Introduction to Unified Mechanics Theory with Applications [2nd ed. 2022] 3031186206, 9783031186202

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Cemal Basaran

Introduction to Unified Mechanics Theory with Applications Second Edition

Introduction to Unified Mechanics Theory with Applications

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Thermodynamic State Index (TSI) axis 0.999 = 5 years old with stage4 cancer

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5 years old

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Cemal Basaran

Introduction to Unified Mechanics Theory with Applications Second Edition

Cemal Basaran Department of Civil, Structural, and Environmental Engineering University at Buffalo, SUNY Buffalo, NY, USA

ISBN 978-3-031-18620-2 ISBN 978-3-031-18621-9 https://doi.org/10.1007/978-3-031-18621-9

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021, 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

A new scientific truth does not triumph by convincing opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.

1918 Nobel Prize in Physics for his work on quantum theory. Friedrich-Wilhelms-Universität Berlin, Germany

Max Planck

v

Preface

This book is the culmination of my research over the last 28 years at the University at Buffalo, The State University of New York. Most of the book relies heavily on the work of my PhD students and our joint publications. This second edition is based on the first edition. However, there are two new chapters in this second edition, Chap. 9 on fatigue and Chap. 10 on corrosion-fatigue interaction. Also, there are more example problems added. Chapter 4 on unified mechanics theory has also been augmented with more and better explanations. The traditional [Newtonian] continuum mechanics portion of this book is primarily based on the seminal textbook by Malvern (1969). Since I consider Prof. Lawrence Malvern’s textbook one of the best written on continuum mechanics, I did not find any reason to change the formulation or his presentation of the basics. Yet, when I was a student, I found his book hard to follow because it was written at a high level. Therefore, I have tried to simplify many of the formulae derivation steps to make them easier for students to understand. While I did change the order of the topics which I copied from Malvern (1969), I have tried to stay true to the message Prof. Malvern had in his formulation. I have tried to explain the unified mechanics theory in the simplest terms possible. To accomplish this objective, I found it necessary to include the entire paper by Sharp and Matschinsky (2015), which is the English translation of Boltzmann’s (1877) original paper. While I acknowledge that including papers from a journal in its entirety is a very unorthodox practice, it was necessary, because Boltzmann’s original paper is misunderstood. Therefore, I had to present his entire derivation annotated with my explanations and comments. To this day, many physics books still refer to Boltzmann’s second law equation to be valid for gasses only, with no reference to his derivation. Boltzmann did not make any such assumption in his formulation that would have invalidated the applicability of his formulation to solids. For over a quarter century, my PhD students and I have provided in the literature the experimental proof of Boltzmann’s second law equation for solids. Sharp and Matschinsky (2015) have also addressed this major misunderstanding in their paper. I quote Sharp and Matschinsky (2015) directly. The fact that Boltzmann vii

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Preface

does not consider the interaction between the particles only increases the possible number of complexions, as Boltzmann defines them. Interparticle interactions, like friction or molecular bonding, would only reduce the number of possible complexions. Therefore, Boltzmann’s formulation can be considered an upper bound limit of possible permutations. However, Boltzmann’s mathematical derivation is valid for solids, gasses, and liquids. It was not possible to include Boltzmann’s paper without Max Planck’s (1900) paper because they complement each other. In introducing the basics of thermodynamics, I have relied heavily on Callen (1985). While there are many books on thermodynamics within the literature, I found the work of Callen (1985) to be the most comprehensive and understandable from a graduate student’s perspective. I also kept in mind that unfortunately most mechanics students never take a basic course on thermodynamics at the undergraduate level. Therefore, the chapter on thermodynamics is based primarily on the work of Callen (1985). The rest of the chapters are all taken from my work with my former PhD students to whom I am eternally grateful. My former PhD students who contributed to the development of this theory over the years are Drs. ChengYong Yan, Rumpa Chandaroy, Hong Tang, Ying Zhao, Yujun Wen, Shihua Nie, Juan Gomez, Eray Mustafa Gunel, Shidong Li, Wei Yao, Hua Ye, Ming Hui Lin, Michael Sellers, Mohammed Abdul Hamid, Yong Chang Lee, Noushad Bin Jamal, and Hsiao Wei Lee. I am also indebted to Ms. Shifan Cheng for the graphics in the book. I am especially grateful to the US Navy, Office of Naval Research, for believing in me and sponsoring most of the work reported in this book. Their continuous support, [starting in 1997 with the ONR Young Investigator Award Program under program director Dr. Roshdy Barsoum], has been an immense help, and without it, this idea would have never come to a realization. Buffalo, NY, USA

Cemal Basaran

References Callen, H. B. (1985). Thermodynamics and an introduction to thermostatistics (2nd ed.). Wiley. Malvern L. E. (1969). Introduction to the mechanics of continuous medium. Prentice-Hall. Planck, M. (1900). Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum. Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 237; Translated in ter Haar, D. (1967). On the theory of the energy distribution law of the normal spectrum (PDF). The old quantum theory (pp. 82). Pergamon Press. LCCN 66029628. Sharp and Matschinsky, “Boltzmann, L. (1877). Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI, 373–435 (Wien. Ber. 1877, 76:373–435)”. Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, pp. 164–223, Barth, Leipzig, 1909 [Kim Sharp* and Franz Matschinsky, Translation of Ludwig Boltzmann’s Paper “On the relationship between the second fundamental theorem of the mechanical theory of heat and probability calculations regarding the conditions for thermal equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI, 1877, 373–435 (Wien. Ber. 1877, 76:373–435)]. Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, pp. 164–223, Barth, Leipzig, 1909. Entropy, 2015, 17, 1971–2009.

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 What Is the Mechanics of Continuous Medium? . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3

2

Stress and Strain in Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Newton’s Universal Laws of Motion . . . . . . . . . . . . . . . . . . . . 2.1.1 First Universal Law of Motion . . . . . . . . . . . . . . . . . . 2.1.2 Second Universal Law of Motion . . . . . . . . . . . . . . . . 2.1.3 Third Universal Law of Motion . . . . . . . . . . . . . . . . . . 2.1.4 Range of Validity of Newton’s Universal Laws of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Relation to the Thermodynamics and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Definitions of Stress and Traction . . . . . . . . . . . . . . . . 2.2.2 Stress Vector on an Arbitrary Plane . . . . . . . . . . . . . . . 2.2.3 Symmetry of Stress Tensor . . . . . . . . . . . . . . . . . . . . . 2.2.4 Couple Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Principal Stresses and Principal Axes . . . . . . . . . . . . . . 2.2.6 Stress Tensor Invariants . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Octahedral Plane and Octahedral Stresses . . . . . . . . . . 2.3 Deformation and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Small Strain Definition . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Small Strain and Small Rotation Formulation . . . . . . . . 2.3.3 Small Strain and Rotation in 3-D . . . . . . . . . . . . . . . . . 2.4 Kinematics of Continuous Medium . . . . . . . . . . . . . . . . . . . . . 2.4.1 Material (Local) Description . . . . . . . . . . . . . . . . . . . . 2.4.2 Referential Description (Lagrangian Description) . . . . . 2.4.3 Spatial Description (Eulerian Description) . . . . . . . . . . 2.4.4 Material Time Derivative in Spatial Coordinates (Substantial Derivative) . . . . . . . . . . . . . . . . . . . . . . .

5 5 5 8 9 11 12 12 12 15 19 21 24 27 30 33 34 38 47 48 48 48 49 49 ix

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2.5

Rate of Deformation Tensor and Rate of Spin Tensor . . . . . . . . 2.5.1 Comparison of Rate of Deformation Tensor, D, and Time Derivative of the Strain Tensor, ε_ . . . . . . . . . 2.5.2 True Strain (Natural Strain) (Logarithmic Strain) . . . . . 2.6 Finite Strain and Deformation . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Green Deformation Tensor, C, Cauchy Deformation Tensor, B-1 . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Relation Between Deformation, Strain, and Deformation–Gradient Tensors . . . . . . . . . . . . . . . 2.6.3 Comparing Small Strain and Large (Finite) Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Strain Rate and Rate of Deformation Tensor Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Relation Between, the Spatial Gradient of Velocity Tensor, L and the Deformation–Gradient Tensor, F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Rotation and Stretch Tensors in Finite Strain . . . . . . . . . . . . . . . 2.8 Compatibility Conditions in Continuum Mechanics . . . . . . . . . . 2.9 Piola–Kirchhoff Stress Tensors . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 First Piola–Kirchhoff Stress Tensor σ 0 . . . . . . . . . . . . . 2.9.2 Second Piola–Kirchhoff Stress Tensor σ~ . . . . . . . . . . . 2.10 Direct Relation Between Cauchy Stress Tensor and Piola–Kirchhoff Stress Tensors . . . . . . . . . . . . . . . . . . . . . 2.11 Conservation of Mass Principle . . . . . . . . . . . . . . . . . . . . . . . . 2.12 The Incompressible Materials . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Conservation of Momentum Principle . . . . . . . . . . . . . . . . . . . 2.14 Conservation of Moment of Momentum Principle . . . . . . . . . . . 2.15 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Hamilton’s Principle: The Principle of Stationary Action . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Work Done on the System (Power Input) . . . . . . . . . . . 3.2.2 Heat Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Quantification of Entropy in Thermodynamics . . . . . . . 3.3.3 Gibbs–Duhem Relation . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Entropy Production in Irreversible Process . . . . . . . . . . 3.3.6 Clausius–Duhem Inequality . . . . . . . . . . . . . . . . . . . . 3.3.7 Traditional Use of Entropy as a Functional in Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . 3.3.8 Entropy as a Measure of Disorder . . . . . . . . . . . . . . . .

53 54 55 57 63 64 66 70

72 72 73 76 76 77 78 79 80 81 84 86 88 89 91 92 93 94 96 98 100 103 104 106 108 109 112 114

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3.3.9 3.3.10

Thermodynamic Potential . . . . . . . . . . . . . . . . . . . . Time-Independent Dissipation (Instantaneous Dissipation) . . . . . . . . . . . . . . . . . . . 3.3.11 Dissipation Power and Onsager Reciprocal Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

. . 118 . . 127 . . 128 . . 130

Unified Mechanics Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Literature Review of Use of Thermodynamics in Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Laws of Unified Mechanics Theory . . . . . . . . . . . . . . . . . . . . . 4.2.1 Second Law of Unified Mechanics Theory . . . . . . . . . . 4.2.2 Third Law of Unified Mechanics Theory . . . . . . . . . . . 4.3 Evolution of Thermodynamic State Index (Φ) . . . . . . . . . . . . . . 4.3.1 On the Relationship Between the Second Fundamental Theorem of The Mechanical Theory of Heat and Probability Calculations Regarding the Conditions For Thermal Equilibrium, by Ludwig Boltzmann (1877) . . . . . . . . . . . . . . . . . . . 4.3.2 Critic of Boltzmann’s Mathematical Derivation . . . . . . 4.3.3 On the Law of Distribution of Energy in the Normal Spectrum, [Max Planck, (1901) Annalen Der Physik, Vol. 4, p. 553 Ff, 1901] . . . . . . . . 4.3.4 Thermodynamic State Index (TSI) in Unified Mechanics Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Experimental Verification Example . . . . . . . . . . . . . . . . . . . . . 4.4.1 Tension-Compression Cyclic Loading . . . . . . . . . . . . . 4.4.2 Monotonic Loading Test . . . . . . . . . . . . . . . . . . . . . . . 4.5 Dynamic Equilibrium Equations in Unified Mechanics Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Derivation of the Dynamic Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Vibration of a 1-DOF Mass-Spring System . . . . . . . . . 4.5.3 Entropy Generation in Sliding Friction Contact: Thermodynamic Fundamental Equation . . . . . . . . . . . . 4.5.4 Comparison with Newtonian Mechanics Results . . . . . . 4.5.5 Comparison with Experimental Data . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unified Mechanics of Thermomechanical Analysis . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Unified Mechanics Theory-Based Constitutive Modeling . . . . . . 5.2.1 Flow Theory and Yield Criteria . . . . . . . . . . . . . . . . . . 5.2.2 Effective Stress Concept and Strain Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Unified Mechanics Theory Implementation . . . . . . . . .

133 133 150 151 153 153

156 199

201 210 215 215 217 217 218 221 224 226 229 231 237 237 237 237 242 243

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5.3

Return Mapping Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Linearization (Consistent Jacobian) . . . . . . . . . . . . . . . 5.4 Thermodynamic Fundamental Equation in Thermomechanical Problems . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Momentum Principle in Newtonian Mechanics . . . . . . . 5.5.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . 5.6 Entropy Law: Second Law of Thermodynamics . . . . . . . . . . . . 5.7 Fully Coupled Thermomechanical Equations . . . . . . . . . . . . . . 5.8 Numerical Validation of the Thermomechanical Constitutive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Thin Layer Solder Joint: Monotonic and Fatigue Shear Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Thermomechanical Analysis of Cosserat Continuum: Length Scale Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Cosserat Couple Stress Theory . . . . . . . . . . . . . . . . . . 5.9.2 Toupin-Mindlin Higher-Order Stress Theory . . . . . . . . 5.9.3 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . 5.9.4 Finite Element Method Implementation . . . . . . . . . . . . 5.9.5 General Couple Stress Theory: Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.6 Reduced Couple Stress Theory: Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.7 Reduced Couple Stress Theory: Mixed Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.8 General Couple Stress Theory Implementation . . . . . . . 5.10 Cosserat Continuum Implementation in Unified Mechanics Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.1 Rate-Independent Plasticity Without Degradation . . . . . 5.10.2 Rate-Dependent Plasticity (Viscoplasticity) Without Degradation . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.3 Introducing the Thermodynamic State Index . . . . . . . . 5.10.4 Entropy Generation Rate in Cosserat Continuum . . . . . 5.10.5 Integration Algorithms . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Thermomechanical Analysis of Particle-Filled Composites . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Ensemble-Volume Averaged Micromechanical Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Noninteracting Solution for Two-Phase Composites . . . . . . . . . 6.3.1 Average Stress Norm in Matrix . . . . . . . . . . . . . . . . . . Average Stress in Filler Particles . . . . . . . . . . . . . . . . . 6.3.2

243 246 249 251 251 253 255 257 263 267 267 272 274 276 278 281 283 284 284 285 288 289 295 296 298 299 306 309 309 312 316 316 322

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6.4

323 323 327 329 331

Pairwise Interacting Solution for Two-Phase Composites . . . . . . 6.4.1 Approximate Solution of Two-Phase Interaction . . . . . . 6.4.2 Ensemble-Average Stress Norm in the Matrix . . . . . . . 6.4.3 Ensemble-Average Stress in the Filler Particles . . . . . . 6.5 Noninteracting Solution for Three-Phase Composites . . . . . . . . 6.5.1 Effective Elastic Modulus of Multiphase Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Ensemble-Average Stress Norm in the Matrix . . . . . . . 6.5.3 Ensemble-Average Stress in Filler Particles . . . . . . . . . 6.6 Effective Thermomechanical Properties . . . . . . . . . . . . . . . . . . 6.6.1 Effective Bulk Modulus . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Effective Coefficient of Thermal Expansion (ECTE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Effective Shear Modulus . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Effective Young’s Modulus and Effective Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Micromechanical Constitutive Model of the Particulate Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Modeling Procedures for Particulate Composites . . . . . 6.7.2 Elastic Properties of Particulate Composites . . . . . . . . . 6.7.3 A Viscoplasticity Model . . . . . . . . . . . . . . . . . . . . . . . 6.7.4 Thermodynamic State Index . . . . . . . . . . . . . . . . . . . . 6.7.5 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.6 Consistent Elastic-Viscoplastic Tangent Modulus . . . . . 6.8 Verification Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Material Properties of ATH . . . . . . . . . . . . . . . . . . . . . 6.8.2 Properties of PMMA . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.3 Properties of Matrix-Filler Interphase . . . . . . . . . . . . . . 6.8.4 Properties of Particulate Composites . . . . . . . . . . . . . . 6.8.5 Finite Element Simulation Results . . . . . . . . . . . . . . . . 6.8.6 Cyclic Stress-Strain Response . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Unified Micromechanics of Finite Deformations . . . . . . . . . . . . . . . 7.1 Introduction to Finite Deformations . . . . . . . . . . . . . . . . . . . . . 7.2 Frame of Reference Indifference . . . . . . . . . . . . . . . . . . . . . . . 7.3 Unified Mechanics Theory Formulation for Finite Strain . . . . . . 7.3.1 Thermodynamic Restrictions . . . . . . . . . . . . . . . . . . . . 7.3.2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Thermodynamics State Index . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Definition of Material Properties . . . . . . . . . . . . . . . . . . . . . . . 7.6 Applications of Finite Deformation Models . . . . . . . . . . . . . . . 7.6.1 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . .

332 332 336 338 339 340 340 342 342 345 346 348 350 354 354 359 361 361 362 362 362 366 366 371 375 375 379 383 384 392 403 405 409 409

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7.7

Numerical Implementation of Dual-Mechanism Model . . . . . . 7.7.1 Simulating Isothermal Stretching of PMMA . . . . . . . . 7.7.2 Simulating Non-isothermal Stretching of PMMA . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

9

. . . .

Unified Mechanics of Metals Under High Electrical Current Density: Electromigration and Thermomigration . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Physics of Electromigration Process . . . . . . . . . . . . . . . . . . . . . 8.2.1 Driving Forces of Electromigration Process . . . . . . . . . 8.2.2 Laws Governing Electromigration and Thermomigration . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Electromigration Electron Wind Force . . . . . . . . . . . . . 8.2.4 Temperature Gradient Diffusion Driving Force . . . . . . . 8.2.5 Stress Gradient Diffusion Driving Force . . . . . . . . . . . 8.3 Laws of Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Vacancy Conservation . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Newtonian Mechanics Force Equilibrium . . . . . . . . . . . . . . . . . 8.5 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Electrical Conduction Equations . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Thermodynamic Fundamental Equation for Electromigration and Thermomigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Entropy Balance Equations . . . . . . . . . . . . . . . . . . . . . 8.8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fatigue of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Predicting High Cycle Fatigue Life of Metals . . . . . . . . . . . . . . 9.1.1 Thermodynamic Fundamental Equation . . . . . . . . . . . . 9.1.2 Entropy Generation Mechanisms . . . . . . . . . . . . . . . . . 9.1.3 Comparison Between Different Entropy Generation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Temperature Evolution due to Mechanical Work . . . . . 9.1.5 Entropy and TSI Calculations . . . . . . . . . . . . . . . . . . . 9.2 Predicting Ultrasonic Vibration Fatigue Life . . . . . . . . . . . . . . . 9.2.1 Thermodynamic Fundamental Equation . . . . . . . . . . . . 9.2.2 Temperature Evolution due to Mechanical Work . . . . . 9.2.3 Comparison Between Different Entropy Generation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Computing Thermodynamic State Index . . . . . . . . . . . 9.3 Predicting Low Cycle Fatigue Life . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

415 416 422 425 427 427 427 428 429 432 435 436 438 439 443 444 447 449 450 456 457 459 460 460 461 473 475 477 480 481 489 492 494 496 501

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Corrosion-Fatigue Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Thermodynamic Fundamental Equation of Corrosion-Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Entropy Generation Mechanisms During Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Entropy Generation Mechanisms: Fatigue . . . . . . . . . . 10.3 Thermodynamic State Index (TSI) . . . . . . . . . . . . . . . . . . . . . . 10.4 Comparing Simulation Results and Test Data . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

505 505 506 506 511 513 513 515

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Abbreviations1

γ δij ε W ε_ p ρ σ S σO σ~ φ Φ TSI Ψ b B-1 C D E E ETotal F F ~ F g

Shear strain Kronecker delta Total strain tensor Vorticity tensor Plastic strain rate tensor Mass density Total stress tensor Deviatoric stress tensor First Piola-Kirchhoff stress tensor Second Piola-Kirchhoff tensor Dissipation potential Thermodynamic state index Thermodynamic state index Helmholtz-free energy Body force per unit mass Cauchy deformation tensor Green deformation tensor Rate of deformation tensor Lagrange strain tensor Euler strain tensor Total energy Deformation gradient tensor Plasticity yield surface Second Piola-Kirchhoff force Gibbs function

1

Most of the time I tried to use the same character to define a variable. However, due to the large number of variables, some characters were used more than once to define different variables. To clarify the variables, they are re-defined after each equation. xvii

xviii

h I I 1, I 2, I 3 J2D J3D k, or kB K V L m mij nij p q Q Q s t u, v, w i, j, k U ~v A V

Abbreviations

Enthalpy Impulse Total stress tensor invariants Second invariant of the deviatoric stress tensor Third invariant of the deviatoric stress tensor Boltzmann’s constant Kinetic energy Potential energy Spatial gradient of velocity mass the distributed moment in couple stress theory Direction cosines Momentum Heat flux vector Plastic potential Total heat input Entropy Time Displacements along spatial coordinates, x, y, z, respectively Unit vectors along spatial coordinates, x, y, z, respectively Strain energy Velocity Area Volume

Chapter 1

Introduction

1.1

What Is the Mechanics of Continuous Medium?

Probably one of the best definitions of mechanics of a continuous medium is provided by Malvern (1969) in his seminal work. “Continuum mechanics is a branch of mechanics concerned with the deformation or flow of solids, liquids, and gasses.” In continuum mechanics, molecular structure and electronic structure of the material are ignored. When molecular structure must be taken into account molecular dynamics and when the electronic structure must be considered quantum mechanics are used. In continuum mechanics, it is assumed that the material is continuous without empty spaces. It is also assumed that all differential equations governing the continuous medium are continuous functions, except at boundaries between continuous regions. It is also assumed that the derivatives of the differential equations are continuous as well. A material satisfying these requirements is considered a continuous medium. It is important to point out that “space” does not mean there cannot be atomic vacancies. There will always be atomic vacancies. It just means that there cannot be a space proportional to the size of the object being studied. The concept of a continuous medium allows us to define stress and strain at an “imaginary point,” a geometric point in space assumed to be occupying no volume. When we say a point in continuum mechanics, we do not mean an atomic point. It is an imaginary point with no volume. Continuum stresses and strains are defined at this point. Atomic stresses are defined differently, which is outside the scope of this book. This approach allows us to use differential calculus to study non-uniform distributions of strain. This assumption of the continuous medium allows us to study deformation in most engineering problems but not all. When the continuous medium assumption is not satisfied or molecular dynamics [movement of atoms] or electronics structure influences the mechanical behavior, then continuum mechanics cannot be used. In these latter instances, a multi-scale mechanics analysis becomes essential.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_1

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Introduction

The traditional definition of the continuous medium also makes three assumptions (Malvern, 1969): Continuity A material is continuous if it fills the space that it occupies, leaving no pores or empty spaces, and if furthermore, its properties are describable by continuous functions. Homogeneity A homogenous material has identical properties at all points. The size is larger than the Representative Volume Element (RVE) of the material. RVE is the smallest volume over which a property measurement can be made that will yield a value representative of the whole (Hill, 1965). Isotropy A material is isotropic with respect to certain properties if these properties are the same in all directions in space. Only the first assumption is needed to define concepts of stress and strain. The last two assumptions are needed when we introduce stress-strain relations (constitutive relations) into continuum mechanics equations. Of course, anisotropic, inhomogeneous, and discontinuous systems can also be analyzed by continuum mechanics by representing them as discrete pieces, as it is done in computational mechanics, such as the finite element method. Chapter 2 of this book covers only the general principles of continuum mechanics, and associated definitions, such as stress and strain. The fundamentals of continuum mechanics, presented in Chap. 2, have been well established by Isaac Newton (1687), Leonhard Euler (1736), Cauchy, Green, Truesdell, and Toupin (1960), Truesdell and Noll (1965), Truesdell (1965, 1966), and others. Maugin (2016) published a comprehensive account of the foundations of continuum mechanics as well as a complete list of references to earlier work. Therefore, there is no attempt in this book to cover earlier fundamental developments. Continuum mechanics cannot be discussed without including material constitutive equations. While material behavior modeling is a very important topic of major current research efforts, it is not the focus of this book. It is covered in the applications of the unified mechanics theory sections, Chaps. 5, 6, 7, 8, and 9. Chapter 3 of the book covers the basics of thermodynamics. This chapter starts with a comprehensive literature survey of the use of thermodynamics in continuum mechanics in the last 150 years. While we do not claim to have included every paper published on the topic, the survey covers all significant developments. Energy dissipation and degradation of materials and structures are discussed in the context of thermodynamics. Since not all engineering students are required to take the course in thermodynamics, the reader has assumed a beginner, as such a piece of elementary information is included. Chapter 4 presents the formulation of the unified mechanics theory. Great effort has been made to include all the details of the formulation. Chapter 4 also discusses the motivation behind the development of the unified mechanics theory. Essentially, the present-day mechanics equations are all based on the three laws of motion of Isaac Newton. However, these three laws are “incomplete,” because they do not include the laws of thermodynamics. The following examples can help us explain

References

3

what we mean by “incomplete.” For example, if a soccer ball is given an initial acceleration by a kick according to Newton’s second law, that acceleration is constant; it does not degrade. The ball will travel with that acceleration forever according to F = ma. However, that is not true. In the same fashion, Newton’s third law (also Hooke’s law, which was published by Robert Hooke 10 years before Newton’s Principia) assumes that reaction due to an applied load will be constant, without degradation of the material. Of course, materials degrade, and response displacement does not remain constant. Fortunately, energy loss in the ball and material degradation can be modeled by the laws of thermodynamics. Basaran and Yan published the first paper in (1998) using Boltzmann’s entropy generation rate as a metric for degradation in microelectronics solder joints. They implemented it by modifying Newton’s laws with Boltzmann’s second law of thermodynamics formulation. Since then, the theory has gone through significant consolidation by experimental verifications and mathematical derivations by many researchers around the world. These developments are included in Chap. 4. Chapter 5 presents a formulation for thermo-mechanical analysis using unified mechanics theory. The chapter includes non-local and local mechanics formulations. Cosserat continuum is explained in great detail and used for introducing length scale into the continuum mechanics formulation. It is presented with simple steps from a beginner graduate student’s perspective Chapter 6 is about particle-filled composite materials’ formulation using the unified mechanics theory. Particle-filled acrylic composite formulation and implementation are presented in the context of small strain formulation. Chapter 7 covers the finite strain formulation of unified mechanics theory. Constitutive modeling of polymer is presented in detail. Chapter 8 presents the electro-thermo-mechanical loading application of the unified mechanics theory. Electromigration and thermomigration are covered in detail. Chapter 9 covers the derivation of thermodynamic fundamental equations for fatigue in metals with examples. Chapter 10 includes a discussion on corrosion-fatigue interaction and derivation of the fundamental equation for corrosion in metals. I provided a list of references at the end of each chapter. While they are not complete, they can help a researcher in the right direction.

References Hill, R. (1965). Continuum micro-mechanics of elastoplastic polycrystals. Journal of the Mechanics and Physics of Solids, 13(2), 89–101. Malvern, L. E. (1969). Introduction to the mechanics of continuous medium. Prentice-Hall. Basaran, C., & Yan, C. Y. (1998). A thermodynamic framework for damage mechanics of solder joints. Transaction of ASME, Journal of Electronic Packaging, 120, 379–384. Maugin, G. A. (2016). Continuum mechanics through the ages - from the renaissance to the twentieth century. Springer.

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Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica, . Euler, L. (1736). Mechanica sive motus scientia analytice exposita Trusdell, C., & Toupin, R. (1960). The classical field theories. In S. Flügge (Ed.), Principles of classical mechanics and field theory / Prinzipien der Klassischen Mechanik und Feldtheorie. Springer. Truesdell, C., & Walter Noll, W. (1965). The non-linear field theories of mechanics. Springer. Truesdell, C. (1965). The elements of continuum mechanics. Springer-Verlag. Truesdell, C. (1966). Rational mechanics of materials. In Six lectures on modern natural philosophy. Springer.

Chapter 2

Stress and Strain in Continuum

2.1

Newton’s Universal Laws of Motion

It is essential to start the discussion about continuum mechanics, with the universal laws of motion as given by Isaac Newton. Understanding their implications in historical context is essential to understanding these simple and probably the most famous physics equations. Of course, Newton’s universal laws of motion are very well known to most readers of this book; however, their historical context and evolution are not. Therefore, we find it necessary to quote these basics from Sir Isaac Newton’s own Wikipedia website with their sources. The information below is quoted directly with little or no editing. Philosophiæ Naturalis Principia Mathematica (Latin for Mathematical Principles of Natural Philosophy), often referred to as simply the Principia, is a work in three books by Isaac Newton, in Latin, first published 5 July 1687. After annotating and correcting his copy of the first edition, Newton published two further editions, in 1713 and 1726 (Harvard UP, 1972). The Principia states Newton’s universal laws of motion, forming the foundation of classical mechanics; Newton’s law of universal gravitation; and a derivation of Kepler’s laws of planetary motion (which Kepler first obtained empirically) (Cohen, 2002; Galili & Tseitlin, 2003; Kleppner & Kolenkow, 1973; Plastino & Muzzio, 1992; Resnick et al., 1992) (Fig. 2.1).

2.1.1

First Universal Law of Motion

The original text in Latin reads: Newton’s original Latin text in Principia states: Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_2

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Fig. 2.1 Original 1687 Principia Mathematica, where Sir Isaac Newton published the three laws, in Latin

Translated to English, this reads: Law I: Everybody persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.

(Modern Translation) Law I: In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force. The Greek philosopher Aristotle (384–322 BC) articulated that all objects have a natural place in the universe. He believed that heavy objects like boulders wanted to be at rest on the Earth and that light objects like smoke wanted to be at rest in the sky and the stars wanted to remain in the heavens. He also believed that a body was in its natural state when it was at rest, and for the body to move in a straight line at a constant speed an external force was needed continually to propel it; otherwise, it would stop moving. Galileo Galilei (1564–1642) however, realized that a force is necessary to change the velocity of a body, i.e., acceleration, but no force is needed to maintain its velocity. In other words, Galileo stated that, in the absence of a force, a moving object will continue moving. The tendency of objects to resist changes in motion was what Johannes Kepler (1571–1630) had called inertia. Kepler’s insight was refined by Newton, who made it into his first law, also known as the “law of inertia”—no force means no acceleration, and hence the body will maintain its velocity. Newton’s first law is a restatement of the law of inertia which Galileo had already described; hence, Newton appropriately gave credit to Galileo.

2.1

Newton’s Universal Laws of Motion

7

The law of inertia was discovered by different philosophers and scientists independently, including Thomas Hobbes (1588–1679) in his Leviathan (Hobbes, 1651). The seventeenth-century philosopher and mathematician René Descartes (1596–1650) also formulated the law, although he did not perform any experiments to confirm it (Hellingman, 1992; Resnick & Halliday, 1977).

2.1.1.1

Formulation of the First Law

The first law, also known as the law of inertia, states that if the net force (the vector sum of all forces acting on an object) is zero, then the velocity of the object is constant. Velocity is a vector quantity, which expresses both the object’s speed and the direction of its motion; therefore, the statement that the object’s velocity is constant is a statement that both its speed and the direction of its motion are constant. The first law can be stated mathematically, when the mass is a non-zero constant, as, F = 0,

d~v =0 dt

where F is the total net force acting on the object, d~v=dt is the time derivative of velocity. Consequently, an object that is at rest will stay at rest unless an external force acts upon it. An object that is in motion will not change its velocity unless an external force acts upon it. This is known as the uniform motion. An object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest (demonstrated when a tablecloth is skillfully whipped from under dishes on a tabletop and the dishes remain in their initial state of rest), where the inertia of dishes balances the force exerted by the table cloth. If an object is moving, it continues to move without turning or changing its speed. This is evident in space probes that continuously move in outer space. Change in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of a total net force, a moving object tends to move along a straight-line path indefinitely. Newton stated the first law of motion to establish a frame of reference for which the other laws are applicable. The first law of motion postulates the existence of at least one frame of reference called a Newtonian or inertial frame of reference, relative to which the motion of a particle, not subjected to a net force, is a straight line at a constant speed (Newton, 1999). Newton’s first law is often referred to as the law of inertia. Thus, a condition necessary for the uniform motion of a particle relative to an inertial frame of reference is that the total net force acting on it is zero. In this sense, the first law can be restated as (Motte’s 1729 translation): In every material universe, the motion of a particle in a preferential reference frame X is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in X. That is, a particle initially at rest or in uniform motion in the preferential frame X continues in that state unless compelled by forces to change it.

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Stress and Strain in Continuum

“Newton’s first and second laws are valid only in an inertial reference frame. Any reference frame that is in uniform motion with respect to an inertial frame is also inertial, i.e., Galilean invariance or the principle of Newtonian relativity” (Resnick & Halliday, 1977). If someone kicks a ball on a moving train car, the acceleration the ball will gain will be with respect to the train not with respect to the ground.

2.1.2

Second Universal Law of Motion

Isaac Newton’s original Latin text reads: Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

This was translated by Motte in 1729 as: Law II: The alteration of motion is ever proportional to the motive force impress’d; and is made in the direction of the right line in which that force is impress’d.

(Modern translation) Law II: In an inertial reference frame, the vector sum of the forces “F” on an object is equal to the mass “m” of that object multiplied by the acceleration “a” of the object: According to modern ideas of how Newton was using his terminology, this is understood, in modern terms, as an equivalent of: The change of momentum, ½d ðm~vÞ, of a body, is proportional to the impulse (Fdt) impressed on the body and happens along the straight line on which that impulse (Fdt) is impressed.

This may be expressed by the formula F = p′ where p′ is the time derivative of the momentum ½p = d ðm~vÞ. This equation can be seen in the Wren Library of Trinity College, Cambridge, UK, in a glass case in which Newton’s manuscript is open to the relevant page. Motte’s 1729 translation of Newton’s Latin manuscript continued with Newton’s commentary on the second law of motion, reading: If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according to as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

The scientific intention in which Newton used his terminology, and how he defined the second law and intended it to be understood, has been extensively studied by historians of science.

2.1

Newton’s Universal Laws of Motion

2.1.2.1

9

Formulation of Newton’s Second Law

The second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in dp dt momentum takes place in the direction of the applied force, F, F=

dp dðm~vÞ = dt dt

The second law can also be stated in terms of an object’s acceleration. It is important to note that a general expression for Newton’s second law for variable mass systems by treating the mass as a variable cannot be derived. However, F = dp dt can be used to analyze variable mass systems only if we apply it to an entire system of constant mass having parts among which there is an interchange of mass. Since Newton’s second law is valid only for constant-mass systems (Beatty, 2006; Thornton, 2004; Hellingman, 1992), mass, m, can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus, F=m

d~v = ma dt

An impulse is defined by a force F acting on an object over an interval of time Δt and can be expressed by Z I=

Fdt Δt

Since force is the time derivative of momentum, it follows that, I = Δp = mΔ~v this relation between impulse and momentum is closer to Newton’s definition of the second law.

2.1.3

Third Universal Law of Motion

Newton’s original Latin text reads: Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi.

Direct translation to English, this reads: Law III: To every action, there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal and directed to contrary parts.

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(Modern translation) Law III: When one body exerts a force on a second body F12 = F21, the second body simultaneously exerts a force F21 equal in magnitude and opposite in direction to the first body. Newton’s explanatory comment to this law: Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavor to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another, and by its force changes the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say if the bodies are not hindered by any other impediments. For, as the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium (Fairlie & Cayley, 1965).

F 12 = F 21 In the above, as usual, motion is Newton’s term for momentum, hence his careful distinction between motion and velocity. Newton used the third law to derive the law of conservation of momentum (Cohen, 1995), from a deeper perspective; however, conservation of momentum is the more fundamental idea (derived via Noether’s theorem from Galilean invariance), and holds in cases where Newton’s third law appears to fail, for instance, when force fields, as well as particles, carry momentum, and in quantum mechanics. An example illustration of Newton’s third law is in which two people push against each other. The first person on the left exerts a normal force F12 on the second person directed towards the right, and the second person exerts a normal force F21 on the first person directed towards the left. The magnitudes of both forces are equal, but they have opposite directions, as dictated by Newton’s third law. The third law states that all forces between two objects exist in equal magnitude and opposite direction: if one object A exerts a force FA on a second object B, then B simultaneously exerts a force FB on A, and the two forces are equal in magnitude and opposite in direction: FA = - FB. The third law means that all forces are interactions between different bodies, or different regions within one body, and thus that there is no such thing as a force that is not accompanied by an equal and opposite force. In some situations, the magnitude and direction of the forces are determined entirely by one of the two bodies, say Body A; the force exerted by Body A on Body B is called the “action,” and the force exerted by Body B on Body A is called the “reaction.” This law is sometimes referred to as the action–reaction law, with FA called the “action” and FB the “reaction.” In other situations, the magnitude and directions of the forces are determined jointly by both bodies, and it is not necessary to identify one force as the “action” and the other as the “reaction.” The action and the reaction are simultaneous, and it does not matter which is called the action and which is called the reaction; both forces are part of a single interaction, and neither force exists without the other (Cohen, 1967).

2.1

Newton’s Universal Laws of Motion

11

The two forces in Newton’s third law are of the same type (e.g., if the road exerts a forward frictional force on an accelerating car’s tires, then it is also a frictional force that Newton’s third law predicts for the tires pushing backward on the road). From a conceptual standpoint, Newton’s third law is seen when a person walks: they push against the floor, and the floor pushes against the person. Similarly, the tires of a car apply a shear force on the road while the road pushes back on the tires with a re-action shear force—the tires and road simultaneously push against each other. The reaction force is responsible for the motion of the person or the car. These forces depend on friction; however, if a person or car is on ice, there will be no motion because of failure to produce the needed reaction force. It is important to point out that Robert Hooke published his law in 1676. Hooke’s law which is more sophisticated than Newton’s third law, for deformable bodies as F = ku where k is the stiffness of a one-dimensional spring and u is the displacement response in the direction of the force. It is easy to see the action–reaction concept in Hooke’s law. “Hooke stated in his 1678 work that he was aware of the law in 1660, which is 17 years before Newton published his Principia in 1687” (Robert Hooke’s Wikipedia page). In their original form, Newton’s laws of motion are not adequate to characterize the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalization of Newton’s laws of motion for rigid bodies called Euler’s laws of motion, later applied as well for deformable bodies assumed as a continuum. If a body is represented as an assemblage of discrete particles, each governed by Newton’s laws of motion, then Euler’s laws can be derived from Newton’s laws. Euler’s laws can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle structure (Lubliner, 2008; Leonard Euler’s Wikipedia page).

2.1.4

Range of Validity of Newton’s Universal Laws of Motion

Newton’s laws were verified by experiments and observations for over 200 years, and they are excellent approximations at the scales and speeds of most of the motions we observe in daily life. Newton’s laws of motion, together with his law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena. These three laws hold for macroscopic objects under everyday conditions. However, Newton’s laws are inappropriate for use in certain circumstances, most notably, at very high speeds (in special relativity, the Lorentz factor must be included in the expression for momentum along with the rest mass and velocity) or in very strong gravitational fields. Therefore, Newton’s laws cannot be used to explain phenomena such as conduction of electricity in a semiconductor, optical properties of materials, errors in non-relativistically corrected GPS systems, and superconductivity. Explanation of these phenomena requires more sophisticated physical theories, including general relativity and quantum field theory. However, in quantum mechanics, concepts such as force, momentum, and position are defined by linear operators that operate on the quantum state; at speeds that are much lower than the speed of light, Newton’s laws are just as exact for these operators as they are for classical objects.

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Stress and Strain in Continuum

Relation to the Thermodynamics and Conservation Laws

In modern physics, the laws of conservation of momentum, energy, and angular momentum are of more general validity than Newton’s laws, since they apply to both light and matter, and both classical and non-classical physics. Because force is the time derivative of momentum, the concept of force is redundant and subordinate to the conservation of momentum, and is not used in fundamental theories (e.g., quantum mechanics, quantum electrodynamics, general relativity, etc.). Other forces, such as gravity, also arise from momentum conservation. Newton stated the third law within a world-view that assumed instantaneous action at a distance between material particles. However, he was prepared for philosophical criticism of this action at a distance, and it was in this context that he stated the famous phrase ‘I feign no hypotheses.’ The discovery of the second law of thermodynamics in the 19th century showed that not every physical quantity is conserved over time, thus disproving the validity of inducing the opposite metaphysical view from Newton’s laws. Hence, a ‘steady-state’ worldview based solely on Newton’s laws and the conservation laws does not take entropy into account.

This brings us to the main topic of this book. In unified mechanics, theory entropy generation is incorporated into Newton’s universal laws of motion.

2.2 2.2.1

Stress Definitions of Stress and Traction

The external surface force per unit area applied to the body is called the traction, t. It is a distributed external load. The vector ΔF acting at an imaginary point in the interior surface is called the internal force vector (Fig. 2.2). ΔA is the area of the internal imaginary point Q; ΔF is the force vector acting on this area as a result of external loads applied to the body. We choose to use the term imaginary point intentionally because later we will assume that the area of this point goes to zero to be able to define stress at a point.

Fig. 2.2 Definition of stress at an imaginary point Q in the internal surface at cross-section cut a-a

2.2

Stress

13

The average stress σ acting on ΔA is the vector ΔF ΔA , ΔF is the vector sum of the forces acting at point Q, and ΔA is the area of the chosen imaginary point. The average stress is also a vector having the same direction as ΔF. Assuming that ΔA is selected so that it represents a point with zero volume as the value of ΔA approaches zero. lim

ΔA → 0

of course, ΔA can never be equal to zero. If it were zero, stress value would be infinite. We let ΔA approach zero to be able to define stress at an imaginary point.   ΔF ΔA → 0 ΔA

σ = lim

Because the area corresponding to internal force is approaching zero, the numerator, ΔF also approaches zero. However, the fraction, in general, approaches a finite limit   ΔF k → 1 ΔA k

σ = lim

where k represents the number of points in the body. This is a basic postulate of continuum mechanics that such a limit exists and is independent of the area used (Malvern, 1969). It is important to explain what we mean by an “imaginary point” with zero volume, as the area ΔA, it is natural to assume that this point represents an atom. It cannot be smaller than an atom. However, this point does not represent an atom. Because at the atomic level fundamental definitions of continuum do not exist, and this definition of stress at the atomic level is not true. This “imaginary point” therefore represents a point in the continuum, but not an atom or any other quantum mechanics’ matter. Since quantum mechanics is outside the scope of this book, we will refer the reader to a textbook on quantum mechanics. In x, y, z Cartesian coordinate system stresses at point Q on the cross-section that has x axis as its normal can be defined by ΔF x

dF x dAx ΔF y dF y σ xy = lim = dAx ΔAx → 0 ΔAx ΔF z dF z σ xz = lim = dAx ΔAx → 0 ΔAx σ xx = lim

ΔAx → 0 ΔAx

=

14

2

Stress and Strain in Continuum

Fig. 2.3 Definition of stress at point Q

Subscript x in dAx indicates that x axis is normal to area dAx (Fig. 2.3). In the same fashion, we can define stress vectors acting on other surfaces, y, and z. As a result, the state of stress at a point is defined by a second-order tensor with nine vector components 2

σ xx

6 σ = 4 σ yx σ zx

σ xy

σ xz

3

σ yy

7 σ yz 5

σ zy

σ zz

Normal stresses, σ ii, are positive when the vector is in the positive axis direction. Shear stresses σ ij are positive when the vector component is in the positive direction on the positive face of the block and the negative direction on the negative face of the block. Shear stresses σ ij are negative when the vector component is in the negative direction on the positive face of the block and the positive direction on the negative face of the block. In this book, tensile stress is defined to be positive, and compression is defined as negative. However, we realize that this definition is not universal. For instance, in geo-mechanics tension is negative and compression is positive. While tension and compression lead to completely different results when applied to any object, negative shear stress and positive shear stress are considered to be the same kind of loading in opposite directions from a physical point of view. There is no compression shear or tension shear.

2.2

Stress

2.2.2

15

Stress Vector on an Arbitrary Plane

Let’s define a plane ABC at an arbitrary slope passing through point Q (Figs. 2.4 and 2.5). To obtain the equations governing the state of stress at an arbitrarily oriented plane ABC, we will use the conservation of momentum principle. Forces acting on the tetrahedron are five vectors representing the resultant force on each of the faces (ΔAx, ΔAy, ΔAz, ΔA) and the resultant body force ρbΔV, where ρ is mass density, b is body force per unit mass, and ΔV is the volume of the tetrahedron. σ (n) is the average value of stress on the oblique face which has n as its normal. (x) σ is the average stress on the area ΔAx that has x axis as its normal, similarly σ ( y) and σ (z) are defined. Equilibrium of the tetrahedron and stress vector components on the inclined plane can be derived from the conservation of momentum principle of a collection of particles. While conservation of momentum is discussed later in the book, assuming students have a rudimentary knowledge of the principle we will use it at this stage to derive the equilibrium equations. The momentum principle of a collection of particles states that the vector sum of all external forces acting on the free body is equal to the rate of change of the total momentum. The total momentum of the collection of particles in a given volume is given by

Fig. 2.4 Stresses acting at a point in Cartesian coordinates

16

2

Stress and Strain in Continuum

Fig. 2.5 (a) An arbitrary tetrahedron QABC at point Q (b) free-body diagram of tetrahedron QABC

Z

Z ΔV

~vρdV =

Δm

~vdm

where dm is the mass of the tetrahedron, dV is the volume of ρ is mass density, and ~v is the velocity of the particle. The time rate of change of the total momentum is given by Δm

d~v d~v = ðρΔV Þ dt dt

It is assumed that Δm does not change over time. The conservation of momentum principle yields the following equilibrium equation for the free body shown in Fig. 2.6. σ ðnÞ ΔA þ ρbΔV - σ ðxÞ ΔAx - σ ðyÞ ΔAy - σ ðzÞ ΔAz = ðρΔV Þ

d~v dt

We need to calculate the height and volume of the arbitrary tetrahedron at point Q. Then we let the height h go to zero, to be able to define the imaginary point Q, cos α =

h , OA

nx = cosα,

cos β =

h h , cos γ = OB OC

ny = cosβ,

nz = cosγ

Three direction cosines, nx, ny, nz, are also bound by the following relation: n2x þ n2y þ n2z = 1

2.2

Stress

17

Fig. 2.6 Geometry of the tetrahedron at a point

As a result h = OA  nx = OB  ny = OC  nz The volume of the pyramid is given by ΔV =

1 hΔA 3

substituting h in the previous equation yields, ΔV =

1 1 1 ðOA  cos αÞΔA = ðOB  cos βÞΔA = ðOC  cos γ ÞΔA 3 3 3

Note that we can define the direction cosines also using the relationship between the areas of the faces of the tetrahedron cosα =

ΔAx ΔA

because ΔAx is the projection of the inclined area of interest ΔA on y - z plane with x axis its normal. Hence ΔV =

1 OA ΔAx 3

18

2

Stress and Strain in Continuum

Similarly, we can write ΔV =

1 1 OB ΔAy and ΔV = OC ΔAz 3 3

substituting ΔV, ΔAx, ΔAy, and ΔAz in the conservation of momentum equation, we obtain σ ðnÞ ðΔAÞ þ ρb



   1 1 d~v hΔA = σ ðxÞ ΔA nx þ σ ðyÞ ΔA ny þ σ ðzÞ ΔA nz þ ρ hΔA 3 3 dt

Eliminating ΔA from each term leads to the general equilibrium equation. 1 1 d~v σ ðnÞ þ ρb h = σ ðxÞ nx þ σ ðyÞ ny þ σ ðzÞ nz þ ρ h 3 3 dt v Let’s assume, there is no body force b = 0, and no acceleration d~ dt = 0, then, we can write

σ ðnÞ = σ ðxÞ nx þ σ ðyÞ ny þ σ ðzÞ nz

ð2:1Þ

In some textbooks, Eq. (2.1) is obtained by allowing the tetrahedron h to go to zero in the equilibrium equation, to define an imaginary point. Equation (2.1) defines the stress at a point on an arbitrary oblique plane passing through point Q. This equation was derived from the conservation of momentum principle of a collection of particles. Hence it applies to solid mechanics as well as fluid mechanics. However, it is important to point out that this is not an equilibrium of stresses equation. There is an equilibrium of forces but not stresses. If direction cosines are replaced by their respective equations in terms of areas, this equation becomes equilibrium of forces at point Q. We should again emphasize that this equation is only in terms of stresses and direction cosines. It is not possible to write equilibrium of stresses. The resultant stress on any inclined plane can be determined from the stress tensor of the point and the direction cosines of the inclined plane. Eq. (2.1) is a vector equation because stresses are vectors. The corresponding algebraic equations yield, ðnÞ

σx = σxx nx þ σyx ny þ σzx nz ðnÞ

σy = σxy nx þ σyy ny þ σzy nz

ð2:2Þ

σðnÞ z = σxz nx þ σyz ny þ σzz nz where σ ðxnÞ is resultant vector component on n plane in the (x) direction, and we have the same definition for y and z directions. We can write a second-order tensor, σ ij, Cauchy stress tensor from these algebraic equations. In indicial notation algebraic equations can be written as

2.2

Stress

19 ðnÞ

σ j = σ ij ni i = x,y,z j = x,y,z in matrix notation {σ (n)} = {n}[σ]. Remember that { } defines a vector and [ ] defines a matrix. Cauchy stress tensor in any other Cartesian coordinate system can easily be obtained. Assume that the new coordinate system is rotated with respect to the original coordinate system by α, β, γ angles. The components of the new rotated stress tensor σ ij can be defined in terms of the three stress vectors acting across the three planes normal to the new coordinates. Components of the new stress tensor σ ij will be ðnÞ

σ i = σ ij nj ∙ σ ij is a second-order tensor (linear vector function); therefore the components of the rotated stress tensor σ ij can be obtained by the tensor transformation equations. σ ij = nki nlj σ kl or in matrix notation ½σ  = ½N T ½σ ½N  ½σ  = ½N ½σ ½N T where [N] is the matrix of direction cosines nki = cos ðxi , xk Þ of the angles between the new and old axes. [N]T is its transpose. The matrix of direction cosine is given by 2

cos ðx, xÞ 6 ½N  = 4 cos ðy, xÞ cos ðz, xÞ

2.2.3

cos ðx, yÞ cos ðy, yÞ cos ðz, yÞ

3 cos ðx, zÞ 7 cos ðy, zÞ 5 cos ðz, zÞ

Symmetry of Stress Tensor

Normal stresses act orthogonal to the surface they are acting on. However, shear stresses act parallel to the plane on each surface of the cubic control volume, which represents a point in space with zero volume. Shear stresses are assumed to be positive when they are on the positive face of the cube and acting in the positive coordinate axis direction. On the negative face, shear stress is positive if the direction of the stress is in the negative direction of the axis.

