The effective crack takes into consideration the inelastic phenomena that take place in the FPZ ahead of the crack. First, the relationship between crack length and compliance (defined as the value of crack mouth opening displacement per unit load) for the specimen type used (for example three-point-notched specimens) is established. The unloading compliance of the specimen is determined by unloading the specimen after reaching the peak load. From the unloading compliance and the crack length versus compliance relationship the effective crack length is calculated. The value of K Ic is then calculated from the peak load and the effective crack length using LEFM formulas. When K Ic is calculated using an effective crack length, instead of the initial crack length, a valid size-independent value for K Ic is obtained. Note that for the determination of K Ic two parameters, the effective crack length and the peak load, are required. (d) The Jenq–Shah method In an effort to provide reliable critical fracture toughness values of cementitious materials, Jenq and Shah [16] proposed a methodology based on the compliance of the notched specimen. The basic idea behind the method is to determine an effective, not the initial, crack length. This effective crack length takes into consideration the inelastic phenomena that take place in the fracture process zone ahead of the crack tip. The effective crack length is equal to the actual crack length plus the length of the fracture process zone. For the application of the method the relationship between the compliance, defined as the value of the crack-tip opening displacement per unit load, and the crack length for a three-point notched bend specimen is established. The unloading compliance of the specimen is determined by unloading the specimen after reaching the peak load. The effective crack length is calculated from the relationship between the compliance and the crack length using the unloading compliance. The critical value of strain energy release rate, GIc , or stress intensity factor, K Ic , is calculated from the peak load and the effective crack length using linear elastic fracture mechanics formulas. It has been established that K Ic is independent of the specimen type and size. Thus, the determination of K Ic needs two quantities measured from the test, the unloading compliance and the peak load. The modulus of elasticity and the critical value of the crack-tip opening displacement are also determined.

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247

The experimental determination of the above quantities is based on the load versus crack mouth opening displacement (CMOD) curve for a loading–unloading cycle of a three-point bend specimen (Fig. 9.13). A schematic form of the curve is shown in Fig. 9.14. From the curve the following quantities are measured: The compliances for the loading and unloading parts and the maximum load. The modulus of elasticity, E, is calculated by: E=

6Sa o g2 (αo ) , Ci b2 t

(9.15)

where C i = the compliance of the loading part of the load-CMOD curve. α 0 = (a0 + HO)/(b + HO)

Fig. 9.13 Experimental setup of fracture mechanics test

Fig. 9.14 Typical load-CMOD curve from loading and unloading procedure for 28 days mortar reinforced with well-dispersed MWCNTs at amount of 0.2 wt% of cement

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9 Testing of Concrete

a0 = crack length. S = span length. b = specimen depth. t = specimen thickness. HO = length between the two points of measuring CMOD g2 (α 0 ) = geometric function given by g2 (α0 ) = 0.76 − 2.28α0 + 3.87α20 − 2.04α30 +

0.66 . (1 − α0 )2

(9.16)

In a similar way, the modulus of elasticity E is calculated from the compliance of the unloading part of the load-CMOD curve as E=

6Sac g2 (αc ) , C u b2 t

(9.17)

where C u = the compliance of the unloading part of the load—CMOD curve ac = the effective crack length α c = (ac + HO)/(b + HO). The unloading compliance is taken within 95% of the peak load calculated from the load-CMOD curve. The value of the effective crack length ac is calculated by equating the values of the modulus of elasticity E defined from Eqs. (9.15) and (9.17). This results in the following equation αc = αo

Cu g2 (αo ) . Ci g2 (αc )

(9.18)

Equation (9.18) is solved numerically for the determination of the critical crack length ac . The critical stress intensity factor is calculated as S K IC

√ S πac g1 (ac /b) , = 3(Pcr + Wh ) 2b2 t

(9.19)

where Pc = the peak load W h = W ho S/ L W ho = the weight of the beam and [ ( a )( (a ) ( a )2 ] ac ) c c c ( a ) 1.99 − b 1 − b 2.15 − 3.93 b + 2.70 b c = g1 . √ ( ac ) 23 ac )( b 1− π 1+2 b b

(9.20)

9.4 Fracture Mechanics of Concrete

249

The critical crack-tip opening displacement is calculated by: 6(Pc + 0.5Wh )Sac g2 (ac /b) Eb2 t [ ( )]1/2 ac )( βo − βo2 · (1 − βo )2 + 1.081 − 1.149 , b

CTODc =

(9.21)

where β 0 = a0 /ac g2 (ac /b) is calculated from Eq. (9.16) with α 0 = ac /b. Based on the values of K s Ic and CMODc a material length, Q, is defined by ( Q=

E CTODc S K IC

)2 .

