Structural Components

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Author: Dominique Francois

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Structural Components


Structural Components Mechanical Tests and Behavioral Laws

Edited by Dominique François

First published in France in 2001 by Hermes Science Publications entitled: “Essais mécaniques et lois de comportement” First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 6 Fitzroy Square London W1T 5DX UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd, 2008 © HERMES Science Ltd, 2001 The rights of Dominique François to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Essais mécaniques et lois de comportement. English Structural components: mechanical tests and behavioral laws/edited by Dominique Francois. p. cm. Includes index. ISBN: 978-1-84821-015-8 1. Materials--Mechanical properties. 2. Deformations (Mechanics) I. François, Dominique, 1948- II. Title. TA404.8.E87 2007 620.1'123--dc22 2007021380 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-015-8 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.

Table of Contents

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dominique FRANÇOIS

1

Chapter 2. Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Louis CHABOCHE

5

2.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Domain of elasticity . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Normality rules . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Unified theory of viscoplasticity . . . . . . . . . . . . . . . . . 2.3.1. General form of the constitutive law . . . . . . . . . . . . 2.3.2. Choice of viscosity law . . . . . . . . . . . . . . . . . . . . 2.3.3. Isotropic hardening laws. . . . . . . . . . . . . . . . . . . . 2.3.4. Kinematic hardening laws. . . . . . . . . . . . . . . . . . . 2.3.5. Cyclic hardening and softening. . . . . . . . . . . . . . . . 2.3.6. Static recovery . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7. Time-independent limit case . . . . . . . . . . . . . . . . . 2.3.8. Methods of determination . . . . . . . . . . . . . . . . . . . 2.3.8.1. Determination of hardening laws within independent time-scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.8.2. Determination of the viscosity law. . . . . . . . . . . . 2.3.8.3. Determination of static recovery effects . . . . . . . . 2.3.9. Other unified approaches . . . . . . . . . . . . . . . . . . . 2.4. Other types of modeling . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Plasticity-creep partition. . . . . . . . . . . . . . . . . . . . 2.4.2. Methods by means of micro-macro transposition . . . . . 2.4.3. More advanced hardening laws. . . . . . . . . . . . . . . . 2.4.4. Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4.5. Damage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 35

Chapter 3. Measurement of Elastic Constants . . . . . . . . . . . . . . . . . . Pascal GADAUD

41

3.1. Elastic constants . . . . . . . . . . . . . . . . . . . 3.1.1. The perfect crystal – elastic constants . . . . 3.1.2. Isotropic solid – elastic moduli . . . . . . . . 3.1.3. From isotropic solid to real material . . . . . 3.1.4. Dynamic modulus . . . . . . . . . . . . . . . 3.2. Quasi-static mechanical tests . . . . . . . . . . . 3.2.1. Uni-axial tensile and compression tests . . . 3.2.2. Torsion and bending tests . . . . . . . . . . . 3.2.3. Hydrostatic compression tests . . . . . . . . 3.3. Ultrasonic methods . . . . . . . . . . . . . . . . . 3.3.1. Principle . . . . . . . . . . . . . . . . . . . . . 3.3.2. Measurement error sources . . . . . . . . . . 3.3.3. Measurements at high temperatures . . . . . 3.3.4. Immersion-bath ultrasound interferometry . 3.4. Resonant methods . . . . . . . . . . . . . . . . . . 3.4.1. Introduction to resonant methods . . . . . . 3.4.2. Various experimental methods . . . . . . . . 3.4.3. Bar and disk tests . . . . . . . . . . . . . . . . 3.4.4. Bending tests on foil . . . . . . . . . . . . . . 3.4.5. Torsion tests . . . . . . . . . . . . . . . . . . . 3.4.6. Other tests . . . . . . . . . . . . . . . . . . . . 3.5. Modulus measurements of coatings . . . . . . . 3.5.1. Vibratory methods . . . . . . . . . . . . . . . 3.5.2. Instrumented indentation . . . . . . . . . . . 3.6. Bibliography . . . . . . . . . . . . . . . . . . . . .

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42 42 44 47 48 50 51 52 54 55 55 56 57 59 60 60 60 61 63 64 65 65 66 67 69

Chapter 4. Tensile and Compression Tests . . . . . . . . . . . . . . . . . . . . Dominique FRANÇOIS

73

4.1. Introduction. . . . . . . . . . . 4.2. Description of the tensile test 4.2.1. Test piece . . . . . . . . . 4.2.2. Gripping . . . . . . . . . . 4.2.3. Tensile testing machine . 4.2.3.1. Arrangements . . . . . 4.2.3.2. Stiffness . . . . . . . .

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73 73 73 74 74 74 76

Table of Contents

4.3. Standard data . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Determination of constitutive equations . . . . . . . . . . 4.4.1. True stress and strain. . . . . . . . . . . . . . . . . . . 4.4.2. Empirical expressions of the work hardening curve 4.4.3. Necking. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3.1. Condition with no strain rate effect . . . . . . . . 4.4.3.2. Strain rate sensitivity . . . . . . . . . . . . . . . . . 4.4.3.3. Yield drops. . . . . . . . . . . . . . . . . . . . . . . 4.5. Damage determination . . . . . . . . . . . . . . . . . . . . 4.6. Compression test. . . . . . . . . . . . . . . . . . . . . . . . 4.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .

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77 79 79 80 83 83 84 85 85 87 87 88 88

Chapter 5. Hardness Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sylvie POMMIER

89

5.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Standard hardness tests . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Vickers hardness tests . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Micro-hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3. Nano-hardness (Berkovich type indenter) . . . . . . . . . . . . 5.2.4. Brinell and Rockwell ball tests . . . . . . . . . . . . . . . . . . . 5.3. Analytical approaches of hardness tests . . . . . . . . . . . . . . . . 5.3.1. Identification of the modulus of elasticity (Hertz contact) . . . 5.3.2. Identification of the yield strength (Hill’s analysis). . . . . . . 5.4. Finite element analysis of hardness test . . . . . . . . . . . . . . . . 5.4.1. Finite element method . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Effect of work-hardening amplitude . . . . . . . . . . . . . . . . 5.4.3. Effect of the type of hardening . . . . . . . . . . . . . . . . . . . 5.4.4. Pile-up method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5. Viscous material . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6. Porous materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.7. Films and surface coatings . . . . . . . . . . . . . . . . . . . . . 5.4.8. Measurement of fracture toughness of brittle materials . . . . 5.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1. Appendix 1: formulary (Hertz contact) . . . . . . . . . . . . . . 5.6.2. Appendix 2: slip line method (Hill) . . . . . . . . . . . . . . . . 5.6.3. Appendix 3: equivalences between tensile and hardness tests. 5.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 90 91 92 92 93 96 96 100 103 103 105 106 107 109 110 111 112 112 113 113 118 121 121

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Chapter 6. Fatigue Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Henri-Paul LIEURADE, Suzanne DEGALLAIX, Gérard DEGALLAIX and Jean-Pierre GAUTHIER 6.1. Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2. Objective of fatigue tests . . . . . . . . . . . . . . . . . . . . 6.1.3. Classification of fatigue tests . . . . . . . . . . . . . . . . . . 6.1.4. Classification of loading modes . . . . . . . . . . . . . . . . 6.1.5. Test pieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5.1. Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5.2. Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5.3. Machining of test pieces . . . . . . . . . . . . . . . . . . . 6.1.6. Calibration of testing machines. . . . . . . . . . . . . . . . . 6.1.6.1. Definition of a dynamometric bar . . . . . . . . . . . . . 6.1.6.2. Determination of cyclic loads . . . . . . . . . . . . . . . 6.1.6.3. Dynamic calibration of testing machines by axial load 6.2. High-cycle fatigue tests – endurance limit . . . . . . . . . . . . 6.2.1. Classification of loadings . . . . . . . . . . . . . . . . . . . . 6.2.2. Stress-number of cycles to failure curve (Wöhler curve). . 6.2.3. Nature and dispersion of fatigue test results . . . . . . . . . 6.2.4. Determination of endurance limit of metallic materials . . 6.2.4.1. Estimation of fatigue resistance, VD . . . . . . . . . . . . 6.2.4.2. Statistical methods . . . . . . . . . . . . . . . . . . . . . . 6.2.5. Expression of the S-N curve . . . . . . . . . . . . . . . . . . 6.2.6. Estimation of the number of cycles N50 by the Henry straight line method . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6.1. Field of application . . . . . . . . . . . . . . . . . . . . . . 6.2.6.2. Principle of the method . . . . . . . . . . . . . . . . . . . 6.2.6.3. Implementation of the method . . . . . . . . . . . . . . . 6.2.7. Accounting for the main parameters of influence . . . . . . 6.2.7.1. Mechanical parameters . . . . . . . . . . . . . . . . . . . 6.2.7.2. Geometric parameters . . . . . . . . . . . . . . . . . . . . 6.2.7.3. Parameters of material processing . . . . . . . . . . . . . 6.2.7.4. Environmental parameters. . . . . . . . . . . . . . . . . . 6.2.8. Examination of the fracture facies of the test pieces . . . . 6.3. Low-cycle fatigue tests . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2. Physical mechanisms of low-cycle fatigue . . . . . . . . . . 6.3.3. Methods for low-cycle fatigue tests . . . . . . . . . . . . . . 6.3.4. Experimental apparatus . . . . . . . . . . . . . . . . . . . . . 6.3.4.1. Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4.2. Test pieces . . . . . . . . . . . . . . . . . . . . . . . . . . .

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125 125 126 127 127 128 128 128 128 130 130 130 130 131 131 132 133 135 135 135 139

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140 140 140 141 141 141 144 146 146 146 147 147 149 152 153 153 154

Table of Contents

6.3.4.3. Instrumentation . . . . . . . . . . . . . . . . . . . . . . . 6.3.5. Data processing . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5.1. Stress response: cyclic hardening/softening curves . 6.3.5.2. Stress-strain hysteresis loops . . . . . . . . . . . . . . . 6.3.5.3. Resistance to low-cycle fatigue curves . . . . . . . . . 6.3.6. Ratcheting effect . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Measurement of the crack propagation rate in fatigue . . . . . 6.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1.1. Fatigue crack initiation and propagation . . . . . . . . 6.4.1.2. Recalling the notion of plastic zone at the crack tip. . 6.4.1.3. The Paris law . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1.4. Crack closure phenomenon . . . . . . . . . . . . . . . . 6.4.2. Implementation of crack propagation rate measurements 6.4.2.1. Principle of the test . . . . . . . . . . . . . . . . . . . . . 6.4.2.2. Test pieces . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2.3. Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2.4. Performing the test . . . . . . . . . . . . . . . . . . . . . 6.4.2.5. Processing and presentation of results. . . . . . . . . . 6.4.2.6. Main information to deliver after the test . . . . . . . . 6.4.3. Other types of tests . . . . . . . . . . . . . . . . . . . . . . . 6.4.3.1. Devices for testing outside ambient environment . . . 6.4.3.2. Variable loading tests . . . . . . . . . . . . . . . . . . . 6.4.3.3. Tests on non-metallic materials . . . . . . . . . . . . . 6.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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154 156 156 163 165 167 168 168 168 170 171 172 173 173 174 176 183 185 188 188 188 188 188 189

Chapter 7. Impact Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Michel FRUND

193

7.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Some history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Description of the impact test . . . . . . . . . . . . . . . . . . . . 7.3.1. Test piece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Charpy impact pendulum . . . . . . . . . . . . . . . . . . . . 7.3.3. Charpy instrumented impact pendulum . . . . . . . . . . . . 7.3.3.1. Measurement of the load . . . . . . . . . . . . . . . . . . 7.3.3.2. Measurement of the displacement . . . . . . . . . . . . . 7.3.3.3. Different time-displacement curves . . . . . . . . . . . . 7.3.3.4. Determinations of the characteristic values of the load. 7.3.3.5. Determination of the global fracture energy . . . . . . . 7.3.3.6. Determination of the percentage of brittle appearance from the load-time curve . . . . . . . . . . . . . . . . . . . . . . . 7.4. Determination of transition curves . . . . . . . . . . . . . . . . .

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7.5. Transition temperature and upper shelf . . . . . . . . . . . . . . . . . . . 7.6. Impact fracture energy-fracture toughness empirical correlations . . . . 7.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 202 205

Chapter 8. Fracture Toughness Measurement . . . . . . . . . . . . . . . . . . Dominique FRANÇOIS

207

8.1. Introduction. . . . . . . . . . . . . . . . . . . . . 8.2. Fracture mechanics bases. . . . . . . . . . . . . 8.2.1. Rice Cherepanov J integral . . . . . . . . . 8.2.2. Stress intensity factor K . . . . . . . . . . . 8.2.3. Plastic zone at the crack tip . . . . . . . . . 8.2.3.1. Plastic zone correction . . . . . . . . . . 8.2.3.2. Plane stress plastic zone . . . . . . . . . 8.2.3.3. Plane strain plastic zone . . . . . . . . . 8.3. Implementation of fracture toughness tests . . 8.3.1. Test pieces . . . . . . . . . . . . . . . . . . . 8.3.2. Data processing . . . . . . . . . . . . . . . . 8.3.3. Validity of measurements . . . . . . . . . . 8.4. Measurement of fracture toughness JIc . . . . . 8.4.1. Basis of elastoplastic fracture mechanics . 8.4.2. JIc testing . . . . . . . . . . . . . . . . . . . . 8.4.2.1. Interrupted loadings method . . . . . . 8.4.2.2. Partial unloading method . . . . . . . . 8.4.2.3. Potential drop method . . . . . . . . . . 8.4.2.4. Other methods. . . . . . . . . . . . . . . 8.4.2.5. Crack tip blunting. . . . . . . . . . . . . 8.4.3. Data processing . . . . . . . . . . . . . . . . 8.4.4. Conditions of validity . . . . . . . . . . . . 8.5. CTOD measurement . . . . . . . . . . . . . . . 8.6. Conclusion . . . . . . . . . . . . . . . . . . . . . 8.7. Notations . . . . . . . . . . . . . . . . . . . . . . 8.8. Bibliography . . . . . . . . . . . . . . . . . . . .

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207 209 209 210 211 211 212 213 213 213 215 216 217 217 217 218 218 219 219 219 219 220 221 221 222 222

Chapter 9. Dynamic Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Luc LATAILLADE

225

9.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Test methods relying on propagation techniques and on the Hopkinson bar . . . . . . . . . . . . . . . . . . . . . 9.2.1. Compression device and uni-dimensional theory . . 9.2.1.1. Radial inertia . . . . . . . . . . . . . . . . . . . . . 9.2.1.2. Calculation of applied load and of displacement 9.2.1.3. Dynamic compression Hopkinson test . . . . . .

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225

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226 228 229 230 231

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Table of Contents

9.2.2. Torsion with Hopkinson bars . . . . . . . . . . . . . . . . . . . . 9.2.2.1. Overall considerations . . . . . . . . . . . . . . . . . . . . . . 9.2.2.2. Case of very ductile materials . . . . . . . . . . . . . . . . . 9.2.3. Tension with Hopkinson bars. . . . . . . . . . . . . . . . . . . . 9.2.3.1. Low resistance specimen adaptation. . . . . . . . . . . . . . 9.2.3.2. Adaptation of a tensile test bench line for interrupted tests 9.2.3.3. Brittle material case: alternatives. . . . . . . . . . . . . . . . 9.2.4. Taylor test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4.1. Overall considerations . . . . . . . . . . . . . . . . . . . . . . 9.2.4.2. Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Dynamic fracture mechanics tests . . . . . . . . . . . . . . . . . . . 9.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Charpy test analysis according to the Kishimoto method . . . 9.3.3. Charpy test with Hopkinson bars. . . . . . . . . . . . . . . . . . 9.4. Plate against plate test. . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1. Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1.1. Traveling diagram . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1.2. Rankine-Hugoniot conservation equation . . . . . . . . . . 9.4.1.3. Lagrangian analysis . . . . . . . . . . . . . . . . . . . . . . . 9.4.1.4. Application to the characterization of an elastomer . . . . . 9.5. Collision tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1. Inertia wheel instrumented impacting device . . . . . . . . . . 9.5.1.1. Overall considerations . . . . . . . . . . . . . . . . . . . . . . 9.5.1.2. Discussion and application to an adhesive joint . . . . . . . 9.5.1.3. Application to an adhesive joint . . . . . . . . . . . . . . . . 9.5.2. Drop weight tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2.1. Introduction: the critical mass-velocity couple . . . . . . . 9.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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246 246 249 251 251 255 256 258 258 259 259 259 260 262 264 264 265 267 270 271 277 277 277 278 281 284 284 289

Chapter 10. Notched Axi-symmetric Test Pieces . . . . . . . . . . . . . . . . . Jacques BESSON

293

10.1. Introduction . . . . . . . . . . . . . 10.2. Geometry and notations . . . . . . 10.3. Notch test piece testing . . . . . . 10.3.1. Mechanical test . . . . . . . . 10.3.2. Observation of damage . . . . 10.4. Specimen elastic analysis . . . . . 10.5. Plastic analysis of specimens. . . 10.5.1. Bridgman analysis. . . . . . . 10.5.2. Finite element analysis . . . . 10.5.3. Small deformations analysis.

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xi

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293 294 296 296 297 297 299 299 303 304

xii


10.5.4. Large deformations analysis. . . . . . . 10.5.4.1. Necked tensile specimen . . . . . . . 10.5.4.2. AEF specimens. . . . . . . . . . . . . 10.5.4.3. Plastic anisotropy . . . . . . . . . . . 10.6. Damage analysis . . . . . . . . . . . . . . . . 10.6.1. Case study: brittle fracture . . . . . . . . 10.6.2. Case study: uncoupled ductile fracture 10.6.3. Case study: coupled ductile fracture . . 10.7. Viscoplasticity and creep damage . . . . . . 10.8. From notch to crack? . . . . . . . . . . . . . 10.9. Bibliography . . . . . . . . . . . . . . . . . .

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305 305 305 306 308 310 313 316 319 320 321

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

325

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327

Chapter 1

Introduction

Structural components must resist various applied loadings. Their nature is very diverse: aggressive environments, temperature and mechanical loads. However, we will only consider this last area in this book. Engineers must design parts so their deformations under these loads remain acceptable and they are not damaged or broken. In many cases structures need to be as light as possible, in order to save materials on the one hand, but above all to decrease energy consumption on the other hand. It is thus important to optimize the shape of parts and to choose the materials they are made of so that they perform without excessive deformations and without loss of integrity. Following design, problems can arise in service, such as crack initiation, which require calculations of the stress and strain distributions. Finally, failure assessments also require such analyses. To reach these objectives, more and more sophisticated design tools are available for engineers, provided that they are suitably supplied with reliable data concerning the mechanical behavior of materials. This is the aim of mechanical testing. The design of a part requires the knowledge of the relations between applied loads and deformations, as well as the limits not to be exceeded at the risk of damage or fracture. This will be the case, for example, with the stiffness of a spring and with its yield load. More generally, these relations involve the stresses and the strains, and more precisely the two corresponding tensors. Solving the problem is achieved by integrating the three stress equilibrium equations together with the boundary conditions. As there are six unknowns, the six components of the stress tensor, there is a lack of equations to reach a solution. They are provided precisely by the constitutive equations between the six components of the stress tensor and Chapter written by Dominique FRANÇOIS.

2


the six components of the strain tensor, which, it must not be forgotten, derive from the displacement field, which itself includes three components. The problem thus involves nine unknowns: the six components of the stress tensor and the three components of the displacement, and nine equations: the three equilibrium equations and the six constitutive equations. These constitutive equations, as a general rule, involve not only the stresses and the strains at a given time, but also the elapsed time. Without even considering the aging effects, the strain state preserves the memory of past deformations. It is only for an elastic material, for which there is no residual deformation after unloading, that the past does not operate. The constitutive laws are empirical laws, which need to be determined experimentally. Nevertheless, various theoretical considerations, which are deduced from the knowledge of deformation mechanisms, enable us to formulate hypotheses regarding the structure of the constitutive equations. If, for example, it is permissible to assume that the material is isotropic, the stress and strain invariants can only appear in those constitutive equations, as the orientation of the coordinates has no influence. Influence of past time can appear explicitly in the formulation of constitutive equations. More generally, it appears implicitly through the derivatives of the components of the stress and strain tensors, and the strain rate only in most cases. In this way, the influence of the loading path, i.e. the evolution of the eigen stresses and directions at a given point, can be introduced before being followed by the present instant. The cyclic character of the loads needs to be introduced, namely in the study of fatigue. The number of variables then becomes large: amplitude and mean level of stresses, frequency, loading paths. The constitutive equations also depend on temperature. In this way, their most general formulations are rather complex and involve a very large number of parameters. Their identification can then require a significant number of tests. A major difficulty immediately appears in the determination of constitutive equations: the deformation of any test specimen under a given load can be calculated only if those equations are known, which assumes that the problem is solved. It is only for very simple shapes for which the stress and strain fields are homogeneous that direct determination is possible. This is the case for the gage of a tensile specimen, in an approximate way for the wall of a thin tube in torsion or under internal pressure. It is nevertheless necessary to perform more complicated tests to explore various three-dimensional stress and strain states. In that case, it is absolutely vital to couple experimentation with theoretical hypotheses concerning the form of the constitutive equations and with calculations. As underlined above, the task of an engineer is far from being achieved when he has determined the deformation of a part; he still needs, and this is often the main

Introduction

3

part of the work, to fix the limits not to be exceeded, in term of stresses and strains, so as to avoid excessive yielding or worse fracture, or in order to reach a given life. It is then required to express these conditions at the level of the basic components of the stress and strain tensors. More or less solid hypotheses allow us to do this using simple tests results: for example, the Von Mises yield criterion depending on the yield strength as measured in a tensile test, and the Goodman diagram for the endurance limit as determined from rotary bending tests results and from the ultimate tensile stress. Otherwise, transposition at the level of parts of tests results of not so easy an interpretation, such as Charpy or fracture toughness KIc tests, needs to be possible. The development of underlying theories is out of the question here. The reader will need a sufficient knowledge of resistance of materials, solid mechanics and fracture mechanics, concepts which are developed in other books. This book is also not intended to cover the entire scope of tests which are performed routinely for the complete determination of constitutive equations in their full generality. Specific tests keep being designed for the determination of a particular parameter. On the contrary, the emphasis will be placed on the most widely used tests, those which are common practice in industrial laboratories, or otherwise, those which can be obtained from various specialized laboratories, without new adjustments and at reasonable costs. A number of them provide access directly to the parameters of the constitutive equations, whereas others do so only indirectly: for instance, hardness tests. Nevertheless, they will be discussed inasmuch that they are of current use, although we will not go into all the information, which they are able to provide. A number of tests are useful not only for the determination of the constitutive equations but also for the determination of the damage and fracture limits. Some essentially deal with these last aspects. The majority of the tests to be discussed are normalized. Of course, it is important to conform rigorously to the standards, in order to avoid any dispute between client and supplier, and also to achieve the most meaningful results as possible. The provisions of standards result partly from compromises based on considerations that are barely scientific, but also on sound theoretical considerations and on round-robins. It is not always easy to understand the reason for which a particular requirement is imposed, and its importance is sometimes underestimated, particularly when difficulties are found in the literal application of procedures. Actually, as far as possible, explanations for the reasons behind the introduction of various provisions in a standard will be given, so that their scope can be suitably appreciated, and the validity limits of these tests can be understood. However, the purpose of this book is not to provide an exhaustive description of the various tests when they are standardized. For this, it will be more useful to refer directly to the standards.

4


One of the drawbacks of normalized tests is that they often require the entry of only a few specific pieces of data: for example, engineering yield strength, ultimate tensile strength, elongation at fracture. Merely settling with those, a large part of the information that could be extracted from the tests is lost for the determination of constitutive equations. This impoverishment can paradoxically be increased by the computerized entry of data, when what is not explicitly required is deleted. Paper records do not have this drawback. It must not be forgotten either that the specimen themselves, which were used in the test, constitute results to be carefully processed by various inspections such as dimensional measurements and macro and microfractographies. One of the aims of this book is to provide the means to exploit as completely as possible the whole test dataset. Following this introduction, a chapter will be devoted to constitutive equations. Without entering in the entire justification of the forms which they take up, it will allow us to understand their structure and to envisage the parameters to be determined, which are more or less numerous according to the complexity of the problem to be solved. Then, various mechanical tests will be discussed in turn, beginning with those, which are mostly used for the determination of the parameters of these constitutive equations and finishing with those which are specifically intended for the study of damage and fracture.

Chapter 2

Constitutive Equations

2.1. Introduction The constitutive law of the material is an essential ingredient in any structural design analysis. It provides the indispensable relation between strains and stresses, a linear relation in the case of elastic analyses (Hooke’s law), and a much more complex non-linear relation in inelastic analyses, involving time and additional internal variables. This book is limited to the traditional continuous medium approach, i.e. the representative volume element (RVE) of the material is considered under quasiuniform macroscopic strain or stress. This continuous medium hypothesis amounts to neglecting the local heterogenity of stresses and strains within the RVE, by working on averaged quantities, the effects of the heterogenities operating only indirectly through a certain number of internal variables. Furthermore, within the framework of the local state method of thermo-mechanics of continuous media, it is assumed that the state of a material point (or of its immediate neighborhood in the sense of RVE) is independent of the state of the neighboring material point, and that the stress or strain gradients do not operate in the constitutive equations. This hypothesis is obviously questioned in recent theories on the mechanics of generalized continuous media, which will not be addressed here.

Chapter written by Jean-Louis CHABOCHE.

6


This entire study is limited to quasi-static movements, which are considered to be sufficiently slow, within the framework of small perturbations (small strains, less than 20% for example). Furthermore, the indicated laws will be formulated without introducing the influence of temperature (though it can be very significant in some cases). In other words, in accordance with common practice in the determination of the constitutive equations of solid materials, a constant temperature (uniform in the RVE) will be assumed. The influence of the temperature will operate only through the evolution of the material parameters defining the constitutive equations. Nothing will be said about the linear elasticity law, three-dimensional Hooke’s law (see Chapter 3), or about viscoelasticity, whether linear or not. Thus, the book focuses more directly on metallic materials, with elastoplastic or elasto-viscoplastic behaviors, even if, in a certain manner, viscoelasticity, i.e. the influence of viscosity on elasticity, could be modeled based on a viscoplastic model. Therefore, among the effects to be considered will be: irreversible deformation, or plastic deformation, associated hardening phenomena, and the effects of time, whether they occur via the influence of the loading rate or through slow temporal evolutions of the various variables (time recovery for instance). Aging phenomena (associated with possible modifications of the metallurgical structure) and the effects of the damage will be mentioned only briefly. The anelasticity of metals (very low viscous hysteresis in the so-called domain of elasticity), which corresponds to reversible movements of dislocations, will not be discussed either. It may be observed immediately that summarizing constitutive laws in a single chapter will not allow us to be exhaustive at all in terms of the presentation of the various theories, the various specific models, their advantages/drawbacks, or to give all the necessary details. We only hope to provide the indispensable general elements as well as the main types of modeling. The interested reader should refer to more complete specialized books [FRA 91, KRA 96, LEM 85b, MIL 87].

2.2. Fundamental concepts 2.2.1. Domain of elasticity Before discussing inelastic phenomena, it is appropriate to remind the existence (more or less real) of a domain of elasticity, typically the domain of the stress space (Figure 2.1) within which stress changes cause only reversible strains (no plasticity).


7

Figure 2.1. (a) Domain of elasticity in the stress space); (b) definition of the plastic strain and conventional yield limit

In general, this domain is defined in a 6-component space (the six components of the symmetric stress tensor). By convention, it is assumed that the boundary of the domain of elasticity is defined by f V 0 (where V is the stress tensor). If f < 0, the stress state is inside the domain. Of course, the definition of the yield limit is often subjective, or conventional, defined by the loss of linearity of the stressstrain response (through an offset), low for a correct characterization, 10–4 for example, higher for a conventional definition (for example, 0.2% for the yield strength Rp0.2 which would correspond to an irreversible strain of 0.2%). The notion of macroscopic plastic strain already appears, which supposes, in small perturbations, the partition of the total strain (Figure 2.1b):

H

[2.1]

He H p

where H e is the linear elastic strain (proportional to V by Hooke’s law) and H p the irreversible strain or plastic strain (residual strain after an almost instantaneous unloading). Remember that this is a macroscopic definition, of continuous medium type. This plastic strain can be negligible macroscopically and not so at the local scale (within some misoriented grains for instance). If the material is assumed to be isotropic, i.e. without any preferred material direction, it is known that the domain of elasticity can be expressed using only the three invariants of the stress tensor. The criterion that is most commonly used and which provides a very good approximation for a large number of materials is the Von Mises criterion: f

V eq k

V k

3 V ' :V k d 0 2

1 3

V ' V TrV 1

[2.2]

where V ' denotes the deviator of the stress tensor and TrV V11 V 22 V 33 is the first invariant, or hydrostatic pressure. V eq is the equivalent Von Mises stress or

8


second stress invariant. The third invariant does not operate in this criterion. A parenthesis on the notations: the symbol “.” between two tensors denotes the product contracted once ( V ik V kj V ij2 , with Einstein’s summation, represents the square of the tensor V ); the symbol “:” denotes the product contracted twice (for example, scalar V ijV ji =TrV 2 ). In Von Mises criterion [2.2], the parameter k plays the role of yield strength, a parameter dependent on the material and temperature. This criterion, represented in the orthonormal space of the principal stresses (V1, V2, V3), is delimited by a cylinder oriented to the tri-sector. On the deviator plane V1+ V2 + V3 = 0, it is brought down to a circle. In one of the principal planes (V3 = 0) an ellipse is found, illustrated in Figure 2.2. The Von Mises criterion is independent of the hydrostatic pressure, which is consistent with the fact that the plastic strain occurs while keeping the volume constant, at least in metals. In the case of an anisotropic material, the Von Mises criterion is generalized by the Hill criterion: f

V

H

k

V : H :V k d 0

[2.3]

where H is a fourth-order tensor respecting material symmetries. If the material is orthotropic (for example, rolled sheet, composites, etc.), the tensor H displays only 9 independent constants (in fact 8 as, giving directions, it can be normalized, and most often, assuming iso-volume, reduces to only 7 constants). Figure 2.2 gives the example of the orthotropic Hill criterion determined for a Zircalloy alloy, and the comparison with the Von Mises criterion. 2.2.2. Hardening Hardening is the manifestation of the evolution of the domain of elasticity with prior plastic strain. Normally, after tensile loading, an increase of the yield strength can be observed: a new plastic flow will occur only at a stress level close to the one previously applied for pre-deforming. In the same manner, during a tensile test, to increase the plastic strain, an increase of the stress is necessary. This phenomenon is associated with the inhomogenities of the material existing at various scales. For instance, for a polycrystal, some misoriented grains will be deformed plastically and others will not. This inhomogenity of plastic strain at the micro-scale can be accommodated only by elastic strains, thus (internal) stresses, which results in global hardening. A very simple convincing model about the source of this effect consists of an assembly of skids and springs associated in parallel. Each of the branches is elastic-plastic (without hardening) but their parallel functioning causes


9

the kinematic hardening effect. At lower scales, the initial inter-granular inhomogenities (inclusions, defects, etc.), as well as those produced by the displacements of the dislocations, the increase in the density of dislocations and the creation of substructures (walls, cells, twins, etc.), will also contribute to the hardening by globally similar effects (here this can be a combination of isotropic hardening and kinematic hardening).

Figure 2.2. Example of a plastically anisotropic material (Hill criterion)

The terms isotropic hardening and kinematic hardening have been mentioned. What do they mean exactly? Isotropic hardening refers to an increase of the yield limit identical in all the directions (in particular, an initially isotropic material remains isotropic). In the stress space, this corresponds to an expansion, or proportional transformation (illustrated in Figure 2.3 for an initially isotropic material). Just one scalar parameter is sufficient to express this change, the increase R of the yield strength: f

V

H

[2.4]

Rk d 0

It can be shown that the state variable associated with the (additional) stress R is the cumulated plastic strain p, defined by integration of the rate modulus: p

H p

H 1

where p

2 H p : H p in the isotropic case. 3

[2.5]

Thus, p is the length of the plastic strain path (in the plastic strain space). Some theories instead consider the cumulated plastic work W p

³ V : d H

p.

10


Current surface

Initial surface

Figure 2.3. Isotropic hardening: schematic representation in the stress space and under traction-compression

Kinematic hardening refers to a translation of the domain of elasticity in the stress space. Thus, the increase of the yield strength in traction after a tensile predeformation is associated with a decrease (in modulus) of the yield strength in compression, this effect being commonly called the Bauschinger effect. To express kinematic hardening, it suffices to use a second order tensor, X , defining the position of the new center of the domain of elasticity: f

VX

H

k d 0

[2.6]

Figure 2.4 illustrates this transformation in the case of an initially isotropic material and with a linear kinematic hardening.

Initial surface

Figure 2.4. Kinematic hardening: schematic representation in the stress space and under traction-compression


11

Of course, hardening can have more complex manifestations. Apart from the two basic types described above, the evolution of a fourth-order tensor (the H tensor) or

even of tensors of a greater order could be introduced to create distortions of the surface. For practical purposes, the combination of kinematic hardening and isotropic hardening only is used. The rest of the chapter is restricted to the case of an initially isotropic material (with 2 H = I d , the fourth order deviatoric identity 3 tensor, such that V c I d : V ), so that: f

V X Rk

3 V ' X ' : V ' X ' R k d 0 2

[2.7]

It may be noticed that X and R are often called internal stresses. They correspond to the macroscopic manifestation of the existence of internal stresses (or residual stresses) at the microscopic scale, but, by definition, the mean value of those is zero. In fact, X and R effectively transfer the effects of the microscopic internal stresses to the macroscopic scale. It can also be noticed that what has been discussed above is presented within the framework of time-independent plasticity, for which the stress state cannot move out of the domain of elasticity (f d 0). The expression of the viscoplasticity effect will be discussed below. 2.2.3. Normality rules Hill’s maximum work principle, not stated here, implies that in the case of plastic flow, the direction of the rate of plastic strain will be normal to the plastic load surface at the current stress point. In the case of associated plasticity, the (current) boundary of the domain of elasticity f d 0 is the same as the load surface, and consequently:

H p

O

wf wV

On

[2.8]

where n represents the normal to the surface (or otherwise the gradient of the surface f = 0) and O is a scalar multiplier which, in time-independent plasticity, is determined by the consistency condition f f 0. In the case of non-associated plasticity, the two surfaces are not superimposed and more complex descriptions arise, used for instance in soil mechanics. Here, for metallic materials, the

12


framework of associated plasticity is accepted or, more precisely, it is generalized in viscoplasticity by: wǷ( f ) wV

H p

wǷ wf wf wV

p

wf wV

[2.9]

p n

where :(f) is the viscoplastic potential, assumed to depend directly on the function f. In this case, the stress state can move out of the domain of elasticity, with f t 0 (Figure 2.5), and the scalar quantity Vv = f (or part of the stress dependent on the rate) can be called the viscous stress. It can then be noted that all the stress states located on the same equipotential line : = :* have the same viscous stress and the same rate modulus p & H p &, given by [2.5]. Note also that n is not 1, n : n = 3/2.

Figure 2.5. Schematic representation of the domain of elasticity and the viscoplasticity equipotential surfaces

Thus, any stress state can be decomposed in the following manner, in which the function V v ( p ) will be deduced by inverting the relation p w: / wf .

V

[2.10]

X + n( R k V v ( p ))

We may note that consistency between the choice of the Von Mises criterion, independent of the first invariant TrV and the isovolume of the plastic strain (the latter, produced by slips of the atomic planes, at least in metallic materials, is achieved at constant volume). In fact, according to [2.2] and [2.9], we obtain:

TrH p

p Tr

wf wV

p Tr

wf wV '

p Tr

3 V' 2 V

0

[2.11]


13

2.3. Unified theory of viscoplasticity

To simplify, the presentation will fall directly in the scheme of viscoplasticity. The case of time-independent plasticity will be deduced simply in the limit. A relatively general form of constitutive laws will first be given, then the most common specific choices for the viscosity function and for isotropic and kinematic hardening will be considered. Thereafter, the case of time-independent plasticity will be covered and a few indications on the determination, based on experiments, of the parameters of the equations will conclude. 2.3.1. General form of the constitutive law

First of all, it can be observed that the constitutive law can be stated in the general formal framework of thermodynamics of continuous media. This subject will not be covered here. The interested reader can refer to [CHA 96, GER 73, HAL 75], for example. The expression of the viscoplastic constitutive law essentially comprises two aspects: – choice of the viscosity function (see section 2.3.2), or choice of the viscoplastic potential :, which will operate in the expression of the plastic strain rate (its dependence on the viscous stress) via the normality rule expressed above [2.9]; – choice of hardening laws for the whole internal variables. These are provisionally denoted by aj, (j =1, 2, …, N); they can be scalar or tensorial. The proposed general form deals with a strain hardening term, a dynamic recovery term and a static recovery term: a j

h j ! H p r jD ! a j H p r jS ! a j

[2.12]

The first term produces an (increasing) evolution of aj with plastic strain. The second, on the contrary, expresses the restoring force, or evanescent memory, but it still interplays (instantaneously) with plastic strain, hence the term “dynamic recovery”. The third is qualified as static recovery or time recovery as it can occur independently of any plastic strain. This is very clear in an incremental expression such as da h d H p r D a d H p r S a dt . The functions hj, r jD , r jS need to be defined (see below). It can be noticed at once that the static recovery mechanism is thermally activated and that the influence of temperature in the functions r jS plays a crucial role. Schematically, this term allows expressing the effects of thermal agitation, inducing mechanisms of climb of dislocations and their annihilations, or even in some cases, effects of recrystallization. We can also note a strong similarity

14


with equations of physical origin [EST 84, GAR 65, KOC 76], relying on the dislocation density U , for example following Estrin [EST 96], in uni-axial: dU

M (k0 k1 U k2 U )d H p r S ( U , T )dt

[2.13]

2.3.2. Choice of viscosity law

This relation between the viscous part of the stress and the modulus of the plastic strain rate is most often highly non-linear. Thus, in a large range of rates, it can be approximated by a power function: p

f D

n

Vv

n

[2.14]

D

The < . > bracket is used here to ensure that when f < 0, i.e. within the elastic domain, p cancels continuously. This expression refers to the Norton law (or the Odqvist law in three dimensions) for secondary creep, when hardening is ignored (see Chapter 7). The exponent n varies according to the materials, according to the strain rate domain considered and according to the temperature, from a purely theoretical value n = 1 for diffusion creep of a perfect alloy, to sometimes very high values when coming closer to the domain of low viscosity of the material (at low temperatures). In practice, 3 d n d 30 is most often considered. The advantage of expression [2.14] is that it can be derived easily from the viscoplastic potential: :

D Vv n 1 D

n 1

[2.15]

For certain materials, a saturation effect of the influence of the rate is observed in the field of high rates. Figure 2.6 shows the example of 316L stainless steel at 550°C. The field of intermediate rates, where the relationship between log10 V v and log10 H p appears approximately linear, with a slope of n = 24, extends to low rates by a sharp fall in the stress (owing to the phenomena of static recovery that will be studied further below) and by a saturation of the stress at high rates, between 10-3 and 10-1 s-1. Various expressions have been proposed to express such a saturation effect in the viscosity function. They are given below, not in any order, without other explanation. The interested reader can refer to [CHA 89a, 89c]:

p

n ª § V · n 1 º § Vv · exp «D ¨ v ¸ » ¨D¸ © ¹ «¬ © D ¹ »¼

p

D ª §V · º q «sinh ¨ v ¸ » © D ¹ »¼ «¬

n

[2.16]


p

§ Vv · ¨D¸ © ¹

n

ª § V ·D º «1 ¨ v ¸ » «¬ © q ¹ »¼

p

ªV º q exp « v 1» D ¬ ¼

15

[2.17]

In all these expressions, the exponent n is highly temperature-dependent, the viscosity phenomenon being thermally activated (n becomes low at high temperatures). Another, not incompatible, manner of defining the viscosity function is given by the Zener-Hollomon formulation [ZEN 44], which combines the influence of temperature and the influence of velocity in a single master curve. This approach consists of writing: § Vv · p T T Z ¨¨ ¸¸ © DV o T ¹

[2.18]

where Z is a unique monotonic function to be defined and where T T and V o T are two functions of the temperature to be defined. The advantage of this formulation, illustrated in Figure 2.7 [FRE 93], is to avoid the high non-linearity of the power function in which the exponent is highly temperature-dependent. The function Z being defined for a large number of decades in rate (24 for instance), the role of the function T T is then to shift by normalization the useful rate field (in practice, limited to 6 to 8 decades in rate). The equivalent exponent (the slope of the Z function in the bilogarithmic diagram) thus moves from a very low value in a certain area of the curve (low values of p / T T ) to a very high value in the opposite area (high values of p / T T ).

Figure 2.6. Viscous stress – plastic strain rate relationship: 316L stainless steel (17-12 SPH) at 600°C

16


2.3.3. Isotropic hardening laws Considering the expression of the modulus of the strain rate [2.14], by replacing f with [2.7], we obtain:

p

V X Rk

n

D

[2.19]

Figure 2.7. Behavior in stationary creep: (a) data on aluminum; (b) data on copper (reproduced from the International Journal of Plasticity, Volume 9, A. Freed & K.P. Walker, with the permission of Elsevier Science Ltd, Kidlington, UK)

Three possibilities to express an isotropic hardening are noted: – through the variable R, by increasing the size of the domain of elasticity; – by increasing the drag stress D; – by a coupling with the law of evolution of the kinematic variable X . In the first two cases, which alone are considered here, it suffices to define the biunivocal function of the dependence between R (or D) and the state variable of the


17

isotropic hardening, namely, the cumulated plastic strain p (possibly the cumulated plastic work Wp). R

R p

D

D p

[2.20]

One possibility is to assume the two evolutions to be proportional. The R p function alone then needs to be defined and from it can be deduced: D p

K ] R p

[2.21]

where K is the initial value of the drag stress and ] is a weighting parameter. A special case corresponding to the Perzyna law [PER 64] is the one obtained with K = k and ] = 1. By the decomposition of the equivalent Von Mises stress (in the case without kinematic hardening, X = 0), the different roles of the two types of isotropic hardening can be noticed:

V eq

k R p V v p , p

k R p D p p 1/ n

[2.22]

In the first case, with R, the increase of the domain of elasticity will take place identically irrespective of the strain rate; in the second case, the increase in D will result in an increase of V eq which will be all the more significant the higher the strain rate. A form of viscoplasticity law with isotropic hardening, the simplest and often used law, is deduced from the combination of the law of secondary creep (Norton law, with a power function between the secondary creep rate and the applied stress) and the law of primary creep (power relationship between strain and time) [LEM 71, RAB 69]. This consists of ignoring in [2.19] any domain of elasticity (k = 0), the corresponding hardening R p , and in assuming a power function for the drag stress D. Hence:

V eq

Kp1/ m p 1/ n

[2.23]

This multiplicative form of hardening is very easy to determine [LEM 71] and provides good results in a relatively large range, at least for quasi-proportional monotonic loadings.

18


2.3.4. Kinematic hardening laws

Kinematic hardening being a quite general manifestation, at least in the domain of small strains, it will be needed to resort to the corresponding models as soon as it is desirable to express correctly either non-proportional monotonic loadings (variation of the direction of the loading or variation of relationships between independent loadings, or thermo-mechanical loadings, etc.) or cyclic loadings. The most common kinematic hardening models are indicated here in an increasing order of complexity. More advanced models for the expression of particular effects will be discussed in section 2.4.3. At present, strain hardening only will be dealt with, the effects of time recovery being discussed in section 2.3.6. The simplest model is Prager’s non-linear kinematic hardening [PRA 49], in which the evolution of the kinematic variable X (called back-stress) is collinear to the evolution of the plastic strain. Thus: X

2 CH p 3

or

X

2 CH p 3

[2.24]

The associated linearity of the stress-strain response (Figure 2.4b) is rarely observed (except perhaps in the significant strains regime). The model initially proposed by Armstrong and Frederick [ARM 66], introducing a feedback term, called dynamic recovery, gives a better description: X

2 CH p J Xp 3

[2.25]

The feedback term is collinear to X (as in general equation [2.12]) and rate. The evolution of X , instead of proportional to the modulus of plastic strain saturation being linear, is thus exponential for a monotonic uni-axial loading, with (suppression of the hardening) for a value C/J. In fact, the integration of [2.25] depending on H p , for uni-axial loading, gives: X

v

C

§ C· ¨ X o v ¸ exp vJ H p H po J © J ¹

[2.26]

where v = ±1 gives the direction of the flow and where X o and H po are the values of X and H p at the start of the loading branch considered.


19

For a strain controlled cyclic loading, it is demonstrated that the stabilization takes place when Xmax + Xmin = 0: 'X 2

§ 'H p · tanh ¨ J ¸ J © 2 ¹

C

Xo

[2.27]

Figure 2.8 shows the example of a few materials, treated as time-independent, in which the cyclic curve (see Chapter 7) is described with [2.27] and 'V / 2 'X / 2 k . A better approximation consists of adding several models such as [2.25], with significantly different spring constants J i (5 to 20 factors between each of them): M

X

¦ X i 1

i

X i

2 Ci H p J i Xi p 3

[2.28]

which helps to express a more extended strain range, ensuring a better description of the smooth transition between elasticity and the beginning of the plastic flow. Figure 2.8 shows the significant improvement obtained in the case where only two variables are superimposed, one being linear, with J 2 0. Other more complex combinations can be used [CAI 95] instead of [2.28], but they do not provide an analytical uni-axial expression.

Figure 2.8. Cyclic curves of a few materials and interpretation using the non-linear kinematic hardening model

20


2.3.5. Cyclic hardening and softening

Within the framework of kinematic hardening, isotropic hardening is used to express the evolution of the mechanical resistance of the material to plastic flow. This phenomenon of cyclic hardening (increase in resistance) or cyclic softening (decrease) is relatively slow, typically taking between 10 and 1,000 cycles of Hp = ±0.2% for example before stabilization. We can vary the dimension of the domain of elasticity, with a law such as:

R

cp g Rp

[2.29]

which is the direct transposition of [2.25] to isotropic hardening, c and g being two coefficients dependent on the material and on temperature (g will range between 50 and 0.5 to ensure the typical saturation mentioned above in 10 and 1,000 cycles respectively). The integration of [2.29] yields an expression R p c / g 1 exp gp , which can also be used in the context of monotonic loadings (but a significantly higher value for g is then needed). Note that in the case of cyclic softening, c < 0 can be chosen. Note also that the drag stress D can be used instead of the threshold stress R, or the two combined, or a coupling between kinematic hardening and isotropic hardening can even be introduced [MAR 79], with a function I (p) to be defined: X i

2 Ci H p J iI p Xi p 3

[2.30]

A possible choice of I(p) consists of using the variable R, with a dependence that is deduced from an endochronic theory [CHA 91b, VAL 80, WAT 86]: I p 1/ 1 Z R p . 2.3.6. Static recovery

The recovery of hardening over time, whether kinematic or isotropic, will generally occur at high temperatures. These mechanisms, thermally activated, are described macroscopically by relations such as [2.12]. Thus, for kinematic hardening, power functions for instance will be used in the feedback term operating according to time: X i

J § Xi · 2 Ci H p J i Xi p i ¨ ¸ W i T ¨© M i ¸¹ 3

mi 1

Xi

[2.31]


21

where mi, Wi, Mi depend on the material and temperature. In practice, Mi = Ci/Ji will be chosen and the time constant Wi will be highly time-dependent. For the restoration of isotropic hardening, any function can be used, for instance [CHA 89b, NOU 83]: R

cp g Rp J r R Qr

m 1

R Qr

[2.32]

which provides accurate results for 316L stainless steel. Figure 2.9 illustrates this, using results from cyclic relaxation tests in strain control ('H = 1.2%), in which a shorter or longer hold time under traction was incorporated. The greater the hold time, the lower the stabilized maximal stress, which demonstrates a decrease in the effect of cyclic hardening, achieved by the compromise of relation [2.32] between strain hardening (the first two terms) and time recovery (the last term). Furthermore, the relaxed stress (difference between the maximum value and the value Vrel after relaxation) decreases significantly, which also requires taking into account the static recovery of kinematic variables with [2.31], whose parameters were identified by long duration creep tests [CHA 89b]. 2.3.7. Time-independent limit case

So far the case of viscoplasticity has been considered, with a part of the stress that is rate-dependent (relations in section 2.3.2). When the temperature is sufficiently low, the viscosity effect can be ignored. For some applications, even at high temperatures, the time-independent plasticity scheme can be preferred.