20

2

Stress and Strain in Continuum

Fig. 2.7 (a) Linear shear stresses acting at point Q. (b) x-y plane view of point Q

When there are no distributed couple-stresses (moments) (body or surface) or the material has no length scale effect, all off-diagonal terms of the stress tensor are equal, due to moment equilibrium at point Q σ xy = σ yx , σ xz = σ zx , σ yz = σ yz Writing equilibrium of moment about z axis leads to ∑Mz = 0 

   σ xy dydz dx - σ yx dxdz dy = 0 σ xy = σ yx

Similarly writing ∑My = 0 and ∑Mx = 0 equilibrium equations lead to σ xz = σ zx and σ zy = σ yz However, we should make it clear that we obtained this result because we assumed that at point Q only linear stress vectors are acting and there are no couple-stress vectors (distributed moments) acting at point Q or material does not exhibit a length-scale effect that leads to differential shear stresses on opposite sides of the point Q. Therefore, in the absence of distributed moment acting at a point, the Cauchy stress tensor is symmetric (Fig. 2.7).

2.2

Stress

2.2.4

21

Couple Stresses

Couple stress at a point can exist due to external loads, like a magnetic field, due to materials microstructure, or due to the small size of the structure which can cause very large deformation gradients. If there are the couple-stresses mxz on x plane and myz on y plane and we write the moment equilibrium with respect to z axis, 

   σ xy dydz dx þ mxz - σ yx dxdz dy þ myz = 0

Therefore σ xy ≠ σ yx, similarly σ xz ≠ σ zx, σ yz ≠ σ zy. The skew-symmetric second-order couple stress tensor can be given by 2

1 m 2 xy

0

6 6 6 1 ½m = 6 - myx 6 2 4 1 m 2 zx

-

0 -

1 m 2 zy

3 1 mxz 7 2 7 7 1 myz 7 7 2 5 0

It is important to point out that the ½ multiplier in front of the couple-stress (moment) is due to the relation between shear stresses and normal stresses [2εxy = γ xy] (Fig. 2.8). y

Fig. 2.8 Couple stresses acting in conjunction with linear shear stresses in the x-y plane

myz σyx σxy

mxz

dy

mxz

σxy

x

σyx myz dz dx

z

22

2

Stress and Strain in Continuum

Example 2.1 Figure 2.1E

Fig. 2.1E State of stress at a point

Equilibrium of forces acting on the wedge on the left can be given by Aσ ðnÞ = σ ðxÞ nx A þ σ ðyÞ ny A = σ ðxÞ Ax þ σ ðyÞ Ay For the contribution of forces in x direction to the resultant force on the inclined plane (n), we can write the following relation, where A represents the area along the inclined surface: σðnÞ x = σxx nx þ σyx ny Aσ ðxnÞ = - 5 MPa ðA cos 45Þ - 2 MPa ðA cos 45Þ σðnÞ x = - 5 MPa cos 45 - 2 MPa cos 45 = - 4:95 MPa Example 2.2 Figure 2.2E Find the stresses acting on the ABC [111] plane [i.e., plane intercepts x = 1, y = 1, z = 1]. 2

σ xx 6 σ = 4 σ yx

σ xy σ yy

σ zx

σ zy

3 2 σ xz 16 7 6 σ yz 5 = 4 - 40 σ zz 25

- 40 42 20

3 25 7 20 5 MPa 25

Projection of ABC plane on x, y, z planes is the same shown below, OBC, OAB, OAC

2.2

Stress

23 y y

σ yy σy

σ yz x

σ zy

σ zz

B

σ xy σ xx

x

σ xz

σ zx

A

O

x

C z

z

Fig. 2.2E Stresses acting on the point

B

B

√2

√2

C

1 A

O

ABC =

Area of

nx =

1 2

1 ∙1 =

1 2

OBC = OAB = OAC

A

1

√2

1 2

OBC =

Area of

√2

√ 2 ∙ √ 2 Cos 30 = 0.866

OBC = 0.5774 , ABC

ny =

OAC = 0.5774 , ABC

nz =

OAB = 0.5774 ABC

3 3 2 0:5774 nx 7 6 7 6 Direction Cosines are n = 4 ny 5 = 4 0:5774 5 which is normal to the [111] 2

0:5774 nz plane. Therefore, we can define the stresses on the [111] plane utilizing its normal n. The dot product between the tensor and the unit vector gives us the components of σ on the surface defined by the normal n 2

σnx

3

2

16

6 6 σn 7 4 y 5 = σ ∙ n = 4 - 40 σnz 25

- 40 42 20

25

3 2

0:5774

3

2

0:5774

3

7 7 6 7 6 20 5 ∙ 4 0:5774 5 = 4 12:703 5 MPa 40:418 0:5774 25

24

2

Stress and Strain in Continuum

y ( )

B

( )

ABC Plane

A

x

C z

2

ðnÞ

σx

3

2

0:5774

3

6 7 6 7 σðnÞ = nT ∙ 4 σðnÞ y 5 = ½0:5774 0:5774 0:5774 ∙ 4 12:703 5 = 31:00 MPa σðnÞ z

40:418 2

2

þ σðnÞ σðnÞ = σðnÞ n s 2

2

2

2

2

þ σðnÞ þ σðnÞ = 1795:5685 σðnÞ = σðnÞ x y z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σs = σðnÞ2 - σ2n = 28:879 MPa

2.2.5

Principal Stresses and Principal Axes

The stress tensor given in the original Cartesian coordinate system changes values if we change the orientation of the coordinate system, i.e., simply rotating it around the origin O. The inclined plane (Fig. 2.9) represents the orientation of a new coordinate system. Therefore, the stress tensor becomes a function of the rotation angle with respect to the original coordinate system. One of these orientations will have the maximum normal stress values, also called principal stresses. For a three-dimensional stress tensor, these principal values and principal orientation can be identified by eigenvalue analysis, Normal and shear stress components acting on the inclined face of the tetrahedron are a function of the orientation of the inclined plane. If the stress tensor is symmetric, we can choose an orientation of the new coordinate axes such that the shear stress components vanish in this new coordinate system. These special axes are called principal axes or principal directions. The three planes that are orthogonal to three principal axes are called the principal planes. On principal planes, all shear stresses are zero. There are only normal stresses on the principal planes. Maximum

2.2

Stress

25

Fig. 2.9 Arbitrary tetrahedron in a stress cube

normal stress is defined as the algebraically largest of the three principal stresses. The minimum normal stress is defined as the algebraically smallest of the three principal stresses. There is always such a set of three mutually orthogonal directions at any point if the state of stress at a point is a symmetric second-order tensor. This is a property common to all symmetric second-order tensors. The principal planes and principal stresses can be found by traditional eigenvalue analysis. Let [I] be a matrix of unit vectors in one of the arbitrary orientations of the coordinate system. Let λ be the principal stress components in the new orientation, whose normals are given by ½n. Since there is no shear stress on the principal planes, the stress vector on the principal plane will be parallel to the normal ðnÞ

σ i = λni where i = x, y, z. In matrix notation n

o σ ðnÞ = λfng

Since n o σ ðnÞ = fng½σ 

26

2

Stress and Strain in Continuum

Therefore, we can write, fng  ½σ  = λfng This can be written as fng½½σ  - λ½I  = 0

ð2:3Þ

where [I] is an identity matrix 2

1

6 I =40 0

0

0

3

1

7 05

0

1

fng is a row matrix {n1, n2, n3} where ni defines the direction cosine. λ is the principal stress value sought. Thus 2

λ 6 λ½I  = 4 0

0 λ

3 0 7 05

0

0

λ

Equation (3.3) has a solution only if the determinant of the matrix is equal to zero since the direction cosines vector cannot be zero, and because n21 þ n22 þ n23 = 1 must be satisfied. Hence,   ðσ xx - λÞ    σ yx   σ zx

½½σ  - λ½I  = 0 σ xy   σ yy - λ σ zy

    =0  ðσ zz - λÞ  σ xz σ yz

This is a cubic equation for the unknown λ. If the stress matrix is symmetric and has real stress values (as opposed to imaginary values), the three roots of the cubic equation λ3 þ aλ2 þ bλ þ c = 0 are all real numbers. The three roots of the cubic equation λ1, λ2, λ3 are the three principal stresses λ1 = σ 1 , λ2 = σ 2 , and λ3 = σ 3 :

2.2

Stress

27

To find the principal directions, we can use Eq. (2.3). fng½½σ  - λ½I  = 0 If we substitute σ 1 for λ in the above equation, these three equations reduce to only two linearly independent equations. These two equations can be solved with the to determine the direction cosines of the help of the n21 þ nn22 þ n23 = 1 requirement, o  normal nð1Þ = nxð1Þ nðy1Þ nðz1Þ to the plane which σ 1 acts on. h When we arei solving for direction cosines, since one of the equations n2x þ n2y þ n2z = 1 is quadratic, two solutions will be found representing two oppositely directed normals to the same plane. The choice of which one is positive is n arbitrary. Ino the same manner, n the processo is repeated for σ 2 to find nðx2Þ , nðy2Þ , nðz2Þ and for σ 3 to find nðx3Þ , nðy3Þ , nðz3Þ . The third principal direction can also be found by taking a direction orthogonal to the first two directions. When all the roots of the characteristic equation are different, then these are three unique orthogonal principal directions. However, when we find the roots of the characteristic equation, if two of them are equal and one is different, then the direction of the different root is unique; however, the other two must be arbitrarily chosen as any two perpendicular axes to each other and perpendicular to the unique axis. Three axes must define a right-handed system. If all three principal stresses are equal, the state of stress is said to be the hydrostatic state of stress. Because this is the only state of stress that can exist in fluids.

2.2.6

Stress Tensor Invariants

Stress invariant is a scalar quantity, and it is an intrinsic property of a stress tensor that does not depend on the coordinate system. When a stress tensor is rotated, each vector entry in the tensor matrix changes its value; however stress invariants stay the same. This is because in Eigen Eq. (2.4), three roots are independent of the coordinate system. The determinant of the characteristic equation can be found by   ðσ xx - λÞ    σ yx   σ zx

    =0  ðσ zz - λÞ 

σ xy σ xz   σ yy - λ σ yz σ zy

λ 3 - I 1 λ2 þ I 2 λ - I 3 = 0

ð2:4Þ

28

2

Stress and Strain in Continuum

where I1, I2, and I3 are the first, second, and third invariants of the stress tensor. Invariants are given by the following equations: I 1 = σ xx þ σ yy þ σ zz = trσ

ð2:5Þ

Or in indicial notation I 1 = σ ii   I 2 = σ xx σ yy þ σ yy σ zz þ σ zz σ xx - σ 2xy - σ 2xz - σ 2yz Or in indicial notation I2 =

 1 σ ii σ jj - σ ij σ ij 2

ð2:6Þ

where i = x, y, z j = x, y, z h i I 3 = σ xx σ yy σ zz þ 2σ xy σ yz σ xz - σ 2xz σ yy - σ 2yz σ xx - σ 2xy σ zz or 2

2.2.6.1

σxx 6 I 3 = 4 σyx

σxy σyy

σzx

σzy

3 σxz 7 σyz 5 = det j σ j σzz

ð2:7Þ

Stress Invariants in Principal Axes

In the principal planes, these are no shear stresses; therefore the invariant terms become simpler: I 1 = σ1 þ σ2 þ σ3

ð2:8Þ

I 2 = σ1σ2 þ σ2σ3 þ σ3σ1

ð2:9Þ

I 3 = σ1σ2σ3

ð2:10Þ

Stress invariants are mostly used in the constitutive modeling of materials. Any function constructed with invariants is also invariant with respect to coordinate systems.

2.2

Stress

2.2.6.2

29

Representation of Stress Tensor in Spherical and Deviatoric Components

We can represent any stress tensor as a summation of the hydrostatic stress tensor and the remainder, which is called the deviatoric stress tensor. The hydrostatic stress part for any stress tensor is given by p=

 1 σ þ σ yy þ σ zz 3 xx

ð2:11Þ

Using this definition, we can split the stress tensor into two parts. 2

p 0 6 ½σ  = 4 0 p 0 0

3 3 2 ðσ xx - pÞ σ xy σ xz 0   7 7 6 σ yy - p σ yz 5 0 5 þ 4 σ yx σ zx σ zy ðσ zz - pÞ p

ð2:12Þ

In Eq. (2.12) the first tensor is called the hydrostatic stress tensor, and the second tensor is called the deviatoric stress tensor. Hydrostatic tensor is also called spherical stress tensor. We will use Sij to represent deviatoric stress tensor, 2 Sxx

 6 Sij = 4 Syx Szx

Sxy Syy

3 Sxz 7 Syz 5

Szy

Szz

where Sxx = σ xx -

 1 σ þ σ yy þ σ zz Sxy = σ xy , Sxz = σ xz , Syz = σ yz 3 xx

In the same fashion Syy and Szz can be written. In Cartesian form, the deviatoric stress tensor is given by Sij = σ ij -

1 σ δ 3 kk ij

where δij is Kronecker delta, when if i = j δij = 1, if i ≠ j δik = 0 Summation over repeated indices rule is utilized, σ kk = σ 11 + σ 22 + σ 33 Hydrostatic stress is the same in all three directions; therefore, in isotropic materials, hydrostatic stress causes elastic volumetric change only. However, deviatoric stress is different in all directions; as a result it can cause shape change and inelastic (irreversible) deformation. In anisotropic materials both hydrostatic and deviatoric stresses can lead to inelastic shape change.

30

2

2.2.6.3

Stress and Strain in Continuum

Invariants of the Deviatoric Stress Tensor

Principal deviatoric stresses are in the same planes as the total stress tensor principal planes. Following the same procedure we used for the total stress tensor, we can obtain the characteristic [eigen] equation of the deviatoric stress tensor as follows: 2

ðSxx - λÞ Sxy   6 Syy - λ 4 Syx Szx

Szy

Sxz Syz

3 7 5=0

ðSzz - λÞ

This equation can be expanded to the following form: λ3 - J 2D λ - J 3D = 0 where J2D and J3D are the second and third invariants of the deviatoric stress tensor, respectively. The first invariant of the deviatoric stress tensor is zero J1D = 0, 1 J 2D = Sij Sij 2 h i 2  2 1  = σ xx - σ yy þ σ yy - σ zz þ ðσ xx - σ zz Þ2 þ σ 2xy þ σ 2xz þ σ 2yz 6 h i 1 J 2D = S2xx þ S2yy þ S2zz þ S2xy þ S2xz þ S2yz 2 1 J 3D = Sij Sjk Ski 3    Sxx Sxy Sxz      J 3D =  Syx Syy Syz     Szx Szy Szz  It is important to point out that the second invariant of the deviatoric stress tensor is a representation of shear stress; however physical interpretation of the third invariant is more complex.

2.2.7

Octahedral Plane and Octahedral Stresses

The plane that makes equal angles with the three principal directions is called the octahedral plane. The projection of the stress vector on the octahedral plane is called the octahedral shear stress. The magnitude of the normal stress on the octahedral plane is constant.

2.2

Stress

31

Octahedral shear stress is given by the following relation: τ2oct =

h i h 2  2  2 i 2  2 1  σ xy þ ðσ xz Þ2 þ σ yz σ xx - σ yy þ σ yy - σ zz þ ðσ xx - σ zz Þ2 þ 3 9

Normal stress on the octahedral plane is given by σ oct =

 1  σ þ σ yy þ σ zz 3 xx

The second invariant of the deviatoric stress tensor is related to octahedral shear stress by the following relation: J 2D =

3 2 τ 2 oct

In most metals plasticity happens mainly due to shear stresses; hence deviatoric stress tensor invariants and octahedral stresses are utilized often in defining yield surfaces. Example 2.3 Figure 2.3E Stresses acting on the point shown above are given below: σ xx = 4 MPa, σ xy = 3 MPa, σ xz = 2 MPa σ yy = 5 MPa, σ yz = 1 MPa, σ zz = 6 MPa,

Fig. 2.3E (a) Stresses acting on the point. (b) Principal directions

32

2

Stress and Strain in Continuum

(i) Calculate the normal and shear stresses acting on the octahedral plane [i.e., 1,1,1 plane with respect to the principal axes] using the stress transformation equations, and then compare those values with the formula given below: σ oct =

τ2oct =

 1

σ þ σ yy þ σ zz 3 xx

h i h i 2  2 1  2  2  2 σ xx - σ yy þ σ yy - σ zz þ ðσ xx - σ zz Þ2 þ σ xy þ σ yz þ ðσ zx Þ2 9 3

(ii) Verify that J 2D = 32 τ2: oct First, we need to calculate principal stresses and principal directions using the Eigen equations,   ð4 - λÞ 3   ð5 - λÞ  3   2 1

    1 =0  ð6 - λÞ  2

This yields a third-order polynomial in λ. The principal stresses are these roots. Principal stresses are λ1 = σ 1 = 9.0000 MPa, λ2 = σ 2 = 4.7321 MPa, λ3 = σ 3 = 1.2679 MPa Direction cosines for octahedral plane [111] normal with respect to the principal directions were calculated earlier in Example 2.2. 2

0:5774

3

7 6 n = 4 0:5774 5 0:5774 Using transformation equations σ oct = nT nσ = n21 σ 1 þ n22 σ 2 þ n23 σ 3 σ oct = 0:57742 9:0000 þ 0:57742 4:7321 þ 0:57742 1:2679 = 5:0000 MPa The norm of the stress vector acting on the octahedral plane is 2

σðnÞ = ðσ1 n1 Þ2 þ ðσ2 n2 Þ2 þ ðσ3 n3 Þ2 = 0:57742 9:00002 þ 0:57742 4:73212 þ 0:57742 1:26792 = 35:0061

2.3

Deformation and Strain

33

Also 2

σðnÞ = σ2oct þ τ2oct

Hence τoct =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ 2 - σ 2oct = 3:1623 MPa

And now using the formula given above we can recalculate the stresses on the octahedral plane rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i h i 2 1 τoct = ð 4 - 5 Þ 2 þ ð 5 - 6Þ 2 þ ð 4 - 6Þ 2 þ ð 3 Þ 2 þ ð 1 Þ 2 þ ð 2 Þ 2 3 9 = 3:1623 MPa σ oct =

1 ½4 þ 5 þ 6 = 5:0000 MPa 3

h i 2  2 1  σ xx - σ yy þ σ yy - σ zz þ ðσ xx - σ zz Þ2 þ σ 2xy þ σ 2xz þ σ 2yz 6 h i 1 = ð4 - 5Þ2 þ ð5 - 6Þ2 þ ð4 - 6Þ2 þ 32 þ 22 þ 12 = 15 6

J 2D = (iii)

Verify 3 2 τ 2 oct 3 15 = 3:16232 = 15 2

J 2D =

2.3

Deformation and Strain

Let’s assume, we have a deformable solid object defined by corners A to H. Under external loading, the corner points displace to new locations given by points A′ to H′ (Fig. 2.10). Using Cartesian coordinates x, y, z, we can easily measure the relative displacement of corner points (or any point in the object) with respect to their original location. However, just knowing the relative displacement dr of each point in the deformable body does not help us to understand the strain imposed on the body due

34

2

Stress and Strain in Continuum

Fig. 2.10 Definition of strain deformation and rigid body motion

to the new configuration. Therefore, we must establish a strain definition. There are many definitions of strain. We can choose any definition of strain and calculate the strain in the deformable body. The point we are trying to emphasize is that strain is a definition (a human construct) calculated using measured displacement quantities. Therefore, strain cannot be measured directly. Strain formulation without any constraints on the magnitude of the strain gives a more accurate representation of the deformation. However, the mathematical description of finite strain (large strain) is quite complex. Therefore, for many engineering problems, the small-strain theory is used, which is satisfactory with a reasonable degree of error. First, we will present the small strain formulation and then a more complex large strain (finite strain) formulation. Rigid body motion does not lead to any strain.

2.3.1

Small Strain Definition

There is no magic number that defines the boundary between small-strain theory and large strain theory. However, any strain value less than 2% is usually, but not always, considered a small strain. The decision to use the small or large strain formulation depends on the problem at hand and the material properties. Now, we will define strains with respect to the initial (undeformed) configuration.

2.3.1.1

Elementary Definition of Pure Uniaxial Strain

At the unit defined by ABCD, corners B and C are stretched to new locations B′ and C′ (Fig. 2.11). As a result, small-strain Exx is the change in length per unit of initial length in the x-direction. In the same fashion, we can define the unit extension per unit length for the y and z directions Eyy and Ezz

2.3

Deformation and Strain

35

Fig. 2.11 Uniaxial extension in one dimension

εxx =

Δu Δv Δw ε = ε = Δx yy Δy zz Δz

where Δv and Δw are extensions in the y and z directions, respectively. We let Δx, Δy, and Δz approach zero to be able to define the strain at an imaginary point. Strain and stress are always defined at a point of zero volume. The term imaginary is coined to reflect the zero-volume feature of the point.

2.3.1.2

Pure Shear Strain

The shear strain is defined as the change in the initial right angle at point A (Fig. 2.12). γ xy =

π - Ψ = θ1 þ θ2 2

for small angles, [θ < 0.5 radian] tan θ = θ tan θ1 = θ1 =

Δv Δu and tan θ2 = θ2 = Δx Δy

As a result, we can write,

Δv Δu radian γ xy = θ1 þ θ2 = þ Δx Δy

36

2

Stress and Strain in Continuum

Fig. 2.12 Shear strain definition in two dimensions

Since we define our strain at a point in the limit dimensions, zero volume of the differential element Δx and Δy approach zero. Hence, we can define strain in differential form as Exx =

∂u ∂v ∂u ∂v Eyy = γ = þ ∂x ∂y xy ∂y ∂x

where u is the displacement in x axis and v is the displacement in y axis. It is important to point out that the initial (undeformed) position is the independent variable and the reference state in these small strain calculations. In other words, the change in per unit length is with respect to the initial length. If we utilize the deformed position as the reference state and as the independent variable, we obtain a different formulation of strain. Of, course these two reference states will lead to two numerically different strain values. Example 2.4 Calculate the shear strain at point B in the given triangle-shaped plate after deformation. Before deformation, the lengths of edges AB and BC are 300 mm and 400 mm, respectively. Point B moves u = 1 mm in x axis direction and v = 3 mm in y axis direction to point B′. Calculate the shear strain at point B using the change in angle and using the formula.

2.3

Deformation and Strain

37

1 mm 3 mm α = 0:01 rad tanβ = ð400 mm þ 3 mmÞ ð300 mm þ 1 mmÞ β = 0:0025 rad h   i h i π π 400 π 300 γ xy = - π - - atan - 0:01 - π - - atan - 0:0025 2 2 300 2 400 = α þ β = 0:0125 rad tanα =

γ xy =

2.3.1.3

∂u ∂v 1 mm 3 mm þ = þ = 0:0125 rad ∂y ∂x 400 mm 300 mm

Pure Rigid Body Motion

Let us assume the cantilever beam shown in Fig. 2.13 is supported at point A and subjected to vertical and axial loads at point B. Point C in the beam will be subjected to rigid-body displacement and rigid-body rotation, but it will not experience any strain, because point C undergoes rigid body motion, only. The point C undergoes displacement in x and y axes directions as well as rotation around z axis. However, these motions will not lead to any strain or stress at point C (Fig. 2.14). Fig. 2.13 Definition of pure rigid body motion

38

2

Stress and Strain in Continuum

Fig. 2.14 Pure rigid body rotation

Fig. 2.15 Spatial description of the new location of point A

Fig. 2.16 Material (local) coordinate system description

B’

B

A’

A

2.3.2

Small Strain and Small Rotation Formulation

We will start with defining spatial coordinate and material coordinate systems. Let’s say f(u, t) is a function that defines the displacement of point A located at a particular point A′ in space at time t. This spatial description is called Eulerian description. Another description is material field description, also called Lagrangian description in many textbooks. In this case function g(r, t) defines the displacement of point A at time t regardless of where the point A is located in space. This is called material or local description (Figs. 2.15 and 2.16).

2.3

Deformation and Strain

39

The coordinates of a point in a local reference coordinate system (configuration) r are referred to as material coordinates. In three dimensions we will use r, s, t for material coordinates. The coordinates of a point in the spatial coordinate system x, y, z are referred to as spatial coordinates. Of course, we can link the material coordinate system and spatial coordinate system. In computational solid mechanics, such as in the finite element method, we prefer to use material coordinates because it is more convenient.

2.3.2.1

Definition of Material (Local) Coordinates

In Fig. 2.17 we define a coordinate system that is shaped in the original shape of the line AB. We place the origin of our local coordinate axis r at point A, where r = 0, and at point B the value of the local coordinate is r = 1. By normalizing the coordinate between zero and one, we can define the displacement of any point between A and B with respect to initial coordinates easily. Of course, we can define displacement of point A using the global Cartesian coordinate system x, y and time, or we can use the material (local) coordinate system (which is usually called the local coordinate system in finite element method terminology). Local coordinate r is in the axis direction of the initial one-dimensional element shown in Fig. 2.18. The origin of the local coordinate system is at point A, or it can be located at any point between A and B. As a result, if point A moves with respect to the initial location due to loading origin A also moves. Therefore coordinate r will define the relative displacement of point A with respect to B at any time regardless of where the point A is. We will use the terminology local coordinates rather than material coordinates. For a two-dimensional case let x - y axes define our global (spatial) Cartesian coordinate system and let r define the local coordinate system. Let dr define an arbitrary infinitesimal line element vector (Fig. 2.18). Where r is the material (local) coordinate axis. It is always in the natural direction of the member, u is the displacement vector along the spatial axis x, and v is the displacement vector along the spatial y axis. Fig. 2.17 Material (local) coordinate system

40

2

Stress and Strain in Continuum

Fig. 2.18 Local (material) coordinate system, r

The local coordinate relative displacement dr in spatial Cartesian coordinate system x, y can be given by du du dx du dy = þ dr dx dr dy dr dv dv dx dv dy = þ dr dx dr dy dr

ð2:13Þ

It is important to point out that the left-hand side of this equation represents strain components with respect to the initial length. Equation (2.13) can be written in a matrix form: 2

du 6 dr 6 6 6 4 dv dr

3

2

du dx 7 6 7 6 7=6 7 6 5 6 4 dv dx

3 du 2 dx dy 7 dr 76 76 6 76 74 dv 5 dy dr dy

3 7 7 7 7 5

Or du ~ =J  n dr where du dr is the column vector on the left-hand side containing relative displacement vector components, ñ is the column vector on the right-hand side containing the dy direction cosines, Cosα = dx dr , Cosβ = dr . J is called the displacement–gradient matrix or the Jacobian matrix, or the unit relative displacement matrix. It may be considered as an operator which operates on a direction vector ~n to yield the unit relative displacement components for an infinitesimal line AB. This process may be considered akin to the transformation from spatial coordinate system x, y to local

2.3

Deformation and Strain

41

(material) coordinate system r. Assume that length of AB → 0; as a result, all derivatives are evaluated at point B, because we need to define strains at a point. We can split the Jacobian matrix into two components. One component will include deformations around point B, and the other will include rigid-body relative displacements. It is assumed that rigid body relative displacements do not cause any strain on the body. This splitting process can be accomplished in the following manner. 2

 3 3 2 ∂u 1 ∂u ∂v ∂u þ 6 2 ∂y ∂x 7 ∂x ∂y 7 7 7=6   6 7 5 ∂v 5 4 1 ∂v ∂u ∂v þ 2 ∂x ∂y ∂y ∂y 2  3 1 ∂u ∂v 0 6 2 ∂y ∂x 7 6 7  þ6  7 4 1 ∂v ∂u 5 0 2 ∂x ∂y

∂u 6 ∂x 6 4 ∂v ∂x

Earlier we defined small strain components as follows:   ∂u ∂v ∂u ∂v Exx = , Eyy = , γ = þ ∂x ∂y xy ∂y ∂x For convenience, we will define shear strain Exy as half the decrease in the angle γ xy to be able to define strain vectors as components of a second-order strain tensor ε. Exy =

1 γ 2 xy

As a result, the first matrix is the strain matrix, and the second matrix is the rotation matrix: Ωxy =

    1 ∂u ∂v 1 ∂v ∂u Ωyx = = - Ωxy 2 ∂y ∂x 2 ∂x ∂y

As a result, the Jacobian matrix can be written as

Exx J= Eyx

0 Exy þ Eyy Ωyx

Ωxy 0

In our stress and strain definitions, the first subscript defines the plane they belong to and the second subscript defines the direction. Now we can show that the strain matrix leads to deformations (shape change) and the rotation matrix leads to rigid body motion only with no shape change (strain) (Fig. 2.19). We can write earlier transformation equations between local (material) coordinates and spatial coordinates to find the unit relative displacement of B with respect to A.

42

2

Stress and Strain in Continuum

Fig. 2.19 Deformed differential element without rigid body motion

2 3 du du 6 dx 6 dr 7 6  4 5=6  dv 4 1 dv du þ dr 2 dx dy 2

 3 2 3 1 du dv dx þ 7 2 dy dx 76 dr 7 74 5 5 dy dv dr dy

Assume γ 1 is very small cos γ 1 =

dx = cos 0 = 1, dr

and dy π = cos = 0 dr 2 As a result, we can write, 3 du Exx 6 dr 7 4 5= dv Eyx dr 2

Exy Eyy

1



Exx = Eyx 0

Thus for AB Exx =

du dv , E = dr yx dr

It is important to point out that these equations are only valid for small deformations. Similarly, for AD we can write Assuming γ 2 is very small. dx π dy = cos = 0, = cos 0 = 1 dr 2 dr

2.3

Deformation and Strain

43

3 du

Exx Exy 0 Exy 6 dr 7 = 4 5= dv Eyx Eyy 1 Eyy dr du dv = Exy = Eyy dr dr 2

2.3.2.2

Small Strain in Local Coordinates

Displacement of corner point C to point C′ is plotted in detail Figs. 2.20, 2.21, and 2.22. Small strain along local axis r can be written as Fig. 2.20 Deformation of ABCD into A′B′C′D′ in local and global coordinates

Fig. 2.21 Displacement of point C to C′

44

2

Stress and Strain in Continuum

Fig. 2.22 Displacement of a point in material (local) and spatial coordinates



 Exx dx þ γ xy dy cos θ þ Eyy dy sin θ Err = dr where dr is the initial length of AC in local coordinates   dx dy dy cos θ þ Eyy sin θ Err = Exx þ γ xy dr dr dr dy Knowing that cos θ = dx dr sin θ = dr Inserting these equations in εrr yields

Err = Exx cos 2 θ þ γ xy sin θ cos θ þ Eyy sin 2 θ Using trigonometric relations, 1 ð1 þ cos 2θÞ 2 1 sin 2 θ = ð1 - cos 2θÞ 2

cos 2 θ =

2 sin θ cos θ = sin 2θ Strain along r axis can also be expressed as Err =

γ xy Exx þ Eyy Exx - Eyy þ cos 2θ þ sin 2θ 2 2 2

We could also obtain components of displacement CC′ in r and s local coordinates by using the dot product of vector CC′ and unit vectors along r and s.

2.3

Deformation and Strain

45

Fig. 2.23 Unit vectors in local (material) r-s coordinates. And spatial coordinates x, y

Material coordinate system unit vectors can be represented in terms of spatial coordinates’ unit vectors as follows (Fig. 2.23): l = i cos θ þ j sin θ m = - i sin θ þ j cos θ

The displacement in r and s directions can be found by the dot product. In r direction, CC0  l =



   Exx dx þ γ xy dy i þ Eyy dy j  ½icosθ þ jsinθ

 = Exx dx þ γ xy dy cosθ þ Eyy dysinθ

In s direction, CC0  m =



    Exx dx þ γ xy dy i þ Eyy dy j ∙ ½ - i sinθ þ jcosθ   = - Exx dx þ γ xy dy sinθ þ Eyy dycosθ

From here we can obtain, Err =

CC0  l dr

Ess =

CC 0  m ds

and

46

2

Stress and Strain in Continuum

Fig. 2.24 Shear strain in local (material) coordinates

2.3.2.3

Shear Strain in Local (Material) Coordinates

Displacement along s axis can be formulated by the following summation (Fig. 2.24):   Eyy dy cos θ - Exx dx þ γxy sin θ

We can also obtain the displacement of point C along s axis from the arc length. Assuming small deformations and small strains   αdr = Eyy dy cos θ - Exx dx þ γ xy dy sin θ   dy dx dy sin θ α = Eyy cos θ - Exx þ γ xy dr dr dr   α = Eyy sin θ cos θ - Exx cosθ þ γ xy sin θ sin θ   Rotation of s axis (β) can be found by substituting θ þ π2 in the equation for α. β = - Eyy sin θ cos θ þ Exx sin θ cos θ - γ xy cos 2 θ In Fig. 2.25, it is assumed that the shear strain is positive. Therefore α is positive when it is counterclockwise, and β is positive when it is clockwise.

2.3

Deformation and Strain

47

Fig. 2.25 Relations between local (material) and spatial coordinates

γ rs = ðα - βÞ 

= Eyy sin θ cos θ - Exx sin θ cos θ - γ xy sin 2 θ 

- - Eyy sin θ cos θ þ Exx sin θ cos θ - γ xy cos 2 θ     γ rs = - 2 Exx - Eyy sin θ cos θ þ γ xy cos 2 θ - sin 2 θ

2.3.3

Small Strain and Rotation in 3-D

Using the derivation, we used for a two-dimensional case we can derive the formulation for a three-dimensional case. In three dimensions, we will use u for displacement along x, v for displacement along y, and w for displacement along z axis. r axis is again our local (material) coordinate axis 2 9 8 ∂u du > > > > 6 ∂x > > > 6 > > = 6 < dr > 6 ∂v dv =6 > 6 ∂x dr > > > > > > 6 > > ; 4 ∂w : dw > dr ∂x

∂u ∂y ∂v ∂y ∂w ∂y

Again Jacobian matrix J can be written as J=E þ Ω

3 ∂u 8 dx 9 > > > > ∂z 7 > > dr > 7> = < > 7> ∂v 7 dy 7 ∂z 7> dr > > > 7> > > dz > ∂w 5> ; : > dr ∂z

48

2

Stress and Strain in Continuum

where  3 1 ∂u ∂w þ 6 2 ∂z ∂x 7 6    7 7 6 ∂v 1 ∂v ∂w 7 6 1 ∂u ∂v E=6 þ þ 7 2 ∂z ∂y 7 6 2 ∂y ∂x ∂y 7 6     5 4 1 ∂u ∂w 1 ∂w ∂v ∂w þ þ 2 ∂z ∂x 2 ∂y ∂z ∂z    3 2 1 ∂u ∂v 1 ∂u ∂w 0 6 2 ∂y ∂x 2 ∂z ∂x 7 6    7 6 7 1 ∂u ∂v 1 ∂v ∂w 7 6 Ω=6 0 7 2 ∂y ∂x 2 ∂z 6 ∂y 7 6 7   4 1 ∂u ∂w 5 1 ∂v ∂w 0 2 ∂z 2 ∂z ∂x ∂y 2

∂u ∂x

  1 ∂u ∂v þ 2 ∂y ∂x

When components of Ω are small, compared to 1 radian, it represents rigid body rotation.

2.4

Kinematics of Continuous Medium

There are three distinct descriptions of the displacements in a continuum.

2.4.1

Material (Local) Description

The variables of this coordinate system are the particle at time t, regardless of where the point is in spatial coordinates. Note that not the position of the particle but the particle itself is the variable.

2.4.2

Referential Description (Lagrangian Description)

This description usually utilizes a reference configuration for description. The independent variables are the r, s, t coordinates and time with respect to a reference configuration and reference state. The chosen reference configuration is usually the initial (unstressed) state at t = 0. When the reference configuration is chosen to be the initial undeformed shape at t = 0, the referential description is also called the Lagrangian description. Many mechanics researchers consider Lagrangian and

2.4

Kinematics of Continuous Medium

49

material description to be the same. It is assumed that the origin of the coordinate system is attached to the particle.

2.4.3

Spatial Description (Eulerian Description)

The spatial description is defined by the current location x, y, z of the object at time t, which are considered the independent variables. This is the most common coordinate description used in fluid mechanics. Because spatial description uses a fixed region of space rather than a body of matter [a particle]. For most students, it is usually difficult to visualize the different descriptions. Here we will quote an explanation from Malvern (1969) “The spatial description is especially useful in fluid mechanics, where we may observe a flow in a wind tunnel or a channel, a fixed region [window] in space. We can imagine perfect instrumentation with which an observer records the fluid velocity at a field point in space [a window] as a function of time. If this is done for every point in the region, we have the spatial description of the velocity ~v as a function of position x,y,z and time.” A Comparison of Lagrangian and Eulerian Descriptions “In computational solid mechanics, all particles in a body must be defined by their location in space from the initial unstressed (undeformed) state to the final configuration. Therefore, Lagrangian (material) description is preferred. This approach stands in contrast to Eulerian formulation, which is usually used in the analysis of fluid mechanics problems, in which attention is focused on the motion of the fluid through a stationary control volume (a fixed window). Considering the analysis of solids and structures, a Lagrangian formulation usually represents a more natural and effective analysis approach than an Eulerian formulation. For example, using an Eulerian formulation of a structural problem with large displacements, new control volumes have to be created (because the boundaries of the solid change continuously [due to large displacements])” (Bathe, 1996). Of course, the transformation of coordinates between these descriptions can easily be accomplished. Of course, the transformation between these descriptions can be done with a simple transformation formulation (Fig. 2.28).

2.4.4

Material Time Derivative in Spatial Coordinates (Substantial Derivative)

The material time derivative is the time derivative with the material coordinates held constant. If we know the complete motion of all particles, x = f(r, t), we can calculate any particle’s velocity simply by taking the partial time derivative with material coordinates r, s, t held constant.

50

2

~v = Gðr, t Þ =

∂ f ðr, t Þ  ∂t



∂x ∂t

 and a  r

Stress and Strain in Continuum

  2 ∂~v ∂ = f ðr, t Þ ∂t r ∂t

The partial time derivative with the material coordinates r, s, t held constant is completely different than the partial time derivative with spatial coordinates x, y, z held constant. In spatial description velocity ~v = gðx, t Þ is given by, hence 

 ∂ ∂~v  gðx, t Þ ∂t x ∂t

And this is different from the acceleration given by 

 ∂ ∂~v  Gðr, t Þ ∂t r ∂t

∂~v

is called the local rate of change of velocity, because spatial coordinates are ∂t x fixed. It is the rate of change of the reading of a velocity meter located at the fixed x, y, z coordinates, which is different than the acceleration of the particle just now passing through the fixed window with x, y, x coordinates. In a steady-state flow, the local rate of change is everywhere zero, but this does not mean that the acceleration of the particle is zero. Even in a steady-state flow, the velocity varies in general from point to point and a particle changes its velocity as it moves from one point of constant velocity to another point of different constant velocity, [like due to changing pipe diameter]. Since the universal laws of motion are written for particle accelerations and not for local rates of change, we need the particle accelerations in the spatial description. This can be accomplished by utilizing the material derivatives. The material time derivative is the time derivative with the material coordinates held constant. However, if the spatial description ~v= gðx ,t) is known but not the to calculate the material referential description ~v = Gðr ,t), we cannot use ∂x ∂t r derivative. We assume that there exists a differentiable and unique function defining the relations between spatial and material coordinates, x(r, t). Then by substituting this function, x(r, t) into the spatial description of the velocity ~v = gðx,t) we obtain, ~v = g½xðr, t Þ, t 

Or in rectangular Cartesian coordinates ~vi = gi ½xðr, s, t, timeÞ, yðr, s, t, timeÞ, zðr, s, t, timeÞ, time where  iis particle   number.  And  by the chain rule ∂xj ∂~vi ∂~vi ∂~vi = ∂t þ ∂xj ∂t (summation on repeated indice j only) ∂t r

x

r

2.4

Kinematics of Continuous Medium

Knowing that ~v =

∂x

∂t r

in Cartesian coordinates ~vj =

51

 

∂xj , ∂t r

substituting it above

in the chain rule yields vi vi vi ai = ð∂~ Þ = ð∂~ Þ þ ~vj ∂~ (summation on repeated indice j only) ∂t r ∂t x ∂xj This equation can be written in matrix notation as a=

d~v ∂~v  þ v~  grad~v dt ∂t

where grad~v is the gradient with respect to the spatial coordinates, (x, y, z) which is sometimes denoted ∇x~v to distinguish it from the gradient ∇r ~v which is with respect to the material coordinates (r, s, t) This material derivative can also be used for any other properties. For example, if the spatial description of density is given by the scalar function ρ(x, t), then the rate of change of the density in the neighborhood of the particle instantaneously at x is given by     ∂ρ dρ = þ v~  gradρ dt r ∂t x   ∂ρ ∂ρ þ ~vj = ∂t x ∂xj The first term

 

∂ρ ∂t x

gives the local rate of change of the density in the neigh-

∂ρ borhood of the point (x, y, z), while the second term ~vj ∂x gives the convective rate of j

change of the density in the neighborhood of a particle as it moves to a place with a different density. Hence we can write the following equation as the material derivative operator:   d ∂ = þ ~v  grad dt ∂t which can be applied to a scalar, a vector, or a tensor function of the spatial position (x, y, z and t). The material derivative has also been described very succinctly, [in MIT 2.20 Marine Hydrodynamics lecture notes], by quantifying the time rate of change of a fluid property as it travels through a given flow field [flow window]. The material derivative is the time derivative of a property x of a particle; hence it is Lagrangian. However, we can write the material derivative in the Eulerian frame of reference. The time rate of change of property, x defined in the Eulerian frame of reference can be given by

52

2

Stress and Strain in Continuum

Fig. 2.26 Material (local), spatial, and referential coordinate system description

df ðxðtÞ,tÞ f ðx þ ~v δt,t þ δtÞ - f ðx,tÞ = lim dt δt δt → 0 Writing Taylor’s series expansion of this variation and considering that ṽδt = δx yields f ðx þ ~vδt,t þ δtÞ = f ðx,tÞ þ δt

∂f ðx,tÞ þ δx  ∇ fðx,tÞ þ higher order terms ∂t

From these two equations, we conclude that the material derivative of a property f as experienced by a particle traveling with a velocity of ṽ is df ∂f = þ ~v ∙ ∇f dt ∂t Generalized notation is given in the following form: d ∂  þ ~v  ∇ dt ∂t ½Lagrangian ½Eulerian

2.5

Rate of Deformation Tensor and Rate of Spin Tensor

53

Fig. 2.27 Relative velocity, d~v, of point A relative to point B

Where r and s are local (material) coordinate system axes that move with the particle. O′ is the origin of the referential (Lagrangian) coordinate system. O is the origin of the spatial description (Eulerian) coordinate system (Fig. 2.26).

2.5

Rate of Deformation Tensor and Rate of Spin Tensor

The rate of deformation tensor, D, is also called the stretching tensor or velocity of strain (strain rate) tensor. The spin tensor, W, is also called the vorticity tensor. We realize that the rate of deformation tensor is defined differently by different authors. We will subscribe to Malvern’s (1969) definition, and use his formulation and derivation with few modifications (Fig. 2.27). The relative velocity components of d~vi of point, A relative to point B can be given by d v~i =

∂~vi dxm ∂xm

or in matrix form ½dv~i  = ½~vi,m ½dxm 

ð2:14Þ

or in tensorial notation d~v = L  dx = dx  LT   where L is Lim = ~vi,m LT im = ~vm,i m and i can have values of 1, 2, and 3. The components of tensor L are spatial gradients of the velocity. Lim = ~vi,m can be written as the sum of a symmetric tensor D, which we will call the rate-of-

54

2

Stress and Strain in Continuum

deformation tensor (also called stretching tensor), and a skew-symmetric tensor W called the spin tensor (also called vorticity) tensor. Hence we can write L=D þ W   1 1 D = L þ LT and W = L - LT 2 2 or in Cartesian coordinates 1 1 Dim = ð~vi,m þ ~vm,i Þ and W im = ð~vi,m - ~vm,i Þ 2 2 W spin matrix is different from the Ω rotation matrix. They should not be confused with each other. The same thing is true, for the rate of deformation tensor D which is not a strain-rate matrix. The rate of deformation tensor, D, allows us to define the relative velocity of point A with respect to point B. As a result, Eq. (2.14) can be given by d~vi = Dim dxm þ W im dxm

2.5.1

Comparison of Rate of Deformation Tensor, D, and Time Derivative of the Strain Tensor, ε_

The small strain tensor ε is defined in terms of local coordinates (material or Lagrangian description) (initial coordinates). Therefore, it is given by   1 ∂ui ∂uj þ Eij = 2 ∂r j ∂r i where ri represents local (material) coordinates [Lagrangian coordinates], ui, represents the amount of deformation in each axis. Therefore, time derivative of the small strain tensor would be given by   dEij 1 ∂~vi ∂~ vj þ = dt 2 ∂r j ∂r i

Since we define the spatial gradient of velocity as follows d ∂u ∂~vi = ð iÞ ∂xj dt ∂xj

2.5

Rate of Deformation Tensor and Rate of Spin Tensor

55

On the other hand, the rate of deformation tensor, D, is given by Dij =

  1 ∂~vi ∂~vj þ 2 ∂xj ∂xi

The time derivative of the strain tensor is with respect to local coordinates (material coordinates-Lagrangian description), while the rate of deformation tensor is defined with respect to spatial coordinates x, y, z (Eulerian description). For small displacement and small strain problems, both tensors are the same. However, D, the rate of deformation tensor is necessary for large displacement and large strain problems. Where i = 1, 2, 3, j = 1, 2, 3. It is important to point out that for indicial notation axes are represented by x, or by xi. However, when convenient x, y, z spatial coordinates and r, s, t are also used throughout the book.

2.5.2

True Strain (Natural Strain) (Logarithmic Strain)

Engineering strain εE is defined by elongation divided by the initial length dEE =

L - L0 L0

in incremental form as dEE =

dL E 1 E = L0 L0

Z

L

dL L0

However, if we use the instantaneous length (new length after deformation) in the denominator, we obtain the true strain (natural or logarithmic strain): dεT =

dL L

Here the increment of true strain is defined by the change in length per unit of instantaneous (new) length. To find the total true strain, we can integrate the increment ZL

ZL dE = E = T

L0

T

L0

    dL L L0 þ ΔL = ln 1 þ EE = ln = ln L L0 L0

56

2

Stress and Strain in Continuum

where ln is the natural logarithm. The relation between true strain and engineering strain can also be written as   T T eE = 1 þ εE or εE = eε - 1 From the Taylor series, we can write the following relation: ex =

1 X x2 x3 xk =1 þ x þ þ þ ⋯... k! 2! 3! k=0

As a result, we can write the following relation between true strain and engineering strain: 2

εT þ

3

εT ET þ þ ⋯:: = εE 2! 3!