(9.22)

The material length Q can be used to characterize the brittleness of the material. The smaller the value of Q and more brittle the material is. It was found that values of Q are in the range of 12.5–50 mm for hardened cement paste, 50–150 for mortar and 150–350 for concrete. (e) The work-to-fracture (RILEM) method The method is based on the experimental determination of the work of fracture in a precracked specimen. It was developed by Hilleborg [13–15] and was proposed by RILEM [21] as the first method of testing for fracture properties of concrete. Conceptually, the method can be applied to various specimen geometries but the proposed RILEM standard uses a beam with a central edge notch loaded in three-point bending. The specimen is a rectangular bar notched to a depth equal half the beam height. The dimensions of the beam are selected in relation to maximum aggregate. The length to height ratio varies from 4 to 8. The smallest recommended beam height is 100 mm. During the test the load-point deflection (P-δ) of the bam is measured and plotted along with the applied load. The test is performed in a closed-loop system under strain control conditions or in a stiff testing machine to produce a stable crack growth. The critical fracture energy GIc is calculated as G Ic =

W0 + mgδ0 , b(d − a0 )

(9.23)

where W 0 = area under P–δ curve up to displacement δ 0 where the load returns to zero mg = weight of beam and fixtures carried by the beam d = height of beam a0 = notch length

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9 Testing of Concrete

b = beam thickness. Note in Eq. (9.23) that the denominator represents the area of the specimen ligament. The work-to-fracture method is a simple and practical method for the experimental determination of fracture toughness, GIc . However, the method does not give sizeindependent values of GIc . Due to its simplicity it has been widely used for measuring fracture energy.

9.5 Size Effect In the previous section it was mentioned that the fracture toughness of concrete as determined by the work-to-fracture method depends on the size of the specimen. This is the so-called size effect. It appears in all structures, but it is more pronounced in structures made of concrete due to the large fracture process zone. The size effect is defined in terms of the nominal stress σ N at maximum (ultimate) load Pu of geometrically similar structures of different sizes. The nominal stress in a structure needs not to represent an actual stress and is defined as σ N = Pu /bd or σ N = Pu /d 2 for a two- or three-dimensional structure, respectively, where d is a characteristic dimension of the structure (e.g., the depth or the span of a beam, the length of the FPZ, etc.) and b is the thickness of the two-dimensional structure. A dependence of σ N on the size of the structure is called size effect. If σ N does not depend on the size of the structure we say that there is no size effect. The size effect in concrete structures can easily be established if we consider as characteristic length the length of the FPZ. If we take that this length is approximately five to six times the size of the aggregates, then the length of the FPZ is constant, whereas the size of the structure changes. Thus, the size of the FPZ compared to the size of the structure is negligible for large structures, while it is appreciable for small structures. This explains the rather brittle behavior of large structures, as opposed to the ductile behavior of small structures. Classical theories, such as elastic analysis with allowable stress or plastic limit analysis, cannot take into consideration the size effect. Contrary, linear elastic fracture mechanics exhibits a strong size effect dependence described by the dependence of stress intensity factor on the crack length. An approximate formula for the prediction of the size effect was proposed by Bazant [3–5]. The formula takes the form ) ( W −1/2 , (σ N )u = A f t 1 + B where

(9.24)

9.6 ASTM Standards

251

Fig. 9.15 Size effect law on the strength in a bi-logarithmic plot

(σ N )u = nominal stress at failure of a structure of specific shape and loading condition. W =Characteristic length of the structure. A, B =positive constants that depend on the fracture properties of the material and on the shape of the structure, but not on the size of the structure. f t =tensile strength of the material introduced for dimensional purposes. Equation (9.24) combines limit analysis for small structures and linear elastic fracture mechanics for large structures. A typical curve showing the size effect is shown in Fig. 9.15. The horizontal dashed line represents the failure status according to the strength or yield criterion. The inclined dashed line exhibits a strong size effect predicted by linear elastic fracture mechanics. The solid curve between the two limiting curves represents the real situation for most structures. From Fig. 9.15 we can observe that for very small structures the curve approaches the horizontal line and, therefore, the failure of these structures can be predicted by a strength theory. On the other hand, for large structures the curve approaches the inclined line and, therefore, the failure of these structures can be predicted by linear elastic fracture mechanics.

9.6 ASTM Standards ASTM has published the following standards for testing of concrete for strength, testing of fresh concrete and time of set: Testing Concrete for Strength

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• C470/C470M-15, “Standard Specification for Molds for Forming Concrete Test Cylinders Vertically” • C512/C512M-15, “Standard Test Method for Creep of Concrete in Compression” • C873/C873M-15, “Standard Test Method for Compressive Strength of Concrete Cylinders Cast in Place in Cylindrical Molds” • C1768/C1768M-12(2017), “Standard Practice for Accelerated Curing of Concrete Cylinders” • C174/C174M-17, “Standard Test Method for Measuring Thickness of Concrete Elements Using Drilled Concrete Cores” • C1856/C1856M-17, “Standard Practice for Fabricating and Testing Specimens of Ultra-High Performance Concrete” • C42/C42M-20, “Standard Test Method for Obtaining and Testing Drilled Cores and Sawed Beams of Concrete” • C39/C39M-21, “Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens” • C192/C192M-19, “Standard Practice for Making and Curing Concrete Test Specimens in the Laboratory” • C1231/C1231M-15, “Standard Practice for Use of Unbonded Caps in Determination of Compressive Strength of Hardened Cylindrical Concrete Specimens” • C617/C617M-15, “Standard Practice for Capping Cylindrical Concrete Specimens” • C1542/C1542M-19, “Standard Test Method for Measuring Length of Concrete Cores” • C496/C496M-17, “Standard Test Method for Splitting Tensile Strength of Cylindrical Concrete Specimens” • C918/C918M-20, “Standard Test Method for Measuring Early-Age Compressive Strength and Projecting Later-Age Strength” • C293/C293M-16, “Standard Test Method for Flexural Strength of Concrete (Using Simple Beam With Center-Point Loading)” • C78/C78M-22, “Standard Test Method for Flexural Strength of Concrete (Using Simple Beam with Third-Point Loading)” • C469/C469M-22, “Standard Test Method for Static Modulus of Elasticity and Poisson’s Ratio of Concrete in Compression” • C31/C31M-22, “Standard Practice for Making and Curing Concrete Test Specimens in the Field”. Testing of Fresh Concrete • C173/C173M-16, “Standard Test Method for Air Content of Freshly Mixed Concrete by the Volumetric Method” • C172/C172M-17, “Standard Practice for Sampling Freshly Mixed Concrete” • C1064/C1064M-17, “Standard Test Method for Temperature of Freshly Mixed Hydraulic-Cement Concrete” • C138/C138M-17a, “Standard Test Method for Density (Unit Weight), Yield, and Air Content (Gravimetric) of Concrete”