Figure 2.9. Modeling of cyclic relaxation tests on 316L stainless steel (17-12 SPH) at 600°C: ı max: maximal stress at the stabilized cycle; ı rel: stress after relaxation; tH: hold time

22


For this, it is only necessary to make the viscous stress tend towards zero, by decreasing the value of the drag stress ( D o 0) in an expression like [2.14] or [2.22]. It necessarily follows that ıv ĺ 0 and the criterion f d 0 will be automatically met. Of course, in an expression like [2.14], an indetermination (0/0) is reached, but it can be eliminated by a consistency condition f f 0 in the case of plastic flow ( f is indeterminate in pure elasticity, i.e. if f < 0). The formal treatment of timeindependent plasticity is a little more complex than that of viscoplasticity, introducing a loading-unloading condition and additional difficulties when the material has negative hardening. These aspects will not be discussed here. The laws of monotonic or cyclic viscoplasticity, with the associated hardening models, simply degenerate into the time-independent case, without any other change than the dimension of the pure domain of elasticity (see section 2.3.8). We may mention the special case of isotropic hardening, for which relation [2.22] becomes:

V eq

k R p

[2.33]

Quite often in the applications, the relation R p can be considered as defined point by point, based on the expression V

k R H p , which is equivalent in the

uni-axial case. This function is then directly drawn from the experimental tensile curve. Quite often it can be assimilated into a power function:

V eq

k Kp1/ m

[2.34]

2.3.8. Methods of determination

The identification of unified viscoplasticity models, combining isotropic hardening, kinematics, viscosity and recovery effects, may be relatively difficult. Here, a step-by-step determination approach, which has often proven applicable, is proposed. 2.3.8.1. Determination of hardening laws within independent time-scheme It is assumed that monotonous and cyclic uni-axial tests are available, for example low cycle fatigue tests up to the stabilized cycle, with ı – İp recordings (see Chapter 6). It is also assumed that these tests are achieved for rather constant ( H | Cte) and relatively high (for example H = 10-3 or 10-2s-1) rates. Starting from the cyclic curve, considering that H p | H | Cte at cycle maxima, the following


23

relation will be identified; this will be valid after cyclic hardening or softening effects: 'V 2

M

k * RS

i 1

§ 'H p · tanh ¨ J i ¸ 2 ¹ ©

Ci

¦J

i

[2.35]

in which k* represents the sum K Ds p 1/ n , assumed to be pretty much constant. Rs is the stabilized value of R (normally Rs = c/g), and J i possibly takes into account the isotropic hardening coupling ( J i is replaced with J iIsat ). In practice, if the number of kinematic models is sufficient (3 for example), k* + Rs will be adjusted in order to have the weakest possible value. It will then be possible to suppose the third variable as linear, and the cyclic curve slope in the strongest amplitudes area (2-3%) will supply the value of C3. Subsequently, the determination of (time-independent) hardening laws is completed according to the available data in monotonous tensile loading, or possibly in compression loading (see Chapter 4); the corresponding experimental curve must be expressed by: M

V

k * R p

C ¦ J 1 exp J H i 1

i

i p

[2.36]

i

The isotropic hardening rate coefficient g will be supplied by the number of cycles necessary to saturate the cyclic hardening or softening with fixed amplitude 'H p / 2 : 2 gN 'H p | 5 is a good criterion for exponential function saturation. A more precise manner is that of tracing out the succession of normalized maxima (Vmax (N) – Vmax (0))/ (Vmax (Nsat) – Vmax (0)) according to p | 2 N 'H p for some low cycle fatigue tests. An iterative treatment of the set of these data, with some readjustments, provides k*, Ci, Ȗi, c, g (and function I ( p )) . 2.3.8.2. Determination of the viscosity law The data in the variable strain rate domains H p between, say, 10-6 s-1 and 10 s , is now used in order to determine the viscosity law, for example exponent n, constant K, and final value k of the true domain of elasticity. The following necessity of readjustment is noted: -2 -1

k * o k Kp *1/ n

[2.37]

24


R p o ]p *1/ n R p

[2.38]

between the already determined version in the time-independent approximation (with a rate pretty much equal to p * ) and the complete version, taking into account the choice [2.21] for the isotropic hardening associated with the evolution of the drag. If monotonous or cyclic relaxation tests are achieved, the determination of n and K will be greatly facilitated by the possible use of a graphic determination method [LEM 85b]. Certain iterations are necessary in order to reach a satisfactory solution (in all these data processing, the parameters determined in stage 1 are taken into account). 2.3.8.3. Determination of static recovery effects Available data is used with a very low strain rate ( H p 106 s 1 ), in creep tests or in long-term relaxation tests. As illustrated in Figure 2.6 for the 316L steel, the influence of the recovery mechanism appears to be directly visible through the appreciable reduction of the stress supported for a given strain rate. By successive approximations, while all the other parameters remain fixed, the static recovery parameters of the envisaged models (mi, Wi, Qr, mr, Jr) are relatively easily obtained. If specific recovery tests are available, these effects and the corresponding parameters will be much more directly measured. For example, such tests consist of normal cyclic loading, until stabilization, then a partial unloading and a hold time, in temperature, of significant duration (100 hours, for example), then cyclic loading again. Recovery must be achieved at a sufficiently low, but possibly non-zero, strain or stress level, chosen in such a manner that partial recovery of the plastic deformation cannot be produced. The comparison and identification of responses before and after recovery thus furnish, quite directly, the values of the sought parameters. 2.3.9. Other unified approaches

For the past 20 years, numerous unified theories have been developed [BRU 94, DEL 88, FRE 88, KRE 86, MIL 76, ROB 83, WAL 81]. They have almost the same ingredients in common, the essential differences being the choice of expressions of functions, for example the viscosity function. The kinematic hardening form, with a hardening term, a dynamic recovery term and a static recovery term, is almost a constant feature, save for Bodner’s approach [CHA 89c], who rejects, despite experimental evidence, the very notion of kinematic hardening. Other minor differences may be signaled: – Robinson’s model [ROB 83] presupposes a non-linear relation between the kinematic hardening module and the kinematic variable itself (power function). This procedure, which implies a biunivocal relationship between the plastic strain and the


25

hardening variable, requires a change of the origin to be made at each cycle change, by producing discontinuities in the tangent moduli, which is of delicate numerical treatment; – Krempl’s model [KRE 86] considers a direct hardening term (the first term of relation [2.12]) proportional to the total strain rate, instead of the plastic strain rate. Although it presents a certain advantage for the description of ratcheting phenomena, this approach is considered to be inadmissible from the thermodynamic point of view [CHA 93, 96]; – Miller’s model [SCH 81] involves a dragging phenomenon by the atoms in solution in the law of isotropic hardening, a phenomenon which corresponds to dynamic aging and to the Portevin-Le Chatelier effects; – other ways of expressing non-linear kinematic hardening have been proposed, involving the simultaneous use of two surfaces (plastic load surface and yield surface). Let us mention the Dafalias and Popov model [DAF 76] in a timeindependent plasticity context and McDowell’s work [MDO 90] in a more general framework. 2.4. Other types of modeling 2.4.1. Plasticity-creep partition

This is the oldest way of describing plasticity and viscosity phenomena simultaneously by adding two independent inelastic deformations. Equation [2.1] is then replaced by:

H

[2.39]

He H p Hc

First of all it is observed that the plastic strain H p of the preceding sections contained both plasticity and creep effects in a unified manner. On the contrary, here it is considered that they are dissociated and, in general, independent. The description of the evolutions of the two inelastic deformations will thus be made: – by means of time-independent plasticity theory for H p , with a normality rule are convenient to the such as [2.8], and in association with the hardening laws that envisaged type of application: isotropic hardening for the applications under quasimonotonous loading, and kinematic hardening or combinations of the two for the applications under cyclic loading, or when non-proportional multi-axial effects can operate. Without explaining them further, they can be formally written: a jp

h jp H p r jp a jp ,! a jp H p

[2.40]

26


The interest of the dissociation between plasticity and creep is that of enabling a simple determination of the material parameters starting either from the monotonous tensile curve, or from the cyclic curve (stage 1 of section 2.3.8); – by means of a creep-type law for H c , incorporating primary creep and secondary creep, or in an integrated form such that:

H ceq V eq , t

A1 V eq t1/ p A2 V eq t

[2.41]

or in a differential form, which is more correct because it implies strain hardening:

Hceq

g c V eq , H ceq

[2.42]

for which it is also possible to again use a multiplicative-type hardening law such as [2.23], replacing p by Hceq. It is also possible to adopt a form similar to [2.12], with additional hardening variables (which combine isotropic and kinematic hardenings): a cj

hcj Hc r jc a cj ,! a cj

[2.43]

This form of evolution law combines strain hardening and static recovery (time effect, which is important in creep). In addition, in this case, regardless of the chosen form of hardening law, the dissociation with plasticity allows a quite simple determination, starting either from pure creep tests or from relaxation tests. This method by means of partition of inelastic deformation has been currently used, up to the point of experimental evidence, which has been reported on numerous occasions, consisting of an obvious association between plastic and creep deformation, by means of associated hardening effects. Figure 2.10 shows schematically the type of observation made on a high rate tensile test, interrupted by a long-duration creep period (constant stress). Clearly there is a quasi-immediate lapse of the creep period and experimental evidence regarding the fact that hardening is correlated with the sum Hp + Hc (and not defined in an independent manner with Hp or Hc). This type of observation, like many others pointing to a similar conclusion, has led to the development of non-unified approaches, but with coupled hardening effects [CAI 95, CON 89], by writing, for example:

a jp

h jp H p h jpc Hc r jp a jp ,! a jp

a cj

c c c c hcp j H p h j Hc r j a j ,! a j

[2.44] [2.45]


27

with all types of possible variations. These approaches are relatively seldom used because they present complications or difficulties of determination that are analogous to those of the unified theories.

Figure 2.10. Schematic results of tensile-creep-tensile tests in comparison with a constant strain rate tensile test

2.4.2. Methods by means of micro-macro transposition

These methods consist of resorting to basic laws of crystalline plasticity, by inscribing directly in the modeling the various slip systems that can be activated for the various grain directions under consideration within the RVE of the polycrystal. Figure 2.11 gives the functional diagram of such an approach, being limited (formally) to a situation of imposed macroscopic stress V (diagram realized on a time increment); the output is the macroscopic plastic strain H p . The method stages: involves the intervention of two localization stages and two averaging macro l grain “g” l slip system Here Cailletaud’s [CAI 92] and Pilvin’s [PIL 94] formulation is followed; it is easy enough to use, but at the same time is sufficiently rigorous (even if it maintains a marked phenomenological character).

28


Figure 2.11. Functional diagram of a model for polycrystalline materials with a constitutive law written on the slip systems of each grain (on average)

The passage from the macro level to that of the (average) stress in each grain is achieved by the following Kröner-type [KRÖ 61] localization rule, but corrected, valid for a polycrystal with grains of identical nature and with macroscopically isotropic elasticity:

V g V

2 P a E g B

[2.46]

where μ is the shear modulus of elasticity, a an adjustment parameter, E g a state variable for each grain, which is analogous to the average plastic strain of the grain H pg , and B the corresponding average value:

V

¦ c V g

g

g g p

¦ c H

Hp

g

g

B

¦ c E g

g

[2.47]

g

where cg is the concentration corresponding to each direction under consideration. Kröner’s elastic localization rule, which is known to be too rigid [ZAO 93], would consist of replacing E g with H pg and B with H p . The originality of Cailletaud’s and Pilvin’s approach is that of adapting this rule, so that it is quasi-elastic in the regime of low plastic deformations, in order to tend towards a tangent rule for the higher strains. This is supplied by the following evolution law for E g , which is very similar to a non-linear kinematic hardening law (combined to linear hardening):

E g

H pg D E g GH pg

H pg

[2.48]


29

D and G being global adjustment coefficients, which serve to make the model close to a self-consistent type model, starting from numerical finite element analyses [CAI 94, PIL 97]. The transposition at the slip system level is performed, for example, with the stress resolved for a slip in the plane of normal n s and in the direction i s :

Ws

V g : ns is

nisV ijs l sj

[2.49]

where n s and i s are the normal to the slip plane and the slip direction respectively. Although it is formally present, the index g of the grain is omitted for the quantities associated with the slip system. The use of Schmidt’s law leads to a yield criterion on each system: fs

W s xs r s k d 0

[2.50]

The slip velocity on the system s is given by a viscoplastic law. Note that this consists of the sum on the grain of all the slips of the same direction: n

fs K

J s

Sign W s x s

[2.51]

xs being a kinematic hardening scalar variable associated with each system and expressing the presence or development of intra-granular non-homogenities (precipitates, inclusions, walls, dislocation cells, etc.). Its evolution law obeys the non-linear kinematic format already mentioned several times: x s CJ s dx s J s

[2.52]

rs being a scalar variable expressing the variation of the criterion dimension on each system. It obeys a non-linear law by implying interactions between various systems; this law is of the type: r s

¦h

sr

r

exp qJ r J r

or other similar versions, where J r is the cumulated slip on the system r (J r

[2.53]

J r ) .

In addition, hsr is the interaction matrix, built according to crystallographic data and knowledge on the physics of slips (it has the dimension of the number of the

30


directions of the systems that can be activated in a grain, 12 for octahedral, 6 for cubic, a total of 18 for a FCC grain, etc.). We observe that the material coefficients k (initial threshold of the Schmidt criterion), n, K, C, d, q may depend on the type of system to which the corresponding relation is applied (octahedral, cubic or others for the non-FCC media). Once the quantities Ȗs are defined, by integration on a time increment, it remains to return to the average plastic deformation of the grain:

H pg

¦ J n s

s

is

sg

[2.54]

then to the macroscopic plastic strain by the third term of [2.47]. The functional diagram of controlled macroscopic deformation is the same, although it requires an iterative resolution, with the law of macroscopic elasticity V C : (H H p ) . NOTE.– Although this is a micro-mechanical approach guided by physical mechanisms, the model thus developed could also be qualified as macroscopic, being part of the multi-criteria-type of approaches [MAN 65]. In fact, the two localization rules applied remain quite close and they refer only to the average quantities on each grain, without precise geographic localization. In fact, we can assimilate the above set of equations to the use of an f gs criterion:

H

p

X gs

¦c g ,s

f gs s

n

K gs

§ x gs m gs 2 PD ¨ E g ¨ ©

f gs

¦ g'

m gs : V X gs

· cg' E g ' ¸ ¸¹

[2.55]

[2.56]

where m gs n s i s for the slip systems of grain g. This approach is included in the class of connected multiple criteria [MOR 69]. 2.4.3. More advanced hardening laws Returning to the framework of macroscopic and phenomenological approaches, there are a certain number of hardening models that are more advanced, but also more complex, and which are worth mentioning.


31

Multi-kinematic models with thresholds: these have been proposed in order to improve the description of ratcheting or progressive deformation effects for loadings implying a significant mean stress (or primary stress). The aim is to describe the existence of a shakedown limit or threshold stress, below which the ratcheting effect is brought to a standstill and reaches plastic shakedown. This consists of introducing a strongly non-linear factor, or a factor with threshold, in the dynamic recovery term. For example [CHA 91a]:

X i

X 2 Ci H p J i 1 li 3 Xi

mi

Xi p

[2.57]

where Xli plays the role of shakedown limit (on each variable): when Xi is low,

Xi d X li , the corresponding kinematic model, behaves in a linear manner (with high hardening module Ci); beyond that, we find a behavior analogous to that of standard model [2.28], with a hardening module which tends towards zero when deformation increases. This makes it possible to combine the properties of Prager’s linear hardening (which completely stops the ratcheting) with those of non-linear kinematics (which often predicts a ratcheting which is too rapid). Slightly different versions have been proposed since then by Ohno and Wang [OHN 93]:

X i

§ Xi · 2 Ci H p J i ¨ ¸ ¨ a ¸ 3 © i ¹

mi

Xi : H p Xi

[2.58]

or more recently by Jiang [JIA 94]. These models provide pretty much the same possibilities, with the same operating domains and the same determination difficulties [CHA 94]. Memory models enable us to express the dependence between the cyclic hardening extend and cyclic plastic strain amplitude suffered by the material. This dependence, which is quite strong in low-energy stacking-fault materials, such as annealed copper or stainless steel, requires the use of additional hardening variables or memory variables [CHA 79, OHN 86], expressing the stability and persistence of certain dislocation sub-structures. It is also possible to express, in macroscopic models, the cyclic over-hardening effects induced by non-proportional multi-axial loading trajectories [CAL 97] (see Chapter 8). Among further possibilities can be mentioned the model developed by Teodosiu and Hu [TEO 95], in which the limit values of kinematic hardening variables and the coupling effects with isotropic hardening are introduced in a specific manner, which is a little more complex, but justified by physical considerations implying the dislocation sub-structures.

32


Discrete surfaces models generalize the notion of two-surface models (load surface and limit surface). The oldest, Mroz’s model [MRO 67], developed in timeindependent plasticity, reproduces in multi-axial mode a behavior analogous to that of an assembly of basic shoes + springs models (or otherwise a multi-layer type of behavior). More elaborate versions were developed later [CHA 89a; MRO 81, RIS 95], which enable us in a discretized manner to associate memory surfaces with the successive peaks of cyclic loadings, which can be proportional or not. 2.4.4. Aging In the preceding section only stable materials, for which the micro-structural evolutions were negligible or mechanically imperceptible, were considered. The temperature effect occurred in the constitutive equations, but in a biunivocal manner, for example by means of the dependence of material parameters according to temperature. On the contrary, under certain temperature conditions, metallurgical evolutions can be produced, phase changes, dissolutions, precipitations, etc., which modify the mechanical behavior significantly. The generic term aging covers all unstable situations, distinguishing the following types: – dynamic aging is connected with dragging of atoms in solution by dislocations; it leads to an inverse relation in rate (exponent of viscosity that would be negative in a certain rate domain). This non-biunivocity of the relationship between ıv and p is a source of instabilities (dislocation avalanches localized in bands), in particular those of the Portevin-Le Chatelier phenomenon, for which the very notion of RVE and mechanics of continuous media becomes debatable at the macroscopic level; – static aging is the increase in time of the resistance of the material (from the mechanical response point of view, it is the inverse of static recovery); it can be expressed by an equation of type dR = h(.)dt. This phenomenon is visible, for example, on certain aluminum alloys at ambient temperature. More or less sophisticated mechanical models for this phenomenon have been proposed [EL 94, MAR 89]; – phase changes appear during thermal treatment or sometimes in service. In terms of models seeking to express the mechanical consequences of these phenomena, Cailletaud’s model [CAI 79] should be mentioned to express the dissolution-precipitation-growth of Ȗ' precipitates in super-alloys for turbo-machine blades; these phenomena occur under certain temperature cycles. This model uses two additional state variables, one connected to the volume fraction of the precipitates, the other to their size. It is obviously impossible to go further in the presentation of these phenomena and various modeling possibilities.


33

2.4.5. Damage Another type of phenomenon can have an influence on the mechanical behavior: damage or progressive deterioration of matter, by cleavage or de-cohesion of inclusions, by nucleation followed by growth and coalescence of cavities, either around the inclusions or at the grain boundaries in a high-temperature creep mechanism, by initiation and propagation of micro-cracks in fatigue, etc. In a certain number of situations, it may be necessary to take into account the effect of damage on the material constitutive equations: for example, in analyses of lifetime in creep, or for a correct description of ductile fracture of metallic parts (see Chapter 10), or even for the description of the behavior in many composites, etc. Among the different types of approaches that have been developed, the following are mentioned in particular: – continuum damage mechanics (CDM), initiated for the creep of metal by Kachanov [KAC 58] and Rabotnov [RAB 69]; it is now used in many situations and with all sorts of materials. It is practically a prerequisite for a correct macroscopic description of brittle materials, such as concrete, and for numerous types of composites, in which the non-linearity of the stress-strain response (in uni-axial traction or under other types of solicitations) is essentially due to de-cohesions and micro-cracks at the re-inforcements scale, with quite significant reductions in the material modulus of elasticity. Damage mechanics has also been developed for ductile fracture [LEM 85a], fatigue, and creep of metals [LEM 85b]. It is underlain by the notion of effective stress, expressed here by an isotropic and scalar damage, d.

V

V

1 d

[2.59]

The effective stress V is the one that should be applied to the virgin (non that it should produce the same deformations as those damaged) material in order observed on the already damaged material submitted to a stress V . This concept makes a coherent macroscopic description of the behavior/damage coupling possible, both in elasticity and plasticity or viscoplasticity; this can be done without having to alter the constitutive equations themselves; – ductile damage models, developed from Gurson’s model [GUR 77], which find an important field of application with ductile fracture of metallic materials submitted to large deformations. The particularity of these models is that they are based from the start on micromechanical analysis of cavity growth, in direct connection with the volume component of macroscopic plastic deformation. Chapter 10 will give further elements. Among the modifications of Gurson’s models, we will mention Rousselier’s model [ROU 80] in France, based on a consistent thermodynamic formalism, which makes it possible to produce quite similar results. Ductile fracture

34


models, or the approaches using damage mechanics for that matter, are confronted with further difficulties in relation to the description of the plastic deformation process and hardening: the dimension of defects, which is often too important to make possible the definition of a representative volume element at the scale of the analysis of the structure. This confusion of scales leads to the introduction of new concepts in the mechanics of generalized continuous media, with non-local approaches or gradient media, which imply characteristic micro-structural lengths. 2.5. Conclusion It is impossible to conclude this introduction on constitutive laws, except perhaps by taking an inventory of the omissions from this chapter: – first, as far as materials are concerned: the focus was put here mainly on the types of constitutive equations in plasticity and viscoplasticity, and on the phenomena taking place in metallic materials. Of course, certain theories and certain aspects have a sufficiently general character and they will be found, with nuances, in other types of materials; – another field is almost absent; this is the field of damage, with its applications to the constitutive equations of brittle damageable materials, concretes, ceramics, composites, etc., but also metallic materials. Similarly, aspects relating to initial or induced anisotropy have merely been touched upon; – the non-elastic behavior of materials submitted to very large deformations has not been considered, nor have applications to metal shaping or to polymer, plastic materials, and rubber behaviors; these require viscoelastic approaches; – constitutive equations in the dynamic regime, in the domains of very high deformation velocities (1 to 105 s-1), even if they can be handled with unified viscoplasticity formalism, have not been considered in a specific manner. There still remains much to say, both on the thermodynamic foundations of constitutive equations and on the way to determine and exploit them in a wide range of temperatures, particularly in the context of cyclic thermo-mechanical loadings. Finally, even if other theories have been invoked or repositioned in relation to those presented in this chapter, it is obvious that this has to do with a certain vision of reality, which is fragmented (perhaps even biased), and that there is still enough room for new theoretical research and applications.


35

2.6. Bibliography [ARM 66] ARMSTRONG P.G. and FREDERICK C.O., A mathematical representation of the multiaxial Bauschinger effect, Report RD/B/N731, CEGB, Central Electricity Generating Board, Berkeley, UK, 1996. [BRU 94] BRUHNS O.T. and ROTT U., “A viscoplastic model with a smooth transition to describe rate-independent plasticity”, Int. J. of Plasticity, 10:347-362, 1994. [CAI 79] CAILLETAUD G., Modélisation mécanique d’instabilités microstructurales en viscoplasticité cyclique à température variable, PhD Thesis, University of Pierre et Marie Curie, Paris 6, 1979. [CAI 92] CAILLETAUD G., “A micromechanical approach to inelastic behaviour of metals”, Int. J. of Plasticity, 8:55-73, 1992. [CAI 94] CAILLETAUD G. and PILVIN P., “Utilisation de modèles polycristallins pour le calcul par éléments finis”, REEF, 3(4):515-541, 1994. [CAI 95] CAILLETAUD G. and SAI K., “Study of plastic/viscoplastic models with various inelastic mechanisms”, Int. J. of Plasticity, 11:991-1005, 1995. [CAL 97] CALLOCH S. and MARQUIS D., Additional hardening due to tension-torsion non-proportional loadings: Influence of the loading path shape, STP 1280, p. 113-130, ASTM, 1997. [CHA 79] CHABOCHE J.L., DANG-VAN K. and CORDIER G., Modeling of the strain memory effect on the cyclic hardening of 316L stainless steel, volume Division L, Berlin, SMIRT 5 1979. [CHA 89a] CHABOCHE J.L., “Constitutive equations for cyclic plasticity and cyclic viscoplasticity”, Int. J. of Plasticity, 5:247-302, 1989. [CHA 89b] CHABOCHE J.L. and NOUAILHAS D., “A unified constitutive model for cyclic viscoplasticity and its applications to various stainless steels”, J. of Engng. Mat. Technol., 111:424-430, 1989. [CHA 89c] CHAN K.S., LINDHOLM U.S., BODNER S.R. and WALKER K.P., “High temperature inelastic deformation under uniaxial loading: theory and experiment”, J. of Engng. Mat. Technol., 111:345-353, 1989. [CHA 91a] CHABOCHE J.L., NAOUILHAS D., PACOU D. and PAULMIER P., “Modeling of the cyclic response and ratcheting effects on inconel 718 alloy”, Eur. J. Mech., A/Solids, 10(1):101-121, 1991. [CHA 91b] CHABOCHE J.L., NOUAILHAS D. and SAVALLE S., “Agice: automatic/graphic identification of constitutive equations”, Rech. Aérosp (Engl. edition 1991), 3:59-76, 1991. [CHA 93] CHABOCHE J.L., “Development of Continuum Damage Mechanics for elastic solids sustaining anisotropic and unilateral damage”, Int. J. Damage Mechanics, 2:311329, 1993.

36


[CHA 94] CHABOCHE J.L., “Modelling of ratchetting: evaluation of various approaches”, Eur. J. Mech., A/Solids, 13(4):501-518, 1994. [CHA 96] CHABOCHE J.L., “Unified cyclic viscoplastic constitutive equations: development, capabilities and thermodynamic framework”, in A.S. KRAUSS and K. KRAUSS (eds), Unified Constitutive Laws of Plastic Deformation, p. 1-68, Academic Press Inc., 1996. [CON 89] CONTESTI E. and CAILLETAUD G., “Description of creep-plasticity interaction with non-unified constitutive equations, application to an austenitic stainless steel”, Nuclear Engng. Design, 116:265-280, 1989. [DAF 76] DAFALIAS Y.F. and POPOV E.P., “Plastic internal variables formalism of cyclic plasticity”, J. of Applied Mechanics, 98:645, 1976. [DEL 88] DELOBELLE P., “Sur les lois de comportement viscoplastique à variables internes”, Rev. Phys Appl., 23:1-61, 1988. [EL 94] EL MAYAS N., Modélisation microscopique et macroscopique du comportement d’un composite à matrice métallique, PhD Thesis, ENPC, 1994. [EST 84] ESTRIN Y. and MECKING H., “A unified phenomenological description of work hardening and creep based on one-parameter models”, Acta Metall., 32:57-70, 1984. [EST 96] ESTRIN Y., “Dislocation-density-related constitutive modeling”, in A.S. KRAUSS and K. KRAUSS (eds), Unified Constitutive Laws of Plastic Deformation, p. 69-106, Academic Press Inc., 1996. [FRA 91] FRANÇOIS D., PINEAU A. and ZAOUI A., Comportement mécanique des matériaux: élasticité et plasticité, Paris, Hermes, 1991. [FRE 88] FREED A.D. and VERILLI M.J., “A viscoplastic theory applied to copper”, in C. OYTANA (ed.), Int. Seminar on Inelastic Behaviour of Solids, p. 27-39, Besançon, MECAMAT, 1988. [FRE 93] FREED A.D. and WALKER K., “Viscoplasticity with creep and plasticity bounds”, Int. J. of Plasticity, 9:213-242, 1993. [GAR 65] GAROFALO F., Fundamentals of Creep and Creep Rupture in Metals, New York, Macmillan, 1965. [GER 73] GERMAIN P., Cours de mécanique des milieux continus, Volume I, Paris, Masson, 1973. [GUR 77] GURSON A.L., “Continuum theory of ductile rupture by void nucleation and growth: Part I: yield criteria and flow rules for porous ductile media”, J. of Engng. Mat. Technol., 44:1-15, 1977. [HAL 75] HALPHEN B. and NGUYEN Q.S., “Sur les matériaux standards généralisés”, J. de Mécanique, 14(1):39-63, 1975. [JIA 94] JIANG Y. and SEHITOGLU H., “Multiaxial cyclic ratcheting under multiple step loading”, Int. J. of Plasticity, 10:849-870, 1994.


37

[KAC 58] KACHANOV L.M., “Time of the rupture process under creep conditions”, Isv. Akad, Nauk. SSR. Otd Tekh. Nauk., 8:26-31, 1958. [KOC 76] KOCKS U.F., “Laws for workhardening and low-temperature creep”, J. of Engng. Mat. Technol., 98:76-85, 1976. [KRA 96] KRAUSS A.S. and KRAUSS K. (eds), Unified Constitutive Laws of Plastic Deformation, Academic Press Inc., 1996. [KRE 86] KREMPLE E., MCMAHON J.J. and YAO D., “Viscoplasticity based on overstress with a differential growth law for the equilibrium stress”, Mech. Mater., 5:35-48, 1986. [KRÖ 61] KRÖNER, “Zur Plastischen Verformung des Vielkristalls”, Acta Metall., 9:155161, 1961. [LEM 71] LEMAITRE J., Sur la détermination des lois de comportement des matériaux élasto-viscoplastiques, PhD Thesis, University of Paris 6, 1971. [LEM 85a] LEMAÎTRE J., “A Continuum Damage Mechanics model for ductile fracture”, J. of Engng. Mat. Technol., 107:83-89, 1985. [LEM 85b] LEMAITRE J. and CHABOCHE J.L., Mécanique des Matériaux Solides, Dunod, Paris, 1985. [MAN 65] MANDEL J., “Une généralisation de la théorie de la plasticité de W.T. Koiter”, Int. J. Solids Struct., 1:273-295, 1965. [MAR 79] MARQUIS D., Etude théorique et vérification expérimentale d’un modèle de plasticité cyclique, Thesis, University of Pierre et Marie Curie, Paris 6, 1979. [MAR 89] MARQUIS D., Phénoménologie et thermodynamique: couplage entre thermoélasticité, plasticité, vieillissement et endommagement, PhD Thesis, University of Pierre et Marie Curie, Paris 6, 1989. [MIL 76] MILLER A., “An inelastic constitutive model for monotonic, cyclic, and creep deformation: Part I: Equations development and analytical procedures”, J. of Engng. Mat Technol., 98(2):97-105, 1976. [MIL 87] MILLER A.K. (ed.), Unified Constitutive Equations for Plastic Deformation and Creep of Engineering Alloys, New York, Elsevier Applied Science, 1987. [MOO 90] MOOSBRUGGER J.C. and MCDOWELL D.L., “A rate-dependent bounding surface model with a generalized image point for non-proportional cyclic plasticity”, J. Mech. Phys. Solids, 38:627, 1990. [MOR 69] MORCHOISNE Y., Contribution à l’étude de l’écrouissage des matériaux métalliques, PhD Thesis, Faculté des sciences d’Orsay, 1969. [MRO 67] MROZ Z., “On the description of the work-hardening”, J. Mech. Phys. Solids, 15:163-175, 1967. [MRO 81] MROZ Z., “On generalized kinematic hardening rule with memory of maximal prestress”, J. Mécanique Appliquée, 5:241-260, 1981.

38


[NOU 83] NOUILHAS D., POLICELLA H. and KACZMAREK H., “On the description of cyclic hardening under complex loading histories”, in C.S. DESAI and R.H. GALLAGHER (eds), Constitutive Laws for Engineering Materials, Theory and Applications, Elsevier, 1983. [OHN 86] OHNO N. and KACHI Y., “A constitutive model of cyclic plasticity for nonlinear hardening materials”, J. of Applied Mechanics, 53:395-403, 1986. [OHN 93] OHNO N. and WANG J.D., “Kinematic hardening rules with critical state of dynamic recovery, parts I and II”, Int. J. of Plasticity, 9:375-403, 1993. [PER 64] PERZINA P., “On the constitutive equations for hardening and rate sensitive plastic materials”, Bull. Acad. Polon. Sci., Série Sci. Tech., 12(4):199-206, 1964. [PIL 94] PILVIN P., “The contribution of micromechanical approaches to the modeling of inelastic behaviour”, in A. PINEAU, G. CAILLETAUD and T. LINDLEY (eds), 4th Int. Conf. on Biaxial/multiaxial Fatigue, vol. 1, p. 31-46, ESIS, May 31-June 3, 1994, SaintGermain, 1994. [PIL 97] PILVIN P., “Une approche inverse pour l’identification d’un modèle polycristallin evp”, in Actes du 3ème Colloque National en Calcul de Structures, p. 207-212, Giens, 20-23 May 1997. [PRA 49] PRAGER W., “Recent developments in the mathematical theory of plasticity”, J. Appl. Phys., 20(3):235-241, 1949. [RAB 69] RABOTNOV Y.N., Creep Problems in Structural Members, North-Holland, 1969. [RIS 95] RISTINMAA M., “Cyclic plasticity model using one yield surface only”, Int. J. of Plasticity, 11(2):163-181, 1995. [ROB 83] ROBINSON D.N., Constitutive relationships for anisotropic high-temperature alloys, Report NASA TM-83437, NASA, 1983. [ROU 80] ROUSSELIER G., “Finite deformation constitutive relations including ductile fracture damage”, in NEMAT-NASSER (ed.), Three-Dimensional Constitutive Relations and Ductile Fracture, p. 331-355, Dourdan, North-Holland Publishing, 1980. [SCH 81] SCHMIDT C.G. and MILLER A., “A unified phenomenological model for non elastic deformation of type 316 stainless steel, parts I and II”, Res. Mechanica, 3:109-129, 175-193, 1981. [TEO 95] TEODOSIU C. and HU Z., Simulation of Materials Processing: Theory, Methods and Applications, in SHAN-FU SHEN and DAWSON P. (eds), p. 173-182, Swansea, Balkema, 1995. [VAL 80] VALANIS K.C., “Fundamental consequences of a new intrinsic time measure: plasticity as a limit of the endochronic theory”, Arch. Mech. Stosow, 32:171-191, 1980. [WAL 81] WALKER K.P., Research and development program for non-linear structural modelling with advanced time-temperature dependent constitutive relationships, Report PWA-5700-50, NASA CR-165533, 1981. [WAT 86] WATANABE O. and ATLURI S.N., “Constitutive modeling of cyclic plasticity and creep, using an internal time concept”, Int. J. of Plasticity, 2:107-134, 1986.


39

[ZAO 93] ZAOUI A. and RAPHANEL J.L., “On the nature of intergranular accommodation in the modeling of elasto-viscoplastic behaviour of polycrystalline aggregates”, in J.L. RAPHANEL, C. TEODOSIU and F. SIDOROFF (eds), Large Plastic Deformations, p. 27-38, Rotterdam, 1993. [ZEN 44] ZENER C. and HOLLOMON J.H., “Effect of strain rate upon plastic flow of steel”, J. Appl. Phys., 15:22, 1944.


Chapter 3

Measurement of Elastic Constants

The very first parameter governing the constitutive law of a material is its stiffness connected with its elasticity. If elasticity laws have been known for quite a long time, the measurement of elastic characteristics itself covers many different realities, from the perfect crystal to polymer, as well as from the isotropic solid to composites, or even coatings. This chapter is not intended as an exhaustive list of methods for measuring elastic constants, but as a panorama of the various techniques used, from standard mechanical tests to more specific tests; comparing the different methods is not an object in itself, but elements are given regarding the implementation of tests and the specimen geometry, the precision and the temperature limitations of the type of measurement, the choice in relation to the type of material (stiffness range or brittleness); all these elements will make it possible to appreciate the complementarities of various techniques and to indicate their possibilities for the user. The presentation will be limited to “mechanical” methods in the sense that they use the application of a load and the indirect or direct reading of a deformation; Xray diffraction methods and neutron or magnetic diffraction methods will not be dealt with here (see [HAU 97]). In order to discriminate between these methods, standard mechanical tests are distinguished as quasi-static, ultrasound and resonance methods as dynamic, and more recently, with the development of coatings, the instrumented indentation methods as local and surface methods.

Chapter written by Pascal GADAUD.

42


3.1. Elastic constants 3.1.1. The perfect crystal – elastic constants The origin of elasticity derives directly from the inter-atomic cohesion energy. Since the equilibrium of a crystal corresponds to a constant inter-atomic distance (equilibrium between attractive and repulsive forces), the application of mechanical loads is absorbed in a reversible manner by the variation of this distance. Elastic constants are obtained by means of double derivation of inter-atomic potential. At macroscopic level and within the framework of isothermal linear elasticity, the elastic constants Cijkl directly connect stresses V ij and strains Hkl through the relations:

V ij

[3.1]

CijklH kl

The energy and symmetry criteria bring the number of these components down to 21, in the most general case by simplifying the indexation ( Cij ). The crystal symmetry also reduces the number of independent constants to 9 for the orthotropic solid, 5 for hexagonal symmetry, and 3 for cubic symmetry (C11, C12, C44). In an analogous manner, the compliance Sijkl can be defined by the relations:

H ij

[3.2]

SijklV kl

Knowing the constants in relation to the crystal symmetry axes, the modulus can be deduced in any loading orientation (Hooke’s law). Take the case G of a cubic single crystal; Young’s modulus (see section 3.1.2), in direction X , whose direction cosines in relation to the principal directions of the crystal are l1, m1, and n1, is written: E x1

ª S11 S12 º ªl 2 m2 m2 n2 n2l 2 º S11 « S44 »¬1 1 1 1 1 1 ¼ 2 ¬« ¼»

[3.3]

from which, for the simple directions of the crystal, the following is deduced: 1 E100

S11

1 E110

S11

[3.4] A 4

[3.5]


1 E111

43

A 3

[3.6]

2 S11 S12 S44

[3.7]

S11

with A

This formulation makes it possible to see the anisotropic elastic character of a single crystal immediately. Consequently, the anisotropy deviation for this symmetry is defined by means of the ratio a 2 S11 S12 / S44 varying from 1 for tungsten to 8.75 for brass. Figure 3.1 illustrates this example with experimental results on a super-alloy AM1 single crystal (coherent precipitates J Ni3Al in a matrix J Ni3Al). The tests were performed according to the traction-compression resonant method (see section 3.4.3) on three bars cut up in the directions , , and (according to Mazot et al. [MAZ 92a]). Notice the comparison with the values obtained using a standard tensile test (see section 3.2.1).

Figure 3.1. Anisotropy of Young’s modulus for super-alloy AM1 single crystal

44


3.1.2. Isotropic solid – elastic moduli The isotropic solid corresponds to a material mechanically equivalent in all directions; in practice, this denomination covers a metal whose sufficiently small grains are randomly oriented as well as an amorphous material. The generalized Hooke laws define the three elastic moduli for simple loading conditions: – Young’s modulus E defined by the longitudinal deformation İ under uni-axial loading ı: E

V H

[3.8]

– the shear modulus ȝ (also called the Coulomb modulus or denoted G) defined by the shear J induced by torsion stress Ĳ:

P

W W | tgJ J

– the bulk modulus K (or B) defined by the relative volume change '

[3.9]

'V V

under hydrostatic pressure p: p

K '

[3.10]

Also of current use is the Poisson ratio Ȟ (or ı) which, in a uni-axial test, is defined by the ratio of the radial deformation Hr and the axial deformation Hx: v

H r

Hx

[3.11]

A fifth and last constant, rather less employed, is the Lamé constant Ȝ defined by:

V ij

2PH ij O'

[3.12]

In fact, the isotropic solid is characterized by two independent elastic constants, the others being deduced by the relations given in Table 3.1. The purpose here is not to give a list of the values of these constants but to establish the orders of magnitude. Thus, Young’s modulus can be depicted over a range of six decades, from polymers to diamond (see Figure 3.3).


45

As far as the Poisson coefficient is concerned, the value 0.3 injected hastily in a constitutive equation is far from being a reality (see Figure 3.4): already for metals, the value of Ȟ may vary considerably (this is important for the relations from Table 3.1). The molecular structure of the polymers implies high values of Ȟ (very low volume variations under stress), as opposed to that of oxides or ceramics (for example for silicon carbide Į) or also to certain composites.

Figure 3.2. Definition of three moduli E, K and G (according to Schreiber et al. [SCH 73])

related to constant

E, Q

E, P

B, Q

3(1 - 2Q)B

E Q

P

E 2(1 + Q)

B

E 3(1 - 2Q)

E 2P

B, P 9B 1 + 3(B/P) 1 - (2P%) 2 + (2P%)

3B(1 - 2Q 2(1 + Q) E 9 - 3(E/P)

Table 3.1. Relations between isotropic elastic constants

46


T

Figure 3.3. Order of magnitude for the Young’s moduli for various categories of materials (according to Ashby et al. [ASH 80])


47

Figure 3.4. Values indicating the Poisson coefficients for various types of materials (according to Kurtz et al. [KUR 87])

3.1.3. From isotropic solid to real material Experimental moduli of real materials incorporate, on the one hand, the presence of defects on a micro-structural scale, but, on the other hand, mainly structural heterogenities; in the elastic sense, this concerns metallic polycrystals, partially recrystallized polymers, as well as composites (fibers or inclusions). Therefore, only “global stiffness” is measured, which is representative of the composite structure and of its constituents. From the mechanical point of view, any case can be envisaged: the material can be macroscopically isotropic or anisotropic or constituted of elastic units that are isotropic (or not) themselves, which are distributed randomly or in a correlated manner. The estimation of the equivalent modulus must therefore be achieved by models, taking structural parameters into account. The simplest example of heterogenous medium modeling concerns isotropic composites with two isotropic components: the laws of Voigt and Reuss, corresponding to in series and parallel elastic models respectively (uniformity of strain or of stress), define Hill’s bounds for the real value of the modulus of the composite. Based on Eshelby’s first contributions [ESH 57] concerning the problem of inclusions, other models enable us to refine the estimation of the modulus; the most current are the Hashin-Strickman bounds [HAS 67], which take into account the spatial distribution of constituents (degree of disorder connected to anisotropy), and the self-consistent models (notion of equivalent homogenous medium which presupposes perfect disorder of inclusions [KRÖ 55]). The specific case of polycrystals has also been addressed (see Ledbetter et al. [LED 90] for the comparison of models applied to copper), as well as the specific

48


case of fiber-reinforced composites (refer for example to [DAT 90] or of ceramics [RIC 77, WAT 68]. For textured metallic materials, it is very difficult to predict the anisotropic characteristics: if we take the example of a cubic metal textured according to in the direction of rolling, Young’s modulus in this direction is very close to E110, the intermediate value for cubic symmetry (see Figure 3.1); it results from this that the modulus measured in a transverse direction (and considering a complete disorientation of grains) also has an order of magnitude of E110. Only a multiplication of measurements according to the orientations of the test samples will bring some precision. 3.1.4. Dynamic modulus If for rigid materials the stress-strain linearity exists in a certain domain (up to 0.2% deformation for metals), this is not the case for flexible materials or those with strong damping. In the case of cast irons, the notions of tangent or secant moduli were introduced in order to approximate the value of the modulus from the curves ı = f(H). This non-linearity is also a general rule in high-temperature tests. In fact, if a macroscopic elastic limit is defined, structural defects can be mobile under very low load and absorb energy in a non-elastic manner; this is called anelasticity [ZEN 48] and the damping coefficient is defined:

Q 1

1 'W 2S W

[3.13]

where ǻW is the energy dissipated by these defects and W the stored elastic energy. This coefficient is generally quite sensitive to the loading frequency as well as to the level of loading in relation to dissipation mechanisms that are generally thermally activated. When this coefficient is high, particularly for high temperatures or for polymers, it is preferable to use methods of cyclic tests under low load, ultrasound or resonant methods (see [3.3] and [3.4]), because, for quasi-static tests, the modulus depends strongly on test frequency. In the case of cyclic tests, the notion of a complex or dynamic modulus is then introduced: G T , iZ

G ' T iG '' Z , T

[3.14]

where Ȧ is the loading angular frequency and T the temperature. The angle of mechanical loss ĳ is defined as: tgM

G '' G'

Q 1

[3.15]


49

Figure 3.5. Evolution of the shear modulus of oxynitride glass with SiC volume fraction (according to Baron [BAR 96])

Figure 3.6 illustrates the notion of dynamic modulus over more than four decades of loading frequency from mechanical spectrometry tests (see section 3.4.6).

Figure 3.6. Dynamic module and damping of oxynitride glass near to the vitreous transition (according to Doën [DOË 98])

50


Returning to the case of polymers, for which elasticity is mainly entropic (rearrangement of macromolecular chain configurations), Perez [PER 92] defines the modulus related to the cohesion of connections as a phonon modulus (measurable by hypersonic velocities) in relation to the experimental modulus taking into account this entropic elasticity. For reinforced materials (1-5 GPa range), Table 3.1 shows that there is a concordance between resonance (1-10 kHz range) and ultrasonic measurements. Dynamic Young’s moduli (GPa) Resonant method

Ultrasound

PC

2.5

2.8

PMMA

5.5

5.4

PP

3.0

3.0

Table 3.2. Comparison between dynamic moduli measured by means of ultrasound and resonant methods (according to Lagattu et al. [LAG 99])

3.2. Quasi-static mechanical tests The simplest method for measuring a modulus is the direct test in which, after the application of a controlled load, the induced deformation will be read; this will be done before attaining the macroscopic yield strength of the material. If the mechanical tests that will be developed in the following chapters determine the parameters of the constitutive equations, the first information they can provide concerns the elastic character. If the tensile test is currently employed, loadings in torsion, in flexion, in uni-axial or hydrostatic compression are also used, because each type of material has its limitations of investigation, such as its brittleness and the difficulty of machining tests pieces with grips and fixtures for tensile testing; such limitations also derive from the concern of reproducing experimentally the real or simplified in-service conditions. The main sources of measurement error is connected with the finite dimension of the samples; clamping, contact, geometry and boundary problems induce certain non-homogenities of the global behavior, which must be optimized by the standardization of test conditions. Only a broad outline of the various types of tests is given here.


51

3.2.1. Uni-axial tensile and compression tests

The tensile test is the typical test for Young’s modulus measurement on cylindrical or flat test pieces. This is directly given by the ratio ǻı/ǻH of the uniaxial stress and strain increments from the linear part of the tensile curve. In a practical manner, E is determined by:

E

'F / S o 'L / Lo

[3.16]

where ǻF and ǻL are the applied force and macroscopic extension increments and S0 and L0 are the initial gauge section and length. In the case of compression, only the sign of increments is changed. The first important standardization is to read the deformation by extensometer (classes A and B-1). The second important point concerns the test velocity: it must be neither too fast, in order to minimize the expansion or contraction problems connected with heating, nor too slow, in order to limit creep. The influence of the test velocity is quite important for polymers or for polymer composites. The experimental fitting alignment must also be the best possible in order to avoid transverse stresses and a bad reading of the deformation. Compression tests achieved on cylinders are more delicate to perform because of the risk of buckling, which can be solved by force transmission with spherical bearing rod ends and an adequate parallelism of support facets. Contact friction problems also involve a barreling effect, which could be neglected through a gauge length adapted to the part of the specimen deformed in a homogenous manner. Tensile tests also enable us to determine the Poisson’s ratio of the materials through simultaneous reading of longitudinal (Hl) and transversal (Ht) deformations under load F, as illustrated in Figure 3.7 (absolute deformations).

v

d H t / dF d H l / dF

[3.17]

Geometry and alignment conditions are important here in order to transversally measure a deformation on a relatively large gauge length so as to limit errors on Ht.

52


Figure 3.7. Measurement principle of Poisson ratio by tensile test

3.2.2. Torsion and bending tests

These tests can be brought together in the sense that, unlike in traction, deformation is not uniform in the entire material, which requires great precautions when it comes to the choice of the range of imposed stress. Torsion tests on cylinders or tubes enable us to determine the shear modulus P using the relation:

P

CL JT

[3.18]

where C is the applied torque, L is the gauge length, T the deformation angle and J the moment of torsion inertia, defined by: J

J

S D4 32

S 32

for a cylinder of diameter D

[3.19]

De4 Di4 for a tube of inner diameter Di and outer diameter De [3.20]

As the grips also involve deformation non-homogenities here, the gauge length must be adapted to the geometry. It is recommended to vary the gauge length or the applied torque level in order to improve measurement range.


53

Bending tests are currently employed for ceramic-type or glass-type brittle materials, as well as for composites, because they require simple geometries of headless test pieces. If the cantilever-type test (fixed support) exits, the three or four point bending methods, as schematized in Figure 3.8, will be preferred, which do not require recess gripping that is detrimental for brittle materials.