In three dimensions for the small strain, this can be generalized as follows: dεTij = Dij dt where Dij is the rate-of-deformation tensor, and dt is the time increment. It is important to point out that increments of natural strain dεTij , are components of a Cartesian tensor; as a result the transformation formulas and principal axis theory all apply. The quantities εTij defined by integration are not components of a Cartesian tensor, because during the deformation principal axes continuously rotate at each increment (Malvern, 1969).

Fig. 2.28 Description of Lagrangian and Eulerian descriptions of a displacement vector

2.6

2.6

Finite Strain and Deformation

57

Finite Strain and Deformation

There are many definitions of finite (large) strain. They can be categorized into two classes: 1. Defining strain with respect to undeformed original configuration and geometry. This approach is called Lagrangian formulation. 2. Defining strain with respect to deformed configuration and geometry. This approach is called the Eulerian formulation. Large (finite) strain formulation is the easiest to define in terms of a deformation– gradient tensor, F. However, the deformation gradient tensor includes the strain tensor and the rotation tensor. As a result, this can make it tricky to be used in material modeling. On the other hand, strain tensor only includes strains where x, y, z are global Cartesian, coordinates (referential description) and r, s, t is the local coordinate (material description) system (Fig. 2.28). In Fig. 2.28 the vector joining point A in the undeformed location and deformed location is the displacement vector. The displacement vector can be defined in the Lagrangian description as follows: uðr, t Þ = cðt Þ þ xðr, t Þ - r If we want to define the displacement vector in the Eulerian description, then we can write uðx, t Þ = cðt Þ þ x - rðx, t Þ where c(t) is the rigid body motion. In the finite element method, the origin of the material (local) coordinate system is usually in the center or at a corner of an element. However, for some special-purpose elements, it is placed at a special location, but always inside the element. Fig. 2.29 Normalized local (material) coordinates system and spatial coordinates

58

2

Stress and Strain in Continuum

The partial derivative of the displacement vector with respect to material coordinates yields the material displacement gradient tensor, ∂x ∂ui = i - δik = F ik - δik ∂r ∂r k where Fik is the deformation–gradient tensor and δik is Kronecker’s delta. If we take the partial derivative of the displacement vector with respect to spatial coordinates, it yields the spatial displacement–gradient tensor ∂uj ∂rj = δjk - F jk- 1 = δjk ∂x ∂xk The deformation gradient tensor, F, describes the stretches and rotations that the material point has undergone from time t0 to t. The deformation equation for a point can be defined in the local (material) coordinate system and can be mapped onto the spatial coordinate system. For example, we can assume the following relationship (Figs. 2.28 and 2.29), between the material and spatial coordinates, x = xA þ ðxB - xA Þr y = yA þ ðyB - yA Þr we will derive the finite strain formulation using Malvern’s (1969) formulation and notation with few changes if any. We will define deformation–gradient tensor F as a tensor whose components are the partial derivatives with respect to material coordinates in the Lagrangian formulation. F, deformation–gradient tensor is defined in terms of the undeformed configuration. x, y, z spatial coordinates are represented by bold x. Local (material) coordinates r, s, t are represented by bold r. The relation between the spatial and local (material) description deformations is given by employing the deformation–gradient tensor F, which is defined as the tensor whose rectangular Cartesian components are the partial derivatives ∂xi=∂ri and which operates on an arbitrary infinitesimal material vector dr located at r, to associate it with a vector dx located at point x as follows: dx = F  dr or dx = dr  FT

In indicial matrix notation

∂xi  dr j fdxi g = ∂r j

2.6

Finite Strain and Deformation

59

Therefore 2

∂x 6 ∂r 6 6 ∂y

 ∂xi =6 F ij = 6 ∂r ∂r j 6 4 ∂z ∂r

∂x ∂s ∂y ∂s ∂z ∂s

3 ∂x ∂t 7 7 ∂y 7 7 ∂t 7 7 ∂z 5 ∂t

In Fij the first index identifies the row, and the second index identifies the column. In Fig. 2.28 we see that the deformation is the difference between the position vector before and after deformation, where c is the rigid body motion. 2

x=r þ u

∂u 6 ∂r 6 6 ∂v F=6 6 ∂r 6 4 ∂w ∂r

∂u ∂s ∂v ∂s ∂w ∂s

3 ∂u ∂t 7 7 ∂v 7 7þI ∂t 7 7 ∂w 5 ∂t

where I is the third-order unity matrix. When Eulerian description is used, and deformation gradient is defined with respect to deformed configuration (the spatial deformation gradient F-1) relations between spatial coordinate x and local (material) coordinates r are given by dr = F - 1  dx or in indicial matrix notation fdr i g =

∂r i dxj ∂xj

From the last equation, we can infer that the spatial deformation gradient, F-1 at point x is the inverse of the material deformation–gradient tensor F, Therefore, F  F-1 = I can be written between the material deformation–gradient tensor and the spatial deformation–gradient tensor, where I is the unit matrix.

60

2

Stress and Strain in Continuum

Fig. 2.30 Axial strain definition in one-dimensional line element

Strain is a human construct. It is not a real physical quantity that can be measured directly, like displacement or time. It is defined to be able to formulate the response of a continuum using mathematics. Because strain is not a direct physical quantity, it can be defined in many ways. For large strain formulation the strain will be defined as one-half the change in squared length of the local (material) vector AB as follows (Fig. 2.30): "  " # " 2 # #     2 0 2 2 2 1 jAB þ dr j - jABj 1 2drAB þ dr  1 AB j - jAB  dr  1  dr 2 = = =  þ   2 2 2 2 2 AB 2 AB 2 jABj jABj jABj

Axial strain = Er þ

1 2 E 2 r

For Lagrangian formulation (referential description) in three dimensions, the strain will be defined as ðdsÞ2 - ðdSÞ2 = 2dr  E  dr where dS = AB is the initial undeformed length and ds = AB′ is the deformed length (Fig. 2.30). In indicial notation the last equation can be given by ðdsÞ2 - ðdSÞ2 = 2dr i Eij dr j

For Eulerian formulation (spatial description) ðdsÞ2 - ðdSÞ2 = 2dx  E  dx in indicial notation ðdsÞ2 - ðdSÞ2 = 2dxi Eij dxj

2.6

Finite Strain and Deformation

61

Example 2-5: Deformation–Gradient Tensor In the following examples, it is assumed that the relation between material (local) coordinates and spatial coordinates are given by linear interpolation functions.

For the quadrilateral element shown above material (local) coordinates of the nodes [1, 2, 3, 4] are given as follows: r 1 = 1,s1 = 1,r 2 = - 1,s2 = 1,r 3 = - 1,s3 = - 1,r 4 = 1,s4 = - 1 Nodal linear interpolation functions are given by Ni =

1 ð1 þ r i r Þð1 þ si sÞ 4

where i is the node number and ri, and si are the local coordinates of node i. Hence, we can define the relation between spatial and material coordinates by xðr, sÞ =

Xi = 4

N x , yðr, sÞ = i=1 i i

Xi = 4 i=1

N i yi

Pure Stretching Deformation–Gradient Tensor Pure stretching in the x and y directions. Assume 100% elongation in the x-direction and a 50% elongation in the y-direction.

62

2

xðr, sÞ =

yðr, sÞ =

Stress and Strain in Continuum

1 1 ð1 þ r Þð1 þ sÞ 4 þ 0 þ 0 þ ð1 þ r Þð1 - sÞ4 = 2 þ 2r 4 4

1 1 3 3 ð1 þ r Þð1 þ sÞ 3 þ ð1 - r Þð1 þ sÞ 3 þ 0 þ 0 = þ s 4 4 2 2

The deformation–gradient tensor is 3 ∂x

∂s 7 = 2 0 5 ∂y 0 1:5 ∂s

2

∂x 6 ∂r F=4 ∂y ∂r

Simple Shear Deformation–Gradient Tensor

1 1 ð 1 þ r Þ ð 1 þ sÞ 2 þ 0 þ 0 þ ð 1 þ r Þ ð 1 - sÞ 2 = 1 þ r 4 4 1 1 1 3 r yðr, sÞ = ð1 þ r Þð1 þ sÞ 3 þ ð1 - rÞð1 þ sÞ 2 þ 0 þ ð1 þ r Þð1 - sÞ 1 = þ þ s 4 4 4 2 2 xðr, sÞ =

The deformation–gradient tensor is 2

∂x 6 ∂r F=4 ∂y ∂r

3 ∂x

∂s 7 = 1:0 0:0 5 ∂y 0:5 1:0 ∂s

The non-zero off-diagonal terms reflect shear, and 1.0 diagonal terms reflect zero normal strain.

2.6

Finite Strain and Deformation

63

Pure Shear Deformation–Gradient Tensorr

xðr, sÞ =

1 1 1 3 1 ð1 þ r Þð1 þ sÞ 3 þ ð1 - r Þð1 þ sÞ 1 þ 0 þ ð1 þ r Þð1 - sÞ2 = þ r þ s 2 2 4 4 4

yðr, sÞ =

1 1 1 3 r ð1 þ r Þð1 þ sÞ 3 þ ð1 - r Þð1 þ sÞ 2 þ 0 þ ð1 þ r Þð1 - sÞ 1 = þ þ s 4 4 4 2 2

The deformation–gradient tensor is 2

∂x 6 ∂r F=4 ∂y ∂r

3 ∂x ∂s 7 = 1:0 5 ∂y 0:5 ∂s

0:5 1:0

The non-zero off-diagonal terms reflect shear, and 1.0 diagonal terms reflect zero normal strain.

2.6.1

Green Deformation Tensor, C, Cauchy Deformation Tensor, B-1

Green deformation tensor C and Cauchy deformation tensor B-1 are related to the strain tensors. Instead of giving one-half the change in squared length per unit squared initial length, Green deformation tensor, C, refers to the undeformed configuration, and its tensor entries provide the new squared length (ds)2 of the element into which the given initial vector dr is deformed, while the Cauchy deformation tensor, B-1, gives the initial squared length (dS)2 of a vector dx defined in the deformed configuration. As a result, we can establish the Green deformation, C tensor, C, as follows: ðdsÞ2 = dr  C  dr

64

2

Stress and Strain in Continuum

or in indicial notation as ðdsÞ2 = dr i Cij dr j In the same manner Cauchy deformation tensor B-1 can be given by ðdSÞ2 = dx  B - 1  dx or in indicial notation as  -1 ðdSÞ2 = dxi Bij dxj comparing Lagrangian strain tensor, E, and Green deformation tensor, C, we observe that 2E = C - I or 2Eij = C ij - δij In the same manner, comparing Eulerian strain tensor E and Cauchy deformation tensor B-1 we observe that  -1 2E = I - B - 1 or 2E ij = δij - Bij There is no special reason for defining the Cauchy deformation tensor as B-1. It could be named just about any character. However, we are following the notation used by Cauchy in 1827. In the formulation given above, both the Green deformation tensor, C, and Cauchy deformation tensor, B-1, reduce to unit tensor when strain is zero (Malvern, 1969).

2.6.2

Relation Between Deformation, Strain, and Deformation–Gradient Tensors

The square of the new length (ds)2 can be written as ðdsÞ2 = dx  dx where dx is the vector in the deformed configuration. On the other hand relation between the dx and the undeformed vector dr in the local (material) coordinates is given by dx = F  dr

2.6

Finite Strain and Deformation

65

Hence, we can write

   ðdsÞ2 = dr  FT ðF  drÞ = dr  FT  F  dr Similarly, the square of the original length (dS)2 can be written as ðdSÞ2 = dr  dr Because dr = F-1  dx, we can write h h   T i - 1 T  i ðdSÞ2 = dx  F - 1  F  dx = dx  F - 1 F - 1 dx Let’s compare these later equations with our strain definition equations where Green deformation tensor C is given by ðdsÞ2 = dr  C  dr Therefore, the Green deformation tensor is also defined by C = FT  F and Cauchy deformation tensor is defined by ðdSÞ2 = dx B - 1 dx Therefore, we can write B-1 =

h

F-1

T  - 1 i  F

in indicial notation, we can write the following relation: C ij =

∂xk ∂xk ∂r i r j

and Bij- 1 =

∂r k ∂r k ∂xi ∂xj

Summation on repeated indices applies in both of these equations.

66

2

Stress and Strain in Continuum

The Lagrangian strain tensor can then be obtained as follows: 2E = C - I C = FT  F

Since E=

 1 T F F-I 2

or indicial notation

1 ∂xk ∂xk Eij = - δij 2 ∂r i ∂r j and Eulerian strain tensor can be given by 2E = I - B - 1 h T  i B-1 = F-1  F-1

Since E =

h  T  i 1 I - F-1  F-1 2

In indicial notation E⋆ ij



1 ∂r k ∂r k = δ 2 ij ∂xi ∂xj

Both Green deformation tensor C and Lagrangian strain tensor E are symmetric tensors. Therefore, they both have three eigenvalues (principal values) in Eigen directions (principal directions). Also, principal directions of C and E coincide, because in principal directions off-diagonal terms are zero in Green deformation tensor C. Then they have to be zero in strain tensor E. Of course, the same arguments can be made between Cauchy deformation tensor B-1 and Euler strain tensor E. However, principal (Eigen) directions of [B-1 and E] and [C and E] will not coincide.

2.6.3

Comparing Small Strain and Large (Finite) Strain

For the sake of simplicity, we will assume that material (local) coordinate axes and spatial coordinate axes are parallel (Fig. 2.31). xi = spatial coordinate, ri = local (material) coordinate. Displacement ui is expressed in local (material) coordinates, ui = ui(r, s, t, time).

2.6

Finite Strain and Deformation

67

Fig. 2.31 Spatial and local (material) coordinates

We can define the Lagrangian strain as follows:

1 ∂xk ∂xk - δij Eij = 2 ∂r i ∂r j Substituting xi = ri + ui where ui = ui(r, s, t, time)

1 ∂ui ∂uj ∂uk ∂uk þ þ 2 ∂r j ∂r i ∂r i ∂r j  

 1 ∂u ∂u ∂v ∂v ∂w ∂w 1þ þ E 11 = 1þ þ -1 2 ∂r ∂r ∂r ∂r ∂r ∂r Eij =

or ∂u 1 E11 = þ ∂r 2

"   2  2 # 2 ∂u ∂v ∂w þ þ ∂r ∂r ∂r

Similarly "

 2  2 # ∂v ∂w þ þ ∂s ∂s "   2   2 # 2 ∂w 1 ∂u ∂v ∂w þ þ E 33 = þ 2 ∂t ∂t ∂t ∂t ∂v 1 E 22 = þ ∂s 2

∂u ∂s

2

68

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Stress and Strain in Continuum

off-diagonal (shear-strain terms) E 12 =

1 2

 1þ

  

∂u ∂u ∂v ∂v ∂w ∂w þ þ 1þ -0 ∂r ∂s ∂r ∂s ∂r ∂s

or   1 ∂u ∂v 1 ∂u þ þ E12 = 2 ∂r 2 ∂s ∂r   1 ∂u ∂w 1 ∂u E13 = þ þ 2 ∂t ∂r 2 ∂r   1 ∂v ∂w 1 ∂u E 23 = þ þ 2 ∂t ∂s 2 ∂s

∂u ∂v ∂v ∂w ∂w þ þ ∂s ∂r ∂s ∂r ∂s



∂u ∂v ∂v ∂w ∂w þ þ ∂t ∂r ∂t ∂r ∂t

∂u ∂v ∂v ∂w ∂w þ þ ∂t ∂s ∂t ∂s ∂t

In indicial notation rj refers to r, s, t local coordinates; ui refers to u, v, w displacement vectors in x, y, z spatial coordinates. The first term in large (finite) strain formulation gives the small strain formulation. In the same way, the Euler strain formulation can be given by Eij =



1 ∂ui ∂uj ∂uk ∂uk þ 2 ∂xj ∂xi ∂xi ∂xj

Again ui represents u, v, w for subscripts 1,2,3. xi represents x, y, z spatial coordinates for 1, 2, and 3. The only difference between Lagrangian strain tensor and Eulerian strain tensor is the fact that in Lagrangian strain tensor all derivatives are with respect to local (material) (undeformed) coordinates; on the other hand Eulerian strain tensor components are with respect to spatial (deformed) coordinates. For small displacement and small strain cases, the difference between the Lagrangian strain and Eulerian strain is small. Example 2.6 One corner, node 1, of the thin, t = 1 cm, steel plate shown below is deformed. All other three corners are fixed. Calculate the Lagrangian strain tensor, E, and the Eulerian strain tensor, E for the element, as a function of material (local) coordinates using the deformation–gradient matrix, F. Assume node 3 coordinates are y = 1 cm, x = 1 cm (Fig. 2.6E). We assume that deformation has a linear distribution in the element. As a result, the relations, between the material (local) coordinates and the spatial coordinates, can be given by the following relations:

2.6

Finite Strain and Deformation

69

Fig. 2.6E Rectangleshaped thin plate elementt

1 1 1 ð1 þ rÞð1 þ sÞ 4 þ ð1 - r Þð1 þ sÞð1Þ þ ð1 - r Þð1 - sÞð1Þ 4 4 4 1 þ ð1 þ r Þð1 - sÞ ð3Þ 4 1 1 1 y = ð1 þ r Þð1 þ sÞ ð3:5Þ þ ð1 - r Þð1 þ sÞ ð3Þ þ ð1 - r Þð1 - sÞ ð1Þ 4 4 4 1 þ ð 1 þ r Þ ð 1 - s Þ ð 1Þ 4 x=

Lagrangian strain tensor E is given by E=

 1 T F ∙F-I 2

Eulerian strain tensor E is given by E =

h  T  i 1 I - F-1 ∙ F-1 2

Deformation–gradient matrix F is given by 3 ∂x ∂s 7 5 ∂y ∂s

2

∂x 6 ∂r F=4 ∂y ∂r 2

x=u þ r

∂u 6 ∂r 6 F=6 4 ∂v ∂r

3 ∂u ∂s 7 7 7þI ∂v 5 ∂s

70

2

x=

X4

xN i=1 i i

y=

X4

yN i=1 i i

∂x X4 ∂N i x = i = 1 i ∂r ∂r ∂x X4 ∂N i x = i = 1 i ∂s ∂s

u=

Stress and Strain in Continuum

X4

uN i=1 i i

v=

X4

vN i=1 i i

∂y X4 ∂N i y = i = 1 i ∂r ∂r ∂y X4 ∂N i y = i = 1 i ∂s ∂s

where xi and yi are coordinates of a point after deformation. As a result, deformation–gradient matrix F is F=



1 ð10 þ 2sÞ ð2 þ 2r Þ 8 ð 1 þ sÞ ð9 þ r Þ

Note that deformation–gradient matrix F is a function of material coordinates, r and s. Because strain is a property of a point, we use material coordinates of the point to define the strain at that point. Then we can calculate our strain tensors using the following relations. Lagrangian strain tensor is given by E=

 1 T F  F-I 2

Eulerian strain tensor is given by E =

h  T  i 1 I - F-1  F-1 2

We can find strain at any point by substituting the local coordinates of the point r = ri, s = si, for example, if we want to know the strain at node 1 where r = 1 and s = 1, we substitute these coordinates for r and s.

2.6.4

Strain Rate and Rate of Deformation Tensor Relations

In Lagrangian formulation strain rate tensor E_ is given by d d d ðdsÞ2 - ðdSÞ2 = ð2dr  E  drÞ dt dt dt dr and dS are constant with respect to time because they represent the initial dimensions. Therefore, we can write dE d  dr ðdsÞ2 = 2dr  dt dt

2.6

Finite Strain and Deformation

71

On the other hand, the rate of deformation tensor D is given by d ðdsÞ2 = 2dx  D  dx dt Also, we have already shown that dx = F  dr. Hence, we write the following relation: d ðdsÞ2 = 2ðdrFT Þ  D  ðF  drÞ dt = 2dr  ðFT  D  FÞ  dr subtracting the strain rate equation from the rate of deformation   dE  dr - 2dr  FT  D  F  dr = 0 dt

 dE  T 2dr  - F  D  F  dr = 0 dt

2dr 

of course, dr = 0 is the trivial solution. Hence the following relation must be satisfied: dE = FT  D  F dt must be satisfied or in indicial notation dE ij ∂xm ∂x Dmn n = dt ∂r i ∂r j The relationship between Green deformation rate tensor C and strain rate tensor E_ can be given by dC dE =2 dt dt When displacement–gradient components are small compared to unity, the strain rate is equal to the rate of deformation. dE =D dt in indicial notation dE ij = Dij dt

72

2

2.6.5

Stress and Strain in Continuum

Relation Between, the Spatial Gradient of Velocity Tensor, L and the Deformation–Gradient Tensor, F

The definition of deformation–gradient tensor F is given by or

dx = F  dr dxi =

∂xi dr j ∂r j

The rate of change of the deformation gradient is F_ . The spatial gradient of velocity tensor L is given by dv = L  dx Hence Lij = ~vi,j = If we write a time derivative of F =

∂xi ∂r j

∂~vi ∂xj

we obtain

  _F = d ∂xi = ∂_xi = ∂_xi ∂xm dt ∂r j ∂r j ∂xm ∂r j    ∂r j _F = ∂vi ∂xm and since ∂vi = d ∂xi ∂xm ∂r j ∂xm dt ∂r j ∂xm F_ = L  F and L = F_  F - 1 Relations between Euler strain rate tensor E, the rate of deformation tensor D, and the spatial gradients of velocity tensor L can be given by    E_ = D - E  L þ LT  E Proof of this equation is provided by Malvern (1969) on page 163.

2.7

Rotation and Stretch Tensors in Finite Strain

When strain is finite (large), the symmetric and skew-symmetric parts of the displacement–gradient matrix, J, cannot be represented by additive decomposition of a pure strain matrix and a pure rotation matrix. However, other types of

2.8

Compatibility Conditions in Continuum Mechanics

73

multiplication decompositions are possible, where one of the two tensors will represent a rigid body rotation and the second tensor will be a symmetric positivedefinite.

2.8

Compatibility Conditions in Continuum Mechanics

When the displacements are known, the strain field can easily be calculated. For the case of small strain, relations are given by Eij =

  1 ∂ui ∂uj þ 2 ∂r j ∂r i

where ui represent displacement vectors u, v, w, and ri represents local (material) coordinate axes r, s, t. There are nine strain equations, but because of symmetry only six of these are linearly independent. For any given strain field to be admissible, certain compatibility conditions that guarantee the continuum character of the medium must be satisfied. This requirement is also due to mathematics. If there are six known strain equations, it is not possible to obtain three unknown displacement components as unique values. The number of unknowns and the number of linearly independent equations must be the same. St. Venant’s compatibility equations must be satisfied by the six strain equations to find unique displacement values. In a three-dimensional solid mechanics boundary value problem, St. Venant’s compatibility equation provides us with six equations. There are also three force equilibrium equations. In addition, there are six stress-strain constitutive relations. As a result, in total there are 15 equations and 12 unknowns (6 stresses and 6 strains). However, only three of the compatibility equations are linearly independent. Hence, we have 12 equations and 12 unknowns. However, if the displacements are unknown then compatibility equations are not needed, because there are 15 equations (6 strain-displacement relations, 6 stressstrain constitutive relations, and 3 force equilibrium equations). On the unknown side, there are 3 unknown displacements, 6 unknown stresses, and 6 unknown strains. Therefore, the number of equations is equal to the number of unknowns. In most computational solid mechanics boundary value problems, usually, the nodal displacements are the primary unknowns, as in the displacement-based finite element method. As a result, we do not need the compatibility equations. However, in force-based analysis methods, compatibility functions become necessary. In summary, when displacements are not explicitly retained as the primary unknowns, the compatibility conditions (this should be called the compatibility of the displacement field) are needed to make sure that the strain field yields a continuous displacement field that has a single displacement value at any point in the continuum (Fig. 2.32).

74

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Stress and Strain in Continuum

Fig. 2.32 Straindisplacement relation

When displacements and strains are small, the distinction between referential and spatial descriptions is negligible for small strain calculations. Assuming small strain and small deformations, we can define small strain in spatial x, y, z coordinate system as follows: Exx =

∂u ∂x

Eyy =

∂v ∂y

Ezz =

∂w ∂x

  1 ∂u ∂v þ 2 ∂y ∂x   1 ∂u ∂w Exz = þ 2 ∂z ∂x   1 ∂v ∂w Eyz = þ 2 ∂z ∂y Exy =

2

∂ E

If we assume that ∂x∂yxy exists " # 2 3 3 ∂ Exy 1 ∂ u ∂ v þ = ∂x∂y 2 ∂2 y∂x ∂x2 ∂y we can also write second derivatives of Exx and Eyy and sum them up 2

2 3 3 ∂ v ∂ Exx ∂ Eyy ∂ u þ = 2 þ 2 2 2 ∂y ∂x ∂y ∂x ∂x ∂y

2.8

Compatibility Conditions in Continuum Mechanics

75

Hence 2

2

2 ∂ Exy ∂ Exx ∂ Eyy þ =2 2 2 ∂y ∂x ∂x∂y

We can repeat this process for Exz and Eyz. Then we have "

# 2    2 2  ∂ Exy ∂2 Ezx 1 ∂ ∂u ∂v ∂ ∂w ∂u þ þ = þ þ 2 ∂x∂z ∂y ∂x ∂x∂z ∂y∂x ∂y∂x ∂x ∂z 3

3 3 3 ∂ w ∂ u 1 ∂ u ∂ v = þ þ þ 2 ∂x∂y∂z ∂x2 ∂y ∂x2 ∂y ∂x∂y∂z   2  2  ∂ ∂u ∂v ∂w 1 ∂ þ 2 = þ 2 2 ∂y∂z ∂x ∂x ∂y ∂y " # 2 2 1 ∂ Exx ∂ Eyz = : 2 þ 2 ∂y∂z ∂x2   2 ∂ ∂Exy ∂ 1 ∂Eyz ∂ Exx = þ ∂y∂z ∂x ∂z ∂y 2 ∂x

A similar process can be repeated for other shear strain pairs Eyz and Exy, Eyz and Exz. These equations prove the integrability of the strain field. Based on this derivation the following six St. Venant’s compatibility equations can be given: 2

2

2 ∂ Exy ∂ Exx ∂ Eyy þ -2 =0 2 2 ∂y ∂x ∂x∂y 2

2

∂ Eyz ∂ Eyy ∂2 Ezz þ -2 =0 2 2 ∂z ∂y ∂y∂z 2

2

2

∂ Ezx ∂ Ezz ∂ Exx =0 þ -2 ∂x2 ∂z2 ∂z∂x   2 ∂Eyz ∂Ezx ∂Exy ∂ Exx ∂ =0 þ þ þ ∂y∂z ∂x ∂x ∂y ∂z   2 ∂ Eyy ∂ ∂Eyz ∂Ezx ∂Exy =0 þ þ ∂z∂x ∂y ∂x ∂y ∂z   2 ∂ ∂Eyz ∂Ezx ∂Exy ∂ Ezz þ þ =0 ∂x∂y ∂z ∂x ∂y ∂z

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Stress and Strain in Continuum

These six compatibility equations must be satisfied to ensure a compatible strain field and to ensure that only a single-valued displacement field exists at any point. Therefore, the compatibility conditions are necessary and sufficient conditions for unique displacement values. While there are six compatibility equations, however only three of them can be linearly independent because they are obtained from three independent displacements u, v, w.

2.9

Piola–Kirchhoff Stress Tensors

Cauchy stress tensor is defined in the spatial coordinates x, y, z (in the deformed configuration). Euler strain is also defined in spatial position in the deformed configuration. Therefore, using the Euler strain tensor definition with the Cauchy stress tensor is appropriate. On the other hand, the Lagrangian formulation is defined in material coordinates r, s, t (in the undeformed configuration) or any other reference state configuration that does not change. In this case, it is also more appropriate to define the Piola– Kirchhoff stress tensor in local (material) reference coordinates. Two Piola–Kirchhoff stress definitions are using the undeformed (or any other stationary reference) state as the basis for formulation.

2.9.1

First Piola–Kirchhoff Stress Tensor σ0

The First Piola–Kirchhoff stress tensor is also called the Lagrangian stress tensor. The derivatives are defined with respect to material (local) coordinates. However, this stress tensor is not symmetric, even when there is no distributed body moment (non-polar case) in the system. Stress is defined as a force per unit undeformed area. Keep in mind that stress is a human construct not a directly measurable physical quantity. Thus, it is possible to define it as we please. The first Piola–Kirchhoff stress tensor is defined by the following equation (Fig. 2.33): 

 dF n0  σ 0 = dA0

where σ 0 is the first Piola–Kirchhoff stress tensor and dF is the actual force acting on the deformed configuration dA; however, the force is divided by the initial undeformed area dA0.

2.9

Piola–Kirchhoff Stress Tensors

77

Fig. 2.33 Piola–Kirchhoff stress definition

dF can also be written using Cauchy stress tensor σ as dF = (n  σ)dA. Hence we can write the following equation between the first Piola–Kirchhoff stress tensor and Cauchy stress tensor, ðn0 σ 0 ÞdA0 = dF = ðn  σ ÞdA

2.9.2

Second Piola–Kirchhoff Stress Tensor σ~

~ that is related to the Second Piola–Kirchhoff stress tensor, σ~ uses a pseudo-force d F actual force dF by the inverse of the deformation–gradient tensor F-1. ~ = F - 1  dF dF Remember that material (local) coordinates and spatial coordinates are also related by dr = F - 1  dx Then we can write ðn0  σ~Þ =

~ dF dA0

Using the relation, we defined above, we can write ðn0  σ~Þ =

F - 1  dF dA0

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At the same time, dF can be also defined in terms of Cauchy stress tensor as  T F - 1  ðn  σ ÞdA ðn  σ Þ  F - 1 dA ðn0  σ~Þ = = dA0 dA0 The Piola–Kirchhoff stress tensors are sometimes called pseudo-stress tensors.

2.10

Direct Relation Between Cauchy Stress Tensor and Piola–Kirchhoff Stress Tensors

The following relation can be given: σ~ =

 T  T ρ0 - 1 F  σ  F - 1 and σ~ = σ 0  F - 1 ρ

Details of the long derivations for these relations are provided by Malvern (1969). The second Piola–Kirchhoff stress tensor is usually preferred in finite-strain elasticity problems. Newtonian equation of motion in undeformed (reference) state using the first Piola–Kirchhoff stress tensor is given by Z

Z n0  σ dA þ A0

Z ρ0 b0 dV =

0

V0

ρ0

d2 x dV dt 2

V0

Transforming the surface integral to volume integral by the Gauss theorem (divergence theorem) in indicial notation, we can write ∂σ 0ij d2 x þ ρ0 b0j = ρ0 2i dt ∂r i where ri is the material (reference) coordinate system axis. Since the relation between the first and second Piola–Kirchhoff stress tensors is given by σ 0 = σ~  FT or in indicial notation σ0ij = ˜σik xi,k

2.11

Conservation of Mass Principle

79

by substituting these equations in the equilibrium equations given above, we obtain the equation of motion in terms of the second Piola–Kirchhoff stress tensor: ½˜σij xk,j i þ ρ0 b0k = ρ0

2.11

d 2 xk dt 2

Conservation of Mass Principle

The rate of change in mass is given by ∂m = ∂t

Z

∂ρ dV ∂t

V

where ρ is the material mass density, which can be a function of x, y, z coordinates and time. The flux is the rate of mass flowing through an area dA with a velocity of ~vn = ~v  n can be given by Z

Z ρ~vn dA =

A

ρ~v  n dA A

Using the divergence theorem (Gauss’s theorem) the integral over a closed surface can be converted to an integral over a volume bounded by the closed surface by the following equation: Z

Z n × ~v dA =

— × ~vdV V

A

where n is the vector normal to the surface, ~v is velocity vector — × ~v is given by    i j k   ∂ ∂ ∂   ~ —×v=   ∂x ∂y ∂z   ~v ~vy ~vz  x where i, j, k are the unit vectors in Cartesian coordinates x, y, z. Hence, Z

Z ρ~v  n dA =

A

—  ðρ~vÞ dV V

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Stress and Strain in Continuum

The time rate of change in mass is defined by Z V

∂ρ dV ∂t

Equating the time rate of change in mass to the rate of inflow through area A yields Z

∂ρ dV = ∂t

V

Z —  ðρ~vÞ dV V

The negative on the right-hand side is because normal of the surface is defined as positive outward. Therefore, inflow is in the negative normal direction. Hence, we can write

Z ∂ρ þ —  ðρ~vÞ dV = 0 ∂t V

Here the integral must be zero for an arbitrary volume. Therefore the integrand must be equal to zero. ∂ρ þ —  ðρ~vÞ = 0 ∂t This equation is called the continuity equation, as a result of the conservation of mass principle. Or in indicial notation, it can be given as

∂ρ ∂~vx ∂~vy ∂~vz =0 þ þ þρ ∂x ∂y ∂z ∂t

2.12

The Incompressible Materials

When the material is incompressible, we assume that density does not change over time, dρ =0 dt

2.13

Conservation of Momentum Principle

81

Therefore ∂~vx ∂~vx ∂~vz þ þ =0 ∂x ∂y ∂z must be satisfied. It is important to point out that in the theory of plasticity during plastic deformation (flow), some materials, like metals, are considered to be incompressible.

2.13

Conservation of Momentum Principle

The equilibrium of forces equation is obtained by using the conservation of momentum principle. This principle applies to a collection of particles as well as a continuous medium. This principle states that the vector sum of all the external forces acting on the free-body is equal to the rate of change of the total momentum (Malvern, 1969).

Force is the time derivative of momentum; therefore the concept of force is redundant and subordinate to the conservation of momentum. In this section, we will derive the equilibrium of forces from the conservation of momentum. According to the conservation of momentum principle for a collection of particles the time rate of change of the total momentum of a given set of particles equals the vector sum of all the external forces acting on the particles is considered to be the modern interpretation of Newton’s second law. It is assumed that Newton’s third law of action and reaction and Hooke’s law govern the internal forces. The continuum form of conservation of momentum principle is the basis for the Newtonian continuum mechanics. In this book, the term “Newtonian Mechanics” refers to all mechanics formulations based on Newton’s universal laws of motion. Consider the cube shown in Fig. 2.34, occupying volume V and bounded by surface A. There is an external surface loading vector σ (n), per unit area on any arbitrary plane, and b is the body force per unit mass. Fig. 2.34 Conservation of momentum

82

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Stress and Strain in Continuum

Momentum, p, is defined by Z ρ ~v dV

p=

where ρ is mass density, ~v is velocity vector, and dV is the volume. The time rate of change of the total momentum of a portion of the particles can be given by the material derivative of the integral, which is defined as the time rate of change of any quantity for a portion of material. Hence, the time rate of change of the total momentum is dp d = dt dt

Z ρ~vdV

External forces acting on the particles can be given by the summation of surface tractions and body forces Z

σ ðnÞ dA þ A

Z ρbdV V

According to the conservation of momentum principle, the time rate of change of the total momentum equals the vector sum of all the external forces Z

σ ðnÞ dA þ

Z ρbdV =

A

V

d dt

Z ρ~vdV

Using a Cartesian coordinate system where x, y, and z are represented by indices i. Z

ðnÞ

Z

σ i dA þ A

ρbi dV = V

d dt

Z ρ~ vi dV

Components of external surface loading vector σ (n) can be given by ðnÞ

σ i = σ ij nj Or in Cartesian coordinates σ ðxnÞ = σ xx nx þ σ xy ny þ σ xz nz σ ðynÞ = σ yx nx þ σ yy ny þ σ yz nz σ ðznÞ = σ zx nx þ σ zy nz þ σ zz nz

2.13

Conservation of Momentum Principle

83

where nx,nx,nx, are direction cosines between the normal to the arbitrary plane and spatial coordinates. In matrix notation, this can also be written as n o σ ðnÞ = fng½σ  2 σxx n oT n oT 6 ðnÞ ðnÞ ðnÞ ðnÞ = σx σy σz σ fng = fnx ny nz g ½σ = 4 σyx σzx

σxy σyy

3 σxz 7 σyz 5

σzy

σzz

[σ] is the second-order Cauchy stress tensor, which is a linear vector function. Using the divergence theorem, we can write the following conversion: Z

ðnÞ σ i dA =

Z

A

∂σ ji dV V ∂xj

Meanwhile, due to the conservation of mass material time derivative of volume integral can be written as Malvern (1969): d dt

Z

Z ρ~vi dV = V

ρ

d~vi dV dt

Substituting the last two relations in the conservation of momentum equation yields  Z Z  ∂σ ji d~v þ ρbi dV = ρ i dV V ∂xj V dt Hence the conservation of momentum principle in Newtonian mechanics takes the final form in the following equation:  Z  ∂σ ji d~vi dV = þ ρbi - ρ dt V ∂xj For any arbitrary volume, we can write Cauchy’s equation of motion in Newtonian mechanics as follows: ∂σ ji d~v þ ρbi = ρ i dt ∂xj Of course, for static equilibrium right-hand side of this equation is equal to zero. Hence ∂σ ji þ ρbi = 0 ∂xj

84

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Stress and Strain in Continuum

In any other Cartesian coordinate system rotated with respect to the original coordinate system stress tensor can be defined using the tensor transformation equations. ½σ  = ½N T ½σ ½N  where [N] is the matrix of direction cosines nik = cos ðxr , xi Þ of the angles between the axes of the original and rotated coordinate systems. Superscript T is used to denote transpose. A bold character indicates a matrix or a vector. Sometimes brackets are used to distinguish a matrix from a vector.

2.14

Conservation of Moment of Momentum Principle

The time rate of change of the total moment of momentum for a collection of masses is equal to the vector sum of the moments of the external forces acting on these masses. Assuming there are no distributed moment couples, we can write d dt

Z

Z  Z  ðnÞ ðr × ρ~vÞdV = r×σ dA þ ðr × ρbÞdV

v

V

A

using vector product definition a × b = eijr aj br = ejri aj br we can re-write the conservation of momentum of moment principle in indicial notation as follows: d dt

Z

Z eijr xj ρ~vr dV =

V

eijr xj σ ðrnÞ dA

Z þ

eijr xj br ρdV V

A

We have defined σ ðrnÞ previously for an arbitrary direction as ðnÞ

σ i = σ ij nj We can also transform the surface integral to a volume integral using the relation given by the divergence (Gauss) theorem Z

Z eijk nj~vk dA = A

eijk V

∂ ð~vk ÞdV ∂xj

2.14

Conservation of Moment of Momentum Principle

85

where ρ is the mass density of the material per unit volume. We can transform the moment of momentum principle into the following form: Z

d  eijr x ~v ρdV = dt j r

V

Z V

 

∂ xj σ kr eijr þ xj ρbr dV ∂xk

The time derivative of displacement is velocity; hence we can substitute ~vi = Z V

dxi dt

 

  Z d~v ∂σ kr eijr ~vj~vr þ xj r ρdV = eijr xj þ ρbr þ σ kr δjk dV dt ∂xk V

eijr ~vj~vr = 0 because ~vj~vr is symmetric for indices jr, while eijr is anti-symmetric. Knowing that Cauchy’s equation of motion is given by d~v ∂σ kr þ ρbr = ρ r dt ∂xk Substituting this equilibrium equation of motion above yields Z eijr σ kr δjk dV = 0 V

which is equal to Z eijr σ jr dV = 0 V

However, the last equation is equal to zero. Because for an arbitrary volume, V, at each point eijr σ jr = 0 for • i = 1: σ 23 = σ 32 • i = 2: σ 31 = σ 13 • i = 3: σ 12 = σ 21 This proves that the stress tensor is symmetric when there are no couple (moment) stresses. This proof also establishes the symmetry of the stress tensor in general without any assumption of equilibrium or uniformity of the stress distribution.

86

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2.15

Stress and Strain in Continuum

Lagrangian Mechanics

In this section, we intend to introduce Lagrangian mechanics briefly. Lagrangian mechanics introduced in 1788 by Joseph-Louis Lagrange is a different formulation of classical Newtonian mechanics. Lagrangian mechanics allows the inclusion of non-conservative forces. However, it does not establish any additional laws of mechanics in addition to Newton’s. Lagrangian mechanics uses energy as the variable rather than forces. The main equation in non-relativistic Lagrangian mechanics is the Lagrangian, defined by the following equation: L=T -V where T is the kinetic energy of the system, and V is the potential energy of the system. The minus sign in the equation is counterintuitive to the total energy of the system. So Lagrangian is not the total energy. There is no unique Lagrangian for all physical systems. Any function which yields the equation of motion can be defined as a Lagrangian. Also, for dissipative forces, another function must be introduced alongside L, if Newtonian mechanics is used that dissipative force is usually an empirical function. In unified mechanics theory, we will introduce the thermodynamic state index into the Lagrangian; hence, there is no need for an empirical dissipation/degradation constraint functional alongside the Lagrangian. Lagrangian mechanics can only be applied to systems whose constraints, if any, are all holonomic. Three examples of nonholonomic constraints are, when the constraint equations are nonintegrable when the constraints have inequalities, or with complicated non-conservative forces like friction. In contrast, the unified mechanics theory does not have this restriction. Unified mechanics theory applies to all systems, holonomic and nonholonomic. Because there are no constraint functionals in the unified mechanics theory. Nonholonomic constraints require special treatment, and one may have to revert to Newtonian mechanics, or use other methods (Hand & Finch, 2008).

The Lagrange equations of the first kind for a single particle are given by   X n ∂f i ∂L d ∂L λi þ =0 ∂x ∂x dt ∂x_ i where coordinates x, y, z are represented by x, dot over x_ indicates time derivative, λi are the Lagrange multiplier for the constraint (dissipation) function fi, and n is the number of constraints needed to satisfy the equilibrium. The kinetic energy is defined by the velocity using the material time derivative of its position with respect to time. The Lagrange equation of the second type is referred to as the Euler-Lagrange equation, given by   ∂L d ∂L =0 dt ∂_r ∂r

2.15

Lagrangian Mechanics

87

where r is the vector of a point in the material configuration space. The spatial position coordinates x, y, z, can be represented as functions of the material (generalized) coordinates r, s, t. . . . and time. Generalized coordinates can be more than 3 that is the reason for additional dots. However, for a simple force deformation mechanics problem, they can be the minimum r, s, t. Example 2.7 For a mass m attached to the tip of a linear-elastic 1-dimensional spring, Fig. 2.7E, obtain the equilibrium equation using the Lagrangian mechanics. For this problem, we can use the Euler-Lagrange equation. Assume that generalized coordinates and position coordinates have the same origin and we have small deformations. Then we can write   ∂L d ∂L =0 ∂x dt ∂x_ L=T -V =

1 1 m_x2 - kx2 2 2

Substituting Lagrangian in the Euler-Lagrange equation above yields - kx - m€x = 0 which is the un-damped dynamic equilibrium equation of motion. We can of course obtain the same equation using the second and third laws of Newton and Hooke’s law. If we use the Lagrange equation of the first kind and add empirical damping [dissipation] constraint function as follows: f = λc_xx

Fig. 2.7E Single degree of freedom system

K

m

X

88

2

Stress and Strain in Continuum

Taking Lagrange multiplier λ = - 1, using the Lagrange equations of the first kind we arrive at - kx - m€x - c_x = 0 It is important to point out that the constraint function is not unique but empirical based on test data.

2.16

Hamilton’s Principle: The Principle of Stationary Action

We do not intend to make this section comprehensive, but just introductory. The concept has been discussed in detail by Morin (2007). We intend to quote a summary from Morin (2007). Let’s define a quantity called the action, S, with dimensions of (energy) × (time) and given by the following equation: Z S=

t2

Ldt t1

where L is the Lagrangian and dt is the time interval. Let’s assume we are dealing with a simple one-dimensional problem, and x is the only coordinate. Let’s pose the following question: What function x(t) yields a stationary value of S. A stationary value is a local minimum, the minimum, or a saddle point. The answer is given by the following theorem. Theorem If the function x0(t) yields a stationary value of S, then the following relation must be satisfied.   d ∂L ∂L = dt ∂_x0 ∂x0 It is assumed that we are considering only functions that have their endpoints fixed. Meaning xðt 1 Þ = x1 and xðt 2 Þ = x2 Finally, Hamilton’s principle (the principle of stationary action) can be given by the following definition: The path of a particle is the one that yields a stationary value of the action, S. This principle is equivalent to F = ma because the above theorem shows that if and only if we have a stationary value of S, then the Euler-Lagrange equation holds. And EulerLagrange equations are equivalent to F = ma. Therefore. “Stationary action” is equivalent to F = ma (Morin, 2007).

References

89

Fig. 2.8E Ball dropped from a platform

Example 2.8 Consider a ball being thrown from a balcony to the ground. There are many possible paths the ball can take as shown in Fig. 2.8E. The vertical distance of the ball from the balcony at any time is defined by x(t). Each path has a different function x(t). We can substitute each one of the x(t) functions in Lagrangian to yield an S value. Z

t2

S=

Ldt t1

However, only one of the paths xðt Þ = 12 gt 2 will yield a stationary value of the action, S. Here we conclude the discussion on basic concepts of continuum mechanics.

References [In Latin] Isaac Newton’s Philosophiae Naturalis Principia Mathematica: the Third edition (1726) with variant readings, assembled and ed. by Alexandre Koyré and I Bernard Cohen with the assistance of Anne Whitman (Cambridge, MA, 1972, Harvard UP). Among versions of the Principia online: Volume 1 of the 1729 English translation is available as an online scan; limited parts of the 1729 translation (misidentified as based on the 1687 edition) have also been transcribed online. Bathe, K. J. (1996). Finite element procedures. Prentice-Hall. Beatty, M. F. (2006). Principles of engineering mechanics volume 2 of principles of engineering mechanics: Dynamics-the analysis of motion (p. 24). Springer. ISBN 0-387-23704-6. Cohen, I. B. (1967). Newton’s Second Law and the concept of force in the principia. In The annus Mirabilis of Sir Isaac Newton 1666–1966. The MIT Press. Cohen, I. B. (1995). Science and the Founding fathers: Science in the political thought of Jefferson, Franklin, Adams, and Madison (p. 117). W.W. Norton. ISBN 978-0-393-24715-2. Cohen, I. B. (P. M. Harman, & A. E. Shapiro, Eds.). (2002). The investigation of difficult things: Essays on Newton and the history of the exact sciences in honor of D.T. Whiteside (p. 353). Cambridge University Press. ISBN 0-521-89266-X. Fairlie, G., & Cayley, E. (1965). The life of a genius (p. 163). Hodder and Stoughton.

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Galili, I., & Tseitlin, M. (2003). Newton’s first law: Text, translations, interpretations and physics education. Science & Education., 12(1), 45–73. Bibcode:2003Sc&Ed..12...45G. https://doi.org/ 10.1023/A:1022632600805 Hand, L. N., & Finch, J. D. (2008). Analytical mechanics. Cambridge University Press. Hellingman, C. (1992). Newton’s third law revisited. Physics Education, 27(2), 112–115. Bibcode:1992PhyEd..27..112H. https://doi.org/10.1088/0031-9120/27/2/011 Hobbes, T. (1651). Leviathan. Oxford University Press. Kleppner, D., & Kolenkow, R. (1973). An introduction to mechanics (pp. 133–134). McGraw-Hill. ISBN 0-07-035048-5. Lubliner, J. (2008). Plasticity theory (Rev. ed.) (PDF). Dover Publications. ISBN 0-86-46290-0. Archived from the original (PDF) on 31 March 2010. Malvern, L. (1969). E introduction to the mechanics of continuous medium. Prentice-Hall. Morin, D. (2007). Introduction to classical mechanics: With problems and solutions. Cambridge University Press. Newton, I. (1999). The principia. A new translation by I. B. Cohen and A. Whitman, University of California Press, Berkeley. Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica (Newton’s personally annotated 1st edition) Newton, I. Principia, Corollary III to the laws of motion Plastino, A. R., & Muzzio, J. C. (1992). On the use and abuse of Newton’s second law for variable mass problems. Celestial Mechanics and Dynamical Astronomy. Netherlands: Kluwer Academic Publishers, 53(3), 227–232. Bibcode:1992CeMDA..53..227P. Resnick, & Halliday. (1977). Physics (3rd ed., pp. 78–79). Wiley. Any single force is only one aspect of mutual interaction between two bodies. Resnick, R., Halliday, D., & Krane, K. S. (1992). Physics (Vol. 1, 4th ed., p. 83). Wiley. The Mathematical Principles of Natural Philosophy. (1687). Encyclopædia Britannica Thornton, M. (2004). Classical dynamics of particles and systems (5th ed., p. 53). Brooks/Cole. ISBN 0-534-40896-6. Wikipedia Sir Isaac Newton webpage. (2001). https://en.wikipedia.org/wiki/Isaac_Newton

Chapter 3

Thermodynamics

Newtonian mechanics is a study of the determination of the location of points in space-time (x, y, z axes and time) coordinate system after being subjected to external forces. Since Newton’s 1687 formulations, many other mechanics theories were proposed. However, they are all based on Newton’s universal laws of motion and are referred to as Newtonian mechanics. In this book term “Newtonian mechanics” is used to refer to all mechanics theories that use Newton’s universal laws of motion. In the universal laws of motion of Newtonian mechanics, an object being studied is assumed to be ageless, and energy loss (entropy generation) is not considered. Lagrangian mechanics considers dissipation by adding an empirical constraint function to the Lagrangian. Thermodynamics is about the past, present, and future of matter, under a given set of initial conditions, boundary conditions, and loading paths. Directly quoting Callen’s (1985) excellent description “Thermodynamics is concerned with the macroscopic consequences of the myriads of atomic coordinates [events] that, by the coarseness of macroscopic observations, do not appear explicitly in a macroscopic description of the system.” Therefore, thermodynamics studies of nature at the macroscopic scale reflect what happens at the atomic scale. However, “Among the many consequences of the ‘hidden’ atomic modes of motion, the most evident is the ability of these modes to act as a repository for energy” Callen (1985). Thermodynamics is a universal subject. As such, it is assumed that laws of thermodynamics are universally valid for all systems at the macro level under equilibrium conditions. Therefore, we can treat any structure or system or any piece of material as a thermodynamic system. The size of the system and time scale is very important for the laws of thermodynamics. In this book, we are only concerned with continuum mechanics where the universal laws of motion of Newton are valid. The boundary between quantum mechanics and continuum mechanics is not an easily quantifiable one. There is a gray area. However, continuum mechanics is the primary topic in this book.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_3

91

92

3

Thermodynamics

When we treat a continuum system (solid or fluid), we will assume that is a closed system. This means that for a given period (time increment) there is no exchange of matter or energy with its surroundings. This can be considered as a fixed control box in space and time. The control box is insulated, to be able to satisfy thermodynamic conservation laws.