Further Readings

253

• C143/C143M-20, “Standard Test Method for Slump of Hydraulic-Cement Concrete” • C1890-19, “Standard Test Method for K-slump of Freshly Mixed Concrete” • C232/C232M-21, “Standard Test Method for Bleeding of Concrete” • C231/C231M-22, “Standard Test Method for Air Content of Freshly Mixed Concrete by the Pressure Method”. Time of Set • C191-21, “Standard Test Methods for Time of Setting of Hydraulic Cement by Vicat Needle” • C266-21, “Standard Test Method for Time of Setting”. Nondestructive and In-Place Testing • C597-16, “Standard Test Method for Pulse Velocity Through Concrete” • C1740-16, “Standard Practice for Evaluating the Condition of Concrete Plates Using the Impulse-Response Method” • C803/C803M-18, “Standard Test Method for Penetration Resistance of Hardened Concrete” • C805/C805M-18, “Standard Test Method for Rebound Number of Hardened Concrete” • C215-19, “Standard Test Method for Fundamental Transverse, Longitudinal, and Torsional Resonant Frequencies of Concrete Specimens” • C900-19, “Standard Test Method for Pullout Strength of Hardened Concrete” • C1074-19e1, “Standard Practice for Estimating Concrete Strength by the Maturity Method” • C1383-15(2022), “Standard Test Method for Measuring the P-Wave Speed and the Thickness of Concrete Plates Using the Impact-Echo Method”.

Further Readings 1. Aitcin PC (2007) High performance concrete. E & FN SPON 2. Balaguru, PN, Shah, SP (1992) Fiber-reinforced cement composites. McGraw-Hill Inc 3. Bazant ZP (ed) (1991) Current trends in concrete fracture research. Kluwer Academic Publishers 4. Bazant ZP (ed) (1992) Fracture mechanics of concrete structures. Elsevier Applied Science 5. Bazant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press 6. Bhavikatti SS (2021) Concrete technology. Wiley, Hoboken 7. Carpinteri A, Ingraffea R (eds) (1984) Fracture mechanics of concrete: material characterization and testing. Martinus Nijhoff Publishers 8. Carpinteri A (1986) Mechanical damage and crack growth in concrete: plastic collapse to brittle fracture. Martinus Nijhoff Publishers 9. Cotterell B, Mai YW (1996) Fracture mechanics of cementitious materials. Blackie Academic & Professional 10. Elfgren L (ed) (1989) Fracture mechanics of concrete structures. Chapman and Hall

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9 Testing of Concrete

11. Elfgren L, Shah SP (eds) (1991) Analysis of concrete structures by fracture mechanics. Chapman and Hall 12. Gdoutos EE (2022) Fracture mechanics, 3rd edn. Springer, Heidelberg 13. Hillerborg A, Modeer M, Peterson PE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 6:773–782 14. Hillerborg A (1980) Analysis of fracture by means of the fictitious crack model, particularly for fiber reinforced concrete. Int J Cem Comp 2:177–185 15. Hillerborg A (1985) The theoretical basis of method to determine the fracture energy Gf of concrete. Mater Struct 18:291–296 16. Jenq YS, Shah SP (1985) Nonlinear fracture parameters for cement based composites: theory and experiments. In: Shah SP (ed) Application of fracture mechanics to cementitious composites. Martinus Nijhoff Publishers 17. Jenq YS, Shah SP (1985) Two-parameter fracture model for concrete. ASCE J Eng Mat 111:1227–1241 18. Neville AM (1997) Properties of concrete, 4th edn. Addison Wesley Longman Ltd 19. Neville AM, Brooks JJ (2010) Concrete technology, 2nd edn. Pearson 20. Popovics S (1998) Strength and related properties of concrete. Wiley, Hoboken 21. RILEM, 1985-TC 50 FMC, Fracture mechanics of concrete. Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams, RILEM Recommendations, Mat & Struct 18, No 106. 22. Rossmanith HP (ed) (1990) Fracture and damage of concrete and rock. Pergamon Press 23. Shah SP (ed) (1985) Application of fracture mechanics to cementitious composites. Martinus Nijhoff Publishers 24. Shah SP, Swartz SE, Ouyang C (1995) Fracture mechanics of concrete: applications of fracture mechanics to concrete, rock and other quasi-brittle materials. Wiley, Hoboken 25. Shah SP, Swartz SE (eds) (1989) Fracture of concrete and rock. Springer, Heidelberg 26. Shah SP, Carpinteri A (eds) (1991) Fracture mechanics test methods for concrete. Chapman and Hall 27. Sih GC, DiTommasso A (eds) (1984) Fracture mechanics of concrete: structural application and numerical calculation. Martinus Nijhoff Publishers 28. Vipulanandan C, Gerstle WH (eds) (2001) Fracture mechanics for concrete materials: testing and applications. American Concrete Institute 29. Wittmann FH (ed) (1983) Fracture mechanics of concrete. Elsevier 30. Wittmann FH (ed) (1986) Fracture toughness and fracture energy of concrete. Elsevier 31. Wittmann FH (ed) (1993) Numerical models in fracture mechanics of concrete. Balkema 32. Zhou X, Li Z, Ma H, Hou D (2022) Advanced concrete technology, 2nd edn. Wiley, Hoboken