Figure 3.8. Flexion tests principle (according to Mencik [MEN 92])

For three-point bending: E

Fl 3 48 yJ

[3.21]

where F is the applied force, y the deflection and J the moment of inertia. For rectangular section samples (b and h), J

bh3 12

[3.22]

and for a circular section D, J

S D4 64

[3.23]

54


These formulae are accurate only when the section is small as to length. For thicker brittle samples, used for the study of other mechanical parameters, the relation becomes:

E

Fl 3 ª 12 JED º «l » 48 yJ ¬ Sl 2 P ¼

[3.24]

where S is the sample section and Į a factor connected with geometry. In this case, it is necessary to complete the tests on at least two different geometries in order to go back to E and P. In order to get rid of the shear effect, it is preferable to use four-point bending (see Figure 3.8b). The relation giving Young’s modulus is then: E

F l1 ª3l02 4l12 º ¼ 48 Jy ¬

[3.25]

In the majority of cases, deflection is measured by the displacement of the crossbeam of the test machine (correction achieved for the rigidity effects of the machine); direct reading of deflection is possible only for adapted assemblies. 3.2.3. Hydrostatic compression tests

These tests are a particular case of confinement-pressure tri-axial compression tests developed for the study of brittle and porous materials such as rocks, for which any other type of test is unacceptable. They can be performed on cylindrical samples of relatively significant size in order to integrate the heterogenities of the structure. Tests of the same type are used for the study of metal plasticity at temperatures where these are also brittle. The experimental set-up consists of an axial mechanical piston coupled with a radial confinement pressure. When controlled to obtain a hydrostatic pressure 'ı, the volume compressibility module K is directly given by: K

'V 'H

as illustrated in Figure 3.9.

[3.26]


55

Figure 3.9. Measurement of K in Lavoux limestone by means of hydrostatic compression (according to Charliez [CHA 91])

3.3. Ultrasonic methods 3.3.1. Principle

Use of ultrasounds for non-destructive testing or for measurement of elastic constants is the simplest method to determine the characteristics of a material at ambient temperature. The calculation of the velocity of an elastic wave in a homogeneous medium derives from the equation of propagation of a longitudinal plane wave of displacement u: w 2u wt 2

E or P w 2u

U

wx 2

where ȡ is the density

[3.27]

For transversal waves (shear), velocity VT is written in an isotropic medium: 1/ 2

VT

§P· ¨ ¸ ©U¹

[3.28]

56


and for longitudinal waves (traction-compression), velocity V1 obeys the inequality: E

U

V1

1 v U 1 v 1 2v

E

[3.29]

Since the velocity depends on the lateral dimensions of the sample, two experimental cases are distinguished: – the “long bar” mode, where the significant wavelength is propagated in the entire relatively small section of the sample. In this case: V1

E

U

[3.30]

– the mode which corresponds to the infinite medium in the case of low wavelengths and of a large section of the sample: V1

1 v U 1 v 1 2v

E

[3.31]

Technically speaking, a piezoelectric transducer which delivers a mechanical impulse is applied to a sample of thickness e and either a second transducer in transmission or the same transducer in reflection records, after a time Ĳ, the passage of the wave which is propagated with velocity V, in such a way that: V

1 or 2 e W

[3.32]

This method is called an “impulse-based method” and in practice it relies on sending wave trains where the time between each impulse can be modulated; in this manner, the emission and reception spectra can be superimposed and the time-offlight Ĳ can be measured in continuum in a more precise manner [MAT 67]. 3.3.2. Measurement error sources

The precision of measurements carried out at ambient temperature depends first of all on geometry conditions: parallelism of emission and reception surfaces, border effects. Any variation of the waves phase modifies the acoustic pressure locally by interference of these waves and introduces an error into the global velocity measurement.


57

The sample-transducer contact being ensured by an adhesive, its thickness introduces a parasite propagation time and the contact parallelism must also be the best possible. The use of a transducer of finite dimension introduces a diffraction of the emitted wave as the plane wave is transformed to give surface waves during the propagation. In the case of heterogenous materials (grains or inclusions), wave diffusion takes place on the defects and for dampening materials, the medium becomes dispersive, enlarging the echoes. In fact, if these limitations are mainly preponderant on the measurement of wave attenuation, they make the measurement of velocity more delicate: abatement and deformation of echoes. Whatever the case, with a minimum of assembly precautions and with the actual possibilities of signal processing in order to refine the measurement of Ĳ, relatively precise modulus measurements are obtained. 3.3.3. Measurements at high temperatures

For measurements at high temperature, the problem is estimating the dilatation of the sample, which the measurement of the sound path duration takes into account. The equation giving the true modulus variation becomes: 'E E

'U

U

2'W

W

2'l t

2'W

W

1 'U 3 U

2 'W

W

D (T )dT

³

t

[3.33]

where D is the thermal expansion coefficient. Concerning experiments, bonding of the transducer is suitable for low temperatures and only at average temperatures: less than 100°C for adhesives; certain cements may resist at higher temperatures (a400ºC). The choice of this binding material also depends on the attenuation, which it introduces, within the range of temperature measurements. In order to compensate for the dilatations of the measuring device, the transducer is stuck to the sample and assembled elastically with springs in order to remain in contact [GRE 81, SPE 70]. For higher temperatures, it is necessary to take the transducer out from the warm region and to transmit the measurement via a waveguide [GAU 89]. Intermediary materials that have dimensions and adequate qualities for wave reflection and transmission should be chosen; a line of measurement is illustrated in Figure 3.10. Signal processing is necessary in order to correctly analyze the sound path durations through various parts of the assembly. This type of assembly is well adapted for the study of glass and ceramics of low thermal expansion coefficient (see Figure 3.11).

58


A1: thorium tungsten: length: 590 mm; propagation velocity: 4,660 m/s A2: aluminum: length: 590 mm; propagation velocity: 10,000 m/s A3: sample: length: 150 mm; propagation velocity: 10,000 m/s Figure 3.10. Line of high-temperature ultrasound measurement (according to [HUG 92])

Figure 3.11. Modulus measurement by the ultrasonic method on two SiAlON glasses (according to Baron [BAR 96])


59

3.3.4. Immersion-bath ultrasound interferometry

In the case of anisotropic materials such as single crystals, the relationships between propagation velocity and elastic constants can also be established; in the following case of [100] propagation for cubic symmetry: V1

C11

U

and VT

C44

U

[3.34]

When the number of independent elastic constants is higher (particularly for orthotropic materials such as longitudinal-fiber composites), rather than multiplying the cutting directions and the number of measurements, immersion ultrasound interferometry (illustrated in Figure 3.12) can be used.

Figure 3.12. Immersion ultrasound interferometry principle (according to [BAU 96])

This technique is characterized by the absence of sample-transducer contact and requires a comparison of propagation time with and without sample. In addition, the possibility of rotating the sample allows the continuous analysis of the signal transmitted in all incident directions (by refraction, the plane wave is decomposed into three waves, a longitudinal and two transverse waves). The measurement of velocities according to the principal symmetry directions of the material is sufficient, but when the number of independent constants becomes high, the propagation measurements are performed outside these axes, and a mathematical treatment makes it possible to go back to the constants [BAS 90].

60


3.4. Resonant methods 3.4.1. Introduction to resonant methods

The principle of resonant methods is based on the relation existing between the resonance frequency N of a sample of density U and its elastic constant(s): N

F dimension, v

E or P

[3.35]

U

The resolution of Navier’s equation (stationary mode of an elastic wave of G displacement u in a medium in the absence of volume force) makes it possible to establish this relation:

O P

JJJJG G JJG grad div u P ('u )

U

G w2 u wt 2

[3.36]

In fact, various resonance frequencies corresponding to various deformation modes and their harmonics can exist. Each case of geometry, deformation mode and boundary conditions leads to a particular analytical or approximate solution. If, historically speaking, the vibration calculations and the measurement principle are ancient, the measurement acquisition and precision are nowadays facilitated by the development of instrumentation (signal analyzers and processing). Here again, bending, torsion or traction-compression tests on simple geometries are used, making the determination of one or two elastic constants possible. The main characteristic of these tests is to impose very low deformation levels, of about 10-6, 10-5 at resonance, in relatively high-frequency ranges: of the order of 1 kHz in bending, several kHz in torsion, over 10 kHz in traction-compression. Thus, we are freed from the problems shown in section 3.1.4. 3.4.2. Various experimental methods

There are many ways of causing the samples to vibrate and of detecting these vibrations. We will distinguish the methods in which there is multi-frequency stimulation, by shock, for example, and where a spectral analysis of the response must be performed in order to detect the resonance and the methods in which the stimulation frequency can be modulated in order to obtain the resonance conditions. The simplest method to perform is the shock and acoustic detection test, as shown in Figure 3.13. A variant of this type of test involves suspending the sample by two threads linked to the stimulation and the acoustic detection [SCH 73].

Measurement of Elastic Constants Impact hammer

Force transducer

61

Microphone

Sample Elastic supports

Amplifiers Response signal

Excitation signal Two-channel FFT analyzer

Figure 3.13. Demonstration of a Young’s modulus measurement device (according to Sanchette et al. [SAN 99])

There are many methods reported by Nuovo [NUO 61], which can be classified as piezoelectric, electromagnetic, electro-dynamic, magneto-restrictive and electrostatic. The main differences between them are that some are limited to magnetic materials or conductors; it is thus sometimes necessary to coat part of the sample with metal, which can disturb the measurement (see section 3.5.2). On the other hand, measurements in temperature are better adapted to tests without contact, in particular using electrostatic methods. 3.4.3. Bar and disk tests These two methods can be brought together because they make it possible to measure E and v simultaneously. First take the case of a bar of length L and diameter d, supported by three supports in its middle; these two constants can be measured by using the traction-compression harmonic frequencies Nn [LOV 27, RAY 45], n being the order of harmonic frequency: 1/ 2

Nn

n §E· ¨ ¸ 2L © U ¹

ª § S nvd ·2 º «1 ¨ ¸ » «¬ © 4 L ¹ »¼

[3.37]

There are other hypotheses of vibration calculation in the literature, but they lead to the same formalism when d a) relation [5.13]: u z1 u z2 h G1 G 2

[5.12]

u z1 u z2 h G1 G 2

[5.13]

Observing that for r = 0, u = Gyields: u z10 u z1 r u z2 0 u z2 r a a

r 2 § 1 1 · ¨ ¸ 2a ©R1 R2 ¹

[5.14]

If Hooke’s law is applied on a state of approximate average deformation, then by noting Pm the average contact pressure: u z 0 u z r Pm | a E

[5.15]

Replacing the first two terms of equation [5.14] with this approximate relation [5.20] yields the following equation for the contact of two solids: Pm Pm a § 1 1 · | ¨ ¸ E1 E2 2 ©R1 R2 ¹

[5.16]

Hardness Tests

or: Pm |

· a § 1 1 ·§ 1 ¨ ¸¨ ¸ 2 ©R1 R2 ¹© 1 E1 1 E2 ¹

115

[5.17]

In the contact of a rigid sphere (of diameter D) on an elastic infinite plane of Young’s modulus E (Brinell test), [5.17] yields a simpler relation: Pm |

aE D

4P Sa 2 this is written:

As Pm

§ P E ·2 3 Pm | ¨¨2 ¸¸ © S D ¹

[5.18]

Thus, the average contact pressure (i.e., hardness) is obtained according to the applied load and diameter of the ball based on Young’s modulus of the material. This approximate method shows that the average contact pressure changes in (E2/3) and in (P1/3). To rigorously determine the relationship between average contact pressure and the modulus of elasticity and the stress state of indented solid, Hertz assumed a distribution of pressure on the contact area. If the two solids are of revolution and the contact remains plane, the contact area is circular of radius a and the pressure distribution p(r) is necessarily written as a sum of terms of the type: §a 2 r 2 ·n p0 ¨¨ 2 ¸¸ © a ¹

pr

[5.19]

It was shown [JOH 85] that n = 1 corresponds to a uniform pressure and n = -1/2 to a uniform displacement. The case of n = 1/2 corresponds to Hertz pressure: §a 2 r 2 ·1 2 p0 ¨¨ 2 ¸¸ © a ¹

pr

with Pm

P

Sa

2

[5.20]

2 p0 3

where P is the applied force and Pm the average contact pressure. From this distribution of pressure, the state of stresses and strains in the indented body is calculated using the following relations [BOU 85, CER 82, LOV 52], which give

116


stresses and deformations of a semi-infinite body subjected to a point-wise loading P, normal on the surface and positioned at the origin (Figure 5.21). The procedure consists of moving the origin of these equations to apply a pointwise load at (x,y); then in multiplying by applied the load at (x,y), i.e. p(x,y)dxdy; then finally in integrating these equations on the entire contact surface, in a numerical or analytical way. For Hertz distribution of pressure, displacements were calculated for z = 0: If r < a u z

1 Q 2 S p0 2a 2 r 2 E 4a

If r > a u r

uT

0

3 2 º ª § r 2 · » ¨1 ¸ p0«1 « ¨© a 2 ¸¹ » r ¬ ¼

1 2Q 1 Q a 2 3E

[5.21]

[5.22]

Figure 5.21. Point-wise load positioned at the origin of a semi-infinite elastic medium

In the same way for stresses, for r < a and z = 0

Vr p0

3 2 º 12 ª 1 2Q a 2 « § r 2 · » § r 2 · ¨ ¸ ¨ ¸ 1 1 2 ¸ ¨1 2 ¸ 3 r 2 « ¨ a ¹ » © a ¹ ¬ © ¼

[5.23]

Hardness Tests

VT p0

Vz p0

117

3 2 º ª § r 2 ·1 2 1 2Q a 2 « § r 2 · » 1 ¨1 2 ¸ ¸ 2Q ¨ ¨1 2 ¸ ¸ 3 r 2 « ¨ a ¹ » © a ¹ ¬ © ¼

[5.24]

§ r 2 ·1 2 ¨ ¨1 2 ¸ ¸ © a ¹

[5.25]

Finally, along the axis (x = y = 0):

Vz p0

§ z 2 ·1 ¨ ¨1 2 ¸ ¸ © a ¹

[5.26]

The maximum shear stress appears in under-layers at z = 0.57 a, and its value is

Wmax = 0.31 p0 = 0.47(P/Sa2). This allows us to explain why plasticity starts in underlayers in contact loadings. This effect is responsible for the initiation of under-layer fatigue cracks contact fatigue.

Relation [5.21] gives the displacement of points of the contact surface in the normal direction to this surface. Now, according to the conditions of contact indicated in Figure 5.5 it was determined that: u z10 u z1 r a

u z2 0 u z2 r a

r 2 § 1 1 · ¨ ¸ 2a ©R1 R2 ¹

[5.27]

Replacing the first two terms of equation [5.27] with displacements calculated by the superposition method [5.21] yields: §1 Q 2 1 Q 2 · S ¨ ¨ E E ¸ ¸ p0 2 ¹2a © 1

1 1 R1 R2

[5.28]

from where comes the expression of average contact pressure: Pm

with E *

4a 1 R1 1 R2 1 R1 1 R2 | 0.59a * * 3S 1 E1 1 E2 1 E1* 1 E2* E 1 Q 2

[5.29]

118


Notice that this exact analytical solution obtained from the Hertz distribution of pressure is very close to relation [5.17], where the calculation had been made using an approximate state of deformation average. For a rigid sphere of diameter D on an elastic plane: 13

a

1 §3PD · ¨ ¸ 2 © E * ¹

and Pm

23 4 § E * · ¸ with h ¨ P S ¨ D ¸ ¹ ©

3 P 4 aE *

The exact formulation of average contact pressure gives a relation in E2/3 between hardness and modulus and in P1/3 between hardness and applied load, as the approximate formulation. The indentation of an elastic plane body by a rigid sphere thus enables us to theoretically determine the modulus of elasticity of the body or rather its modulus in “plane strain”: E*. 5.6.2. Appendix 2: slip line method (Hill)

A part of analysis carried out by Hill [HIL 47] by the slip line method is recalled below. The material is assumed to be rigid perfectly plastic with shear yield strength k. If, at a given point, the direction of principal stresses Vp is known, deformation will be described as a flow along slip line D and E at 45° of principal stresses. The hydrostatic pressure denoted here V is given in plane strain by:

V

1 V p max V z V p min 3

12 V p max V p min

Writing the equilibrium of volume element included between the lines D and Eas indicated in Figure 5.22, the evolution of Valong the slip line is deduced. To solve the problem, it is necessary to determine the geometry of the slip line and the boundary conditions of the problem. In Figure 5.8, a slip line field is proposed for the problem of indentation of a plane by an infinite wedge whose angle between the two sides is denoted as 2D. Usually for Vickers tests 2D = 136°. If plane strain and absence of friction between the indenter and the plane are assumed, no shear stress appears along the faces of the indenter. The principal axes are thus axis z, a normal axis to the indenter surface, and a tangential axis. Hence, slip line emerge at 45° of the sides of the indenter. In the same way, along the free surface, the stress tensor is reduced to the tangential component on the surface and to its component along z, and consequently the principal axes are

Hardness Tests

119

tangential and normal to the free surface, and slip line emerge at 45° of the surface. In addition, the matter pushed back by the indenter turns around point B and forms a pile-up on the surface. Consequently, the areas of triangles OAF and BFC are equal. As the material is supposed to be rigid, the flow follows the slip line without deformation of elements of volume inside the lines and thus the lengths of BF and BC are equal. To determine the geometry of the slip line field, it is necessary to determine Mthe angle between (BD) and (BE). As all the angles of the two triangles (OAF) and (BFC) are known, two relations A(OAF) = A(BFC) and (BF) = (BC) = l, make it possible to connect D and Mby the following equation: sin2D 2M sin 2D sin D sinD M , which can be solved in a numerical way. The equation put forward by Johnson [JOH 85] is a little different and depends on the analysis of velocities field, but the results obtained by both methods are rigorously the same. In the BCD region, by applying the Tresca criterion, being in plane strain and Vz always being the intermediate principal stress, can be written:

V p max V p min

2k

Now, one of the principal directions is normal to the surface and along this direction Vpmin = 0. Knowing that loading in the tangential direction is compressive, Vpmax= -2k, Vz= -k and hydrostatic pressure is V= -k. Applying the relations obtained by resolution of the equilibrium equations along the slip line (Figure 5.22) yields along the line (CDEA): V2kI = Cst. Between C and D, I= 0 and therefore V= -k remains constant. Between D and E, as I= M, V 2kM = -k to ensure continuity at D. Then between E and A again I= 0, and so Vremains constant and to ensure continuity at E, V = k + 2kM. From this, the value of pressure pm on the side of the indenter can be deduced; indeed, the Tresca criterion on the side of indenter gives: |pm - Vpmin| = 2k, and as V= (pm + Vpmin )/2, is found: pm =2k (1+ M)

120


Figure 5.22. Calculation principle of the slip line method in plane strain

Figure 5.23. (a) Slip line field for indentation without friction in plane strain, (b) determination of angle M

Hardness Tests

121

5.6.3. Appendix 3: equivalences between tensile and hardness tests [BAR 95]

HBS: Steel ball HBW: Tungsten carbon ball HV: Diamond pyramid HRC: Diamond cone Rm (MPa)

HV

Rm/HV

HBS or HBW

Rm/HB

HRC

280

80

3.5

76

3.68

-

380

110

3.45

104.5

3.63

-

480

145

3.31

137.8

3.49

-

580

180

3.22

171

3.39

-

680

210

3.23

199.5

3.40

-

780

245

3.18

232.8

3.35

21.3

880

275

3.20

261.2

3.36

26.4

990

310

3.19

294.5

3.36

31.0

1120

350

3.20

332.5

3.36

35.5

1290

400

3.22

380.0

3.39

40.8

1400

430

3.35

408.5

3.42

43.6

1500

460

3.26

432.0

3.47

46.1

2180

640

3.40

536

3.72

57.3

Table 5.2. Comparative table of Rm, HV, HB and HRC for carbon or low alloy steels

5.7. Bibliography [BAR 95] BARRALIS J., MAEDER G., Précis de métallurgie, élaboration, structures propriétés et normalisation, Nathan, 1995. [BIW 95] BIWA S., STORAKERS B., “An analysis of fully plastic Brinell indentation”, J. Mech. Solids, vol. 43, no. 8, pp. 1303-1333, 1995. [BOU 85] BOUSSINESQ J., Application des potentiels à l’étude de l’équilibre et du mouvement des solides élastiques, Paris, Gauthier-Villars, pp. 45-108, 1885. [CER 82] CERRUTI V., Roma, Acc. Lincei., Mem. Fis., mat., p. 45, 1882. [FIV 98] FIVEL M.C., ROBERTSON C.F., CANOVA G.R., BOULANGER L., “Three dimensional modeling of indent-induced plastic zone at mesoscale”, Acta Mater, vol. 46, no.17, pp. 6183-6193, 1998.

122


[HAI 95] HAIDER M.A., HOLMES M.H., “Indentation of a thin compressible elastic layer: approximate analytic and numerical solutions for rigid flat indenters”, J. Mech. Solids, vol. 43, no. 8, pp. 1199-1219, 1995. [HAN 94] HAN H.N., KIM H.S., OH K.H., LEE D.N., “Elastoplastic finite element analysis for porous metal”, Powder Metallurgy, vol. 37, no. 2, 1994. [HER 82] HERTZ H., Miscellaneous Papers, H. Hertz, Eds. Jones and Schott, London, Macmillan, pp. 90-156, 1896. [HIL 47] HILL R., LEE E.H., TUPPER S.J., “Theory of wedge indentation of ductile metals”, Proceedings of the Royal Society, A188, 273, p. 162, 1947. [JOH 85] JOHNSON K.L., Contact Mechanics, Cambridge University Press, 1985. [LAR 96] LARSSON P.L., GIANNAPOULOS A.E., SODERLUND E., ROWCLIFFE D.J., VESTERGAARD R., “Analysis of Berkovich indentation”, Int. J., Solids. Structures, vol. 33, no. 2, pp. 221-248, 1996. [LOV 52] LOVE A.E.H., A Treatise on the Mathematical Theory of Elasticity, 4th ed. Cambridge University Press, 1952. [MAT 94a] Matériaux métalliques, Essai de dureté Brinell, Partie 1, Méthode d’essai, NF IN 10003-1, 1994. [MAT 94b] Matériaux métalliques, Essai de dureté Rockwell et superficiel Rockwell, NF IN 10109-1, 1994. [MAT 98] Matériaux métalliques, Essai de dureté Vickers, Partie 1, Méthode d’essai, NF IN ISO 6507-1, 1998. [MUR 92] MURAKAMI Y., YUAN L.P., “Finite elements method (FEM) analysis of elastic linear hardening materials and comparison with measurements on commercial materials”, Journal of Testing and Evaluation, JTEVA, vol. 20, no. 1, pp.15-24, 1992. [MUS 91] MUSIKANT S., “What every engineer should know about ceramics”, What Every Engineer Should Know, vol. 28, pp. 106-107, ed. by Middendorf W.H., Marcel Dekker, Inc, New York, 1991. [POI 99] POISSONNET S., Mesures de dureté sub-micronique instrumentée. Méthode et applications, Thesis, Paris Sud University, UFRS Orsay, 1998. [PRA 96] PRAKASH O., JONES D.R.H., “Creep of metal-type organic compounds-II, indentation creep”, Acta Mater., vol. 44, no. 3, pp. 891-897, 1996. [PRO 80] Produits sidérurgiques, Valeurs de conversion de dureté de l’acier, NF Ao3-172, November 1980. [PRO 84] Produits sidérurgiques, Valeurs de conversion dureté-résistance à la traction de l’acier, NF Ao3-173, November 1984. [ROB 97] ROBERTSON C., POISSONNET S., BOULANGER L., “Plasticity in ionirradiated austenitic stainless steels”, J. Mater. Res., vol. 13, no. 8, pp. 2123-2131. [SCH 99] SCHNEIDER J.M., BIGERELLE M., IOST A., “Statistical Analysis of the Vickers Hardness”, Mat. Scien. Engng., A262, pp. 256-263, 1999.

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123

[STO 94] STORAKERS B., LARSSON P.L., “On Brinell and Boussinesq indentation of creeping solids”, J. Mech. Solids., vol. 42, no. 2, pp. 307-332, 1994. [TAB 51] TABOR D., Hardness of Metals, Oxford University Press, pp. 176-199, 1951. [WAN 93] WANG A.F., BANGERT H., “Three dimensional finite element simulation of Vickers indentation on coated systems”, Mat. Science Engineering, A163, pp. 43-50, 1993. [ZEN 96] ZENG K., SÖDERLUND E., GIANNAPOULOS A.E., ROWCLIFFE D.J., “Controlled indentation: a general approach to determine mechanical properties of brittle materials”, Acta. Mater., vol. 44, no. 3, pp. 1127-1141, 1996.


Chapter 6

Fatigue Tests

The objective of this chapter is to present the key fatigue testing methods that are developed either to compare materials and fabrication processes or to obtain necessary data to predict fatigue life. Strict application of these methods allows us to accurately take into account the effects of the numerous parameters, which influence the behavior of materials and mechanical components. 6.1. Principles 6.1.1. Definition If the amplitude and number of periodical loads are sufficient, they lead to progressive cracking and eventually to failure by fatigue. Generally, failure occurs suddenly, without precursory elongation, and the fracture surface presents a smooth zone, the fatigue fracture itself, and a fibrous or brittle fracture zone, which corresponds to the final instant rupture. Failure can occur for applied stresses lower than the tensile strength and even the proof strength. Generally, the term fatigue is used to denote the cause of this failure in service, and the term endurance for the capacity to resist fatigue. General recommendations concerning the terms used and the presentation of the results of the tests are given in standards [NFA 400].

Chapter written by Henri-Paul LIEURADE, Suzanne DEGALLAIX, Gérard DEGALLAIX and Jean-Pierre GAUTHIER.

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6.1.2. Objective of fatigue tests The objective of fatigue tests is to determine the behavior of test pieces, parts or mechanical structures, under periodical loadings, simulating those which, in practice, are experienced in service. Tests on parts or mechanical structures generally correspond to one of the following three objectives [RAB 00]: – to check the conformity of a product to a standard or specification; – to validate or optimize a new product; – to seek constructive solutions following the damage of a component in service. In all these three objectives, most often the task is to check that the part or the structure can withstand a fixed number of cycles of given amplitude without cracking or fracturing. The difficulties in implementing the tests corresponding to the three objectives are as follows: – the loading method: points of application and direction of efforts or displacements; – the amplitude level of loading, which has to be representative of the loadings encountered in service. The tests on specimens correspond to: – optimizing a mode of fabrication: in this case, the difficulty lies in reproducing the realistic effect of the conditions of fabrication in a test piece, which is generally of small dimensions (see section 6.2.7.2); – validating new technological solutions: it is a case of taking into account the scaling effect in an appropriate way; – determining the basic fatigue properties by applying the standard conditions for implementation of tests. This chapter mainly concerns the determination of the basic fatigue characteristics of structural materials by tests on specimens. However, its application can be extended to other cases, provided certain necessary additions are made. Moreover, tests under constant amplitude loading will be considered, excluding tests under variable amplitude loading, which leads to the application of particular procedures for tests and analysis of results [LIE 97].

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6.1.3. Classification of fatigue tests A distinction can be made between endurance tests, which allow us to determine the endurance limit or the fatigue life, and fatigue crack propagation tests, which concern the measurement of the propagation velocity of fatigue cracks. The endurance or the high-cycle fatigue regime can also be distinguished from the lowcycle fatigue regime corresponding to a small number of cycles to failure, as a consequence of generalized or localized plastic strain (notch) of the part or of the test piece (plastic fatigue) [NFA 401]. In addition, distinction can be made between load controlled tests, for example rotational bending tests on cantilever specimens where the load is applied by weights and displacement controlled tests, for example plane bending tests on a specimen with the help of a connecting rod and an eccentric. In general, high-cycle fatigue tests are load-controlled, whereas low-cycle fatigue tests are straincontrolled, the machine being controlled by the measurement of the cyclic deformation of the specimen by an extensometer. 6.1.4. Classification of loading modes Cyclic loadings can be classified according to the nature of the efforts on the parts or on the test pieces. Thus, the following are distinguished: – plane bending tests where all fibers at the same distance from the neutral fiber are simultaneously in the same tensile or compression state; – rotational bending tests where all the fibers (except the neutral fiber) are successively extended and then compressed, the value of the stress exercised on each fiber varying linearly, in the domain of elasticity, with its distance to the surface, where the external fiber supports the maximum stress; – tests under axial, tensile or compression loads, where, in the absence of geometric discontinuity, the stress is constant in any cross-section; – torsion tests; – tests under combined stresses, static and dynamic stresses, torsion and bending, tension and torsion, etc. Different types of machines make these tests possible: bending machines, rotational bending machines, torsion machines, pulsators, etc.

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6.1.5. Test pieces The choice of test pieces is very important and depends on the objectives of the tests: – investigation on the influence of the forming operations of the material during the fabrication of a product requires either to reproduce, on the surface of the specimens, thermo-mechanical conditions as close as possible to those imposed on the product, or to take out specimens from the product itself in such a way as to not modify the state of the surface (roughness, hardening, residual stress) introduced by the fabrication conditions of the product; – the knowledge of the effect of a surface treatment applied to a component also necessitates the processing of the test pieces in similarly adapted conditions. In particular, in this case, the dimensions of the samples have to be such that they enable us to obtain a residual stress field comparable to that of the real component; – obtaining the fatigue endurance characteristics of the material in a given microstructural state requires, on the other hand, the fabrication of the test pieces under the most neutral conditions, i.e. without generating a surface condition that modifies the properties being studied. The following sections cover the last objective in detail. 6.1.5.1. Shape A distinction is made between parallelepiped test pieces (Figure 6.1) of constant thickness and constant or variable cross-section and axi-symmetric test pieces of circular cross-section (Figure 6.2), either cylindrical with constant cross-section, or torus-shaped with variable cross-section. The latter model is the most widely used and is thus recommended when other conditions do not impose the choice of another test piece (minimum material, machining and finishing facility, elimination of buckling, dissipation of heat, localization of rupture, etc.). In all cases, test pieces of constant cross-section are distinguished from notched test pieces. 6.1.5.2. Dimensions The dimensions depend on the capacity of the machine (maximum load test on modern machines at least 10% of the capacity of the machine) and its characteristics (eigenfrequency of the test piece two times higher than the test frequency). Recommended dimensions of test pieces for fatigue tests and the allowed machining tolerances are specified in standards [NFA 401, NFA 402]. 6.1.5.3. Machining of test pieces When the problem is to determine the basic characteristics of a metallic material in fatigue, it is necessary to reduce, as much as possible, the influence of the surface

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conditions of the test piece on the fatigue process which leads to failure. For this reason, the surface condition of the test pieces has to be particularly precise and repetitive for the same batch of test pieces. Hence, it is advisable is to use the types of test pieces and machining methods, which assure maximum regularity and reproducibility of the surface condition and introduce only negligible residual stresses and superficial hardening.

For test pieces whose gage length is limited by a transition curve of radius r, the condition r 5d is imposed to obtain a theoretical stress concentration factor Kt close to 1. For parallelepiped test pieces with gage length Lc, a ratio Lc /d < 3 is imposed in the case where the stress cycle includes a part in compression in order to avoid the risk of buckling. Figure 6.1. Parallelepiped test pieces with rectangular section

For test pieces whose gage length is limited by a transition curve of radius r, the condition r 5d is imposed to obtain a theoretical stress concentration factor Kt close to 1. For cylindrical test pieces of gage length Lc, a ratio Lc /d < 3 is imposed in the case where the stress cycle includes a part in compression in order to avoid the risk of buckling. Figure 6.2. Axi-symmetric test pieces with circular section

A check of the superficial roughness after machining is advised. In the case of a large batch of specimens, the sampling of specimens to be checked have to be done in a statistical way in order to have a mean value of roughness, representative of the batch, and indication on the dispersion of measurements. However, in determining the fatigue properties, certain influential factors can be integrated, such as surface conditions or residual stresses, by fabricating specimens representative of the part to be designed. It is also important that the test piece has the same properties and thus the same micro-structural state as the zone of the part where the calculation of fatigue is carried out. In the case of tests on flat specimens, the angles have to be rounded, either by a grinding operation, or by Ayr stone polishing in order to avoid

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any initiation from the corners. Finally in the case of anisotropic materials, the removal direction of the test pieces is particularly important; it has to be chosen according to the directions of service loads. 6.1.6. Calibration of testing machines For a long time, mere static calibration for fatigue testing machines was considered to be good enough. However, it is known that for the same indication of the dynamometer of the machine, the static and dynamic loads applied to the test piece can noticeably differ. That is why it is indispensable to carry out not only static but also a dynamic calibration of fatigue testing machines. The calibration process consists of the following stages as described in standard NF A 03-509 [NFA 509]. 6.1.6.1. Definition of a dynamometric bar Because of their very small delay time, dynamometric bars with bonded strain gages are used. In general, four self-compensated gages are pasted on the four faces of the bar. They are connected to a Wheatstone bridge by compensation wires of the same length. The bar is then calibrated in traction as well as compression on a static testing machine, which in turn is calibrated with a standard dynamometer. 6.1.6.2. Determination of cyclic loads Determination of the cyclic loads, applied to a dynamometric bar, is obtained by measuring the cyclic deformation of the strain gages bonded on the bar. To obtain the best sensitivity, it is necessary to use a null method. The principle of this method consists of canceling the differential signal of the imbalance voltage of the gages of the bridge and a reference voltage that can be varied in a continuous manner. Before actually proceeding with calibration, it is advisable to check the stability of the dynamic load. 6.1.6.3. Dynamic calibration of testing machines by axial load The first stage in the process of calibration of a fatigue testing machine by axial load consists of verifying the co-axiality of the grip ends. Indeed, good alignment of the grip ends is imperative to obtain a homogenous stress distribution in the test piece. The dynamometric bar has to be fixed to a fatigue machine with the same precautions as on a static calibration machine. In particular, it is necessary to connect the same reading instruments that were used for static calibration. As a general rule, the loads and frequencies are varied in the entire extent of the possible use of the machine. The results are obtained in the form of calibration curves giving the dynamic load, indicated by the bar according to the load shown by the machine.

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Time t

Figure 6.3. Classification of loadings

6.2. High-cycle fatigue tests – endurance limit 6.2.1. Classification of loadings Periodical stresses which exist in service are generally sinusoidal, and in accordance with the value of the superimposed static stress Vs, we distinguish: – pure or symmetric alternating stress Vs = 0 (Vm = 0); – dissymmetric alternating stress Vs < Va (0 < | Vm | < Va); – repeated stress Vs = Va (Vm = Va); – fluctuating stress Va < Vs (Va < Vm). with Va as the variable stress amplitude and Vm the mean stress (equal to the static stress, Vs) (Figure 6.3).

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The stress ratio R is also defined as the algebraic ratio of minimum stress to maximum stress: R=Vmin/Vmax

[6.1]

Figure 6.4. Endurance diagram and Wöhler or S-N curve

6.2.2. Stress-number of cycles to failure curve (Wöhler curve) The simplest fatigue test consists of subjecting each specimen to load cycles of constant maximum amplitude and frequencies and noting the number of cycles Nr at which failure occurs. The number Nr is plotted generally on a logarithmic scale, according to the maximum stress of the cycles. Each specimen corresponds to a point on the plane (V, Nr) and, from a batch of specimens subjected to different stress amplitudes, a curve is obtained, which has a shape that is represented in Figure 6.4. On this curve, known either as the Wöhler curve or the S-N curve (stress-number of cycles), three regions are classified: 1) low-cycle fatigue regime under high stress; the failure occurs after a small number of alternations and is preceded by a noticeable plastic deformation (see section 6.3); 2) fatigue or limited endurance regime, where the failure is reached after a limited number of cycles, a number that increases when the stress decreases; 3) unlimited endurance regime or safety zone, under low stress, for which the failure does not occur before a given number of cycles Nr: 107, 3.107 and even 108, higher than the lifetime envisaged for the part.

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In many cases, an asymptotic branch can be traced on the S-N curve (the asymptote being the endurance limit or fatigue limit VD). On the other hand, in certain cases, there does not seem to be a horizontal asymptote. For example, this is the case when fatigue and corrosion occur simultaneously.

Figure 6.5. Statistical aspects of the endurance curve

6.2.3. Nature and dispersion of fatigue test results Nowadays dispersion of fatigue test results is accepted both as an experimental fact as well as a natural fact. Earlier attributed to imperfections of the test conditions, dispersion is now considered to be a physical aspect of the phenomenon, whose origins are related to: – the material (inclusions, heterogenity of structure, etc.); – the samples (difference of surface conditions, dimensional tolerances, etc.); – test conditions (centering of specimens on the machine, monitoring of the load, cycle frequency, heating of the sample, environmental effects, etc.). Experience has shown that the result of these combined actions express themselves generally by a characteristic form of the response curves to the stress: the number of cycles having been fixed and the observed event being rupture (or non-rupture) of the specimen before this number of cycles, the representative curve of the probability of this event, according to the amplitude of the cycles, is called the response curve to the stress (Figure 6.5). These curves are normal sigmoid, depending on two parameters ȝ and s. Under these conditions, statistical analysis of fatigue is related to the phenomena of all or none, and enables the application of various statistical techniques. The objective is to estimate the parameters of the response curve: ȝ, the amplitude of stress, for which the failure probability is 0.5 and s, the standard deviation of the dispersion in stress. Each of the response curves to the stress makes it possible to define a value of the stress amplitude VU for which the failure probability UVU is obviously a function of the number of cycles N and defines the equal-probability curve of failure corresponding to U (Figure 6.5b).

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In general, each equal-probability failure curve presents an asymptote (Figure 6.6); in particular, the equal-probability curve 0.5 tends towards a limit. This is the endurance limit VD. In practice, this limit is inaccessible and it is replaced with a conventional endurance limit VD(Nr) relative to the number of cycles chosen (Nr), 5.107, 108, etc.

Figure 6.6. Equal-probability failure curves for a given number of cycles or for a given stress

Figure 6.7. Wöhler or S-N curve of a XC10 carbon steel

This value is associated with a standard deviation of the endurance zone, the standard deviation of the response curve in stress (see Figure 6.7). The statistical processing of the fatigue data is described in standard A03-405 [NFA 405].

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6.2.4. Determination of endurance limit of metallic materials Different statistical methods can be used to perform the tests and analyze the results obtained in order to determine the endurance limit or fatigue resistance of metallic materials. The conditions under which these methods can be applied are specified in the documentation booklet NFA 03.405 [NFA 405]. However, the application of these methods depends on the knowledge of the order of magnitude of the expected limit. 6.2.4.1. Estimation of fatigue resistance, VD This estimation can be done on the basis of the tensile characteristics of the material envisaged, tensile strength Rm (MPa) and elongation after fracture A (%). For steels, we can cite: VD = 0.32 Rm+121 [BRA 80]:

VD = 0.37 Rm + 77 VD = 0.41 Rm + 25

[6.2]

(in rotational bending at 107 cycles and for steels with tensile strength between 300 and 2,200 MPa) [LIE 82]. The value of standard deviation s of the endurance zone can be evaluated: – in rotational bending on steel, by s = 0.045 Rm -12.3 [LIE 82]; – or, more generally, by s = 0.02 Rm. For aluminum alloys for fatigue resistance at 108 cycles (VD8) we can cite: – wrought aluminum alloys 0.40 Rm VD8 0.50 Rm; – wrought age hardening aluminum alloys 0.25Rm VD8 0.30 Rm; – aluminum casting alloys 0.3 Rm VD8 0.45 Rm and s is of the order of 5 to 10 MPa. 6.2.4.2. Statistical methods A number of statistical methods, such as the probits method, the staircase method and the iteration method, can be used to determine the endurance limit of metallic materials [LIE 82]: – the probits method consists of choosing the amplitude levels of stress (four or five) enclosing the presumed endurance limit and leading, at the same time, to ruptures and non-ruptures. The same number of samples n (for example, 10 per level) are tested at each level, up to the number of cycles fixed beforehand (for

136


example, 107 cycles for the endurance limit of steel); the variation of the reported rupture frequency on a Gaussian scale, according to the stress amplitude, makes it possible to trace a Henry straight line from which the mean and standard deviation are calculated; – the staircase method is much simpler than the probits method for the choice of levels, which is automatically carried out; it can be achieved with a low number of tests. However, the precision obtained on the standard deviation depends on the number selected. Having fixed the maximum number of cycles N (107, for example), grading steps for the stress levels, which are of the order of the assumed standard deviation s, are defined. The specimens are successively tested in the following manner. The first test is carried out at a level close to the estimated fatigue limit. If the specimen breaks, the second specimen will be tested at the next lower level; if it does not break, the second specimen will be tested at the next upper level. This process is then continued until all the available specimens are used by applying the rule:

Vi+1 = Vi ±d

[6.3]

with d spacing of stress levels, +d if the specimen i is not broken and –d if the specimen i is broken. The type of event, rupture or non-rupture, which occurred least often is determined. Let ni be the number of least frequent events at each test level. The test levels are numbered by attributing the value i = 0 to the lowest of those for which the event was observed at least once, i denoting the test level. The estimation of the median VD is given by:

V D

§A 1 · S0 ¨ r ¸ ©N 2 ¹

>@

(+ if the calculation is based on non-rupture tests, – if it is based on rupture tests). In addition, the estimation of the standard deviation s is given by: §NB A 2 · ¸ s 1.62d ¨ 0.029 ¨ ¸ 2 © N ¹

>6.5@

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Figure 6.8. Example for determining the fatigue limit using the staircase method

Stress (MPa)

i

ni

i.ni

i2.ni

i

ni

i.ni

i2.ni

360

1

3

3

3

3

0

0

0

340

0

4

0

0

2

10

20

40

320

1

11

11

11

300

0

1

0

0

N = 22

A = 31

B = 51

N=7 Fatigue limit Standard deviation of the endurance region

A=3

B=3

S0 = 340 MPa; d = 20 MPa VD = 339 MPa

NB A 2 N2

0.24 0.3

S0 = 300 MPa; d = 20 MPa VD = 338 MPa

NB A 2 N2

s: calculation not possible

0.33 ! 0.3

s = 12 MPa

Table 6.1. Calculations related to the example in Figure 6.8, in order to determine the fatigue limit VD and the associated standard deviation s (either with 15 specimens or 45 specimens)

with: S0

the lowest level used with the least frequent event,

d

spacing of levels,

N

ni,

A

ini,

B

i2ni.

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It should be noted that s can be calculated only if [(NB – A2)/N2] > 0.3. This estimation is considered valid only if the number of test pieces is large enough (50), whereas the method can be used to estimate VD as soon as about 10 specimens have been tested, as shown in the example of Figure 6.8 and the adjoined table, provided 1/2 < d/s < 2; – the iteration method: this consists of an application to the particular case of response curves. This method uses the least number of test pieces, but its implementation is rather critical owing to the influence of the choice of the initial level. In practice, nine or 10 specimens are used, but the method is applicable as soon as four or six specimens are available. Having fixed the maximum number of cycles N, the first level of the test is chosen. The choice of the next stress level depends on the results obtained in the previous level, but the distance between levels is variable and defined by the order number of the test. The choice of the first level S1 is important. It has to be as close as possible to the expected value. The tests can be carried out by groups of any number, including 1, i.e., specimen by specimen. Let f1 be the failure frequency under S1. The second level S2 is determined by the equation: S2 = S1 + 2.5 s (0.5 – f1 )

[6.6]

with: s

standard deviation of the endurance zone,

0.5

failure probability for fatigue limit VD.

In a more general manner: Si1

Si

A s p f i i

[6.7]

which converges towards S(p), with: A

a constant,

p

the chosen failure probability which can be different from 0.5.

For p = 0.5, A = 2S § 2.5. The formula usually used is thus: Si1

Si

2.5 s0.5 f i i

which converges towards S(0.5) = VD.

[6.8]

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Determination of the fatigue limit is obtained without calculation since the sequence of applied stress amplitudes Si tends towards VD. If Sn is the last level of the tests, the result obtained makes it possible to calculate Sn+1, which would be tested if an additional specimen were available. This latter value is used to estimate VD. 6.2.5. Expression of the S-N curve To describe the form of the Wöhler curve (S-N curve), various expressions were proposed. The oldest, proposed by Wöhler, is written as: log Nr = a – bV

[6.9]

where a and b are two constants. This relation treats as a straight line the portion corresponding to limited fatigue life. To account for the curvature and the horizontal asymptote where VD is the fatigue limit, Stromeyer proposed (1914): log N r

a b logV V D

6.10]

or: 1

V with: A

§ A · b V D ¨ ¸ ©N r ¹

[6.11]

exp2.303a and Palmgren (1924):

logN r B a b logV V D

[6.12]

which also accounts for an inflection point; then Bastenaire (1971):

Nr

A exp>C V V D @

V V D

[6.13]

and Nr B

A exp>C V V D @

V V D

with constants A, B, C determined from the experimental data.

[6.14]

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An example of a curve corresponding to the latter equation is given in Figure 6.9. The adjustment of parameters from experimental results is performed by a repetitive process with the help of a computer [ESO 93]. Curve 1 is the median, curves 3 and 3’ are equal-probability failure curves at 0.10 and 0.90 respectively. Curves 2 and 2’ are the confidence limits at 0.95 of the median 1.

Figure 6.9. Steel XC10: experimental results in rotational bending with 1,000 specimens and adjustment of equal-probability curves using the Bastenaire formula [lb.10]

6.2.6. Estimation of the number of cycles N50 by the Henry straight line method 6.2.6.1. Field of application The method is applied to determine the points of the S-N curve in the limited fatigue life region. It provides an estimation of N50 (failure probability of 50%) and dispersion from a series of tests carried out at the same stress Vi. 6.2.6.2. Principle of the method The method assumes that the distribution of the logarithms of the number of cycles at failure for a given stress level is normal. This is generally observed in the limited fatigue life regime. In a Gausso-arithmetic reference frame (Gaussian scale

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141

of ordinates and arithmetic scale of x coordinates), the representative points of the lifetime (expressed in logN) and of the proportion of broken specimens below the considered lifetime are plotted. When there is alignment of the experimental points (Henry straight line), the distribution of logN can be considered normal. In these conditions, the best estimation of the Henry straight line is obtained from the mean and standard deviation estimations. 6.2.6.3. Implementation of the method A minimum of 10 specimens is recommended. The n values of Nr are arranged in ascending order values: Nr1, Nr2,......Nri,....Nrn

[6.15]

The couples Ni, Fi are then plotted on a diagram with Gausso-logarithmic coordinates (Figure 6.10) and the Henry straight line is traced from the estimations, log N r of the mean and, s, of standard deviation

¦ log N ri

log N r

n

ª «¦ log N ri log N ri s « n 1 « ¬

1 2 º 2

»

[6.16]

» » ¼

The representative distribution line is obtained by making it pass through the abscissa and ordinate points: log N r - s, 16% and log N r - s, 84%

[6.17]

6.2.7. Accounting for the main parameters of influence Fatigue resistance of metallic materials depends on a large number of parameters. The absence or the oversight of consideration for one of these in the design of a part can lead to catastrophic failure [LIE 87]. In order to highlight the effect of each of these parameters on a given material, appropriate experimental methods and representations of the results are proposed [BAU 97, LIE 94]. 6.2.7.1. Mechanical parameters Loading modes Loads applied to a mechanical component can be tensile, bending or torsion loads. The fatigue strength depends on the loading mode; that is why special test

142


pieces and machines or test assemblies are used to determine this resistance. The test results are plotted according to the amplitude, either of stress Va (case of tensile or cyclic bending) or shear stress Wa (case of cyclic torsion), corresponding to the fibers of the test piece under the highest stress.

Figure 6.10. Henry straight line – numerical example for n = 10

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Combined alternating and static stress along the same axis The level of static or mean stress Vm plays an important role on the fatigue limit of a part Va. According to the value of Vm the stress is known as alternating, repeated or fluctuating (section 6.2.1). The idea is to represent the fatigue limit corresponding to a given number cycles and failure probability according to the mean stress Vm (or static stress V). In general, endurance limits are obtained for a failure probability of 50%. Experience shows that when the mean component of the stress increases, the amplitude admissible by the material at failure decreases. Different representations were proposed: Haigh, Goodman-Smith, Ros, Moore-Kommers and Jasper diagrams.