3.1

Thermodynamic Equilibrium

All macroscopic systems have “memory.” This memory is always with respect to their stress-free state, which is the minimum energy state. All macroscopic systems tend to evolve toward a minimum energy point, which is defined by their intrinsic properties. Evolution towards these minimum energy points can be slow or fast, depending on internal and external factors. These minimum energy points of the system are called thermodynamic equilibrium states. These states usually are asymptotic states. Hence, they are static. Callen (1985) provides a good description of thermodynamic equilibrium from the atomic point of view, “the macroscopic thermodynamic equilibrium state is associated with incessant and rapid transitions among all the atomic states consistent with the given boundary conditions. If the transition mechanism among the atomic states is sufficiently effective, the system passes rapidly through all representative atomic states in the course of a macroscopic observation; such a system is in thermodynamic equilibrium. In actuality, few systems are in absolute and true thermodynamic equilibrium. In absolute thermodynamic equilibrium, from the atomic point of view, all radioactive materials would have decayed completely and nuclear reactions would have transmuted all nuclei to the most stable of isotopes. Such processes would take cosmic times to complete, and generally can be ignored. A system that has completed the relevant [significant] processes of spontaneous evolution and that can be described by a reasonably small number of parameters can be considered to be in metastable thermodynamic equilibrium. Such a limited thermodynamic equilibrium is sufficient for the application of [laws of] thermodynamics.” This is the definition of thermodynamics equilibrium we use in the rest of the book. From a practical point of view, a system is in equilibrium if it satisfies the laws of thermodynamics. However, this justification is circular. That is to say if a system satisfies the laws of thermodynamics it is said to be in equilibrium. A succinct definition of thermodynamics with an example is given by Callen (1985) “Thermodynamics is the determination of the equilibrium state that eventually results in a closed system” (Fig. 3.1). Let us assume we have a container separated into two sections with a moveable rigid wall. The container wall is assumed to be impermeable to matter, and adiabatic (heat does not enter or leave the container). This container is the definition of a closed system in thermodynamics. Initially, the separation wall is fixed. If we release the separation wall, it will move to a new location due to the gradient of pressure on each side of the wall.

3.2

First Law of Thermodynamics

93

Fig. 3.1 Thermodynamic chambers. P pressure, V volume, N number of particles, T temperature, C concentration

Then if we remove the adiabatic coating from the separation wall, the heat can freely flow between two sections of the container. Now we drill holes in the separation wall, the matter can flow freely between two sections depending on the concentration gradient. Every time a constraint is removed, a spontaneous process will take place that will result in a new equilibrium state. Initial values of pressure, volume, temperature, chemical concentration, and all other parameters in each chamber will take on new thermodynamic equilibrium values. The primary problem in thermodynamics is the computation of these parameters at the thermodynamic equilibrium. It is important to point out again, what we mean by a closed system. Because our thermodynamic equilibrium is defined for a closed system. According to thermodynamics, a system is considered to be closed if it cannot exchange any energy, matter, or anything whatsoever with its surrounding. This assumption does not prevent us from solving any engineering problem with fluctuating loads or masses. Because the problem is always solved in incremental time steps. At each step, equilibrium must be satisfied (Callen, 1985).

3.2

First Law of Thermodynamics

This law of thermodynamics states that total energy is conserved in every process. Therefore, it is sometimes referred to as the law of conservation of energy. The law is independent of the path taken between the initial and final states of the system. This law is generalized from experimental observations. Therefore, it is an empirical conclusion. While mathematically it may be difficult to prove this law universally, we assume its validity because there is no way to disprove it mathematically or experimentally [at least until now]. Now, we can formulate the first law of thermodynamics for application in continuum mechanics. The conservation of energy can be written as d ðu þ kÞ = W input þ Qinput dt

ð3:1Þ

94

3

Thermodynamics

where u is the internal potential energy, k is macro-level kinetic energy, Winput is the work done by the system, due to external loads, and Qinput is energy added to the system by heat transfer. Both Winput and Qinput do not have exact differentials. Therefore, I W input dt ≠ 0

ð3:2Þ

Qinput dt ≠ 0

ð3:3Þ

 W input þ Qinput dt = 0:

ð3:4Þ

and I

However I



Here, exact differential means that there exists a continuously differentiable function, called the potential function. In physical terms, if a system is loaded heated, and H returned to its initial state, Winput and Qinput, will not be recovered completely. dt denotes the integral through the loading cycle. However, the total energy of the system is an exact differential I ðdu þ dk Þdt = 0

3.2.1

ð3:5Þ

Work Done on the System (Power Input)

The work done on the system by external (forces) surface tractions and body forces can be given by Z W input =

σ ðnÞ  ~v dA þ

Z ρb  ~v dV

ð3:6Þ

V

A

where σ (n) is the external surface traction per unit area, ~v is the velocity field in the body, and b is the per unit mass body forces, and ρ is the mass density. Concentrated point loads are also represented by surface tractions. We can write this equation in indicial notation as Z

ðnÞ

Z

σ j ~vj dA þ

W input = A

ρ bj~vj dV V

ð3:7Þ

3.2

First Law of Thermodynamics

95 ðnÞ

The relation between external surface traction σ j given by

and internal stresses σ ij are

ðnÞ

σ j = σ ij ni

ð3:8Þ

The surface integral can be transformed to volume integral by the divergence theorem: Z

ðnÞ

σ j ~vj dA =

 Z  ∂σ ij ∂~vj ~vj dV þ σ ij ∂xi ∂xi

ð3:9Þ

V

A

Hence Winput can be written as   Z   ∂σ ij ∂~vj ~vj dV W input = þ ρbj þ σ ij ∂xi ∂xi

ð3:10Þ

V

where xj denotes Cartesian coordinate system axes. Cauchy’s equation of motion is given by 

 d~vj ∂σ ij þ ρbj = ρ dt ∂xi

ð3:11Þ

We can substitute this equilibrium equation in Winput, which yields W input =

   Z   ∂~vj d~vj ~vj ρ þ σ ij dV dt ∂xi

ð3:12Þ

V

 d~v vj ρ dtj can be written as ~vj ρ

   d~vj d 1 = ρ~vj~vj dt dt 2

ð3:13Þ

Z h i Z ∂~vj 1 σ ij dV ρ~v ~v dV þ 2 j j ∂xi

ð3:14Þ

Hence, d W input = dt

V

∂v

where ∂xji is the spatial gradient of velocity.

V

96

3

Thermodynamics

If we assume that there are no distributed moments (couple stresses) and/or polarized fields acting on the system, then, the stress tensor σ ij is symmetric. Earlier spatial gradient of velocity tensor L was given by ∂~vi =L=D þ W ∂xj

ð3:15Þ

where D is the rate of deformation tensor and W is the rate of spin tensor. We have shown that for non-polar cases the stress matrix σ ij is symmetric; on the other hand, the rate of spin tensor Wij is skew-symmetric. Therefore, we can write the following relations: σ ij W ij = 0 and σ ij Dij ≠ 0

ð3:16Þ

Substituting these equations into Winput leads to d W input = dt

Z

1 ρ~v ~v dV þ 2 j j

V

Z σ ij Dij dV

ð3:17Þ

V

The first term represents the kinetic energy, and the second term represents the internal (strain) energy of the system. Here we should point out that this equation is based on Cauchy’s equilibrium equation, which is based on Newtonian mechanics laws.

3.2.2

Heat Input

The heat input, Q, has two parts. One is heat conduction through the surface coming from outside, and the second part is the distributed internal heat generation with a source strength of r per unit mass. Internal heat generation is due to internal scattering, friction, or chemical reactions. These two components of the heat travel in opposite directions. One is coming from outside through the surface, and the second part is directly generated by the material Z Qinput =

Z q  ndA -

A

ρrdV

ð3:18Þ

V

where q is the heat flux vector, and n is the surface normal vector. The negative sign due to internal heat generation is in the opposite direction of the heat coming from outside, and it is outward. More importantly, r is generated internally by the system due to atomic level scattering and internal friction.

3.2

First Law of Thermodynamics

97

We can substitute all these terms in the first law of thermodynamics equation. For the general case, the total energy of the system will be the summation of the kinetic energy and internal potential energy. Kinetic energy here is associated with the macroscopic level velocity of the matter. It does not refer to the kinetic energy of the atoms. That is included in the internal potential energy, which refers to many forms of energy stored in the lattice. For arbitrary values of Winput and Qinput we can write the following relations:

 dE = dU þ dK = Qinput þ W input dt

ð3:19Þ

Total energy rate dE_ total can be given by dE dU dK d = þ = dt dt dt dt

Z h V

i 1 ρ u þ ρ ~v  ~v dV 2

ð3:20Þ

where u is the internal strain energy per unit mass. We can write the first law of thermodynamics by remembering that Qinput is negative when heat enters the system from outside and Winput is positive when the work is done by the system. dK Since d U_ = Qinput þ W input dt 2 3 Z Z Z d 6 7 ρudV = 4 - q  ndA þ ρrdV 5 dt V

2

6d þ4 dt

V

A

Z

1 ρ~v  ~vdV þ 2

Z

3

7 d σ : DdV 5 dt

V

V

Z

1 ρ ~v  ~vdV 2

ð3:21Þ

ð3:22Þ

V

Kinetic energy terms cancel each other. Using the divergence theorem, we can transform surface integral to volume integral: Z

Z q  ndA =

A

∇  qdV

ð3:23Þ

V

We finally collect all terms on one side and set it to equal to zero: Z  ρ V

 du þ ∇  q - ρr - σ : D dV = 0 dt

ð3:24Þ

98

3

Thermodynamics

Finally, the energy conservation law in indicial notation can be written as   ∂qj du ρ = σ ij Dij þ ρr xj dt

ð3:25Þ

where u is the internal potential energy per unit mass. The left side of the equation is the rate of increase in the internal potential energy. The right side of the equation represents the input power that is converted into internal work as mechanical work and as heat generation, however, excluding the kinetic energy at the macro level. The last term is negative because it represents heat inflow per unit volume through the boundaries. We should point out that this equation is a very simple form of the first law of thermodynamics because it assumes that only Winput is due to surface traction (distributed loads). If we have distributed fields such as electro-magnetic and chemical loads, they must be included in this equation. These load cases will be covered in Chap. 8. Moreover, the first law of thermodynamics given above does not include energy loss terms, because mechanical work terms are defined according to Newtonian mechanics. In the next chapter, the same equation will be derived for unified mechanics theory where energy loss will be included.

3.3

Second Law of Thermodynamics

There are several ways to define the second law; however, we believe KelvinPlanck’s statement is the most concise and classical description. According to this statement, “it is impossible to construct an engine that will produce no other effect than the extraction of heat from a single heat reservoir and the performance of an equivalent amount of work.” In other words, we can state that it is not possible to construct an engine that has an efficiency of 100%, meaning that the input energy and output energy for the intended work cannot be equal. There will always be energy loss for unintended work. This energy loss will only occur in the positive direction, meaning that we can only dissipate energy, but cannot gain energy in a closed system. This law of nature is expressed mathematically as an inequality stating that the internal entropy production is always nonnegative and is positive for an irreversible process. This inequality is called Clausius–Duhem inequality. Over simplified definition of entropy is that entropy quantifies how much energy is unavailable for work. Another way of stating the second law of thermodynamics is that the entropy of all-natural processes increases. The concept of entropy will allow us to tie probability and statistics to continuum mechanics. In the first law of thermodynamics, we saw interconvertibility of energy between different forms such as heat and mechanical work. In Newtonian mechanics, kinetic energy and potential energy may be converted from one to other with no energy loss [Fdt = d(mv)]. The transformation can proceed in either direction, like a pendulum

3.3

Second Law of Thermodynamics

99

Fig. 3.2 Twin chambers

with no friction that swings until eternity. Of course, in addition to dissipation of kinetic energy due to friction, there is also degradation of the pendulum material due to stresses induced by self-weight. On August 5, 1976, the Great Westminster clock in London, UK, also known as Big Ben, experienced fatigue failure after 117 years in use. The second law of thermodynamics states that a fatigue-free pendulum is not possible. Each time a pendulum swings, it loses some of its energy; as a result there is less total energy to continue swinging. Complete reversal is not possible. Frictional dissipation and degradation in materials are irreversible processes. The second law of thermodynamics also defines a preferred direction in any natural process. This is clearly stated by the definition of the second law given by Clausius (1850). He stated that the heat never flows on its own from the colder system to the warmer. This defines a preferred direction in the flow of heat between two bodies at different temperatures (Fig. 3.2). At this point, it is important to define what temperature means. While at the macroscale in continuum mechanics, we consider objects stationary, at the atomic scale this is not true. Even if an object is not subjected to any dynamic loading, if the temperature of the object is above zero Kelvin, all its atoms are vibrating continuously. This phenomenon is called atomic vibrations. At any given point in time, not all atoms vibrate at the same frequency and amplitude, nor with the same energy. Vibrational modes of atoms are called phonons. All modes can be determined by solving the eigenvalue equation, just like in continuum mechanics. Atoms will be vibrating at different levels of energy; however, the average vibrational energy of all atoms in the object will be fluctuating around a constant energy level. This constant average energy level will increase when the temperature of the object increases. Therefore, the temperature is just a measure of the average vibrational energy of all atoms in a body. Going back to the second law of thermodynamics, consider two containers filled with the same gas. Assume the temperature in A is lower and in B is higher. Low energy atoms (colder atoms) can’t move towards B. Because atoms in B have higher energy levels, their movement towards A is possible. This directional preference is another statement of the second law of thermodynamics.

100

3.3.1

3

Thermodynamics

Entropy

The term “entropy” was first used by Clausius (1850). The second law of thermodynamics is related to a variable called entropy, s, which is a mathematical quantity. Therefore, entropy allows us to quantify the second law of thermodynamics. While the concept of entropy is not easily understood from a continuum mechanics point of view, it can be explained by utilizing molecular behavior. First, we should clarify that the second law of thermodynamics cannot be proven mathematically, but it is a generalization from observations (inductive reasoning). There are several simplified descriptions of entropy while they are not mathematical; however, they help in explaining an abstract quantity (Fig. 3.3). Assume we have a ball at point A. It has a potential energy of mgh. Once the ball is slightly pushed, it rolls down and travels along the flat surface. At the point, B the ball will have no potential energy but just kinetic energy. Finally, at some point C the ball will come to a stop. It will have zero energy. As the ball is traveling from point A to point C, the amount of energy available for work continually decreases. Hence, the amount of energy unavailable for work (entropy) increases. This latter one is called entropy. In other words, “production of entropy is associated with dissipation (or consumption) of the capacity for spontaneous change when any process occurs” (DeHoff, 1993). The entropy of the closed system increases or remains constant in all processes. Therefore, entropy always increases. Energy and entropy are governed by different laws. Energy can enter or exit a body but it cannot be created inside a body. However, entropy can be created inside the body. Callen (1985) formulates the interpretation of entropy in a set of postulates depending upon a posteriori justification. These postulates are based on the basic principle that the minimization of energy function happens at equilibrium. In the following section, we will quote these postulates as defined by Callen (1985). Postulate I There exist particular states, called equilibrium states, of simple systems that macroscopically are characterized completely by the internal energy and mole numbers of the chemical components.

Fig. 3.3 Different states of energy

3.3

Second Law of Thermodynamics

101

Postulate II There exists a function, called the entropy (s), of the extensive parameters of any composite system, defined for all equilibrium states and having the following property. The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states. An extensive parameter in a homogeneous system is proportional to the total mass of the system. An intensive variable is defined as a variable that has the same value everywhere in a homogeneous system. The densities of extensive variables (e.g. u,s,~v ) are intensive variables. This postulate assumes the existence of a function called entropy only for the thermodynamic equilibrium states. In the absence of constraints, the system is free to select anyone number of states; however, the state of maximum entropy is always selected by the system. As stated earlier, the basic problem of thermodynamics is “The single, all-encompassing problem of thermodynamic is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed composite system” Callen (1985). This basic problem can be solved if the entropy of the system is known as a function of the intrinsic parameters of the system. The relation that gives the entropy as a function of the system parameters is known as a thermodynamic fundamental relation. It, therefore, follows that if the fundamental relation of a particular system is known, all conceivable thermodynamic information about the system can be obtained from it. Essentially if the fundamental relation of a system is known, every thermodynamic attribute is completely and precisely determined.” Callen (1985) did an outstanding job in establishing the postulator basis for thermodynamics. Postulate III The entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and differentiable and is a monotonically increasing function of the energy. The additive property of entropy is just a summation of the entropies of the constituent subsystems. s=

Xn

s = i=1 i

Xn

s ðU i , i=1 i

V i, Ni, . . . Nr Þ

ð3:26Þ

where n is the number of subsystems. The entropy of each subsystem is a function of extensive parameters of that subsystem alone when the additive property is used for spatially separate subsystems. The following property must be satisfied. “The entropy of a simple system is a homogeneous first-order function of the extensive parameters.” In other words, if all

102

3

Thermodynamics

the extensive parameters are multiplied by a constant, say λ, the entropy is multiplied by the same constant. sðλU, λV, λN . . . λN r Þ = λsðU, V, N 1 . . . N r Þ

ð3:27Þ

Postulate II implies that the partial derivative of entropy with respect to internal energy is a positive quantity. 

∂s ∂U

 >0

ð3:28Þ

V,N i

The possibility of negative values of this derivative is a topic of research. It was first mentioned by Ramsey (1956). However, such instances are non-equilibrium states. They can only be produced in unique systems, and most importantly, they decay away very quickly, [usually at the atomic level time scale]. Therefore, these non-equilibrium transient points do not invalidate the positive value of the partial derivative for steady-state equilibrium states. Since entropy is a continuous, differentiable, and monotonic function of energy, the opposite is also true. We can express energy as a single-valued, continuous, and differentiable function of entropy and other system parameters, as follows. U = U ðs, V, N 1 , . . . N r Þ

ð3:29Þ

where V is the volume and N1, . . .Nr are the entire system parameters. Postulate IV At zero Kelvin temperature, the entropy of any system vanishes in the state for which ∂U =0 ∂s

ð3:30Þ

Essentially this postulate states that when atoms have no vibrational energy, it is not possible to generate entropy. Therefore, at zero Kelvin temperature systems have zero entropy generation. Postulate IV is an extension of the Nernst postulate or also known as the third law of thermodynamics. These four postulates are the rational basis for the development of thermodynamics. Based on these postulates, we can solve any thermodynamics problem. Since everything organic or inorganic is a thermodynamics system, these postulates are applicable. Using these postulates, we can solve any problem that we can derive the thermodynamic fundamental relation, in the following procedure. Let us assume the thermodynamic fundamental equations governing each of the constituent systems (mechanisms) are known. These fundamental governing equa-

3.3

Second Law of Thermodynamics

103

tions allow us to calculate the entropy generation in each sub-system (mechanisms) when these mechanisms are in equilibrium. If the entire system is in a constrained equilibrium state, the total entropy is obtained by the addition of the individual entropies for each subsystem (a mechanism). Entropy generation in each subsystem (a micro mechanism) is calculated. Therefore, the total entropy is a function of the various parameters of the subsystems (micro-mechanisms). Taking straightforward differentiation of the entropy function, we can compute the extrema of the function. Of course, extrema can be minimum, maximum, or a horizontal inflection point. Then by taking the second derivative, we classify these equilibrium states as stable equilibrium, unstable equilibrium, or meta-stable equilibrium. When a system is in a stable equilibrium point for energy, it is also in a stable equilibrium state for entropy function (fundamental relation). Minimization of energy function corresponds to maximization of entropy function because energy is a function of the system parameters and entropy (Fig. 3.3).

3.3.2

Quantification of Entropy in Thermodynamics

Carnot in 1824 defined entropy for a hypothetical reversible engine, where he shows the mechanical equivalence of heat energy. Entropy is a state variable defined by ds = -

  dQ ðfor a reversible processÞ T

ð3:31Þ

where Q is heat energy in Joule or Calorie and T is the temperature in Kelvin. In 1851, William Thomson (Baron Kelvin) and Clausius published Carnot’s work posthumously, where it is shown that entropy is a state variable. “If the integral of a quantity around any closed path is zero, that quantity is called a state variable, that it has a value that is characteristic only of the state of the system regardless of how that state was arrived at” (Halliday & Resnick, 1966). I ds = 0 ðfor a reversible processÞ

ð3:32Þ

We should point out that neither heat energy Q nor temperature T is a perfect differential of any function. However, ds = dQ T is a perfect differential of a function. The fact that entropy is a state function can easily be proven for an ideal gas. Entropy will return to its initial value whenever the temperature returns to its initial value, for any reversible process. Of course, a reversible process does not exist in real life. It is just an imaginary idealization created for expedience in the formulation.

104

3

3.3.3

Thermodynamics

Gibbs–Duhem Relation

The fundamental equation for total internal energy U is a function of entropy, volume, and extensive parameters given by U = U ðS, V, N 1 , . . . . . . . . . N r Þ

ð3:33Þ

Taking the first differential of the total energy, we can write         ∂U ∂U ∂U ∂U dV þ dU = dS þ dN 1 þ ⋯ þ dN r ∂N r ∂S ∂V ∂N 1

ð3:34Þ

Partial derivatives for individual terms are defined as follows:



  ∂U = T, ðtemperatureÞ ∂S   ∂U = P, ðpressureÞ ∂V



 ∂U = μi , electro - chemical - mechanical potential of the ith component ∂N i

ð3:35Þ ð3:36Þ ð3:37Þ

Temperature, T, pressure, P, and electro-chemical potential of any subcomponent, μi, are called intensive parameters. Because they have the same value at all points in a homogeneous system, of course, this does not mean that we cannot use them for heterogeneous systems. In computational mechanics, we discretize a heterogeneous system into small pieces of homogeneous elements. Based on these definitions, for a homogeneous system in equilibrium, the differential of total internal energy can be given by dU = TdS - PdV þ μ1 dN 1 þ ⋯ þ μr dN r

ð3:38Þ

The formal thermodynamic definition of temperature (at the macro level) agrees = T). We should point out that this is the with the intuitive definition given by (∂U ∂S macro definition of temperature since at the atomic level temperature is defined by the amplitude of intensity of vibrational energy of atoms. In addition, the definition of pressure given above agrees with the mechanics’ definition of pressure. [-PdV] is called quasi-static work. dW m = - PdV

ð3:39Þ

The term quasi-static here used is in contrast to a dynamic load where a load is applied very fast; as a result, inertia forces are not negligible. It is necessary to discuss the negative sign in quasi-static work. The quasi-static work is assumed positive if it increases the total energy of the system.

3.3

Second Law of Thermodynamics

105

If the change in volume dV is negative, work done on the system is positive increasing its energy, because of the negative sign in the equation. dW M = - Pð - dV Þ = þ PdV

ð3:40Þ

Heat flux can be defined quantitatively with the help of quasi-static work. For a quasi-static process at a constant number of moles, the heat energy dQ can be defined by dQ = dU - dW m

ð3:41aÞ

dQ = dU þ PdV

ð3:41bÞ

or

In thermodynamics terms, heat and heat flux are used interchangeably. In thermodynamics terminology “heat” like “work” is only a form of energy. It is assumed that once the energy is transferred to a system either in the form of heat or as mechanical work, it is indistinguishable from any energy that may have been transferred directly to the system. It is important to remind that Joule is the unit for energy and work. Therefore, there cannot be a distinction between variables with the same units. The total energy U of a state is not just a sum of work (dW) and heat (dQ). Because dWM and dQ are imperfect differentials, which means that the differential is path-dependent. [Exact differential is path independent.] The integrals of dWM and dQ is the work and heat for that particular path of the process. Their sum is the total energy difference ΔU. However, ΔU is independent of the path taken. We should point out that when the fundamental equation is written as U = U(S, V, N1, . . .Nr) the total energy U is a dependent variable, and entropy S is an independent variable. This is the fundamental equation unified mechanics theory is based on. Since entropy is an independent variable, it can only be defined in space on its axis. The relation U = U(S, V, N1, . . .Nr) is also named the energetic fundamental relation. However, S = S(U, V, N1, . . .Nr) is said to be entropic fundamental relation, which is referred to as thermodynamic fundamental relation in this book. The remaining terms in dU equation represent an increase of internal energy associated with the addition of matter to a system. These terms are called quasi-static electro-chemical work. The term matter here does not exclude electron flow as a matter due to an electrical bias. The quasi-static electro-chemical work is given by dW c =

Xr

μN i=1 i i

ð3:42Þ

Therefore, energy equilibrium energy becomes a summation of heat, mechanical work, and electro-chemical work.

106

3

Thermodynamics

dU = dQ þ dW M þ dW C

3.3.4

ð3:43Þ

Euler Equation

Quoting Callen’s (1985) formulation, the homogeneous first-order property of the energetic fundamental relation permits that equation be written in a convenient form called the Euler form. U ðλS, λX 1 , . . . , λX r Þ = λU ðS, X 1 , . . . X r Þ

ð3:44Þ

Following the same differentiation process and using the same definitions we used earlier in this section for the energy equilibrium equation and differentiating with respect to λ yields   ∂U ∂ðλSÞ ∂U ∂λX i þ ⋯ = U ðS, X 1 , . . . X r Þ þ ∂ðλSÞ ∂λ ∂ðλX i Þ ∂λ

ð3:45Þ

Assuming λ = 1. Then Xt ∂U ∂U X =U Sþ i = 1 ∂X i i ∂S Xr PX U = TS þ i=1 i i

ð3:46Þ ð3:47Þ

For a simple system, this equation becomes U = TS - PV þ μ1 N 1 þ ⋯ þ μr N r

ð3:48Þ

The last two equations are called Euler relations. A differential form of the relation among the intensive parameters can be obtained directly from the Euler relation and is known as the Gibbs–Duhem relation. Taking differentiation of Euler relation given by Eq. (3.47) leads to dU = TdS þ SdT þ

Xr

P dX i þ i=1 i

Xr i=1

X i dPi

ð3:49Þ

Earlier we derived that, dU = TdS þ

Xr

P dX i i=1 i

ð3:50Þ

We should note that in the derivation of the last equation, we separated quasistatic mechanical work and quasi-static electro-chemical work. Here they are both represented in one term:

3.3

Second Law of Thermodynamics

107

dU = dQ þ dW M þ dW C Xr P dX i i=1 i

ð3:51Þ ð3:52Þ

Substituting this last equation in the differential above, we end up with SdT þ

Xr i=1

X i dPi = 0

ð3:53Þ

This equation is referred to as the Gibbs–Duhem relation. For a single component system, this equation can be written as SdT - VdP þ Ndμ = 0

ð3:54Þ

The Gibbs–Duhem relation presents the relationship among the intensive parameters in differential form. The number of linearly independent intensive parameters that have independent variations is called the number of thermodynamic degrees of freedom of any system, which are mapped onto the thermodynamic state index axis in the unified mechanics theory, which is discussed in the next chapter. At this point, it is necessary to restate the thermodynamic formalism in energy representation. This fundamental equation is U = U ðS, V, N 1 , . . . N r Þ

ð3:55Þ

which contains all thermodynamic information of any system. Relying on our definitions the equations of thermodynamic state can be given by T = T ðS, V, N 1 , . . . N r Þ P = PðS, V, N 1 , . . . N r Þ μ1 = μ1 ðS, V, N 1 , . . . N r Þ μr = μr ðS, V, N 1 , . . . N r Þ

ð3:56Þ

When all equations of thermodynamic state are combined, it is equivalent to the thermodynamic fundamental equation. Assuming we have a simple single-component system in which two equations of the state are known, the Gibbs–Duhem relation can be integrated to obtain the third equation of state. As pointed out by Callen (1985), it is possible to express internal energy U without entropy S, term. However, such a total energy equation is not a fundamental energetic relation since it does not contain all thermodynamic information about the system. Many textbooks refer to Gibb’s relation in the following form, which Gibbs defined only for the case of a fluid,

108

3

du = Tds - Pdv

Thermodynamics

ð3:57Þ

where u is the internal energy density (specific internal energy), s is entropy for unit volume, P is the thermodynamic pressure, and v is the specific volume of the fluid.

3.3.5

Entropy Production in Irreversible Process

The cantilever beam shown in Fig. 3.4 is subjected to linear elastic static cyclic loading. Let us assume that the force is small enough to produce a stress level well below the macro yield stress. A student observing this beam will see that after every load removal, the beam returns to the original shape. To the observer, this gives the impression that deformation under elastic loading is a perpetual reversible process. However, after a certain number of cycles, the cantilever beam will fatigue and fail regardless of how small the stresses are. It will probably break away from the support where stresses are maximum. This simple experiment shows that while we observe macro behavior to be reversible, however, our observation at the macro level is not accurate. If the linear elastic loading-unloading cycle were reversible, the beam would never fatigue, and deflections from one cycle to another cycle would not change in magnitude. In earlier pages, we cited the fatigue failure of Big Ben [clock in London] after 117 years in service. According to the second law of thermodynamics, reversible processes are imaginary. They cannot happen in the real domain. What happens in the lattice is when the beam is mechanically loaded, there is strain in the lattice. However, when the load is removed, atoms do not go back to their original lattice site. They go to a lattice site with a lower energy level, to minimize their energy level. In the process, they leave behind a vacancy. As a result, the process is not reversible. Keep in mind that the thermodynamically reversible process is where the initial and the final states Fig. 3.4 A cantilever beam subjected to cycling loading

3.3

Second Law of Thermodynamics

109

are the same. If an atom moves to a new lattice site with lower energy, state and is left behind a vacancy that is a different thermodynamic state than the initial one. There is a positive entropy generation, which is a quantitative measure of dissipation in the system. DeHoff (1993) states that “changes in the real world are always accompanied by friction and something [energy] that is dissipated. When the pebble is dropped into a pond its kinetic energy at impact is dispersed by the wave motion throughout the pond and the [initial kinetic energy] is dissipated in the absorbing water of the pond.” In this irreversible process, the energy cannot be recovered. “The entropy production in a system is a quantitative measure of this energy dissipation [mechanisms].” Eddington (1958) referred to entropy as “entropy is time’s arrow,” The total entropy of any system plus its surroundings always increases with increasing time. However, the entropy-time relationship is not linear, and each one is linearly independent. Time can change without a change in entropy, as in at zero Kelvin or in an undisturbed system. However, time cannot change unless there is a change in entropy. If we had a process where entropy is destroyed, then the direction of the time arrow must be reversed for that process to happen. However, this is not possible in a real system. A reader may question the accuracy of the theory of elasticity where all mechanisms are assumed to be reversible. Because of the second law of thermodynamics, the theory of elasticity provides an approximate solution to an imaginary system that cannot exist in reality. In the words of DeHoff (1993) “All real processes have a finite rate in response to finite influences; such real processes are called irreversible to emphasize the contrast with imaginary reversible processes. Real processes are irreversible and suffer dissipations that result in the production of entropy and thus a permanent (irreversible) change in the system.” Therefore, for an irreversible process, we can write Z2   dQ >0 ΔS = S2 - S1 = T irreversible

ð3:58Þ

1

Of course, here it is assumed that it is a closed system and it is an adiabatic process. While we acknowledge that all real processes are irreversible, sometimes for the sake of simplicity a system is modeled as an imaginary reversible process, to be able to get an understanding of the system near time = 0.

3.3.6

Clausius–Duhem Inequality

There are different approaches to deriving this inequality. We will quote the derivation given by Malvern (1969). This inequality is considered a simple quantitative expression of the second law of thermodynamics. The thermodynamic theory

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explains the relationship between energy flowing into a system and the entropy generation in the system and its surroundings (Fig. 3.5). Heat input in a system is given by the difference between the heat conduction coming from outside through the surface A and distributed internal heat generation r per unit mass: Z Qinput =

Z - q  ndA þ

Z

dQ T.

ð3:59Þ

V

A

Entropy was defined by dS =

ρrdV

Hence entropy input rate can be given by

q -  ndA þ T

Z

ρr dV T

ð3:60Þ

V

A

We are assuming that the system is closed. If the system is open in a control surface, [fixed in space], additional entropy input due to mass flux would be accounted for by Z - ρsV  ndA

ð3:61Þ

A

where s is entropy per unit mass. According to the second law of thermodynamics, entropy increase rate in the system ≥ entropy input rate d dt

Z

Z ρsdV ≥ V

A

q -  ndA þ T

Z

r ρdV T

ð3:62Þ

V

This is the integral form of the Clausius–Duhem inequality. This equation means that in an irreversible process internal entropy production is always greater than input. In this equation, outward normal is a positive sign. An equal sign ensures that the equation holds for the reversible [imaginary] process.

Fig. 3.5 Heat exchange for a system

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We can transform the surface integral into a volume integral. Since volume can be an arbitrary value, we can write the local version of the Clausius–Duhem inequality that must be satisfied at each point in any given volume as follows:  ds r 1 q ≥ - div dt T ρ T

ð3:63Þ

or γ

ds r 1 q - þ divq - 2  gradT ≥ 0 dt T ρT ρT

ð3:64Þ

where γ is the internal entropy production rate per unit mass. A stronger assumption of inequality was proposed by Truesdell and Noll (1985): ds r 1 divq ≥ 0 - þ dt T ρT

-

1 q  gradT ≥ 0 ρT 2

ð3:65Þ

where the first inequality represents the local entropy production by the system and the second inequality represents entropy production by heat conduction. The Truesdell and Noll (1985) interpretation of the second law of thermodynamics is important for the unified mechanics theory because it is assumed that disorder in the system increases only due to internal entropy production. Entropy production by heat conduction in a free (unconstrained) system, Fig. 3.6, cannot lead to irreversible disorder in the system because the system travels between stress-free equilibrium states. Of course, the entropy of the entire surrounding including the system increases. The interatomic equilibrium distance in a free (unconstrained) system is a function of temperature; therefore due to the heat conduction temperature of the system increase (atoms have higher vibrational energy) the equilibrium inter-atomic distance increases. However, there will be no strain (or stress) exerted on the atoms. Therefore, internal work will be zero; as a result, internal entropy production will be zero. If the atoms move from one thermal equilibrium to another thermal equilibrium, it is assumed that there will be no internal entropy generation. Of course, this assumption is not 100% true. When

Fig. 3.6 Expansion of a cantilever beam due to increasing ambient temperature

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Fig. 3.7 Repulsive and attractive bonding energy between two isolated atoms as a function of interatomic distance. (After Callister Jr & Rethwisch, 2010)

Thermodynamics

Repulsive energy

r0

Interatomic disctance r

Net Energy EN Attractive energy

there is heat transport in a lattice, there is always scattering with phonons. We are assuming this scattering-induced internal entropy generation is so small that we can ignore it. When there are no external constraints blocking atoms moving freely at each temperature, they have a stress-free equilibrium interatomic separation distance r0. When the interatomic distance is r0, attraction/repulsion atomic bond energy is at an extremum (maximum), and the force acting on the atoms is zero (Fig. 3.7). Therefore, internal work is not possible, because internal work and internal entropy production are proportional to the dislocation of an atom from the equilibrium lattice site and force acting on the atom. Of course, when a system is at equilibrium and the force acting on the atom is zero, the work done must be zero. However, this argument assumes that the system is free to move and boundary conditions do not impose any constraint on the system that prevents it from moving freely. If the cantilever shown in Fig. 3.6 is fixed on both sides, of course, there will be internal work and internal entropy generation. Clausius–Duhem inequality is just a quantitative statement of the second law of thermodynamics; hence, it must be satisfied by every process in organic or inorganic systems. The coordinate system assumed in the derivation of the formulation in previous pages is arbitrary; however the positive definite property of the inequality is not arbitrary. It must be satisfying for all and any coordinate system.

3.3.7

Traditional Use of Entropy as a Functional in Continuum Mechanics

The unified mechanics theory uses entropy as a linearly independent state variable. Furthermore, entropy (thermodynamics state index) defines a new additional 5th linearly independent axis in addition to Newtonian space-time axes. This is needed to be able to define the location of a system in a space-time-thermodynamic state index coordinate system. This is needed to be able to define the coordinates of a system in a space-time-thermodynamic state index coordinate system. While there are similarities between the traditional interpretation of entropy in Newtonian (continuum) mechanics and unified mechanics, there are also major differences. In Newtonian (continuum) mechanics, the derivative of displacement with respect to

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entropy is taken as zero. In the unified mechanics theory, derivatives with respect to entropy are not zero. When derivatives with respect to entropy are taken as zero, unified mechanics theory collapses to Newtonian mechanics. We assume that the derivative of time with respect to entropy is small enough to be considered zero for most engineering problems. The traditional use of entropy in Newtonian (continuum) Mechanics is considered extensively by Malvern (1969), Coleman (1964), and Coleman and Mizel (1964, 1967). Here we summarize their work. “In Newtonian continuum mechanics, the use of the Clausius-Duhem inequality differs from the usual second law of thermodynamics, where the entropy is a state function determined by the instantaneous values of the other state variables. In Newtonian continuum mechanics, entropy is not explicitly postulated to be a state function. As a result, in Newtonian continuum mechanics entropy maximization (and minimization of entropy generation rate) does not affect system state variables or system properties. In simple terms, in Newtonian continuum mechanics, dissipation and degradation of the system are not directly included in the laws of Newton, but as empirical test data curve-fitting constraints. Traditionally many researchers and authors of thermodynamics have postulated that entropy is a state function of all the state variables including possibly some variables that are not observable by macro-scale specimen testing.” We must clarify what we mean by “state variables.” For a simple ideal gas, pressure, temperature, and volume are thermodynamic variables. However, the internal energy and the entropy are state variables (also called state functions). State variables are expressed employing thermodynamics variables. For a variable to be classified as a state variable (also called state function), it must have an exact differential, based on a vanishing integral in an arbitrary cycle. This definition assumes that the material is capable of going through an arbitrary cycle and being restored to its initial state. As an example, we can plastically deform a steel beam, and then melt it or anneal it and return it to its original state. Therefore, the total energy of the system is a state variable because it has an exact differential: I

  Pinput þ Qinput dt = 0

ð3:66Þ

Malvern (1969) states that if the “state” of a continuum is taken to be defined by a limited number of explicitly enumerated macroscopic state variables, observable at least in principle, and then the entropy must in general depend on the history of this limited number of state variables and not merely on their current values. This is of course the case for any solid that experiences inelastic deformations. Because plastic strain is path-dependent, therefore, entropy is also of course path-dependent. Coleman (1964) and Malvern (1969) define entropy as a function of the history of deformation gradient tensor F, temperature, and the instantaneous value of temperature gradient. However, this definition is for thermo-mechanical loading only. Malvern (1969) makes the distinction between function and functional by defining that a function depends only on the instantaneous values of its variables. Functional is a function of the history of the variables. Malvern (1969) citing a derivation by Coleman and Mizel (1964) defines the caloric equation of state [called thermodynamic fundamental equation in this book], in the following form:

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3

s = f ðu, F Þ

Thermodynamics

ð3:67Þ

where entropy is a function of internal energy and deformation gradient. However, this definition ignores other electro-thermo-chemical-radiation mechanisms that could contribute to entropy generation. Earlier Truesdell and Toupin (1960) postulated an energetic equation of state as follows: u = uðs, V i , . . . , V n Þ

ð3:68Þ

Here s is entropy per unit mass and Vi is a thermodynamic substate variable accounting for all thermos-electro-mechanical-chemical-radiation processes. Of course, the inverse of the last Eq. (3.68) also holds true: s = sðu, V 1 , . . . , V n Þ

ð3:69Þ

which is a more general thermodynamic fundamental [caloric] equation of the state. The terminology naming these equations is not uniform in the literature. However, they have the same meaning.

3.3.8

Entropy as a Measure of Disorder

Order and disorder are relative terms, meaning that they are defined with respect to an initial reference state. For example, in Fig. 3.8a, the initial location of each piece, such as A7, can be considered in the ordered state. In general, the initial or strain-free state of a system is considered as ordered. Of course, this does not mean that the strain-free state has no disorder at the atomic scale. It is just a benchmark [reference] state. Any change from the initial reference state is an increase in disorder. For a quantifiable disorder to happen in a system, there must be some inhomogeneity. For example, if we have a box (Fig 3.8b) with four identical baseballs, regardless of how much we shake the box, the final “disordered” state will be identical to the initially ordered state. As a result, there is no way to quantify a disorder. There are 24 distinct configurations possible for these balls in the container. However, the positions the balls could take are not observable at the macroscale, because the balls are identical. Fortunately, most systems and materials at the micro aggregate scale are not made up of identical crystals. Now assume that we mark the balls as A, B, C, D (Fig. 3.8c), and initially we place them in the box in alphabetical order. We assume that the initial alphabetical order configuration is a reference “ordered” state. In this case, after we shake the box among 4 ! = 24 possible configurations, the initially ordered state is only possible 1/24 of the time. That means 23/24 times it will be a disordered state. If we have 26 baseballs marked A to Z, the possibility of getting the ordered state is one in millions. All organic and inorganic systems are made up of a large number of atoms (or molecules). When subjected to any external load (disturbance), they move from their initial ordered

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Fig. 3.8 Description of order and disorder

state to a new configuration that we call “disordered state,” of course, there is an energy cost associated with changing the ordered state. It does not happen on its own with no external energy input. “In statistical mechanics, the entropy of a state is related to the probability of the occurrence of that state among all the possible states that could occur” (Malvern, 1969). This is only rational because entropy generation requires work. As more entropy is generated, other configurations that are possible are achieved. Malvern (1969) states “Thus, increasing entropy is associated with increasing disorder. The second law of thermodynamics seems to imply an almost metaphysical principle of preference for disorder.” In nature, it is observed that changes of state are more likely to occur in the direction of greater disorder. However, the second law of thermodynamics also implies that when the disorder is maximum, entropy is also at maximum, and the entropy generation rate is zero. The balls in a box example are an illustration of this argument. An engineeringrelated example would be having a standard 15 cm in diameter and 39 cm in height concrete cylinder, where each gravel, a sand particle is numbered. Under compression loading, this concrete cylinder will go through many disordered states before failure. Callen (1985) explains eloquently why entropy is a measure of disorder, quantitatively. We will summarize his approach. Callen (1985) states that in statistical mechanics the number of microstates [arrangement of A, B, C, D balls in our example, Fig. 3.8] among which the system undergoes transitions and which thereby share a uniform probability of occupation, increases to the maximum permitted by

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the constraints. This statement is strikingly reminiscent of the entropy postulate of thermodynamics, according to which the entropy increases to the maximum permitted by the imposed constraints, for a closed system. It is possible to conclude that entropy can be identified with the number of microstates in a closed system. Callen (1985) points out that entropy is additive, but the number of microstates is multiplicative. The number of microstates available to two systems is the product of the number of microstates available to each system. If we have two dices, each has six microstates. But two dices rolled together have 6 × 6 = 36 microstates. Callen (1985) points out that to interpret the entropy, then we require an additive quantity that measures the number of microstates available to a closed [isolated] system. Boltzmann (1877) [English translation by Sharp and Matschinsky (2015)] and Planck (1900a, b, c, d) suggested that this problem could be solved by identifying the entropy with the logarithm of the number of available microstates because the logarithm of a product being the sum of the logarithms: ln ða  bÞ = ln a þ ln b

ð3:70Þ

Thus, the following equation resulted: s = k ln w

ð3:71Þ

where w is the number of microstates consistent with the macroscopic constraints of the closed [isolated] system. The confusion about w being the number of microstates or the probability of a microstate is discussed in the next chapter. Boltzmann constant k is defined by k = R=N A = 1:3807 × 10 - 23 J=K

ð3:72Þ

is used to define temperature on the Kelvin scale and to ensure consistency of units Joule on both sides of the equation. R = 8:31 molK is the gas constant; NA = 6.022 × 1023 molecules/mol is the Avogadros number. Equation (3.71) is considered the basis for statistical mechanics. Callen (1985) refers to Eq. (3.71) as a postulate that is dramatic in its brevity, simplicity, and completeness. According to Boltzmann, the equation calculating the natural logarithm of the number of microstates available to the system and multiplying with a constant k yields the entropy. However, the opposite is also true to calculate the number of microstates. In Unified Mechanics Theory, this second approach is used, since entropy is a function of internal state variables and all active mechanisms; it can directly be calculated because entropy is the most general caloric equation [fundamental equation] of state. Callen (1985) refers to the Boltzmann equation as the statistical mechanics in the microcanonical ensemble. Entropy calculations in continuum mechanics are not readily available for all mechanisms. Of course, very often we do not know what these mechanisms are. This is expected to be an active research field soon. Entropy in thermodynamics and statistical physics are the same thing. Statistical mechanics interpretation of entropy where natural progress towards more disorder establishes a concrete and understandable concept of entropy. However, in statistical mechanics to be able to calculate the

3.3

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entropy all active mechanisms and processes taking place in the material must be accounted for. Then entropy generation for each process must be calculated. Unfortunately, this is a very primitive field in the mechanics of materials, because all material constitutive models at continuum scale are based on empirical functions obtained by testing of materials. There are no explicit formulas for entropy generation due to thermo-mechanical-electrical-chemical-radiation loads. This is a wideopen research topic and a very challenging one, to say the least, because the material modeling field has always been based on empirical observations. For example, we crash a concrete cylinder, and model it with macro measurement variables, such as stress, strain, and curve fitting. However, macroscale testing does not give us much information about the actual micro-mechanisms that are responsible for degradation and final failure. These are the micro-mechanisms that we need to know in detail to compute entropy generation. It is always much easier to curve fit to a test data and use it than to understand the actual mechanisms and processes leading to failure. Theoretical physicist and mathematician Freeman Dyson in Scientific American September (1954) suggested that heat is disordered energy. The author gave an example of a flying rifle bullet. The bullet has kinetic energy but no disorder. After the bullet hits a steel plate target, its kinetic energy is transferred to random motions of the atoms in the plate and bullet. This disordered energy makes itself felt in the form of heat. However, the author ignores other mechanisms, because heat is not the only product of the initial kinetic energy. There are plastic deformations, melting, phase change, thermomigration, and other entropy generation mechanisms that could result from an impact. Freeman Dyson (1954) also adds that the quantity of disorder is measured in terms of the mathematical concept called entropy; of course, entropy is a function of energy. Therefore, we can compute disorder in terms of energy using Boltzmann’s equation. It is important to point out that disorder is the perfect way to express the degradation of all organic and inorganic systems. Because as the entropy increases the amount of disorder from the initial “ordered,” reference state increases. The degradation of the material is nothing but another disordered state. Realignment of atoms in any system under loading just takes one of the possible disordered states. Of course, because of a very large number of possible disordered states, the system cannot get into every possible disordered configuration before entropy becomes maximum, because of the closed-isolated system requirement. This is easier to explain with an example. Say we have a box with 26 baseballs marked A to Z. Initially they are all in alphabetical order. Say the box is 10 kg. We are allowed to spend 100 Joule energy to shake the box. Assume we have probably 10 chances to shake the box. Of course, for our closed system when 100 joules are consumed, the maximum entropy is reached. However, we could not discover all possible disorder states that 26 baseballs could take. In the same fashion when a material fails in a fatigue loading or monotonic loading, the failure is always through a different path. Yet the maximum entropy, the value to reach the failure is always the same. Experimental verifications have been provided by Naderi et al. (2009), Yun and Modarres (2019), and Tu and Gusak (2019). Failure or increase in disorder in a system can be considered travel over an energy terrain. A soccer ball sitting in a

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Fig. 3.9 Energy terrain along different paths

valley after a kick will travel. However, there are many paths it can take. Each one of them is possible. Each one represents a disorder path. However, the ball will finally come to stop at a valley. We can write the fundamental relation. Of course, we have to know what path the ball will take to use the properties relevant to that path. Then we can exactly predict in which valley the ball will stop (Fig. 3.9).