Chapter 10

Testing of Composites

Abstract Composites are produced from two or more constituent materials with dissimilar chemical or physical properties that are emerged to create a material with properties unlike the individual elements. Of the various types of composites the most important category is the fiber composites, in which a matrix is reinforced with continuous fibers. Fiber composites are anisotropic materials that need special methods for the study of their mechanical behavior, much more complicated than those of isotropic materials. In this chapter we present test methods for the mechanical characterization of a unidirectional lamina of fiber reinforced composite materials under tension, compression and shear. We also present methods for the determination of the interlaminar fracture toughness of laminates under mode-I, II, III and mixedmode loading. Finally, we introduce the sandwich materials, and study their failure mechanisms.

10.1 Introduction A composite is a material made from two or more constituent materials with significantly different physical or chemical properties that, when combined, produce a material with different characteristics from the individual components. The constituent materials remain separate and distinct within the composite, differentiating composites from mixtures and solid solutions. The mechanical performance and properties of the composite are superior to those of the constituent materials. Of the constituent materials of the composite one is distinct, stiffer and stronger and is called reinforcement, whereas the other is continuous and weaker. It is called matrix. The properties of the composite depend on the properties of the constituent materials and the volume (or weight) fraction of the reinforcement, called fiber volume ratio. There are two main forms of composites: particulate and fiber reinforced. A particulate composite is composed of particles embedded in a matrix. Fiber reinforced composites are classed as either continuous (long fibers) or discontinuous (short fibers). When the fibers are aligned they provide maximum strength and stiffness along the direction of alignment. The composite is considerably weaker along

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 E. Gdoutos and M. Konsta-Gdoutos, Mechanical Testing of Materials, Solid Mechanics and Its Applications 275, https://doi.org/10.1007/978-3-031-45990-0_10

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10 Testing of Composites

other directions. Particle reinforcing in composites is less effective of strengthening than fiber reinforcement. Particulate composites achieve gains in stiffness primarily, but also in strength and toughness. In all cases the improvements are less than in fiber reinforced composites. In this chapter we will deal only with continuous fiber reinforced composites.

10.2 Fiber Reinforced Composites Fiber reinforced composites are anisotropic. Their properties vary with direction. The basic building block of a fiber composite is the lamina or ply. It consists of unidirectional fibers embedded in a matrix. The lamina is an orthotropic material that has three mutually perpendicular planes of symmetry. The principal axes of the lamina are in the direction of the fibers (longitudinal), normal to the fibers in the plane of the lamina (in-plane transverse) and normal to the plane of the lamina (out-ofplane transverse). The three principal axes are designated as 1, 2 and 3, respectively. A composite material may consist of laminae stacked together at different orientations. The assembly is called laminate. The basic elastic properties of the lamina are: The moduli of elasticity (Young’s moduli) E 1 , E 2 , E 3 , along the directions 1, 2, 3; the shear moduli G12 , G23 , G31 in the 1–2, 2–3, 3–1 planes; Poisson’s ratios ν 12 , ν 23 , ν 31 (the first subscript denotes the loading direction and the second denotes the strain direction). The basic strength properties of the lamina are: The tensile strengths F 1t , F 2t , F 3t along the directions 1, 2, 3; the compressive strengths F 1c , F 2c , F 3c along the directions 1, 2, 3; the shear strengths F 12 , F 23 , F 31 in the 1–2, 2–3, 3–1 planes. Characterization of the elastic behavior of a lamina needs nine elastic constants, while for isotropic materials only two constants are needed (Young’s modulus and Poisson’s ratio). Similarly, characterization of the strength behavior of a lamina needs nine strength constants, while for isotropic materials only three constants are needed (tensile, compressive and shear strength). Thus, the mechanical characterization of a composite is much more complicated than that of an isotropic material. A great number of different test methods have been developed to determine the above mechanical constants of a composite material. The size and shape of the specimen, the manner of application of the load and the followed procedure are different in the different methods. In all methods small rectangular specimens are desired. For example, cylindrical or ring specimens are difficult to manufacture, and methods based on these types of specimens are not preferable. Furthermore, the method should provide a pure (without other stresses) and uniform state of stress in the test piece. In this chapter we present the basic test methods for the characterization of a unidirectional lamina in the three primary loading modes: tension, compression and shear. Furthermore, we present methods for the determination of the interlaminar fracture toughness of laminates. Finally, we introduce the sandwich materials and study their failure mechanisms.