Figure 6.11. Haigh diagram

Haigh diagram In this diagram (Figure 6.11), the stress amplitude Va is plotted as a function of the mean stress Vm under which the fatigue test was carried out. Two particular points are to be considered: – point A, which represents the fatigue limit VD in symmetric alternating loading (Vm = 0); – point B, which represents the limit strength of the material for a zero alternating stress (Va = 0). This point corresponds to the tensile strength Rm of the material in a static test. The endurance limits obtained for various values of the mean stress fall on a curve AB adjusted according to the material. The field limited by this AB curve and the two co-ordinate axes represents the authorized functional field of the material in order to have a lifetime longer than or equal to the number of cycles for which the

144


endurance limit was determined. Several models were proposed to allow the construction of an approximate endurance diagram when the only things known are the fatigue limit in purely alternating loading and the mechanical static characteristics (Rp, Rm) of the material. The use of the Goodman line ensures that we are always on the safe side:

§

V a V D ¨1 ©

V m · ¸ Rm ¹

[6.18]

Goodman-Smith diagram Maximum stress Vmax and minimum stress Vmin are represented according to the mean stress Vm (Figure 6.12). Particular points are: – point A where Vm = 0 (purely alternated stress); – point B where Va = 0 (tensile strength in a tensile static test); – point C where Vmin = 0 (repeated stress). As for the Haigh diagram, different curve forms can connect these particular points, in particular the Goodman line. Rös diagram Maximum stress Vmax is reported according to the minimum stress Vmin. This diagram is principally used in English-speaking countries. Moore-Kommers-Jasper diagram Maximum stress Vmax is represented according to the ratio of stress R = Vmin /Vmax. This diagram is mostly used in German-speaking countries. 6.2.7.2. Geometric parameters 6.2.7.2.1. Notch effect The presence of geometric discontinuities induces zones of stress concentration, whose role is crucial in the fatigue resistance of a cyclically loaded part. A safe way of taking into account such a discontinuity consists, at first, of evaluating its coefficient of stress concentration, Kt = Vmax /Vn, where Vmax is the stress at the notch root estimated in elasticity and Vn, the nominal stress, calculated as the mean stress in the resistant section [LU 98]. The endurance limit in the notched part is then VDe = VDl/Kt, where VDl is the endurance limit of a smooth test piece of the same material.

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Figure 6.12. Goodmann-Smith diagram

In order to take into account the adaptation capacity of the material at the notch root, the reduction coefficient Kf = VDl /VDe, (Kf Kt) can be evaluated from tests on two batches of test pieces, either of constant cross-section, or notched, and possessing the same Kt as the considered part. The test results, which are then expressed according to the nominal stress (or mean) in the resistant section of the test pieces, make it possible to determine the respective fatigue limit, VDl and VDe (see Chapter 10). 6.2.7.2.2. Scaling effect Usually, the fatigue strength decreases when the size of the test pieces or of the parts increases. Several reasons are given for this result, among which are the evolution of the stress gradient in the case of bending or torsion loadings, surface conditions (roughness, hardening, residual stress) and the probability of the presence of defects [LIE 87]. Particular attention must be given to the selected conditions of similitude.

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6.2.7.2.3. Role of roughness Similar to the notch effect, the micro-geometry created in the processing of a given part can influence the level of fatigue resistance. This effect is greater the higher the resistance. In this case, the finishing conditions of the test pieces have to necessarily reflect industrial practice. 6.2.7.3. Parameters of material processing In addition to the level of roughness created on the part, the hardening rate, the microstructure and the profile of residual stress induced by a given processing play a decisive role on the characteristics of fatigue strength. Thus, it is necessary to try to achieve correct reproduction of the desired surface condition on the test pieces and to control it. 6.2.7.4. Environmental parameters The presence of a corrosive environment, whether it is aqueous or gaseous, conjointly with the application of a cyclic mechanical loading can lead to a damage mechanism by corrosion-fatigue [LIE 98]. In this case, a certain number of precautions have to be taken in order to correctly account for the phenomena encountered in service: – before beginning the test, a sufficient period of time must be respected so that a stabilized electrochemical condition can be reached; – the corrosive medium has to be representative of the one present in service; it has to be periodically controlled so as to maintain its characteristics throughout the test; – the frequency of the loading cycles has to be as close as possible to the one observed in service. 6.2.8. Examination of the fracture facies of the test pieces In order to obtain the maximum information from the tests carried out, a comprehensive examination of the broken test pieces has to be done to complement the statistical analysis of the data; this examination begins by a careful visual observation, which is focused on the following aspects [RAB 00]: – Position of the initiation site of the fatigue crack: in the case of a cylindrical or parallelepiped test piece, an initiation situated at the transition curve with the end of the test piece indicates either a bad choice of geometry (too small radius of the transition curve) or a bad alignment of the grips. The test is thus not valid and cannot be taken into consideration in the statistical analysis; in the case of a test

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piece with toroidal profile, an initiation situated away from the thinner section is also incorrect; it can denote either a defective quality of machining or nonhomogenous quality of the material; the plan of procedure or the type of test piece should thus be re-examined. The presence of under-layer initiation leads us to consider more precisely the role of the microstructure and to the evaluation of the elements, such as an unforeseen strain hardening or a particular hardening in the neighborhood of the surface, the presence of inclusions or blowholes, and the existence of residual compressive stress. – Number of initiation sites: when the initiation of the fatigue crack corresponds to several sites, radial lines (following the direction of propagation) are observed on the fracture surface; observation of the number of these lines enables us to account for the level of the material homogenity, particularly in the case of high-cycle fatigue. A single initiation makes it possible, with the help of additional microscopic examination, to specify the origin of initiation (inclusion, blowholes, machining defects, etc.) and to propose solutions for improvement. Multiple initiations lead to the conclusion of the existence of a number of defects with the same harmfulness or existence of an abnormally high loading. Considering the generally high cost of the fatigue tests, these observations and examinations at different scales thus enable us to obtain the best information contained in each broken test piece, either to improve the test conditions or to take into account the characteristics of fatigue strength of the tested material. 6.3. Low-cycle fatigue tests 6.3.1. Introduction As seen in section 6.2, the fatigue phenomenon of materials consists of a progressive damaging, resulting from cyclic loading and leading to fracture. “Lowcycle fatigue” corresponds to a lifetime Nr resulting from the application of a cyclic loading less than 105 cycles. For a majority of metallic materials, such a lifetime corresponds to a strain level higher than the yield strength of the tested material. Thus, at each cycle, the cyclically loaded material is subjected to a plastic deformation in the bulk and it is the accumulation of plastic deformation, successively in traction and compression, which leads to progressive damage. This is why the term “plastic fatigue” is often used. Low-cycle plastic fatigue damage is essentially found in structures designed with low yield strength materials (austenitic stainless steel for example, at ambient temperature as well as at high temperature) and in structures loaded at high

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temperatures (aeronautical structures made of light alloys or super alloys for example). In these structures, loading in low-cycle plastic fatigue particularly concerns stress concentration zones at notches, where there is a cyclic plastic zone at the notch root; however, very often this zone remains confined in an elastic surrounding medium. A study of the behavior and fatigue strength of such structures can be approached in two ways: tests, most of the time uni-axial and tensile/compression, are carried out either on stress controlled loaded notched test pieces or on strain controlled loaded test pieces of constant cross-section. The second type of test is called the “low-cycle fatigue test”. This second method of estimating the plastic fatigue behavior of a notched structure is similar to a local approach method. When the structures are subjected to cyclic variable temperatures, they are also damaged in low-cycle plastic fatigue. Then, three types of plastic fatigue tests are defined: – low-cycle fatigue at constant temperature; – thermal fatigue, during which the structure is not subjected to any exterior force, stress and strain then resulting from restrain conditions of the fatigue zone; – thermo-mechanical fatigue, during which the temperature and stress in the structure vary cyclically at the same time, in phase, out of phase or in a more complex manner. Here only low-cycle fatigue at constant temperature will be considered. Nowadays, low-cycle fatigue tests at constant temperatures are most often performed under strain control; however, sometimes, they are performed under stress control (we will refer to this later). All these tests have the existence of a plasticization in the bulk at each cycle in common. Initial studies of low-cycle fatigue date back to 1950, owing particularly to the introduction of servo-controlled hydraulic testing machines. They were then quickly developed and have become increasingly important for the following two main reasons: – they allow us to take into account the low-cycle fatigue phenomenon in the design of the mostly concerned structures (aeronautic, energy production, etc.); – they enable a better understanding of the mechanical and metallurgical phenomena operating during such solicitations. The first point is here the most interesting one. Consequently the second point will be presented in less detail in the following section. More details in this field can be found in [ELL 97, PIN 76, 97, POL 91].

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6.3.2. Physical mechanisms of low-cycle fatigue Damage in low-cycle fatigue of a material takes place in several stages. The operating physical processes mainly depend on the nature of the material (composition, micro-structural condition), on the imposed loading level, strain rate and temperature. Consider a low-cycle fatigue test under cyclic uni-axial loading of constant tensile/compression amplitude. At first, micro-structural evolutions take place in the bulk due to cumulative plastic deformations: creation of micro-structural defects (specific defects, dislocations, etc.), then multiplication and rearrangement of these and eventually phase transformations. At the same time, plastic deformation concentrates gradually in bands (generally persistent slip bands), which will emerge on the surface to create a relief consisting of intrusions and extrusions (Figure 6.13). In fact, the internal structure of the material and its evolution is narrowly linked to the initial crystallographic structure (FCC, BCC, HC, etc.) and to the gliding mode of dislocations related to the stacking fault energy. A material with low stacking fault energy will have a rather planar dislocation structure, while that with high stacking fault energy will favor cross slip, rapidly leading to a more complex cellular type of dislocations structure [DEG 84, NIL 83]. In all cases, the structure evolves towards a more stable condition. At the same time, intrusions/extrusions will gradually intensify.

a) Persistent slip bands

b) Schematic representation of surface relief Figure 6.13. Surface relief in low-cycle plastic fatigue: a) emergence of persistent slip bands on the surface of a fatigued Cu single crystal [POL 93], b) diagram of a surface relief consisting of intrusions/extrusions

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Figure 6.14. Dislocation structures in low-cycle plastic fatigue [KLE 80]: a) on the surface, b) in the bulk

In 1973 Lukas and Klesnil [KLE 80] summarized their observations of dislocation structures after fatigue in BCC and FCC metals in the diagrams in Figure 6.14. These are often quoted even today. When stress concentrations at the root of intrusions are sufficient, microscopic cracks initiate there. These microscopic cracks will then at first propagate in neighboring grains, especially on the surface and a little deeper. This propagation occurs by sliding along the active slip planes of the grains, that is to say propagation in mode II (perpendicular sliding method). Next these microscopic cracks will coalesce and gradually propagate in mixed mode (I+II). During this first stage, the cracks are called “short”. Their propagation is slow and highly influenced by the microstructure. Indeed, in the course of their propagation, the short cracks are likely to be stopped at first, then only slowed down, by “micro-structural barriers” (grain boundaries, phase boundaries, etc.) which they are lead to encounter [MIL 86]. After propagation at a distance of several grains, the sensitivity to the microstructure gradually disappears. Finally one of the cracks thus created will become the main one (sometimes two). Henceforth, it will propagate in the depth in mode I (opening mode). The crack will then be termed “long”. Its propagation will be slow at first, then faster, until the conditions of instant fracture at the tip of the crack are satisfied, and the ligament is broken.

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Figure 6.15. Diagrammatic stress curves-number of initiation cycles and stress-number of propagation cycles

The moment of the appearance of the first microscopic cracks is defined by a “number of cycles at initiation” Na. The determination of Na largely depends on the resolution of the method used to detect microscopic cracks. The first observable cracks to the naked eye are about a tenth of a millimeter long; with a binocular magnifier they are a few tens of a micron; under a scanning electron microscope, they are about the size of a micron. However, the latter method in particular requires a dismantling of the test piece, and hence there is an interruption of the test, and the propagation velocity of short cracks (length of some grain sizes) being very slow, the number of initiation cycles can only be determined with great uncertainty. The propagation phase, properly speaking, covers, in the physical sense of the term, the rest of the lifetime, that is, a number of cycles equal to: Np =Nr - Na

[6.19]

called “number of propagation cycles”. The relative duration of the initiation and propagation phases vary with the level of the imposed strain. The ratio Np/Nr increases when this level increases, as is diagrammatically represented in Figure 6.15. If the major part of the lifetime is covered by the initiation phase in high-cycle fatigue, the propagation phase is generally the longest in low-cycle fatigue. Transmission electron microscopes (introduced in 1939) and scanning electron microscopes (introduced in 1964) have allowed the detection, analysis and understanding of the micro-mechanisms of damage and fracture in low-cycle plastic fatigue. Examination of thin foils under transmission electron microscopes enables us to follow the micro-structural evolution related to the accumulation of plastic deformation. Examination of the surfaces under a scanning electron microscope

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allows us to locate the initiation sites and understanding the processes of initiation, of cracks propagation, and final fracture. Nowadays development of in situ testing machines in scanning electron microscopes enables us to follow the damage micromechanisms and fatigue propagation in real time and at a microscopic scale. 6.3.3. Methods for low-cycle fatigue tests To be representative of the behavior in service of a real structure, low-cycle fatigue tests have to be conducted with the same loading ratio (minimum load/maximum load), at the same temperature, the same frequency and in the same environment as the real structure. In addition to be representative of the intrinsic behavior of the material, it has to be conducted on test pieces of constant crosssection; moreover, they would have to be carried out in vacuum to be particularly freed from any interaction between the cyclic plastic deformation and environmental corrosion. However, tests in vacuum or in a specific environment are more expensive than those carried out in the normal atmosphere. For more details in the low-cycle plastic regime, see [FOU 97]. The majority of low-cycle fatigue tests are uni-axial tests; nevertheless, multi-axial tests are also performed nowadays, but they also require equipments which is much more expensive than for uni-axial tests. The cyclic plastic deformation process being the origin of the damage in lowcycle fatigue of materials, the tests are generally deformation controlled. Controlled plastic deformation would be ideal; however, most of the time, a controlled total deformation is used. Certain laboratories perform controlled low-cycle fatigue tests with controlled load. In all cases the imposed signal can be sinusoidal, but it is more often in triangular form, which presents the advantage of a constant imposed loading rate (plastic deformation, total deformation or stress). A consultation held within the framework of a working group VAMAS-FranceFatigue oligocyclique/SF2MGT7 “Informatisation” issued a proposition at the origin of the standard AFNOR NF A 03-403 entitled “Practice of low-cycle fatigue tests” published in 1990 [AFN 90]. There was no corresponding European standard at this time. The standard uni-axial low-cycle fatigue test recommended by the French standard consists of submitting a test piece of constant cross-section to an imposed constant amplitude cyclic deformation, at a constant uniform temperature. The controlled signal is usually purely alternated. It is the test which will serve here as a reference (see Figure 6.16), while other types of tests will be compared to it. It is described in detail in [LIE 82].

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a) Geometric parameters of the test piece (1). 10: initial length of the gage. S0: initial cross-section of the gage. 1 = 10 + 'l. S: section for elongation 'l. b) Mechanical characteristics (1, 2). E: modulus of elasticity. F: applied load. V= F/S0: engineering stress. H = 'l/l0: engineering strain. Qe, Qp: elastic, plastic Poisson’s ratios. Vv= F/S: true stress. Hv = Ln(l/l0) = Ln(1 + H): true strain. c) Parameters for hysteresis loop (1, 2). H t : total strain rate. Values of the reference cycle (3). 'V 'F/S: stress variation. 'He, 'Hp,'Ht: elastic, plastic, total strain variation. Va = stress amplitude, (Ha = 'H/2)e,p,t: elastic, plastic, total strain amplitude. Values of the first quarter of the cycle. Va0 = 'Vstress amplitude. Hpa0 = 'Hp/2: plastic strain amplitude. d) Low-cycle fatigue laws. NR: number of cycles until fracture. Cp and c: coefficient and exponent of Manson-Coffin relation ('Hp= Cp.NR-c). Ce and b: coefficient and exponent of Basquin relation ('He= Ce .NR-b). K’ and n’: coefficient and exponent of cyclic hardening law (Va = K’Hpan’). K and n: coefficient and exponent of monotonous hardening law (1st quarter cycle) (Va0 = KHpa0n). Figure 6.16. Symbols and vocabulary used in low-cycle fatigue, according to standard AFNOR NF A 03-403

6.3.4. Experimental apparatus 6.3.4.1. Machines Low-cycle fatigue tests have to be performed on a hydraulic or electromechanical tensile-compression machine, controlled in a closed loop. The testing machine has to be controlled, either by displacement of the hydraulic cylinder (hydraulic machine) or of the crosshead (electromechanical machine), either in force or in deformation. The grips have to be adapted to the machine and to the shape of the test pieces. The machine and the grips have to be sufficiently rigid and be able to function smoothly during the change from tensile to compression and vice versa. The grips must enable a good alignment of the axis of the test piece with the loading line, in order to avoid buckling of the test piece in compression. Nowadays most machines are controlled by a computer, which also ensures the acquisition of measurements.

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6.3.4.2. Test pieces Low-cycle fatigue test pieces have to possess a gage length, representative of the volume element of the material which is studied. Their shape has to be adapted to the product of origin and to the shape of the grips. They generally consist of machined ends, which can be inserted according to need. The gage length, preferably cylindrical, should not be too slender to avoid the risk of buckling, but should be sufficient in order to ensure a uniform distribution of stress and strains and the possibility of adaptation of a sufficiently precise extensometer device. To determine the intrinsic properties of the material, the surface of the gage length of the test piece has to be polished after machining. The low-cycle fatigue test pieces can be of cylindrical (Figure 6.17) or flat (Figure 6.18), following the shape or dimensions of the initial product. In case of flat test pieces with low thickness, a stiffener device can be used in order to minimize the risks of buckling. Certain laboratories use diabolo shaped test pieces; the profile of the diabolo has to be sufficiently soft in such a way that the stress distribution can be considered uniform in any cross-section of the gage length.

Figure 6.17. Geometry of low-cycle fatigue cylindrical test piece drawn from a thick product, as recommended by standard NF A 03-403

Figure 6.18. Example of geometry of a low-cycle fatigue flat test piece drawn from a product of thickness 2.5 < e < 5 mm, as recommended by standard NF A 03-403

6.3.4.3. Instrumentation Investigation of the behavior in low-cycle fatigue of a material consists of analyzing the response of this material to the imposed cyclic loading: stress response

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for strain control, whether it is total or plastic, strain response for stress control. The lifetime, or the number of cycles until failure, is the other main result of a fatigue test. Hence, it is necessary to measure the load and the deformation of the gage length of the test piece and count the number of cycles. The load is measured using a cell with load calibrated strain gages. Deformation is measured with an extensometer, most often longitudinal, sometimes diametrical. At an ambient temperature, the extensometer generally consists of blades and strain gages. The ends of the blade are fixed to the test piece using spring systems and it is advisable to control the contact pressure in such a way that there is no sliding of the extensometer, but there is no marking either of the test piece, susceptible to be a preferred area for initiation of a crack. At high temperatures, an extensometer with extension rods and LVDT sensor or blades and strain gages is generally used, in such a way as to carry out the measurement sufficiently far from the furnace. Contact between the test tube and extension rods is often made using flanges machined on the test piece or small holes in the gage length. These solutions present the disadvantage of the existence of stress concentrations at the foot of the flange or at the bottom of the hole, defined by their geometry. A cycles counter makes it possible to follow the number of cycles. The system of acquisition, most often computerized nowadays, thus makes it possible to record in time load, deformation and number of the current cycle. Detection of stress peaks in strain control, and of deformation peaks in stress control, is highly valuable. Closed loop control of a low-cycle fatigue test machine from deformation or from load signals does not pose major problems (total strain or load controlled tests). On the other hand a plastic strain controlled test requires a calculator; closed loop control is then achieved with the result of plastic deformation calculation from total deformation and load, after entry of Young’s modulus E in the calculator: Hp

Ht

V E

[6.20]

Such a control poses the problem of the value of the Young’s modulus entered in the calculator. The Young’s modulus can be measured in the first quarter of the cycle or by preliminary cycling of the test piece in purely alternating controlled stress of low amplitude (strictly in the elastic field). The same type of problem is encountered in low-cycle fatigue tests on diabolo shaped test pieces where it is the diametrical strain which is measured. Strain control, whether it be total or plastic, requires a calculator and data of elastic (E, Qe) and plastic (Qp, generally taken as 0.5 owing to the hypothesis of iso-volume plastic deformation) characteristics of the material.

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6.3.5. Data processing During a standard low-cycle fatigue test, a test piece is loaded with total imposed deformation in a triangular purely alternated cycle, thus with a ratio of strain loading: RH

H min H max

[6.21]

1

under total constant imposed strain rate Ht , and total deformation variation per cycle:

'Ht

Ht max Ht min

const.

[6.22]

The reaction of the material to such a loading is a stress response; it results in a number of cycles at fracture Nr, called “lifetime”. The continuous recording of the total deformation Ht, of the stress V and the number of cycles N leading to an interpretation of the results in terms of: – evolution of the stress and stress-strain hysteresis loop in the cycle; and – fatigue resistance, expressed more often as the lifetime according to the applied loading amplitude. If the test is stress controlled, for cyclic hardening materials, the amplitude of stress at the beginning of the test must increase but gradually up to the set point in order to avoid damage or even premature fracture of the test piece due to overloading. The control signal may be chosen as sinusoidal. Nevertheless, this type of signal, of variable strain rate during a cycle, is to be avoided at high temperatures, when the dynamic of dislocations is strongly influenced by the strain rate. 6.3.5.1. Stress response: cyclic hardening/softening curves The stress response of the material evolves in the course of cycling. This response is then often represented by curvesVmax and Vmin, or Va = (Vmax - Vmin)/2 (stress amplitude) according to the number of cycles N, in bilinear and/or semilogarithmic diagram. The bilinear diagram shows clearly, with the proper time scale, the various stages of the response of the material; however, it has the disadvantage of “crushing” the evolution of stress amplitudes in the first cycles: it is then hardly discernible. The N logarithmic diagram makes it possible to clearly display this evolution during the first cycles, but it distorts the image of the relative durations of the various stages. These two diagrams are then used in a complementary way. Figures 6.19 and 6.20 give two examples of Va – N and Va – log10N curves respectively relative to ambient temperature fatigue tests on an

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austenitic X2CrNiMo17-12-2 stainless steel (Figure 6.19) and 350°C fatigue test on a martensitic X20CrMoV12-1 stainless steel (Figure 6.20).

Figure 6.19. Cyclic hardening/softening curves of austenitic stainless steel X2CrNiMo17-12-2 fatigue tested at ambient temperature [DEG 86]

Figure 6.20. Hardening/softening curves of a martensitic stainless steel X20CrMoV12-1 fatigue tested at 350°C [DEC 82]

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Two typical behaviors usually observed will now be described (refer to the experimental curves in Figures 6.19 and 6.20). Numerous variants exist, particularly concerning the first stage, called the “accommodation” stage. Consider a purely alternate total strain-controlled test and the stress response. During a first stage, which can last several tens and even several hundreds of cycles, the maximum and minimum stresses increase (Figure 6.21a) or decrease (Figure 6.21b) at each cycle according to the material and its micro-structural state. During this stage, the material is progressively adapted to the amplitude of the imposed total strain, either by hardening (in the case of annealed solid solutions in particular) or by softening (in the case of strain hardened materials in particular), or in a more complicated manner, which is not shown on Figure 6.21. This transitional stage is called the “cyclic hardening/softening stage”, or otherwise the “accommodation stage”; it gives its name to the Va – N and Va – log10N curves, called “cyclic hardening/softening curves”. It is during this stage that most of the micro-structural evolutions of the material take place: creation of defects, arrangements of those defects, etc., until a quasi-stabilized arrangement is reached. The level of the stress peaks is quasi-stable afterwards for a large number of cycles: it is the stage of stress stabilization, or “saturation” stage. During this stage, dislocations continue to rearrange themselves but much more slowly. The plastic strain is then essentially accommodated by the displacement of few dislocations between stable walls of high dislocation density (wall, cells, etc.). The existence of this stabilization allows us to associate stress amplitude with the amplitude of the imposed strain level. In low-cycle fatigue purely alternate stress-controlled tests, the response is strain. A strain hardening is then expressed by a decrease of the minimum and maximum total and plastic strains, whereas a softening is expressed in an increase of these quantities. Usually, the stress response of a material, which is stabilized in strain control, corresponds to a strain response, which is stabilized in stress control: the observed couple stress amplitude-strain amplitude is identical in both cases. Some materials do not display any stabilization of the response. The adopted reference response will then be chosen as the level obtained at mid life. At the end of the saturation stage, a drop of the stress amplitude in traction is displayed, owing to the initiation and propagation of microscopic cracks, which will end up in the final fracture. It is the stage of macroscopic crack propagation; its duration is relatively short (< 20% of life usually). Recall that crack initiation took place in the saturation stage, long before the drop of the stress. While cracks are short, the reduction in the area of the cross-section is negligible and it does not affect stress. On the contrary, when the crack is long, it propagates in the depth; the

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net section is reduced in an appreciable way; the stiffness of the test piece becomes smaller, and the maximum stress drops progressively up to the final fracture in a quarter of cycle in traction. The real propagation stage is thus much longer than the macroscopic propagation stage defined here.

Figure 6.21. Diagrammatic representation of the response of a material to a purely alternated low-cycle fatigue loading: a) cyclic hardening; b) cyclic softening

Considering that after a certain drop of stiffness the integrity of the structure is no longer ensured, the lifetime is defined conventionally as the number of cycles when a x% drop of the maximum tensile stress is observed. This number of cycles is also denoted Nx. More generally, the lifetime is denoted Nr (number of cycles to fracture); in that case the adopted fracture criterion must be defined before hand (separation of the test piece in two parts, x% drop of the maximum stress amplitude, etc.). The Va - HDcurve characteristic of stabilized state is defined point by point, each point corresponding to a low-cycle fatigue test at constant total or plastic strain amplitude or constant stress amplitude. The Nr/2 cycle is generally adopted as the reference stabilized cycle. This curve is called the “cyclic hardening curve” or the “cyclic consolidation curve” of the material. To be obtained, a test series must be performed with the same conditions of cycle form, of RHV ratioor of mean level ofthe control parameter) and of temperature.

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The cyclic hardening curve is often compared with the monotonous strainhardening curve (stress-strain curve recorded in a tensile monotonous test or given by the first quarter of cycle of the low-cycle fatigue test); the difference between the two curved is characteristic of the accommodation of the material, hardening and/or softening (Figure 6.22). For a material, which does not display a stress stabilization stage, the amplitude of the reference response (stress or strain) is also defined at mid-life.

or

Figure 6.22. Monotonous and cyclic consolidation curves: a) cyclic hardening material; b) cyclic softening material

6.3.5.1.1. Methods for obtaining the cyclic consolidation curve As has just been explained, obtaining the cyclic consolidation curve requires a certain number of low-cycle fatigue tests to have been performed at different levels of strain amplitude; this method of deriving the curve, called the “one test piece per level method”, is therefore time consuming and costly. It is for this reason that several other methods of obtaining the cyclic consolidation curve were proposed from one single test (thus using a single test piece). The name of these methods and the corresponding loading mode are described in Figure 6.23.

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By the one test piece per level method, the cyclic consolidation curve is defined as the locus of the reference stress amplitude points (stabilized or at 50% lifetime) according to the total imposed strain amplitude, each point corresponding to a lowcycle fatigue standard test, so to one tested specimen. In the successive increasing levels method, each level is carried out up to stabilization of the stress response. The cyclic consolidation curve is then defined as the locus of the points of stress amplitude at the last cycle of the level according to the total imposed strain amplitude.

Figure 6.23. Methods of obtaining the cyclic consolidation curve

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The successive decreasing levels method is analogous to the preceding one, the order of levels being reversed. For incremental and random methods, which both use blocks of identical successive loadings, incremental for some and random for others, it is necessary to perform a Rain-Flow analysis [AFN 93] to identify stress-strain closed cycles which are supposed to be purely alternated (mean stress and strain equal to zero). In both methods, the cyclic consolidation curve is defined as the locus of the points stress amplitude-strain amplitude related to closed cycles defined from a stabilized stress loading block. Using the tensile testing after cycling method, the cyclic consolidation curve is defined as the monotonous tensile curve obtained after successive incremental loading blocks applied up to stabilization of the stress response. Using the half hysteresis loop method, the cyclic consolidation curve is defined, from a constant total strain amplitude controlled test, as the ½ ratio proportional curve of the ascending portion of the stabilized hysteresis loop. In fact, these methods do not usually yield a single cyclic consolidation curve. Materials are indeed more less affected by the loading history (memory effect) and, of course, the more they are affected, the more the cyclic consolidation curves differ from one another. When the imposed maximum strain amplitude is the same for all methods, all cyclic consolidation curves converge towards the same maximum point. The incremental and random methods yield almost the same cyclic consolidation curve, which is often the same as the monotonous tensile curve obtained after cycling. At last, the relative positions of the different curves in comparison with one another and with the monotonous tensile curve depend on the initial metallurgical structure of the material and of its evolution during cycling. The two following figures show as an example the cyclic consolidation curves obtained by different methods on a X2CrNIMo17-12-2 austenitic stainless steel (Figure 6.24) [MOU 97] and for a X2CrNiMoN22-5-2 duplex stainless steel (Figure 6.25) [DEG 99]. 6.3.5.1.2. Cyclic consolidation law By analogy with the monotonous work hardening curve, the cyclic consolidation curve is often represented mathematically by a power law:

'V 2

Va

§'H p ·N c K c¨ ¸ © 2 ¹

[6.23]

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Figure 6.24. Cyclic consolidation curves obtained by different methods for an austenitic stainless steel X2CrNiMo17-12-2

This is also known as the cyclic consolidation (or hardening) law. The coefficient K’ and exponent N’ of the cyclic consolidation law are respectively called the coefficient of cyclic resistance and the cyclic hardening exponent as standard. Each cyclic consolidation curve associated with one of the preceding methods may correspond a cyclic hardening law characteristic of the associated loading history. Moreover, it is necessary, in the design of a variable amplitude loaded structure, to consider the cyclic consolidation curve characteristic of the real service loading history of the structure. On the other hand, it should be recalled that the cyclic consolidation curves, which have been defined, concern the uni-axial behavior of the material. Under multiaxial loadings, the behavior of the material can be much more complex. 6.3.5.2. Stress-strain hysteresis loops The cyclic consolidation curve should not be confused with the stress-strain relation in each cycle: the latter is defined by the stress-strain hysteresis loop. Continuous stress-strain recording at the beginning of a low-cycle fatigue test are schematically given in Figure 6.21, for materials which harden (Figure 6.21a), or soften cyclically (Figure 6.21b). A low-cycle fatigue constant deformation amplitude test is generally characterized in terms of characterizing parameters of the stabilized hysteresis loop (or hysteresis loop at 50% Nr for non-stabilizing materials) defined in Figure 6.16.

164


Figure 6.25. Cyclic consolidation curves obtained by different methods for duplex stainless steel X2CrNiMoN22-5-2

The total strain variation 'Ht, breaks down into plastic strain variation 'Hp, and elastic strain variation 'He: 'H t

'H e 'H p

[6.24]

'Hp is often measured by the width of hysteresis loop at zero stress. This assumes that the linear portions VH extend down to complete unloading of the test piece (V 0), which is far from always being the case. In the opposite case, the following relation is used: 'H p

'H t

'V E

[6.25]

where E is the modulus of elasticity measured initially (by preliminary cycling of the test piece in the elastic range for example). It is logical to use 'He (or 'V and especially 'Hp as damage parameters of the material in the expression of low-cycle fatigue resistance laws (see section 6.3.5.3). The analysis of hysteresis loop allows us on the other hand to investigate the evolution of the possible stress and cycles departure from symmetry during cycling. It thus enables an analysis of the Bauschinger effect of the material.

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The area of the stabilized hysteresis loop , 'Ws is also an interesting parameter. It represents the mechanical energy dissipated per cycle and it amounts to a damage parameter, which combines stress and strain. When the cyclic consolidation curve can be well defined by the half hysteresis loop method explained earlier, then the cyclic strain hardening exponent N’ can be used to calculate the area of the hysteresis loop by the formula, which was proposed by Halford and Manson [HAL 67]: 'W s

'V'H

1 N c 1 N c

[6.26]

6.3.5.3. Resistance to low-cycle fatigue curves The low-cycle fatigue consolidation curves previously defined are needed to understand the service behavior of structures under low-cycle fatigue loadings. As for the resistance to low-cycle fatigue curves of materials, they are needed to predict the service durability of structures. The diagrams of resistance to low-cycle fatigue are expressed in terms of the total strain variation 'Ht, plastic strain variation 'Hp and elastic strain variation 'He according to the life-time Nr. In a bi-logarithmic diagram the curves 'Hp- Nr and 'He- Nr can be adjusted by straight lines; from which power law relations can be written: 'H p

C p N rc

[6.27]

known as the Manson-Coffin relation [COF 54, MAN 54], and:

'H e

C e N rb

[6.28]

issued from Basquin relation [BAS 10]: 'V 2

Va

DN rb

[6.29]

From which, the relation total strain variation-fatigue life: 'H t

'H p 'H e

C p N rc C e N rb

[6.30]

represented by a concave curve directed upwards (Figure 6.26). Relations [6.27], [6.28] and [6.30] are often expressed not in terms of variations of deformations

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according to number of cycles to fracture Nr but in terms of strain amplitude according to number of alternations to fracture (2 Nr):

'H t 2

'H p 2

'H e 2

c §Vc · b Hcf 2 N r ¨ f ¸ 2N r

[6.31]

© E ¹

In this expression, H’f is called the fatigue ductility coefficient and –c the fatigue ductility exponent; V’f is called the fatigue resistance coefficient and –b the fatigue resistance exponent. The Basquin relation is then written: b

V a Vcf 2N r

[6.32]

It was also proposed to express the low-cycle fatigue resistance law in terms of the plastic strain energy. The total energy dissipated Wt in a low-cycle fatigue test can be evaluated with very good approximation by: Wt

[6.33]

N rW s

On the other hand, it was shown that the following relation exits between the dissipated energy at the stabilized cycle and the plastic strain variation [ELL 97]: 'W s D'H Ep

[6.34]

1,000

10,000

100,000

1,000,000

Figure 6.26. Low-cycle fatigue resistance curves relative to martensitic stainless steel X20CrMoV12-1 tested at 350°C [DEG 82]

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a) adaptation

b) accommodation

167

c) ratcheting

Figure 6.27. Schematic representation of low-cycle fatigue behaviors in loadings at non-zero mean stress: a) adaptation; b) accommodation; c) ratcheting

Considering Manson-Coffin relation [6.27], the total dissipated energy is related to the fatigue life by: Wt

CW N re

[6.35]

The low-cycle fatigue resistance curves and laws are characteristics of the material, of the imposed loading rate, of the imposed loading ratio (RH Hmin/Hmax or RV Vmin/Vmax according to the loading mode), of the temperature and of the environment. If one of these parameters is modified, these characteristics can also be modified. 6.3.6. Ratcheting effect If required by service loading representativeness, low-cycle fatigue tests can be strain or stress constant amplitude controlled with zero mean level. When the test is strain controlled with a mean strain, which is not zero, in many materials the initial absence of symmetry of the stress response disappears progressively by relaxation of the mean stress. The material behavior is then identical to the one previously described; it is akin to that of a pre-strained material. When the test is stress controlled with a mean stress, which is not zero, the response of the material can be more complex. Three types of response can be observed: – after a few cycles (or a few tens of cycles), the behavior can become purely elastic (Figure 6.27a): it is said that there is “adaptation”;

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– after a few cycles (or a few tens of cycles), the behavior of the material can remain elastoplastic, but after an initial increase of the mean strain level at each stress cycle, a stabilized cycle is reached (Figure 6.27b); it is said that there is “accommodation”; – even after a large number of cycles, the mean strain level at each stress cycle keeps increasing (Figure 6.27c): it said that there is progressive deformation or that the material display a ratcheting effect. The progressive deformation phenomenon is akin to a creep phenomenon. 6.4. Measurement of the crack propagation rate in fatigue Preliminary remarks This section presents the most widely used standard fatigue crack propagation test. It essentially concerns metallic materials, constant amplitude fatigue loadings and mode I of crack propagation. The end of the section will nevertheless address, in a sketchier manner, more elaborate or specific tests (tests in environment, under random loading, tests on nonmetallic materials, etc.), which are generally not so much used except in particular industries and laboratories. In the discussion, it will be assumed that the main concepts of fracture mechanics are known (refer in particular to [FRA 93, SUR 91]). 6.4.1. Introduction 6.4.1.1. Fatigue crack initiation and propagation Applying cyclic loading (of constant or variable amplitude) to a part leads to damage, which includes two stages. The first stage, called the initiation stage, corresponds most often to the initiation of micro-cracks on the surface of the part from persistent slip bands; these cracks propagate across a few grains following crystallographic directions [MIL 97, SUR 91]. The second stage corresponds to the propagation of those cracks in a plane perpendicular to the applied load direction, propagation which continues generally until final fracture. a) Initiation Most often (in most transportation industries, hoisting engines, etc.) parts are designed so as to be in the domain where micro-crack propagation does not create macro-cracks. This domain corresponds to the endurance (below the fatigue limit).

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The test on which the design rests is the fatigue limit test on test pieces of constant cross-section. This test is presented in section 6.2. b) Propagation However, parts can include defects (welding defects, casting defects, etc.) and they could fracture in service for loading amplitudes below the fatigue limit. This situation gives rise to an evaluation of the fatigue resistance of parts based on crack propagation data (obtained from crack propagation rate tests). This approach is justified as far as, considering the dimension of real structures, the duration of crack propagation covers an important part of the lifetime (aeronautics, nuclear industry, etc.).

Figure 6.28. Schematic representation of the various stages of fatigue crack propagation (according to [MAS 97])

In the case of a pre-existing defect in a part or of the initiation of a crack (which is easier and easier to detect owing to the improvements of non-destructive testing), two types of problems may be treated: – evaluation of the crack propagation rate (or of the defect considered as a crack) so as to estimate the residual lifetime of the part. This problem is well solved using linear elastic fracture mechanics and stress intensity factor K [FRA 93, SUR 91]; – evaluation of the loading for which cracks do not propagate: in that case the stress intensity threshold 'K0 is determined. The threshold corresponding to “long” cracks is only considered here. The case of so-called “short” cracks or “micro-structural” cracks requires the use of elastoplastic fracture mechanics. Approaches to handle these problems are described in several books and papers [MIL 97, SUR 91].

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Three stages of (“long”) crack propagation are generally distinguished (Figure 6.28): – stage I: when loading is below a certain threshold, cracks do not propagate at the macroscopic level. In that case the fatigue crack propagation threshold determination test is performed. The considered quantity is in some way the equivalent of the endurance limit determined with test pieces of constant crosssection; – stage II: cracks propagate (trans-granular in polycrystalline materials) most often in the plane perpendicular to the applied load. The crack propagation rate test corresponds mainly to stage II; – stage III: this corresponds to high loadings for which tearing occurs at the crack tip. The fast rates observed in this stage quickly lead to fracture of the test piece. The rate measurements in this domain are implemented only in some specific cases (leak before break in tubes, etc.). In the following the testing methodologies related to stages I and II will be presented. Particular attention will be given to questions of test and measured quantities validity. Notions to facilitate the presentation of the fatigue crack propagation rate are presented. For more details, refer to the books and papers mentioned above. 6.4.1.2. Recalling the notion of plastic zone at the crack tip The elastic limit in the immediate vicinity of the crack is exceeded under monotonous loading. A plastic zone is created the dimension of which is related to the level of applied load and to the crack length. Irwin [IRW 60] was the first to propose a model giving the dimension of the plastic zone (using fracture mechanics concepts). For zones which are circular he obtained in mode I: r

K I2

2SR 2p

[6.36]

in plane stress; r

K I2

6SR 2p

[6.37]

in plane strain. The finite element method and consideration of realistic plastic flow rules enabled more precise determinations of the plastic zone size, which consists of two wings. Two factors are considered: rx plastic zone size in the direction of the crack

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171

and ry in the direction perpendicular to the crack. The model mostly used appears to be that of Rice [LEV 97]. Several relations are given in the literature for different situations (see [BAT 97] for a summary). In cyclic loadings the plastic zone is more complicated. In particular, it includes a second plastic zone, resulting from the local compression, which takes place at crack closure upon unloading, even when in traction. Rice [RIC 67] developed a model to account for the creation of a plastic zone at the crack tip under cyclic loading. In this model the size of the peripheral plastic zone (the monotonous plastic zone) is of the form (Kmax/Rp)2 and that of the central zone (the cyclic plastic zone) is of the form (Kmax/2Vf’)2. Vf’ is the cyclic yield strength determined from the cyclic hardening curve (section 6.3). For more details concerning this section, see [BAT 97, FRA 93, SUR 91]. These few recollections will help in introducing the concept of crack closure referring to the cyclic plastic zone (section 6.4.1.4) and the concept of validity referring to the monotonous plastic zone (section 6.4.2.2.3). 6.4.1.3. The Paris law Models were elaborated to describe at best fatigue crack propagation. They aim to relate the crack propagation rate da/dN (increase of crack length per cycle) to parameters, which influence this propagation. “Theoretical” models relying on various hypotheses concerning cracking mechanisms were introduced [MAS 97]: models based on dislocations theory, on the material behavior at the crack tip, on low-cycle fatigue laws, etc. Those models do not describe all the encountered situations well and are not easy to implement, so that empirical laws are preferred in design. Among these, the Paris law prevailed [PAR 61, 63]: da dN

C'K m

[6.38]

where 'K is the variation of the stress intensity factor, which is a function of the loading (Fmax – Fmin), of the crack length, of the geometry of the part and of the crack. C and m are material parameters. This relation is applicable in the case where linear elastic fracture mechanics is valid. It concerns stage II of crack propagation (see section 6.4.1.1). Most parameters which operate on crack propagation have no or little influence. The parameters which are most influential are in particular the Young’s modulus (the higher the Young’s modulus, the slower the propagation: variation in 1/En with n close to 2) and corrosion, provided the loading frequency is low [MAS 97]. Tests for

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fatigue crack propagation rate measurements concern the determination of the coefficients of the Paris law. 6.4.1.4. Crack closure phenomenon Elber [ELB 70, 71] demonstrated that a fatigue crack in an aluminum plate could close when load decreased even when it remained tensile. This phenomenon is due to residual strains, which are created during crack propagation (see section 6.4.1.2). To highlight this phenomenon, Elber recorded the displacement G at the crack tip according to load with the help of an extensometer (Figure 6.29).

Figure 6.29. Opening and closing of fatigue crack (according to [BAT 97])

The crack is opened between C and D, partially closed between B and C and closed between B and A. The value of the opening load Pop (increasing load) is deduced from this recording; the closure load Pcl (decreasing load) can also be measured. Assuming that the crack does not propagate when closed, Elber proposed replacing 'K K max K min (corresponding to a complete cycle) in the Paris law by the variation of the effective stress intensity factor 'K eff K max K op (corresponding to the part of the cycle when the crack is opened).

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The measurements of crack closure during fatigue crack propagation tests is highly recommended as there are other phenomena than those mentioned by Elber at the origin of crack closure: – fabrication processes can introduce complex residual stress fields in industrial parts; this is the case for instance for heat treated parts (quenching as an example) or welded parts; – account must be taken of residual stresses, which are not completely relaxed following machining; – when a test piece is in an aggressive environment during fatigue crack propagation testing, an oxide deposit can occur at the crack tip, which induces closure on part of the load cycle. The following section tests will present the means to be used for the measurement of crack closure with a good precision and the methodology for the presentation of test results in terms of 'Keff.

Figure 6.30. Principle of fatigue crack propagation rate test

6.4.2. Implementation of crack propagation rate measurements 6.4.2.1. Principle of the test This test consists of applying cyclic loading to a notched test piece (Figure 6.31). The crack initiated at the notch root grows during cycling. Its length is measured using one of the several measuring techniques at the disposal of the user. From the values of a and N (once the pre-cracking stage is passed: see section 6.4.2.4.2), the values of the crack propagation rate are deduced. Using constant amplitude cyclic loading tests, the Paris law expressing the propagation rate according to the variation of the stress intensity factor 'K is determined. With the help of decreasing amplitude tests, the fatigue propagation threshold 'K0 is determined.

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The test is described in several standards, which should be respected: ASTM [AST 647], AFNOR [AFN 91], ISO [ISO 99]. In the following text, reference is made to the AFNOR standard, the details of which will be found by consultation. 6.4.2.2. Test pieces 6.4.2.2.1. Geometry The test piece must possess the same metallurgical structure as the material for which crack propagation rates are sought. Several types of test pieces can be used, because, when the material is homogenous and the conditions of validity of linear elastic fracture mechanics are respected, the results are independent of the geometry of the test piece. The recommended test pieces are the compact tension test piece (CT), the central notched test piece (CCT), the three points bend test piece (SENB3), the four points bend test piece (SENB4), the eight points bend test piece (SENB8) and the side notched test piece (SENT). The choice depends on requirements to be respected (type of product, amount of available material, type of available equipment, test cost, experience of the laboratory, etc.). Figure 6.31 shows the geometry of the most frequently used test pieces.

a) CT specimen

b) 3 points bend specimen (SENB3) Figure 6.31. Compact tension and 3 points bend specimens (according to [AFN 91])

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175

The AFNOR standard provides information about notch fabrication as well as about tolerances and recommended surface qualities. To obtain a straight enough crack front (see the conditions of validity in section 6.4.2.4), side grooves are often machined. The depth of these side grooves is generally equal to B/10 for most steels. The direction of removal in the product needs to be specified. The conventions used are given in the standard [ISO 95]. 6.4.2.2.2. Stress intensity factors The calculation of 'K ('Kmax – 'Kmin) requires the knowledge of the relation between K and the loading and the test piece geometry. This is expressed in the form [AFN 91]: K

FY

[6.39]

BW 1 2

for F > 0 K

0

[6.40]

for F 0 The relations between Y and geometry as well as the domain of validity of Y for the most widely used test pieces are the following: a) compact tension test piece (CT):

Y

2 D 0.886 4.64D 13.32D 2 14.72D 3 5.6D 4 1 D 3 2

[6.41]

D= a/W, domain of validity a/W > 0.2 b) 3 points bend test piece (SENB3):

Y

ª º 6D 1 2«1.99 D 1 D 2.15 3.93D 2.7D 2 » ¬ ¼

1 2D 1 D 3 2

[6.42]

D a/W domain of validity: any value of a/W For other types of test pieces, refer to the standard. For test pieces with side grooves, the same formulae are used, replacing B by Bnet denoting the net section area.

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6.4.2.2.3. Test piece size: domain of validity of results The test pieces used must have minimum dimensions to respect conditions of linear elastic fracture mechanics (global elastic behavior of the test piece, plastic zone size of sufficiently small dimension with respect to the dimensions of the test piece, for the propagation distance considered). The conditions of validity define a ligament width (W – a) not to be exceeded. The relation is expressed as a function of Kmax (or Fmax) and of the flow strength Rf Rf = Rp0.2 if Rm/ Rp0.2 1.3

[6.43]

Rf = (Rp0.2 + Rm)/2 if Rm/ Rp0.2 > 1.3

[6.44]

Conditions of validity for the most widely used test pieces are given below: a) compact tension test piece (CT) [HOE 74]: 2 4 §K max · ¸ W a t ¨ ¸ S ¨ © R f ¹

[6.45]

b) 3 points bend test piece: ·1 2 §3SF max ¸ W a t ¨ ¨ 2BR ¸ f ¹ ©

[6.46]

(B or Bnet), with S (distance between supports) = 4W 6.4.2.3. Equipment 6.4.2.3.1. Testing machine The machine may be electromechanical or electrohydraulic. It must meet several requirements: the axes of the part fixed on the load cell and of the part fixed on the hydraulic cylinder must be aligned in the area where the test piece and the grips with the desired precision will take place to avoid a crack front which would not be symmetric. Several methods exist to achieve this alignment [LIE 82]. A simple method consists of using a comparator (Figure 6.32). All these methods require the possibility of precise slight displacements of the load cell with respect to its mean position, which is not easy on all machines. It must provide a reproducible fixing of the test piece with a convenient gripping device, minimizing the number of mechanical interfaces. Several gripping devices

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177

exist: hydraulic grips or making use of Wood metal in particular for performing tension-compression tests, single ball and socket joint, double clevis mounting [LIE 82]. For CT specimens a system using a fixed clevis mounting and a free rotating clevis mounting (hydraulic jack) seems to be the most satisfactory. Rigorous machining tolerances should allow us to keep the alignment in the space devoted to the test piece. The material for the grips should be chosen according to test temperature so as not to experience any plastic deformation while testing. The machine must maintain a constant loading during the test duration (load control). It is generally useful to be able to generate cycles of various forms (triangular, sinusoidal, etc.) within a frequency range between 0.1 Hz for tests in a corrosive environment and 100 Hz to keep down the test duration in air.