3.3.9

Thermodynamic Potential

In Newtonian continuum mechanics, the choice of thermodynamic potential is subjective. One can use different thermodynamic potentials depending on the problem. However, there are some mathematical conditions that all thermodynamic potentials must satisfy. To be able to define thermodynamic potentials, first we must define thermodynamic variables that are also called state variables or independent variables. Thermodynamic Variables The thermodynamic local state of a system at any given time and space can be defined by the thermodynamic variables using the thermodynamic fundamental relations. The evolution of a thermodynamic state under external loading can be defined by a succession of equilibrium states. Each valley [microstate] in the energy landscape can be an equilibrium point. However, that does not mean that it is at a maximum entropy point. We should point out that, as Lemaitre and Chaboche (1990) suggest, the ultra-rapid phenomenon for which the time scale of the evolution is in the same order as the atomic relaxation time for a return to thermodynamic equilibrium (atomic vibrations) is excluded from this theory’s field of applications because in continuum mechanics time scale to reach thermodynamic equilibrium is much longer than atomic vibration periods.

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Any physical phenomenon can be defined with the appropriate choice of thermodynamic variables. Any physical process defined by thermodynamics variables is admissible if it satisfies all laws of thermodynamics. It is essential to point out that when degradation is introduced into Newton’s laws of motion, Lemaitre and Chaboche (1990) opined that there is no objective way of choosing the internal variables best suited to study a phenomenon. This is an opinion in which the choice is dictated by empirical experience (laboratory observations), “physical feeling,” and very often by the type of application. This approach is the only option when Newtonian mechanics is used for Kachanov-type empirical damage mechanics models. However, in the unified mechanics theory, all thermodynamics variables must be chosen objectively to be able to assemble the fundamental equation based on physics and mathematics. They must represent all active micromechanisms responsible for internal entropy production. In Lemaitre and Chaboche’s (1990) approach, which is based on Newtonian Mechanics, thermodynamics variables are up to the scientist/engineer to decide so that experimental data can be fit into the same curve. In unified mechanics theory, all micro-mechanisms that are responsible for entropy generation must be included, and they must be physically and mathematically formulated in the fundamental equation without a curve fitting a function to a test data. Observable Thermodynamics Variables 1. Temperature 2. Space coordinates While some authors define total strain or stress as observable variables, we do not subscribe to this school of thought, because strain and stress are human constructs, and definitions, not physical quantities. We can only measure displacement and then calculate strain and stress. If something is calculated, it is not a directly observable variable. Independent Internal Thermodynamic Variables 1. Entropy 2. Internal energy 3. Crystal structure (phase) Formulation of Thermodynamic Potential Postulate: There exists a thermodynamic potential function defined by thermodynamic variables. The thermodynamic state laws are derived from this potential. Thermodynamic potential must be concave [f ′′(X) < 0] with respect to temperature and convex [f ′′(X) > 0] with respect to other variables. These convex and concave requirements ensure the thermodynamic stability requirement imposed by the Clausius–Duhem inequality. Malvern (1969) proposes the following thermodynamic potentials given in Table 3.1. The Helmholtz free specific energy potential Ψ is the portion of the internal energy that is available for doing work at a constant temperature. In continuum

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Table 3.1 Thermodynamic potentials (Malvern 1969) Potential Internal energy Helmholtz free energy Enthalpy Free enthalpy, or Gibbs function

u Ψ h g

Relation to u u Ψ = u - sT h = u - τjvj g = u - sT - τjvj =h - sT

Thermodynamic independent variables s, vj T, vj s, τj T, τj

mechanics, free energy Ψ is a function of thermodynamic state variables defined earlier Ψ = u - sT

ð3:73Þ

Enthalpy, h, is the portion of the internal energy, u, that can be released as heat when the thermodynamic tensions are held constant. Enthalpy can also be given by h = u - τ i vi

ð3:74Þ

where and vi is a thermodynamic tension conjugate and τi is a thermodynamic tension defined by 

∂u τi = ∂vi

 ð3:75Þ S

In differential form, thermodynamic potentials can be given by du = Tds þ τi dvi

ð3:76Þ

dΨ = - sdT þ τi dvi

ð3:77Þ

dh = Tds - vi dτi

ð3:78Þ

dg = - sdT - vi dτi

ð3:79Þ

Assuming state variables defined by the subscripts are held constant, the following relationships can be written:   ∂u T= ∂s vi   ∂u τi = ∂vi s   ∂Ψ s= ∂T vi

ð3:80Þ ð3:81Þ ð3:82Þ

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  ∂Ψ τi = ∂vi T   ∂h T= ∂s τi   ∂h vi = ∂τi s   ∂g s= ∂T τi   ∂g vi = ∂τi T

ð3:83Þ ð3:84Þ ð3:85Þ ð3:86Þ ð3:87Þ

It is assumed that thermodynamic tensions can be identified with the Piola– Kirchhoff stress tensors, and with recoverable work assumption. Malvern (1969) states that “The concept of recoverable work is bound to the existence of a caloric equation [fundamental equation] of state and only thermodynamic tensions derived from a potential defined by such an equation do work not contributing to entropy production. The assumed existence of such a caloric equation [fundamental equation] of state does not imply our knowledge of such a formula for it.” We subscribe to the school of thought that if there is such a recoverable work, it would violate the second law of thermodynamics. Hence, fully recoverable work [reversible thermodynamic process] and thermodynamic tensions are just abstract concepts used for expediency in our mathematical framework. Here, it is assumed that internal energy u is a potential for the thermodynamic tensions when entropy is constant. Of course, this assumption eliminates using internal energy as a thermodynamic potential in any real process, because the derivative of internal energy with respect to entropy is not zero. However, Helmholtz’s free energy density is a thermodynamic potential in an isothermal process. Since most engineering problems are solved in incremental format, Helmholtz free energy is more appropriate as a thermodynamic potential. It should be clarified that the term thermodynamic tension is not the same as universal thermodynamic force or just any stress tensor. Helmholtz free energy can be implemented as a thermodynamic potential, for an elastic-plastic solid under thermo-mechanical loads. Helmholtz free energy for an elastic-plastic solid is given by Ψ = Ψð½D - Dp , T Þ = ΨðDe , T Þ

ð3:88Þ

Then we can write the following relation based on the deformation gradient tensor which is a summation of elastic and plastic parts: ∂Ψ ∂Ψ ∂Ψ = ∂De ∂D ∂Dp

ð3:89Þ

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We can also write _ = ∂Ψ : De þ ∂Ψ T_ Ψ ∂De ∂T

ð3:90Þ

Thermodynamic potential must satisfy Clausius–Duhem inequality, which is given by ds r 1 q gradT - þ divq - : 2 ≥ 0 dt T ρT ρ T

ð3:91Þ

If we substitute r from the conservation of energy equation, we obtain the fundamental inequality that contains both the first and second laws of thermodynamics    ds 1 q 1 du þ div - σ : D þ divq ≥ 0 dt ρ T T dt

ð3:92Þ

Since div

 divq q  gradT q = T T T2

ð3:93Þ

Multiplying each term with T,   ds du gradT þ σ : D-q  ρ T ≥0 dt dt T

ð3:94Þ

Helmholtz-specific free energy is given by Ψ = u - Ts

ð3:95Þ

Differentiating Eq. (3.95) we obtain dΨ du ds dT = -T -s dt dt dt dt

ð3:96Þ

We can write  -

 dΨ dT ds du =T þs dt dt dt dt

If we substitute this new relationship with fundamental inequality, we get

ð3:97Þ

3.3

Second Law of Thermodynamics

123

  ∂Ψ dT gradT -q  σ : D-ρ þs ≥0 dt dt T

ð3:98Þ

We should point out that in the conservation of energy time derivative of internal energy is multiplied by density (ρ), which is dropped in this derivative for consistency of units. Now we can substitute dΨ dt in the fundamental inequality. Meanwhile, we can split the deformation–gradient tensor, D, into elastic and plastic components: D = De þ D p

ð3:99Þ

Hence, we can write  ∂Ψ ∂Ψ _ gradT e _ : D þ T þ s T -q ≥0 T ∂De ∂T     ∂Ψ _ gradT ∂Ψ e p : D þ σ : D ρ s þ T -q ≥0 σ -ρ T ∂T ∂De

σ : ðDp þ De Þ - ρ



ð3:100Þ ð3:101Þ

If we assume a small strain formulation under pseudo-static loading, a deformation–gradient tensor is equal to a small strain rate tensor: De þ DP = ε_ e þ ε_ p

ð3:102Þ

In many continuum mechanics textbooks for thermo-elastic materials, the following assumption is made for T_ = 0: sþ

∂Ψ =0 ∂T

ð3:103Þ

However, Eq. (3.103) assumes that τidvi= 0, which is not true. Because entropy generation happens due to many mechanisms, not just temperature change. Moreover, the reversible process is imaginary; it cannot happen in real life, without violating the second law of thermodynamics. There is always entropy generation. Therefore, thermo-elastic laws are defined as an imaginary process where τidvi= 0 ε_ p = 0,grad T = 0 and T is arbitrary. Then the following relations define thermoelastic laws in Newtonian mechanics. σ =ρ

∂Ψ ∂Ψ and s = ∂T ∂εe

ð3:104Þ

The stress-Helmholtz free energy relation in Eq. (3.104) assumes that stress is independent of change in entropy, of course, which is not true.

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Thermodynamic Forces Thermodynamic potential Ψ(D, T, Vi, . . .Vr) is a function of thermodynamic state variables. In constitutive modeling, the concept of thermodynamic force is very convenient for material modeling. It is defined by  Ai = ρ

∂Ψ ∂V i

 ð3:105Þ

where Ψ is the thermodynamic specific free energy potential. Ψ = Ψðε, s, T, V k Þ

ð3:106Þ

Ψ is a function of observable state variables and internal variables. Of course, the vector of this force is normal to the thermodynamic potential Ψ surface. Dissipation Potential In Newtonian mechanics-based constitutive modeling, a phenomenological dissipation potential is defined. This is also the basis for yield surface or plastic potential used in the theory of plasticity. Lemaitre and Chaboche (1990) indicate correctly that the whole problem of modeling a phenomenon (material behavior) lies in the empirical determination of the analytical expressions for the thermodynamic potential Ψ and dissipation potential φ or its dual φ, and their identification in characteristic experiments. Therefore, constitutive modeling of Newtonian mechanics of materials is an empirical process. In Newtonian mechanics, it is postulated that there exists a dissipation potential (also called pseudopotential in some books). This dissipation potential is expressed as a continuous and convex scalar-valued function of the state variables E_ P , → V_ k , - q : Of course, this definition limits the potential to be valid for thermoT

mechanical loading only. Hence, it can be given by

 _ V_k , q=T φ e,

ð3:107Þ

The state variable Vk is used to define any internal state variable. Dissipation potential is assumed to have a zero value at the origin of the state variables. However, this is not a mathematical requirement. It is possible to define a dissipation function with a non-zero value at the origin. The complementary relationships between internal state variables and associated variables can be defined by employing dissipation potential. The normality rule requires that the first derivative of the dissipation potential with respect to state variables is normal to the potential surface and directed outwards: σ=

∂φ ∂φ → ∂φ g=Ak = ∂_E ∂Vk ∂ðq=T Þ

ð3:108Þ

3.3

Second Law of Thermodynamics

Table 3.2 Dissipation flux variables and dual variables (Lemaitre and Chaboche 1990)

125

The flux of state variables e_ -Vk -

→

q T

Dual variables σ Ak g = grad T

We should point out that in most continuum mechanics textbooks dissipation potential is a function of plastic strain, not elastic strain. However, this approach assumes that elastic strain does not cause dissipation and is completely recoverable. However, this approach is not correct, because there is dissipation during elastic response. If there were no dissipation during elastic response, there would not be fatigue under elastic loading. According to Lemaitre and Chaboche (1990), the thermodynamics forces are the ! components of the vector gradφ , which are normal to the φ surface in the state variable space. The term “thermodynamic force” is an abstract force concept that is supposed to satisfy the laws of thermodynamics. However, when elastic deformations, all entropy generating mechanisms are ignored they do not satisfy the laws of thermodynamics, in the strict sense. Therefore, it is more appropriate to call these forces pseudo-thermodynamics forces. It is easier to formulate material models that constitute laws in state variables that can easily be calculated or measured. Therefore, the Legendre–Fenchel transformation can be used to define the complementary potential corresponding to dissipation potential. Dissipation variables for the thermomechanical system are given in Table 3.2.  → The corresponding potential is φ σ, Ak , g , which is the dual of the dissipation  → potential φ E_ , V_k , q =T . For details of the Legendre–Fenchel transformation of the dissipation, potential readers are referred to Lemaitre and Chaboche (1990). The corresponding potential must be differentiable. The normality rule also applies to the corresponding potential. Therefore, constitute relation can be given, e_ p =

∂φ ∂σ

ð3:109Þ

∂φ - V_k = ∂Ak

ð3:110Þ

q ∂φ = T ∂g

ð3:111Þ

-

The potentials φ and φ must be non-negative, convex, and zero at the origin to satisfy Clausius–Duhem inequality. According to Lemaitre and Chaboche (1990), the normality rule is sufficient to ensure the satisfaction of the second principle of

126

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Thermodynamics

thermodynamics, (assuming all entropy generating mechanisms are represented in the potential) but it is not a necessary condition. However, this assumption cannot be substantiated mathematically because it ignores entropy generation due to other mechanisms. This rule applies to “generalized standard materials” under thermomechanical loads only. Lemaitre and Chaboche (1985) define standard material as  applies. This first that for which only the first of the above three rules E_ P = ∂φ ∂σ relation yields the plasticity or viscoplasticity. The standard material here would only include metals under thermo-mechanical loading. Determination of dissipation potential in Newtonian mechanics is an empirical process. The dissipation potential represents the energy dissipated in the system by heat, mechanical work at the lattice level, and all other mechanisms. While potential cannot be directly measured, state variables that define the potential can be measured or indirectly calculated. Therefore, these potentials are defined in terms of state _ V_k , -T q or dual variables σ, Ak, g. variables e, The dissipation potentials can be written as a function of the rate of state variables or as a function of state variables themselves: [Note that in Newtonian mechanics dissipation potential is not a function of entropy directly. It is usually a function of strain, or stress.]  q _ V_k , φ e, or φðe, T, V k Þ T

ð3:112aÞ

φ ðσ, Ak , gÞ or φ ðE, T, V k Þ

ð3:112bÞ

Decoupling of Intrinsic and Thermal Dissipation The dissipation process involves reversal mechanisms, because we assume that thermal loading in an unconstrained system does not lead to any irreversible dissipation. Thermal dissipation is usually treated separately. Therefore, the dissipation potential can be written as the sum of two terms one representing intrinsic dissipation and the other thermal dissipation. φ = φ1 ðσ, Ak Þ þ φ2 ðqÞ

ð3:113Þ

The second law of thermodynamics must be satisfied by each dissipation term. The following inequalities now must be satisfied: ∂φ1 ∂φ1 ≥0 þ Ak σ : eP - Ak V_ k = σ : ∂σ ∂Ak -g

∂φ q - -g 2 ≥0 T ∂g

ð3:114aÞ ð3:114bÞ

3.3

Second Law of Thermodynamics

3.3.10

127

Time-Independent Dissipation (Instantaneous Dissipation)

When we were discussing the laws of thermodynamics, it was stipulated that energy exchange could not happen instantaneously. However, in constitutive modeling of materials assuming that the material response is independent of the rate simplifies the modeling and analysis significantly. Therefore, such a simplification yields the theory of plasticity. Asstated by Lemaitre and Chaboche (1990) when the dissipa

tion potential φ e_ p , V_k is a positive, homogeneous function of degree one, its dual function φ(σ, Ak) is non-differentiable. Convexity of the dissipation potential φ(σ, Ak) is checked with an indicator function: When φ = 0 if f < 0 E_ P = 0

ð3:115aÞ

φ = þ 1 if f = 0 E_ P ≠ 0

ð3:115bÞ

The indicator function f (also referred to as yield criteria) must be convex and defined by the same dual variables,Ak f(σ, Ak). Based on the normality rule, it is possible to show that ∂F _ e_ = λ if ∂σ p

(

f_ = 0 f =0

ð3:116Þ

where F is a potential function (yield surface/function) that is equal to f in the case of associative plasticity theories and λ_ is a scalar multiplier determined by the consistency condition of f_ = 0: Equations describing the normality rule are then given by e_ P = λ_

∂F _ ∂f =λ ∂σ ∂σ

∂F ∂f - V_k = λ_ = λ_ ∂Ak ∂Ak

ð3:117Þ ð3:118Þ

when the plastic strain increment is not normal to F function (yield surface). Then the plastic strain is normal to a new function Q, plastic potential. However, this is called non-associative plasticity, because the increment of the plastic strain is not normal to the yield function F but to a plastic potential function Q.

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Non-associative plasticity is governed by ∂Q e_ = λ_ if ∂σ P

Q=0 Q_ = 0, if < 0 → E_ P = 0

ð3:119Þ

The theory of plasticity fundamentals and implementation details are discussed in the following chapters.

3.3.11

Dissipation Power and Onsager Reciprocal Relations

Earlier in the chapter, we discussed separating internal entropy production into two parts, namely, mechanical work dissipation and the dissipation due to heat conduction. Specific entropy, s, the equation was derived earlier as    ds 1 q 1 du þ div - σ ij Dij þ qi,i ≥ 0 dt ρ T T dt

ð3:120Þ

Remembering the first law of thermodynamics ρ

du = σ ij Dij þ ρr - qi,i dt

ð3:121Þ

We can write specific entropy rates as r 1 ds 1 q = σ D þ ρT i,i dt ρT ij ij T

ð3:122Þ

Earlier we also derived specific entropy production as γ

q 1 ds r - þ q - i  gradT dt T ρT i,i ρT 2

ð3:123Þ

1 q  gradT σ D ρT ij ij ρT 2

ð3:124Þ

Hence, we can write γ=

Thus, internal entropy generation is separated into two parts; the first part is due to mechanical work dissipation, and the second part is due to irreversible heat conduction. The strong form of Clausius–Duhem inequality requires 1 σ D >0 ρT ij ij

ð3:125Þ

3.3

Second Law of Thermodynamics

-

129

qi  gradT > 0 ρT 2

ð3:126Þ

From here onward, we will use Malvern’s (1969) description of Onsager reciprocal relations, as follows: “The terms in the internal entropy production are called generalized irreversible forces X and fluxes J. Selection of force and flux term is arbitrary. However, it is assumed that the dot product of the generalized irreversible force X and the generalized flux vector J must give the dissipation power, per unit mass”. Applying this concept to the formulation above, we can assign: 1 σ ij and - ρT1 2  gradT: Generalized irreversible forces, ρT Generalized fluxes Dij and qi Hence, we can write γ = X i ∙ J = X im Jm

ð3:127Þ

Constitutive equations give the fluxes as functions of the forces or vice versa. Hence, we can write a phenomenological relation between generalized irreversible fluxes and forces as follows: J m = Lmk X ik

ð3:128aÞ

X ik = akm J m

ð3:128bÞ

where it is assumed that coefficients satisfy the Onsager reciprocal relations (Onsager 1931) Lmk = Lkm and akm = amk

ð3:129Þ

“It is pointed out that in certain cases the Onsager reciprocal relations must be replaced by amk(b) = akm(-b), or amk = - akm, for example in systems affected by a magnetic field b” (de Groot 1952; Malvern 1969). The symmetry of the constitutive matrix may also be affected by other factors such as material anisotropy or length scale affects. Substitution of the phenomenological constitutive relations into the entropy production yields two quadratic equations γ=

1 L X i X i ≥ 0 and γ = akm J k J m ≥ 0 T mk m k

ð3:130Þ

These quadratic forms must be positive-definite. A necessary and sufficient condition for the positive definiteness of a quadratic equation having an asymmetric coefficient matrix with real elements is that all the eigenvalues |Lmk - λδmk| = 0 or | akm - λδkm| = 0 must be positive. The second quadratic equation is called a dissipation function, Q

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Thermodynamics

Table 3.3 Examples of different thermodynamic forces and corresponding fluxes in some dissipative processes Primary mechanism Heat conduction Plastic deformation of solids Chemical reaction Mass diffusion Electrochemical reaction

Thermodynamic force, X Temperature gradient

Thermodynamic flux Heat flux

Stress

Plastic strain

Reaction affinity Chemical potential Electrochemical potential

Reaction rate Diffusion flux Corrosion rate density

Examples Fatigue, creep, wear Fatigue, creep, wear Corrosion, wear Wear, creep Corrosion

After Imaninan and Modarres (2016)

Q = akm J k J m

ð3:131Þ

Based on the above definition, the generalized irreversible force can be given by the following relation: X ik =

1 ∂Q 2 ∂J k

ð3:132Þ

Ziegler (1963) has shown that these phenomenological relations, Eqs. (3.128a, 3.128b, 3.131) and Onsager relation if they have symmetric dissipation function, follow from a principle of the maximum rate of entropy production or maximum dissipation power, which he deduces for “quasi-static processes” by a statistical mechanics approach modified to include irreversible processes” (Malvern 1969). Table 3.3 lists some of the thermodynamic forces and fluxes for some dissipation mechanisms. Here we conclude our discussion of some fundamental concepts of thermodynamics as it relates to continuum mechanics. For further understanding of thermodynamics concepts, readers are encouraged to refer to thermodynamics books listed in the References section of this chapter.

References Anahita Imanian, Mohammad Modarres (2016) Thermodynamics as a fundamental science of reliability, 2016/12, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, Volume 230, Issue 6, Pages 598–608, Publisher SAGE Publications Boltzmann, L. (1877). Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI, 373–435 (Wien. Ber. 1877, 76:373–435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, pp. 164–223, Barth, Leipzig, 1909 [Kim Sharp* and Franz Matschinsky, Translation of Ludwig Boltzmann’s Paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability

References

131

Calculations Regarding the Conditions for Thermal Equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, 1877, LXXVI, 373–435 (Wien. Ber. 1877, 76:373–435)]. Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, pp. 164–223, Barth, Leipzig, 1909 Entropy, 2015, 17, 1971–2009. Callen, H. B. (1985). Thermodynamics and an introduction to thermostatistics (2nd ed.). Wiley. Callister, W. D., Jr., & Rethwisch, D. (2010). Materials science and engineering and introduction. Wiley. Clausius, R. (1850). Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen. Annalen der Physik. 79 (4): 368–397, 500–524. Coleman, B. D. (1964). Thermodynamics of materials with memory. Archive for Rational Mechanics and Analysis, 17, 1–46. Coleman, B. D., & Mizel, V. J. (1964). Existence of caloric equations of state in thermodynamics. The Journal of Chemical Physics, 40, 1116–1125. Coleman, B. D., & Mizel, V. J. (1967). Existence of entropy as a consequence of asymptotic stability. Archive for Rational Mechanics and Analysis, 25, 243. De Groot, S. R. (1952), Thermodynamics of Irreversible Processes, Published by North Holland/ Interscience, Amsterdam/NY, 1952, First Reprint, 1952 DeHoff, R. T. (1993). Thermodynamics in materials science. McGraw Hill. Eddington, A. S. (1958). The nature of the physical world. University of Michigan Press. Freeman J. Dyson (1954), Scientific American, Vol. 191, No. 3 (September 1954), pp. 58–63 (6 pages) Halliday, D., & Resnick, R. (1966). Physics. Wiley. Jean Lemaitre, Jean-Louis Chaboche, Mechanics of Solid Materials, Cambridge University Press, 1985 Karl Ziegler (1963), The Nobel Prize in Chemistry 1963, Prize motivation: for their discoveries in the field of the chemistry and technology of high polymers, https://www.nobelprize.org/prizes/ chemistry/1963/ziegler/facts/ Lars Onsager (1931), Reciprocal Relations in Irreversible Processes. I., Phys. Rev. 37, 405, 15 February 1931 Lemaitre, J., & Chaboche, J. L. (1990). Mechanics of solid materials. Cambridge University Press. Malvern, L. E. (1969). Introduction to the mechanics of continuous medium. Prentice-Hall. Naderi, M., Amiri, M., & Khonsari, M. M. (2009). On the thermodynamic entropy of fatigue fracture. Proceedings of the Royal Society A: Mathematical, Physical and Engineering, 466(2114), 423–438. Planck, M. (1900a). Über eine Verbesserung der Wienschen Spektralgleichung. Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 202–204. Translated in ter Haar, D. (1967). On an improvement of Wien’s equation for the spectrum (PDF). The old quantum theory (pp. 79–81). Pergamon Press. LCCN 66029628. Planck, M. (1900b). Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum. Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 237. Translated in ter Haar, D. (1967). On the theory of the energy distribution law of the normal spectrum (PDF). The old quantum theory (pp. 82). Pergamon Press. LCCN 66029628. Planck, M. (1900c). Entropie und Temperatur strahlender Wärme (Entropy and temperature of radiant heat). Annalen der Physik, 306(4), 719–737. Bibcode:1900AnP...306..719P. https://doi. org/10.1002/andp.19003060410 Planck, M. (1900d). Über irreversible Strahlungsvorgänge (On irreversible radiation processes) (PDF). Annalen der Physik, 306(1), 69–122. Bibcode:1900AnP...306...69P. https://doi.org/10. 1002/andp.19003060105 Ramsey, N. F. (1956, July 1). Thermodynamics and statistical mechanics at negative absolute temperatures. Physics Review, 103, 20.

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Truesdell, C., & Noll, W. (1985). The non-linear field theories of mechanics. Springer. Truesdell, C. and Toupin, R.A. (1960) The Classical Field Theories. In: Flugge, S., Ed., Handbuck der Physik, Vol. III/I, Springer-Verlag, Berlin. Tu, K. N., & Gusak, A. M. (2019). A unified model of mean-time-to-failure for electromigration, thermomigration, and stress-migration based on entropy production. Journal of Applied Physics, 126, 075109. Yun, H., & Modarres, M. (2019). Measures of entropy to characterize fatigue damage in metallic materials. Entropy, 21(8), 804.

Chapter 4

Unified Mechanics Theory

The term unified is used to describe the unification of universal laws of motion of Newton and laws of thermodynamics ab initio level. As we discussed in earlier chapters, Newton’s laws do not account for energy loss or degradation. They only govern what happens to a system in the initial moment a load is applied to a brandnew structure. However, the laws of thermodynamics control what happens after the initial moment over time. Historically, continuum mechanics is based on the laws of Newton only, and phenomenological test data fit empirical models that are supposed to satisfy thermodynamics’ laws introduce energy loss, dissipation, and degradation. In the next section, we will discuss the earlier work on using thermodynamics in continuum mechanics.

4.1

Literature Review of Use of Thermodynamics in Continuum Mechanics

Efforts to unify Newtonian mechanics and thermodynamics have been attempted since the first formulation of the laws of thermodynamics, because Newton’s laws do not account for dissipation of energy and degradation researchers used different methods to incorporate energy dissipation into mechanics formulations to be able to formulate degradation, fracture, fatigue, and life span predictions. Most of these efforts have been based on phenomenological curve fitting methods, where an empirical potential is used to curve fit the dissipation, failure, or degradation test data. There is no unifying theme among these models other than fitting a function to a test data and using it in conjunction with Newton’s laws. For the sake of completeness, we will summarize some of them. While we do not claim to include every effort, we will summarize the most well-known work in the field. Lagrangian mechanics (1788) [long before Thermodynamics laws were written] can be considered the first attempt to include non-conservative forces, into Newton’s © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_4

133

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laws. Quoting from Haddad’s (2017) seminal work on the literature review of thermodynamics, “In an attempt to generalize classical thermodynamics to non-equilibrium thermodynamics, Onsager (1931, 1932) developed reciprocity theorems for irreversible processes based on the concept of a local equilibrium that can be described in terms of state variables that are predicated on linear approximations of thermodynamic equilibrium variables. Onsager’s theorem pertains to the thermodynamics of linear systems, wherein a symmetric reciprocal relation applies between forces and fluxes. In particular, the force exerted by the thermal gradient causes a flow or flux of matter in thermo-diffusion. Conversely, a concentration gradient causes a heat flow, an effect that has been experimentally verified for linear transport processes involving thermo-diffusion, thermo-electric, and thermo-magnetic effects. Classical irreversible thermodynamics as originally developed by Onsager characterizes the rate of entropy production of irreversible processes as a sum of the product of fluxes with their associated forces, postulating a linear relationship between the fluxes and forces. The thermodynamic fluxes in the Onsager formulation include the effects of heat conduction, the flow of matter (i.e., diffusion), mechanical dissipation (i.e., viscosity), and chemical reactions. This thermodynamic theory, however, is only correct for near thermodynamic equilibrium processes wherein a local and linear instantaneous relation between the fluxes and forces holds.” “Building on Onsager’s classical irreversible thermodynamic theory, Prigogine (1955, 1961, 1968) developed a thermodynamic theory of dissipative non-equilibrium structures. This theory involves kinetics describing the behavior of systems that are far away from thermodynamic equilibrium states. Prigogine’s thermodynamics lacks the thermodynamic fundamental equation] function of the system, and hence, his concept of entropy for a system away from [thermodynamic] equilibrium does not have a total differential. Furthermore, Prigogine’s characterization of dissipative structures is predicated on a linear expansion of the entropy function about a particular [thermodynamic] equilibrium, and hence, is limited to the neighborhood of the [thermodynamic] equilibrium. This is a severe restriction on the applicability of this theory. In addition, his entropy cannot be calculated nor determined” (Prigogine & Herman, 1971; Prigogine & Nicolis, 1977; Haddad, 2017). Prigogine’s work (Prigogine & Defay, 1954; Prigogine, 1955, 1957, 1961; Prigogine & Herman, 1971; Prigogine & Nicolis, 1977) on dissipative structures and their role in thermodynamic systems far from equilibrium won him the Nobel Prize in Chemistry in 1977. In engineering mechanics, most of our states are near the thermodynamic equilibrium point. Therefore, Prigogine’s work is important to understand. Quoting from Prigogine’s own Wikipedia webpage, “Prigogine proved that dissipation of energy in chemical systems results in the emergence of new structures due to internal self-re-organization [in the previous sentence if we replace “chemical” with “mechanical”, it still holds]. Prigogine (1955) drew connections between dissipative structures and the Rayleigh-Bénard instability, and the Turing mechanism. Turing’s (1952) mechanism describes how patterns in nature such as stripes and spots can arise naturally out of a homogeneous uniform state. RayleighBénard instability is a type of natural convection, occurring in a plane horizontal

4.1

Literature Review of Use of Thermodynamics in Continuum Mechanics

135

layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells (Getling, 1998; Koschmieder, 1993).” “Prigogine’s dissipative structure theory led to pioneering research in selforganizing systems, as well as philosophical inquiries into the formation of complexity on biological entities and the quest for a creative and irreversible role of time in the natural sciences. Prigogine’s formal concept of self-organization was used also as a “complementary bridge” between General Systems Theory and thermodynamics, conciliating the cloudiness of some important systems theory concepts with scientific rigor.” “Definitions of self-organization and general systems theory are described as, ‘self-organization’, also called (in the social sciences) spontaneous order, is a process where some form of overall order arises from local interactions between parts of an initially ‘disordered’ system. The term ‘disorder’ here defines the initial state. The process can be spontaneous when sufficient energy is available, not need control by any external agent. It is often triggered by random fluctuations, amplified by positive feedback. The resulting organization is wholly decentralized, distributed over all the components of the system, [i.e., cooling molten metal yields a solid polycrystal metal is a good example]. As such, the organization is typically robust and able to survive or self-repair substantial perturbations. Chaos theory discusses self-organization in terms of islands of predictability in a sea of chaotic unpredictability. General systems theory is the interdisciplinary study of systems. A system is a cohesive conglomeration of interrelated and interdependent parts that is either natural or fabricated. Every system is delineated by its spatial and temporal boundaries, surrounded, and influenced by its environment, described by its structure and purpose or nature, and expressed in its functioning. In terms of its effects, a system can be more than the sum of its parts if it expresses synergy or emergent behavior. Changing one part of the system usually affects other parts and the whole system, with predictable patterns of behavior. For systems that are self-learning and self-adapting, the growth and adaptation depend upon how well the system is adjusted to its environment. Some systems function mainly to support other systems by aiding in the maintenance of the other system to prevent failure. The goal of systems theory is systematically discovering a system’s dynamics, constraints, and conditions and elucidating principles (purpose, measure, methods, tools, etc.) that can be discerned and applied to systems at every level of nesting, and in every field for achieving optimized equifinality” (Beven, 2006) quoting from Wikipedia to define, “equifinality is the principle that in open systems a given end state can be reached by many potential means. It also means that a goal can be reached in many ways. The same final state may be achieved via many different loading paths. However, in closed systems, a direct cause-and-effect relationship exists between the initial condition and the final state of the system. Biological and social systems are open systems, however, operate quite differently.” Simply, the idea of equifinality implies that the same results may be achieved with different initial conditions and in many ways. Entropy has also been used beyond positive sciences. The seminal paper on the topic was published by Jaynes (1957) who proved that statistical mechanics, which is based on Boltzmann’s entropy formulation, could be generalized to information

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theory independent of experimental verification. Jaynes (1957), using von-Neumann–Shannon’s definition of entropy as a measure of uncertainty represented by a probability distribution, showed that entropy becomes a primitive concept more fundamental than energy. Jaynes (1957) also proved that thermodynamic entropy is identical to the information-theory entropy of the probability distribution except for the presence of Boltzmann’s constant. Valanis (1971) established the irreversibility and existence of entropy as a state function in Newtonian mechanics formulation. Valanis (1971) was able to prove this for reversible and irreversible systems and processes, irrespective of the constitutive equations of the system. He postulated that, in the case of a reversible system, entropy is a function of the deformation gradients (strains) and temperature; in the case of an irreversible system, it is also a function of n internal variables necessary to describe the irreversibility of the system. Valanis (1971) based his proof of the existence of entropy as a consequence of the integrability of the differential form of the first law using an extended form of the Caratheodory conjecture, to the effect that in the neighborhood of a thermodynamic state there exist other states, which are not accessible by processes, which are reversible and adiabatic. Valanis (1971) proposed the endochronic plasticity theory, which deals with the plastic response of materials employing memory integrals, expressed in terms of memory kernels. Formulation of this theory is based on thermodynamics concepts and provides a unified point of view to describe the elastic-plastic behavior of materials since it places no requirement for a yield surface and “loading function” to distinguish between loading and unloading. A key ingredient of the theory is that the deformation history is defined with respect to a deformation memory scale called intrinsic time. In the original version of the endochronic theory, proposed by Valanis (1971), the intrinsic time was defined as the path length in the total strain space. The so-called endochronic theory violates the second law of thermodynamics and leads to constitutive relations, which characterize inherently unstable materials (Rivli, 1981). Aiming the correction of this deficiency, a new version of the endochronic theory was developed by Valanis and Komkov (1980) in which the intrinsic time was defined as the path length in the plastic strain space. Rice (1971) used an internal-variable thermodynamic formalism for the description of the microstructural rearrangements to characterize metal plasticity. Rice (1971) postulated that the theoretical foundations of constitutive relations at finite strain for metals exhibiting inelasticity are a consequence of specific structural rearrangements on the microscale of the material, such as metals deforming plastically through dislocation motion. Rice (1971) is a generalization of his earlier work by Kestin and Rice (1970), where it is assumed that each microstructural arrangement proceeds at a rate governed by its associated thermodynamic force. This work became the framework for the time-dependent inelastic behavior in terms of a flow potential and reduces to statements on the normality of plastic strain increment to yield surface in the time-independent case. Rice’s (1971) formulation assumes inelastic deformation under macroscopically homogeneous strain and temperature as a sequence of constrained equilibrium states. Using equilibrium thermodynamic formalism and thermodynamic potentials, Rice (1971) relates changes in the local

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Literature Review of Use of Thermodynamics in Continuum Mechanics

137

structural rearrangements to corresponding changes in the macroscopic stress or strain states. However, Rice (1971) assumes that a discrete set of scalar internal variables characterize the state of internal rearrangement. Each internal variable characterizes a specific structural rearrangement. Unfortunately, this classification does not satisfy the second law of thermodynamics. According to the second law of thermodynamics, not the individual internal variables like stress or strain, but only the fundamental relation [entropy generation rate] determines the new structural rearrangement. Rice (1971) also assumes that if various equilibrium states are considered each corresponding to the same set of values for the thermodynamic internal variables, then neighboring states are related by the usual laws of thermoelasticity. This later assumption also violates the second law of thermodynamics, because according to the second law of thermodynamics even thermo-elastic deformation leads to irreversible entropy generation. The easiest way to explain this is fatigue under elastic loading or molecular dynamics simulations under elastic loading. According to the traditional thermo-elasticity assumption, if there is no inelastic deformation, thermodynamics internal state variables do not change. As a result, materials could never fatigue under elastic loading. Of course, this violates the second law of thermodynamics, because there is no reversible thermodynamic process in metals or nature in general. Rice (1971) postulates that “if neighboring constrained equilibrium states corresponding to different sets of internal variables are considered, we must write, [using his variables without adopting to our notation]

 V 0 S : δE - f α δξα þ θδ V 0 η = δ V 0 u

ð4:1Þ

In the formulation, V0 denotes volume at some reference state at a temperature θ0, S is Kirchoff stress (symmetric), δE is the increment of macroscopic Lagrange (or material) strain tensor, fα defines thermodynamic forces ( f1, f2, . . ., fn acting on internal variables, δξα set of (total number is unspecified as n) thermodynamic internal state variables that characterize the state of internal rearrangement, θ is temperature, δ(V0η) is the increment of entropy, and δ(V0u) is the increment of internal energy. This is the most important equation in Rice’s (1971) theoretical framework. The rest of the theoretical framework is based on this equation. This equation finally leads to defining a yield surface as a potential, which is the most important ingredient for the theory of incremental plasticity. Unfortunately, in this equation given by Rice (1971), there is no relation between entropy and thermodynamic forces ( f1, f2, . . ., fn ) acting on internal variables in the formulation. Derivatives of these forces with respect to entropy are considered zero. As a result, in practice defining these thermodynamic forces and internal variables becomes a trialerror process (or even art). This point was also emphasized in Chap. 3 when we discussed thermodynamic potential. Because of this problem, yield surface is an empirical function with different coefficients for different materials for different loading paths for different temperatures, different strain rates, and different length scales or geometries of the materials. Of course, it is possible to come up with a

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different yield surface for the same material, using different constants. Therefore, it would be more accurate to call these thermodynamic forces pseudo-thermodynamics forces, since they are empirical, and many are not directly related to the laws of thermodynamics. While Rice (1971) interprets, fαδξα as the work of increments of internal variables, in practice when we obtain a yield surface from experiments many of our exponents, or “material constants” have no physical meaning to justify as an internal thermodynamic variable, because two different scientists can have significantly different number of constants to define a yield surface for the same material. Our argument becomes clearer in the following formulation by Rice (1971).” Free energy ϕ and its Legendre transform ψ (complementary energy) are given by the following relations: ϕ = ϕðE, θ, ξÞ = u - θη ψ = ψ ðS, θ, ξÞ = E :

∂ϕ ∂ϕ -ϕ=E : - u þ θη ∂E ∂E

ð4:2Þ

Based on definitions given in Eq. (4.2) and assuming that entropy is constant, the variation of complementary energy can be written as δψ = E : δS þ

1 f α δξα þ ηδθ V0

ð4:3Þ

Of course, in real life when a variation is applied, in addition, to stress δS internal variables δξα and temperature δθ, and entropy will of course change. Nevertheless, here entropy change is assumed zero, θδη = 0. If the internal variables δξα = 0 are also constant, Rice (1971) derives the thermo-elastic constitutive equation based on Newtonian Mechanics, where it is assumed that no elastic deformation can generate entropy and all derivatives with respect to entropy are zero: E=

∂ψ ðS, θ, ξÞ ∂S

and S =

∂ϕðE, θ, ξÞ ∂E

ð4:4Þ

Thermodynamics forces associated with internal variables are defined by f α = V0

∂ψ ðS, θ, ξÞ ∂ϕðE, θ, ξÞ = - V0 ∂ξα ∂ξα

ð4:5Þ

Using Maxwell’s relations, which are a set of equations derivable from the symmetry of second derivatives of potentials. Rice (1971) arrives at ∂E ðS, θ, ξÞ 1 ∂f α ðS, θ, ξÞ = 0 ∂ξα ∂S V

ð4:6Þ

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Equation (4.6) serves as the central equation in the rest of Rice (1971) in the development of the theoretical foundations of inelastic constitutive theory. A flow potential for the inelastic strain rate is proposed by postulating that at any given temperature and pattern of internal rearrangement within the material, the rate at which any specific structural rearrangement occurs is fully determined by the thermodynamic forces. The concept, used by Rice (1971), is identical to the basis for the unified mechanics theory where pseudo-dynamic forces are replaced with the thermodynamic fundamental equation of the material. It is assumed that any specific microstructural configuration can be fully determined by the thermodynamic force associated with that microstructural rearrangement, as follows: ξ_ β is a function of f β ,θ,ξ ðfor β = 1, 2, . . . , nÞ

ð4:7Þ

Rice (1971) further postulates that the current temperature and pattern of internal microstructural rearrangement may enter the kinetic equations as parameters, but the influence of the macroscopic stress state on a given microstructural rearrangement appears only through the fact that the associated force is dependent on stress. The author acknowledges that this is not the most general class of kinetic equations; however, it does represent conventional metal plasticity behavior where the associated shear stress governs slip on a crystallographic system or at the discrete dislocation. Based on Newtonian mechanics, it is assumed that the force on a given segment of the dislocation line governs its motion. Rice (1971) recast the kinematic equations in an integral form as follows: ∂ ξ_ β = ∂f β

Z

f

0

ξ_ α ðf , θ, ξÞ df α

ð4:8Þ

where the integral is carried out at fixed values of θ and ξ and defines a point function since each term in the integrand is an exact differential. Rice (1971) further postulates that thermodynamic forces may be viewed as functions of the macroscopic stress, S, temperature θ,and internal variables ξ and then defines a function ΩðS, θ, ξÞ =

1 V0

Z 0

f

ξ_ α ðf , θ, ξÞ df α

ð4:9Þ

The stress derivative of this function is given by ∂f ðS, θ, ξÞ ∂ΩðS, θ, ξÞ 1 = 0 ξ_ α ðf , θ, ξÞ α ∂S ∂S V

ð4:10Þ

Rice (1971) elegantly shows that the right-hand side of this equation is equal to the inelastic strain rate in the following way. Let δE be the difference (variation) in strain between the neighboring constrained equilibrium states, differing by δS, δθ, δξ. It is assumed that the variations in the

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internal variables correspond to variations in the macroscopic strain. An inelastic or plastic portion (δE)p of the strain difference is defined as that part which would result from the change in internal variables if stress and temperature were held fixed: ðδE Þp =

∂EðS, θ, ξÞ 1 ∂f α ðS, θ, ξÞ δξα = 0 δξα ∂S ∂ξα V

ð4:11Þ

Similarly, an elastic (or thermo-elastic) portion (δE)e is defined as that which would result from the change in stress and temperature, if the other internal variables were held fixed. As a result, the following relations can be written: δE = ðδE Þe þ ðδE Þp

ð4:12Þ

From the complementary energy, we can write ðδE Þe =

2

2

∂ ψ ∂ ψ : δS þ δθ ∂S∂S ∂S∂θ

ð4:13Þ

Based on the viewpoint of the classical theory of irreversible processes, e.g., De Groot and Mazur (1962a), Rice (1971) assumes that macroscopically homogeneous deformation processes may be suitably approximated as the sequence of constrained thermodynamic equilibrium states, each fully characterized by values for E, θ, ξ at the corresponding instant. As a result, Rice (1971) assumes that all the preceding relations are valid during this process. Thus, the following relations can be written:
e
p E_ = E_ þ E_

ð4:14Þ


p 1 ∂f α ðS, θ, ξÞ _ E_ = 0 δ ξα ∂S V

ð4:15Þ

An analogous expression for the elastic portion can be written from Eq. (4.13). Therefore, Ω is called a flow potential, (or more commonly known as a yield surface). According to the normality rule, it can be shown that the inelastic portion of the strain-rate vector is normal to the surface of constant flow potential in stress space.
p ∂ΩðS, θ, ξÞ E_ = ∂S

ð4:16Þ

Rice (1971) defines thermodynamics restriction on the formulation in the following way. Following the earlier stated assumption that material is taken from one constrained thermodynamic equilibrium state to another by an irreversible process extending from time t1 to t2. Then the first and second laws of thermodynamics in classical form dealing exclusively with the comparison of constrained equilibrium states, are given by

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Literature Review of Use of Thermodynamics in Continuum Mechanics

Z

t 2 Z

t1

   qdA dt = V 0 u2 - u1

Z T  z_ dA þ A

t1

ð4:17Þ

A

t 2 Z

Z

141

   ðq=θÞdA dt ≤ V 0 η2 - η1

ð4:18Þ

A

where T is the surface stress vector (traction), q is the heat supplied to the surface per unit area (positive inward), z_ is the deformation rate, u is the internal energy, and η is the entropy. Time integral from t1 to t2 follows the irreversible process. It is important to point out that elastic deformation here is considered a reversible process; hence, fatigue under elastic stresses is not possible. Alternatively, it is assumed that during a molecular dynamics simulation under elastic loading atoms return to their original lattice site. Based on earlier work by Kestin and Rice (1970), following the classical approach to irreversible processes for a macroscopically homogeneous system with internal variables, Rice (1971) views all actual processes as a sequence of constrained equilibrium states. The work of surface stresses can be written in terms of internal stresses and strains as V 0 S : E_ for the constrained equilibrium states and the temperature is assumed uniform in the material. Then the first and second laws of equilibrium thermodynamics can be generalized for any actual homogenous process as follows: S : E_ þ Q_ = u_ Z 1 Q = 0 q dA ≤ θη_ V A

ð4:19Þ

where Q is the total heat supply rate per unit reference volume. Rice (1971) further defines entropy generation rate due to internal variables and conjugate thermodynamic forces as σ=

1 f α ξ_ α ≥ 0 θV 0

ð4:20Þ

Then, Rice (1971) defines the total change in entropy as, the summation of Q, the total heat supply rate per unit reference volume plus the entropy generation due to internal variables and thermodynamic forces as Q þ θσ = θη_

ð4:21Þ

Q is defined as positive inward, in Rice’s (1971) formulation, which we are quoting without modification. It is assumed that heat entering from outside does not increase the irreversible entropy generation. On the other hand, heat generated in the system and dissipating outward leads to irreversible entropy generation. The second law requires that the non-negative work rate of associated forces on internal variables is given by