10.3 Tension Testing

257

10.3 Tension Testing 10.3.1 Introduction Tensile testing is the easiest and more straightforward testing of the three (tension, compression, shear) broad classes of mechanical test methods. The test is performed according to ASTM D3039 for high-strength composites and according to ASTM D638 for lower-strength composites.

10.3.2 Specimens According to ASTM D3039 specimens are straight-sided flat coupons of constant cross section with adhesively bonded beveled tabs (Fig. 10.1a). The longitudinal (0°) specimen is 1.27 cm (0.50 in.) wide, while the transverse (90°) specimen is 2.54 cm (1 in.) wide. Usually, specimens of six laminae for the longitudinal specimen and at least eight laminae for the transverse specimen are used. According to ASTM D638 the specimens are untabbed, dog-boned flat coupons (Fig. 10.1b). The purpose of the enlarged ends is to increase the cross-sectional area at the grips, so that failure of the specimen takes place in the gage length. At the grips the load is transferred to the specimen via shear forces. To avoid shear failure Fig. 10.1 a Tabbed straight-sided flat specimen, b untabbed dog-boned flat specimen

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10 Testing of Composites

at the grips the shear strength of the material should be sufficiently high relative to the tensile strength. This is also the case of tabbed specimens, wherever for the tabbed specimens the contact of stress transfer is much higher than for the dog-boned specimens.

10.3.3 Experimental The specimen is subjected to an increasing load up to failure. Axial and transverse strains are obtained by means of a pair of two-gage rosettes mounted on both sides of the specimen to check undesirable bending of the specimen.

10.3.4 Data Reduction The following properties are determined from the uniaxial tension test: E 1 , E 2 : Longitudinal and transverse Young’s moduli, respectively. ν 12 , ν 21 : Major and minor Poisson’s ratios, respectively. F 1t , F 2t : Longitudinal and transverse tensile strengths, respectively. u u ε1t , ε2t : Longitudinal and transverse ultimate tensile strains, respectively.

10.4 Compression Testing 10.4.1 Introduction Compression testing of composites is more difficult and complicated than tension testing. The test should be designed in such a way to avoid premature failure due to buckling or crushing. Compression loading along the fibers may cause fiber buckling, while perpendicular to the fibers involves failure of the matrix and the fibermatrix interface. Many test methods incorporating a variety of specimen designs and loading fixtures have been developed. The method should produce compression failure without introducing eccentricity, stress concentrations at the loaded ends and buckling instability. The test methods can be classified into three groups: shear loading methods, end loading methods and sandwich methods. We will briefly present the above three groups of methods.

10.4 Compression Testing

259

10.4.2 Shear Loading Methods The load is introduced via shear through end tabs, in a similar way loads are introduced in tensing testing, except that the loads are of opposite sign and the wedges in the grips are inverted. The so-called Celanese test was the first method of transferring the compressive load to the specimen via shear trough friction. The test is described in ASTM D-3410. The fixture of the test requires a perfect cone-to-cone contact, which is not normally achieved. Higher values of the compressive strength are obtained. A modification of the Celanese test is the IITRI test method (Fig. 10.2). Trapezoidal wedges are used instead of conical grips of the Celanese test. A guidance system consisting of two parallel roller bushings is used. Strain gages are mounted on both sides of the specimen to verify that it fails in compression and not by buckling. The fixture allows for testing of wider and thicker specimens. The IITRI fixture is larger and heavier (about 46 kg) than the Celanese fixture (about 4.6 kg). The Wyoming-modified Celanese compression test fixture was developed to improve limitations of the Celanese fixture. The fixture can accommodate a specimen up to 12.7 mm (0.50 in.) wide and 7.6 mm (0.30 in.) thick (twice those of the Celanese fixture).

Fig. 10.2 Sketch of the IITRI compression test fixture

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10 Testing of Composites

10.4.3 End Loading Methods The specimen is loaded in compression directly through its ends than using wedge grips. To avoid premature end crushing tabs can be bonded to the specimen ends. The test is described in ASTM D695. Dog-boned specimens without tabs can be used. The Wyoming end-loaded side-supported fixture and the ASTM D695 test fixture are examples of end-loaded fixtures. The specimen is similar to the tensile specimen, slightly shorter with longer tabs. The specimen is supported over the entire gage length.