Figure 6.32. Checking the alignment of the gripping device (according to [LIE 82])

It must be equipped with a load cell with mechanical and electronic qualities required for correct application of load on the test piece, and this is generally the case with modern machines. The user should nevertheless ensure that the capacity of the load cell is adapted to the applied loads. In practice, using the load cell below 20% of its capacity is not recommended. Regular calibration and inspection of the load cell must be implemented. The load measuring system must be connected to the calibration channel of the National Bureau of Metrology (NBM). Cycle counting must meet precise requirements [AFN 91]; in particular it must have different counting ranges adapted to the expected number of cycles, connected to the test frequency.

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6.4.2.3.2. Methods to measure the crack length The crack length must be measured with sufficient precision. This length must be followed while testing. The most widely used methods are the following: – optical method; – compliance method; – Foucault current method; – ultrasonic method; – electric method. Information concerning these methods can be found in [LIE 82]. Other not so widely used methods, such as interferometric methods, are also presented in this book. The optical method was the first to be used; other methods appeared later. Its implementation is demanding, as the technician must be present during the whole test duration. It also requires that the crack tip be accessible at all times. Also, the crack tip is not always easy to detect even using a stroboscope (measurement at the upper part of the cycle so as to better see the crack tip). It is nevertheless useful for controls (sections 6.4.2.4.2 and 6.4.2.5). The most widely used methods are the compliance and the potential drop methods. The compliance method, which is very useful when measuring crack closure, is less suitable for high temperature measurements: the extensometer (used for opening measurement) must be protected from the temperature elevation. The standard potential drop method can be used for a wide range of temperatures and frequencies and in various environments. These methods are described later. They are indirect measurements. Indications will also be given about the ultrasonic method (applicable to ambient temperature tests), whose implementation is demanding but which allows more results to be obtained. a) Compliance method This method is based on the relation between the stiffness of the test piece and the crack length. Practically, the compliance of the test piece, which is the inverse of its stiffness, is measured. It requires the determination of the crack mouth opening displacement with the help of various sensors. The strain gages sensor is the most widely used. These sensors must meet several conditions to ensure a good reproducibility of results [AFN 91]. In particular, they must be adapted to the test frequency (not entering in resonance). The sensor measure the opening either on the front end of the test piece or on the loading line. The load F is recorded according to the mouth opening V with an analog recorder or, more and more frequently, load and displacement acquisition is

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performed (Figure 6.33). The slope 'V/'F is determined along the linear portion of the cycle. Computerized measurement is recommended, as the measurement on a flatbed plotter is possible only at relatively low frequencies, generally lower than the test frequency. Also, even at low frequency, effects due to inertia and undue use of electrical filtering lead to a possibly distorted cycle recording [SF2 93]. The relations between the normalized compliance EBV/F (in the case of grooved test pieces Be = B – (B – BN)2/B is used) and crack length were determined for various types of test pieces.

Figure 6.33. Compliance method (according to [AFN 91])

The results are given here for the compact tension (CT) specimen and the 3 points bend specimen: – CT specimen [HUD 78]: a W

Ux

C 0 C1U x C 2U x2 C 3U x3 C 4U x4 C 5U x5

1 § 'Vx ·1 2 ¨EB ¸ 1 © 'F ¹

[6.47] [6.48]

'Vx/'F is the compliance measured, at a distance x from the loading line. The values of the coefficients C0 to C5 are given for measurements on the load line and on the front face of the test piece [AFN 91]. – 3 points bend specimen [WU 84]: a W

0.999748 3.9504U x 2.982U x2 3.21408U x3

51.51564U x4

113.03U x5

[6.49]

180


Ux

1 ª º1 2 'V WEB x S 4» 1 « ¬ ¼ 'F

[6.50]

with Vx the crack opening measured on the front face, S the distance between supports = 4W NOTE.– To ensure quality of measurements, readjustment of theoretical measurements with respect to the experimental measurements must be performed by determining the crack length at the end of pre-cracking and at the end of the test on the fracture surface of the broken test piece (measurement at 9 points along the crack front; see section 6.4.2.5.1). The adjusting parameter is the Young’s modulus. The final crack front is visualized by marking at the end of the test: it is obtained by cycling (to break the test piece) at a different temperature from that of the test and taking care to avoid excessive deformation. b) Potential drop method The method consists of passing a direct electric current in the test piece (which is isolated from the machine) and to measure the resulting voltage across both sides of the notch. This voltage increases as the crack propagates (Ohm’s law). Preliminary calibration (depending on the geometry of the test piece but not on its size) allows us to determine the relation between the electric voltage and the crack length. The material is assumed to be electrically isotropic. Two types of technique can be used: the direct current technique and the alternating current technique. The second gives a potential field, which is identical to that of the first, except for frequencies above 100 Hz for which a skin effect takes place. The signals given by these techniques being very weak, they must be amplified (factor of 103 to 105). DC measurements can be disturbed by noises, electrical drifts and thermo-electrical effects. Chopped direct current eliminates these superimposed effects on the measuring current. When such a method is not available, corrective measures can be implemented [AFN 91]. Alternating current technique eliminates thermo-electric effects, noises, drifts and moreover gives a better sensibility than the DC technique. On the other hand, its implementation is more exacting and when the skin effect is too strong, it provides the surface crack length only. Calibration provides the relation between the measured voltage and crack length. It can be expressed in the following way:

1)

U U0

§ a · f ¨ ¸ ©a 0 ¹

[6.51]

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a W

§ a · f ¨ ¸ ©W ¹ §U · f ¨ ¸ ©U 0 ¹

a W

·1 2 §U f ¨ 1¸ ©U 0 ¹

2)

U U0

181

[6.52]

U0 is the reference potential and a0 is either the length of the notch or the length of the initial crack (measured on the fracture surface of the broken test piece). U0 is generally the voltage measured on the test piece for initial crack length a0. U0 can be the voltage measured on the test piece between two points distinct from the measuring points. U0 can be the voltage measured on a test piece of the same material for which environment conditions are the same as those of the tested test piece. The use of a reference potential enables us to get rid of variations due to the material and to measuring instruments and thus to improve the sensibility of crack length measurement. The relation U/U0 = f(a/W) is the most widely used. Note that the calibration curve for a given type of test piece is independent of size, of current intensity and material, provided the dimensions of the given test piece, the positions of feeding and measuring plugs are proportional. Calibration curves are usually given in a polynomial form. Examples of such curves for CT and CCT specimens with specified positions of electrical connection are given in the AFNOR standard [AFN 91]. Calibration needs crack length variations. This can be achieved in two ways (ensuring that the proportionalities mentioned above are respected): – using several test pieces identical to those used in the test, on which several front marks are put; – using mock-ups of large dimensions, which may have a low thickness (saving material), and in which crack propagation is simulated by machining notches of various lengths. Whatever the method, it is advisable to achieve several markings on real test pieces in order to perform controls. In particular, it will be useful to measure the final crack length on the fracture surface of the broken test piece, in order to eventually readjust it. The implementation of this technique requires us to respect several conditions and some precautions needed for a successful test (precision and sensibility of measuring instruments, choice of the position of current plugs, etc.). Technical details are not described in this book; it will be useful to refer to the standard [AFN 91].

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c) Ultrasonic method Several techniques using this method were developed [LIE 82]. The MU3F method, presented briefly here, is one of the most interesting [VAD 95]. The method is based on the diffraction of an ultrasonic beam focused on the crack tip (Figure 6.34). The observed echo-dynamic peak is linked with the crack front. It is used to detect and follow the crack tip. For that purpose, an immersion transceiver can be moved in two perpendicular directions X and Y associated with the planes of symmetry of the test piece, thanks to an appropriate mechanical device. The displacements of the transceiver are controlled by a micro-computer, which records the displacement and amplitude of ultrasonic signals. The CT specimen is well adapted for this kind of measurement. Adjustment enables us to give adequate incidence angle i to the ultrasonic beam. The method makes it possible to detect a mere 0.05 mm crack.

Figure 6.34. MU3F method (according to [VAD 95])

Following the crack is easy to perform: the echo diminishes to the right of the transceiver when the crack grows. What is needed is to move the transceiver so as to recover the maximum amplitude thanks to the control. The displacement of the transceiver may be triggered every N cycle or when a predetermined amount of decrease of the echo amplitude has been reached. The precision of the measurement is 0.1 mm. The method allows us to determine the position of the crack front along two perpendicular directions, which can bring useful information in particular when using large test pieces. The crack front position is determined with a resolution of 0.03 mm in the propagation direction and of 0.8 mm in the transverse direction. The

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method also allows to measure crack closure. It can be performed thanks to the fact that the closed part of the crack is transparent to ultrasounds. The measurement is achieved by cycling interruption and monotonous loading of the test piece. This method is not adapted for high temperature testing. On the other hand, its implementation is rather demanding and for this reason it is not used as much as the preceding methods, but it allows us to obtain useful information about cracking in case detailed analyses are needed. 6.4.2.4. Performing the test 6.4.2.4.1. Preliminary checks The technician must ensure good quality of the testing environment: constant temperature, hygrometric degree, etc. [AFN 91]. In the case of standard tests the form of the cycles is sinusoidal. It must be generated by a good adjustment and a good reliability of the control so that it remains constant during the test duration [AFN 91]. Dispersions generally observed on metallic materials lead us to perform a minimum of two tests for predetermined conditions. 6.4.2.4.2. Pre-cracking Pre-cracking is a must because the crack needs to be far enough from the notch perturbed zone, which exclude the performing of any measurement in this zone. Precracking imposes several loading steps. The loading must be high enough at the beginning to initiate a crack at the notch root, and low enough at the end of precracking so as not to disturb the crack propagation rate at the beginning of the measurement proper. The choice of the first loading results from a compromise: not too low, which could produce excessive initiation durations and not too high, which could deform the test piece and multiply the pre-loading steps. In practice, a slightly conservative loading might be chosen at first and increased if initiation is not produce in 30,000 cycles and so on. The standard [AFN 91] gives recommendations to achieve the (decreasing) pre-cracking loading steps in a satisfactory way. The indications concern the load reduction at each step and the level of crack growth at each step depending on the crack tip plastic zone size (see section 6.4.1.2 concerning the plastic zone concept). The aim in pre-cracking is to reach a minimum crack length (larger than 0.1 B, than e width of the grooves and than 2.5 mm) and a final value of Kmax smaller than or equal to the required Kmax value for the test (the choice of Kmax and of Kmin gives a propagation rate at the beginning of testing of about 10-5 mm per cycle corresponding to the beginning of stage II). At the end of pre-cracking the technician measures (optical method) the crack length. It must meet several requirements: a difference between the measurements on the two faces of the test piece less than a given value, and an angle between the crack plane and the notch plane less than a given value [AFN 91]. If these requirements are not met, the technician must look for the reason of the imperfection: misalignment of the machine, gripping defects, poor machining of the test piece, etc.

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6.4.2.4.3. Test at constant load amplitude The loading amplitude remains constant for the whole duration of the test. It is characterized by a constant frequency, by Fmean and 'F: (Fmax - Fmin) constant so that the load ratio Fmin/Fmax remains constant. It results that the test is performed with increasing 'K (refer to the formulae of section 6.4.2.2.2). The loading is established so as to achieve a crack propagation rate of 10-5 mm/cycle (see above). The test is stopped when the crack propagation rate is at most equal to that of the beginning of stage III in standard tests. Computerization of the test allows us to detect quickly when the law of crack propagation is not that corresponding to stage II anymore. When crack closure measurement is not foreseen, a sufficiently high R ratio ought to be chosen so as to avoid crack closure. A value of the R ratio exists for which (above this value) the curve in term of 'K coincides with the one in term of 'Keff. To measure crack closure using the compliance method, a flatbed plotter recording should be provided, or better the computerized data should be extracted. Decreasing the loading frequency during measurement, the values of Pop are determined, enabling us to calculate Kop (see section 6.4.1.4). Particular devices exist [ROB 77, TRU 77] to detect with great precision the change of slope that can be observed on the loading cycles. Note that it is also possible to measure closure using the ultrasonic method (see section 6.4.2.3.2) and by interferometric methods. An interrupted test may be resumed provided it is checked that the propagation rate measured after interruption is identical with the one measured before. 6.4.2.4.4. Decreasing loading amplitude tests This is the test which corresponds to the measurement of the crack propagation threshold stress intensity factor (see section 6.4.1.1). The range of propagation rates concerned corresponds to rates below 10-5 mm/cycle (stage I). The test, which requires no interruption, is performed by decreasing the load amplitude, so that 'K is decreasing, and the load ratio R and the frequency are kept constant. The initial loading is larger than the one achieved at the end of pre-cracking. It is possible to decrease the load by steps or in a continuous manner by computer control. The conditions for decreasing the steps are: lowering of the load less than or equal to 10% and crack growth less than or equal to 0.5 mm. The test is stopped when the value of 'K or of da/dN is the one desired (see section 6.4.2.5.4) 6.4.2.4.5. Measurement of crack length a) Direct method Optical crack length measurements, which are not recommended to follow the crack while testing, are useful to check whether standard recommendations are respected during pre-cracking and during testing. Recommendations concerning the implementation of this method (in particular desired increments between each measurement) are described in the standard [AFN 91].

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b) Indirect method It is necessary to choose well the increment of 'a between each crack length measurement (in analog fashion or by data acquisition). In fact, too low a value of 'a might induce a high dispersion of the da/dN values (linked to the choice of 'a and not to the material). In practice, the value of the minimum of 'a must be equal to 10 times the precision of the measurement or to 0.01 W (the larger one of these values). In the case of automatic testing (compliance method, potential drop method, etc.) this condition is included in the acquisition program. The same requirements exist as for pre-cracking concerning the difference of measurements on the two faces of the test piece and the crack plane. 6.4.2.5. Processing and presentation of results The test data to be processed are the values of a (measured on the surface by optical method or including the shape of the crack front for the compliance method or the potential drop method), the values of N and loading. Several types of processing are needed. 6.4.2.5.1. Crack front curvature correction Crack front curvature correction (usually bulging) is usually needed. a) Optical measurements The crack fronts are measured (materialized by markings) at the end of precracking and at the end of cracking on the fracture surface of the broken test piece by the 9 points method [AFN91]. The mean of the two measurements is determined. The curvature correction is equal to the difference between this mean value and the mean value of the measurements on the two faces of the test piece. When the curvature changes during crack propagation, an interpolation method may be used. b) Compliance method measurements The method in theory provides the exact mean value of a. However, differences might be observed between the values given by the compliance method and those measured (at 9 points) on the fracture surface. The adjustment is achieved by a modification of the Young’s modulus. c) Potential drop measurements The method provides a value of a since it lies on calibration. However, it is often necessary to make an adjustment after measurement of the final pre-crack length and final crack length on the test piece.

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6.4.2.5.2. Determination of crack propagation rate The crack propagation rate is determined from pairs of ai – Ni values. The values of the increment of the crack length must cope with the requirements in section 6.4.2.4.5. The recommended method is the secant method; it implies the following relations: a

a i a i i 2

[6.53]

da dN

a i1 a i N i1 N i

[6.54]

'K

f a

[6.55]

6.4.2.5.3. Determination of 'K a) Case where crack closure measurement is not carried out 'K R

K max K min K min K max

[6.56]

The Paris law determined by 'K depends on ratio R except for high R values (no crack closure). b) Case where crack closure is determined 'K eff R

K max K op

K min K max

[6.57]

Kop being determined from Pop. The Paris law determined using 'Keff does not depend on ratio R. c) Particular case of residual stresses In the case where it is believed that there exist residual stresses (test piece extracted from a welded joint, etc.), it is necessary to measure or to estimate the residual stresses KR along the crack path. The associated stress intensity factor KR is calculated (method of Bruckner weight functions). Two cases are then distinguished: 1) Kmin + KR 0, then 'Keff = Kmax + KR and R = 0

[6.58]

2) Kmin + KR > 0, then 'Keff = Kmax - Kmin and R = (Kmin + KR)/( Kmax + KR)

[6.59]

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6.4.2.5.4. Test results presentation a) Determination of Paris law In the case of constant load tests, the points da/dN – 'K ('Keff) are plotted on a diagram with log – log scales (usually da/dN in Y axis and 'K in X axis).

Figure 6.35. Example of results of fatigue crack propagation rate test (according to [AFN 91])

It is usual to perform linear regression on those points so as to determine coefficients C and m of the Paris law. To avoid forgetting important information concerning the test performed, a recommendation is made to fill a table of results whose model is in the AFNOR standard [AFN 91] and which will be included in the test report. b) Determination of the crack propagation threshold In the decreasing 'K tests, the threshold value 'K0 is determined, being theoretically the asymptotic value of 'K for which da/dN tends to zero. In practice, the threshold value is defined as the 'K value for which the crack propagation rate is equal to 10-7 mm/cycle: to find this, a linear regression is performed on 5 pairs of points between the 10-6 and 10-7 mm/cycle.

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6.4.2.6. Main information to deliver after the test The reader will find in the AFNOR standard [AFN 91] the list of information which should be given in the test report to easily transfer data to data banks. This report is an element of quality, as it accounts for conformity to standard specifications and possible deviations with respect to them. Its structure must ensure the easy transfer to data banks. 6.4.3. Other types of tests The development of test techniques and methods nowadays enables us to perform complex crack propagation tests in order to investigate fatigue cracking of materials or to approach real industrial loading situations. Various themes are mentioned here and the reader should refer to the literature cited. 6.4.3.1. Devices for testing outside ambient environment Testing in vacuum, in gaseous controlled environment, in liquid environment (corrosion), under pressure or at high temperatures can be envisaged. Devices with adequate instrumentation are fabricated for that purpose [FOU 97, LIE 82, PEL 97]. 6.4.3.2. Variable loading tests Real structures in which cracks can initiate and propagate are not generally loaded at constant amplitude. The loading amplitude can vary in a deterministic manner (combination of small and large cycles, over-loading, etc.) or in a random manner. Transitory loadings (stopping-starting of installations) can dangerously influence the lifetime of structures. Laboratory tests generally use the same types of test pieces as those for constant amplitude loading. Applied loadings are representative of the real loadings. The over-loading delay time phenomenon is taken into account using models based on the crack closure concept [LIN 97]. Results of random crack propagation loading tests seem to be rather well represented by the concept of equivalent loading (Miner rule) in the case of Gaussian loadings [AMZ 94]. 6.4.3.3. Tests on non-metallic materials In spite of the very different damage mechanisms of metallic matrix composites and of non-metallic materials from those of metallic materials, fatigue crack propagation tests on those materials use the same type of test pieces and techniques [BAT 97, SFM 90].

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6.5. Bibliography [AFN 90] AFNOR NF A 03-403, Pratique des essais de fatigue oligocyclique, 1990. [AFN 91] AFNOR NF A 03-404, Pratique des essais de vitesse de propagation de fissure en fatigue, June 1991. [AFN 93] AFNOR NF A 03-406, Fatigue sous sollicitation d’amplitude variable. Méthode Rainflow de comptage des cycles, 1993. [AMZ 94] AMZALLAG C., Le DUFF J.A., ROBIN C., MOTTET G., “Crack closure measurements and analysis of fatigue crack propagation under variable amplitude loading”, Automation in Fatigue and Fracture: Testing and Analysis, ASTM STP 1231, p. 311-333, 1994. [AST 647] ASTM designation: E 647, Standard Test Method for Measurement of Fatigue Crack Growth Rates. [BAS 10] BASQUIN C.H., “The exponential law of endurance tests”, American Society for Testing and Materials, PV ac. 10, p. 625, 1910. [BAT 97] BATHIAS C., “La fatigue des polymères et des élastomères”, La Fatigue des Matériaux et des Structures, 2nd edition revised and expanded, Hermes, p. 665-684, 1997. [BAT 97] BATHIAS C., “Mécanismes de la déformation plastique en relation avec la fissuration par fatigue”, La Fatigue des Matériaux et des Structures, 2nd edition revised and expanded, Hermes, p. 215, 1997. [BAU 97] BAUDRY B., FLAMAND L., FOUGERES R., HUNTER C., LIEURADE H.P., PINEAU A., “La fatigue”, Les Aciers Spéciaux, Lavoisier TEC-DOC p. 477-560, 1997. [BRA 80] BRAND A., FLAVENOT J.F., GREGOIRE R., TOURNIER C., Recueil de données technologiques sur la fatigue, CETIM, 1980. [COF 54] COFFIN L.F., “The problem of thermal stress fatigue in austenitic steels at elevated temperatures”, American Society for Testing and Materials, STP 165, Philadelphia, p. 31, 1954. [DEG 82] DEGALLAIX G., FOCT J., GABREL C. and VANDERSCHAEGHE A., “Comportement d’un acier martensitique à 12 % Cr en fatigue oligocyclique à haute température”, Mémoires & Etudes Scientifiques Revue de Métallurgie, vol. 79, no. 1, p. 21, 1982. [DEG 84] DEGALLAIX S., TAILLARD R., FOCT J., “Role of nitrogen interstitials in plastic fatigue of austenitic stainless steels”, Fatigue 84, Beevers C.J. et al. (eds.), EMAS, London, vol. 1, p. 49, 1984. [DEG 86] DEGALLAIX S., “Low-cycle fatigue behaviour of duplex stainless steels”, Progress in Mechanical Behaviour of Materials – ICM8, Ellyin F. and Provan J.W. (eds.), vol. I, p. 214, 1999. [DEG 99] DEGALLAIX S., Rôle de l’azote interstitiel et combiné sur le comportement en plasticité cyclique et monotone d’aciers inoxydables austénitiques, PhD Thesis, Lille, 1986.

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[ELB 70] ELBER W., “Fatigue crack closure under cyclic tension”, Eng. Fract. Mech., vol. 2, p. 37-45, 1970. [ELB 71] ELBER W., ASTM STP 486, p. 230-242, 1971. [ELL 97] ELLYN F., Fatigue Damage, Crack Growth and Life Prediction, Chapman and Hall, London, 1997. [ESO 93] ESOPE software for Windows, Manuel d’utilisation, IRSID, IMSA 93/55, 1993. [FOU 97] DE FOUQUET J., MENDEZ J., PETIT J., “Influence de l’environnement”, La Fatigue des Matériaux et des Structures, 2nd edition revised and expanded, Hermes p. 387-428, 1997. [FRA 93] FRANÇOIS D., PINEAU A., ZAOUI A., Comportement mécanique des matériaux, Hermes, Paris, 1993. [HAL 67] HALFORD G.R., MANSON S.S., “A method of estimating high-temperature lowcycle fatigue behavior of materials”, Metals and Metallurgy Trust, London, p. 154, 1967. [HOE 74] HOEPPNER D.W., KRUPP W.E., “Prediction of component life by application of fatigue crack growth knowledge”, Engineering Fracture Mechanics, vol. 6, p. 47-70, 1974. [HUD 78] HUDAK S.J. Jr., SAXENA A., “Review and extension of compliance information for common crack growth test pieces”, International Journal of Fracture, vol. 14, no. 5, p. 453-478, 1978. [IRW 60] IRWIN G., “Plastic zone near a crack and fracture toughness”, Proceedings of the Seventh Sagamore Ordonance Materials Conference, vol. IV, New York, Syracuse University, p. 63-78, 1960. [ISO 95] Désignation des axes des éprouvettes, NF EN norm ISO 3785, July 1995. [ISO 99] Matériaux métalliques, Essais de fatigue, Méthode d’essai de propagation de fissure en fatigue, project of international norm ISO/DIS 12108, 1999. [KLE 80] KLESNIL M. and LUKAS P., Fatigue of Metallic Materials, Elsevier, Amsterdam, 1980. [LEV 71] LEVY N., MARCAL P.V., OSTERGREN W.J., RICE J.R., Internat. J. of Fract. Mechanics, 7, p. 143-156, 1971. [LIE 82] LIEURADE H.P. and la Commission Fatigue des Métaux de la S.F.M., La pratique des essais de fatigue, Pyc Edition, p. 209-280, 1982. [LIE 87] LIEURADE H.P., La rupture par fatigue des aciers, OTUA, 1987. [LIE 94] LIEURADE H.P., LEMAITRE J., LACHAT J.C., “La résistance à la fatigue”, in Le Livre de l’Acier, Lavoisier Tec-Doc, p. 276-329, 1994. [LIE 97] LIEURADE H.P., FLAVENOT J.P., Les essais de fatigue sous sollicitations d’amplitude variable. La fatigue des métaux, Hermes, p. 429-470, 1997. [LIE 98] LIEURADE H.P., “Essais de fatigue-corrosion”, Techniques de l’Ingénieur, Fascicules M 135, 6, 1998.

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[LIN 97] LINDLEY T.C., “Croissance des fissures de fatigue sous chargement d’amplitude variable”, La Fatigue des Matériaux et des Structures, 2nd edition revised and expanded, Hermes, p. 471-506, 1997. [LU 98] LU J. and LIEURADE H.P., “Concentration de contrainte”, Techniques de l’Ingénieur, Fascicules BM 5.040, 4, 1998. [MAN 54] MANSON S.S., Behavior of materials under conditions of thermal stresses, NACA Technical Note 2933, 1954. [MAS 97] MASOUNAVE J., BAÏLON J.P., DICKSON J.I., “Les lois de fissuration par fatigue”, La Fatigue des Matériaux et des Structures, 2nd edition revised and expanded, Hermes, p. 255-288, 1997. [MIL 86] MILLER K.J., DE LOS RIOS E.R., The Behaviour of Short Fatigue Cracks, Mechanical Engineering Publications, London, 1986. [MIL 97] MILLER K.J., “The three thresholds for fatigue crack propagation”, ASTM STP 1296, American Society for Testing Materials, p. 267-286, 1997. [MIL 97] MILLER K.J., “Materials science perspective of metal fatigue resistance”, Développements récents en fatigue des matériaux et des structures, Ecole d’été, SaintPierre d’Oléron, p. 19-28, 1997. [MIL 97] MILLER K.J., “Retrospective and prospective views of fatigue research”, Développements récents en fatigue des matériaux et des structures, Ecole d’été, SaintPierre d’Oléron, p. 1-7, 1997. [MOU 97] MOUSAVI E., Ecrouissage cyclique d’aciers inoxydables austénitique, ferritique et austéno-ferritique: influence de l’histoire du chargement, PhD Thesis, Lille, 1986. [NFA 400] Produits sidérurgiques, Essais de fatigue, Principes généraux. [NFA 401] Produits sidérurgiques, Essais de fatigue sous charge axiale. [NFA 402] Produits sidérurgiques, Essais de fatigue par flexion rotative. [NFA 405] Produits métalliques, Essais de fatigue, Traitement statistique des données. [NFA 509] Produits sidérurgiques, Etalonnage des machines d’essais de fatigue. [NIL 83] NILSSON J-O., “The effect of slip behaviour on the low-cycle fatigue behaviour of two austenitic stainless steels”, Scripta Metallurgica, vol. 17, p. 593, 1983. [PAR 61] PARIS P.C., GOMEZ M.P., ANDERSON W.P., “A rational analytic theory of fatigue”, The Trend in Engineering, 13, p. 9-14, 1961. [PAR 63] PARIS P.C., ERDOGAN F., “A critical analysis of crack propagation laws”, Journal of Basic Engineering, 85, p. 528-534, 1963. [PEL 97] PELLOUX R.M., GENKIN J.M., “Fatigue-Corrosion”, La Fatigue des Matériaux et des Structures, 2nd edition revised and expanded, Hermes, p. 365-386, 1997. [PIN 76] PINEAU A., “Mécanismes d’accommodation et de fissuration en fatigue oligocyclique”, Mécanique - Matériaux - Electricité, no. 323-324, p. 6, 1976.

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[PIN 97] PINEAU A., “La fatigue plastique oligocyclique”, in La Fatigue des Matériaux et des Structures, Chap. 4, C. Bathias and J-P. Baïlon (eds.), Hermes, Paris, 1997. [POL 91] POLAK J., Cyclic Plasticity and Low-cycle Fatigue Life of Metals, Materials Science Monographs no. 63, Elsevier, Amsterdam, 1991. [POL 93] POLAK J., DEGALLAIX S. and DEGALLAIX G., “The role of cyclic slip localization in fatigue damage of materials”, Journal de Physique IV, vol. 3, p. 679, 1993. [RAB 00] RABBE P., LIEURADE P., GALTIER A., “Essais de fatigue”, Techniques de l’Ingénieur, Fascicules, M 4 170 and 171, 3, 2000. [RIC 67] RICE J.R., “Mechanics of crack tip deformation and extension by fatigue”, Fatigue Crack Propagation, Special Technical Publication 415, American Society for Testing Materials, Philadelphia, p. 247-309, 1967. [ROB 77] ROBIN C., DOMINIAK S., PLUVINAGE G., “Variation of crack opening – load diagram with fatigue crack growth rate”, Materials Science and Engineering, 29, p. 145150, 1977. [SF2 93] SF2M, GT3, Mesures de fermeture de fissure, 1993. [SFM 90] SOCIETE FRANÇAISE DE METALLURGIE, “Commission de Fatigue des Métaux”, 4e Journées Internationales de Printemps, Fatigue des Composites à Matrice Métallique et des Multimatériaux, Editions de la Revue de Métallurgie, Paris, 12-13 June 1990. [SUR 91] SURESH S., Fatigue of Materials, Cambridge University Press, 1991. [TRU 77] TRUCHON M., “La fermeture à la pointe des fissures de fatigue”, Mécanique – Matériaux – Electricité, 328, p. 121-130, 1977. [VAD 95] DE VADDER D., PARK Y., FRANÇOIS D., “Apparatus for Ultrasonic in situ Accurate Crack Size Measurement on Laboratory Test Pieces”, Special Applications and Advanced Techniques for Crack Size Determination, ASTM STP 1251, p. 114-129, 1995. [WU 84] WU S.X., “Crack length calculation formula for three-point-bend test pieces”, International Journal of Fracture, vol. 24, p. R33-R35, 1984.

Chapter 7

Impact Tests

7.1. Introduction Knowing the mechanical characteristics deduced from tensile tests may be insufficient since fractures can take place below the yield strength in conditions such that the material is brittle. A common factor of these conditions is stress concentration linked to a defect which can vary according to temperature, strain rate and dimensions of loaded parts. One of the most traditional mean to characterize steel brittleness is the impact test [BAR 93]. 7.2. Some history [MAN 99, TOT 02] The oldest known publications concerning impact effect on a material date back to early 19th century and they are the work of Tredgold who was interested in the dynamic loading resistance of steel. In 1857, Captain Rodman designed an impact testing machine to characterize weapon steel and for 30 years it was used to qualify steels, in particular those for railroads. This first impact device used homogenous rectangular bar test pieces, without any notch or crack. Thus, this test allowed the implementation of results for quite brittle materials, whereas tests on ductile materials could not be interpreted. Le Chatelier introduced notched test pieces in 1892, and he demonstrated that materials which displayed ductile behavior in tests using un-notched test pieces could behave in an extremely brittle manner as soon as a notch was introduced. In 1898, Russel designed a testing machine based on the concept of an impact pendulum very similar to the devices which are used Chapter written by Jean-Michel FRUND.

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nowadays. Impact tests were frequently investigated at the beginning of the 20th century and the works of the association IATM (International Association for Testing Materials) chaired by G. Charpy from 1906 are at the root of present standards and procedures. This is why the name of Charpy is always associated with the impact test on notched test pieces. 7.3. Description of the impact test The impact test consists of breaking in one blow of an impact pendulum a test piece notched in the mid-section resting on two supports (Figure 7.1). The absorbed energy in joules W is determined from which the notched impact strength is deduced (energy expressed in joules/cm2 needed to break the test piece). Other characteristics can be deduced from this test, in particular the fracture appearance (percentage of brittle fracture) or lateral expansion measured on the test piece after testing. It will be seen that the development of transitory analyzers allowed us to instrument the impact test and thus obtain other constitutive characteristics of materials at the test temperature. 7.3.1. Test piece The test piece is a rectangular bar of square cross-section (usually 10 x 10 mm2) notched in the middle. The shape of the notch varied over the course of time following the developments of machining possibilities. Thus, the first test pieces to be used were the so-called KUF with a keyhole shaped notch. In fact, Charpy introduced this notch shape to avoid machining scratches. Nowadays, the two test pieces adopted by international standards are the V shaped notch test piece (the so-called KCV) and the U shaped notch (the so-called KCU). The main differences between the two test pieces are the amount of material tested, or working cross-section, (0.8 cm2 for the KCV specimen and 0.5 cm2 for the KCU specimen), and the notch sharpness and thus the stress tri-axiality. It must be noted that whereas KCU test pieces are still being used today, it is only at ambient temperature and for comparison with older results (to follow thermal aging in particular).

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Figure 7.1. Description of the impact fracture energy test

Denominations: – KU: impact energy in joules (J) absorbed by fracture of a KCU specimen; – KV: impact energy in joules (J) absorbed by fracture of a KCV specimen; – KCU: impact energy in joules per square centimeter (J/cm2) absorbed by fracture of a KCU specimen; – KCV: impact energy in joules per square centimeter (J/cm2) absorbed by fracture of a KCV specimen. Note that since 1983, the notion of impact resistance (resilience), which was defined as the absorbed energy divided by the initial working cross-section of the test piece, was abandoned. The present standards define KU and KV energies only.

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7.3.2. Charpy impact pendulum The Charpy pendulum consists of a striker fixed on a hammer, which oscillates in a vertical plane around an axis. Part of the energy accumulated in the drop is used to fracture the test piece. The center of gravity of the pendulum is always situated near the striker edge. To perform the test, the striker is moved from the vertical by a height h0, which corresponds to a certain initial energy (usually close to W0 = 300 J). The striker is freed which, dropping while crossing the vertical position, breaks the test piece. The height h1 at which the pendulum climbs up is used to calculate the energy which was not absorbed W1. The energy absorbed by the test piece is represented by the difference W0 – W1. An important point to be noticed is the difference of the radius of the striker edge according to the ISO standard and that recommended by the ASTM. In the first case the radius of the striker edge varies from 2 to 2.5 mm, whereas in the ASTM procedure the radius of the striker edge is 0.25 mm. The consequence of this difference is for example for A533B steel a translation of about +10°C of the transition curves and a drop of 15 to 20% of the ductile upper shelf given by an ISO striker with respect to the ASTM [EDF 78]. For materials displaying a brittle-ductile transition, the evolution of the percentage of brittle fracture appearance according to temperature can be determined. This is achieved on the fracture surface of test pieces either by direct measurement by the technician or by more sophisticated means such as image analysis. The last parameter usually determined is the lateral expansion, which is the increase in the width of the specimen, measured with a thickness indicator, at the level of the fracture surface. 7.3.3. Charpy instrumented impact pendulum Strain gages stuck in a Wheatstone bridge on the striker allow us to record the load-time curve during the impact test. Different types of curves are obtained according to the regions of the transition curve. These recordings also allow us to characterize the percentage of brittle fracture and the absorbed energy in the various stages of the test. 7.3.3.1. Measurement of the load A complete bridge circuits includes two equally loaded strain gages stuck on the opposite faces of the striker and two compensating strain gages or substituting electrical resistances. The compensating strain gages must not be stuck on any part

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of the machine shock loaded or vibrating. Any other kind of load recorder may be used provided it meets the required performance level. The measuring device of the load (transducer, amplifier, recording system) must have a response of at least 100 kHz. The dynamic evaluation of the load measuring channel can be simplified by measuring the first initial peak value. Experience shows that the dynamic response of a measuring channel may be judged to be satisfactory when the first initial peak for a V shaped steel specimen is above 8 kN with an impact velocity of about 5.5 m/s. This is valid if the centers of the active strain gages are located at 10 to 15 mm from the contact point of the striker. Calibration of the measuring and recording system can be achieved in practice under static conditions. It is recommended that the calibration of the striker is included in the hammer. The force is applied on the striker with the help of a special frame equipped with a calibrated load cell and using a special block of high rigidity replacing the test piece. The contact conditions must be representative of an impact test. The static linearity and the error of reversibility of the instrumented striker, including all parts of the measuring system up to the recorder (printer, graphic recorder, etc.), must be within a tolerance of ±2% of the recorded force, between 50 and 100% of the nominal force, and of ±1% of the recorded force, between 10 and 50% of the nominal force. 7.3.3.2. Measurement of the displacement The displacement is determined from the load-time measurements; the relation between these two parameters is proportional to acceleration. For a striker assumed to be rigid of mass m, an initial velocity v0, and time t following the beginning of deformation tO, the bending displacement s(t) of the specimen is calculated by double integration: v t v 0

st

1 t ³ F t dt m 0

³ tt0 vt dt

[7.1]

[7.2]

Displacement can also be measured by a contact-less measurement of the striker displacement with respect to the supports of the specimen using an optical, inductive or capacitive method. The transfer characteristics of the signal of the displacement measuring system must correspond to those of the load measurement so as to synchronize the recording systems. The displacement measuring device must be designed for nominal values up to 30 mm; the linearity errors of the measuring system must provide measured values at ±2% in the 1 to 30 mm range. A dynamical

198


calibration of the displacement-measuring device can be obtained by dropping the pendulum in the absence of the specimen when the velocity is estimated to be v0 2gh0 . The velocity signal recorded when the pendulum goes through the lower position should correspond to velocity v0. 7.3.3.3. Different time-displacement curves The ISO standard defines 6 types of time-displacement curves (Figure 7.2). The first type is characterized by unstable crack propagation alone (completely brittle behavior of the material), while the last one is characterized by stable crack propagation alone (completely ductile behavior of the material). The intermediate types are different mixtures of these two extreme behaviors. Note that on these curves are superimposed the vibrations linked to the impact waves reflections in the test piece. 7.3.3.4. Determinations of the characteristic values of the load Four characteristic values of the load are usually determined from the instrumented Charpy impact test. These values are the following (Figure 7.2): – FgY, called the flow load, is determined as the load corresponding to the end of elastic behavior of the material on the curve obtained after smoothing out the oscillations; – Fm, called the maximum load, is defined as the maximum of the load obtained after smoothing out the oscillations; – Fiu, is defined as the load at unstable fracture of the material; – Fa is the load at crack arrest. Note that in the case of a very brittle material, FgY = Fm = Fiu. 7.3.3.5. Determination of the global fracture energy The determination of the global fracture energy is obtained from the determination of the area, for appropriate portions, under the time-displacement curve. If the energy determined on the dial of the Charpy machine is to be compared, the entire area under the curve needs to be evaluated; on the other hand, if initiation energy is sought, the area under the curve between the start and the value Fiu of the load will be determined.

Impact Tests

Figure 7.2. Different types of load displacement curves [STE 00]

Figure 7.3. Conventional load values for instrumented Charpy pendulum

199

200


7.3.3.6. Determination of the percentage of brittle appearance from the load-time curve It was stated that when, on the load-time curve, the maximum load is within the domain of elasticity and corresponds to the unstable fracture of the test piece, the material behavior is perfectly brittle. Thus, in that case, the percentage of brittle fracture appearance on the fracture surface is 100%. On the other hand, if the loadtime curve shows no unstable crack propagation, the proportion of brittle fracture appearance on the fracture surface is 0%; the material behavior is perfectly ductile. Thus, it is possible to determine the proportion of brittle fracture appearance on the fracture surface from the conventional parameters of the load-time curve. The working group TC5 of ESIS listed different empirical formulae proposed in the literature: F Fa C1 100 iu Fm

C 2 100

C 3 100

Fiu Fa

Fm Fm Fgy

Fiu Fa

Fm 0.5 Fm Fgy

ª 1 F 2F Fiu Fa º gy m » C 4 100« Fm Fm Fm »¼ « ¬ 3

[7.3]

7.4. Determination of transition curves The drawing of transition curves of the impact fracture energy, of brittle fracture appearance and of lateral expansion from experimental data from a number of KCV specimens of a single material tested at different temperatures leaves room for subjective interpretation, especially when dispersion is large and the number of tests is small. Methods for the determination of the scatter band at certain temperatures (mainly in the brittle-ductile transition region), as well as of the mean curve of this band, were developed and enable us to reduce subjectivity. However, there must exist as many methods as there are laboratories specialized in impact testing. For this reason, one method only is explained here, without forgetting that other ones are just as justified. This method is based on least squares analysis and on the choice of a hyperbolic tangent function. A corrective factor was added to this basic function to

Impact Tests

201

take care of the lower part of some curves (brittle lower shelf). At the end the following was kept: y x th x agx

[7.4]

with x = (T – T0)/T1 (Figure 7.4) and § gx 3¨x 0.5 ©

·

x 0.5 2 0.1 ¸ ¹

for impact fracture energy and lateral expansion:

>

2

@ >

2

gx exp 2x 1 exp 2x 1

@

for brittle fracture appearance. 7.5. Transition temperature and upper shelf Smoothing of transition curves allows us to determine transition temperatures and upper shelf energy. These values, determined from the mean fracture resistance, fracture appearance or lateral expansion curves often allow us to follow the aging evolution of the material (thermal aging, strain aging, irradiation, etc.). The reference temperature TKX is defined as the temperature for which an impact energy, a brittle fracture appearance percentage or a lateral expansion are evaluated as X daJ/cm2, % or mm. The nuclear industry as an example frequently uses the reference temperature TK7 (the temperature for which the mean transition curve evaluates an impact fracture energy equal to 7 daJ/cm2), TK50 (the temperature for which the transition curve evaluates a brittle fracture appearance percentage of 50%) or TK0.9 (the temperature for which the transition curve evaluates a lateral expansion of 0.9 mm). For impact fracture energy transition curves, TK3.5 and TK8.5 are also frequently used. The fracture energy at the upper shelf denoted by USE is frequently extracted from the mean impact fracture energy transition curves. This parameter is often correlated with chemical contents (sulfur for instance for ferritic steels) or to mechanical parameters such as the resistance to ductile tearing JIc or J0.2.

202


Figure 7.4. Impact fracture energy transition curve of steel 16MBD 5

7.6. Impact fracture energy-fracture toughness empirical correlations Numerous correlations between the impact fracture energy KV and fracture toughness KIc have been published [PHA 91]. However, owing to the important differences between fracture energy impact tests and fracture toughness tests, these correlations must be used with great precautions. Indeed, measured energies are not the same for the following reasons: – KCV test pieces have a notch root radius of 0.25 mm whereas CT specimens are fatigue pre-cracked: the crack root radius is then much less than 0.25 mm. This aspect is important because crack initiation depends on the crack tip stresses; – in order to ensure plane strain conditions at the crack tip, standard NFA 03 180 requires a thickness for test pieces according to the yield strength and fracture toughness. In the case of impact testing specimens, this condition is only met for low fracture toughness level; – strain rates are very different in impact fracture energy tests and in fracture toughness tests (10-1 to 103 s-1 for the first ones, 10-4 s-1. for the second ones). Now, the yield strength is very strain rate-dependent. Thus, it is possible that at a given temperature, a CT specimen would display ductile fracture, whereas a KV specimen would display brittle fracture; – fracture toughness tests allow us to determine crack initiation only, whereas propagation is also measured in impact fracture energy tests. The literature quoted in the bibliography show that the best correlations KIc-KCV are obtained for the lower level of the transition curve. Table 7.1 gives the main correlations available in the literature.

Upper shelf

Transition – Upper shelf

Region

Girenko, Lyndin

1985

1983

1982

Priest, Charnock, Stewart

Witt

1981

1971

1970

1976

Year

Norris, Reaugh, Server

Ault, Wald, Bertolo

Barsom and Rolfe

Thorby and Fergusson

Names

200 to 1700

414 to 848

290 to 450

190 to 320

447 to 1696

1,610 to 1,980

250 to 1700

400 to 560

Yield strength range (MPa)

-

47 to 203

102 to 203

5 to 120

22 to 192

15 to 29

4 to 82

5 to 125

Energy range (J)

09G25, 18Ni, A527F, ...

A533B, A508 Cl.2, A302B, Ni-Cr-Mo, A516 Grade 70

Parent metal

SAW

A533B

-

A302B, A517F, A36, ...

HY 60

Materials

)

(

MPa m MPa J, mm²

J

KIC = (0.068 KV + 9.90)² KIC = (0.1 E/(1-Q²) KV/A)1/2

MPa m

J

MPa

MPa m

kN/m J MPa

KIC = 2.78 KV – 3.93

K IC )² = 1.622 ( KV ) – 0.011 Vy Vy

( K IC )² = 0.65 ( KV ) – 0.00637 Vy Vy

V y 1600 1300

m, J, MPa

( K IC )² = 0.18 ( KV ) – 0.0011 Vy Vy JIC = KV (

kPa.m J

J/cm²

MPa m

Units

KIC2 = 0.222 E KV1.5

A is the surface of the section of KCV samples

KIC = 18.2 (KV/A)0.534

Expression

Impact Tests 203

Transition

Lower shelf – Transition – Upper shelf

Lower shelf – Transition

Region

1991

PD 6493-1

1971

Logan and Crossland Thorby and Fergusson 820 to 1,420

274 to 820

-

< 480

250 to 1,700

| 630

410 to 815

Yield strength range (MPa)

5 to 54

5 to 50

-

-

7 to 68

5 to 200

7 to 70

Energy range (J)

3%Ni, Cr-Mo-V, ...

A533 Grad B, E36, 10 CD 9 10, ...

16MND5 (A508Cl.3)

A533B

A302B, A517F, A36, ...

Welded joint with 9% of Ni

A533B, A542, A517F, A302B, ABS-C

Materials

= 2.5

KV

KV

(0.0261(T - T

K5

+ 88.9))

KV A A is the surface of the section of KCV samples

KIC = 18.2

with TKIC = 9 + 1.37 TK3.5

KIC = 19 KV0.5

(0.036 (T - TK + KIC = 36.5 + 3.1 e 3.5 55.5))

KIC = 29.18 + 1.333 e

KIC = 8.47 KV0.63

Vy

KIC ²

KIC = 14.6

Expression

Table 7.1. Some empirical correlations impact fracture energy – fracture toughness found in the literature

1977

Marandet and Sanz

1998

1981

Roberts and Newton

Frund

1984

1973

Sailors and Corten

Imai, Nakaya, Yajima, Tada, Yamagata, Deguchi

Year

Names

J/cm²

MPa m

MPa m J

°C

MPa m

°C

MPa m

MPa m J

MPa m MPa J

J

MPa m

Units

204 Structural Components

Impact Tests

205

7.7. Bibliography [AUL 71] AULT R.T., WALD G.M., BERTOLO R.B., Development of an improved ultraright strength steel for forged aircraft components; AFML-TR-71-27, Air Force Materials Laboratory, 1971. [BAR 70] BARSOM J.M., ROLFE S.T., “Correlations between KT C and Charpy V notch test results in the transition temperature range”, ASTM-STP-466, p. 281-302, 1970. [BAR 93] BARRALIS J., MAEDER G., Précis de métallurgie, Nathan Editions, 1993. [BSI 91] BSI PD 6493, Draft published document for guidance on methods for assessing the acceptability of flaws in welded structures, British Standards Institution, London, 1991. [EDF 78] Internal document EDF, August 1978. [FRA 02] FRANÇOIS D., PINEAU A., From Charpy to Present Impact Testing, ESIS pub. 30, Elsevier, p.3-19, 2002. [GIR 85] GIRENKOV. S., LYNDIN V.P., “Relationship between the impact strength and fracture mechanics criteria SIC and KIC of structural steels and welded joints in them”, Automatic Welding, p. 13-19, September 1985. [IMA 84] IMAI J., NAKAYA I., YAJIMA H., TADA M., YAMAGATA S., DEGUCHI A., “Safety evaluation of LNG storage tanks of 9% Ni steel based on fracture analysis”, Transport and Storage of LPG and LNG, p. 91-100, Bruges, 7-10 May 1984. [LOG 71] LOGAN J.-G., GROSSLAND B., “The fracture toughness of (En 25) and a 3% NiCr-Mo-V steel at various strength levels together with Charpy impact data”, Practical Applications of Fracture Mechanics to Pressure Vessel Technology, Institution of Mechanical Engineers, p. 148-155, 1971. [MAN 99] MANAHAN M.P., MC COWAN C.N., SIEVERT T.A., HOLT J.M., “For 100 years, notched bar impact testing standards have yielded widespread benefits for industry”, ASTM Standardisation News, February 1999. [MAR 77] MARANDET B., SANZ G., “Evaluation of the toughness of thick medium strength steels by LEFM and correlations between KTC and CVN”, ASTM-STP-631, p. 72-95, 1997. [NOR 81] NORRIS D.M., REAUGH J.E., SERVER W.L., “A fracture-toughness correlation based on Charpy initiation energy”, Fracture Mechanics, 13th Conference, ASTM-STP743, American Society for Testing and Materials, p. 207-217, 1981. [PHA 91] PHAAL R., MACDONALD K.A., Critical examination of correlations between fracture toughness and Charpy impact energy, Report TWI no. 5605, March 1991. [PRI 82] PRIEST R.H., CHARNOCK W., STEWARD A.T., “The effect of accelerated irradiation on fracture behaviour”, Effects of Radiation on Materials, 11th Conference, ASTM-STP-782, American Society for Testing and Materials, p. 475-491, 1982. [ROB 81] ROBERT S.R., NEWTON C., “Interpretive report on small-scale test correlations with KIC data”, Welding Research Council Bulletin, 265, p. 1-18, 1981.