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∂ϕðE, θ, ξÞ ξ_ α ≥0 ∂ξα

Unified Mechanics Theory

ð4:22Þ

Remember in Eq. (4.5) fα was defined as f α = - V0

∂ϕðE, θ, ξÞ ∂ξα

ð4:23Þ

Rice (1971) concludes “That is, the rates of internal arrangement occurring during a process must be such that the free energy would decrease if strain and temperature were held fixed at current values.” In Newtonian mechanics, Eqs. (4.22) and (4.23) are primarily used to require that flow potential must be convex and smooth surface but are not included in Newton’s universal laws to degrade the energy or material. Degradation is imposed by empirical formulation utilizing a damage potential obtained from experimental data. Dissipation is imposed by an empirical constant, like damping, or by an empirical function. Later Rice (1977) derived thermodynamic restrictions on the quasi-static growth or healing of Griffith cracks. However, as presented they were nothing beyond global restrictions on the detailed molecular kinetics of crack growth. Therefore, there was no attempt to integrate these restrictions into Newton’s universal laws of motion. Other work by Rice (1977) on the use of thermodynamics in the theory of plasticity and fracture mechanics was based on or a derivative of his work discussed here. Bazant (1972) used thermodynamics for modeling the mechanics of interfaces in concrete structures, where a thermodynamically consistent formulation of interacting continua with surfaces based on surface thermodynamics was proposed. Creep, shrinkage, and delayed thermal dilation and their interaction with adsorbed water layers confined between two solid adsorbent surfaces were accounted for in the formulation. Kijalbaev and Chudnovsky (1970) and Chudnovsky (1973, 1984) proposed a probabilistic model of the fracture process unifying the phenomenological study of long-term strength degradation of materials, fracture mechanics, and statistical approach to fracture mechanics. Chudnovsky (1984) used irreversible thermodynamics to model the deterministic side of the failure phenomenon and stochastic calculus to account for the failure mechanisms controlled by “chance,” particularly the random roughness of fracture surfaces. Kijalbaev and Chudnovsky (1970) and Chudnovsky (1973, 1984) derived the entropy production for an elastic medium with damage. Citing Swalin (1972) on the invariance of the entropy jump during a phase transition with respect to stresses and temperature, Chudnovsky (1973) hypothesized that the entropy jump invariance can be used to predict the local failure. The author proposed pairing of a phenomenological critical damage parameter and entropy jump, ΔS, requirement as a local failure criterion. It is important to point out that Chudnovsky proposed calculating entropy generation in a fracture

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mechanics problem for all mechanisms, damage, translation, rotation isotropic expansion, and distortion of the active crack zone. Klamecki (1980a, b, 1984) proposed an entropy-based model of plastic deformation energy dissipation in sliding. In the entropy production rate author included both microstructural changes and internal heat generation. This is the first paper in the literature we could find where a partial derivative of entropy was taken with respect to internal energy, microstructure generated entropy, surface area, and mass. As a result, Klamecki (1980a, 1984) postulated that the temperature component of internal energy U, microstructural energy, G, surface energy, and chemical potential are represented in entropy calculations, and defined entropy S by S = SðU, G, A, M Þ dS =

ð4:24Þ

∂S ∂S ∂S ∂S dU þ dG þ dA þ dM ∂U ∂G ∂A ∂M

ð4:25Þ

TdS = dU þ ϕdG þ γdA þ μdM

ð4:26Þ

And

[Note: In the derivation of Eq. (4.26) Klamecki assumes U = ST.] where the temperature, internal energy, microstructural energy, surface energy, and chemical potential are represented by T, U, ϕ, γ, and μ, respectively. Klamecki (1980a, b, 1984) defined the conservation of energy by dU = δQ þ δW

ð4:27Þ

where Q represents heat and W represents mechanical work. δ is used to imply the equation is for a process [variation]. Work, W, is assumed to be due to plastic deformation only, and then taken as stress multiplied by plastic strain. The heat, Q, has two components: one is assumed to be due to internal heat generation, R, in the system, and the other one heat flows into the system due to temperature gradient. As a result, internal energy can be written as dU = K i ð∇T Þi þ R þ Mρ - 1 τij dεij

ð4:28Þ

where K is the thermal conductivity and ρ is the mass density. Substituting internal energy given by Eq. (4.28) in entropy Eq. (4.26) leads to TdS = K i ð∇T Þi þ R þ Mρ - 1 τij dεij þ ϕdG þ γdA þ μdM

ð4:29Þ

As Klamecki (1984) points out this definition of entropy is very valuable since it contains, or may be further generalized to contain, all the energy dissipation mechanisms in sliding bodies. Of course, there is nothing in the equation that limits it to siding bodies only. Klamecki (1984) speculated that there are three regimes of

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system behavior, which are of interest when the intent is to describe changes in the system by the use of entropy generation. At thermodynamic equilibrium, the system entropy is at maximum, and the change in entropy with time is zero. When the system is not in equilibrium because of a continual supply of energy to it, it can be near equilibrium or far from the equilibrium state. Klamecki (1984) asserted that the process occurring in the nonequilibrium states could be analyzed by studying the entropy production in the system. The entropy created in the system is given by TdS = R þ Mρ - 1 τij dεij þ ϕdG þ γdA þ μdM

ð4:30Þ

These terms have the general form of a thermodynamic force F, multiplied by a corresponding flux, J. Then entropy production can be expressed using the Onsager relation S=

n X

Fi J i

ð4:31Þ

i=1

where n is the number of internal entropy generation mechanisms. Entropy production rate is defined by n  dJ dF dS X S_ = Fi i þ J i i = dt dt dt i=1

ð4:32Þ

Here, Klamecki (1984) postulates that the first term in the summation represents the entropy generation rate far from the equilibrium state where thermodynamic force does not change over time. The second term represents the entropy generation near equilibrium. As a result, the summation can be summarized as follows: S_ = S_ J þ S_ F

ð4:33Þ

In near equilibrium, entropy production rate S_ F will continuously get smaller, according to the second law of thermodynamics. Thermodynamic forces change in such a way that a stationary state of the minimum rate of entropy generation results in equilibrium, as stipulated by the second law of thermodynamics. S_ F = 0

ð4:34Þ

Klamecki (1984) attributes this principle, Eq. (4.33), to Glansdorff and Prigogine (1971a, b). As stated by Klamecki (1984) far from the thermodynamic equilibrium, no widely accepted entropy production evolution criterion has been formulated so far. Laws of thermodynamics are valid around equilibrium points only. Since in engineering mechanics most of our problems are around equilibrium points and

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solved as a sequence of constrained equilibrium points, not having laws governing far from equilibrium is not a concern. Evans et al. (1993), proposed the fluctuation theorem (FT). A laboratory experiment that verified the validity of the FT was carried out in 2002, where a plastic bead was pulled through a solution by a laser. Fluctuations in the velocity were recorded that were opposite to what the second law of thermodynamics would dictate for macroscopic systems (Wang et al., 2002; Chalmers, 2016; Gerstner, 2002; Searles & Evans, 2004). The fluctuation theorem does not state or prove that the second law of thermodynamics is wrong or invalid. The second law of thermodynamics is valid for macroscopic systems at equilibrium or near equilibrium. Rivas and Martin-Delgado (2017) have also encountered a partial violation of the second law of thermodynamics in a quantum system known as the Hofstadter lattice. Ostoja-Starzewski (2016), Ostoja-Starzewski and Raghavan (2016) proved violations of the second law are relevant as the length and/or time scales become very small. The second law then needs to be replaced by the fluctuation theorem, and mathematically, the irreversible entropy is a sub-martingale. As indicated above, these are far from thermodynamic equilibrium states. This partial violation has no place within the framework of classical thermodynamics because it is a spontaneous event that does not affect the laws of thermodynamics at the macroscale at equilibrium or near equilibrium states. Klamecki (1984) summarized these concepts in a very simple figure. The author postulates that entropy production is a function of l thermodynamic variables and so can be represented by a surface in (l + 1) dimensional space. For ease of plotting only two variables are used in Fig. 4.1. The equilibrium exists at the state of maximum entropy and zero entropy generation rate and is represented by point E in Fig. 4.1. If the system is maintained in a non-equilibrium state by supplying energy to it, two processes are usually considered. Near equilibrium, the system will evolve from its initial state A to state B at the minimum rate of entropy generation. If the system is far from equilibrium, it may move from its initial state of C to state D but at the present, no description of this state or the process in terms of entropy production is available in the literature. The only accepted governing principle is energy conservation for systems far from equilibrium. Whaley (1983) proposed a model for fatigue crack nucleation using irreversible thermodynamics to quantify the damage caused by plastic strain. Whaley’s (1983) model is based on the hypothesis that entropy gain which results from dynamic irreversible plastic strain is a material constant and the plastic strain can be used as a parameter for an empirical function curve-fit to fatigue test data to simulate fatigue behavior of metals. Whaley (1983) also postulated that structural damage by fatigue could be quantified by a single material parameter, the critical entropy threshold of fracture. Whaley (1983) postulated that the critical entropy threshold of fracture is therefore related to the irreversible part of the fracture energy. The critical entropy threshold of static fracture is a random variable and the variability can be quantified by a confidence interval. According to Whaley (1983), the confidence interval for the entropy threshold just comes from the variance of the plastic strain. It is noteworthy that Whaley (1983) assumed that only plastic strain generates irreversible entropy; of

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Fig. 4.1 Graphical representation of entropy generation associated with system states at equilibrium (state E) near equilibrium (state B) and far from equilibrium (state D) as a function of thermodynamic state variables 1 and 2. (After Klamecki (1984))

course, that is not accurate, because using plastic strain alone as a metric violates the second law since if there is no increase in plastic strain but loading under the same stress level, there will be no entropy generation according to this model. Naderi et al. (2010) were able to prove experimentally that total cumulative entropy generation is constant at the time of failure and is independent of sample geometry, load, and loading frequency. They named this critical entropy value Fatigue Fracture Entropy (FFE). Figure 4.2 shows experimental fatigue fracture entropy invariance with respect to geometry and loading frequency. Total accumulated entropy of metals, undergoing repeated cyclic load as it reaches the point of fracture is a constant value, independent of load amplitude, geometry, size of the specimen, frequency, and stress state; Naderi et al. (2010) experimentally validated the invariance of fatigue fracture entropy with respect to loading path and displacement loading amplitude for Al 6061-T6 samples. In Fig. 4.2 fracture fatigue entropy remains at about 4 MJm-3 K-1 for both tension-compression and bending fatigue. Displacement amplitude is varied from 25 to 50 mm. A filled square represents tension-compression; a filled circle represents bending; a filled star represents torsion data. Amiri and Khonsari (2012) and Jang and Khonsari (2018) also did extensive experiments to show that fatigue fracture entropy remains constant independent of loading condition, loading frequency, and sample geometry. Their samples were AISI 1018 carbon steel and Al 7075-T6. Liakat and Khonsari (2015) measured fatigue fracture entropy of un-notched and V-notched specimens. They observed that fatigue fracture entropy remains constant. Imanian and Modarres (2015, 2018) and Yun and Modarres (2019) experimentally proved the concept of using entropy as a degradation metric; authors discussed the entropic characterization of the corrosion-fatigue degradation mechanism. They proposed an entropy-based damage prognostics and health management technique for integrity assessment and remaining useful life prediction of aluminum 7075T651 specimens. Their experimental validations proved that using entropy as a

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Fig. 4.2 Fatigue fracture entropy versus the number of cycles to failure for different bending fatigue tests of Al 6061-T6 with different specimen thickness, frequencies, and displacement amplitudes. Fatigue fracture entropy remains at roughly 4 MJm-3K-1, regardless of thickness, load, and frequency. Displacement amplitude varied from 25 to 50 mm. Filled circle, thickness = 6.35 mm, f = 10 Hz; filled diamond, thickness = 3.00 mm, f = 10 Hz; filled star, thickness = 4.82 mm, f = 10 Hz; unfilled circle, thickness = 6.35 mm, f = 6.5 Hz; unfilled triangle, thickness = 4.82 mm, f = 12.5 Hz; unfilled star, thickness = 6.35 mm, f = 6.5 Hz; unfilled diamond, thickness = 6.35 mm, f = 12.5 Hz. (After Naderi et al. (2010))

thermodynamic state function for damage characterization is an effective way of handling the endurance threshold uncertainties for life prediction purposes. The authors also derived the thermodynamic fundamental equation for corrosion fatigue. Imanian and Modarres (2015, 2018) and Yun and Modarres (2019) also performed experiments where multiple entropy generation mechanisms were contributing to the damage evolution of corrosion-fatigue total entropy for different loading conditions. They measured the cumulative final value of fracture corrosionfatigue entropy. The final entropy value is between 0.7 and 1.5 MJm-3 K-1. The authors observed that there is a narrow band of distribution of entropy to failure data points [fracture entropies] irrespective of loading condition. They concluded that entropy can quantify the uncertainties associated with microstate variabilities. Furthermore, they stated that it reveals the independence of entropy to the loading condition (i.e., failure path). Fracture entropy’s slim distribution band can be interpreted due to uncertainties, such as instrumental measurement errors, the legitimacy of the assumptions considered in entropy evaluation, weak control of the experimental, operational, and environmental conditions, and human error. Sosnovskiy and Sherbakov (2015, 2016) postulated the existence of a generalized theory of evolution based on the concept of tribo-fatigue entropy. The essence of the proposed approach is that tribo-fatigue entropy is determined by the process of degradation of any system due to thermodynamic mechanical effects causing the change in the state. Sosnovskiy and Sherbakov (2016) derived a framework for the

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law of entropy increase in the general form. They also provided extensive experimental validation for the theory of what they called “mechanothermodynamics.” They stated, “It is shown that mechanothermodynamics—a generalized physical discipline—is possible by constructing a bridge between Newtonian mechanics and thermodynamics. The entropy is the bridge between thermodynamics to mechanics.” The first and the second principles of mechanothermodynamics were presented. They formulated the foundation of the general theory of degradation evolution of mechanothermodynamic systems for: • Energy theory of limiting states • Energy theory of damage • Foundations of the theory of electrochemical damage They provided mathematical fundamentals of the theory of interaction between damage caused by loads of different nature (mechanical, thermal, etc.). The authors proposed a single function for the critical damage (limiting) state of metals and polymeric materials operating in different conditions. The analysis of 136 laboratory experiment results showed that logarithmic function is fundamental: it is valid for low-, average-, and high-strength pure metals, alloys, and polymers over a wide range of temperatures of medium (from helium temperature to 0.8 TS, where TS is the material melting temperature) and mechanical stresses (up to the strength limit for single static loading) while the fatigue life was of the order of 106–108 cycles. Mechanothermodynamics uses the same idea as the unified mechanics theory to unify Newtonian mechanics and thermodynamics. That is using entropy as a bridge to connect Newtonian mechanics and thermodynamics, which was first published in Basaran and Yan (1998). Sosnovskiy and Sherbakov (2015, 2016) used an empirical logarithmic function to fit the experimental evolution of degradation. It is not clear why they do not use the Boltzmann equation, since their logarithmic function can be obtained directly from the Boltzmann equation. The term tribo-fatigue-entropy in their work refers to entropy generation in tribology [which is the focus of their research] and the fatigue process. They surmise that if an analogy between light and strain energy is justified, then the strain energy absorption law may be like Bouguer’s light absorption law. This law, which is also exponential, becomes the basis for their empirical degradation function formulation. Haddad (2017) published probably one of the most comprehensive reviews of the history of thermodynamics from its classical to its postmodern forms. Haddad et al. (2005) and Haddad (2019) also provided a general systems theory framework for thermodynamics which attempts to harmonize thermodynamics with classical Newtonian mechanics. The main idea used by Haddad (2019) to unify mechanics and thermodynamics is attributed to Basaran and Yan (1998). Haddad stated, “The dynamical system’s notion of entropy proposed by Haddad et al. (2005, Haddad (2019), Basaran and Yan (1998), Basaran and Nie (2004)), Sosnovskiy and Sherbakov (2016) involving an analytical description of an objective property of matter can potentially offer a conceptual advantage over the subjective quantum expressions for entropy proposed in the literature (e.g., Daróczy entropy, Hartley entropy, Rényi entropy, von Neumann entropy, infinite-norm entropy) involving a

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measure of information. An even more important benefit of the dynamical systems representation of thermodynamics is the potential for developing a unified classical and quantum theory that encompasses both mechanics and thermodynamics without the need for statistical (subjective or informational) probabilities.” Cuadras et al. (2015) proposed a method to characterize electrical resistor damage based on entropy measurements. They postulated that irreversible entropy and the rate at which it is generated are more convenient parameters than resistance for describing damage because they are essentially positive in virtue of the second law of thermodynamics, whereas resistance may increase or decrease depending on the degradation mechanism. They tested commercial resistors to characterize the damage induced by power surges. Resistors were biased with constant and pulsed voltage signals, leading to power dissipation in the range of 4–8 W, which is well above the 0.25 W nominal power to initiate failure. Entropy was inferred from the added power and temperature evolution. They studied the relationship between resistance, entropy, and damage. They stated that the power surge dissipates into heat (Joule effect) and damages the resistor. They observed that there is a correlation between entropy generation rate and resistor failure. Cuadras et al. (2015) concluded that damage could be conveniently assessed from irreversible entropy generation. Cuadras et al. (2016) proposed a method to monitor the aging and damage of capacitors based on their irreversible entropy generation rate (Fig. 4.3). The authors overstressed several electrolytic capacitors in the range of 33 me – 100 mF and monitored their entropy generation rate. A strong relationship between capacitor degradation and entropy generation rate was observed. Therefore, they proposed a threshold entropy generation rate as an indicator of capacitor time-to-failure. This magnitude is related to both capacitor parameters and a damage indicator such as entropy. They validated the model as a function of capacitance, geometry, and rated voltage. Moreover, they identified different failure modes, such as heating, electrolyte dry-up, and gasification from the dependence of entropy generation rate with temperature.

0.100

4

S

P(W)

3

0.075

rate

2 1

Srate(WK−1)

Fig. 4.3 Capacitor entropy generation rate versus time (seconds) for different voltage levels. (After Cuadras et al. (2016))

0 280

0.050

300 320 340 T(K)

360 380

20 V 25 V 30 V 35 V 40 V

0.025

0.000

0

10

20

30 t(s)

40

50

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Cuadras et al. (2017) proposed a method to assess the degradation and aging of light-emitting diodes (LEDs) based on an irreversible entropy generation rate. Researchers degraded several LEDs and monitored their entropy generation rate in accelerated tests. They compared the thermo-electrical results with the optical light emission evolution during degradation. They found a good relationship between aging and entropy generation rate because they both are related to device parameters and optical performance. They proposed a threshold of entropy generation rate as a reliable damage indicator of LED end-of-life that can avoid the need to perform optical measurements to assess optical aging. The method is far more physics-based and beyond the typical empirical statistical models based on curve fitting to test data for lifetime prediction provided in the literature. Cuadras et al. (2017) tested different LED colors and electrical stresses to validate the electrical LED model and analyzed the degradation mechanisms of the devices to validate the model. There has been significant interest in using entropy generation rate as a damage metric to predict degradation, fracture, and fatigue life prediction, such as Basaran and Chandaroy (2002); Basaran and Tang (2002); Basaran et al. (2003, 2005, 2008a, b); Tang and Basaran (2003); Gomez and Basaran (2005, 2006); Lin and Basaran (2005); Gomez et al. (2006); Li et al. (2008); Li and Basaran (2009); Gunel and Basaran (2010, 2011a, b); Basaran and Lin (2007a, b, 2008); Basaran and Nie (2007); Bin Jamal et al. (2020); Sherbakov and Sosnovskiy (2010); Sosnovskiy (1987, 1999, 2004, 2005, 2007, 2009); Sosnovskiy and Sherbakov (2012, 2017, 2019); Temfack and Basaran (2015); Wang and Yao (2019); Yao and Basaran (2012, 2013a, b, c); Yun and Modarres (2019); Wang and Yao (2017), Guo et al. (2018), Zhang et al. (2018), Wang et al. (2019), and Osara and Bryant (2019a, b). Young and Subbarayan (2019a, b) and Suhir (2019) used the Boltzmann-ArrheniusZhurkov equation to predict the evolution of time to failure, while this approach essentially is based on Basaran and Yan’s (1998) concept. Suhir advocates using the Boltzmann equation independent of entropy and Newtonian mechanics. Unfortunately, this approach reduces to using Boltzmann distribution just as an evolution function for an empirical model curve-fit to test data. Hsiao and Liang (2018) developed a sensor that monitors entropy generation in real-time and that can give real-time information on system aging and prediction for further estimating the failure of electrical or any mechanical system. We do not claim to have included a comprehensive survey of the literature on the topic; however, we tried to give a brief historical perspective of the efforts.

4.2

Laws of Unified Mechanics Theory

Laws of unified mechanics theory are not new laws of nature. They are the unification of the existing laws of Newton and thermodynamics at the ab initio level. The first law of thermodynamics is also known as the law of conservation of energy. Newton’s first law is about a state where the summation of all externally applied forces is zero. Newton’s universal first law of motion can be summarized as

4.2

Laws of Unified Mechanics Theory

151

“An object at rest stays at rest and an object in motion stays in motion with the same speed and the same direction unless acted upon by an unbalanced force.” Newton’s first law of motion is also known as the law of inertia, exemplified by the simple example of pulling the tablecloth swiftly from the table while leaving all cups and plates in place due to their inertia forces. The first law is intended for a system with no dissipative, or unbalanced forces acting on the system. Therefore, we will start numerating laws of unified mechanics with the second law.

4.2.1

Second Law of Unified Mechanics Theory

The second law of unified mechanics theory is the ab initio level unification of the second law of Newton and the second law of thermodynamics in Boltzmann’s formulation. “The change of momentum of a body is proportional to the impulse impressed on the body and happens along the straight line on which that impulse is impressed. Degradation of the input impulse takes place according to the second law of thermodynamics. The rate of degradation of impulse is directly proportional to the entropy generation rate in the system along the path chosen. The entropy generation rate of the system can be mapped onto a linearly independent non-dimensional axis called the Thermodynamic State Index (TSI) which can have coordinate values only between 0 and 1 (Fig. 4.4). Thermodynamic state index (1 - Φ) is the normalized non-dimensional form of the second law of thermodynamics in Boltzmann entropy formulation.” As a result, unification of the second laws in the Newtonian sense can be given by F dt ð1 - ΦÞ = dðmvÞ

ð4:35Þ

Generalizing Eq. (4.35) for multiple sources of impulse, [i.e., mechanical, magnetic, electric, chemical, etc.] and assuming that the mass is not a function of time, and there are n number of different impulses acting on the body, including both conservative and non-conservative, then the second law of the unified mechanics theory becomes n Z X

½F ð1 - ΦÞdt i = mdv

ð4:36Þ

i

Combining the second laws of Newton and thermodynamics requires the modification of the Newtonian space-time coordinate system. A new axis must be added to be able to define the thermodynamic state. As a result, the motion of any particle in the universe can be defined only in a five-dimensional space that has five linearly independent axes. None of these axes can represent the information of other axes.

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Fig. 4.4 The coordinate system in unified mechanics theory

The thermodynamic state index is the normalized form of the second law of thermodynamics as given by the Boltzmann entropy formulation. Derivation of the (1 - Φ) term is provided in the following section. The additional axes Thermodynamics State Index (TSI) is necessary to locate the thermodynamic state of a particle. Space-time coordinates of a point can be defined by Newton’s universal laws of motion. However, thermodynamic state index axis coordinate cannot be defined by a space-time coordinate system. We find it necessary to give the following example. Let us assume there is a 5-year-old boy in Istanbul and a 100-year-old man in New York. Using the space-time Cartesian coordinate system, we can define their location by x, y, z coordinates and their age on the time axis. However, this does not give any information about their thermodynamic state. Let us assume that a 5-yearold boy has a terminal illness and is expected to die in a few days and 100-year-old is expected to die in a few days. This information cannot be represented in a space-time coordinate system. However, on the TSI axis, both persons will be at Φ = 0.999 coordinate. Another example is, if a soccer ball is given an initial acceleration with a kick, F, it will have an initial acceleration of a = F/m but eventually, it will come to a stop. Let’s assume the ball was stationary initially in the terrain profile shown in Fig. 4.6. Depending on the path it follows, it will come to a stop at one of the valleys. The amplitude and direction of the initial acceleration of the ball are governed by the second law of Newton and the slowing down process of the ball is governed by the laws of thermodynamics, which is represented by (1 - Φ) term.

4.3

Evolution of Thermodynamic State Index (Φ)

4.2.2

153

Third Law of Unified Mechanics Theory

“To every action, there is always opposed an equal reaction. The initial reaction of a body in response to action will change over time as the stiffness of the reacting system degrades over time according to the second law of thermodynamics and its thermodynamic fundamental equation.” The reaction will be governed by the third law of the unified mechanics theory, given by F 12 = F 21 =

h i 1 dU 21 d = k 21 ½1 - Φu221 du21 du21 2

ð4:37Þ

dΦ is assumed to be negligible in each increment, then we For problems where du 21 can write the following simplified form of the third law:

F 12 = F 21 = ½k21 ð1 - ΦÞ  u21

ð4:38Þ

where U21 is the strain energy of the reacting member, Φ is the Thermodynamic State Index, k21 is the stiffness of the reacting member, and u21 is the displacement in the reacting member. It is noteworthy that the third law of the unified mechanics theory includes both Hooke’s law (1678) and Newton’s third law (1687).

4.3

Evolution of Thermodynamic State Index (Φ)

Materials, under externally applied loading, [i.e., mechanical, thermal, electrical, chemical, etc.], change their thermodynamic state. This process will follow the laws of thermodynamics. Simply put the first law of thermodynamics will govern the conservation of energy and the second law of thermodynamics will govern the entropy generation rate according to the thermodynamic fundamental equation of the system. The evolution of entropy [disorder] in the system is given by Boltzmann’s entropy equation. When the entropy is maximum and the entropy generation rate is zero for a closed and isolated system, the change in thermodynamic state index will come to a stop. While the author of this book and many others have proven the validity of Boltzmann’s equation in solids in the last 30 + years, extensively, Boltzmann’s (1877) original work and its interpretation of disorder were misinterpreted until recently as being only applicable to gasses. Recently, there has been a great deal of interest in translating Boltzmann’s original papers into English. One recent such paper is by Sharp and Matschinsky (2015). We find it essential to include the entirety of their translation in this section. It is important to point out that, Sharp and Matschinsky (2015) also state that “what Boltzmann wrote on these subjects is rarely quoted directly, his methods are not fully appreciated, and key concepts

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have been misinterpreted.” The following section about Boltzmann’s work is a direct quotation from Sharp and Matschinsky (2015): The translation provided here is for Boltzmann (1877). Previous work of Maxwell and Boltzmann’s derivations was based on mechanical laws of motion and particle interaction of gasses. However, this work by Boltzmann is much more general. His formulation requires only that particles can exchange kinetic energy, but they do not specify the mechanism. As Boltzmann predicted in the final sentences of this paper, his approach applied to not just gasses, but liquids and solids of any composition. Indeed, the Boltzmann distribution has also passed almost unchanged into the quantum world.

Boltzmann established the theoretical basis for statistical mechanics. Boltzmann used three levels of hierarchy to describe the processes. At the highest level, there is the macro-state, where the thermodynamic state variables such as temperature and pressure can be observed directly. The second level in the hierarchy is where the energy or velocity components of each molecule can be specified. He calls this [energy or momentum of particles] Komplexion, which Sharp and Matschinky (2015) translate as complexion. Finally, there is the third level at which the number of molecules with each its energy level, velocity, and their [spatial] position is specified. Here Boltzmann does not make any assumptions about the type [or the state] of the molecules (Boltzmann’s w0, w1, etc.). Boltzmann calls a particular set of w-values a Zustandeverteilung translated as a “state distribution” or “distribution of states.” It is important to point out, as discussed by Sharp and Matschinsky (2015), that a microstate is divided into two separate hierarchies in Boltzmann’s formulation, both complexion and state distribution. “Boltzmann then uses permutation mathematics to determine the distribution of states, which he denotes by p.” “Boltzmann then shows how to find the distribution of states (w0max ,w1max , . . .) with the largest number of complexions pmax subject to constraints on the temperature and number of molecules. Boltzmann’s postulate is that (w0max ,w1max , . . .) is the most probable state distribution and that this corresponds to thermal equilibrium.” Although Boltzmann introduces discrete energy levels in Section I of his paper as a convenience and considers them [discrete energy levels] unphysical, in Section II he shows that there is no material difference compared to using continuous energy levels. The statistical mechanics’ formulation of entropy is presented in Section V of Boltzmann (1877). “Boltzmann shows that the statistical mechanics quantity he denotes by Ω (multiplied by 2/3) is equal to the thermodynamic quantity entropy (S) as defined by Clausius, with an additive constant. Boltzmann called, Ω, the permutabilities translated here literally as ‘permutability measure.’ Boltzmann defines permutability measure Ω as follows: A state distribution is specified by the number of molecules having velocity components within some small interval u and u + du, v and v + dv, w and w + dw for every u, v and w velocity values in the 1 , + 1 range, and having [spatial] position coordinates in each small volume (x, x + dx), (y+, y + dy), (z, z + dz) ranging over the total volume V. For each state distribution, there are several possible complexions. [Combination of molecules with a certain level of energy or velocity]. One state distribution has the most complexions and so, therefore, is the most probable. Ω Is given by the [natural] logarithm of the number of complexions for that state with the most complexions.”

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155

“Boltzmann (1877) also clearly demonstrates that these are two distinct contributions to entropy generation, one arising from the distribution of heat (kinetic energy of atoms) and the distribution in space of atoms or molecules.” In the initial effort to understand the nature of entropy, Carnot, Clausius, Maxwell Kelvin, and others focused almost entirely on the contribution from heat only. “Boltzmann unified the entropy due to thermal aspects with spatial distribution entropy into one statistical mechanic’s formulation.” Boltzmann also discovered the third fundamental contribution to entropy, namely, radiation by deriving the Stefan-Boltzmann Law (1884). “Careful reading of Boltzmann (1877) is enlightening about some apparent paradoxes subsequently encountered in the development of statistical mechanics.” Unfortunately, many terms in Boltzmann’s equations and derivation have been misinterpreted in many physics textbooks. “First, the variable, Ω, is not the logarithm of a probability.” That would be obtained by dividing the number of complexions p for a given state distribution, by the total number of complexions, J. Boltzmann gives this a different symbol, w, from the first letter of the German word for probability [wahrscheinlichkeit], but he does not use it. Confusingly, Planck later chose to write Boltzmann’s equation for entropy as Planck (1901). S = k ln w þ constant

ð4:39Þ

k is Boltzmann’s constant, w is the probability that the system will exist in the state it is in relative to all the possible states it could be in (Halliday & Resnick, 1966). On the other hand, in most thermodynamics’ textbooks, Boltzmann’s equation is given by S = k ln Ω

ð4:40Þ

where k is the Boltzmann’s constant and Ω is the number of microstates corresponding to a given state with the macroscopic constraints (Callen, 1985; De Hoff, 1993). Sharp and Matschinky (2015) point out that Boltzmann’s Permutabilitätmass (measure of permutability) method for counting possible micro states, there is no need for a posteriori division by N! “to correct” the derivation using the “somewhat mystical arguments of Gibbs and Planck.” Ehrenfest and Trakal (1921), van Kampen (1984), Jaynes (1992), and Swendsen (2006) have pointed out that the correct counting of micro states ~ a la Boltzmann precludes the need for the spurious indistinguishability term (/N!). “This fact has been ignored in most textbooks.” Translation by Sharp and Matchinsky (2015) also clarifies one very important point about Boltzmann’s derivation regarding non-equilibrium states. “lnΩ is not the logarithm of a volume in momentum-coordinate-phase space occupied by the system [Boltzmann here refers to a 5-dimensional space, the 5th axis being momentum]. Boltzmann notes, using Liouville’s theorem, that dV [differential volume in momentum-coordinate phases space] remains constant in time, and so it cannot describe the entropy increase upon approach to the equilibrium that Boltzmann

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was so concerned with. He thus avoids at the outset the considerable difficulty Gibbs had accounting for changes in entropy with time. Boltzmann gave us, for the first time, a definition of entropy applicable to every state (distribution) at equilibrium or not. “Then the entropy of the initial and final states is not defined, but one can still calculate the quantity which we have called the permutability measure. By extension, every complexion can then be assigned an entropy, using the permutability measure of the state distribution to which that complexion belongs, Lebowitz (1993), opening the door to the statistical mechanics of non-equilibrium states and irreversible processes” (Sharp & Matchinsky, 2015). Boltzmann’s work has historically been misinterpreted, assumed to apply to gasses only, and ignored in the continuum mechanics field. Therefore, we feel compelled to include his original paper in this book. The following section is the English translation of Boltzmann (1877) by Sharp and Matschinky (2015) in its entirety.

4.3.1

On the Relationship Between the Second Fundamental Theorem of The Mechanical Theory of Heat and Probability Calculations Regarding the Conditions For Thermal Equilibrium, by Ludwig Boltzmann (1877)

The relationship between the second fundamental theorem [i.e., the second law of thermodynamics] and calculations of probability became clear for the first time when I demonstrated that the theorem’s analytical proof is only possible based on probability calculations. (I refer to my publication “Analytical proof of the second fundamental theorem of the mechanical theory of heat derived from the laws of equilibrium for kinetic energy” Wien. Ber. 63, p 8 reprinted ass Wiss. Abhand. Vol I, reprint 20, pp 295 and my “Remarks about several problems in the mechanical theory of heat” 3rd paragraph, Wiss. Abhand. Vol II, reprint 39.) [Boltzmann refers to thermodynamics by the term “The mechanical theory of heat.”] [Wiss. Abhand refers to Boltzmann’s collected works.] This relationship is also confirmed by demonstrating that an exact proof of the fundamental theorem of the equilibrium of heat is most easily obtained if one demonstrates that a certain quantity—which I wish to define again as E—has to decrease as a result of the exchange of the kinetic energy among the gaseous molecules and therefore reaching its minimum value for the state of the equilibrium of heat. (Compare my “Additional studies about the equilibrium of heat among gaseous molecules” Wiss. Abhand. Vol I, reprint 22, p 316.) The relationship between the second fundamental theorem and the laws of the equilibrium of heat is made even more compelling in light of the developments in the second paragraph of my “Remarks about several problems of the mechanical theory of heat.” There, I mentioned for the first time the possibility of a unique way of calculating the equilibrium of heat using the following formulation. It is clear that every single uniform state distribution which establishes itself after a certain time

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157

given a defined initial state is equally as probable as every single non-uniform state distribution. Comparable to the situation in-game of Lotto [a board game] where every single quintet is as improbable as the quintet 12345. The higher probability that the state distribution becomes uniform with time arises only because there are far more uniform than non-uniform state distributions. Furthermore: “It is even possible to calculate the probabilities from the relationships of the number of different state distributions. This approach would perhaps lead to an interesting method for the calculation of the equilibrium of heat.” It is thereby indicated that it is possible to calculate the state of the equilibrium of heat by finding the probability of the different possible states of the system. The initial state in most cases is bound to be highly improbable and from it, the system will always rapidly approach a more probable state until it finally reaches the most probable state, i.e., that of the heat equilibrium. If we apply this to the second basic theorem [i.e., the second law of thermodynamics], we will be able to identify that quantity which is usually called entropy with the probability of the particular state. Let’s assume a system of bodies that are in a state of isolation with no interaction with other bodies, e.g., one body with higher and one body with lower temperature and one so-called intermediate body which accomplishes the heat transfer between the two bodies; or choosing another example by assuming a vessel with absolutely even and rigid walls one half of which is filled with the air of low temperature and pressure whereas the other half is filled with the air of high temperature and pressure. The hypothetical system of particles is assumed to have a certain state at time zero. Through the interaction between the particles, the state is changed. According to the second fundamental theorem, this change must take place in such a way that the total entropy of the particles increases. This means according to our present interpretation that nothing changes except that the probability of the overall state for all particles will get larger and larger. The system of particles always changes from an improbable state to a probable state. It will become clear later what this means. After the publication of my last treatise regarding this topic, the same idea was taken up and developed further by Mr. Oskar Emil Meyer totally independent of me [Die Kinetische Theorie der Gase, Breslau 1877, Seite 262]. He attempts to interpret, in the described manner, the equations of my continued studies concerning the equilibrium of heat particles. However, the line of reasoning of Mr. Meyer remained entirely unclear to me, and I will return to my concerns with his approach on page 172 (Wiss. Abhand. Vol. II). We have to take a different approach because it is our main purpose not to limit our discussion to thermal equilibrium but to explore the relationship of this probabilistic formulation to the second theorem of the mechanical theory of heat. We want first to solve the problem which I referred to above and already defined in my “Remarks on some problems of the mechanical theory of heat” [Wiss. Abhand. reprint 39], namely, to calculate the probability of state distributions from the number of different distributions. We want first to treat as simple a case as possible, namely, a gas of rigid elastic spherical molecules trapped in a container with absolutely elastic walls. (Which interact with central forces only within a certain small distance, but not otherwise, the latter assumption, which includes the former as a special case, does not change the calculations in the least.) Even in this case, the

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application of probability theory is not easy. The number of molecules is not infinite, in a mathematical sense, yet the number of velocities each molecule is capable of is effectively infinite. Given this last condition, the calculations are very difficult; to facilitate understanding, I will, as in earlier work, consider a limiting case. Kinetic Energy Has Discrete Values We assume initially that each molecule is only capable of assuming a finite number of velocities, such as 1 2 3 p 0, , , ,⋯, q q q q

ð4:41Þ

where p and q are arbitrary finite numbers. Upon colliding, two molecules may exchange velocities, but after the collision, both molecules still have one of the above velocities, namely, [The fact that Boltzmann allows the exchange of energy between molecules automatically makes his formulation general, i.e., irrespective of the state of matter gas, liquid, or solid.] 0, or

1 2 p , or , etc till q q q

ð4:42Þ

This assumption does not correspond to any realistic mechanical model, but it is easier to handle mathematically, and the actual problem to be solved is re-established by letting p and q go to infinity. Even if at first sight, this seems a very abstract way of treating the problem, it rapidly leads to the desired objective, and when you consider that in nature all infinities are [nothing] but limiting cases one assumes each molecule can behave in this fashion only in the limiting case where each molecule can assume more and more values of the velocity. To continue, however, we will consider the kinetic energy, rather than the velocity of the molecules. Each molecule can have only a finite number of values for its kinetic energy. As a further simplification, we assume that the kinetic energies of each molecule form an arithmetic progression, such as the following: 0, E, 2E, 3E, . . . , pE

ð4:43Þ

We call P the largest possible value of the kinetic energy, pE. Before impact, each of the two colliding molecules shall have a kinetic energy of 0, or E, or 2E, etc: pE

ð4:44Þ

This means that after the collision, each molecule still has one of the above values of kinetic energy. The number of molecules in the vessel is n. If we know how many of these n molecules have a kinetic energy of zero, how many have a kinetic energy of E, and so on, then we know the kinetic energy distribution. If at the beginning there

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Evolution of Thermodynamic State Index (Φ)

159

is some state distribution among the gas molecules, this will in general be changed by the collisions. The laws governing this change have already been the subject of my previous investigations. But right way, I note that this is not my intention here, instead, I want to establish the probability of state distribution, regardless of how it is created, or more specifically, I want to find all possible combinations of the ( p + 1) kinetic energy values allowed to each of the n molecules and then establish how many of these combinations correspond to each state distribution. [The term “state distribution” is better translated/interpreted as the distribution of states into English. However, I am keeping the original translation. Because the term refers to what Boltzmann refers to as complexion, which is the distribution of [discrete] kinetic energies of molecules]. The latter number [( p + 1)] then determines the likelihood of the relevant state distribution, as I have already stated in my published “Remarks about several problems in the mechanical theory of heat” (Wiss. Abhand. Vol II, reprint 39, p 121). As a preliminary, we will use a simpler schematic approach to the problem, instead of the exact case. Suppose we have n molecules. Each of them can have kinetic energy 0, E, 2E, 3E, . . . , pE

ð4:45Þ

and suppose these energies are distributed in all possible ways among the n molecules, such that the total energy is a constant, e.g., λE = L. Any such distribution, in which the first molecule may have a kinetic energy of, e.g., 2E, the second may have 6E, and so on, up to the last molecule, we call a complexion and so that each individual complexion can be easily enumerated. We write them in sequence (for convenience we divide through by E), specifying the kinetic energy of each molecule. We seek the number P of complexions where w0 [number of] molecules have kinetic energy 0, w1 molecules have kinetic energy E, w2 have kinetic energy 2E, up to the wp which have kinetic energy pε. We said, earlier, that given how many molecules have kinetic energy 0, how many have kinetic energy E, etc., this distribution among the molecules specifies the number of P of complexions for that distribution; in other words, it determines the likelihood of that state distribution. Dividing the number P by the number of all possible complexions, we get the probability of the state distribution. [Boltzmann never performs this division. It is done later by Planck (1901).] Since the distribution of states does not determine kinetic energies exactly, the goal is to describe the state distribution by writing as many zeros as molecules with zero kinetic energy (w0), w1 ones for those with kinetic energy E, etc. All these zeros, ones, etc. are the elements defining the state distribution. It is now immediately clear that the number P for each state distribution is the same as the number of permutations of which the elements of the state distribution are capable and that is why the number P is the desired measure of the permutability of the corresponding distribution of states. Once we have specified every possible complexion, we have also all possible state distributions, the latter differing from the former only by immaterial permutations of molecular labels. All those complexions which contain the same

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Table 4.1 Possible permutations for different states P 1. 2. 3. 4. 5.

0000007 0000016 0000025 0000034 0000115

7 42 42 42 105

6. 7. 8. 9. 10.

0000124 0000133 0000223 0001114 0001123

P 210 105 105 140 420

11. 12. 13. 14. 15.

0001222 0011113 0011122 0111112 1111111a

P 140 105 210 42 1

a The state distributions are so arranged that read as a number, the rows are arranged in increasing order

number of zeros, the same numbers of ones, etc., differing from each other merely by different arrangements of elements, will result in the same state distribution, the number of complexions forming the same state distribution, and which we have denoted by P, must be equal to the number of permutations which the elements of the state distribution are capable of. To give a simple numerical example, take =7, λ = 7 p = 7 so L = 7E, P = 7E. With seven molecules, there are eight possible values for the kinetic energy 0, E, 2E, 3E, 4E, 5E, 6E, 7E to distribute in any possible way such that the total kinetic energy = 7E. There are then 15 possible state distributions. We enumerate each of them in the above manner, producing the numbers listed in the second column of the following table of state distributions (Table 4.1). The numbers in the first column label the different state distributions. In the last column, under the heading P is the number of possible permutations of members for each state. The first state distribution, for example, has six molecules with zero kinetic energy, and the seventh has kinetic energy 7E. So w0 = 6, w7 = 1, w2 = w3 = w4 = w5 = w6 = 0 [w1 = 0 is also zero. However, Boltzmann does not include it]. It is immaterial which molecule has kinetic energy 7E. So, there are seven possible complexions, which represent this state distribution. Denoting the sum of all possible complexions, 1716 by J then the probability of the first state distribution is 7/J, similarly, the probability of the second state distribution is 42/J, and the most probable state distribution is the tenth, as its elements permit the greatest number of permutations. Hereon, we call the number of permutations the relative likelihood of the state distribution; this can be defined in a different way, which we next illustrate with a specific numerical example since generalization is straightforward. Suppose we have an urn containing an infinite number of paper slips. On each slip is one of the numbers 0, 1, 2, 3, 4, 5, 6, 7; each number is on the same number of slips and has the same probability of being picked. We now draw the first septet of slips and note the numbers on them. This septet provides a sample state distribution with a kinetic energy of E times the number written on the first slip for molecule 1 and so forth. We return the slips to the urn and draw a second septet, which gives us a second state distribution, etc. After we draw a very large number of septets, we reject all those for which the total does not equal seven. This still leaves a large number of septets. Since each number has the same probability of occurrence and the same elements in a different order from different complexions, each possible complexion will occur equally often. By ordering the numbers within each septet by

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161

size, we can classify each into one of the 15 cases tabulated above. So, the number of septets that fall into the class 0000007 relative to the 0000016 class will be 7:42, similarly for all the other septets. The most likely state distribution is the one, which produces the most septets, namely, the 10th. (Boltzmann’s footnote: If we divide the number of septets corresponding to a particular state by the total number of septets, we obtain the probability of distribution. Instead of discarding all septets whose total is not 7 we could, after drawing a slip, remove from the urn all those other slips for which a total of 7 is now impossible, e.g., on drawing a slip with 6 on it, all other slips except those with 0 or l would be removed. If the first 6 slips all had 0 on them, only slips with 7 on them would be left in the urn. One more thing should be noted at this point. We construct all possible complexions. If we denote by w0 the arithmetic mean of all values of w0 which belong to different complexions and form analogous expressions w1 , w2 , . . . in the limit these quantities would also form the same state distribution.) [Translators Sharp and Matschinsky’s note: Boltzmann’s comments on the results of Mr. Oskar Meyer beginning “Ich will hier einige Worte €uber die von. Hrn. Oskar Meyer. . .” on p 172 (Wiss. Ab.) and ending with “. . .Bearbeitung des allgemeinen Problems zur€ uckkehren,” on p 175 (Wiss. Ab.) are of historical interest only and are omitted.] The first task is to determine the permutation number, previously designated by P, for any state distribution. Denoting by J the sum of the permutations P for all possible state distributions, the quotient P/J is the state distribution’s probability, henceforth denoted by W. [Boltzmann uses W for the initial of the German word, Wahrscheinlichkeit.] We would first like to calculate the permutations P for the state distribution characterized by w0 molecules with kinetic energy 0, w1 molecules with the kinetic energy E, etc. It must be understood that w0 þ w1 þ w2 þ ⋯ þ wp = n

ð4:46Þ

w1 þ 2w2 þ 3w3 þ ⋯ þ pwp = λ

ð4:47Þ

Because the total number of molecules is n, and the total kinetic energy is λE = L. Describing the state distribution as before, a complexion has w0 molecules with zero energy, w1 [is the number of molecules] with one unit [of energy], and so on. The permutations, P, arise since of the n elements w0 are mutually identical, similarly, with w1, w2 etc., elements. The total number of permutations is well known. P=

n! w0 !w1 ! . . .

ð4:48Þ

The most likely state distribution will be for those w0, w1, . . . values for which P is a maximum or since the numerator is a constant, for which the denominator is a minimum. The values w0, w1 must simultaneously satisfy the two constraints (4.46) and (4.47). Since the denominator of P is a product, it is easiest to determine the minimum of its logarithm, that is the minimum of

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M = ln ½w0 ! þ ln ½w1 ! þ ⋯

ð4:49aÞ

Here ln is the natural logarithm.1 It is natural in our problem that only integer values of w0, w1, . . . are meaningful. However, to apply differential calculus, we will allow non-integer values and so find the minimum of the expression M 1 = ln Γðw0 þ 1Þ þ ln Γðw1 þ 1Þ þ ⋯

ð4:49bÞ

which is identical to (4.49a) for integer values of w0, w1, . . . . We then get the non-integer values which for constraints (4.46) and (4.47) maximize M1.2 The solution to the problem will in any case be obtained if for w0, w1, etc., we select the closest set of integer values. If here and there a deviation of a few integers is required, the nearest complexion is easily found. The minimum of M1 is found by adding to both sides of the equation for M1 equation (4.46) multiplied by the constant h and Eq. (4.47) multiplied by the constant k and setting the partial derivatives with respect to each of the variables w0, w1, w2, . . . to zero. We thus obtain the following equations: d ln Γðw0 þ 1Þ þ h = 0, dw0

ð4:50aÞ

d ln Γðw1 þ 1Þ þ h þ k = 0, dw1

ð4:50bÞ

d ln Γðw2 þ 1Þ þ h þ 2k = 0 dw2

ð4:50cÞ

⋮ ⋮ ⋮
 d ln Γ wp þ 1 þ h þ pk = 0, dwp

ð4:50dÞ

which leads to d ln Γðw1 þ 1Þ d ln Γðw0 þ 1Þ dw1 dw0

ð4:50eÞ

d ln Γðw2 þ 1Þ d ln Γðw1 þ 1Þ dw2 dw1

ð4:50fÞ

Translators’ footnote: The ambiguous symbol “l” [used by Boltzmann] for [natural] logarithm in the original text has been replaced throughout by “ln .” 2 Translators’ footnote: The original text reads as “maximized but should mean minimized” [because Boltzmann’s objective is to maximize P]. 1

4.3

Evolution of Thermodynamic State Index (Φ)

=

163

d ln Γðw3 þ 1Þ d ln Γðw2 þ 1Þ ⋯ dw3 dw2

ð4:50gÞ

The exact solution of the problem through evaluation of the gamma function integral is very difficult; fortunately, the general solution for arbitrary finite values of p and n does not interest us here, but only the solution for the limiting case of a larger and larger number of molecules. Then the numbers w0, w1, w2 etc. become larger and larger, so we introduce the function3 ϕðxÞ = ln Γðx þ 1Þ - xð ln x - 1Þ -

1 ln 2π: 2

ð4:51Þ

Then we can write the first equation of (4.50) as follows: ln w1 þ

dϕðw1 Þ dϕðw0 Þ dϕðw2 Þ dϕðw1 Þ - ln w0 = ln w2 þ - ln w1 ð4:52Þ dw1 dw0 dw2 dw1

Similarly, for the other equations of (4.50). It is also well known that 1 1 þ ... ð4:53aÞ ln x þ 12x 2 pffiffiffiffiffi This series is not valid for x = 0, but here x! and 2π ðx=eÞx should have the same value, and ϕ(x) = 0. Therefore, the problem of finding the minimum of w0 ! w1 ! w2 ! . . . is replaced by the easier problem of finding the minimum of ϕð xÞ = -

pffiffiffiffiffiw0 w0 pffiffiffiffiffiw1 w1 pffiffiffiffiffiw2 w2 2π 2π 2π e e e Providing w is not zero, even at moderately large values of p and n both problems have matching solutions. From Eq. (4.53a) it follows dϕðw0 Þ 1 1 =- ... dw0 2w0 12w20

ð4:53bÞ

which for larger and larger values of w0 or lnw0 vanishes, the same also applies to the other w′s, so Eq. (4.52) can be written as follows: ln w1 - ln w0 = ln w2 - ln w1

ð4:54aÞ

Or

   
 Boltzmann approximates ln x! by x ln x - x þ 12 ln ð2π Þ rather than x þ 12 ln x - x þ 12 ln ð2π Þ as is now usual. For x ≪ 30,the relative difference is small.