10.4.4 Sandwich Methods There are two methods, the compression and the flexure sandwich methods. In the compression sandwich method two composite specimens are bonded to a honeycomb core, as described in ASTM C 364-61 (Fig. 10.3). The specimens have a rectangular cross section. According to ASTM C 364-61 the unsupported length of the specimen should not exceed twelve times the total specimen thickness to avoid buckling. The loaded ends are reinforced and fitted inside grooved cylindrical rods that are loaded between two metal plates. In the flexure sandwich method the specimen consists of a honeycomb core with the composite coupon bonded on the compressive side and a metal sheet bonded on the tensile side. The beam is subjected to four-point bending at two quarter-span points (Fig. 10.4), as described in ASTM C 393. The compressive side of the beam is subjected to nearly uniform compression. The thickness of the metal sheet on the

Fig. 10.3 Sandwich specimen for compression testing of composites

10.4 Compression Testing

261

Fig. 10.4 Sandwich specimen in four-point bending for compression testing of composites. The composite is placed on the compressive side of the specimen and is subjected to nearly uniform compression

tensile side is adjusted to ensure failure of the composite on the compression side of the sandwich beam. The compressive stress by assuming uniform deformation in the facings and neglecting the bending stresses in the core is given by σx =

Nx PL , = h 4bh(2H + h + h ' )

(10.1)

where P Nx L H b h h'

Applied load Compressive force on the composite coupon Span length Thickness of the honeycomb Depth of the beam Composite facing thickness Thickness of the metal sheet on the tensile side.

To ensure that the sandwich beam will fail by compression of the composite coupon and not by shear of the honeycomb core the following condition should be satisfied: Fcs τ ≥ Fxc σ

(10.2)

with τ=

PL P , σ = , 2H b 8H hb

where τ σ Fxc Fcs

Shear stress in the honeycomb core Bending compressive stress in the composite facing Compressive strength of composite facing Core shear strength.

(10.3)

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10 Testing of Composites

10.4.5 Data Reduction The following properties are determined from the uniaxial compression test: E 1 , E 2 : Longitudinal and transverse Young’s moduli, respectively. ν 12 , ν 21 : Major and minor Poisson’s ratios, respectively. F 1c , F 2c : Longitudinal and transverse compressive strengths, respectively. u u ε1c , ε2c : Longitudinal and transverse ultimate compressive strains, respectively.

10.5 Shear Testing 10.5.1 Introduction Shear tests are usually performed in thin-walled tube specimens. However, cylindrical composite specimens are difficult and expensive to produce. Thus, flat specimens are mainly used for shear testing of composites. Shear testing is the most difficult of the three (tension, compression, shear) broad classes of mechanical test methods of a composite lamina. It is not easy to develop a pure and uniform shear stress. In most methods shear is accompanied by normal stresses. There are three shear stresses, the in-plane and two interlaminar (through-the-thickness) stresses. In this section we will present methods for in-plane shear testing. Interlaminar shear testing will be studied in the next section. Of the various methods of in-plane shear testing we will present the rail method, the tensile methods and the Iosipescu and Arcan methods.

10.5.2 Rail Method This is a straightforward method to apply shear stress. Two test configurations, the two-rail and the three-rail test are used. They are described in ASTM D4255. In both tests the rectangular specimen is gripped along its long edges by two or three pairs of rails that are loaded in a direction parallel to the edges. In the two-rail test fixture the rails are pulled in opposite directions. The specimen between the rails is subjected to in-plane shear. In the three-rail configuration (Fig. 10.5) the outer rails are supported and the center rail in loaded in compression to create two shear zones. The three-rail fixture has the advantage of being symmetrical and the disadvantage of requiring a larger specimen. In the three-rail configuration the average shear stress applied to the specimen is τ12 = where

P , 2Lh

(10.4)

10.5 Shear Testing

263

Fig. 10.5 Three-rail shear test fixture

P Compressive load L Specimen length along rails h Specimen thickness. A strain gage is mounted on the exposed face of the specimen at 45° with the rail axes. The shear strain γ 12 is γ12 = 2(ε)θ=45◦ ,

(10.5)

where (ε)θ =45◦ is the normal strain measured by the strain gage. The in-plane shear modulus is obtained from Eqs. (10.4) and (10.5) as G 12 =

τ12 . γ12

(10.6)

10.5.3 Tensile Methods We will present two tensile test methods, the [± 45]ns method and the 10° off-axis method. In the [± 45]ns method a [± 45] ns laminate is loaded in axial tension, as described in ASTM D3518 (Fig. 10.6). The specimen is a tabbed straight-sided coupon. When

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10 Testing of Composites

Fig. 10.6 [± 45] ns laminate is loaded in uniaxial tension for the determination of the in-plane lamina shear properties

the laminate is loaded in a uniaxial tensile stress σ x the stresses σ 1 , σ 2 , τ 6 in an element at ± 45 with the loading axis are given by σ1 =

σx σx σx + τx y , σ2 = − τx y , τ6 = , 2 2 2

(10.7)

where τ xy is the in-plane shear stress generated from the shear coupling mismatch. The in-plane lamina strains are ε1 = ε2 =

εx + ε y , γ6 = ε x − ε y , 2

(10.8)

where ε x and ε y are axial (along the loading direction) and transverse strains. They are measured by two-gage rosettes. The in-plane shear modulus G12 is obtained from Eqs. (10.7) and (10.8) as G 12 =

σx τ6 ). = ( γ6 2 εx − ε y

(10.9)

By dividing the numerator and denominator of this equation by ε x we obtain

10.5 Shear Testing

265

Ex ), G 12 = ( 2 1 + νx y

(10.10)

where E x is the Young’s modulus and νx y is Poisson’s ratio of the [± 45]ns laminate. The primary advantage of this method is its simplicity. A great disadvantage is that the state of stress in each lamina is not pure shear but a biaxial tensile stress state (stresses σ 1 and σ 2 ) coexists with the shear stress τ6 . The method overestimates the shear strength of the lamina. In the 10° off-axis method the unidirectional coupon is subjected to a uniaxial stress σ x at an angle 10° with the direction of the fibers (Fig. 10.7). Two strain gages are mounted on the specimen at angles 45° and − 45° with the fiber direction. The shear strain γ 6 is given by γ6 = εA − εB ,