206


[ROU 77] ROUSSELIER G., Traitement statistique des résultats des essais de résilience KV pour le tracé des courbes de transition, de cristallinité et d’expansion latérale, Procèsverbal EDF/EMA no. D 384, 30 March 1977. [SAI 73] SAILORS R.H., CORTEN H.T., “Relationship between material fracture toughness using fracture mechanics and transition temperature tests”, ASTM-STP-514, p. 164-191, 1973. [STE 00] STEEL–CHARPY V., Notch Pendulum Impact Test Instrumented Test Method BS EN ISO 14556: 2000, July 2000. [THO 76] THORBY P.N., FERGUSON W.G., “The fracture toughness of Hy60”, Materials Science and Engineering, 22, p. 177-184, 1976. [TOT 02] TOTH L., ROSSMANITH H.-P., SIEVERT T.A., “Historical Background and Development of the Charpy Test”, in From Charpy to Present Impact Testing, François D. and Pineau A. (eds), ESIS pub. 30, Elsevier, p. 3-19, 2002. [WIT 83] WITT F.J., “Relationships between Charpy impact-shelf energies and upper-shelf KI C values for reactor pressure vessel steels”, Int. J. Press. Ves & Piping, 11, p. 47-63, 1983.

Chapter 8

Fracture Toughness Measurement

8.1. Introduction The term fracture toughness refers to the fracture resistance of the material. It can be evaluated by various more or less quantitative tests [FRA 07]. However, more specifically, fracture toughness concerns the resistance to crack propagation. It is measured by the energy needed to increase the crack area by one unit. An essential question is to know whether this quantity is a characteristic of the material, independent of the shape and size of the crack itself and of the part in which it is located. This is indeed the case if the zone close to the crack tip, where the stresses and strains are maximum, is autonomous. It is known that this is the case in linear elasticity, the asymptotic stress and strain fields at the tip of a perfectly sharp crack depending on the loading on the part, on the crack size and on their shape through a single parameter: the stress intensity factor K. Under those conditions, provided the stress intensity factors are the same, the crack tip sees identical environments in a test piece and in any part. The test results are transferable. The measurement of fracture toughness should not raise great problems. However, the difficulty arises from the necessity of avoiding too large perturbations with respect to the ideal conditions of linear elastic fracture mechanics; these conditions are never perfectly achieved because irreversible deformations at the crack tip are unavoidable. The standards specify conditions to be respected in order that the linear elasticity approximation is satisfied.

Chapter written by Dominique FRANÇOIS.

208


When the yield strength of the material is not sufficient for these conditions to be fulfilled, use must be made of elastoplastic fracture mechanics. The autonomy of the crack tip zone is no longer as well insured as in linear elasticity. However, the J parameter, calculated by the Rice contour integral, characterizes in a satisfactory manner the intensity of the field of strain energy density at the crack tip and this constitutes a good approximation of the strain energy release rate when the crack starts growing. This parameter then allows the transferability of results, at least as a first approximation. Also, experimental techniques combined with numerical calculations are available for the evaluation of this parameter. A particular measurement of strain intensity at the crack tip is that of the crack tip opening displacement (CTOD). Rather than resorting to J, some people prefer to use this parameter. The determination of the CTOD is described in a British standard. The experimental technique does not differ much from that of J determination. The transferability of results to any part is more problematic. However, this type of test will be mentioned, as it is required by some industries. It will be seen that fracture mechanics tests are costly. It is useless to perform them, if calculating the service tolerance of a part is not envisaged.

Figure 8.1. Definition of the Rice-Cherepanov integral on a contour around the crack tip. ti is the stress vector, ui the displacement vector and ni the vector normal to the contour

In this chapter, after a brief summary of fracture mechanics bases, the main rules existing in standards will be described, inviting those who wish to perform tests to refer to them, as it is important to comply with all requirements. It is useful to understand the reasons behind these requirements, some of which are not completely evident.


209

8.2. Fracture mechanics bases 8.2.1. Rice Cherepanov J integral The Rice Cherepanov J integral (Figure 8.1) [RIC 68-1, RIC 71] is a contour integral, which is path independent in linear as well as in non-linear elasticity. It is demonstrated under the same conditions that it is equal to the strain energy release rate, when, keeping the same direction of propagation, the crack area virtually increases by one unit. It is expressed in the following way:

J

§

³ ¨Wdx 2 t i ©

wu i · ds ¸ wx1 ¹

[8.1]

where W is the strain energy density, ti the stress vector on the contour and ui the displacement vector. Axis Ox1 is in the direction of crack propagation, axis Ox2 is perpendicular to the crack and ds is an element of the contour. The contour of integration surrounds the crack tip, going from one of the faces to reach the other in the counterclockwise direction. No force is applied on the crack faces. The property of path independence enables us to show, choosing as contour a circle of small radius r, that the asymptotic strain energy field is of the form: W

J ˜ W (T ) r

[8.2]

r and T being the cylindrical coordinates at the crack tip. Thus, the strain energy density presents a 1/r singularity. Hence, in linear elasticity stresses and strains present a 1/r1/2 singularity. These asymptotic fields depend only on the J parameter. If this parameter is the same for a test piece and for a part, the situations at the crack tips are identical, and so are the crack behaviors. In particular, the crack, whether in a test piece or in a part, starts propagating when J reaches a critical value, JIc, which depends on the material (index I indicates that propagation is in mode I, that is with displacement of the crack faces perpendicular to the crack). Choosing a contour following the external surface of the part, it is shown that J depends on the loading and on the shape. It also depends on the crack length. J being equal to the strain energy release rate, it is possible to measure the value of this parameter making the difference, for the same deformation, of the energies absorbed by two test pieces containing, for the one, a crack of area A + dA and of area A for the other, divided by dA. In linear elasticity this quantity is designated by G, and its critical value, the fracture toughness, by Gc.

210


8.2.2. Stress intensity factor K In linear elasticity the asymptotic stress field is in (J/r)1/2, multiplied by a constant of elasticity to be homogeneous [IRW 58]. It is written:

V ij

KM 2 Sr

f Mij T

[8.3]

index M being related to the crack opening mode (Figure 8.2). It is demonstrated that, in mode I, the relation between J and KI is written: J

K I2 Ec

[8.4]

Figure 8.2. Reduced stress VRp at the crack tip according to the reduced distance r/rY, rY being the Irwin correction: rY = (1/2S)(KI/Rp)2

where E’ is equal to Young’s modulus E in plane stress, and to E/(1 - Q2) in plane strain, Q being Poisson’s coefficient. It is important to note that the stress intensity factor K is independent of the properties of the material and depends only on loading, crack length and geometry. The critical value JIc corresponds to a critical value Kc, another way of expressing the fracture toughness of the material.


211

8.2.3. Plastic zone at the crack tip 8.2.3.1. Plastic zone correction The theoretical solutions calculated for the asymptotic fields in linear elasticity are obviously perturbed in a zone close to the crack tip by plastic deformations and damages. A sufficiently small plastic zone at the crack tip is denoted the small scale yielding plastic zone when the singular elastic solutions still exist, simply moved away a little by rY in the propagation direction: the crack with its plastic zone behaves like a purely elastic fictitious crack longer than the real crack by rY (Figure 8.3):

rY

§K ·2 D ¨¨ I ¸¸ ©R p ¹

[8.5]

with D = 1/2S in plane stress and D = 1/6S in plane strain. Rp is the proof strength. This is known as the Irwin plastic zone correction.

Figure 8.3. Plastic zone correction. The crack behaves like a fictitious elastic crack moved by a distance rY

212


8.2.3.2. Plane stress plastic zone The state of plane stress exists in thin plates or at the unloaded surface of a part. In mode I, in the plane Ox1-Ox3, which is the extension of the crack plane, the principal stress directions are those of the coordinates axes. As principal stress V3 is zero, and as the radius of the Mohr circle corresponding to principal directions 1 and 2, in the plastic zone, is equal to the yield strength in shear, the maximum principal stress V2 is equal to the tensile proof strength. Plastic deformation, which is achieved by slip along glide planes in the directions of maximum shears, follows bands inclined at 45° to axes Ox2 and Ox3, which emerge at the surface on both sides of the plate (Figure 8.4). The length of the plastic zone ahead of the crack is approximately given by:

R

2 1 §K I · ¨ ¸ S ¨©R p ¸¹

Figure 8.4. Plastic zone in plane stress. Glide planes emerging on the lateral surfaces. Mohr circle

[8.6]


213

For small scale yielding, so that the linear elastic approximation may be used, the length R must be sufficiently small with respect to the ligament length b in front 2

of the crack. It is considered that this condition is satisfied when b t 2.5 K I R p . 8.2.3.3. Plane strain plastic zone The state of plane strain exists at the crack tip in the center of a part, at a

distance to the surfaces where the crack emerges larger than 0.16 K I R p

2

.

In mode I, in the plane Ox1-Ox3, which is the extension of the crack plane, the principal stress directions are those of the coordinates axes. In the plastic zone, stress V3 is equal to the mean of stresses V1 and V2. As, owing to small scale yielding, stress V1 is positive, and as the radius of the Mohr circle corresponding to principal directions 1 and 2 in the plastic zone is equal to the yield strength in shear, the maximum principal stress V2 is somewhat larger than the tensile proof strength. Hence, it is in the plastic zone in plane strain that damage and crack propagation are initiated. Plastic deformation follows slip bands inclined at 45° ahead of directions Ox1 and Ox2; they die down at the plastic zone boundary (Figure 8.5). The length of the plastic zone is worth about a third of that in plane stress. In order to be mostly in plane strain along the crack front, its length must exceed

2

2.5 K I R p .

8.3. Implementation of fracture toughness tests 8.3.1. Test pieces It is evident that pre-cracked test pieces are needed. Cracks are initiated at the root of a machined notches. Standards [NOR 96] propose several types of notched test pieces for which the values of the stress intensity factor KI are given according to crack length a in relation to the test piece width W. Other types of test pieces may be used, provided the source of stress intensity calculation is given. One of the most widely used test pieces is the compact tension (CT) specimen, which requires the least amount of material (Figures 6.31a and 8.6). The choice of the test piece thickness B determines by proportionality all other dimensions. The specimen designation is according to this thickness: for instance a 25 mm thick CT specimen will be denoted CT 25.

214


Figure 8.5. Plastic zone in plane strain. Glide planes towards the front of the crack. Mohr circles

Figure 8.6. Schematic drawing of a CT specimen


215

The machining of the test pieces must be done with care, especially that of the notch root so as to ease fatigue crack initiation. In fact, the crack must be as sharp as possible. Fatigue cracks are quite suitable; however, brittle crack initiation may be accepted provided crack arrest is possible. Most often, test pieces are fatigue precracked, up to a sufficiently large value of a/W, taking care that the stress intensity factor at the end of pre-cracking remains smaller than 60% of the critical stress intensity factor for fracture (denoted KQ in the following section). This specification is aimed at avoiding the presence of too large a fatigue plastic zone, especially because such plastic deformation creates residual compressive stresses, which close the crack tip. Test pieces are equipped before testing with a crack opening sensor, used to check the linearity of loading. 8.3.2. Data processing Usually, the recording of applied load according to sensor opening displacement in the course of the test displays a deviation from linearity a little before instant crack propagation. This phenomenon is partly due to plastic deformation at the crack tip and mostly to stable crack growth before instability. This stable crack propagation results from the fact that resistance to crack propagation is not constant as the crack grows, but on the contrary increases. This variation is denoted the “R curve” of the material. For propagation to become unstable, it is not enough that the strain energy release rate is equal to cracking resistance; it is also necessary that its derivative with respect to crack length is larger than that of the R curve. The point of instability, which corresponds to this condition, depends on the crack length and on the shape of the test piece. Thus, it does not enable us to determine an intrinsic property of the material. Now, the determination of the R curve is difficult because it requires crack length measurement while testing. Hence, it is agreed to measure fracture toughness as close as possible to stable crack growth initiation, a value which itself is an intrinsic property. By convention this is done at a point of the load-opening displacement curve situated at the intersection of a straight line passing through the origin and whose slope is 5% less than the initial slope of the curve (Figure 8.7). This corresponds to a small amount of stable crack propagation, which increases the compliance of the test piece a little, i.e. the inverse of the stiffness. PQ denotes the load which corresponds to that point. To calculate the value of the corresponding stress intensity factor KQ the crack length at the beginning of the test ought to be known. It can be easily measured on the fracture surface, because of the facies difference between the fatigue and instant fracture. The fatigue pre-crack front is thus well distinguished. It is never perfectly straight. Standards specify the way to determine the mean crack length by measurements at various points along the crack front. They also specify the tolerable deviations from a rectilinear crack front and from its non-skewness.

216


8.3.3. Validity of measurements Having determined KQ with the help of tables included in standards for each type of test piece (see section 6.4.2.2.2), a check must first be made that the ratio of the maximum load at instability to load PQ is less than 1.1. This specification ensures that stable crack propagation remains within acceptable limits. It is then required to

2

check that the specimen thickness B is larger than 2.5 K I R p . This specification ensures that the crack front is mostly in plane strain. In the opposite case, the measured fracture toughness would depend on the specimen thickness and thus would not be an intrinsic property of the material. Indeed, the fracture energy along the sides in plane stress is larger than the fracture energy in the center in plane strain, because the plastic zone size is larger in plane stress than in plane strain, and because the stress level is smaller. Hence, the fracture energy depends on the ratio of the crack front length in plane stress to that in plane strain; it becomes independent of thickness once the above condition is fulfilled.

Figure 8.7. Determination of the conventional critical load

When the two preceding conditions are fulfilled, KQ = KIc, which corresponds to the conventional denomination of fracture toughness in plane strain.


217

8.4. Measurement of fracture toughness JIc 8.4.1. Basis of elastoplastic fracture mechanics When the ligament length b is not sufficient to guarantee an acceptable confinement of the plastic zone, it is necessary to rely on elastoplastic mechanics to determine fracture toughness. For that purpose, the two fundamental properties of the J integral demonstrated in elasticity are extended to the case of plasticity: namely, path independence and equality with strain energy release rate. This is acceptable provided loading is monotonous. The 1/r singularity of strain energy density is then recovered. However, as the constitutive behavior of the material is no longer linear, the stress and strain singularities are not in 1/r1/2. There is no such parameter as the stress intensity factor. Nevertheless, it is possible to consider that J plays a similar role, as far as the asymptotic fields depend on this only parameter. In small scale yielding plastic zones, Rice and Rosengren [RIC 68-2] and Hutchinson [HUT 68] calculated these fields and they are known as HRR fields. J measurements can be achieved in the same manner as in elasticity. However, whereas in linear elasticity the stress intensity factor does not depend on the material, so that it is possible to draw up formulae giving various expressions for K in a great many different situations, in elastoplasticity this is difficult to conceive as the strain hardening laws must be introduced in J calculations. Unfortunately, it is found that asymptotic field characterization by the J parameter alone is accurate only in small scale yielding, which brings us back to the case of linear elasticity by applying a crack length plastic correction. When plasticity extends, a stress denoted Q is superimposed on the HRR fields; it is calculated by finite elements1. It depends on the conditions which exist on the external boundaries of parts. Thus, in elastoplasticity it is not really possible, strictly speaking, to characterize fracture energy by JIc alone. A critical curve ought to be drawn in a J – Q diagram. 8.4.2. JIc testing Tests for JIc measurements do not differ much from the preceding ones for KIc measurements [ESI 92]. However, it is here necessary to know the value of J to determine the strain energy release rate at any time during the test. This implies measuring not only the crack mouth opening but also the displacements of the points of application of the load. Hence a slight modification of the shape of the test pieces in order to be able to place the displacement sensor at a convenient location on the load line of CT specimens (denoted CTJ in that case), on the central support 1. In linear elasticity the second term in the stress decomposition at the crack tip is a constant

Vxx stress denoted T. It remains small with respect to the singularity and does not modify much fracture toughness.

218


axis of three points bend specimens. It is not necessary to determine J by measuring the difference of deformation energies absorbed by specimens with cracks of slightly different lengths (this could be a possibly costly experimental measuring method). It can be demonstrated in fact that: J

KU Bb

[8.7]

U being the absorbed energy, Bb the ligament area and K a geometrical parameter, which depends on the test piece used and in some cases on the crack length. For instance, its value is 2 for a bend specimen and 1 for a tensile specimen. A table in the ESIS standard provides this parameter for CTJ specimens. It is then enough to determine the energy U at any time during the test by integrating the load– displacement curve. When the crack propagates, unloading takes place, which makes the validity of J measurement doubtful, as loading is no longer monotonous. It would then be proper to accurately determine the initiation of crack propagation. Usually it does not correspond to any particular feature on the loading curve. Methods are then needed to determine this initiation point. Several such methods exist (see section 6.4.3.2). 8.4.2.1. Interrupted loadings method A batch of test pieces are loaded up to increasing values of deformation, then unloaded and broken after marking the crack front position. This can be achieved for instance by oxidation of the cracked surfaces, or by modification of the fracture facies by low temperature fracture, or by fatigue. Plotting the tearing area beyond pre-cracking according to deformation, or rather to J, enables us to determine the initiation of crack propagation. Yet, the small crack length increase due to blunting of the crack tip has to be taken into account. Theoretically it is proportional to J; the proportionality coefficient is equal to the yield stress multiplied by a numerical factor often chosen to be equal to 4 by convention. It would be better if possible to determine it experimentally in each case, but it is a demanding measurement. Standards provide extrapolation rules in order to determine the point of tearing initiation. 8.4.2.2. Partial unloading method During the test the test piece is periodically partly unloaded in order to then measure the slope of the load-displacement curve. This slope is smaller the longer the crack. This enables us, by extrapolation, to find the initiation point. Unloading is not total so as not to introduce residual crack closure stresses.


219

8.4.2.3. Potential drop method The test piece is fed with a constant high intensity electrical current. Potential connectors are located on either sides of the crack so as to measure the potential drop when it propagates. Plotting the potential drop according to J displays a modification of the slope, which corresponds to the initiation of crack propagation. In fact, plastic deformation itself has an effect on potential drop, even in the absence of crack propagation. It is advisable to mount the potential connectors in a bridge with connectors outside of the zone of influence of the crack. In this way, the interferences due to temperature variations are eliminated. Measurements can use either direct current or alternating current. In this last case additional effects can modify the impedance of the circuit, which sometimes make the determination of the initiation of crack propagation easier. 8.4.2.4. Other methods Other methods are available. It is for instance possible to locate the crack front with ultrasounds. In this category, the most precise consists of using the signal diffracted by the crack tip. Blunting decreases its intensity, whereas it increases noticeably at the tearing initiation. Injecting ultrasonic waves in the test piece requires a transmitting medium, the simplest being water. Solutions exist to work at various temperatures. 8.4.2.5. Crack tip blunting An additional difficulty comes from crack tip blunting owing to plastic deformation inducing a small crack growth. It is proportional to crack tip opening displacement, denoted G or CTOD. It is itself proportional to J/Rp. When the tearing length is measured, it would be necessary, strictly speaking, to subtract this crack growth due to blunting. When there is no direct post-mortem measurement on the fracture surfaces, a tricky measurement requiring the bringing together of the two parts of the broken test piece, a straight line is drawn on the J-'a diagram representing the evolution of crack tip blunting; its conventional slope is equal to 4(J/Rp). It is at the intersection of this line with the J-'a curve that the J value is found at the tearing initiation: Ji. When the slope is very high this determination is tricky and not precise. Thus, it is then preferred to measure a conventional value following a set ductile tearing propagation: 0.2 mm for instance. The result is given subtracting or not the crack tip blunting. 8.4.3. Data processing Various characteristic J values may be determined (Figure 8.8): – fracture toughness at stable crack growth initiation Ji;

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– fracture toughness at unstable crack propagation below 0.2 mm of crack propagation Jc; – fracture toughness at 0.2 mm of propagation including crack tip blunting J0.2, or excluding it J0.2:BL; – fracture toughness at unstable crack propagation over 0.2 mm of crack propagation Ju. Note that Ji is the only value, which would be an intrinsic property, if the Q stress would not operate. J0.2 and J0.2:BL are conventional values, which are satisfactory approximations of JIc. Ju is not at all an intrinsic property.

0.2

Figure 8.8. Data processing of a J-'a curve (figure due to Nicolino)

8.4.4. Conditions of validity The HRR fields are perturbed very near the crack tip by large deformations and by damage. This zone, whose dimension is of the order of the crack tip opening, must not be too large with respect to the ligament width b. As the crack tip opening


221

displacement is proportional, as noted before, to J/Rp, the condition of validity of the test is:

b t 50

J Rp

[8.8]

8.5. CTOD measurement The crack tip opening displacement (CTOD) is proportional to J/Rp. Its direct measurement is quite difficult as it is of the order of a tenth of a millimeter. It is deduced by measuring the crack mouth opening, assuming a rotation of the two branches of the test piece around an instantaneous center of rotation. Its position must be calculated. At the beginning of the test it is located near the crack tip, and it moves away as plastic deformation increases, to end more or less near to the center of the ligament width. The determination of the initiation of crack propagation strictly poses the same problems as those encountered in critical J measurements; the same methods are used. The indices characterizing various values of the CTOD are identical to those defined for J, the CTOD being denoted G. Note that calculation of crack opening displacement for any part, so as to predict cracking, poses more demanding numerical problems than for J calculations. 8.6. Conclusion Fracture toughness measurements in the conditions of validity of linear elastic fracture mechanics approximation are now well solved and in current use. On the other hand, in elastoplasticity, this determination is more dubious. On top of the difficulties linked with the determination of initiation of crack propagation, there are noticeable influences of test piece geometry and loading mode. Nevertheless, fracture toughness measurements in elastoplasticity are performed more and more and standards exist to do so under good conditions. Results must be taken with a certain amount of care. To be as rigorous as possible, finite element calculations of the test piece ought to be achieved. This implies adapted refining of the mesh size near the crack tip. Under those conditions, as far as stresses and strains are known in this region, why not confront them with local fracture criteria at the microscopic level? This is the local approach to fracture mechanics. Chapter 10 will be devoted to this method and especially to measurements with notched test pieces to reach these local criteria.

222


8.7. Notations A a B b E’ G Gc J Ji JIc K KQ Kc KIc P PQ r R Rp rY ti U ui W W

G K T V, Vij

Crack area Crack length Specimen thickness Ligament length Young’s modulus E in plane stress, E/(1-Q2) in plane strain Strain energy release rate in linear elasticity Critical value of G, the fracture toughness of the material Rice integral; intensity parameter of the HRR field; strain energy release rate Critical value of J at the initiation of crack propagation Critical value of J at the initiation of ductile tearing Stress intensity factor Conventional critical value of K Critical value of K: fracture toughness Fracture toughness in plane strain Load Conventional critical value of the load Circle radius; distance to the crack tip Plastic zone size Proof strength Plastic zone correction Stress vector Absorbed energy Displacement vector Strain energy density Specimen width Crack tip opening displacement (CTOD) Shape factor Polar angle at the crack tip Principal stresses Stress tensor

8.8. Bibliography [ESI 92] ESIS Recommendation for determining the fracture toughness of materials, ESIS Procedure P2/92, 1992. [FRA 07] FRANÇOIS D., “Essais de mesure de la ténacité”, Techniques de l’Ingénieur, M 4 166, 2007. [HUT 68] HUTCHINSON J.W., “Singular behavior at the end of a tensile crack in a hardening material”, J. Mech. Phys. Solids, 16, 1968, p. 13-31.


223

[IRW 58] IRWIN G., “Elasticity and plasticity”, in S. Flügge (ed.), Encyclopaedia of Physics, 6, Springer-Verlag, 1958, p. 551-590. [NOR 96] NORME ISO, Matériaux métalliques. Détermination du facteur d’intensité de contrainte critique. Norme ISO 12 737, 1996. [RIC 68-1] RICE J.R., “A path independent integral and the approximate analysis of strain concentration by notches and cracks”, J. Applied Mech., 35, 1968, p. 379-386. [RIC 68-2] RICE J.R., ROSENGREN G.R., Plane strain deformation near a crack tip in power-law hardening material, J. Mech. Phys. Solids, 16, 1968, p. 1-12. [RIC 71] RICE J.R., “Mathematical analysis in the mechanics of fracture”, in H. Liebowitz (ed.), Fracture, an Advanced Treatise, 2, Academic Press, New York, 1971.


Chapter 9

Dynamic Tests

9.1. Introduction The study of dynamic tests on materials constitutes a vast and complex research field, which has very diverse applications: defense, transportation, civil engineering, machining, forming. Properly speaking, dynamic tests assume inertia forces and thus propagation phenomena, which have an influence on the behavior of the test piece: this behavior then results from local effects at the scale of microstructure and from global effects at the scale of the mechanical structure; it is always tricky to dissociate local from structural effects, owing to the complexity of their interactions. Depending on whether the loading is sustained – generally harmonic or more seldom pseudo-random – or transient, the tests are denoted as stationary or short transient, usually performed by shock or impact loadings; loadings achieved in very short times (0.1 ms – 1 s) then produce strain rates from 1 s-1 to 107 s-1. This chapter is devoted to transient cases. Taking into account inertia effects in transient cases is usually difficult, only because the entire kinematics of the test piece are not experimentally accessible. At best, kinematics histories are available at one or two, sometimes a few points. Generally the test must be analyzed by inverse methods assuming the form of constitutive equations and with the help of numerical tools. Nevertheless, there exist experimental techniques, which, beyond the time to create an equilibrium, ensure a Chapter written by Jean-Luc LATAILLADE.

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quasi-static state of the test piece. This allows us, as a first step at least, to get rid of inertia terms. Such tests are nevertheless called dynamic, and consequently behaviors are called dynamic: of course, this is a terminology, which may be criticized; it suggests only that such tests are able to reveal strain rate sensitivity. Kinematic considerations only are then to be considered, as the problem reduces always to the capacity of materials to reorganize themselves at various scales and on time scales, which are controlled by kinetics of various mechanisms allowing this accommodation, including damage and fracture.

Figure 9.1. Experimental devices for high velocity methods

One of the main problems for experimental studies at high strain rates lies in the absence of a unique mean to scan the whole range of engineering or research strain rates (from 1 to 107 s-1). Processing and comparison of data are then not immediate since techniques do not allow us to reach the same stress and strain levels, measure the same mechanical quantities or create the same stress and strain fields. Figure 9.1 enables us to locate the various techniques used according to the strain rates scale. 9.2. Test methods relying on propagation techniques and on the Hopkinson bar The most widely used method is the Kolsky-Hopkinson bar principle: a small size test piece sandwiched between two elastic bars, whose acoustic impedance is Uc (Uvolumetric mass, c non-dispersive bar wave velocity) and mechanical impedance Zb = UcA (A: cross-sectional area of the bar). These quantities may be denoted Ub, cb, Ab when dealing with bar (b).

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Figure 9.2. Different loading modes

Loading (in compression, tension, torsion, bending) is achieved by wave trains propagating along this mechanical line in which the test piece constitutes an impedance discontinuity (Z impedance of the test piece); this discontinuity constitutes the very principle of the technique. The first problem is that of propagation in an infinitely long bar. In compression or in traction, solutions display a solution approached for the phase velocity of the form:

cI c0

2

§R · 1 Q 2 S 2 ¨ ¸ ©O ¹

[9.1]

with: – cI = phase velocity; – c0

E

U

, E being the Young’s modulus;

– R = radius of the bar; –O = wave length; –Q = Poisson ratio. When the second term can be neglected, a non-dispersive bar wave velocity c0 is obtained, in accordance with the uni-axial theory. For this, the wavelength must be large with respect to the bar radius or the Poisson’s ratio must be close to zero, which is the case for beryllium. Of course health hazards linked with this metal forbid its usual application. Simply remember that to avoid dispersion and thus to keep down the Chree-Pochammer oscillations (see section 9.2.1), slender bars should be chosen. This is not always possible: for instance, when heterogenous materials need to be characterized for which the size of the largest heterogenity already reaches about 20 mm (concrete, rocks), so that large bar diameters must be envisaged.

228


Figure 9.3. Wave velocity in quasi-longitudinal conditions

9.2.1. Compression device and uni-dimensional theory The equilibrium of a volume element, assuming the three following conditions: – plane cross-sections remain plane; – cross-sectional stress distribution is homogenous; – radial inertia is neglected; can be written: G wV divV wx

U

w 2u wt 2

[9.2]

U : mass density; u: displacement u(x,t); V: stress vector V(x,t). Hence, for an elastic bar with Young’s modulus E: V

u x,t

c02u,xx

EH

E

wu : wx [9.3]

with c0 = (E/U)1/2.

Figure 9.4. Force equilibrium

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229

The solution of equation [9.3] is the superposition of two waves f and g propagating in reverse directions: f x c 0 t gx c 0 t

u

wu f cx c 0 t gcx c 0 t wx V EH E > f cx c 0 t gcx c 0 t @ H

v

wu wt

[9.4]

c 0 > f cx c 0 t gcx c 0 t @

In the case of a wave propagating in a single direction, a classic result is recovered:

V

Uc 0 v

[9.5]

UEv

v being the particle velocity. It will be retained that the stress level is proportional to the square root of mass density and Young’s modulus, and so that the mechanical 1 energy density carried by the wave is equal to Uv 2 . In practice, for test pieces of 2 high mechanical impedance, a high mass density ought to be chosen; this is obvious. 9.2.1.1. Radial inertia Davies [DAV 63] reformulated the preceding problem taking into account radial inertia. He found:

w 2u wt 2

c 02

w 2u I w 2u Q A wx 2wt 2 wx 2

[9.6]

where I is the moment of inertia with respect to the bar axis, A the cross-sectional area, and Q the Poisson’s ratio. Hence, when a bar of length L, of cross-sectional area A and of moment of inertia I, is instantly submitted to a compression -Vat point x = L, the other end being free, the preceding equation is satisfied by: i ª 2 2V L f 1 V t u « 0 20 ¦ 2 « ¬ 2UL S E i 1 i

V

E

wu wx

2V 0 L

S 2E

f

¦

i 1

1 i §¨1 cos i 2 ¨ ©

º · ¸sin iSx » ¸ L » ¹ ¼

[9.7]

· ¸iS cos iSx 2 2 2 ¸ L L L i D ¹

[9.8]

§ ¨1 cos iSc 0 t ¨ © L2 i 2D 2 iSc 0 t

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Figure 9.5 displays radial inertia effects on a recording during impact on the end of a bar (damped oscillations, known as Chree-Pochammer oscillations).

Figure 9.5. Example of oscillations due to radial inertia

This is a dispersive phenomenon, which needs to be minimized in Hopkinson tests. 9.2.1.2. Calculation of applied load and of displacement The problem is to calculate the applied load and displacement at a cross-section x from measurements of incident HI and reflected HR waves. By convention, a wave propagating in the positive direction (here from left to right) is called an incident wave and a wave propagating in the negative direction is called a reflected wave. But this is an abuse of language: “the wave improperly called a reflected wave is not necessarily the result of the reflection of the incident wave on an interface (for instance refer to the case of a torsion device, or to the case of uneven profiles)” [BAC 93].

Figure 9.6. Measurement of applied load and displacement

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231

From expressions [9.4]:

H t f cx c 0 t gcx c 0 t H I t H R t

[9.9]

Hence the load: Pt AEH t AE>H I t H R t @

[9.10]

where E is the Young’s modulus of the bar of cross-sectional area A. Moreover, the particle velocity v(t) is given by: v t

wu wt

c 0 > f cx c 0 t gcx c 0 t @ c 0 >H R t H I t @

[9.11]

The displacement is then: ut c 0 ³ 0t >H R W H I W @dW

[9.12]

It can be stated that: “relations [9.10] and [9.11] enable to calculate the displacement and the load applied to a bar cross-section provided measurements of waves going one way and another can be decoupled”. Relations [9.10] and [9.11] can be written: Pt i t r t Z 0 vt r t i t

[9.13]

with i(t) = AEHI(t) and r(t) = AEHR(t). Z0 = AUc0 is the impedance of the bar, also denoted AbUbcb. Thus, the problem is to decouple the waves going one way and another. 9.2.1.3. Dynamic compression Hopkinson test 9.2.1.3.1. Principle (Kolsky-Hopkinson device) Hopkinson bars enable the evaluation according to time of the applied load and displacements at the specimen ends sandwiched between the bars, strain rates possibly varying between 102 and 104 s-1.

232


The experimental device, which is represented in Figure 9.7, consists of a projectile, an input bar, the specimen and an output bar, freely co-axially sliding. It is also possible to install a damping bar if it is desired to trap part of the energy injected in the system. This allows us to limit the translation of the whole device. It remains to stop the damping bar.

Figure 9.7. Compression device

Often the input and output bars are identical. They have the following characteristics: – cross-sectional area Ab; – mass density Ub; – Young’s modulus Eb. The projectile can impact an input bar end, thus generating a compression elastic wave HI propagating towards the specimen: this is the most usual mounting. Part of the wave HR is then reflected on the input bar-specimen interface, the other part HT being transmitted to the specimen and output bar. The friction between bars and specimen are reduced as much as possible. In fact, axial compression is accompanied by radial extension, and as the Poisson’s ratios of the bars and of the specimen are not the same, at both interfaces there are differences of displacements of bars and specimen, and hence a gliding tendency, which should be eliminated: at interfaces, friction must be minimized. 9.2.1.3.2. Shock between two bars Assuming a perfectly plane interface, and with self-evident notations, the following can be written: – velocity continuity: V01 v1 V02 v 2 ; – load continuity: F1 = F2 V 1 A1 V 2 A2 . G G It can be assumed that V02 0 and V 01 in the x > 0 direction. The shock thus creates a wave propagating in the x > 0 direction in (2) and in the x < 0 direction in (1). Hence:

V 1 U1c1v1 V 2 U 2c 2 v 2

[9.14]

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233

Z being the acoustic impedance, the following system is deduced:

V1 V 2 Z1

Z2

V01

[9.15]

0

A1V 1 A2V 2

It is then found: A2 Z1Z 2 V1 V01, v1 A1Z1 A2 Z 2 Z1 V2 A1Z1Z 2 V01,v 2 A1Z1 A2 Z 2 Z2

V1 V2

[9.16]

Figure 9.8. Shock between two bars for positive time

PARTICULAR CASE.– A1 = A2, Z1 = Z2, then with V01 = Vimpact:

V 1 Uc V 2 Uc

Vimpact 2 Vimpact 2

Vimpact

,v1 V01 , v2

2

Vimpact

[9.17]

2

9.2.1.3.3. Interface boundary conditions An incident wave (I) propagating in bar (1) is considered. When it meets the interface, a transmitted wave (T) is generated in bar (2), while a reflected wave (R) moves back in bar (1). At any time at the interface, the following is respected: – velocity continuity: vI + vR = vT (condition V01 = V02 = 0 was chosen); – load continuity A1V I V R A2V T .

234


With hypothesis concerning bars orientations: uI

u I x c1t

uR

u R x c1t

uT

uT x c 2 t

V I U1c1v I V R U1c1v R V T U 2c 2 v T

[9.18]

Hence: A1V I V R A2V T

[9.19]

U 2c 2 V I V R U1c1V T

Figure 9.9. Evolution of a wave at interface

This yields:

VT

2 A1Z 2 V I vT A1Z1 A2 Z 2

2 A1Z1 vI A1Z1 A2 Z 2

VR

A2 Z 2 A1Z1 V I vR A1Z1 A2 Z 2

A1Z1 A2 Z 2 vI A1Z1 A2 Z 2

[9.20]

Note that VT.VI > 0. PARTICULAR CASES 1) U1 = Uc1 = c2

VT

2 VI 1 A2 A1

A1 1 VI 1 A2 A1 A2 ! A1 o V RV I ! 0 A2 A1 o V RV I 0

Then: V R

A2

[9.21]

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235

2) VR = 0, A1Z1 = A2Z2 Then: V T

VI

E2 U 2 E1U1

[9.22]

3) VT = 0 U 2c 2 U1c1 free end Then at interface v = vT = 2vI 4) v = vT = vI + vR = 0 A2 Z 2 !! A1Z1 end restrained Then at interface V VT = 2VI if A1 = A2 9.2.1.3.4. Loading a compression test piece It is advantageous to represent conditions at interfaces (1) and (2) in a diagram involving shock polars (load-material velocity curves) of the bars (input and output) and of the specimen (Figure 9.11). ep

Figure 9.10. Loading of a compression specimen

According to the preceding relations, for each wave: 'F DUcA'v DZA'v where D = ±1.

It is assumed here that input and output bars are made of the same material and have the same cross-sectional area. Loading of the input bar by the incident wave is represented by point A on line OA in the diagram (Figure 9.11). Its unloading by the reflected wave at interface 1 must, on the other hand, be located at the intersection of the line AB (the reflected wave propagates with a coefficient D contrasting with that of the incident wave) and of the line representing specimen loading, namely point 1.When the wave reaches interface 2 loading of the output bar takes place by a wave propagating towards positive x, as well as loading of the specimen by reflection. Interface 2 state is then represented by point 2. This last reflected wave propagates in the specimen towards interface 1, which, when reached, goes into state 3, and so on.

236


Thus, there is a set of reverberations in the specimen, which consequently is brought in a quasi-static state: progressively F1 equilibrates F2; inertial forces in the specimen go to zero. This diagrammatic representation merits further comment. First, it can be shown that it is indeed equivalent to equations [9.3]. The diagram shows, in particular, the values of transmitted and reflected waves at interface 1 when the incident wave reaches it for the first time. Furthermore, it is important to note that successive points 1, 2, 3, … represent the successive states of interface 1, while points 2, 4, 6, … represent those of interface 2. These interferences remain in such states during time lapses equal to 2Lsp/csp. Figure 9.11b also qualitatively represents the evolution of input forces (interface 1) and output forces (interface 2) according to geometric progressions. The permanent interaction between specimen and loading device must be noticed. This test is neither rate controlled nor load controlled. This coupling is represented by a differential equation of the apparatus (see section 9.2.1.3.7); mixed boundary conditions are also mentioned. It can be shown, with the help of this diagram, that the total acoustic impedance (A.Z) of specimens must be smaller than that of bars, if the specimen is to be put in compression (v[1] > v[2]). It is also shown that the closer the total acoustic impedances of specimens and bars, the shorter the time taken to reach quasi-static equilibrium. The disequilibrium of specimens for short times is clearly displayed.

Figure 9.11. (a) Load-particle velocity, (b) successive reflections of the wave in the specimen

9.2.1.3.5. Lagrange diagram (or traveling diagram) This concerns representing the evolution of the waves according to the dual variables time and space.

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237

Impact creates two compressive waves, obeying equations [9.3], which propagate respectively in the input bar and in the projectile. For this, at t = Lp/c (Lp length of the projectile, c bar wave velocity), the wave reflects on the free end as a tensile wave so that the input bar is unloaded at t = 2Lp/c by this tensile wave. The input wave traveling in the input bar (VI = UbcbVb/2) reaches the interface 1 at t = L/c. A fraction m only of the wave (m depending on time) is transmitted to the output bar, the other part being reflected as a tensile wave, which is superimposed on the incident wave. When the first unloading due to tension in the projectile reaches the specimen at t = (L + 2Lp)/c, the particle velocity in the input bar, according to the value of coefficient m, can be larger or smaller than the particle velocity in the output bar. Consequently, either the bars separate, or the specimen is loaded again. Measurement of elastic waves can be achieved in the middle of bars. In such a case, when the input bar length is over twice the projectile length, complete decoupling can be obtained between incident and reflected waves. In the output bar, the transmitted signal from the unloading wave can be observed coming from the back of the projectile. Obstruction constraints could make the installation of extensometric bridges in the middle of bars difficult. Other cases of impossibility of wave decoupling can be envisaged, for instance when the projectile is quite long so that the loading duration is very large. However, decoupling remains possible thanks to the two gages technique; knowing waves evolutions at two sections A and B, the wave can be reconstructed in any section E, for instance at the interface.

Figure 9.12. Lagrange diagram

238


Force F(t) and velocity v(t), according to the preceding considerations, obey relations (see Figure 9.13):

F t i t r t

[9.23]

Zvt r t i t

i(t) being the incident wave in term of force, r(t) the retrograde wave, and Z the impedance of the bar. Measurements of FA=AEH$t and FB=AEHB(t) yield: v A t v A t 2T AB

1 >2FB t TAB FA t FA t 2TAB @ Z

[9.24]

where TAB is the transit time from point A to point B. Thus, the particle velocity can be deduced from measurements of strains HA and HB at A and B, as it is known that vA(t 0) = 0. From vA(t) and FA(t) the force FE(t) and the particle velocity vE(t) at interface 1 can be deduced. Considering Figure 9.13b: FE t v E t

1 Z >FA t T AE F t T AE @ 2 >v A t T AE v A t T AE @ 2 1 1 v A t T AE v A t T AE @ > >FA t T AE FA t T AE @ 2 2Z

[9.25]

In the case of very weak signals, for instance with the transmitted wave, because the yield strength or the fracture resistance is low, it could be advantageous to use semi-conductor gages even though technology is very demanding.

Figure 9.13. Elastic wave diagram

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239

9.2.1.3.6. Data processing Strains corresponding to incident, reflected and transmitted signals are measured with strain gages placed on the bars. Taking into account the propagation direction of the various waves, the following relations concerning the specific velocities at interfaces 1 and 2 can be written: v1

cH R H I

v2

cHT

[9.26]

then: u1 c ³ 0t H R H I dt u2

[9.27]

t

c ³ 0 HT dt

The specimen mean strain is:

H

u 2 u1 2Lsp

H

c t ³ HT H R H I dt Lsp 0

[9.28]

Loads at the specimen ends are given by: F1

Eb Ab H I H R

F2

E b A bH T

[9.29]

When the stress equilibrium is assumed in the specimen: F1

F2 H I H R

HT

[9.30]

This yields: c dH 2 b H R dt Lsp c H 2 b ³ 0t H R dt Lsp

V Eb

Ab HT Asp

[9.31]

240


Test data can then be processed from transmitted and reflected signals. Among the limitations of this processing method, the following points can be quoted: – hypothesis of quasi-stress equilibrium in the specimen requires that processing is achieved only for strains larger than a more or less important critical strain (> 4% to 5% for most materials); – above developments assume uni-axial stresses and the possible neglect of radial inertia. It is thus essential that bars and specimen diameters be smaller the higher the strain rates. 9.2.1.3.7. Practical considerations a) Strain gages Wave measurements are generally performed with resistance strain gages in complete bridges. Two diametrically opposite longitudinal gages for the elimination of possible parasitic bending are associated with two transversal gages, diametrically opposite themselves, feeling the Poisson’s effect in the bars. Sensitivity is thus increased since the unbalanced potential e is such that: e|

2k 1 Q V E

[9.32]

where E is the Young’s modulus, Q Poisson’s ratio of bars and k the gage factor. Piezoresistive semi-conductor gages allow us to reach gage factors k between 100 and 200. b) Device equations Equations [9.30] and [9.31] yield the differential equation of the Hopkinson device proper: V t K H t

Ab Vi t Asp

V*i t

[9.33]

with 2K = UbcbLsp(Ab/Asp). The experimental device is analogous to associating in parallel the tested material whose current state is characterized by the pair (V(t), H t ) and a cell whose viscosity is K, loaded by a stress Vi*(t) (see Figure 9.14). If the constitutive behavior of the material is modeled for instance by a Maxwell cell the relaxation time of which is W R K E , the analogous diagram of the specimen-bar system is represented in Figure 9.14.

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241

The differential equation takes the form:

§ V V · V t K ¨ ¸ © E K¹

V*i t

[9.34]

Assuming Heaviside loading (see Figure 9.14), we obtain with K*

V t

ª § 1 K* K V*i «1 exp ¨ ¨ K K «¬ © t WR

·º ¸» ¸ ¹ »¼

H t

§ 1 K* V*i ª «1 K* exp ¨ ¨ K K « © t WR ¬

·º ¸» ¸ ¹ »¼

K K

:

[9.35]

Thus, this example shows that the strain rate cannot remain constant when the incident wave amplitude is constant, i.e. when the projectile cross-section is constant. It decreases in an exponential way faster the higher the reduced viscosity K KK). On the contrary, increasing K reduces the strain rate drop. The practical solution consists of increasing the Ab/Asp ratio, and/or the density Ub. The drawback of a very dense material for the bars is too strong an impedance mismatch. The greater it is, the stronger the reflected wave, as well as the strain rate, all other things being equal. Modifying the geometry of the projectile to counter-balance the strain rate drop owing, at constant Vi*, to viscous effects (or viscoplasticity as well) might be another solution. It simply consists of achieving incident wave profiles such as those shown in Figure 9.15, thanks to well-adapted projectile shapes.

Figure 9.14. Analogy scheme of bar-specimen system

Of course, the best profile results from successive iterations. Procedure heaviness and the cost of tests can be imagined, even when hybrid procedures associating experiments with numerical analyses are performed.

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c) High temperature testing High temperature experiments can pose the problem of an impedance gradient of the bar owing to a temperature gradient, i.e. acoustic dispersion. The difficulty rests in the waves transport formulae along a bar with an impedance gradient. Indeed, impulsive waves disperse in that case. The solution rests in the two gages technique, as it is adapted to accounting for impedance variations in the bar. A practical manner to consider this impedance variation consists of discretization of the bar in a sufficient number of large pieces, each one of them of constant impedance, for a temperature itself considered as piecewise constant. Solving equations such as [9.25] as many times as there are boundaries between pieces, load and velocity at the bar end in contact with the specimen can be deduced, provided the acoustic impedance of each piece is known.

Figure 9.15. Incident wave profile

Figure 9.16. Experimental device

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Figure 9.17. (a) Test bar, (b) counter bar, (c) Z/Z0 according to x at 950°C

In order to appreciate the contribution of this method, the loads and velocities at the end of a refractory steel bar at 950°C, which impacts a counter bar at ambient temperature such as the device shown in Figure 9.16, can be evaluated. The discretization method assumes a determination of acoustic impedance Z(x) (piecewise), which follows temperature recording from which the Young’s modulus profile E(x) is calculated, and hence Z(x) [BAC 91], as indicated in Figure 9.16. The precision of the method can be estimated comparing for a free end either at 950°C or at ambient temperature the load according to product Z0V, V being the particle velocity associated with an impact (see Figure 9.18). The numerical artifact is visible on diagram (a) and the beneficial effect of the acoustic dispersion correction (Figure 9.18b) can be appreciated when referring to the ordinary noise evaluated in a test at ambient temperature.

244


Figure 9.18. (a) Bar with a free end at 950°C and nominal impedance at 24°C, (b) bar with a free end at 950°C and variable impedance, (c) bar at ambient temperature

An additional proof of the interest of the method is provided by the comparison of the wave profiles such as the one reconstructed at point F from measurements at points A and B and the one directly recorded at C (see Figure 9.19). 9.2.1.3.8. Notes about confinements The first type of confinement of the specimen is that of inertia origin; of course it depends on the dimensions of the specimen, which must respect two requirements: – the diameter Imust be smaller than that of the bars so as to insure uniform stress distribution on the cross-section; – the length Lsp must be small enough for the transitory stage leading to quasistatic equilibrium to be as short as possible. An investigation from Davies and Hunter [DAV 63] reveals a geometrical criterion, which allows us to remove radial inertia effects: 2Lsp

I

Q 3

[9.36]

with Poisson’s ratio Q Moreover, it is checked that if PIZ1W Z 2 W @dW

2

JUc b

t

³ 0 T r W dW

[9.43]

Wave recording is achieved by complete Wheatstone bridges fixed up for shear associated to the torsion wave measurements. An usual oscillogram (Figure 9.23b) provides the needed measurements for test data processing: – the incident wave on which the reflected wave is superimposed after time t = d/c0, d being the distance CD between the gage bridge C and the specimen D, i.e. T i t T r t T t t ; – the transmitted wave Tt(t), here translated to the output interface and thus shifted by 't = l/c, c being the wave velocity in the specimen of length l.