3

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Unified Mechanics Theory

ð4:54bÞ

Likewise, the equations for the remaining w′s are w2 w3 w4 = = = ... w1 w2 w3

ð4:54cÞ

One sees immediately that by neglecting the expression (4.53b) the minimum of the denominator of pffiffiffiffiffi
nn 2π e pffiffiffiffiffi
w w0 pffiffiffiffiffi
w 0 2π e 2π we1 1 . . . is found instead of the minimum of the denominator of (4.48). So, for problems involving w!, the use of a well-known papproximation (see Schlömilch’s ffiffiffiffiffi Comp. S. 438) amounts to the substitution of 2π ðw=eÞw for w!. If we denote the common value of the quotient (4.54c) by, x we obtain w1 = w0 x,w2 = w0 x2 ,

w3 = w0 x3 ,etc:

ð4:55Þ

The two Eqs. (4.46) and (4.47) become
 w0 1 þ x þ x2 þ . . . þ xp = n
 w0 x þ 2x2 þ 3x3 þ . . . þ pxp = λ

ð4:56Þ ð4:57Þ

One sees immediately that these equations differ negligibly from Equation (42) and the preceding ones from my earlier work “Study of the thermal equilibrium of gas molecules.” [Boltzmann refers to Eq. (42) in his earlier study, not Eq. (42) in this derivation] We can use the last equation to write xpþ1 - 1 =n x-1

pþ1  d x -1 =λ w0 x  dx x - 1 w0 

ð4:58aÞ ð4:58bÞ

Carrying out the differentiation in the last equation w0 x

pxpþ1 - ðp þ 1Þ xp þ 1 =λ ð x - 1Þ 2

Dividing this equation by Eqs. (4.58a, 4.58b) gives

ð4:59Þ

4.3

Evolution of Thermodynamic State Index (Φ)

165

pxpþ2 - ðp þ 1Þ xpþ1 þ x λ = n ðxpþ1 - 1Þ ðx - 1Þ

ð4:60aÞ

ðpn - λÞxpþ2 - ðpn þ n - λÞxpþ1 þ ðn þ λÞx - λ = 0

ð4:60bÞ

Or

One can see immediately from Descartes’ theorem4 that this equation cannot have more than three real positive roots, of which two are = 1. Again, it is easy to see that both roots are not solutions to Eqs. (4.56) and (4.57) and do not solve the problem, but that they showed up in the final equation merely because of multiplying by the factor (x - 1)2. [This appears to be a typographic error. The ( p + 2)2 power makes mathematical sense, Sharp and Matschinsky (2015)]. To be convinced of this, one needs only to derive the final equation directly by dividing Eqs. (4.56) and (4.57). Following this division and having removed the variable x from the denominator and collecting powers of x throughout, we get the equation ðnp - 1Þxp þ ðnp - n - λÞxp - 1 þ ðnp - 2n - λÞxp - 2 þ . . . ðn - λÞx - λ = 0,

ð4:61Þ

which is an equation of pth degree, and whose roots supply the solution to the problem. Thus, Eqs. (4.60a, 4.60b) cannot have more positive roots than the solution requires. Negative or complex roots have no meaning for the solution to the problem. We note again that the largest allowed kinetic energy P = pE is very large compared to the mean kinetic energy of a molecule L λE = n n

ð4:62Þ

From this, it follows that p is very large compared to λ/n. The polynomial Eq. (4.61), which shares the same real roots with Eqs. (4.60a, 4.60b), is negative for x = 0, x = 1; however, it has the value  p λ , nðp þ 1Þ 2 n

ð4:63Þ

which is positive and very large, since p is very large compared to n. The only positive root occurs for x between 0 and 1, and we obtain it from the more convenient Eq. (4.60b). Since x is a proper fraction, then the pth and ( p + 1)th powers are smaller and can be neglected, in which case we obtain

4

Cardano’s formula.

166

4

x=

Unified Mechanics Theory

λ nþλ

ð4:64Þ

This is the value to which x tends for large p, and one can see the important fact that for reasonably large values of p the value of x depends almost exclusively on the ratio λ/n, and varies little with either λ or n providing their ratio is constant. Once one has found x, it follows from Eq. (4.58b) that w0 =

1-x n 1 - xpþ1

ð4:65Þ

And Eq. (4.55) gives the values of the remaining w′s. It is seen from the quotients w0 w1 w2 , , ,etc: n n n that the probabilities of the various kinetic energy values for larger p are again dependent almost exclusively on the mean energy of the molecule. For infinitely large p we obtain the following limiting values: w0 =

n2 nþλ

w1 =

n2 λ ðn þ λÞ2

w2 =

n2 λ 2 ðn þ λÞ3

etc:

ð4:66Þ

To establish whether we have a maximum or a minimum, we need to examine the second variation of Eq. (4.49b). We note that w0, w1, w2 etc. are very large, so we can use the approximation formula for lnΓ(w + 1) w  ð ln w - 1Þ -

1 1 1 ln w - ln ð2π Þ þ þ etc: 2 2 12w

and neglecting terms, which have second or higher powers of w in the denominator, obtain δ2 M =

ðδw0 Þ2 ðδw1 Þ2 þ þ ... w0 w1

ð4:67Þ

Therefore, we do have a minimum. I also want to remark on the size of the term previously designated J. One easily finds that J is given by the following binomial coefficient: J=



λþn - 1 λ



When you neglect terms that diminish with increasing λ or n

ð4:68Þ

4.3

Evolution of Thermodynamic State Index (Φ)

167

λþn - 2 1 ð λ þ n - 1Þ J = pffiffiffiffiffi 1 1 2π ðn - 1Þn - 2 λλþ2 1

ð4:69Þ

Now λE/n is equal to the average kinetic energy μ of a molecule, therefore λ μ = n E

ð4:70Þ

So, for large numbers one has ðλ þ n - 1Þλþn - 2 = = λλþn - 2 1

1

 1þ2n2λ- 1 1 n-1 λ 1þ = λλþn - 2  en - 1 λ

ð4:71Þ

[Note: double equal is a relational operator used to compare two variable values whether they are equal or not.] Therefore 1 λ n - 1 en - 1 J = pffiffiffiffiffi 1 , 2π ðn - 1Þn - 2

ð4:72Þ

Therefore, neglecting diminishing terms ln J = n ln

λ 1 1 þ n - ln λ þ ln n - 1 - ln ð2π Þ n 2 2

ð4:73Þ

It goes without saying that these formulas are not derived here solely for finite p and n values, because these are unlikely to be of any practical importance, but rather to obtain formulas that provide the correct limiting values when p and n become infinite. Nevertheless, it may help to demonstrate, with specific examples of only moderately large values of p and n that these formulas are quite accurate, and though approximate, are of some value even here. We first consider the earlier example, where n = λ = 7, i.e., the number of molecules is seven, and the total kinetic energy is 7E, and so the mean kinetic energy is E. Suppose first, that p = 7, so each molecule can only have 0, E, 2E, 3E, . . ., 7E of kinetic energy. Then Eq. (4.60b) becomes 6x9 - 7x8 þ 2x - 1 = 0,

ð4:74Þ

From which it follows x=

1 7 8 þ x - 3x9 : 2 2

ð4:75Þ

168

4

Unified Mechanics Theory

Since x is close to 12, we can set x = 12 in the last two very small terms on the righthand side, and obtain x=

1 1 1 1 þ ð7 - 3Þ = þ 7 = 0:5078125 2 2 2 29

ð4:76Þ

You could easily substitute this value for x back into the right side of Eq. (4.75) and obtain a better approximation for x; since we already have an approximate value for x, a more rapid approach is to apply the ordinary Newton iteration method to Eq. (4.74) which results in x = 0:5088742 . . . From this, one finds per Eq. (4.55) w0 = 3:4535

w4 = 0:2316

w1 = 1:7574 w2 = 0:8943

w5 = 0:1178 w6 = 0:0599

w3 = 0:4551

w7 = 0:0304

These numbers satisfy the condition that pffiffiffiffiffi w0 w0 pffiffiffiffiffi w1 w1 2π  2π . . . etc: e e is minimized, while the minimized variables w obey the two constraints w0 þ w1 þ w2 þ w3 þ w4 þ w5 þ w6 þ w7 = 7,

ð4:77aÞ

w1 þ 2w2 þ 3w3 þ 4w4 þ 5w5 þ 6w6 þ 7w7 = 7,

ð4:77bÞ

which are minimum, incidentally because the first of Eq. (4.77a) coincides with the minimum of ðw0 Þw0 ðw1 Þw1 . . .. This provides only an approximate solution to our problem, which asks for so many (w0) zeros, so many (w1) ones, etc. with as many permutations as the resulting complexion permits, while the w′s simultaneously satisfies the constraints Eqs. (4.77a, 4.77b). Since p and n here are very small, one hardly expects any great accuracy, yet you already get the solution to the permutation problem by taking the nearest integer for each w, except for w3, for which you have to assign the value of 1 instead of 0.4551. In this manner, it is apparent w0 = 3,w1 = 2,w2 = w3 = 1,w4 = w5 = w6 = w7 = 0 and we saw in the previous table that the complexion of 0001123 has the most permutations. We now consider the same special case with n = λ = 7, but set p = 1; that is, the molecules may have kinetic energies of 0, 1, 2, 3, . . ., 1. We know then

4.3

Evolution of Thermodynamic State Index (Φ)

169

that the values of the variables w will vary little from those of the former case. In fact, we obtain x=

1 ; 2

w0 =

7 w = 3:5, w1 = 0 = 1:75, 2 2

w2 =

w1 = 0:875,etc: 2

We consider a little more complicated example. Take n = 13, λ = 19, but we only treat the simpler case where p = 1. Then we have x = 19=32

w4 = 0:6560

w0 = 5:28125 w1 = 3:13574

w5 = 0:38950 w6 = 0:23133

w2 = 1:86815

⋮

w3 = 1:10493

⋮

Substituting here for the w′s the nearest integers, we obtain w0 = 5,w1 = 3,w2 = 2,w3 = 1,w4 = 1,w5 = w6 . . . 0, Already from the fact that w0 + w1 + . . . should = 13, it is seen that again one of the w′s must be increased by one unit. From those w′s that is set to = 0, w5 differs least from the next highest integer. We want therefore w5 = 1, and obtain the complexion 0000011122345 whose digit sum is in fact = 19. The number of permutations this complexion is capable of is 13! 13! 1 =  5!3!2! 4!3!2! 5 A complexion whose sum of digits is also = 19, and which one might suppose is capable of very many permutations, would be the following: 0000111222334 The number of permutations is 13! 13! 1 =  4!3!3!2! 4!3!2! 6

170

4

Unified Mechanics Theory

This is less than the number of permutations of the first complexion we found from the approximate formula. Likewise, we expect that the number of permutations of the two complexions 0000111122335 And 0000111122344 is smaller still. This is, for both complexions 13! 13! 1 =  4!4!2!2! 4!3!2! 8 Other possible complexions are capable of still fewer permutations, and it would be quite superfluous to follow these up here. It is seen from the examples given here that the above formula, even for very small values of p and n gives values of w within one or two units of the true values. In the mechanical theory of heat, we are always dealing with extremely large numbers of molecules, so such small differences disappear, and our approximate formula provides an exact solution to the problem. We see also that the most likely state distribution is consistent with that known from gases in thermal equilibrium. According to Eq. (4.66), the probability of having a kinetic energy sE is given by  s n2 λ  ws = nþλ nþλ

ð4:78Þ

Since λE/n is equal to the average kinetic [energy] of a molecule μ, which is finite, so n is very small compared to λ. So, the following approximations E n n2 nE λ n n2 = = , = 1 - = e-λ = e-μ nþλ λ μ nþλ λ

ð4:79Þ

Hold. From which it follows that ws =

nE - Esμ e , μ

ð4:80Þ

To achieve a mechanical theory of heat, these formulas must be developed further, particularly through the introduction of differentials and some additional considerations.

4.3

Evolution of Thermodynamic State Index (Φ)

171

Kinetic Energies Exchange in a Continuous Manner To introduce differentials into our formula, we wish to illustrate the problem in the same manner as indicated on p 171 (Wiss. Abhand. Vol II) because this seems to be the best way to clarify the matter. Here each molecule was only able to have one of 0, E, 2E, . . ., pE values for kinetic energy. We generated all possible complexions, i.e., all the ways of distributing 1 + p values of the kinetic energy among the molecules, yet subject to the constraints of the problem, using a hypothetical urn containing infinitely many paper slips. Equal numbers of paper slips have kinetic energy values 0, E, etc. written on them. To generate the first complexion, we draw a slip of paper for each molecule and note the value of the kinetic energy assigned in this way to each molecule. Very many complexions are generated in the same way, they are assigned to this or that state distribution, and then we determine the most probable one. That state distribution which has the most complexions, we consider as the most likely or corresponding to thermal equilibrium. Proceeding to the continuous kinetic energy case the most natural approach is as follows: Taking E to be some very small value, we assume that in the urn are very many slips of paper labeled with kinetic energy values between 0 and E. In the urn are also equal numbers of paper slips labeled with kinetic energy values between E and 2E, 2E and 3E up to infinity. Since E is very small, we can regard all molecules with kinetic energy between x and x + E as having the same kinetic energy. The rest of the calculation proceeds as in Section I above. We assume some complexion has been drawn; w0 molecules have kinetic energy between 0 and E, w1 molecules have values between E and 2E, w2 have values between 2E and 3E, etc. Here, because the variables w0, w1, w2 etc. will be infinitely small, of the order of magnitude of E, we prefer to write them as w0 = E f ð0Þ;

w1 = E f ðEÞ;

w2 = E f ð2EÞ etc:

ð4:81Þ

The probability of the state distribution in question is given, exactly as in Section I, by the number of permutations that the elements of the state distribution are capable of, e.g., by the number n! w0 !w1 !w2 ! . . . Again, the most likely state distribution, which corresponds to thermal equilibrium, is defined by the maximum of this expression, that is, when the denominator is minimized. We use again the reasonable approximation of Section I, replacing w! with the expression pffiffiffiffiffiw w 2π e

172

4

Unified Mechanics Theory

pffiffiffiffiffi We can omit the term 2π since it is a constant factor in the minimization; the key again is to replace minimization of the denominator with minimization of its logarithm; then we obtain the condition for thermal equilibrium that M = w0 ln w0 þ w1 ln w1 þ w2 ln w2 þ . . . - n

ð4:82Þ

is a minimum, while again satisfying the two constraints n = w0 þ w1 þ w2 þ . . .

ð4:83Þ

L = Ew1 þ 2Ew2 þ 3Ew3 þ . . .

ð4:84Þ

which are identical to Eqs. (4.46) and (4.47) of Section I. Using Eq. (4.81) here, we replace the variables w by the function f and obtain thereby M = E½f ð0Þ ln f ð0Þ þ f ðEÞ ln f ðEÞ þ f ð2EÞ ln f ð2EÞ þ . . . þ E ln E½f ð0Þ þ f ðEÞ þ f ð2EÞ þ . . . - n

ð4:85Þ

and Eqs. (4.83) and (4.84) become n = E½f ð0Þ þ f ðEÞ þ f ð2EÞ þ . . .,

ð4:86Þ

L = E½E f ðEÞ þ 2E f ð2EÞ þ 3E f ð3EÞ . . .:

ð4:87Þ

Using Eq. (4.86) the expression for M can also be written as M = E½f ð0Þ ln f ð0Þ þ f ðEÞ ln f ðEÞ þ f ð2EÞ ln f ð2EÞ þ . . . - n þ n ln E

ð4:88Þ

Since n and E are constant, (because E has the same value for all possible complexions, and is constant between different state distributions), one can minimize M 0 = E½f ð 0Þ ln f ð0Þ þ f ðEÞ ln f ðEÞ þ f ð2EÞ ln f ð2EÞ þ . . .

ð4:89Þ

instead. As E is made still smaller, the allowed values of kinetic energy approach a continuum. For vanishingly small E, various sums in Eqs. (4.86, 4.87, 4.89) can be written in the form of integrals, leading to the following equations: 0

Z

M = 0

1

f ðxÞ ln f ðxÞdx

Z

n= 0

1

f ðxÞdx,

ð4:90Þ ð4:91Þ

4.3

Evolution of Thermodynamic State Index (Φ)

Z

1

L=

xf ðxÞdx,

173

ð4:92Þ

0

The functional form of f(x) is sought which minimizes expression (4.90) subject to the constraints (4.91) and (4.92), so one proceeds as follows: To the right side of Eq. (4.90) one adds Eq. (4.91) multiplied by a constant k, and Eq. (4.92) multiplied by a constant h. The resulting integral is Z

1

½f ðxÞ ln f ðxÞ þ kf ðxÞ þ h xf ðxÞdx

0

where x is the independent variable, and f is the function to be varied. This result in Z

1

½ ln f ðxÞ þ k þ hx δf ðxÞdx

0

Setting the quantity, which has been multiplied by δf(x) in square brackets = 0, and solving for the function f(x), we obtain f ðxÞ = Ce - hx

ð4:93Þ

Here the constant e-k - 1 is denoted by C for brevity. The second variation of M′ 0

Z

δ M = 2

1

0

½δf ðxÞ2  dx, f ðxÞ

ð4:94Þ

is necessarily positive, since f(x) is positive for all values of x lying between 0 and 1. By the calculus of variations M′ is a minimum. From Eq. (4.93), the probability that the kinetic energy of a molecule lies between x and x + dx at thermal equilibrium is f ðxÞdx = Ce - hx dx

ð4:95Þ

The probability that the velocity of a molecule lies between ω and ω + dω would be Ce -

hmω2 2

 mωdω

ð4:96Þ

where m is the mass of a molecule. Equation (4.96) gives the correct state distribution for elastic disks [bold emphasized by this author] moving in two dimensions, for elastic cylinders with parallel axis moving in space, but not for elastic spheres, which move in space. For the latter, the exponential function must be multiplied by ω2dω not ωdω. To get the right state distribution for the latter case, we must set up

174

4

Unified Mechanics Theory

the initial distribution of paper slips in our urn in a different way. To this point, we assumed that the number of paper slips labeled with kinetic energy values between 0 and E is the same as those between E and 2E. As also for slips with kinetic energies between 2E and 3E, 3E and 4E, etc. Now, however, let us assume that the three velocity components along the three coordinate axes, rather than the kinetic energies, are written on the paper slips in the urn. The idea is the same: There are the same number of slips with u between 0 and E, v between 0 and ξ, and w between 0 and η. The number of slips with u between E and 2E, v between 0 and ξ, and w between 0 and η is the same. Similarly, the number for which u is between E and 2E, v is between ξ and 2ξ, w is between 0 and η. Generally, the number of slips for which u, v, w are between the limits u and u + E, v and v + ξ, w and w + η are the same. Here u, v, w have any magnitude, while E, ξ, η are infinitesimal constants. With this one modification of the problem, we end up with the actual state distribution established in gas molecules. (LB footnote: We can of course, instead of using finite quantities E, ξ, η and then taking the limit as they go to zero, write du, dv, dw from the outset, then the distribution of paper slips in the urn must be such that the number for which u, v, w are between u and u + du, v and v + dv, w and w + dw are proportional to the product dudvdw and independent of u, v, and w. The earlier distribution of slips in the urn is characterized by the fact that although E could be replaced by dx, kinetic energies between 0 and dx, dx and 2dx, 2dx and 3dx, etc. occurred on the same number of slips.) If we now define wabc = E ζ η f ðaE, bζ, cηÞ

ð4:97Þ

as the number of molecules of any complexion for which the velocity components lie between the limits aE and (a + 1)E, bζ and (b + 1)ζ, cη and (c + 1)η, the number of permutations, or complexions of these elements for any state distribution, becomes n! P = Qa = þp Qb = þq Qc = þr a= -p

b= -q

c= -r

wabc !

ð4:98Þ

where we first assume u adopts only values between -pE and +pE, v between -qζ and +qζ, w between -rη and +rη. Where again, the most likely state distribution occurs when this expression, or if you will, its logarithm, is maximum. We again substitute pffiffiffiffiffiw w 2π e pffiffiffiffiffi where you can again immediately omit the factors of 2π as they simply contribute additive constants - 12 ln 2π to ln P ; omitting also the constant n ln n the term, the requirement for the most probable state distribution is that the sum n! by

pffiffiffiffiffi n n 2π e

and

w! by

4.3

Evolution of Thermodynamic State Index (Φ)

175

= þq cX aX = þp bX = þr

-

wabc ln wabc

a= -p b= -q c= -r

is a maximum, which only differs from ln P by an additive constant. The constraints that the number of molecules=n and that the total kinetic energy=L, take the following form: n=

aX =p

bX =q

c=r X

wabc

ð4:99Þ

a= -p b= -q c= -r

L=

b=q a=p c=r  m X X X
2 2 a E þ b2 ζ 2 þ c2 η2 wabc  2 a= -p b= -q c= -r

ð4:100Þ

Substituting for wabc using Eq. (4.97), one immediately sees that the triple sums can in the limit be expressed as definite integrals; omitting an additive constant, the quantity to be maximized becomes Zþ1 Zþ1 Zþ1 f ðu, v, wÞ ln f ðu, v, wÞ dudvdw,

Ω= -1

-1

ð4:101Þ

-1

The two constraint equations become Zþ1 Zþ1 Zþ1 f ðu, v, wÞ dudvdw,

n= -1

L=

m 2

Zþ1 -1

Zþ1 -1

Zþ1

-1




ð4:102Þ

-1

 u2 þ v2 þ w2 f ðu, v, wÞ dudvdw,

ð4:103Þ

-1

The variable Ω, which differs from the logarithm of the number of permutations only by an additive constant, is of special importance for this work, and we call it the permutability measure [bold emphasized by L. B.] I note, incidentally, that suppression of the additive constants has the advantage that the total permutability measure of two bodies is equal to the sum of the permutability measures of each body. Thus, it is the maximum of the quantity (4.101), subject to the constraints (4.102) and (4.103), that is sought. No further explanation of this problem is needed here; it is a special case of the problem I have already discussed in my treatise “On the thermal equilibrium of gases on which external forces act”5 in the section which

5

Wien. Ber. (1875) 72:427–457 (Wiss. Abhand. Vol. II, reprint 32).

176

4

Unified Mechanics Theory

immediately precedes the appendix. There I provided evidence that this state distribution corresponds to the condition of thermal equilibrium. [Boltzmann’s use of the term thermal equilibrium is important. In mechanics, this corresponds to the equilibrium of energy, irrespective of the state the matter is in.] Thus, one is justified in saying that the most likely state distribution corresponds with the condition of thermal equilibrium. If an urn is filled with slips of paper labeled in the manner described earlier, the most likely sampling will correspond to the state distribution for thermal equilibrium. We should not take this for granted, however, without first defining what is meant by the most likely state distribution. For example, if the urn were filled with slips labeled originally, then the statement would be incorrect. The reasoning needed to arrive at the correct state distribution will not escape those experienced in working with such problems. The same considerations apply to the following circumstance: If we group all the molecules whose coordinates at a particular time lie between the limits ξ and ξ þ dξ,η and η þ dη,ζ and ζ þ dζ,

ð4:104Þ

In addition, whose velocity components lie between the limits u and u þ du,v and v þ dv,w and w þ dw,

ð4:105Þ

and let these molecules collide with other molecules under specific conditions, after a certain time their coordinates will lie between the limits Ξ and Ξ þ dΞ,H and H þ dH,Z and Z þ dZ,

ð4:106Þ

and their velocity components will lie between the limits U and U þ dU,V and V þ dV,W and W þ dW,

ð4:107Þ

dξ  dη  dζ  du  dv  dw = dΞ  dH  dZ  dU  dV  dW:

ð4:108Þ

Then at any time

This is a general result. If at time zero the coordinates and velocity components of arbitrary molecules (material points) lie between the limits (4.104) and (4.105) and unspecified forces act between these molecules, [the fact that Boltzmann does not make any assumption about the type of forces between the molecules, therefore, shear forces between molecules are not excluded]. So that at time t the coordinates and velocity components lie between the limits (4.106) and (4.107), then Eq. (4.108) is still satisfied. [Note that here Boltzmann uses a 5-axes coordinate system to locate the molecules, the 5th axis being the momentum.] If instead of the velocity components, one uses the kinetic energy x and the velocity direction defined by the two angles α and β, to describe the action of the forces, these variables would initially lie between the limits

4.3

Evolution of Thermodynamic State Index (Φ)

177

ξ and ξ þ dξ,η and η þ dη,ζ and ζ þ dζ, x and x þ dx,α and α þ dα,β and β þ dβ, And then after the action of the forces lie between the limits Ξ and Ξ þ dΞ,H and H þ dH,Z and Z þ dZ, X and X þ dX, A and A þ dA,

B and B þ dB,

And so pffiffiffiffi pffiffiffi dξ  dη  dζ  x  dx  φðα, βÞdα  dβ = dΞ  dH  dZ  X  dX  φðA, BÞdA  dB

ð4:109Þ

So the product of the differentials du  dv  dw becomes dU  dV  dW. Therefore, the list of slips in the urn must be labeled uniformly with velocity components lying between u and u + du, v and v + dv, w and w + dw, whatever values u, v, w have. Given a certain value of the coordinates, the p velocities must be p described by the ffiffiffiffi ffiffiffi corresponding “moments.” On the other hand, x dx goes over to X dX. With the introduction of kinetic energy, slips must be labeled so that you have the same pffiffiffi number with kinetic energy between x and x þ x dx where dx is constant but x is completely arbitrary. This last sentence is in agreement with my “Remarks on some problems of the mechanical theory of heat” (Wiss. Abhand. Vol II, reprint 39, p 121), where I demonstrated that this is the only valid way to find the most likely state distribution corresponding to the actual thermal equilibrium; here we have demonstrated a posteriori that this leads to the correct state distribution for thermal equilibrium, that which is the most likely in our sense. Of course, it is easy to analyze those cases where other conditions exist besides the principle of conservation of kinetic energy. Suppose, for example, a very large number of molecules for whom (1) the total kinetic energy is constant; (2) the net velocity of the center of gravity in the directions of the x-axis; (3) y-axis, and (4) zaxis are given. The question arises, what is the most probable distribution of the velocity components among the molecules, using the term in the previous sense? We then have exactly the same problem, except with four constraints instead of one. The solution gives us the most probable state distribution f ðu, v, wÞ = Ce - h½ðu - αÞ þðv - βÞ 2

2

þðw - γ Þ2 

,

ð4:110Þ

where C, h, β, γ are constants. This is in fact the state distribution for gas at thermal equilibrium at a certain temperature, not at rest, but moving with a constant net velocity. You can treat similar problems such as the rotation of gas in the same manner, by adding in the appropriate constraint equations, which I have discussed in my essay “On the definition and integration of the equations of molecular motion in gases” (Wiss. Abhand. Vol II, reprint 36).

178

4

Unified Mechanics Theory

Some comment regarding the derivation of Eq. (4.101) from Eq. (4.98) is required here. The formula for x! is pffiffiffiffiffiffiffiffix x 1 þ⋯ 2πx e12x : e The substitution of this into Eq. (4.98) gives P=

pffiffiffiffiffi nþ1 2π n 2  ð2π Þ

pþqþrþ32

e12nþ⋯ 1

wabc þ2 þp b = þq c = þr Πaa =  e12wabc = - p Πb = - q Πc = - r ðwabc Þ 1

1

þ

ð4:111Þ

From which it follows  1 1 ln P = n þ ln n þ þ ⋯ - ðp þ q þ r þ 1Þ ln 2π 2 12n  ð4:112Þ = þq cX aX = þp bX = þr  1 1 wabc þ ln wabc þ þ⋯ 2 12wabc a= -p b= -q c= -r First note, that in determining the magnitude of P , in the limit of a very large number of molecules, n (and thus also of wabc), other small quantities such as E, ζ, η can be treated as infinitesimals. So, all terms which have n or wabc in the denominator can be neglected, and the 12 in the term wabc þ 12. The terms containing wabc scale with the total mass of the gas, while the related 12 terms refer only to a single molecule. So, the latter quantities can be neglected as the number of molecules increases. We then get ln P = n ln n - ðp þ q þ r þ 1Þ ln 2π -

= þq cX aX = þp bX = þr

wabc ln wabc : ð4:113Þ

a= -p b= -q c= -r

Substituting [E ζ η f(aE, bζ, cη)], for wabc we obtain ln P = n ln n - ðp þ q þ r þ 1Þ ln 2π - n ln ðEζηÞ-

= þq aX = þp bX a= -p b= -q



cX = þr

Eζηf ðaE, bζ, cηÞ ln f ðaE, bζ, cηÞ:

ð4:114Þ

c= -r

One sees that aside from the triple sum, the terms on the right-hand side are constant, and so can be omitted. We also let E, ζ, η decrease while p, q, r increase infinitely, so the triple sum goes over into a triple integral over limits -1 to +1 and from ln P we arrive immediately at the expression given by Eq. (4.101) for the permutability measure Ω. The critical condition is that the number of molecules is

4.3

Evolution of Thermodynamic State Index (Φ)

179

very large; this means that wabc is large compared to 12 ; also that the velocity components between the limits aE and (a + 1)E, bζ and (b + 1)ζ, cη and (c + 1)η are identical to those between the limits u and u + du, v and v + dv, w and w + dw. This may appear strange at first sight, since the number of gas molecules is finite albeit large, whereas du,dv, dw are mathematical differentials. But, on closer deliberation this assumption is self-evident. All applications of differential calculus to the theory of gases are based on the same assumption, namely, diffusion, internal friction, [bold emphasized by this author], heat conduction, etc. In each infinitesimal volume element dxdydz there are still infinitely many gas molecules whose velocity components lie between the limits u and u + du, v and v + dv, w and w + dw. The above assumption is nothing more than that very many molecules have velocity components lying within these limits for every u, v, w. Consideration of Polyatomic Gas Molecules and External Forces I will now generalize the formulas obtained so far, by first extending them to so-called polyatomic gas molecules and then including external forces and thereby finally beginning to extend the discussion to any solid and liquid. [Bold emphasized by this author] In order not to consider too many examples, I will in each case deal with the most important case, where, aside from the equation for kinetic energy, there is no other constraint. The first generalization can be applied to our formulas without difficulty. So far, we assumed each molecule was an elastic sphere or a material point so that its position in space was entirely defined by three variables (e.g., three orthogonal coordinates). We know that this is not the case with real gas molecules. We shall therefore assume that three coordinates are insufficient to completely specify the position of all parts of a molecule in space; rather r variables will be necessary p1 , p2 , p3 , . . . , pr , the so-called generalized coordinates. Three of them, p1, p2, p3 are the orthogonal coordinates of the center of mass of the molecule; the others can be either the coordinates of the individual atoms relative to the center of mass, the angular direction, or whatever specifies the location of every part of the molecule. We will also remove the restriction that only one type of gas molecule is present. We assume instead, that there exists a second type whose every molecule has the generalized coordinates p´1 , p´ 2 , p´ 3 , . . . , p´ r0 For the third type, the generalized coordinates are p001 , p}2 , p}3 , . . . , p}r}

180

4

Unified Mechanics Theory

If there are v + 1 types of molecules, the generalized coordinates of the final type are ð vÞ

ðvÞ

ðvÞ

ðvÞ

p1 , p 2 , p 3 , . . . , p r ð v Þ : The first three coordinates are always the orthogonal coordinates of the center of mass. Of course, the necessary assumption is that many molecules of each type are present. Let l be the total kinetic energy of the first type of gas; χ is its potential energy6 (so that χ + l is constant if internal forces only are acting). Furthermore q 1 , q2 , q3 , . . . , q r are the momentum coordinates corresponding to p1, p2, . . ., pr. [Bold emphasized by C.B. Note that, here Boltzmann defines momentum as a new additional axis’ coordinates.] We can think of l in terms of the coordinates p1, p2, . . ., pr and their derivatives with respect to time p_ 1 , p_ 2 , p_ 3 , . . . , p_ r ,  and denote the quantities c1

dl dp_ 1



 ,c2

dl dp_ 2



by q1, q2, . . ., where c1, c2, . . . are arbitrary

constants. I would like to note here that in my essay “Remarks on some problems in the mechanical theory of heat” Section III (Wiss. Abhand. Vol II, reprint 39) the indexed variables designated pi referred to coordinate derivatives, while here they have been designated by q; this mistake would probably not have caused any misunderstanding. We denote with the appropriate accents the analogous quantities for other types of molecules. According to the calculations of Maxwell, Watson, and myself, in a state of thermal equilibrium, the number of molecules for which the magnitudes of p4; p5. . .pr, q1, q2, q3, . . ., qr lie between the limits p4 and p4 þ dp4 ; p5

and

p5 þ dp5 etc:qr and qr þ dqr

ð4:115Þ

is given by Ce - hðχþlÞ dp4 dp5 . . . dqr

ð4:116Þ

where C and h are constants, independent of p and q. Analogous expressions hold of course for the other molecular species with the same value of h, but different values of C. Exactly the same equation as (4.116) is also obtained using the methods of Sections I and II. Consider all those molecules of the first type, for which the

6

The quantity χ called “Kraftfunktion” or “Ergal” by Boltzmann is translated as potential energy.

4.3

Evolution of Thermodynamic State Index (Φ)

181

variables p4, p5, . . ., qr at some time zero lies between the limits (4.115), after a lapse of sometime t, the values of the same variables lie between the limits P4 and P4 þ dP4 ,P5 and P5 þ dP5 ,etc: Qr and Qr þ dQr

ð4:117Þ

The general principle already invoked gives the following equation: dp4  dp5 . . . dqr = dP4  dP5 . . . dQr

ð4:118aÞ

dp1  dp2 . . . dp3 = dP1  dP2 . . . dP3

ð4:118bÞ

There is of course also

So that, in fact, for the variables p4 , p5 , . . . , q r The product of their differentials does not change during a constant time interval. Therefore, we must now imagine v + 1 urns. In the first are slips of paper, upon which are written all possible values of the variables p4, p5, . . ., qr; and the number of slips that have values within the limits of Eq. (4.115) is such that when divided by the product dp4,  dp5. . .qr it is a constant. Similarly, for the labeling of the slips with the variables p04 , p05 , . . . , q0r in the second urn, except that for the latter the constant can have a different value. The same applies to the other urns. We draw from the first urn for each molecule of the first type, from the second urn for molecules of the second type, etc. We now suppose that the values of the variables for each molecule are determined by the relevant drawings. It is of course entirely chance that determines the state distributions for the gas molecules, and we must first discard those state distributions, which do not have the prescribed value for total kinetic energy. It will then be most likely that the state distribution described by Eq. (4.116) will be drawn, i.e., that one corresponding to thermal equilibrium. The proof of this is straightforward. So, the results found in the first two sections can be readily generalized to this case. We want to generalize the problem further, assuming that the gas is composed of molecules specified exactly as before. But now so-called external forces are acting, e.g., those like gravity, which originate outside the gas. [Of course, external forces are not limited to gravity; they can be friction forces exerted by other molecules.] For details on the nature of these external forces, and how to treat them, see my treatise “On the thermal equilibrium of gases on which external forces act.”7 The essence of the solution to the problem remains the same. Only now, the state distribution will no longer be the same at all points of the vessel containing the gas; therefore dp1  dp2  dp3 = dP1  dP2  dP3 will no longer hold. We will now understand the 7

Wien. Ber. (1875) 72:427–457.

182

4

Unified Mechanics Theory

generalized coordinates p1, p2, . . ., pr more generally to determine the absolute position of the molecule in space and the relative position of its constituents. The notion that p1, p2, p3 are just the orthogonal coordinates of the center of gravity is dropped. The same is true for the molecules of all the other types of gas. There is one further point to notice. Previously the only necessary condition was that throughout the vessel very many molecules of each type were present; now it is required that even in a small element of space, over which the external forces do not vary significantly in either size or direction, very many molecules are present (a condition, incidentally, which must hold for any theoretical treatment of problems where external forces on gases come into play). This is because our method of sampling presupposes that the states of many molecules can be considered equivalent, in the sense that the state distribution is not changed when the states of these molecules are exchanged. The probability of a state distribution is then determined by the number of complexions of which this state distribution is capable. This is why, for the case just considered, with v + 1 molecular species present, v + 1 urns must be constructed. We assume first that a complexion has been drawn where w000... = f ð0, 0, 0, . . .Þαβγ . . .

ð4:119Þ

molecules whose variables p1, p2, . . ., qr lie between limits 0 and α, 0 and β, 0 and γ, etc. Furthermore, exactly w10000... = f ðα, 0, 0, . . .Þαβγ . . .

ð4:120Þ

Molecules have the same variables within the limits α and 2α, 0 and β, 0 and γ, etc., and generally wabc = f ðaα, bβ, cγ⋯kκÞαβγ⋯κ

ð4:121aÞ

molecules with variables p1, p2, . . ., qr between limits aα and ða þ 1Þα,bβ and ðb þ 1Þβ . . . kκ and ðk þ 1Þκ,

ð4:121bÞ

These limits are so close that we can equate all the values in between, then it is as if the variable p1 could only take the values 0, α, 2α, 3α, etc., variable p2 could take the values 0, β, 2β, 3β, etc. Let n be the total number of molecules of the first type. We again distinguish the variables for the other gases by the corresponding accents, so that P=

n!n0 !n00 ! . . . nðvÞ !

ðvÞ

Πwabc...k !Πw0a0 b0 ...k0 Πw00a00 b00 ...k00 ΠwaðvÞ bðvÞ ...kðvÞ !

ð4:122Þ

4.3

Evolution of Thermodynamic State Index (Φ)

183

is the possible number of permutations of the elements of this complexion, which we call the permutability. The products are to be read so that the indices a, b, . . ., a′, b′, . . . etc. run over all possible values, i.e., -1 to +1 for orthogonal coordinates, 0 to 2π for angular coordinates, and so on. Consider first, the case where p1 really can take only the values 0, α, 2α, 3α, . . ., and similarly with the other variables; then expression (4.122) is just the number of complexions this state distribution could have; this number is, according to the assumptions made above, a measure of the probability of the state distribution. The pffiffiffiffiffiffi ffi variables w and n are all very large; we can again, therefore, replace w! with 2π ðw=eÞw . We also denote the sum n ln n þ n0 ln n0 þ . . . nðvÞ ln nðvÞ By N, so we can also replace n! by logarithm ln P = N - C ln 2π -

hX

pffiffiffiffiffi 2π ðn=eÞn and then immediately take the

wab...  ln wab... þ

X

i w0a0 b0 ...  ln w0a0 b0 ... þ . . . : ð4:123Þ

The sums are to be understood in the same sense as the products above. 2C is the number of factorials in the denominator of Eq. (4.122) minus v + 1. Let us now substitute the expression (4.121) for the variables w into Eq. (4.123) and then take the limit of infinitesimal α, β, γ. . . . Omitting unnecessary constants, the magnitude we obtain for the permutability measure, denoted by Ω, is

Z Z . . . f ðp1 , p2 , . . ., qr Þ ln f ðp1 , p2 , . . ., qr Þdp1 dp2 , . . . ,dqr

Ω=

Z Z þ



 . . . f p01 , p02 , . . ., q0r0 ln f p01 , p02 , . . ., q0r0 dp01 dp02 , . . . ,dq ′ r0 þ . . . ð4:124Þ

The integration is to extend over all possible values of the variables. I have in my paper “On the thermal equilibrium of gases on which external forces act” demonstrated that the expression in the square brackets is at a minimum for a gas in a state of thermal equilibrium, including, of course, the kinetic energy constraint equation. On the Conditions for the Maximum of the Power-Exponent Free Product Determining the State Distribution Function Before I go into the treatment of the second law, I want to concisely treat a problem whose importance I believe I have shown in Section I, in the discussion of the work of Mr. Oskar Emil Meyer on this subject, namely, the problem of finding the maximum of the product of the probabilities of all possible states. However, I want to deal with this problem only for mono-atomic gases, and with no other constraint than the equation for the kinetic energy. We first consider the simplest

184

4

Unified Mechanics Theory

case where only a discrete number of kinetic energy values, 0, E, 2E, . . ., pE are possible, and to start we use kinetic energies, not velocity components, as variables. We again denote by w0, w1, w2, . . ., wv the number of molecules with kinetic energy 0, E, 2E, . . ., pE. If we treat the subject in the usual way, the following relationship holds: The quantity B = w0  w1  w2  ⋯  wp

ð4:125Þ

Or, if you prefer, the quantity ln B = ln w0 þ ln w1 þ ln w2 þ ⋯ þ ln wp

ð4:126Þ

Must be a maximum, with the constraints n = w0 þ w1 þ w2 þ w2 þ ⋯ þ wp

ð4:127Þ


 L = w1 þ 2w2 þ 3w3 þ ⋯ þ pwp E:

ð4:128Þ

And

If to Eq. (4.126) we add Eq. (4.127) multiplied by h and add Eq. (4.128) multiplied by k, then set the partial derivatives of the sum with respect to w0, w1, w2. . . equal to zero, we obtain the equations 1 þ h = 0, w0

1 þ h þ k = 0, w1

1 þ h þ 2k = 0 etc: w2

ð4:129Þ

From which, by elimination of the constants h and k 1 1 1 1 1 1 = = = ... w1 w0 w2 w1 w3 w2

ð4:130Þ

1 1 1 1 = a, = a þ b, = a þ 2b,⋯ = a þ pb: w0 w1 w2 wp

ð4:131Þ

Or

Substituting these values into Eqs. (4.127) and (4.128), the two constants a and b can be determined:

4.3

Evolution of Thermodynamic State Index (Φ)

1 1 1 1 þ þ þ⋯þ a þ pb a a þ b a þ 2b

ð4:132Þ

E 2E 3E pE þ þ þ⋯þ a þ b a þ 2b a þ 3b a þ pb

ð4:133Þ

n= L=

185

The direct determination of the two unknowns a and b from these equations would be extremely lengthy. The method of Regula falsi would provide a more rapid solution for each special case; I have not troubled myself with such calculations but will give here only a general discussion of how the expected solutions can be easily obtained, keeping in mind that these methods can only provide an approximation solution to the problem, since only positive integers are allowed, but fractional values are not. The first point to note is that the problem ceases to have any meaning as soon as the product p  ( p + 1)/2 is greater than L/E. Because then it necessarily follows that one of the w′s, and so also the product B, is zero. Then there is no question of a maximum value for B. For the problem to make any sense, an excessive value for the kinetic energy cannot be possible. If p

pþ1 L = 2 E

ð4:134Þ

Then all the w′s from w0 onwards must be equal to one for B to be non-zero. A greater variation in values can occur only if smaller values of p are chosen. Then, when n is large the above equations provide usable approximations. First, a will be significantly smaller than b, so w0 is very large, and w1 will be much smaller; w2 will be close to w1/2, w3 will be close to 2w2/3, etc. In general, the decrease in the variable w with an increasing index will be fairly insignificant when the maximum of w0  w1  w2. . . is sought, rather than the maximum of ww0 0 ww1 1 ww2 2 . . . . Given much smaller p values, the value of a is not much less than b, so w0 is also not that much larger than the other w′s; then w2 is greater than w1/2, w3 is greater than (2/3)w2, etc. The decrease of w with increasing index is even less. Decreasing p still further, a will dominate, and there will be hardly any decrease in w with increasing index. Finally, b becomes negative, and the size of w will even increase with increasing index. The following cases provide examples, for each of the integer values of the w′s which maximize B are given.

186

4

Unified Mechanics Theory

n = 30,L = 30E,p = 5,w0 = 17,w1 = 5,w2 = 3,w3 = 2,w4 = 2,w5 = 1: n = 31,L = 26E,p = 4,w0 = 18,w1 = 6,w2 = 3,w3 = 2,w4 = 2: n = 40,L = 40E,p = 5,w0 = 23,w1 = 7,w2 = 3,w3 = 3,w4 = 2,w5 = 2: n = 40,L = 40E,p = 6,w0 = 24,w1 = 6,w2 = 3,w3 = 3,w4 = 2,w5 = 1,w6 = 1: n = 18,L = 45E,p = 5,w0 = 3,w1 = 3,w2 = 3,w3 = 3,w4 = 3,w5 = 3: n = 23,L = 86E,p = 5,w0 = 1,w1 = 2,w2 = 2,w3 = 3,w4 = 4,w5 = 11: ð4:135Þ Let us now turn to the case where the value of the kinetic energy is continuous; first, consider the kinetic energy x as the independent variable [bold emphasized by this author], so the problem, in our view is the following: The expression ZP ln f ðxÞdx

Q=

ð4:136Þ

0

Becomes a maximum, while at the same time ZP

ZP f ðxÞdx and

n=

L=

0

xf ðxÞdx

ð4:137Þ

0

are constant. P is also constant. I have purposefully set the upper integration limit to P, not 1. It is then still straightforward to allow P to increase more and more. Proceeding accordingly, we obtain: ZP

ZP δ

½ ln x þ h f ðxÞ þ kx f ðxÞ dx = 0

 1 þ h þ kx dx δf = 0 f

ð4:138Þ

0

From which it follows f=-

1 1 = h þ kx a þ bx

ð4:139Þ

If we set h = - a; k = - b. To determine these two constants, we use the equations

4.3

Evolution of Thermodynamic State Index (Φ)

ZP n=

187

1 a þ bP ln , a b

ð4:140Þ

P a a þ bP ln , b b2 a

ð4:141Þ

 P bn = ln 1 þ α

ð4:142Þ

f ðxÞ dx = 0

ZP xf ðxÞ dx =

L= 0

which, writing a/b as α, leads to L þ αn =

P , b

And also  P ðL þ αnÞ ln 1 þ = Pn: α

ð4:143Þ

From this transcendental equation, α must be determined, from which it is easy to obtain a and b. Since Pn is the kinetic energy that the gas would have if every molecule in it had the maximum possible kinetic energy P, we see immediately that Pn is infinitely greater than L. L/n is the average kinetic energy of a molecule. It is then easy to verify that P/α cannot be finite because then Pn/αn would be finite and in the expression (L + αn), L could be neglected. But then in Eq. (4.143), only P/α terms would remain, and only vanishingly small values of this term could satisfy the equation, which is inconsistent with the original assumption. Nor can P/α be vanishingly small because then L would again be vanishingly small compared to αn. Furthermore  P ln 1 þ α

ð4:144Þ

Could be expanded in powers of P/α, and the Eq. (4.143) would yield a finite value for P/α. There remains only the possibility that P/α is very large. Since αn Pn ln Pn αn vanishes, Eq. (4.143) gives α=

nP a = pe - L , b

ð4:145Þ

188

4

Unified Mechanics Theory

from which follows b=

P p2 nP , a = e- L L L

ð4:146Þ

By the approach used in this section, using the mean kinetic energy of a molecule, these equations show that in the limit of increasing p, W, L the probability of dispersion in kinetic energy remains indeterminate. We now want to consider a second, more realistic problem. We take the three velocity components u, v, w parallel to the three coordinate axes as the independent variables [bold emphasized by this author], and find the maximum of the expression Zþ1 Zþ1 Zþ1 ln f ðu, v, wÞdudvdw,

Q= -1

-1

ð4:147Þ

-1

While simultaneously the two expressions Zþ1 Zþ1 Zþ1 f ðu, v, wÞdudvdw,

ð4:148Þ

 u2 þ v2 þ w2 f ðu, v, wÞ dudvdw

ð4:149Þ

n= -1

Zþ1

Zþ1

-1

Zþ1

L= -1

-1




-1

-1

remain constant, integrating over u, v, w. If the velocity magnitude is ω=

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 þ v2 þ w2

ð4:150Þ

In addition, its direction is given by the two angles θ and φ (length and breadth8); we have as is well known dudvdw = ωdω sin θdθdφ,

ð4:151Þ

Hence ZP Q = 4π

ln f ðϱÞϱ2 dϱ 0

8

Translators note: Altitude and azimuth?