(10.11)

where εA and εB are the strains measured by the strain gages A and B. The intralaminar (in-plane) shear stress τ6 referred to the fiber coordinate system is given by Fig. 10.7 Unidirectional specimen subjected to a uniaxial stress σ x at an angle θ = 10° with the direction of the fibers for the determination of the in-plane shear strain γ 6

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10 Testing of Composites

τ12 = − σx sin(− 10◦ ) cos(− 10◦ ) = 0.171σx .

(10.12)

The shear modulus G12 is obtained from the initial slope of the τ6 versus γ 6 curve.

10.5.4 Iosipescu and Arcan Methods Both the Iosipescu and Arcan test methods are based on the same principle. The specimens are small rectangular coupons with two edge 90° notches, whose purpose is to reduce the cross-sectional area and to produce a nearly uniform state of shear stress in the section between the two notches. The fixtures for the Iosipescu and Arcan test methods are shown schematically in Figs. 10.8 and 10.9, respectively. In the Iosipescu method the specimen is a beam, while in the Arcan method the specimen is a coupon. In both tests the average shear stress in the test section between the two notches is τ6 =

P , lh

(10.13)

where l is the specimen height between the two notches and h is the specimen thickness.

Fig. 10.8 Fixture of the Iosipescu shear test

10.6 Interlaminar Shear Strength

267

Fig. 10.9 Fixture of the Arcan test for pure shear and mixed-mode loading

The shear strain γ 12 is measured with a centrally located ± 45° rosette with the loading direction as in the case of the three-rail shear specimen. Strain rosettes are mounted on both sides of the specimen to cancel the effects of twist. A drawback of both tests is that the state of stress in the section of the specimen between the two notches is neither pure shear nor uniform. An advantage of the test is that they are simple to perform. Furthermore, they can be used for determining all three components of shear by orienting the specimen in an appropriate manner. The Iosipescu test method is standardized as ASTM D5379.

10.6 Interlaminar Shear Strength We will present two test methods for the determination of the interlaminar shear strength: the short beam shear test and the double-notch shear test. The short beam shear test uses a short beam specimen machined from a relatively thick unidirectional laminate with fibers in the axial direction. The beam is loaded in three-point bending perpendicularly to the plies (Fig. 10.10). The test is standardized as ASTM D2344. The principle of the test is based on the strength of materials bending theory according to which for short beams the normal stresses are small and the shear stresses are large. The bending stresses are proportional to the applied load and the length of the beam. The shear stress varies parabolically across the height of the beam and takes its maximum value at the mid-height of the beam. The interlaminar shear strength is given by F31 =

3P , 4bh

(10.14)

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10 Testing of Composites

where P Failure load b Width of the beam h Depth of the beam. To ensure interlaminar shear failure prior to flexural failure the material should be sufficiently stronger in tension/compression than in interlaminar shear. The following condition should be satisfied: F31 Ft < , τmax σmax

(10.15)

where Ft is the tensile strength in fiber direction, τmax is the maximum shear stress at the mid-height of the beam and σmax is the maximum normal stress at the most remote fiber of the beam. These stresses are given by τmax =

3P 3P L , σmax = . 4bh 2bh 2

(10.16)

where L is the beam length. Introducing these values of stresses into Inequality (10.15) we obtain Ft 2L < h F31

(10.17)

During the test because of the small span length-to-thickness ratio it is not practical to measure the beam deflection in order to calculate the shear modulus. From the test only the interlaminar shear strength is obtained. The test is criticized because the shear stress is not uniform within the specimen (varies parabolically through the specimen thickness) and the bending stresses are always present. In the double-notch shear test the specimen is a unidirectional coupon in which two parallel notches are machined (Fig. 10.11). The notches are on opposite faces of the specimen and have a depth equal to half the specimen thickness. When the

Fig. 10.10 Short beam shear test for measuring of interlaminar shear strength

10.6 Interlaminar Shear Strength

269

specimen is loaded in uniaxial tension or compression the part of the specimen along the mid-plane between the two grooves is subjected to shear. The test is described in ASTM specification D3846. The interlaminar shear strength is given by F31 =

P , wl

(10.18)

where P is the failure load, l is the distance between notches and w is the width of the specimen. To ensure interlaminar shear failure prior to tension failure the following condition should be satisfied: F1t F31 < , τ σ

(10.19)

2P P , σ = . wl wh

(10.20)

where τ=

With these values of τ and σ Inequality (10.19) becomes F31 h < . F1t 2l

Fig. 10.11 Double-notch specimen for measuring the interlaminar shear strength

(10.21)

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10 Testing of Composites

10.7 Interlaminar Fracture Toughness 10.7.1 Introduction A common form of failure of laminated composites is the separation of laminae or plies called delamination or interlaminar fracture. A delamination is a plane crack separating adjacent plies. The plane of the crack coincides with the plane of the interface between the plies. A delamination more often extends in a selfsimilar manner. The growth of delamination can be studied by using fracture mechanics. A delamination can grow under three basic modes, opening-mode (modeI), sliding-mode (mode-II) or tearing-mode (mode-III), or under a combination of these modes. The resistance to delamination growth for each of the three basic modes is expressed by the corresponding interlaminar fracture toughness. In the following the three delamination modes and mixed-mode-I and II delamination will be studied separately.