Figure 9.23. a) Gage plane; b) type of torsion test oscillogram

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It is checked that, provided quasi-static equilibrium, thus after a transitory stage, T t t T i t T r t . By graphical construction, is displayed the quantity

Ti + Tr – Tr = 2Tr = 2(Ti – Tt) = J Uc b

1 J t rm

[9.44]

It is indeed found that the shear rate J is proportional to the reflected wave, and it is the greater the stronger the impedance mismatch. 9.2.2.2. Case of very ductile materials In the case of materials that can support extensive deformation, it is necessary to consider Green’s tensor: G

1 C 1 2

C being the left extension tensor defined from the deformation gradient: C

[9.45] FT F .

From Lagrange coordinates Xi (I = 1, 2, 3), the following can be written with usual notation: ª x , x , x º F « 1 2 3 » ¬X1, X 2 , X 3 ¼

[9.46]

xi denoting displacement functions. In torsion, considering simple shear: x1

X1

x2

X 2 K r X1

x3

X3

[9.47]

where 2 is the ortho-radial direction. K r

r J R R

R being the outside radius of the cylindrical specimen.

[9.48]

250


Under those conditions:

>C @1,2,3

ª1 K 2 « « K « 0 ¬

>G @1,2,3

ªJ 2 1 « «J 2 « ¬ 0

K 1 0

0º » 0» 1» ¼

0º » 0 0» 0 0» ¼

[9.49]

J

Thus, a diagonal term appears along the longitudinal direction of the specimen, which, taking into account the inertia counter-reaction of the bars, cannot exist. This amounts to considering in practice the existence of an axial stress wave, i.e. in the stress tensor matrix in the specimen the term V11, which is a counter-reaction term:

>V M @1,2,3

ªV 11 W 0º « » 0 0» « W « ¬ 0 0 0» ¼

[9.50]

Usually, in the analysis of the constitutive behavior of the material, the only term

W of the stress tensor is kept; the slightly biaxial nature of loading is thus neglected, because the amplitude of stress V11 is small enough to consider the phenomenon as secondary. The torsion test is rather appropriate for the investigation of adiabatic shears, in metals as well as in thermoplastic polymers. Adiabatic shear being a self-catalytic phenomenon, temperature measurement is indispensable to establish the thermomechanical balance needed for setting the equation of the processes. Temperature measurement must be understood as measurement of the temperature field. Taking into account the characteristic time operating in such a phenomenon (10-6s to 100.10-6s), it is not possible to use thermocouples; even those which are electric arc welded, and the size of which can be 10-3 mm, offer response times of the order of a millisecond, which is thus much too great. Their use is only justified to validate constitutive laws using a specimen thermal description. The only technique which is able to solve the problem lies on very wide bandwidth infrared radiations (0–1 MHz). Furthermore, it offers the major advantage of being a contact-less method [LAT 87]. The principle simply consists of adapting the optic of the infrared sensor on the specimen surface blackened beforehand with carbon black, in order to increase significantly the factor of emissivity of the material and to make it independent of the strain state. It is then possible to obtain recordings such as that shown in Figure 9.24.

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Figure 9.24. a) Temperature according to strain (case of a polycarbonate); b) synoptic

Test data processing assumes a preliminary dynamic calibration, operating in the reported example, in times of the order of 10 microseconds, and is also able to identify the non-linearity of the measuring assembly [MAR 95]. 9.2.3. Tension with Hopkinson bars The principle remains of course the same as for compression. In practice, the difficulty lies in specimen gripping, particularly when specimens are flat. When they are cylindrical, an easy solution, as suggested in Figure 9.2, consists of joining the specimen to the bars by screwing. For flat specimens, specific tools screwed to threads in the bars and receiving the specimen within appropriate slots must be provided. The liaison at this place is ensured by structural adhesive, as suggested in Figure 9.25a in the case of a glass-epoxy ±45° composite specimen [LAT 96]. 9.2.3.1. Low resistance specimen adaptation It may prove useful, or even necessary, to adapt the impedance of the output bar to the level of fracture strength of the material. In the present case (Figure 9.25b) it consists of a tube the stiffness of which is much smaller than that of the input bar, so that an increase in the precision in the calculation of the stress applied on the material is thus provided. Signals recorded on the input and output bars are displayed in Figures 9.26a and b.

252


Figure 9.25. a) Specimen and specimen support assembly; b) principle scheme of tensile Hopkinson bars device

Figure 9.26. a) Deformation signals of output tube at two different cross-sections; b) input bar deformation signal at a cross-section

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The incoming signal displays an inertia peak due to the heel at the input bar end placed so as to ensure the percussion by the projectile. The Chree-Pochamer oscillations are negligible and they do not interfere with the usual processing. For the output bar, the comparison of the curves do not show any dispersion effect. This is corroborated by double measurement: one according to the wave processing, the other by the direct measurement of the elongation of the specimen with the help of an optical wide bandwidth extensometer, as shown in Figure 9.27.

Figure 9.27. Validation of the longitudinal deformation measurement technique by Hopkinson metrology

NOTE.– It is often the case that a small strength of the material is associated with a low Young’s modulus, thus with a low wave velocity. This occurrence, which is encountered for instance with PMMA shock, limits or even excludes the use of the Hopkinson device, since rather a long time is needed to reach quasi-static equilibrium in the specimen [IBO 99]. Considering n back and forth travels in the specimen of an incident wave of amplitude F, load FI at the input bar-specimen interface and load FII at the specimen-output bar interface are given by:

FI FII

ª 2 4 2n º F«1 RBE REBT BET EB 1 REB REB ... REB » ¬ ¼

2 4 2n FT BET EB 1 REB REB ... REB

[9.51]

254


where RBE, TBE and REB, TEB are respectively the wave reflection and transmission coefficients at the input bar-specimen interface on the one hand and at the specimenoutput bar interface on the other. Considering that the reverberation duration in a PMMA shock is 54.10-6s, it is possible to calculate the difference between loads FI and FII as a function of time. The results are reported in Figure 9.28.

Figure 9.28. Evolution of difference between loads FI and FII according to time

It can be noted that equilibrium is approximately verified beyond 4 ms, the time corresponding to the loading, which a 1,100 mm long steel projectile would provide and for which the bar wave propagation velocity would be 5,400 ms-1. This projectile length corresponds practically to the maximum value, which for technological reasons may be allowable in this kind of device. The limitation of the technique can thus be assessed. The only way to treat the problem then consists of describing the propagation phenomena in the specimen. In order to do so, it is necessary to assume a constitutive law and by an inverse method to adapt it taking into account experimental results. This amounts to the problem of the Taylor test (see section 9.2.4). If necessary, investigations of Zhao [ZHA 92] about polymeric foams may be consulted.

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9.2.3.2. Adaptation of a tensile test bench line for interrupted tests For a number of materials, identifying damage mechanisms according to the strain and the strain rate is an essential objective. Interrupted tests are therefore required, i.e. at stress levels controlled by the loading duration T, thus by the length Lp since T = 2Lp/cp, and by the incident wave level V,thus by the projectile velocity Vp, as noticed in Figure 9.29 as a matter of principle.

Figure 9.29. a) Evolution of maximum applied stresses according to projectile lengths, b) interrupted tensile tests at various loads on stratified materials (±45°) (strain rate = 4.8 s-1)

Of course, the unloading back wave returns at the bar ends are responsible for specimen reloading. Reflected and transmitted waves at the ends must then be trapped to avoid specimen reloading. The waves traps can consist of [DEL 95]: – a bar barely touching the heel, the impedance of which matches that of the input bar (Figure 9.30a and b); – a tube coaxial with the output tube, which reacts to the compression wave return, owing mostly to the reflection of the tensile wave at the output tube end, which traveled through the specimen, as shown in Figure 9.31. Figure 9.31 ascertains the trap efficiency.

256


Figure 9.30. a) Lagrange diagram with trapping device at the input bar; b) validation of wave trapping at the input bar

Figure 9.31. a) wave trapping device at the output tube level; b) tensile test on composite material specimen (±45°)

9.2.3.3. Brittle material case: alternatives The Hopkinson bar technique lends itself quite poorly to investigations on brittle materials. Data processing could necessitate heavy numerical simulations [LAM 97]. Concerning propagation methods, several alternatives can exist. For instance [GRA 87], in the case of rocks or concrete, there is the one which consists in submitting the specimen to triaxial compression in a compression cell, and to release it abruptly. A double tensile release results from specimen ends

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towards the center, which progressively unloads the specimen (Figure 9.32). From situation denoted (e) onwards, tensile stresses appear in the specimen. Of course, this technique can be applied only to materials whose fracture strength in compression is larger (in absolute value) than that in tension.

Figure 9.32. a) Dynamic elastic loading technique with confinement. End pressures are simultaneously and suddenly released; radial pressure is maintained; b) axial strain and stress distribution

Another example consists of impacting a specimen through an interposed bar [KHA 87]. In view of the preceding explanations, it can be understood that following the reflections, tensile waves are created, the controlled combination of which is sought by an appropriate choice of the assembly geometry. Fracture can then be achieved in a perfectly determined zone (Figure 9.33).

258


Figure 9.33. x – t diagram

9.2.4. Taylor test 9.2.4.1. Overall considerations This test is not aimed at the determination of constitutive laws, even by an indirect method. Its only objective is the determination of the yield limit in compression at very high loading rates (103 – 105 s-1). It is thus a valuable validation mean for computer codes relying on constitutive models determined with other tests performed in simple and precise loading conditions: Hopkinson bars and plate impacts, for instance. It can also help in the classification of materials of the same type, for particular loading conditions. A complete numerical simulation of this test associated with precise measurement of the deformation of the specimen with ultra-rapid cinematography allows us to reach the constitutive parameters, postulated a priori. However, this remains quite debatable owing to the influence of non-observable quantities, of difficult to apprehend thermo-mechanical couplings (self heating), or even in some cases to adiabatic shears, which cannot be modeled. Non-uniformity of stresses, strains and strain rates constitutes a major drawback for constitutive parameter determination. On the other hand, they are advantageous in numerical processing, as they allow us to validate a constitutive model, which has been identify in another manner over a large range of strains and strain rates.

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9.2.4.2. Principle The principle is to project a plane faces cylindrical circular specimen (Ifrom 4 to 10 mmL0 from 20 to 70 mm) onto a very hard and massive target, at velocities between 50 and 500 ms-1; very high strain rates are obtained near the impact face (104 to 106 s-1). Of course, fractures must be avoided. When the stress induced in the material exceeds the yield strength, an elastic wave propagates associated with longitudinal dynamic compression and radial flow; the plastic wave is preceded by an elastic wave, which reflects successively between the back end and the plastic wave front, within a zone, which is submitted to elastic strains only. The specimen deforms as an elephant foot, as shown in Figure 9.34.

Figure 9.34. Elephant foot specimen in Taylor test

From quantities L, L0, X and H, Taylor [TAY 48], postulating rigid plastic behavior, deduced the mean yield strength according to impact velocity Vi: Rp

L X L0 L 1 UVi2 0 2 logL0 X

[9.52]

In the literature, models other than the rigid plastic model were used. 9.3. Dynamic fracture mechanics tests 9.3.1. Introduction Investigation of the critical value of stress intensity factor may suggest various situations. Restricting ourselves to mode I: – in dynamic loading, unstable propagation takes place. Dynamic fracture toughness KID corresponds to the value of the stress intensity factor at initiation tf. Usually, the critical value depends on loading rate V and on temperature T:

260


K ID V ,T

K Idyn t f ;

– propagation becomes unstable when crack propagation velocity reaches V at time t: K ID V K Idyn t ;

– crack arrests when stress intensity factor reaches a value KIa (crack arrest toughness); – branching can take place when the stress intensity factor equals KIb. In the following, experimental determination of the dynamic fracture toughness KID will be considered, limited to Charpy testing. 9.3.2. Charpy test analysis according to the Kishimoto method In contrast with static loading (see section 9.7), in dynamic three points bend testing there is no simple relation between applied load and stress intensity factor. It can nevertheless be assumed that the stress intensity factor is proportional to crack tip opening displacement (CTOD) G, i.e. to the deflection at the center of the specimen.

Figure 9.35. Geometrical characteristic of a Charpy specimen

C being the specimen compliance, under the condition that there is no branching: K I t

§ a ·G t Y ¨ ¸ ©W ¹ C 2BW

3S a

2

[9.53]

Y being a calibration polynomial, quantities S, a, B, W being defined in Figure 9.35.

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In the case where S/W is equal to 4: § a · Y ¨ ¸ ©W ¹

ª

§ a ·

§ a ·2

§ a ·3

§ a ·4 º

«¬

©W ¹

©W ¹

©W ¹

©W ¹ »¼

S «1.09 1.735¨ ¸ 8.2¨ ¸ 14.18¨ ¸ 14.57¨ ¸ » [9.54]

However, in a Charpy test it is easier to measure the applied load with a good precision than the deflection. From a theoretical expression due to Nash [NAS 69], Kishimoto [KIS 80] demonstrated: K Id t

ZI

K Ista

F t

t

³ 0 F W sin Z I t W dW

[9.55]

ZI denoting the first eigenmode pulsation. In this equation a convolution integral is recognized allowing the calculation of specimen response to any loading F(t) from the ZI pulsation sinusoidal response to an impulsion. Of course, this constitutes a restriction, since the interaction specimen-striker can a priori activate other response modes of the beam specimen and consequently other fracture modes (modes II and III). On the other hand, this expression relies on the superposition principle, so implicitly on linear elastic behavior. We can state: – the need to determine the first eigenmode and thus of measuring the specimen compliance C; – the need to measure the applied load at fracture since it is the one which yields KIsta according to: K Ista

3S a

§ a · Y ¨ ¸F t . ©W ¹ 2BW 2

[9.56]

Kalthoff and Böhme proposed a different procedure [BÖH 82]. It consists in establishing an experimental response curve to impact detecting the crack initiation time. The method decomposes into two parts [KAL 77, BÖM 86, KAL 85]. A first step consists of a rough estimation of the stress intensity factor obtained from modeling and static analytical calculation; a model of springs in series (see Figure 9.36) allow us to determine a quasi-static coefficient KIqs(t): K Iqs

12 ª EM EB º » « V0Y sin t * « MC s* » ¼ ¬WBC s

[9.57]

262


where V0 is the impact velocity, Y the calibration polynomial, M the mass of the striker, E the Young’s modulus of the material, W and B the height and the width of the specimen and Cs* the specimen compliance possibly corrected to take into account that of the machine.

Figure 9.36. Modeling of striker-specimen system with Csp and Cmac compliances

Figure 9.37. Dynamic correction of Kid according to Kalthoff et al.

A second step consists in taking into account dynamic effects with a correction factor Kdyn so that: K Idyn

K dyn K Iqs

[9.58]

Kdyn is a dynamic correction factor analogous to Kishimoto’s, but this coefficient K was determined experimentally from results of a number of investigations on many different materials. It is a function of time, which depends on the specimen geometry. Figure 9.37 gives the curve for the most frequently used specimen. To determine KID at crack initiation, it is sufficient to know the corresponding time; Kalthoff proposed particular experimental techniques (caustics and magnetic detection). dyn

9.3.3. Charpy test with Hopkinson bars With a traditional Charpy pendulum, load measurement is achieved by strain gages glued on the striker, calibrated in static conditions. The elastokinetic wave

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propagation is neglected, which is the source of additional errors. This difficulty can be overcome by the use of Hopkinson bars, as shown in Figure 9.38. The striker is an intermediate cylindrical bar (instrumented with gages) between the three points bend notched specimen and a projectile, the velocity of which partly fixes the velocity at the contact point with the specimen. This velocity as well as the applied load on the specimen are deduced from the analysis of reflected and transmitted waves in the striker bar from the two gages technique and according to equations [9.25] above. Proportionality between deflection and stress intensity factor can be used directly (equations [9.53] to [9.58] concerning PMMA. As an example, comparison can be made between the stress intensity factor deduced from deflection (KIG) and that determined with a gage placed close to the crack tip (KIgage). The quasi-static stress intensity factor is also calculated (see Figure 9.39).

Figure 9.38. Conventional mounting of Hopkinson bar

Figure 9.39. KI according to time; full line: application point displacement; fine line: applied load; dotted line: deformation as measured on the specimen

Even if the value of KIgage is obtained with a good precision, in the case of high temperature testing, measurement is not possible. The Hopkinson bar technique is then of great interest [LAT 94].

264


9.4. Plate against plate test 9.4.1. Principle The plate impact test creates a uni-axial deformation. It consists of projecting a usually metallic sheet (i.e. a very thin plate of thickness between 0.1 and 1 mm) at very high velocity (200 ms-1 < Vp < 1,500 ms-1) on a plate target to be investigated (see Figure 9.40a) usually instrumented with piezoresistive or piezoelectric strain gages, placed at various planes so as to observe the evolution of the shock wave, and also with electro-magnetic gages to measure specific velocities. In particular, it is this evolution which allows us to determine constitutive laws or criteria: viscoplastic behavior of metals and some polymers (eventually coupled with damage), quasibrittle damaging behavior of rocks and ceramics, and hyper-elastic and viscoelastic behavior of some elastomers. The back face can be aimed by a laser beam associated with a Doppler interferometric assembly; this enables us to measure the free back face particle velocity. The impact creates a longitudinal compression wave, the duration of which depends on the thickness of the flying sheet. In the case of plastic materials, a plastic wave preceded by an elastic precursory can be obtained; this occurs when the stress created in the material exceeds the yield strength. This longitudinal stress is associated with axial stresses, i.e. in the target plane. Oblique impacts can also be considered, so as to obtain simultaneously compression and shear stresses (see Figure 9.41). The biaxiality ratio is adjusted by angle E.

a)

b) Figure 9.40. a) High energy plane shock method; b) schematic representation of plane plate impact

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Figure 9.41. Schematic view of experimental technique used to create compression and shear waves

Performing these tests is always demanding owing to the very small tolerance regarding the parallelism of the projectile and the target, and also owing to the operating metrologies, which is very costly anyway because of heavy instrumentation. They are however unavoidable when it is necessary to identify constitutive behaviors for velocities between 105 and 107 s-1. 9.4.1.1. Traveling diagram The shock wave pressure is proportional to the impact velocity and it depends on the shock impedance of the materials of the two plates. If it is assumed, in a first approach, that the target dimension is infinite in the impact direction, the traveling diagram shown in Figure 9.42 can be established; it displays compression waves (C) and release waves (R). At impact, two compression waves start from the projectile (medium A)/target (medium B) interface. One propagates in the target the other one in the projectile. It is assumed that the shock impedance of the projectile is larger than that of the target. The initial compression wave, which crosses the projectile, reflects on its back face as a release wave. This is transmitted through the projectile/target interface and travels back in the projectile. The same scheme repeats itself afterwards from the reflected shock.

266


Figure 9.42. Traveling diagram

Figure 9.43 allows us, in the (Vx, v) plane, to follow states of media A and B from the shock polars and release curves. Before the shock medium A has a velocity denoted v0, medium B is at rest. The initial shock creates a stress discontinuity equal to V1 and a material velocity discontinuity equal to v1. From operating point P1 located at the intersection of the two shock polars, it is possible to follow the evolution of medium A on the release curve originating at P1. During release, the state of medium A goes from point P1 to point P2. The new operating point between media A and B will be point P3, located at the intersection of the recompression curve of A originating at P2 and of the shock polar B. Thus, the stress history of a point located in the target at a given abscissa x is represented in Figure 9.43.

Figure 9.43. Representation of compression decompression test in the plane stress/particle velocity

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9.4.1.2. Rankine-Hugoniot conservation equation The deformation field xi is uni-axial. Xi denoting Lagrange coordinates, we obtain: x1

X1 u1X1,t

x2

X2

x3

X3

[9.59]

under the condition that the medium can be considered as continuous and that the gradient can be linearized:

H11

wu1 wX1

'V V0

U0 1 U

[9.60]

U0 and U denoting the densities of the unloaded material and of the material under pressure, 'V measuring the volume variation initially equal to V0. The stress field is such that V Vthe principal directions being along the coordinate axes. Thus, p denotes pressure and W the maximum shear stress: 4 3

V 11 p W

[9.61]

2 3

V 22 V 33 p W An ideal shock wave is a stress and velocities discontinuity propagating through the material. For such a surface, the usual conservation equations are replaced by Rankine-Hugoniot equations in Lagrangian coordinates, which express jump conditions. Considering a system reduced to uni-dimensional space and assimilating the discontinuity surface to the propagation of a constant velocity jump C (Figure 9.44), the conservation equations become: Conservation of mass

U v 0 C

0

Conservation of linear momentum

V U 0 vC

0

Conservation of energy

U 0 ¨e v 2 ¸C Vv 0

U

§ ©

1 2

· ¹

268


Provided state S+ = {e0, V0, v0, U0} of the material in which the shock propagates is specified, five unknowns remain, C and S- = {e, Vv, U}, to describe the shock. Boundary conditions for the load, which produces the shock, is one of them. The Rankine-Hugoniot conservation equations provide three equations between four variables. One equation is missing. It is the constitutive equation of the material. Consider the material at rest upstream of the front (v0 = 0 and V0 = 0) and in which the shock velocity is assumed to be positive (Figure 9.44). A large number of materials are accurately modeled by a constitutive linear relation between C and v [DAV 79]:

C

C 0 sv

[9.62]

where C0 and s are material parameters depending on state S+. This choice being made, Rankine-Hugoniot equations enable us to deduce the stress under shock, denoted VH and called the Hugoniot stress. It is given by the equation:

VH

with K 1

U 0C 02K

1 sK 2

[9.63]

U0 U

Figure 9.44. Shock propagation: variables upstream and downstream of the shock

Figure 9.45. Representation of Hugoniot and Rayleigh line and total energy

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Figure 9.46. Shock polar and shock impedance

Curve VH(V), V = 1/Udenoting the specific volume, is called an adiabatic dynamic curve or otherwise a Hugoniot curve. It is the locus of points (VH, V) obtained under shock as shown in Figure 9.45. The chord joining the initial state (V0, V0) to any point on the adiabatic dynamic is called the Rayleigh line. During the shock the material follows non-equilibrium states located along this line. The Hugoniot itself is made of a series of points characterizing equilibrium states. The combination of Rankine-Hugoniot conservation equations and of the constitutive equation of the material allows us to determine the relation between the Hugoniot stress and the velocity:

>V @ V 0 V H U 0 C 0 sv v

[9.64]

This relation enables us to draw the curve VH(v) called the shock polar (see Figure 9.46). The slope of the straight line joining the origin to any point on the shock polar is called the shock impedance. It obeys relation: Z

V 0 V H v v0

U 0C

[9.65]

It is a measure of the difficulty to compress a material. NOTE.– When pressure levels are very high, the effect of shear stresses can become negligible, so that loading can be assimilated to hydrostatic pressure. This is the hydrostatic approximation.

270


9.4.1.3. Lagrangian analysis This analysis is based on measurements with electro-magnetic gages working according to the Laplace law: a magnetic field being produced around the specimen, the displacement of the gage associated with the particle velocity of the point where it is located enables us to measure this particle velocity thanks to the electromotive force produced. Velocity profiles v(x, t) are then available. Incremental formulation of conservation equations yields: §1 · dv d ¨ ¸ ©U ¹ U 0C v

with C v

dV

with CV

dE

[9.66]

wv wt X wv wX t

§wX · ¨ ¸ ©wt ¹v

U 0CV dv

[9.67]

wV wt X wV wX t

§wX · ¨ ¸ ©wt ¹V

Vdv U 0C v

[9.68]

Thus, to calculate the other flow quantities, it is enough to measure the wave propagation velocities at constant particle velocity (Cv) and constant stress (CV). In reality, the experiment does not give direct access to CV; it only provides Cv. Indeed, if particle velocities on several gage planes are available, Cv = ('X/'t)v is easily obtained, as shown in Figure 9.47. However, to obtain CV it can be demonstrated that:

§wC · ¨ v ¸ © wt ¹v

w ª wv º CV C v » « wv ¬ wt ¼

§wv · and as ¨ ¸ is known, it is possible to calculate CV ©wt ¹X

[9.69]

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Figure 9.47. Wave propagation scheme in a material

9.4.1.4. Application to the characterization of an elastomer The example described hereafter is taken from investigations of the Arcueil Technical Center for the optimization of reactive armor [BON 01]. 9.4.1.4.1. Experimental configuration The experimental configuration adopted for these tests is represented in Figure 9.48. The striker is made of either a pyrex disk or a PMMA disk. It is placed on a low-density foam, which reproduces free end conditions. The target is made from a transmitting PMMA medium and the elastomer specimens glued together. The role of the transmitting medium is to make the target stiffer. The choice of PMMA is due to its shock impedance close to that of elastomers. The metrology adopted for those tests is the electromagnetic gages technique, which gives access to particle velocity evolutions inside the target. The gages used were cupper jumpers encapsulated within kapton sheets. Their active length was 9 mm and their thickness 150 Pm. To be able to process the results according to the Lagrangian analysis technique, the gages were placed in four measuring planes 5 mm apart from one another. Two face-to-face gages were glued in the middle of specimens when assembling the target (see Figure 9.48).

Figure 9.48. Schematic representation of experimental configuration for plate impact test

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9.4.1.4.2. Presentation of tests Testing conditions are described in the following table. Test number

Target material

Striker

Striker velocity m/s

Striker

Window

1

AM1

Pyrex

541 + 5

10.05

2.95

2

AM1

PMMA

533 + 5

4.99

3.02

Test number

Target 1 (mm)

Target 2 (mm)

Target 3 (mm)

Target 4 (mm)

1

4.96

4.96

4.96

19.80

2

4.96

4.96

4.96

19.80

The choice of the impacting material (pyrex or PMMA) results from a compromise between, on the one hand, the maximum stress to be attained and on the other hand, observation of release waves down to as low a stress level as possible. To reach stress levels in the target of the order of 2.5 GPa, for a maximum striker velocity of 800 ms-1 (imposed by the thruster), it is necessary to choose for the projectile a material having a shock impedance at least two times larger than that of the elastomer. Pyrex fulfills this condition; however, it does not allow us to generate a release wave bringing back the target to zero stress, during the workable test duration. This is the reason why a PMMA striker was used in test no. 2. Indeed, its impedance is close to that of the elastomer, and its use allows us to have a complete observation of release waves.

Figure 9.49. Particle velocity diagram obtained on AM1 polymer in test no. 1 with pyrex striker

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9.4.1.4.3. Results display Two examples of obtained results are shown in Figures 9.49 and 9.51: test no. 1 with a pyrex striker and test no. 2 with a PMMA striker. All diagrams present similar features: – a compression wave with a rise time between 100 and 300 ns, up to nominal velocity; – a velocity plateau which is increasingly stationary the further away it is from the impact plane; – a release consisting of two waves separated by a plateau when the striker is made of pyrex and of a single monotonous wave when it is made of PMMA. The interpretation of the appearance of particle velocities diagrams is made easier by the construction of the traveling diagram, which schematically visualizes wave propagation in a space-time plane. Two cases are distinguished according to the striker type (Figures 9.50 and 9.52). Before presenting those two cases, it should be noted that the travel diagram construction was simplified, for reasons of clarity. The PMMA behavior during release was approached by a single wave; now, the reality is somewhat different as release of the PMMA is achieved by a cluster of release waves. Moreover, the interface displacement are not taken into account.

Figure 9.50. Traveling diagram scheme for pyrex striker

274


a) Pyrex striker Taking into account the impedance mismatch between the pyrex striker, PMMA transmitting medium and elastomer, a reverberation of the initial compression wave occurs in the PMMA (see Figure 9.50). Part of this wave is reflected at the interface PMMA/AM1, then at interface pyrex/PMMA, to be transmitted afterwards in the AM1 target. The corresponding singularity is marked (1) in the first particle velocity diagram in Figure 9.49. It is located at 1.7 Ps after the first compression front, which corresponds to the duration of wave travel in the PMMA. On the other hand, the initial compression wave generated in the striker reflects on the target surface as a first release wave. It propagates in the PMMA, then in the target and it is marked (3) in the diagram. At the impact interface, part of this wave reflects as compression, which transforms after reflection on the back free end in a second release wave. These two release waves are separated by a velocity plateau of the order of 3 Ps duration. Singularities marked (2) in the three first velocity diagrams correspond to reflected waves on the gage planes. Indeed, when the initial compression wave meets the gage, a release wave is generated owing to the impedance mismatch with the elastomer. This wave propagates in the target and is observed in the upstream measuring plane. Note that this singularity is not observed in the fourth diagram since there is no measuring plane downstream.

Figure 9.51. Particle velocity diagram obtained with AM1 polymer in test no. 2 with PMMA striker

b) PMMA striker In the case of the PMMA striker, the scheme of wave propagation is slightly different (see Figure 9.52). The striker and the transmitting medium being made of identical materials, there are no more reflections at the impact interface and one

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release wave alone marked (3) is observed in the target (see Figures 9.51 and 52). However singularities due to the presence of gages still exist on the first three curves (2).

Figure 9.52. Traveling diagram scheme for PMMA striker

9.4.1.4.4. Data processing Data processing of the particle velocities diagrams presented above is performed by Lagrangian analysis. The results obtained are shown in Figures 9.53 and 9.54 and in the following table. Test number

Particle velocity (ms-1)

Propagation velocity (ms-1)

Deformation (%)

Stress (GPa)

Release wave propagation velocity (ms-1)

1

413

2,989

13.8

1.46

4,200

2

292

2,681

10.9

0.93

3,824

The points in Figure 9.53 correspond to propagation velocities of the compression wave and of the first release wave. For the compression wave, results can be interpolated by a straight line of equation: C

C 0 sv

with C0 = 2,010 ms-1and s = 2.18.

[9.62]

276


Stress (GPa)

Figure 9.53. Evolutions of propagation velocities according to the particle velocity for the AM1 polymer

Figure 9.54. Compression release curves in plane plate impact on AM1

Linearity between propagation velocity C and particle velocities v is observed for a large number of materials. Differing from Figure 9.53 in which equilibrium points only are represented, Figure 9.54 represents compression and release paths followed by the material in the stress/deformation plane, according to two experimental configurations. Results related to the two achieved tests show that the material behavior under shocks of different levels does not present any particular singularities in the stress range investigated, whether in compression or in release. From the behavior point of view, results obtained under shock do not reveal any particular transition from one state to another. The analysis of the curves reveals a hydrostatic behavior. Indeed, release waves are homogenous with return to zero. Thus, the specimen does not

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display any residual deformation after unloading and no trace of deviatoric behavior was shown for stresses between 0.5 and 2.3 GPa. Note furthermore that unloading paths located on an isotrope are very close to the Hugoniot curve characterizing adiabatic dynamic. This means that thermal effects have a small influence on the total pressure. 9.5. Collision tests Collision tests consist of impacting the investigated specimen or structure with a projectile (a striker, or an inertia ram or adjustable masses) and with possibly controlled velocity. 9.5.1. Inertia wheel instrumented impacting device 9.5.1.1. Overall considerations This is a device inspired by the Charpy pendulum. It consists of a massive inertia wheel, motioned by a motor capable of communicating controlled rotation velocities, adjustable in a continuous manner, and providing kinetic energy much larger than that needed to break the test piece. This kinetic energy is according to devices between 500 and 100,000 J. The fork fixed on the wheel can thus pull the free end of a test piece at a practically constant velocity (1 to 20 ms-1), when this test piece, with the help of a pendular device, is put in place at the time the wheel reaches the chosen nominal velocity to obtain the desired strain rate. Between the pendulum and the cross piece a piezoelectric load sensor is placed conditioned by a load amplifier (Figure 9.6a and b).

b)

a) Figure 9.55. a) Principle of the inertia wheel device; b) detail of pendulum device

278


To avoid instrumentation of the test piece itself, a wide bandwidth contact-less extensometer sensor can be used. It is based on the tracking of two optical contrasts located on the test piece, the distance of which defines the reference length, which is usually equal to the gage length. 9.5.1.2. Discussion and application to an adhesive joint Absorbed kinetic energy in the test is distributed in the following way: 'E0

EI Esp EMV EME

[9.70]

with:

– EI the energy needed to motion the specimen end (inertial component); – Esp the total energy needed to fracture the specimen; – EMV the absorbed vibration energy of the machine; – EME the absorbed elastic energy of the machine.

Figure 9.56. a) Detail of loading device; b) load sensor mounting

For brittle fractures, the value of EMV is usually very small compared to 'E0. The inertia force is the one needed to motion the specimen end. It is maximum at impact and then decreases quickly as the end velocity of the specimen increases. The value of EI can be evaluated from the load-time curve and from the following equations,

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replacing in the integral tr by ti, the time associated with h inertia force (see Figure 9.57):

'E0

§ E · Ea ¨1 a ¸ © 4 E0 ¹

[9.71]

where Ea

V0 ³ 0tr Fdt

where V0 is the ram speed just before impact and F the applied load during the test duration tr.

Figure 9.57. Inertia effects on recorded load-time signal

Quantity EME is an elastic energy. It is expressed as: EME

1 2 F CM 2

[9.72]

where CM is the elastic compliance of the machine. The energy needed to break the specimen can thus be written: Esp

1 V ³ 0tr Fdt V0 ³ 0ti Fdt F 2C M 2

[9.73]

The true displacement of the specimen during the test can be evaluated from the load-time curve and the machine compliance CM: d sp

V t FC M

[9.74]

280


According to the fundamental dynamic law:

³ 0tr Fdt M V0 V f

[9.75]

where M is the mass of the striker and VF its velocity after fracture of the specimen. Hence:

Vf

12 §2E0 ·1 2 § 'E0 · ¸ ¨1 ¨ ¸ © M ¹ © E0 ¹

§ 'E ·1 2 0 V0 ¨1 ¸ © E0 ¹

V

1 2 º ª § 'E · 1 « § 'E0 · » 0 1 ¨1 V0 ¨1 ¸ ¸ 2 « © E0 ¹ » 4 E © 0 ¹ ¬ ¼

[9.76]

and [9.77]

The true displacement of the test piece can thus be written:

d sp

§ 'E · 0 FC V0 t ¨1 ¸ M © 4E0 ¹

[9.78]

Characteristics proper to inertia wheel 1) The frame is stiff enough to neglect its deformations (CM > Ea) allows assuming constant loading velocity equal to the initial imposed velocity V0. The acceleration of the specimen is then quasi-instantaneous (ti 0.02 ms-1) the loading velocity derived from the optical sensor measurement is constant during a preponderant stage of the test and in particular when the maximum load is reached, associated with the yield strength of the structural adhesive. This velocity is indeed equal to the nominal velocity of the wheel. The deformation measured by the optical sensor in fact includes the deformation of the adhesive itself, but also the compliance of the assembly. To correct the measurement, it is necessary to proceed with a test piece without adhesive as shown in Figure 9.60. The physical quantities are then calculated according to the following equations, which yield the shear rate, the maximum shear stress and the maximum shear:

J

V0 h

Wmax

Fi Aj

J max

V0 tr t s h

[9.81]

where h is the adhesive thickness and Aj the area of the joint. As an example of experimental results, the variation at 25°C of the ratio Rp/T according to the logarithm of the shear rate (log J ) may be considered on the one hand and that of the yield strength Rp according to temperature on the other for a shear rate J 1.8.103 s 1 , i.e. for a critical value appearing in the first variation. These results reveal a molecular transition in the macroscopic response of the joint, associated with a secondary relaxation E of the adhesive, in term of critical rate J E as well as critical temperature TE. The correspondence between

J E

T 25qC

1.8.103 s 1 and TE

J 1.8.103 s 1

25qC of course is not fortuitous but a

proof of the validity of an equivalence temperature-strain rate principle, which is usual for solid viscoelastic polymers [LAT 97].

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Figure 9.60. Signal processing

Figure 9.61. Behavior transition display of epoxy EP1 expressed in terms of critical strain rate or critical temperature

283

284


9.5.2. Drop weight tests

9.5.2.1. Introduction: the critical mass-velocity couple Although used for fracture mechanics applications [CRO 69] or for material characterization, for example composite materials [HAR 90], dynamic drop weight tests are mostly tests on structures, as shown schematically in Figure 9.62a and b. Beyond simple technological characterization, they cannot only be analyzed from numerical simulations: this is the entire challenge of hybrid techniques for which, when there is damage and/or plasticity, failure scenarii at various scales, constitutive laws, fracture criteria, etc. must be imagined. Indeed their analysis is always exacting, since the global response integrates dimensional factors together at the scale of the structure as well as boundary conditions [WAH 97] and local behavior of constituent materials, at the impact point for instance. In the same way as local and global responses are coupled, the local behaviors can also be the complex resultant of the interactions of mechanisms at various scales, in particular those which may characterize the micro or meso-structure of the investigated material. It is easily conceived that at each scale correspond one or several mechanisms, on which characteristic times depend. The total process is thus poly-kinetic. In other words, the system usually presents a transfer function, which can be expressed in a response time spectrum. This spectrum can be very wide. Considering, without going into detail, a composite plate resting on two supports and loaded by the impact of a projectile as an example, we will intervene at one extremity of the spectrum times of the order of one second (those corresponding to the fundamental bending mode) and at the other extremity times of the order of tens of microseconds, which corresponds for instance to micro-crack propagation under the striker. It is easily conceivable that it is not prudent to replace dynamic tests with static tests. The failure modes could be different. This remark is sufficient to make clear that the energy of the striker considered by many authors as a pertinent experimental parameter is quite insufficient to qualify the test type. Assume a 10 g ball at a velocity of 220 ms-1: its kinetic energy is close to 250 J like that of a 5 kg mass at 10 ms-1. A ball at this velocity will perforate a thick glass/polyester composite plate (6 mm) [AYA 93], whereas a guided 5 kg mass falling from a 6 m height will crush locally, under the hemispherical punch, a sandwich plate glass polyester/fabric/foam, the skin of which are 2 mm thick, together with delaminations.

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Figure 9.62. a) Drop weight Crosley device; b) schematic device and Lagrange diagram for Hopkinson elastic bar

These two examples clearly show that: – in the first case the structure practically does not deform; the damage is quite localized and results from elementary coupled mechanisms. It is the effect of the material which predominates. The development of the damage zone depends on the interaction time between the ball and the plate; – in the second case, further to punching under the striker, damage is much more delocalized. Interfacial fractures between the lower skin and the core can propagate, since they have the time to do so and the necessary damage energy is provided by the plate transitory deflection, which lasts a few tenths of a second. Of course, similar situations are found with other materials. For instance, in the case of a steel beam, the ball effect will be perforation together eventually with adiabatic shears, the development of which depends on the penetration velocity. The effect of the massive projectile on an identical plate will be a vibratory response, together with a plastic hinge eventually, the development of which can be strain rate dependent when the behavior is viscoplastic. Another example [VID 98] concerning

286


laminated glass panels hit by a hard ball underlines that the greater the loading velocity, the more resistant the glass. This stresses that in the case of low velocity shocks it is the couple mass-velocity which must be considered.

Figure 9.63. a) Localized failure mode by ballistic impact perforation; b) failure mode by low velocity impact

9.5.2.1.1. Design matrix and experimental campaigns Bringing to light a mechanical, physical or chemical phenomenon often requires the realization of more or less numerous tests. The design matrix allows us to structure the test campaign so as to minimize the variance of results, together with a reduction in the importance of experimental characterization. The use of this matrix leads to empirical polynomial modeling relating the mechanical system variables to the response of the structure. The fundamental result is that the variance of the model (Y) is expressed as the product of the experimental variance (V2) by a term which depends on the organization of the tests (A). (V2) is estimated by repeating the same test. The estimator will be more precise the larger the number of repetitions. This means that choosing an optimal experimental distribution of tests can minimize the variance of the model. The setting up of a design matrix requires several steps. The nature of variables and of their limit values must be decided first. In a second step, the degree of the polynomial must be chosen. Then the matrix is established following precise rules [BOX 87]. To facilitate the determination of the respective influence of the various parameters, the variables are in centered and reduced coordinates. The amplitude of their variation is thus brought in the interval [-1; +1]. Pre-investigation on a stratified material estimated the influence of mass and velocity on responses such as: contact duration, contact load, deflection in the center of the plate, possible delamination area between plies of various orientations, the number of cracks parallel to the plate width (n1) and number parallel to the length (n2). This first step was not dedicated to the establishment of constitutive models. On the other hand, particular attention was attached to the coupling between the two

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variables mass and velocity. An experiment factorial matrix was used for the investigation of this material. X1

X2

(mm)

(mm)

1

-1

-1

2

+1

-1

3

-1

+1

4

+1

+1

Test number

The model corresponding to this matrix was a second order polynomial: Y

b0 b1 X1 b2 X 2 b12 X1 X 2

[9.82]

Parameter X1 represents the drop height and parameter X2 the mass. Coefficient b12 quantifies coupling. A Doehler matrix [DOE70] was chosen for the sandwich materials. It consists of a design matrix called a response surface matrix. The basic principle consists of examining the surrounding of a given point in the experimental domain (see Figure 9.64). These matrices are of spherical symmetry. Continuous and quantitative factors are assumed. The three variables chosen in this investigation were the span (X1), the projectile velocity (X2) and the mass (X3). The associated design matrix for three variables required 13 tests. Moreover, the associated polynomial had to be second order. This assumed that the responses were linear: Y b0 b1 X1 b2 X 2 b3 X 3 b12 X1 X 2 b23 X 2 X 3 b13 X1 X 3 b11 X1 X1 b22 X 2 X 2 b33 X 3 X 3

[9.83]

The impact tests were performed with a drop weight device. The principle was to drop a guided mass from a given height. Three systems were used. It should be noted that there are as many polynomials to be identified as there are devices for a given response. The following table gives the mass and velocity ranges for each drop weight device.

288


Figure 9.64. Experiment distribution according to Doehler plan

Device

Impact energy (J)

Mass (kg)

Velocity (ms-1)

Span (cm)

1

11-100

4-7

2-6

30-80

2

30-300

9-24

2-6

30-80

3

30-300

4-6

4-11

30-80

The skins of sandwich materials were 2 mm thick fiber glass and polyester resin. Two different 20 mm thick cores of the same density 80 kgm-3 were used. This consisted of a linear foam (called ductile) and a reticulated foam (called stiff). Sandwich materials were more thoroughly investigated since the plate area variable was added to the preceding ones by the consideration of the span. For instance the experiment plan related to no. 2 device was established from a three variable Doehler design matrix (see table below). Span (cm)

Velocity (ms-1)

Mass (kg)

Impact energy (J)

30-67.5

2-6

9.21-23.92

33-298

The two other design plans for the two other devices were built from the same design matrix. In this matrix, the variables were not associated with the same number of values. Three values were connected to the same mass variable, five values to span and seven values to velocity. It is then advisable, in the course of a design plan, to assign to the variable, which changes the least, the most difficult parameter to adjust experimentally.

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9.5.2.1.2. Results The processing of this model allowed us to draw the evolution of the response in the experimental domain, for instance for device no. 1. The damage of sandwich materials is more complex than the damage of stratified composites. Thus, cracking, together with fiber breaking, could be observed on the upper skin (impact side), then perforation when test conditions enabled this. The core was compressed. Figure 9.65 shows the model of this variable established from the design matrix connected with device no. 1. It allows us to examine the joint influence of mass and velocity on the contact load. The iso-value lines are rather perpendicular to the velocity axis for low values of velocities. This suggests that the drop velocity is the dominating factor in this domain. On the other hand, when velocity increases (> 4.5 ms-1), mass becomes the significant variable. It is possible to use models associated with all responses to analyze their evolution in the explored experimental domain and with a relatively low number of tests (a total of 13 for each device).

Figure 9.65. Contact load isovalue curves measured with ductile foam device

9.6. Bibliography [AYA 93] AYAX E., Caractérisations expérimentales et modélisation du comportement d’une plaque composite tissée en régime balistique, PhD Thesis, ENSAM, 1993. [BAC 91] BACON C., CARLSSON J., LATAILLADE J.L., J. of Phys. IV, C3:395, 1991. [BAC 93] BACON C., Mesure de la ténacité dynamique de matériaux fragiles en flexion trois points à hautes températures. Utilisation des barres de Hopkinson, PhD Thesis, University of Bordeaux I, 1993.

290


[BÖH 82] BÖHME W., KALTHOFF J.F., Int. J. of Fract., 20: 139, 1982. [BÖH 86] BÖHME W., PhD Thesis (in German) Darmstadt, 1986. [BON 98] BONNET N., Etude du comportement sous sollicitations intenses d’élastomères. Application au cas du blindage réactif, PhD Thesis, ENSAM, 1998, 98-25. [BON 01] BONNET N., THOMAS T., LATAILLADE J.L., “Study of an engineering elastomer to a curved plate impact loading”, Int. J. of Impact Engng., 2001. [BOU 86] EL BOUNIA N.E., Etude de la rupture et de la fragmentation en torsion d’une alumine technique sous chargement rapide et à haute température, PhD Thesis, University of Bordeaux I, 1986. [BOX 87] BOX G.E.P., HUNTER W.H., HUNTER J.S., Statistics for Experiments, Wiley & Sons, 1987. [CAY 95] CAYSSIALS F., Prise en compte des propriétés visco-élastiques de colles structurales pour l’optimisation de joints adhésifs vis-à-vis de leur résistance au choc, PhD Thesis, ENSAM, 1995. [CRO 69] CROSLEY P.B., RIPLING E.J., J. of Basic Engng., p. 525, 1969. [DAV 63] DAVIES E.D.H., HUNTER S.C., “The dynamic compression testing of solids by the method of the shpb”, J. Mech. of Solids, 11: 155, 1963. [DAV 79] DAVISON L., “Shock compression of solids”, Physics Reports, 55(4), 1979. [DEL 95] DELAET M., Influence des vitesses de chargement et de la température sur l’endommagement et la résistance de composites croisés verre-époxy, PhD Thesis, ENSAM, 1995. [DOE 70] DOEHLER D.H., “Uniform shell designs”, Applied Statistics, 19(3): 231-239, 1970. [GRA 87] GRAN et al., Mechanics of Materials, 6: 113, 1987. [HAR 90] HARDING J., Impact Damage in Composite Materials, in LATAILLADE J.L. and MASSIAH A. (ed.), 3ème Ecole d’Aubonne du CODENAC, Guéthary (France), October 1990. [IBO 99] IBOUCHICHENE-AICHE Y., Contribution à la caractérisation rhéologique du PMMA-choc. Effets des vitesses de chargement et de la température sur la résistance mécanique en traction, PhD Thesis, ENSAM, 1999, 99-11. [KAL 77] KALTHOFF J.F. and SHOKEY D.A., Journal of Applied Physics, 48: 984-992, 1977. [KAL 85] KALTHOFF J.F., WINKLER S., Instrument for detecting the instant of which a crack begins in a mechanical strength test of a ferromagnetic material, Technical Report Serial 66/652 320, U. Patent and Trademark Applications, 1985. [KHA 87] KHAN A.S., IRANI F.K., Mechanics of Materials, 6: 285, 1987. [KIS 80] KISHIMOTO, AOKI, SAKATA, Eng. Fract. Mech., 13: 501-508, 1980.

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[LAM 97] LAMAISON M., Contribution à la compréhension des phénomènes de rupture sous sollicitations dynamiques d’un diborure de titane et d’un tungstène, PhD Thesis, ENSAM, 1997. [LAT 87] LATAILLADE J.L., Infrared Technique in Solid Polymers Testing and High Strain Rates: Application to the Viscoplasticity of Polycarbonate, Elsevier, B.U., North Holland, Thermomechanical Couplings in Solids, 1987. [LAT 94] LATAILLADE J.L., BACON C., COLLOMBET F., DELAERT M., Wave Propagation and Emerging Technologies, AMD, Edit. W.K. KINRA, 188, 1994. [LAT 96] LATAILLADE J.L. et al., Int. J. Imp. Eng., 16(6): 679, 1996. [LAT 97] LATAILLADE J.L., CAYSSALS F., Polym. Engng. Sc., 10: 37, 1997. [LOP 92] LOPEZ J.M., Conception et réalisation d’une machine à choc pour l’étude de la déchirure à grande vitesse de tôles composites, Mémoire d’ingénieur, CNAM (centre associé de Bordeaux), 1992, 249. [MAR 95] MARCHAND A., Détermination des constantes thermo-élastiques de plaques circulaires minces par analyse modale – Application aux céramiques déposées par projection plasma, PhD Thesis, University of Bordeaux I, 1995. [NAS 69] NASH J.E., Int. J. of Fract. Mech., 5(4): 269-286, 1969. [PEC 96] PECAULT S., Tolérance au choc de structures sandwichs et effets d’échelle, PhD Thesis, ENSAM, 1996, 96-37. [SHL 81] SHLADERER M., Conception et réalisation d’une machine d’essai par torsion à haute vitesse de distorsion des matériaux. Mémoire d’ingénieur, CNAM (Centre associé de Bordeaux), 1981, 156. [TAY 48] TAYLOR G.I., Proc. Roy. Soc., A 194: 289, 1948. [VID 98] VIDAL B., Modélisation d’impacts sur vitrages feuilletés, PhD Thesis, ENSAM, 1998. [WAH 97] WAHAD ABDUL M., Etude du comportement de panneaux sandwich à l’impact, PhD Thesis, ENSAM, 1997, 97-02. [ZHA 92] ZHAO H., Analyse de l’essai aux barres d’HOPKINSON. Application à la mesure du comportement dynamique des matériaux, PhD Thesis, Ecole Nationale des Ponts et Chaussées, 1992.