10.7.2 Mode-I Delamination In the mode-I loading the delamination surfaces separate perpendicularly to the plane of delamination. The double cantilever beam (DCB) (Fig. 10.12) is the most commonly used specimen for characterization of mode-I delamination. The two arms of the DCB may be considered to a first approximation as cantilevers with zero rotation at their ends. According to elementary beam theory, the relative

Fig. 10.12 Double cantilever beam subjected to an end point concentrated load, P, or displacement

10.7 Interlaminar Fracture Toughness

271

displacement u of the points of application of the concentrated loads P for conditions of generalized planes stress is u=

4Pa 3 , 3E 1 bh 3

(10.22)

where a 2h B E1

Length of delamination Height of DCB Thickness of DCB Longitudinal modulus (along beam length). The compliance of DCB is C=

8a 3 u = . P 3E 1 bh 3

(10.23)

The strain energy release rate is obtained as ( ) 1 u 2 dC 12P 2 a 2 GI = = . 2B C 2 da u E1 B 2 h3

(10.24)

The critical energy release rate GIc is obtained at maximum applied load at crack extension. When the effect of shear force is taken into consideration GI is given by ] [ 12P 2 ( a )2 E1 , GI = + E1 B 2 h h 10μ31

(10.25)

where μ31 is the transverse shear modulus. For applied total displacement u, GI is given by GI =

3u 2 E 1 h 3 . 16a 4

(10.26)

From Eq. (10.24) we obtain that 24P 2 a dG I = > 0, da E1 B 2 h3

(10.27)

and, therefore, crack growth under fixed load conditions is unstable. For fixed grip conditions we obtain from Eq. (10.26) 3u 2 E 1 h 3 dG I =− < 0, da 4a 3

(10.28)

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10 Testing of Composites

and, therefore, crack growth is stable. For the experimental determination of the critical strain energy release rate GIc most test are performed under fixed grip conditions (displacement controlled) which render to stable crack growth. For the preparation of the specimen a preexisting end delamination is produced by inserting a Teflon film at the mid-plane of the laminate. For the unrestrained rotation at the end of the specimen the load is applied through metallic piano hinges that are bonded to the delaminated end of the specimen. The specimen is loaded at a low crosshead rate in order to produce stable crack growth. The applied displacement is measured from the crosshead displacement of the testing machine or by means of a linear variable differential transformer (LVDT) extensometer. The load–deflection curve is obtained in which incremental crack lengths are marked during stable crack growth. The area method can be used for the determination of GI during stable crack growth. The strain energy release rate GI is calculated by GI =

Pi u j − P j u i ( ), 2B a j − ai

(10.29)

where the indices i and j refer to two consecutive crack growth increments. Monitoring the crack length during stable crack growth becomes difficult at high rates of loading. This problem is alleviated by using the width-tapered double cantilever beam (WTDCB) specimen in which the width varies linearly along the length of the beam (Fig. 10.13). In the WTDCB specimen the rate of change of compliance with respect to crack length is constant. The strain energy release rate is given GI =

12P 2 k 2 , E1 h3

(10.30)

where k=

a b

b = Beam width at crack length a. Typical results of mode-I critical strain energy release rates for various types of carbon/epoxy composites are given in Table 10.1 [5].

10.7.3 Mode-II Delamination For mode-II delamination toughness measurements the same double cantilever beam specimen with an edge delamination as for mode-I is used. The specimen is tested in three-point bending and is called end-notched flexure (ENF) specimen (Fig. 10.14). At the tip of the crack shear stresses are produced since the delamination lies along the

10.7 Interlaminar Fracture Toughness

273

Fig. 10.13 Width-tapered double cantilever beam (WTDCB) specimen

Table 10.1 Mode-I critical strain energy release rates for various carbon fiber composite materials Material

Type of test

Strain energy release rate GIc (J m−2 )

T300/5208

DCB

88–103

AS4/3501-6

DCB

190–198

AS4/3502

DCB

160

T300/F-185

WTDCB

1880

AS4/PEEK

DCB

1460–1750

neutral axis of the uncracked specimen. It has been found that no excessive friction between the crack surfaces is introduced. The strain energy release rate can be determined from the compliance of the specimen using strength-of-materials analysis. The strain energy stored in the ENF specimen of length 2L is given by 2L

U=

M2 dx, 2E 1 I

(10.31)

0

where M is the bending moment, E 1 is the modulus of elasticity and I the moment of inertia. For a beam of width b and height 2h, I is given by

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10 Testing of Composites

Fig. 10.14 End-notched flexure (ENF) specimen

I =

2bh 3 . 3

(10.32)

To determine U we divide the ENF specimen into three regions. We have for the bending moment (x is measured from the left support of Fig. 10.14) M=

Px , 2

M=

P(L − x) , 0< x < L −a 2

M=

P(L − x) , L − a < x < L. 4

0