Chapter 10

Notched Axi-symmetric Test Pieces

10.1. Introduction Notched axi-symmetric test pieces (AE for the French “axisymétrique entaillée”) are laboratory test pieces used to characterize fracture properties (mostly metallic) of materials. The stress level plays an essential role in ductile or brittle fracture initiation. Notched axi-symmetric test pieces with an experimental traction device (which is thus simple) allow us to reach locally stress levels somewhat larger than the yield strength of the material through a geometrical effect resulting in a stress tri-axiality ratio. This ratio is defined as the ratio of mean stress to Von Mises stress. Notched axi-symmetric test pieces constitute small structures. Mechanical tests are rather simple, but it is otherwise necessary to analyze stress and strain states by calculation of the structure. This is usually performed by finite element calculations. In the following, the performance of experiments as well as the mechanical analysis of AE test pieces will be detailed. The two first parts will be devoted to experimental aspects: test pieces geometry, mechanical testing and observation of damage. It will then be appropriate to proceed with mechanical analysis. The elastic case and then various treatments of plasticity will be presented. These treatments will be the basis of brittle or ductile fracture analysis, which will be illustrated by short case studies. Creep damage will also be briefly covered.

Chapter written by Jacques BESSON.

294


10.2. Geometry and notations A schematic representation of notched axi-symmetric test pieces geometry is given in Figure 10.1. The initial notch radius is denoted R, the initial diameter of the minimum cross-section IIthe current value the maximum diameter of the test piece Im. The symbolic denomination of the various test pieces are AEFwith F= 10R/I0. During testing a stress gradient appears in the notch. The mean axial stress in the minimum cross-section is denoted V :

V

8

I

2

³ 0I 2 V zz r rdr

F S

[10.1]

where F is the applied load and S the minimum cross-sectional area. This definition takes into account geometry reactualization during deformation. A load normalized by the initial minimum cross-sectional area S0 will also be used:

V0

F S0

[10.2]

Figure 10.1. (a) Schematic geometry of notched axi-symmetric test pieces, (b) proportional AE10 test piece I0 = 3.75, 6, 10, 15 mm [BAU 96]

Von Mises stress will be denoted Ve and mean stress Vm (1/3 trace of V the stress tensor). The stress tri-axiality ratio is defined as W Vm/Ve. Notched tensile specimens allow us to generate W values much larger than in a smooth specimen. This is the reason why they are widely used in investigations on brittle and ductile fracture of metals. Indeed, these processes are partly controlled by the largest principal stress (brittle) or the stress tri-axiality ratio (ductile). For these reasons, in the following text, characterization of the values of W for various specimens in


295

mechanical loading will be described in particular. The mean strain Hin the notch is defined as: § S ·

H log¨ ¸

©S0 ¹isotrope

§ I · 2 log¨ ¸ ©I 0 ¹

[10.3]

In the following, the diameter variation will be denoted 'I I0 – I. The notch effect is expressed by the fact that for a given H, the macroscopic load F is larger the more severe the notch. In practice, three test pieces denoted AE10, AE4 and AE2 are often used. They are considered as being slightly, average or strongly notched. They allow the generation of higher and higher stress tri-axiality ratios in the smallest cross-section. As a general rule a minimum diameter I0 = 10 mm is chosen; it is quite possible to use proportionally smaller or larger test pieces (Figure 10.1b). This could allow the investigation of scaling effects [DEC 98, DEV 97] or more simply to test small thickness sheet materials. Figure 10.2 gives the drawings of the three specimen types for a minimum diameter I0 = 10 mm.

Figure 10.2. AE2, AE4, AE10 test pieces

296


10.3. Notch test piece testing 10.3.1. Mechanical test A test on a notched axi-symmetric test piece is achieved in the same manner as in a tensile test. However, the measurement of the vertical displacement is not terribly interesting as far as the plastic deformation is highly localized in the notch zone. Measurement of the load (F) and of the axial displacement ('L) must be complemented by measurement of the notch radial displacement ('I). A diametric sensor should then be placed at the minimum cross-section (Figure 10.3). In some cases, the material’s mechanical behavior is anisotropic, so that the deformation in the minimum cross-section is not homogenous; initially circular, it becomes elliptic. It may then be interesting to measure the diameter reduction in two perpendicular directions. When only one sensor can be placed, the test should then be interrupted to achieve manual measurement point after point in the other direction (Figure 10.4a).

Figure 10.3. (a) Set up scheme, (b) placing diameter sensor, (c) complete experimental set up including diameter sensor and longitudinal sensor

The tensile test is usually controlled by vertical displacement. Under those conditions, as soon as large damage develops in the notch, control is lost and the specimen breaks instantly. In the case of ductile materials (as far as the experimental device is stiff enough), it is nevertheless possible to control the test by keeping


297

radial displacement constant. It is then possible to obtain a load-radial displacement curve including the large load drop due to central crack initiation (Figure 10.4b).

Figure 10.4. (a) Example of load-diameter reduction curve obtained for rolled anisotropic steel. Loading direction is the rolling direction (L); displacement measurement is achieved in the transverse direction (T, 'IT) and short transverse direction (S,'IS) [BEN 99]. (b) Example of load-diameter reduction curves obtained with radial displacement control [TAN 99]

10.3.2. Observation of damage AE test pieces can also be used to characterize damage mechanisms. This is achieved either by observation of the fracture facies or of polished longitudinal sections. In this last case, it is most interesting to be able to interrupt tests at various stages of fracture. This type of observation is of course limited to ductile fracture. This approach is illustrated in Figure 10.5 in the case of rolled steel including flattened MnS inclusions in the sheet plane [BEN 99]. The photographs in Figure 10.5(i) and (ii) correspond to the beginning of damage characterized by cavities growth initiated on inclusions in the center of the specimen. The photographs in Figure 10.5 (iii) to (v) illustrate crack propagation initiated in the specimen center. The photograph in Figure 10.5 (vi) illustrates cavities coalescence. 10.4. Specimen elastic analysis There are no exact solutions providing stress and strain fields in a notched test piece deforming in the elastic range. Tabulated solutions are to be found in [MUR 87]

298


and [PET 74]. However, finite element calculations enable case after case calculation of this type of test piece. In the following, all (finite element) calculation was performed with Poisson’s ratio Q = 0.3. Stress Vzz is maximum at the notch side. A stress concentration factor KT is defined as the ratio V zz /V . It is reported according to ratio I/R in Figure 10.6a. The limit I/R = corresponds to an axi-symmetric cracked specimen [MUR 87]. Note that Von Mises stress Ve is also maximum at the notch side. In all cases this is where plasticity will start. Figure 10.6b indicates the stress triaxiality ratio in the specimen center according to notch severity. W is maximum in the center of the smallest cross-section for I/R ratios included between 0 and § 7. Beyond, maximum stress tri-axiality no longer occurs in the specimen center. At the limit I/R = is the case of an axi-symmetric cracked specimen [MUR 87] for which a stress intensity factor KI determines the crack tip stress field.

Figure 10.5. Examples of observation on sections: (i) and (ii) cavity growth in the specimen center, (iii) central crack in advanced stage of coalescence. Cavities are initiated on MnS inclusions flattened in the plane, which is here perpendicular to the loading axis. Cracks resulting from this mechanism can be observed (arrows), (iv) loading direction is here in the plane of inclusions (arrow 1). The cup and cone fracture is readily observed; it ends by shear lips near the surface (arrow 2), (v) central zigzag crack, (vi) coalescence of micro-cavities [BEN 99]


299

10.5. Plastic analysis of specimens To analyze stress and strain fields in notched axi-symmetric test pieces when the material deforms in the plastic regime, simulations by finite element calculations should usually be performed [BER 80]. The Bridgman analysis presented hereafter is not fine enough to allow accurate evaluation of local fields needed in damage analysis. 10.5.1. Bridgman analysis The analysis of the stress state in a necked tensile piece was proposed by Bridgman and revisited by Hill [HIL 50]. The analysis assumes uniform deformation in the minimum cross-section. This hypothesis is rather well verified for slightly notched specimens. Radial displacement ur in the minimum crosssection is thus proportional to radius r. Hence, radial and hoop strains (Hrr and HTT) are equal and constant. Stresses Vrr and VTT are then equal. Mechanical equilibrium is expressed for z = 0 as:

wV rr wV rz wr wz

0

[10.4]

Figure 10.6. Elastic analysis of notched specimens: (a) stress concentration factor K = Vzz/ V at the notch side according to ratio IR. Contours indicate the map of ratio Ve/ V for an AE10 specimen, (b) stress tri-axiality ratio W in the specimen center according to ratio IR. The contours indicate the stress tri-axiality map for an AE10 specimen

300


Owing to symmetry, Vrz = 0 for z = 0. As the minimum cross-section is in plastic deformation, for z = 0, 0 r I/2, the stresses obey:

V zz V rr

[10.5]

Rp

As the deformations are constant in the minimum cross-section, the yield stress Rp is constant. The analysis can be performed without assuming any particular strain hardening law. Figure 10.7 represents description of notch geometry proposed by Bridgman. Denote by \ the angle between the maximum principal stress direction V1 and the vertical axis. V2 is the second principal stress in r – z plane. In the plane z = 0, we obtain V1 = Vzz and V2 = Vrr. Changing coordinates for \

V zz # V 1 ,V rr # V 2 ,V rz # \ V 1 V 2

>@

In a volume element infinitely close to the minimum cross-section, the yield condition is written: V1 – V2 = Rp, hence Vrz = \Rp. Then: §wV rz · ¨ ¸ © wz ¹z 0

§w\ · R p ¨ ¸ ©wz ¹z 0

Rp

U

>10.7@

where U is the radius of curvature of the maximum principal stress direction lines (Lines L1). Equilibrium [10.4] then yields a Vzz differential equation:

wV zz R p wr U

0

>10.8@

Figure 10.7. Bridgman analysis: geometry


301

The main difficulty of Bridgman analysis lies in the calculation of U as a function of r. At the surface, of course U = R; in the center for r = 0, U = . It may be imagined that U varies continuously between these two values. The second principal stress direction lines (lines L2) are perpendicular to lines L1 (these carry principal stress V1). It is then assumed that they are circles, which intersect axis Oz and the surface at right angle. Consider a point T of the structure. The center of line L1 through T is C (on axis Or). Line L2 through T intersects axis Or at N and M. Writing the power of point C with respect to the circle along NTM (see Figure 10.7) yields: CT 2

U2

CM .CN

OC OM .OC OM

OC 2 OM 2

[10.9]

Considering point T infinitely close to axis Or, OC § r + U and OM = constant. Hence: §I ·2 r 2 2rU Cte ¨ ¸ IR ©2 ¹

thus U

I 2 2 IR r 2 2r

[10.10]

It is then possible to integrate with boundary condition Vzz(I/2) = Rp:

V zz r Rp

§I 2 4IR 4r 2 · ¸¸ 1 log¨¨ 4RI © ¹

[10.11]

The value of the mean stress in the minimum cross-section is:

V zz r Rp

r

§ 4R · § I · ¸ ¨1 ¸log¨1 © I ¹ © 4R ¹

[10.12]

The stress tri-axiality ratio in the center of the specimen is equal to:

W center

1 § I ·º 1 ª ¸» «1 log¨1 © 4R ¹¼ 3 ¬

[10.13]

302


Figure 10.8 compares the simplified Bridgman analysis to a finite elements calculation. The strain hardening law of the material is given by K(H0 + p)n, with K = 100 MPa,H0 = 0.001 and n = 0.1. The notch radius is high and is equal to R = 10.I. Bridgman analysis is applied using the mean value of plastic deformation in the minimum cross-section calculated by finite elements. It is first observed that plastic deformation p is not uniform in the minimum cross-section. V rr z V TT for r 0. The stress gradient calculated by finite elements is higher than the one provided by Bridgman analysis. On the other hand, the difference between the mean stress V zz r calculated by finite elements and the stress provided ignoring the notch (i.e. the yield strength corresponding to the mean strain) is somewhat underestimated by Bridgman analysis (about 10%). This relatively good agreement enables us for instance to use [10.12] to analyze tensile tests after the beginning of necking. The radius of curvature of the neck as well as the local variation of the diameter should be measured to evaluate Rp from the applied load. In the absence of measurement, the empirical Bridgman relation [BRI 52] relating I/R and mean notch deformation can be used:

I/4R = 0 when H t H0 I/4R = H H0 when H d H0

[10.14]

H0 corresponds to deformation when the Considere criterion is reached (see section 4.4.3.1).

Figure 10.8. Comparison of Bridgman analysis with finite element simulation


303

10.5.2. Finite element analysis In the following, the analysis of notched axi-symmetric test pieces assumes isotropic hardening plastic behavior. The strain hardening law is given by: n

R p K H 0 p

[10.15]

with K = 200 MPa, H0 = 0.001 and n the strain hardening exponent. Calculations were performed with n = 0.1 and n = 0.2. p is the cumulated plastic deformation. The yield strength Re = KH0n. The material is assumed to obey the Von Mises plasticity criterion. The elastic behavior is isotropic as well: Young’s modulus 200 GPa, Poisson’s ratio Q = 0.3. Finite element analyses were performed using quadratic interpolation elements with reduced Gauss integration. Calculations used the small perturbations hypothesis and finite transformations (Lagrange re-actualized). Although the problem is usually to be treated in large deformations, small deformation analysis allows the characterization of stress and strain states generated by the presence of the notch only. In the case of analysis of a real test, large deformations analysis must be used. In the case when plastic deformation is large, strong pressure variations in the elements are frequently observed. This comes from the fact that plastic deformation involves no volume variation and becomes much larger than elastic deformation. To treat this problem without modifying the constitutive relations, so called pressure selective integration methods may be used requiring special finite elements [HUG 80, NAG 74, ODE 83, SIM 85]. In practice, quite close results are obtained using the following post-treatment: 1) perform the classical calculation; 2) calculate the mean value Vkk per element; 3) calculate a corrected stress V *ij such that ( s stress deviator, 1 unit tensor):

V* s

Vkk 3

1

Taking care of strong variations is particularly important in the case where the calculation is to be post-treated to determine characteristic parameters of fracture behavior: Weibull stress (see section 10.6.1), cavity growth rate (see section 10.6.2). Post-treatment is then achieved using tensor V * . In the case of coupled damage

304


models (see section 10.6.3) in which there is a plastic volume variation (Rousselier and Gurson models), the material is no longer incompressible and the abovementioned problems disappear as soon as porosity is large enough. The objective of this section is to provide some illustrations of calculations to be performed to analyze various specimens and to underline some numerical problems that are frequently encountered. However, in practice, each material constitutes a particular case. 10.5.3. Small deformations analysis Figure 10.9 gives an example of stress profiles Vzz and of plastic deformation p in the minimum cross-section for three specimens AE10, AE4 and AE2 for n = 0.1. The stress profile has a maximum at the notch side when deformation remains elastic. When plasticity starts, this maximum moves progressively towards the specimen center and ends up with a maximum there. This evolution is faster the less acute the notch. In the case of the three notched specimens investigated the stress triaxiality ratio is initially maximum at the center (elastic case in Figure 10.6) and remains there during notch plastic deformation. The fact that the stress Vzz becomes maximum in the center expresses an increase of the stress tri-axiality ratio as indicated in Table 10.1. In this table, the plastic case corresponds to a stabilized state, which does not take into account re-actualization of positions. Plastic deformation profiles indicate that a relatively homogenous plastic deformation is reached rather quickly in the case of specimen AE10. On the other hand, in the case of specimen AE2, the notch side remains the most deformed zone. Similar results are obtained with stress exponent n = 0.2.

We

Wpl

n = 0.1

Wpl

n = 0.2

AE10

0.54

0.79

0.66

AE4

0.81

1.14

0.95

AE2

1.09

1.32

1.13

Table 10.1. Stress tri-axiality ratio in the center of specimens AE10, AE4 and AE2 in elastic case (We) and for plastic behavior (Wpl with n = 0.1 and n = 0.2)


305

10.5.4. Large deformations analysis 10.5.4.1. Necked tensile specimen In smooth specimens, necking takes place so that a notch is formed (see section 4.4.3.1). The occurrence of necking is predicted from the Considere criterion: dF/d'+ = 0 (F applied load, 'H elongation). Figure 10.10a gives the diameter variation 'II0 according to height variation 'H/H0 for a tensile test for which H0/I0 = 12. It can be observed that the finite element calculation indeed obeys the Considere criterion for n = 0.1 and n = 0.2. Figure 10.10b shows the evolution of the stress triaxiality ratio according to diameter reduction. As soon as necking appears, triaxiality increases quickly to reach comparable levels to those obtained with AE specimens. It is maximum in the center of the specimen. Figure 10.10 also gives examples of cumulated plastic strain p distribution and of stress tri-axiality ratios. In this last case, it should be noted that a compression zone is formed (tr V < 0) above the necked zone [NOR 78]. 10.5.4.2. AEF specimens Figure 10.11 gives the evolution of the stress tri-axiality ratio in the center of specimens according to diameter reduction for geometries AE10 … AE1, as well as for a tensile specimen for n = 0.1. In all cases an increase of Wcenter according to deformation is observed. This increase is larger the less acute the initial notch. The values calculated for the stabilized state in small deformations (Table 10.1) underestimate the values obtained in large deformations. In the case of specimen AE2 a stable value is reached rather quickly: Wcenter § 1.5. It is observed numerically as well as experimentally that less acute specimens can be “re-notched” locally (necking effect) and thus generate a stronger curvature. This phenomenon is illustrated in Figure 10.11 in the case of specimen AE10 for which the final curvature ('II0) can be compared to the initial curvature. Figure 10.12 illustrates the same experimentally observed phenomenon. More acute specimens have a tendency to be “de-notched”. This is clearly the case for specimen AE1 for which the stress tri-axiality ratio increases up to 'II0 = 15% (redistribution of stresses and strains) then decreases because of the blunting of the notch.

306


Figure 10.9. Examples of stress (Vzz/Rp) and strain (p) profiles in the notch of AE10, AE4 and AE2 specimens. Dashed curves correspond to stress profile at plasticity initiation at the specimen side (n = 0.1)

10.5.4.3. Plastic anisotropy In the case of plastically anisotropic materials (rolled sheet for instance) it might be interesting to perform 3D calculations to describe the notch deformation well. This type of calculation is illustrated in the case of a rolled steel sheet [RIV 98] (Figure 10.13). The rolling direction (L) is distinguished from transverse (T) and


307

short transverse (S) directions. The material tensile tested in various directions obeys Hill criterion:

VH 2 with V H

[10.16]

Rp 3 2 2 2 2 2 2 hSS sSS hTT sTT hLL sLL 2hST sST 2hTL sTL hSL sSL 2

s is the stress deviator tensor, hSS, …, hSL are anisotropy coefficients. VH is the Hill equivalent stress and Rp the yield strength. Coefficients hIJ are equal to:

hLL = 1.185, hTT = 0.823, hSS = 1.552, hLT = 1.586, hTS = 1.896, hSL = 0.801

[10.17]

Figure 10.10. Necking of smooth specimen: (a) evolution of the diameter 'II0 according to height variation 'H/H0 (calculations were performed with +I = 12). Contours give an example of the plastic deformation map in the necked zone. (b) Evolution of stress tri-axiality ratio in the specimen center according to diameter variations. Contours give an example of W values map. Dashed vertical lines indicate necking initiation according to the Considere criterion (p = n)

These are such that Rp corresponds to the strain hardening curve in direction T. Figure 10.13 compares experiment and simulation in the case of a specimen AE4 loaded in the T direction. It is noted that direction S deforms more than direction L (because hSS > hLL). The load-diameter reduction curves are well reproduced up to the point where fracture is initiated (A).

308


Figure 10.11. Evolution of stress tri-axiality ratio in the AE10 specimen center (resp. AE4, AE2, AE1) according to diameter variations (boxes). The dashed line (T) shows this evolution in the case of a tensile specimen (Figure 10.10). Contours give values of stress tri-axiality ratio of a AE10 specimen for diameter reduction 'II0 = 56%. Initial mesh is also drawn.

10.6. Damage analysis The notched axi-symmetric test pieces are used to investigate fracture mechanisms. Brittle and ductile fracture can be analyzed. Owing to the complexity of stress and strain fields, it is once again necessary to perform numerical analysis of results. Two damage analysis strategies may be used: – post-processing analysis: a purely plastic finite element analysis is performed. From calculations, it is then possible to calculate the fracture criterion. Optimization of the parameters of the damage model is carried out from successive evaluations of the criterion, which usually are not time-consuming in terms of calculation time; – analysis including plasticity and damage: the finite element analysis includes a constitutive law which couples plastic deformation and damage. It is then possible to model initiation of cracks as well as their propagation within the notch. However, optimization of the parameters of the model requires several complete finite element calculations. This strategy is of course much more time-consuming in terms of calculation time. In practice, automatic optimization is seldom achieved.


309

Figure 10.12. Highly deformed AE10 specimen in which an acute notch formed. The dashed lines indicate in an approximate manner the initial shape of the specimen

Brittle fracture, which is controlled by fracture of the weakest element, clearly resorts to the first approach. In the case of ductile fracture, the material softens before fracture, owing to the growth of cavities and possibly to nucleation of new ones. The second approach is then more appropriate, but its complexity being more important, the post-processing approach may be preferred. Three examples of applications of damage and fracture models follow: i) brittle fracture (section 10.6.1), ii) ductile fracture analyzed by a criterion in postprocessing (section 10.6.2) and iii) ductile fracture analyzed with coupled models (section 10.6.3).

z

z

y

x

z

y

x

y

x

Figure 10.13. Load-diameter reduction (along S and T) curves for an AE10 specimen. The maps represent values of stress tri-axiality ratio at Gauss points W , of plastic deformation (p) and of axial stress (VTT) for point A (fracture initiation)

310

Structural Components AE2

AE4

AE10

–196ºC

3

11

3

17

–165ºC

–

2

–

2

–150ºC

9

–

–

9

12

13

3

28

Table 10.2. Tested AEF specimens

10.6.1. Case study: brittle fracture In this investigation, the material was 16MND5 steel used for Pressurized Water Nuclear Reactors (PWR) shell rings. Mechanical tests were performed with tensile specimens and specimens AE10, AE4 and AE2 with an initial diameter I0 = 6 mm. The temperatures were between –196°C and –150°C. Table 10.2 gives the list of AEFspecimens used. The constitutive law used incorporated variable temperature isotropic strain hardening. The strain hardening law was given by: R p p,T V 0 T Q1 >1 expb1 p @ Q2 >1 expb2 p @

[10.18]

(T temperature, p cumulative plastic strain). Rp(T) is the yield strength of the material; it depends on temperature according to:

V 0 T V a Q0 exp>cT T 0 @

[10.19]

Q1 and b1 coefficients are temperature-dependent; Q2 and b2 are constant. The set of coefficients is given in Table 10.3. Yield strength

Strain hardening Q1

3.10–6T3 + 6.10–4T2 – 0.28T + 142.0

1,122

B1

0.144T + 30.7

0.011

Q2

472

273.15

B2

1.72

Va

421

Q0 C T0

Table 10.3. Plastic behavior coefficients for IGMND5 steel for temperatures between 196°C and +200°C (units of measurement: MPa, °C) [TAN 99]


311

In Figure 10.14, pairs ('I,F) corresponding to fracture are plotted over calculated curves. A good agreement between experiment and simulation is then necessary in order to calculate the stresses, which intervene in the evaluation of the fracture criterion, precisely. The brittle fracture analysis of the material was performed for –196°C, assuming that it follows the weakest link model. A Weibull distribution was also postulated. The Weibull stress Vw is defined by a volume integral as:

[10.20]

Strength (kN)

Vw

§ ·1 m m dV ¨³V p V I ¸ V0 ¹ ©

Diameter reduction (MM)

Diameter reduction (MM)

Figure 10.14. ('I, F) simulated curves and experimental points corresponding to fracture instants

Vp represents the volume where plastic deformation took place; it is considered that the cleavage fracture of steel cannot be triggered unless there is plastic deformation [BER 83]. In the case of brittle fracture of ceramics, the integral would be on the entire volume. V0 is a reference volume set a priori. m is an adjustable parameter of the model allowing us to describe experimental dispersion. VI is the largest principal stress. In integrating [10.20] care should be taken to account for the symmetries of the finite element calculation; generally,

312


half a structure is calculated so that the result of integration should be multiplied by 2. Fracture probability is then given by: ª § ·m º V 1 exp«¨ w ¸ » « ©V u ¹ » ¬ ¼

Pr

[10.21]

Vu being the second parameter of the model. The adjustment procedure of parameters m and Vu are standardized [ESI 01]. In [DIF 96] a fairly complete analysis as well as a simplified identification procedure are found. For a pair (m,Vu), the corresponding Vw for each instant of fracture can be calculated. These can then be arranged in increasing order k = 1 … N, where N is the number of tested specimens. The fracture probability corresponding to the kth value of Vw is:

Pr V wk

k 1 2 N

[10.22]

m and Vu are adjusted in order to minimize the difference between the theoretical probability [10.21] and the experimental probability [10.22]. The adjustment for the tests at –196°C are plotted in Figure 10.15a. It is of course possible to use any other law to calculate the stress Vw, or even the fracture probability (for instance, in the case of two populations of defects [YAH 98]). For instance, to try adjusting the entire set of tests (integrating various temperatures) a threshold V0 can be introduced in the calculation of Vw, yielding:

Vw

ª «³V p V I V 0 ¬

1m

m

dV º » V0 ¼

[10.23]

Function . is such that x x if x > 0, and x 0 otherwise. Adjustment is compared with experiments in Figure 10.15b. The agreement looks acceptable; however, it is noted that data obtained at the higher temperatures systemically correspond to the highest values of Vw. The agreement will be mediocre if only the points obtained at –196°C are considered. The solution consists of using different parameters for the different temperatures and so performing new experiments to complete the data.


313

Figure 10.15. Adjustment of the Weibull model parameters: (a) two parameter law (–196°C): Vu = 2,610 MPa, m = 19.9, V0 = 0.001 mm3, (b) three parameter law (–196, –165, –150°C): Vu = 3,865 MPa, m = 9.4, V0 = 367 MPa, V0 = 0.001 mm3

Brittle fracture has a random character, but the most probable fracture locations can be sought. Taking into account the revolution symmetry, stress Vw [10.20] can be expressed in the form:

Vw

§ ·1 m m drdz 2 S V r ¨ ³V p I ¸ V0 ¹ ©

1m § drdz · 2 S Z ¨ ³V p ¸ V0 ¹ ©

[10.24]

with Z = VImr. The most probable fracture locations are those for which Z is maximum. In Figure 10.16 it is observed that fracture rather takes place in a zone between the center and the side of the minimum cross-section. 10.6.2. Case study: uncoupled ductile fracture Uncoupled analysis lies, as the analysis of brittle fracture, on post-processing treatment. Here again, various criteria may be envisaged. A commonly used criterion is based on the analysis of cavity growth in an infinite medium proposed by Rice and Tracey [BER 81, DIF 96, RIC 69]. For a large enough stress tri-axiality ratio, the growth rate is given as:

R R

§3 · 0.283 exp ¨ W ¸ p ©2 ¹

[10.25]

where R is the cavity radius assumed to be spherical. This hypothesis is true for

W > 1. It is considered that fracture occurs when the growth rate reaches a critical

314


value (R/R0)c, a characteristic of the material and a priori independent of the specimen type. Thus, the treatment consists of integrating [10.25] and identifying (R/R0)c as the maximum value calculated for the structure at the instant of experimentally observed fracture. Mean fracture ductility can be defined as:

HR

§I · 2 log¨ 0 ¸ ©I R ¹

[10.26]

where IR is the mean diameter value measured after fracture. This quantity is plotted in Figure 10.17a for a C-Mn steel [BAU 96, DEC 98]. Some dispersion can be noted, which is not analyzed here. In the following, mean quantities are considered. Test simulation is drawn in Figure 10.17b. In Figure 10.18 the evolutions of the maximum value of R/R0 (results of integration of [10.25]) are reported according to the mean deformation in the notch. The values corresponding to HR are identified with (R/R0)c.

Figure 10.16. Map of Z parameter normalized by its maximum value indicating the most probable fracture location

Figure 10.17. C-Mn steel: (a) fracture ductilities for 4 types of specimens, (b) load-reduction of diameter calculated curves


315

Figure 10.18 shows that rather good results are obtained for specimens AE10, AE4 and AE2 with a (R/R0)c mean value of 1.27. On the other hand, in the case of the tensile specimen, a much larger (R/R0)c value is identified. This essentially results due to the fact that [10.25] is adapted to high stress tri-axialities. The Huang growth law may then be used [HUA 91] (which is an improvement of the Rice and Tracey law):

R R

§3 · 14 °0.427W exp ¨ W ¸ p ° ©2 ¹ ® ° 0.427 exp § 3 W · p ¨2 ¸ °¯ © ¹

[10.27]

the first expression being valid if W < 1, the second one if W > 1. A better agreement is obtained in the sense that the identified (R/R0)c value is much less dependent on the specimen type.

Figure 10.18. Evolution of maximum value of R/R0 parameter for 4 types of specimens according to mean deformation in the notch. Points correspond to experimental fracture

It is also possible, as in brittle fracture, to determine the location where fracture was initiated. Figure 10.19 indicates, for the Rice and Tracey model, the localization of the most damaged zones at the instant corresponding to the experimentally observed fracture. It is noted that for specimens TB and AE10 damage is concentrated in the center of the minimum cross-section. For specimen AE4 the notch zone appears to be damaged in a rather homogenous manner. In the case of

316


specimen AE2 damage is concentrated on the notch side. Slightly different results are obtained with the Huang model, although the tendencies are respected.

Figure 10.19. Damage in tensile specimens AE10, AE4 and AE2. The model used is the Rice and Tracey model. Black zones are such that (R/R0) > (R/R0c) = 0.05. R/R0c value is the one determined specimen after specimen in Figure 10.18

10.6.3. Case study: coupled ductile fracture

It is also possible to model the fracture of the material using constitutive laws coupling plastic and cavities damage behaviors. Two models are mostly used: i) the Gurson model [GUR 77] and above all the modified Tvergaard and Needleman version (GTN model) [TVE 90]; ii) the Rousselier model [ROU 87]. Damage is represented by porosity f. The yield surface depends on porosity as well on the trace of the stress tensor. The application of normality rule to determine the direction of flow implies plastic variation of volume, which can express hardening ( f 0 ) or damage ( f ! 0 ). Here the GTN model will be used. Yield surface () is given by:

)

V e2

§3 V 2 f * q1 cosh ¨ q2 m 2 ¨ R 2 R p0 p0 ©

· ¸ 1 q12 f *2 ¸ ¹

[10.28]

Rp0 being the undamaged material yield strength. q1 and q2 are coefficients and f* is a function of porosity such that: f*

f ° ® G f f fc °¯ c

[10.29]

depending on whether f < fc (first expression) or f > fc (second expression. fc and Gare two new coefficients of the model. The function f* is used to represent the coalescence of cavities which begins, according to the model, for a porosity equal to fc. Beyond this value, the effect of porosity on the mechanical response of the


317

material is much larger. This effect is expressed by using a coefficient G larger than 1. The evolution of porosity is given by mass conservation, yielding: f

1 f trace §¨ H p ·¸ ©

¹

[10.30]

where H p is the plastic strain rate tensor. Applying the normality rule:

H p

1 f O

w) wV

[10.31]

An equivalent plastic deformation p is also defined to describe strain hardening of the material. p and O [SHI 76] are related by writing the equality of microscopic plastic work and macroscopic plastic work: H p : V

1 f R p0 p

[10.32]

The material is here the same as in section 10.6.2. The temperature is 20°C. At this temperature the material is ductile and presents a viscoplastic behavior, so that Rp0 depends at the same time on p and p : R p 0 p, p

VYpl p Kp 1 n

[10.33]

where VYpl(p) is given by [10.18] with the coefficients of Table 10.3. At 20°C, K = 25 MPa and n = 3.5. An important parameter, when there is damage and then crack propagation, is the mesh size used for finite element calculations [LIU 94]. Fracture energies are smaller the smaller the mesh size. Many authors then proposed to adjust the mesh size considering it to be an adjustable parameter of the model [ROU 87, XIA 95]. Tests on AE specimens (I0 = 6 mm) were adjusted using the following parameters: – initial porosity

1.75x10-4

– q1

1.5

– q2

1.0

– fc

8.0x10-4

318


– mesh size

250x125 Pm

–G

$($($(

Initial porosity corresponds to the volume fraction of inclusions determined with Franklin’s formula [FRA 69]. Different values of the parameter G were used; in fact, it is usually difficult to adjust the whole set of experiments with a unique set of parameters. A rather good fit between experiment and simulation is then obtained, whether concerning central crack initiation (change of the slope of the curve F/S0-'II) or its propagation (which intervenes after the slope change)

Figure 10.20. GTN model: experiment/simulation comparison

Figure gives maps of axial stresses at Gauss points after central crack propagation over a length corresponding to two elements. Ahead of the crack exists a strongly damaged zone over 2 to 3 elements. For the three types of specimen, the stress profiles are rather similar.


319

Figure 10.21. Vzz axial stresses maps after central crack propagation over two elements. Ahead of crack a much damaged zone is also found

10.7. Viscoplasticity and creep damage

AE specimens can be used to investigate creep damage at high temperature [LEV 86, PIQ 89, PIQ 91, YOS 90]. Also noteworthy is the use of AE specimens in compression to investigate the rheology of porous ceramics [BES 91]. To illustrate the use of AE specimens in the case of creep, the parameters given in [LEV 86] are used. The material was an AISI 316 creep resistant stainless steel. Two geometries were employed: FLE5 I0 = 7.6 mm, R = 5 mm; FLE1 I0 = 7 mm, R = 1 mm. In both cases Im = 14 mm. The constitutive law incorporates plasticity and viscoplasticity, which is the dominating deformation mechanism in the set of experiments achieved at 600°C. The creep rate is decomposed in primary creep (time hardening): H v1

C1t m Ven1

[10.34]

and stationary creep: H v 2

C2 Ven2

[10.35]

with C1 = 4.3.10-14, m = 0.52, n1 = 4.20, C2 = 7.24.10–26, n2 = 8.62 (Ve in MPa and t in h). This constitutive equation allows us to evaluate stress and strain fields according to time. The evolution of the notch mean strain according to time is given in Figure 10.22 for a FLE5 specimen creep tested under an initial mean stress of 330 MPa, and of 404 MPa.

320


Figure 10.22. Notched specimens in creep. Evolution of mean deformation in the notch. Creep deformation maps after 900 hrs (FLE 5) and 500 hrs (FLE 1) at 600°C

Following the same approach as the one used in the case of brittle fracture, an analysis of damage (inter-granular here) can be performed by post-processing treatment. A damage variable D is defined whose evolution is given by: D

AVDI HEv H v

[10.36]

where Hv is the viscous cumulated deformation. A law in which t stress and strain take place simultaneously is recovered. For the given material, D = 4, E = -0.5, A = 1.5.10-12. Figure 10.22 gives illustrations of damage maps obtained for specimens FLE5 and FLE1. It is noted that in the first case damage is maximum in the center of the specimen (less acute notch) whereas in the second case it is maximum between the center and the side. 10.8. From notch to crack?

Axi-symmetric notched test pieces are used for investigation of damage and fracture. They are employed in the case of brittle, ductile and creep fracture. Simple to perform, these tests allow us to identify physical mechanisms of degradation and to fix parameters of numerical damage models. These test pieces are thus important tools in a local approach to fracture. In practice, however, the interest is in the mechanical resistance of structures which contain notches but also cracks. Taking into account the heaviness of mechanical testing on cracked specimens, it would be


321

favorable to be able to characterize materials with the help of AE specimens alone. There are nevertheless important differences between notches and cracks: – stress tri-axiality ratios at crack tips can reach a value of 3. The values obtained in AE specimens do not exceed 1.5. How could it be guaranteed that parameters adjusted in this range of values remain valid? – volumes in which plastic deformation takes place are much larger (particularly in the case of brittle fracture) in AE specimens than in a cracked specimen. Do the scaling laws remain valid when passing from notch to crack? This is not the case when there are defect populations of very different sizes (see, for instance, [YAH 98]). 10.9. Bibliography [BAU 96] BAUVINEAU L., Approche locale de la rupture ductile : application à un acier Carbone-Manganese, Thesis, Ecole des Mines de Paris, 1996. [BEN 99] BENZERGA A., BESSON J., PINEAU A., “Coalescence-controlled anisotropic ductile fracture”, J. Eng. Mat. Eng., 121: 121-229, 1999. [BER 80] BEREMIN F.M., “Calculs élastoplastiques par la méthode des éléments finis d’éprouvettes axisymmétriques entaillées circulairement”, J. Mécanique Appliquée, 4(3): 307-325, 1980. [BER 81] BEREMIN F.M., “Cavity formation from inclusions in ductile fracture”, Met. Trans., 12A: 723-731, 1981. [BER 83] BEREMIN F.M., “A local criterion for cleavage fracture of a nuclear pressure vessel steel”, Met. Trans., 14A: 2277-2287, 1983. [BES 91] BESSON J., ABOUAF M., MAZEROLLE F., SUQUET P., “Compressive creep tests on porous ceramic notched specimens”, in M. Zyckowski (ed.), Creep in Structures, 4th IUTAM Symposium Cracow/Poland, p. 45-53, Springer-Verlag, 1991. [BRI 52] BRIDGMAN P.W., Studies in Large Plastic Flow and Fracture, McGraw-Hill, 1952. [DEC 98] DECAMP K., BAUVINEAU L., BESSON J., PINEAU A., “Size and geometry effects on ductile rupture of notched bars in a C-Mn steel: experiments and modelling”, Int. J. Fracture, 88(1): 1-18, 1998. [DEV 97] DEVILLERS-GUERVILLE L., BESSON J., PINEAU A., “Notch fracture toughness of a cast duplex stainless steel: modelling of experimental scatter and size effect”, Nuclear Engineering and Design, 168: 211-225, 1997. [DIF 96] DIFANT M., LE COQ V., CLEIZERGUES O., CAROLLO G., MUDRY F., BAUVINEAU L., BURLET H., PINEAU A., MARINI B., KOUNDY M., SAINTECATHERINE C., ERIPRET C., Mise en place d’une méthodologie simplifiée pour utiliser des critères locaux de rupture, IRSID, 1996.

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[ESI 01] ESIS, Procedure to measure and calculate material parameters for the local approach to fracture using notched tensile specimens, Technical report, ESIS [FRA 69] FRANKLIN A.G., “Comparison between a quantitative microscope and chemical methods for assessment of non-metallic inclusions”, J. of the Iron and Steel Institute, 181-186, February 1969. [GUR 77] GURSON A.L., “Continuum theory of ductile rupture by void nucleation and growth. Part I. Yield criteria and flow rules for porous ductile media”, J. Eng. Mat. Technol., 99: 2-15, 1977. [HIL 50] HILL R., The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950. [HUA 91] HUANG Y., “Accurate dilatation rates for spherical voids in triaxial stress fields”, J. Appl. Mech., 58: 1084-1086, 1991. [HUG 80] HUGUES J.R., “Generalization of selective integration procedures to anisotropic and non linear media”, Int. J. Numerical Methods in Engineering, 15: 1413-1418, 1980. [LEV 86] LEVAILLANT C., PINEAU A., YOSHIDA M., PIQUES R., “Creep tests on axisymmetric notched bars: global displacement measurements and metallographic determination of local strain and damage”, in Techniques for Multiaxial Creep Testing, p. 199-207, 1986. [LIU 94] LIU Y., MURAKAMI S., KANAGAWA Y., “Mesh-dependence and stress singularity in finite element analysis of creep crack growth by continuum damage mechanics approach”, Eur. J. Mech., A/Solids, 13(3): 395-417, 1994. [MUR 87] MURAKAMI Y., Stress Intensity Factors Handbook, vol. 2, Pergamon Press, 1987. [NAG 74] NAGTEGALL J.C., PARKS D.M., RICE J.R., “On numerically accurate finite element solutions in the fully plastic range”, Comp. Meth. Appl. Mech. Engng., 4: 153177, 1974. [NOR 78] NORRIS D.M., MORAN B., SCUDDER J.K., QUINONES D.F., “A computer simulation of the tension test”, J. Mech. Phys. Solids, 26: 1-19, 1978. [ODE 83] ODEN J.T., CAREY G., Finite Elements. Volume IV. Mathematical Aspects, Prentice-Hall, Englewood Cliffs, N.J., 1983. [PET 74] PETERSON R.E., Stress Concentration Factors, John Wiley & Sons, 1974. [PIQ 89] PIQUES R., PINEAU A., “Global and local approaches of creep crack initiation and growth applied to an austenitic stainless steel and an aluminium alloy”, in Advances in Fracture Research, p. 1707-1714, Pergamon Press, 1989. [PIQ 91] PIQUES R., PINEAU A., “Global and local approaches to creep crack initiation and creep crack growth”, in M. Zyckowski (ed.), Creep in Structures, 4th IUTAM Symposium Cracow/Poland, p. 433-438, Springer-Verlag, 1991. [REG 95] REGLE H., Approche locale de la rupture ductile de deux aciers pour pièces de transport de gaz, Technical report, Armines, 1995.


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[RIC 69] RICE J.R., TRACEY D.M., “On the enlargement of voids in triaxial stress fields”, J. Mech. Phys. Sol., 17: 201-217, 1969. [RIV 98] RIVALIN F., Développement d’aciers pour gazoducs à haute limite d’élasticité et ténacité élevée : mécanique et mécanismes de la rupture ductile a grande vitesse, Thesis, Ecole des Mines de Paris, 1998. [ROU 87] ROUSSELIER G., “Ductile fracture models and their potential in local approach of fracture”, Nuclear Engineering and Design, 105: 97-111, 1987. [SHI 76] SHIMA S., OYANE M., “Plasticity theory for porous metals”, Int. J. Mech. Sci., 18: 285-291, 1976. [SIM 85] SIMO J.C., TAYLOR R.L., PISTER K.S., “Variational and projection methods for the volume constraint in finite deformation elastoplasticity”, Comp. Meth. Appl. Mech. Engng., 51: 177-208, 1985. [TAN 99] TANGUY B., Simulation numérique de l’essai Charpy à l’aide des critères locaux de rupture, Technical report, ENSMP-Centre des Matériaux, 1999. [TVE 90] TVERGAARD V., “Material failure by void growth to coalescence”, Advances in Applied Mechanics, 27: 83-151, 1990. [XIA 95] XIA L., SHIH C.F., HUTCHINSON J.W., “A computational approach to ductile crack growth under large scale yielding conditions”, J. Mech. Phys. Solids, 43(3): 389413, 1995. [YAH 98] YAHYA O.M., BORIT F., PIQUES R., PINEAU A., “Statistical modelling of inter-granular brittle fracture in a low alloy steel”, Fatigue and Fracture of Engineering Materials and Structures, 21: 1485-1502, 1998. [YOS 90] YOSHIDA M., LEVAILLANT C., PIQUES R., PINEAU A., “Quantitative study of intergranular damage in an austenitic stainless steel on smooth and notched bars”, in High Temperature Fracture Mechanisms and Mechanics (ESF6), p. 3-21, MEP, London, 1990.


List of Authors

Jacques BESSON Centre des matériaux Ecole National Supérieure des Mines de Paris France Jean-Louis CHABOCHE ONERA Châtillon LASMIS University of Technology of Troyes France Gérard DEGALLAIX Laboratoire de mécanique de Lille Ecole centrale de Lille France Suzanne DEGALLAIX Laboratoire de mécanique de Lille Ecole centrale de Lille France Dominique FRANÇOIS Ecole centrale de Paris France Jean-Michel FRUND Division recherche et développement Electricité de France France

326


Pascal GADAUD LMPM Ecole nationale supérieure de mécanique et d’aérotechnique Futuroscope France Jean-Pierre GAUTHIER Formerly CEREM Commissariat à l’énergie atomique Saclay France Jean-Luc LATAILLADE LAMEFIP Ecole nationale supérieure des arts et métiers de Bordeaux University of Bordeaux 1 France Henri-Paul LIEURADE CETIM Senlis France Sylvie POMMIER LMT Ecole normale supérieure de Cachan France

Index

A, B Adhesive joint 278 Adiabatic shear 250 Aging 32 Analysis Bridgman 299-302 elastic 297 finite element 303-304 plastic 299 Angle of mechanical loss 48 Average contact pressure 89 Brinell indentation problem 95, 96 Brittle fracture 200

C Calibration 197 Charpy pendulum 196 Chree-Pochammer oscillation 230, 253 Considere’s relation 83 Constant elastic 41 Lamé 44 Crack arrest 198 Crack closure 171 Crack propagation rate in fatigue 168 Crack tip blunting 219 Criterion Considere 302 Hill 8 Von Mises 8

CTOD 208, 219 Curve time-displacement 198 transition 196, 200 Cyclic hardening and softening 20-21

D, E Damage creep 319 ductile 33 Damage parameter 86 Data processing 239 Displacement 197 Dynamic fracture mechanics test 259 Elastomer 271 Extensometer 74

F Flow load 198 Fracture brittle 293, 313 ductile 313 unstable 198 Fracture probability 312 Fracture toughness dynamic 259 of brittle materials 112

328


G, H Geometry defects 62 Gripping 74 Gurson model 304 Hardening isotropic 9, 20-21 kinematic 10, 18 Prager’s non-linear kinematic 18 Hopkinson bar 226, 231, 245, 246, 256, 262 HRR fields 217

I, J, K Initiation 168-169 Instrumented indentation 67 Inverse assembly 74, 76 Kolsky-Hopkinson device 231

L Large deformations 303 Law behavior 5 Cowper-Symonds 82 hardening 13, 16, 18, 22 Hooke’s 5 Ludwick’s 84 Ramberg-Osgood 80 Rice and Tracey 315 Schmidt’s 29 viscosity 14, 24 Load cell 74 Loading method interrupted 218 partial 218

M, N Material isotropic 7 orthotropic 59 Maximum load 198 Mesh size 317 Method compliance 178 optical 178

potential drop 180 sphere 65 ultrasonic 182 vibratory 66 Micro-macro transposition 28 Modulus bulk 44 complex 48 dynamic 48 shear 44, 52 Young’s 42-44, 46, 48, 50-52, 54, 61, 97, 98, 104, 106, 113, 115, 155, 171, 180, 185, 210, 222, 227-229, 231, 232, 240, 243, 253, 262, 303 Necking 83

P Partial unloading method 218 Percentage elongation 79, 87 Phase changes 32 Piober-Lüder bands 85 Plastic anisotropy 306-307 Plastic shakedown 31 Plastic zone correction 211 PMMA shock 253 Poisson’s ratio 51 Portevin-Le Chatelier phenomenon 33, 76 Potential drop method 219

R Rankin-Hugonoit conversation equation 267 Ratcheting 31 Recovery dynamic 18 static 21 Representative volume element (RVE) 5 Rice Cherepanov integral 209 Rice contour integral 208 Rousselier model 304

S Shakedown limit 31 Shock between two bars 232 Standard characteristics 73

Index Steel 16MND5 310 C-Mn 314 Stiffness 76 machine 76 Strength tensile 78 Stress effective 33, 86 Weibull 311 Stress intensity factor 207, 210 Stress-strain relation 73

T Temperature reference 201 transition 201 Test bending 63 Charpy 260 collision 277 compression 51, 87 drop weight 284 Hopkinson 231 hydrostatic compression 54 impact 194 interrupted tensile 255 notch test piece 296 shock 60, 193 torsion 64

Test piece compression 235 KCU 194 KCV 194 notched axi-symmetric 293, 294 Testing machine hydraulic 75 mechanical 75 tensile 74 Thin film 65, 67 Torsion 52 with Hopkinson bars 246 Traveling diagram 247

V, W Viscoplasticity 12, 13 Viscosity function 13 Work hardening exponent 80

Y, Z Yield drops 85 Yield strength peak 76 Zone cyclic plastic 171 plane strain plastic 213 small scale yielding plastic 211

329

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