ð4:152Þ

4.3

Evolution of Thermodynamic State Index (Φ)

189

ZP n = 4π

ϱ2 f ðϱÞ dϱ

ð4:153Þ

ϱ4 f ðϱÞ dϱ

ð4:154Þ

0

ZP L = 4π 0

If there are no external forces [like friction or body forces], then clearly f(u, v, w) is independent of the direction of the velocity. Instead of integrating to infinity, we intentionally integrate to a finite value of P. Evaluating f(ϱ) just as we did for f(x) earlier, we obtain f ð ϱÞ = -

1 1 = h þ kϱ2 a2 þ b2 ϱ2

ð4:155Þ

where we set -h = a2 and -k = b2. The two constants a and b are to be determined from Eqs. (4.153) and (4.154) which become, given the value of f(ϱ), ZP n = 4π 0

 ϱ2 dϱ P a bP = 4π arctg a a 2 þ b2 ϱ2 b 2 b3 ZP

L = 4π 0

ϱ4 dϱ 4πP3 a2 = - 2n 2 3b2 b a2 þ b ϱ2

ð4:156Þ

ð4:157Þ

From the last equation, we get  3 a2 4π = 3 Lþ 2n : P b b2

ð4:158Þ

Substituting this equation into the first equation of (4.157) gives 

L a2 n=3 2 þ 2 2 n P b P

 a bP 1: arctg Pb a

If however b2 is negative, we put -b2 instead of b2 and obtain f ð ϱÞ =

a2

1 , - b2 ϱ2

ð4:159aÞ

190

4

Unified Mechanics Theory



 P a a þ bP n = 4π - 2 þ 3 ln a - Pb b 2b 4πP3 a2 þ 2n 3b2 b    2 a a þ bP an L 1n=3 2 - 2 2 ln 2bP a - Pb P bP L= -

ð4:159bÞ

From Eqs. (4.159a) and (4.159b), one first has to calculate the ratio a/b; and by the same means by which Eq. (4.143) was analyzed, we first determine whether bP/a is infinitely small, finite, or infinitely large, the only difference being in Eq. (4.159a) every infinitesimal variation of L/nP2 occurs. However, I will not discuss the point further, except to note that as n and P grow larger; one also cannot get a result, which depends only on the average kinetic energy. Also, I will not discuss in detail those cases where there are other constraint equations besides the equation for kinetic energy, as this would lead me too far afield. [C.B. Having other constraints would not lead to any more permutations, but actually less number of permutations.] To provide a demonstration of how general the concept of the most probable state distribution of gas molecules is, here I supply another definition for it. Suppose again that each molecule can only have a discrete number of values for the kinetic energy, 0, E, 2E, 3E, . . ., 1. The total kinetic energy is L = λE. We want to determine the kinetic energy of each molecule in the following manner: We have in an urn just as many identical balls (n) as molecules present. Every ball corresponds to a certain molecule. We now make λ draws from this urn, returning the ball to the urn each time. The kinetic energy of the first molecule is now equal to the product of E and the number of times the ball corresponding to this molecule is drawn. The kinetic energies of all other molecules are determined analogously. We have produced a distribution of the kinetic energy L among the molecules (a complexion). We again make λ draws from the urn and produce a second complexion, then a third, etc. many times (J ), and produce J complexions. We can define the most probable state distribution in two ways: First, we find how often in all J complexions a molecule has kinetic energy 0, how often the kinetic energy is E, 2E, etc., and say that the ratios of these numbers should provide the probabilities that a molecule has kinetic energy 0, E, 2E, etc. at the thermal equilibrium. Second, for each complexion, we form the corresponding state distribution. If some state distribution is composed of P complexions, we then denote the quotient P =J as the probability of the state distribution. At first glance, this definition of a state distribution seems very plausible. However, we shall presently see that this should not be used, because, under these conditions, the distribution whose probability is the greatest would not correspond to thermal equilibrium. [C. B.: Boltzmann does not justify or explain this statement. However, later Planck does show that the highest probability is the equilibrium and uses P/J.] It is easy to cast the hypothesis that concerns us into formulas. First of all, we want to discuss the first method of probability determination. We consider the first molecule, and assume that λ draws were made; the probability that the first molecule was

4.3

Evolution of Thermodynamic State Index (Φ)

191

picked in the first draw is 1/n; however, the probability that another ball was drawn is (n - 1)/n. Thus the probability that on the 1st, 2nd, 3rd, . . ., kth draws the molecule corresponding to the first ball has been picked, and then a different ball for each of the following is given by  k    1 n - 1 λ-k n-1 λ 1 k =  n n n n-1

ð4:160Þ

Likewise, the probability that the ball corresponding to the first molecule is picked on the 1st, 2nd, 3rd, . . ., (k - 1)th, and then (k + 1)th draws, etc. The probability that the ball corresponding to the first molecule is picked for any arbitrary k draws and not for the others is wk =

  λ! n-1 λ 1 k : n-1 n ðλ - k Þ!k!

ð4:161Þ

This probability that a molecule has then kinetic energy kE is exactly the same for all the other molecules. Using again the approximation formula for the factorial, we obtain rffiffiffiffiffi 

k 1 n-1 λ 1 λ-k -λ  λ  q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wk =  ffi ðn - 1Þk  ðλ - k Þ , 2π n k 1- λ  k

ð4:162Þ

This shows that the probability of the larger kinetic energies is so disproportionately important that the entire expression does not approach a clearly identifiable limit with increasing k, λ, 1/E, and n. We will now proceed to the second possible definition of the most probable state distribution. We need to consider all J complexions that we have formed by J drawings of λ balls from our urn. One of the various possible complexions consists of λ drawings of the ball corresponding to the first ball. We want to express this complexion symbolically by mλ1  m02  m03 . . . m0n . A second complexion, with λ - 1 draws of the ball corresponding to the first molecule, and one draw of the ball corresponding to the second molecule we want to express as mλ1 - 1  m12  m03  ⋯  m0n We see that the different possible complexions are expressed exactly by various components; the sum of these appears as the power series ðm1 þ m2 þ m3 þ ⋯ þ mn Þλ

ð4:163Þ

192

4

Unified Mechanics Theory

That is developed according to the polynomial theorem. The probability of each such complexion is thus exactly proportional to the coefficient of the corresponding power series term when you first form the product
0 
  ð λÞ ðλÞ m1 þ m02 þ ⋯ þ m0n m001 þ m002 þ ⋯ þ m00n ⋯ m1 þ m2 þ ⋯ þ mðnλÞ Finally omit from this product the upper indexes, which then generate a term exactly proportional to the polynomial coefficient. Then the symbols m01  m003  000 m7 . . . we understand that the first pick corresponded to the first molecule, the second pick corresponded to the third molecule, on the third pick the ball corresponding to the seventh molecule was picked out, etc. All possible products of the variables m01 ,m001 ,m02 etc. represent equi-probable complexions. We want to know how often among all the terms of the power series (4.163) (whose total number is nλ); there occur terms whose coefficients contain any one state distribution. For example, consider the state distribution where one molecule has all the kinetic energy, and all others have zero kinetic energy. This state distribution appears to correspond to the following members of the power series (4.163) mλ1  m02  m03 , . . . ,m01  mλ2  m03 , . . . ,m01  m02  mλ3 . . . etc: with “undivided” λ. Similarly, for the state distribution in which w0 molecules have kinetic energy zero, w1 molecules have kinetic energy E, w2 molecules have kinetic energy 2E, etc., there are λ! w0 !w1 !w2 ! . . . wλ ! members of the power series (4.163). Each of these elements has the same polynomial coefficient, and that is identical to λ! ð0!Þw0 !  ð1!Þw1 !  ð2!Þw2 ! . . . ðλ!Þwλ ! In summary, therefore, according to the now accepted definition, the probability of this state distribution is ðλ!Þ2 1 1   nλ w0 !w1 !w2 ! . . . wλ ! ð0!Þw0  ð1!Þw1  ð2!Þw2 . . . ðλ!Þwλ ! However, the maximization of this quantity also does not lead to the state distribution corresponding to thermal equilibrium.

4.3

Evolution of Thermodynamic State Index (Φ)

193

Relationship of the Entropy to that Quantity which I Have Called the Probability Distribution When considering this relationship, let us initially deal with the simplest and clearest case, by first investigating a monoatomic gas on which no external forces act. In this case, formula (4.101) of Section II applies. To give it full generality, however, this formula must also include the x, y, z coordinates of the position of the molecule. [Note by C.B., LB considers the new axis linearly independent of space-time.] The maximum of such a generalized expression (4.101) then yields, not only the distribution of the velocity components of the gas molecules, which was sufficient for the case considered there, but also the distribution of the whole mass of gas in an enclosing vessel, where there it was taken for granted that the gas mass fills the vessel uniformly. The generalization of (4.101) for the permutability measure can be easily obtained from Eq. (4.124) by substituting x, y, z, u, v, w for p1, p2, . . ., qr and simply omitting the terms with accented variables. It reads as follows: Ω=

Z Z Z Z Z Z

-

f ðx, y, z, u, v, wÞ ln f ðx, y, z, u, v, wÞdxdydzdudvdw, ð4:164Þ

where f(x, y, z, u, v, w)dxdydzdudvdw is the number of gas molecules present for which the six variables x, y, z, u, v, w lie between the limits x and x + dx, y and y + dy, z and z + dz. . . etc. [u and u + du, v and v + dv] w and w + dw and the integration limits for the velocity are between -1 and +1, and for the coordinates over the dimensions of the vessel in which the gas exists. If the gas was not previously in thermal equilibrium, this quantity must grow. We want to compute the value this quantity has when the gas has reached the state of thermal equilibrium. Let V be the total volume of the gas, T be the average kinetic energy of a gas molecule, and N be the total number of molecules of the gas; finally, m is the mass of a gas molecule. There is then for the state of thermal equilibrium f ðx, y, z, u, v, wÞ =

V

N - 3mðu2 þv2 þw2 Þ :
4πT 3=2  e 4T

ð4:165Þ

3m

Substituting this value into Eq. (4.164) gives Ω=

 3  3N 4πT 2 - N ln N þ N ln V 2 3m

If by dQ we denote the differential heat supplied to the gas, where

ð4:166Þ

194

4

Unified Mechanics Theory

dQ = NdT þ pdV

ð4:167Þ

And pV =

2N  T: 3

ð4:168Þ

p is the pressure per unit area, the entropy of the gas is then Z

 3 dQ 2 = N  ln V  T 2 þ C: T 3

ð4:169Þ

Since N is regarded as a constant, with a suitable choice of constant Z

dQ 2 = Ω T 3

ð4:170Þ

It follows that for each so-called reversible change of state, wherein in the infinitesimal limit the gas stays in equilibrium throughout the change R of state, the increase in the Permutability measure Ω multiplied by 23 is equal to dQ/T taken over the state change, i.e., it is equal to the entropy increment. Whereas in fact when a very small amount of heat dQ is supplied to a gas, its temperature and volume increase by dT and dV. Then it follows from Eqs. (4.167) and (4.168) dQ = NdT þ

2N  TdV 3V

ð4:171Þ

while from Eq. (4.166) it is found that dΩ = þ N

dV 3N dT þ  : V 2 T

ð4:172Þ

It is known that if in a system of bodies many reversible changes are taking place, then the total sum of the entropy of all these bodies remains constant. On the other hand, if among these processes are ones that are not reversible, then the total entropy R of all the bodies must necessarily increase, as is well known from the fact that dQ/T integrated over a non-reversible cyclic process is negative. According to Eq. (4.170), the sum of the permutability measures of the bodies ΣΩ and the total permutability measure must have the same increase. Therefore, at thermal equilibrium the magnitude of the permutability measure times a constant is identical to the entropy, within an additive constant; but it also retains meaning during a non-reversible process, continually increasing. We can establish two principles: The first refers to a collection of bodies in which various state changes have occurred, at least some of which are irreversible, e.g., where some of the bodies were not always in thermal equilibrium. If the system was

4.3

Evolution of Thermodynamic State Index (Φ)

195

in a state of thermal equilibrium before and after all these changes, then the sum of the entropies of all the bodies can be calculated before and after those state changes without further ado, and it is always equal to 2/3 times the permutability measure of all the bodies. The first principle is that the total entropy after the state changes is always greater than before it. The same is of course true of the permutability measure. The second principle refers to a gas that changes state without requiring that it begin and end in thermal equilibrium. Then the entropy of the initial and final states is not defined, but one can still calculate the quantity, which we have called the permutability measure; and that is to say that its value is necessarily larger after the state change than before. We shall presently see that the latter proposition can be applied without difficulty to a system of several gases, and it can be extended as well to polyatomic gas molecules, and when external forces are acting. For a system of several gases, the sum of the individual gas permutability measures must be defined to be the permutability measure of the system; if one introduces on the other hand the number of permutations itself, then the number of permutations of the system would be the product of the number of permutations of the constituents. If we assume the latter principle applies to anybody, then the two propositions just discussed are special cases of a single general theorem, which reads as follows: We consider any system of bodies that undergoes some state changes, without requiring the initial and final states to be in thermal equilibrium. Then the total permutability measure for the bodies continually increases during the state changes and can remain constant only so long as all the bodies during the change of state remain infinitely close to thermal equilibrium (reversible state changes). [C.B.: it is important to point out that here Boltzmann drops the term “gas,” in his theorem, because there is nothing in his formulation that restricts it to gasses. The liquid and solid state of the matter will introduce additional constraints but not less than gasses. As a result, his permutability measure is an upper bound for all possibilities.]

To give an example, we consider a vessel divided into two halves by a very thin partition. The remaining walls of the vessel should also be very thin so that the heat they absorb can be neglected, and surrounded by a substantial mass of other gas. One-half of the vessel should be completely filled with gas, while the other is initially completely empty. Suddenly pulling away from the divider, which requires no significant work, causes that gas to spread at once throughout the vessel. Calculating the permutability measure for the gas, we find that this increases during this process, without changes in any other body. Now the gas is compressed in a reversible manner to its old volume by a piston. To achieve this manipulation, we can if we want assume that the piston is created by a surrounding dense gas enclosed in infinitely thin walls. This gas is unchanged except that it moves down in space. Since the permutability measure does not depend on the absolute position in space, the permutability measure of the gas driving the piston does not change. That of the gas inside the vessel decreases to the initial value since this R gas has gone through a cyclic process. However, since this was not reversible, dQ/T integrated over this cycle is not equal to the difference between the initial and final values of the entropy, but it is smaller due to the uncompensated transformation in the expansion.

196

4

Unified Mechanics Theory

In contrast, heat is transferred to the surrounding gas. So, for this process, the permutability measure of the surrounding gas is increased by as much as that of the enclosed gas in the vessel during the first process. Since the latter mass of gas went through a cyclic process, its entropy decreasedR by as much during the second process as it increased during the first, but not by dQ/T; and because the second process was reversible, the entropy of the surrounding gas increased as much as the enclosed gas’s decreased. The result is, as it has to be, that the sum of the permutability measures of all bodies of gas has increased. For a gas that moves at a constant speed in the direction of the x-axis9 2 2 2 N - 3m 4T ððu - αÞ þv þw Þ : f ðx, y, z, u, v, wÞ = V qffiffiffiffiffiffiffiffiffiffiffiffi
 e

4πT 3 3m

ð4:173Þ

If we substitute this expression into Eq. (4.164), we get exactly Eq. (4.166) again. Thus, the translational movement of a mass of gas does not increase its permutability measure. And, the same is true for the kinetic energy arising from any other net mass movement (molar movement), because it arises from the progression of the individual volume elements and their deformations and rotations which are of a higher order—infinitely small—and therefore entirely negligible. Here we obviously ignore the changes in permutability measure due to internal friction or temperature changes connected with those molecular motions. The temperature T of the moving gas is understood to mean half of the average value of m[(u - α)2 + v2 + w2]. So, if frictional [bold emphasis by CB] and temperature changes are not present (e.g., if a gas, together with its enclosing vessel falls freely) a net mass movement does not affect the permutability measure until its kinetic energy is converted into heat, which is why molar motion is known as the heat of infinite temperature. Let us now move on to a mono-atomic gas on which gravity acts. The permutability measure is represented by the same equation (4.124) but instead of the generalized coordinates, we again introduce x, y, z, u, v, w. Equation (4.124) thus gives us a value for Ω which is exactly the same as Eq. (4.164). In the case of thermal equilibrium, one has f ðx, y, z, u, v, wÞ = Ce - 2T 3




gzþmw2

2



ð4:174Þ

where ω2 = u2 + v2 + w2. The constant C is determined by the density of the gas. One has, e.g., a prismatic-shaped vessel of height h, with a flat, horizontal bottom surface with area =q. Further, let N be the total number of gas molecules in the vessel, and z denote the height of a gas molecule from the bottom of the vessel, then

9

Translator note: Here V should appear in the denominator, cf the equation after (4.165).

4.3

Evolution of Thermodynamic State Index (Φ)

C=


4πT 32 3m

197

N N  =
3 Z h 4πT 2 2T - 3gh 3gz 2T  q  1 e 3m 3g  q e 2T dz

ð4:175Þ

0

From which it follows   3gh 3N 4πT 3=2 2T þ N ln q þ N ln þ N ln 1 - e - 2T 2 3m 3g 0 1 3gh 3ghe - 2T A - N ln N þ N @1 -  3gh 2T 1 - e 2T

Ω=

ð4:176Þ

It is immediately apparent from the last formula that when a mass of gas falls a bit lower, without otherwise changing, that does not change its Ω value a bit. (Of course, gravity acts as a constant downward force, and its increase with the approach to the earth’s center is neglected, which is always the case in heat theory problems.) Let us go now to the most general case of an arbitrary gas on which any external force acts, so we again apply Eq. (4.124). However, so that the formulas are not too lengthy, let only one type of gas be present in the vessel. The permutability measure of a gas mixture can then be found without difficulty since it is simply equal to the sum of the permutability measures each component would have if present in the vessel alone. For thermal equilibrium, then Ne f=Z Z

- hðχþLÞ

e - hðχþLÞ

do dw,

ð4:177Þ

where χ is the potential energy, L is the kinetic energy of a molecule, and N is the number of molecules in the vessel, where do = dp1 dp2 . . . dpr ,

dw = dq1 dq2 . . . dqr

ð4:178Þ

It therefore follows Z Z Ω= þ

rN 2

Z Z f ln fdo dw = - N ln N þ N ln

e - hðχþLÞ do dw þ hN χ ð4:179Þ

In the penultimate term, χ is the average potential energy of a molecule, whose magnitude is

198

4

1 N

Z Z

Z Z χf dodw = Z Z

Unified Mechanics Theory

χe - hðχþLÞ dodw e - hðχþLÞ dodw

ð4:180Þ

The last term can be found by taking into account that Z Z L= Z Z

Le - hðχþLÞ dodw e - hðχþLÞ dodw

=

r 2h

ð4:181Þ

(In respect of the above, see the already cited book of Watson, pp 36 and 37). The second term on the right of Eq. (4.179) can be further transformed if one introduces instead of q1, q2, . . ., qr the variables s1, s2, . . ., sr, which have the property that the term L is reduced to s21 þ s22 þ ⋯ þ s2r . We then denote by Δ the Jacobian X

±

dq1 dq2 dqr ⋯ ds1 ds2 dsr

So then Z Z

Z Z Δχe - hχ do  2r π - hðχþLÞ - hχ Z e Δe do dw = do, χ = h Δe - hχ do

ð4:182Þ

And so also Z Ω = N ln

Δe - hχ do -

Nr rN ln h þ hNχ þ ð1 þ ln π Þ - N ln N 2 2

ð4:183Þ

To this expression, one can compare Equation (18) of my paper “Analytical proof of the second law of thermodynamics from the approach to equilibrium in kinetic energy” (Wiss. Abhand. Vol I, reprint 20), or Equation (95) of my “Further Studies” (Wiss. Abhand. Vol I, reprint 22) by replacing p1, p2, . . . with x1, y2, . . ., q1, q2, . . . with u1, v2, . . .; s1, s2, . . . with rffiffiffiffi rffiffiffiffi m m v ..., u1 , 2 1 2 r

4.3

Evolution of Thermodynamic State Index (Φ)

199

with 3r, whereby Δ=

 3r2 2 m

ð4:184Þ

Equation (4.183) is identical to 3N/2 times Equation (18) of the aforementioned paper, except for an additive constant, wherein multiplying by N indicates that Equation (18) applies to one molecule only. Equation (95) of “Further Studies” is in an opposite manner denoted as Ω and thus also as the entropy. Taken with a negative sign, it is however greater than Ω here by N ln N. The former is because in the “Further Studies” I was looking for a value, which must decrease, because of this, introducing the magnitude of f  instead of using f; this was however less clear. From this agreement, it follows that our statement about the relationship of entropy to the permutability measure applies to the general case exactly as it does to a mono-atomic gas [bold emphasis by this author]. Up to this point, these propositions may be demonstrated exactly using the theory of gases. If one tries, however, to generalize to liquid drops and solid bodies, one must dispense with an exact treatment from the outset, since far too little is known about the nature of the latter states of matter, and the mathematical theory is barely developed. But, I have already mentioned reasons in previous papers, in virtue of which it is likely that for these two aggregate states, the thermal equilibrium is achieved when Eq. (4.124) becomes a maximum, and that when thermal equilibrium exists, the entropy is given by the same expression. It can therefore be described as likely that the validity of the principle which I have developed is not just limited to gases, but that the same constitutes a general natural law applicable to solid bodies and liquid droplets, although the exact mathematical treatment of these cases still seems to encounter extraordinary difficulties [bold emphasis by this author].

4.3.2

Critic of Boltzmann’s Mathematical Derivation

Boltzmann assumes that particles are independent rather than an ensemble like in a molecular chain or crystal. He also assumes that there is no group interdependent interaction. However, when his formulation is used in conjunction with conservation of energy, conservation of mass, and Newton’s laws, it automatically makes a solid basis for statistical mechanics for all phases of matter. Because even if the molecules he used in his virtual experiments were no single gas atoms and were interacting chains, the number of microstates could not change for the ensemble. Essentially gas atoms can be replaced by an ensemble of molecules in his formulation and his statistical basis would not change because he does not make any assumptions to preclude forces between molecules or external forces. However, internal friction and inter-dependent group interaction between an ensemble of atoms would make his

200

4

Unified Mechanics Theory

formulation an upper bound for solids. Because of internal inter-dependent interactions and frictions, there may be fewer microstates possible in solids compared to gasses and liquids; however, the number of possible microstates cannot be more than Boltzmann’s analytical derivation has. Finally, Boltzmann’s entropy equation is given by S = k ln Ω

ð4:185Þ

where S is thermodynamic entropy, k is Boltzmann’s constants, and Ω is the number of microstates corresponding to a macrostate. In 1901, Max Planck modified this equation to be S = k ln w þ constant

ð4:186Þ

Here w is the probability (wahrscheinlichkeit) of a microstate. While we discussed the difference between Boltzmann’s equation and Planck’s version of Boltzmann’s equation earlier, unfortunately, except for the physics textbook by Halliday and Resnick (1986), w term defined by Planck (1901) has been misreported in the literature as the number of microstates. Therefore, we find it necessary to include the entire paper by Max Planck (1901) in this chapter. The most important contribution of Max Planck’s (1901) publication is that it establishes the relationship between entropy and disorder. Even though his paper is about the electromagnetic theory of radiation, the concept applies to solids, liquids, and gasses. The fact that Boltzmann does not account for inter-dependent ensemble interaction makes his number of possible permutations bigger and the possibility of complexions higher. Essentially, he calculates an upper bound for the number of complexions. Of course, if we accounted for interaction between the molecules, as spring constraints, the number of complexions may be smaller. Therefore, his permutation gives the upper bound solution for the number of complexions. When we calculate the complexions in solids, of course, we need to take into account the ensemble interaction by all governing laws [Newton and Hooke’s], and conditions imposed on the system. Therefore, Boltzmann’s solution, as he also believes, is perfectly valid for systems with ensemble interactions in any state of the matter. Moreover, Boltzmann assumes that molecules do not have any potential energy. However, he maintains the conservation of energy. Therefore, if we assume that his kinetic energy also includes the potential energy of the molecules, like strain energy, then his mathematics is still accurate.

4.3

Evolution of Thermodynamic State Index (Φ)

4.3.3

201

On the Law of Distribution of Energy in the Normal Spectrum, [Max Planck, (1901) Annalen Der Physik, Vol. 4, p. 553 Ff, 1901]

The recent spectral measurements made by O. Lummer and E. Pringsheim,10 and even more notable those by H. Rubens and F. Kurlbaum,11 which together confirmed an earlier result obtained by H. Beckmann,12 show that the law of energy distribution in the normal spectrum, first derived by W. Wien from molecular-kinetic considerations and later by me from the theory of electromagnetic radiation, is not valid generally. In any case, the theory requires a correction, and I shall attempt in the following to accomplish this based on the theory of electromagnetic radiation, which I developed. For this purpose, it will be necessary first to find in the set of conditions leading to Wien’s energy distribution law that the term can be changed; thereafter it will be a matter of removing this term from the set and making an appropriate substitution for it. In my last article,13 I showed that the physical foundations of the electromagnetic radiation theory, including the hypothesis of “natural radiation,” withstand the most severe criticism; and since to my knowledge there are no errors in the calculations, the principle persists that the law of energy distribution in the normal spectrum is completely determined when one succeeds in calculating the entropy S of an irradiated, monochromatic, vibrating resonator as a function of its vibrational energy U. Since one then obtains from the relationship dS/dU = 1/θ, the dependence of the energy U on the temperature θ, and since the energy is also related to the density of radiation at the corresponding frequency by a simple relation,14 one also obtains the dependence of this density of radiation on the temperature. The normal energy distribution is then the one in which the radiation densities of all different frequencies have the same temperature. Consequently, the entire problem is reduced to determining S as a function of U, and it is to this task that the most essential part of the following analysis is devoted. In my first treatment of this subject, I had expressed S, by definition, as a simple function of U without further foundation, and I was satisfied to show that this form of entropy meets all the requirements imposed on it by thermodynamics. At that time, I believed that this was the only possible expression and that consequently Wien’s law, which follows from it, necessarily had general validity. In a later, closer

10

O. Lummer and E. Pringsheim, Transactions of the German Physical Society 2 (1900), p. 163. H. Rubens and F. Kurlbaum, Proceedings of the Imperial Academy of Science, Berlin, October 25, 1900, p. 929. 12 H. Beckmann, Inaugural dissertation, Tübingen 1898. See also H. Rubens, Weid. Ann. 69 (1899), p. 582. 13 M. Planck, Ann. d. Phys. 1 (1900), p. 719. 14 Compare with Eq. (4.205). 11

202

4

Unified Mechanics Theory

analysis,15 however, it appeared to me that there must be other expressions that yield the same result and that in any case, one needs another condition to be able to calculate S uniquely. I believed I had found such a condition in the principle, which at the time seemed to me perfectly plausible, that in an infinitely small irreversible change in a system, near thermal equilibrium [bold emphasis by this author], of N identical resonators in the same stationary radiation field, the increase in the total entropy SN = NS with which it is associated depends only on its total energy UN = NU and the changes in this quantity, but not on the energy U of individual resonators. This theorem leads again to Wien’s energy distribution law. But since the latter is not confirmed by experience, one is forced to conclude that even this principle cannot be generally valid and thus must be eliminated from the theory.16 Thus, another condition must now be introduced which will allow the calculation of S, and to accomplish this it is necessary to look more deeply into the meaning of the concept of entropy. Consideration of the untenability of the hypothesis made formerly will help to orient our thoughts in the direction indicated by the above discussion. In the following, a method will be described that yields a new, simpler expression for entropy and thus provides also a new radiation equation that does not seem to conflict with any facts so far determined. Calculations of the Entropy of a Resonator as a Function of Its Energy 1. Entropy depends on the disorder and this disorder, according to the electromagnetic theory of radiation for the monochromatic vibrations of a resonator when situated in a permanent stationary radiation field, depends on the irregularity with which it constantly changes its amplitude and phase, provided one considers time intervals large compared to the time of one vibration but small compared to the duration of a measurement. If amplitude and phase both remained absolutely constant, which means completely homogeneous vibrations, no entropy could exist, and the vibrational energy would have to be completely free to be converted into work. The constant energy U of a single stationary vibrating resonator accordingly is to be taken as time average, or what is the same thing, as a simultaneous average of the energies of a large number N of identical resonators, situated in the same stationary radiation field, and which are sufficiently separated so as not to influence each other directly. It is in this sense that we shall refer to the average energy U of a single resonator. Then the total energy U N = NU

ð4:187Þ

Of such a system of N resonators, there corresponds a certain total entropy

15

M. Planck, loc. cit., pp. 730 ff. Moreover one should compare the critiques previously made of this theorem by W. Wien (Report of the Paris Congress 2, 1900, p. 40) and by O. Lummer (loc. cit., 1900, p. 92).

16

4.3

Evolution of Thermodynamic State Index (Φ)

SN = NS

203

ð4:188Þ

In the same system, where S represents the average entropy of a single resonator and the entropy SN depends on the disorder with which the total energy UN is distributed among the individual resonators. 2. We now set the entropy SN of the system proportional to the logarithm of its probability [bold emphasized by CB] w, within an arbitrary additive constant, so that the N resonators together have the energy EN: SN = k log w þ constant

ð4:189Þ

In my opinion, this serves as a definition of the probability [wahrscheinlichkeit] w [bold emphasized by CB], since in the basic assumptions of electromagnetic theory there is no definite evidence for such a probability. The suitability of this expression is evident from the outset, because of its simplicity and close connection with a theorem from kinetic gas theory.17 [C.B. it is not clear why Planck uses log versus ln used by Boltzmann] [C.B. it is important to point out that the term “probability” defined by Planck is different than the modern use of the term. This w is not limited to 0 and 1.] Later we define the thermodynamic state index, which can only have values between 0 and 1. 3. It is now a matter of finding the probability w so that the N resonators together possess the vibrational energy UN. Moreover, it is necessary to interpret UN not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts. Let us call each such part the energy element E; consequently, we must set UN = P E

ð4:190Þ

where P represents a large integer generally, while the value of E is yet uncertain. (The above paragraph is in the original German) Es kommt nun darauf an, die Wahrscheinlichkeit W daf¨ur zu finden, dass die N Resonatoren insgesamt die Schwingungsenergie UN besitzen. Hierzu ist es notwendig, UN nicht als eine stetige, unbeschr¨ankt teilbare, sondern als eine discrete, aus einer ganzen Zahl von endlichen gleichen Teilen zusammengesetzte Gr¨osse aufzufassen. Nennen wireinen solchen Teil ein Energieelement E, so ist mithin zu setzen U N = PE

ð4:191Þ

wobei P eine ganze, im allgemeinen grosse Zahl bedeutet. . .

17

L. Boltzmann, Proceedings of the Imperial Academy of Science, Vienna, (II) 76 (1877), p. 428.

204

4

Unified Mechanics Theory

Now it is evident that any distribution of the P energy elements among the N resonators can result only in a finite, integral, definite number. Every such form of distribution we call, after an expression used by L. Boltzmann for a similar idea, a “complex,” [actually Boltzmann calls this complexion]. If one denotes the resonators by the numbers 1, 2, 3, . . ., N, and writes these side by side, and if one sets under each resonator the number of energy elements assigned to it by some arbitrary distribution, then one obtains for every complex a pattern of the following form:

ð4:192Þ Here we assume N = 10, P = 100. The number R of all possible complexes is obviously equal to the number of arrangements that one can obtain in this fashion for the lower row, for a given N and P. For the sake of clarity, we should note that two complexes must be considered different if the corresponding number patterns contain the same numbers but in a different order. From combination theory, one obtains the number of all possible complexes as R=

N ðN þ 1ÞðN þ 2Þ⋯ðN þ P - 1Þ ðN þ P - 1Þ! = 1  2  3⋯P ðN - 1Þ!P!

ð4:193Þ

Now according to Stirling’s theorem, we have in the first approximation N! = N N

ð4:194Þ

Consequently, the corresponding approximation is R=

ðN þ PÞNþP N N  PP

ð4:195Þ

4. The hypothesis which we want to establish as the basis for further calculation proceeds as follows: for the N resonators to possess collectively the vibrational energy UN, the probability w must be proportional to the number R of all possible complexes formed by the distribution of the energy UN among the N resonators; or in other words, any given complex is just as probable as any other. Whether this actually occurs in nature one can, in the last analysis, prove only by experience. But should experience finally decide in its favor it will be possible to draw further conclusions from the validity of this hypothesis about the particular nature of resonator vibrations, namely, in the interpretation put forth by J. v. Kries18 regarding the character of the “original amplitudes,

18

Joh. v. Kries, The Principles of Probability Calculation (Freiburg, 1886), p. 36.

4.3

Evolution of Thermodynamic State Index (Φ)

205

comparable in magnitude but independent of each other.” As the matter now stands, further development along these lines would appear to be premature. 5. According to the hypothesis introduced in connection with Eq. (4.189), the entropy of the system of resonators under consideration is, after suitable determination of the additive constant, SN = k log R = k fðN þ PÞ log ðN þ PÞ - N log N - P log Pg

ð4:196Þ

Moreover, by considering (4.190) and (4.187) SN = kN

n

1þ

 o U U U U log 1 þ - log E E E E

ð4:197Þ

Thus, according to Eq. (4.188), the entropy S of a resonator as a function of its energy U is given by S=k

n

1þ

 o U U U U log 1 þ - log E E E E

ð4:198Þ

Introduction of Wien’s Displacement Law 6. Next to Kirchhoff’s theorem of the proportionality of emissive and absorptive power, the so-called displacement law, discovered by and named after W. Wien,19 which includes as a special case the Stefan-Boltzmann law of dependence of total radiation on temperature, provides the most valuable contribution to the firmly established foundation of the theory of heat radiation. In the form given by M. Thiesen20 it reads as follows: E  dλ = θ5 ψ ðλθÞ  dλ

ð4:199Þ

where λ is the wavelength, E  dλ represents the volume density of the “blackbody” radiation21 within the spectral region λ to λ + dλ, θ represents temperature, and ψ(x) represents a certain function of the argument x only. 7. We now want to examine what Wien’s displacement law states about the dependence of the entropy S of our resonator on its energy U, and its characteristic period, particularly in the general case where the resonator is situated in an arbitrary diathermic medium. For this purpose, we next generalize Thiesen’s form of the law for the radiation in an arbitrary diathermic medium with the velocity of light c. Since we do not have to consider the total radiation, but only

19

W. Wien, Proceedings of the Imperial Academy of Science, Berlin, February 9, 1893, p. 55. M. Thiesen, Transactions of the German Physical Society 2 (1900), p. 66. 21 Perhaps one should speak more appropriately of a “white” radiation, to generalize what one already understands by total white light. 20

206

4

Unified Mechanics Theory

the monochromatic radiation, it becomes necessary to compare different diathermic media to introduce the frequency n instead of the wavelength λ. Thus, let us denote by u  dν the volume density of the radiation energy belonging to the spectral region ν to ν + dν; then we write u  dν instead of E  dλ; c/ν instead of λ, and c  dν/ν2 instead of dλ. From which we obtain u = θ5

 c cθ  ψ ν ν2

ð4:200Þ

Now according to the well-known Kirchhoff-Clausius law, the energy emitted per unit of time at the frequency ν and temperature θ from a black surface in a diathermic medium is inversely proportional to the square of the velocity of propagation c2; hence the energy density u is inversely proportional to c3 and we have  θ5 θ u= 2 3  f ν ν c

ð4:201Þ

where the constants associated with the function f are independent of c. In place of this, if f represents a new function of a single argument, we can write u=

 θ ν3  f ν c3

ð4:202Þ

In addition, from this, we see, among other things, that as is well known, the radiant energy u  λ3 at a given temperature and frequency is the same for all diathermic media. 8. To go from the energy density u to the energy U of a stationary resonator situated in the radiation field and vibrating with the same frequency, ν, we use the relation expressed in equation (34) of my paper on irreversible radiation processes:22 K=

ν2 U c2

ð4:203Þ

(K is the intensity of a monochromatic linearly, polarized ray), which together with the well-known equation u= yields the relation 22

M. Planck, Ann. d. Phys. 1 (1900), p. 99.

8πK c

ð4:204Þ

4.3

Evolution of Thermodynamic State Index (Φ)

u=

8πν2 U c3

207

ð4:205Þ

From this and Eq. (4.202) follows U =ν  f

 θ ν

ð4:206Þ

where now c does not appear at all. In place of this we may also write θ=ν f

 U ν

ð4:207Þ

9. Finally, we introduce the entropy S of the resonator by setting 1 dS = θ dU

ð4:208Þ

 dS 1 U = f dU ν ν

ð4:209Þ

We then obtain

And integrated S=f

 U ν

ð4:210Þ

That is, the entropy of a resonator vibrating in an arbitrary diathermic medium depends only on the variable U/ν, containing besides this only universal constants. This is the simplest form of Wien’s displacement law known to me. 10. If we apply Wien’s displacement law in the latter form to Eq. (4.198) for the entropy S, we then find that the energy element E must be proportional to the frequency ν, thus: ε = hν

ð4:211Þ

And consequently S=k

n  o U U U U log 1 þ log 1þ hν hν hν hν

ð4:212Þ

Here h and k are universal constants [C. B. Planck’s constant and Boltzmann’s constant]. By substitution into Eq. (4.207) one obtains

208

4

Unified Mechanics Theory

 1 k hν = log 1 þ θ hν U U=

hν hν ekθ - 1

ð4:213Þ ð4:214Þ

and, from Eq. (4.205) there then follows the energy distribution law sought for u=

8π h ν3 1  hν c3 ekθ - 1

ð4:215Þ

or, by introducing the substitutions given in (4.202), in terms of wavelength λ instead of the frequency E=

1 8πch  ch λ5 ekλθ - 1

ð4:216Þ

I plan to derive elsewhere the expressions for the intensity and entropy of radiation progressing in a diathermic medium, as well as the theorem for the increase of total entropy in nonstationary radiation processes. Numerical Values 11. The values of both universal constants h and k may be calculated rather precisely with the aid of available measurements. F. Kurlbaum,23 designating the total energy radiating into the air from 1 square centimeter of a black body at temperature t °C in 1 s by St, found that S100 - S0 = 0:0731  W=cm2 = 7:31  105  erg=cm2 s

ð4:217Þ

From this one can obtain the energy density of the total radiation energy in the air at the absolute temperature 1: 4  7:31  105
 = 7:061  10 - 15  erg=cm3 deg4 3  105  3734 - 2734

ð4:218Þ

On the other hand, according to Eq. (4.215), the energy density of the total radiant energy for θ = 1 is

23

F. Kurlbaum, Wied. Ann. 65 (1898), p. 759.

4.3

Evolution of Thermodynamic State Index (Φ)

Z

1

209

Z

1

v3 dv hv=k - 1 0 e 0 Z 1  hν 2hν 3hν 8πh u = 3 ν3 e - k þ e - k þ e - k þ . . . dν c 0 

u =

8πh u dv = 3 c

ð4:219aÞ ð4:219bÞ

And by termwise integration u =

 4  8πh k 1 1 1  6 1 þ þ þ þ ⋯ h 24 34 44 c3

48πk 4 = 3 3  1:0823 c h

ð4:219cÞ

If we set this equal to 7.061  10-15, then, since c = 3  1010 cm/s, we obtain k4 = 1:1682  1015 h3

ð4:220Þ

12. O. Lummer and E. Pringswim24 determined the product λmθ, where λm is the wavelength of maximum energy in the air at temperature θ, to be 2940 microndegree. Thus, in absolute measure λm = 0:294 cm  deg

ð4:221Þ

On the other hand, it follows from Eq. (4.216), when one sets the derivative of E with respect to θ equal to 0, thereby finding λ = λm  1-

ch  ech=kλm θ = 1 5kλm θ

ð4:222Þ

And from this transcendental equation λm θ =

ch 4:9651  k

ð4:223Þ

Consequently h 4:9561  0:294 = 4:866  10 - 11 = k 3  1010

ð4:224Þ

From this and Eq. (4.220) the values for the universal constants become

24

O. Lummer and E. Pringsheim, Transactions of the German Physical Society 2 (1900), p. 176.

210

4

Unified Mechanics Theory

h = 6:55  10 - 27 erg  s ½today known as Planck0 s constant k = 1:346  10 - 16 erg=deg ½today known as Boltzmann0 s constant These are the same numbers that I indicated in my earlier communication. [There are no references at the end of Planck’s paper.]

4.3.4

Thermodynamic State Index (TSI) in Unified Mechanics Theory

We postulate that at any given temperature and pattern of internal rearrangement within the matter, the rate at which any specific microstructural rearrangement occurs is fully determined by the thermodynamic forces; this is the same concept, postulated by Rice (1971). The thermodynamic state index (TSI) axis coordinate defines, as the name implies, the thermodynamic state of any system between 0 and 1. The TSI axis coordinate value, which can be between 0 and 1, cannot be defined by the Newtonian space-time coordinate system; therefore it is a linearly independent axis. It is like what Boltzmann (1871) defined as the additional momentum axis. Therefore, the unification of Boltzmann’s second law formulation and the laws of Newton requires this additional linearly independent axis. However, in the unified mechanics theory, the thermodynamic state index axis is in the normalized form of Boltzmann’s second law formulation; therefore, it is unitless. The coordinate value on the TSI axis is directly defined by the laws of thermodynamics and according to the thermodynamic fundamental equation of the system/ matter. The thermodynamic fundamental equation must account for all irreversible entropy generation mechanisms according to the “failure” definition chosen. In the last sentence “failure” is in quotation because failure is a vague (subjective) term. For example, if the color change in an acrylic particle-filled polymer is our failure definition, then the fundamental equation will only have mechanisms that contribute to color change, not mechanical failure. All processes, regardless of their result effect, are governed by the laws of thermodynamics. The term “failure” here is used to refer to change in order and increasing disorder, which can be any process when there is irreversible entropy generation. Of course, if the physical collapse of a building is our definition of failure for a problem at hand, the thermodynamic fundamental equation must include all entropy generation mechanisms contributing to physical collapse. We have already presented the second law of thermodynamics in BoltzmannPlanck formulation as S = k ln w

ð4:225Þ

4.3

Evolution of Thermodynamic State Index (Φ)

211

Fig. 4.5 Schematic of a solid continuum undergoing internal entropy change. A sample under uniaxial tension W,S are disorder parameter and entropy, respectively

where S is entropy, k is Boltzmann’s constant, and w is the disorder parameter, which is the probability that the system will exist in the state it is in, relative to all the possible states it could be in. In Fig. 4.5a W, S are the disorder parameter and entropy, respectively, and subscript o is for the initial state and f is for the final state. The definition of “final failure” is problem specific. The problem shown in Fig. 4.5b is identical to the thermodynamic chamber explained in Fig. 3.1. Assume we have a composite material as shown with a schematic in Fig 4.5b. Initially, each particle has a coordinate, such as A3. As a result of external loading particles move to new location and new coordinates. As a result, the disorder will increase proportional to entropy generated. This process is identical to the virtual experiment described by Boltzmann in Sect. 4.3.1. Boltzmann gives each particle a kinetic energy; in the example shown in Fig. 4.5b, assuming a pseudo-static mechanical loading, particles will have strain energy. The second law of thermodynamics states that there is a natural tendency of any isolated (closed) system, living or non-living, to degenerate into a more disordered state. When the irreversible entropy generation rate becomes zero, the system reaches a thermodynamic equilibrium point. This is usually a valley in the energy landscape (Fig. 4.6). It is important to point out that not all specific entropy

212

4

Unified Mechanics Theory

Fig. 4.6 Definition of thermodynamic state index

generation mechanisms contribute to our “failure” definition. Those that do must be identified and must be included in the thermodynamic fundamental equation. Reaching absolute thermodynamic equilibrium is usually not necessary for our formulation to be valid. Because the formulation assumes the closed system is near thermodynamic equilibrium (Fig. 4.1). In statistical mechanics, thermodynamic entropy is considered an intrinsic and independent property of a system. Boltzmann-Planck equations relate entropy to the number of microstates and the probability of a microstate that is consistent with the macroscopic boundary conditions and initial conditions that characterize the system. It is important to point out that the final form of the Boltzmann equation was derived by Planck (1901). This equation is referred to as Boltzmann-Planck equation or just the Boltzmann equation. Therefore, we will use Planck’s definition for variable w (probability or disorder parameter). However, even if we use Ω, the number of microstates, nothing would change in the formulation of TSI, because we normalize the equation with respect to a reference state; as a result, it is a non-dimensional axis. In Eq. (4.225) entropy is for one atom. To convert it to unit-specific mass (g/mol), we need to multiply it by Avogadro’s number NA = 6.022352 × 1023 atoms/mol and divide it by the specific mass (g/mol) ms s=

NA k ln w ms

ð4:226Þ

As a result, the unit for specific entropy s is J/kg K. Initially, let the probability of a matter being in a completely “ordered ground state” [a reference state] be w0. Under any external loading/disturbance, the system will move from the initial [reference] configuration to a new microstate defined by s and w s0 → s ms S0

ms S

w0 = e N A k → w = e N A k

ð4:227Þ

According to the second law, in the final stage, the system will reach maximum entropy and maximum disorder (zero entropy generation rate) state.

4.3

Evolution of Thermodynamic State Index (Φ)

213

s → s max w=e

ms s NA k

→ w max = e

ms s max NA k

ð4:228Þ

During this travel over the energy terrain, we can define the thermodynamics state of the system at any point as a dimensionless variable that defines the distance from the origin, [or any reference state] as follows: 0≤Φ= ms s0

ms s

Φ=

eN A k - e N A k ms s

eN A k

w - w0 ≤1 w h s - s0 i - ms = 1 - e NA ð k Þ

ð4:229Þ ð4:230Þ

Boltzmann constant k can also be given by = NRA , where R is the gas constant. Finally, the thermodynamics state index (TSI) can be given in (Fig. 4.4) h i ms Δs Φ = 1 - e- R

ð4:231Þ

The thermodynamic state index imposes the laws of thermodynamics on Newton’s laws. We will explain this with the following simple examples. For example, we have a soccer ball with an initial acceleration at A (Fig. 4.7). Eventually, the ball will come to a stop and acceleration will be zero at some point B. The energy of the ball between these 2 points will decrease according to the laws of thermodynamics, which is represented by (1 - Φ). Another example would be a simple one-dimensional linear-elastic spring shown in Fig. 4.8. If the spring is subjected to any cyclic or continuous monotonic loading, over time the stiffness of the spring will degrade, and the displacement at the tip of the spring will increase. Finally, the spring will break into two pieces due to fatigue. The degradation of the strain energy capacity [referred to as resilience in some textbooks] of the spring will follow the second law of thermodynamics

Fig. 4.7 Initial acceleration of the ball slows down according to laws of thermodynamics

214

4

Unified Mechanics Theory

Fig. 4.8 One dimensional spring subjected to uni-axial cycling loading

U=

1 2 kδ ð1 - ΦÞ 2

ð4:232Þ

Taking the first derivative of strain energy with respect to deformation and ignoring dΦ dδ for the sake of simplicity, we obtain δ=

F k ð1 - ΦÞ

ð4:233Þ

Of course, the degradation of elastic strain energy storage capacity is due to the degradation of stiffness of the spring material. However, we should point out that in ∂½12kδ2  Eq. (4.233) we assumed that ∂Φ is smaller than ∂δ which is true for high cycle ∂δ fatigue, but it is not true for low cycle fatigue. It is important to remind readers that entropy is essentially energy unavailable for work. Therefore, the total available energy of a closed-isolated system degrades along the TSI axis between 0 and 1. Multiplication of stiffness k with (1 - Φ) is due to the simplicity of the example chosen. Essentially, the TSI coordinate maps the entropy generation rate [thermodynamic fundamental equation] of any system to a linearly independent axis. The question may be asked what happens to a system moving from one stable equilibrium to another and getting an external energy boost during the travel due to an external factor. Because entropy is an additive property, and we have to maintain conservation of energy, we can include the new addition in the system in an incremental form. Actually, in computational mechanics, we solve differential equations in incremental format. Therefore, using the thermodynamic state index works very well for incremental solution procedures. The ramification of the TSI coordinate is the fact that entropy generation rate becomes a nodal unknown in computational mechanics in addition to other nodal unknowns, for example, displacements in a thermo-mechanical analysis. However, to reduce the amount of computation, it is easier to calculate the TSI at Gauss integration points based on the results of the previous step. The amount of error is negligible if the increments are small.

4.4

Experimental Verification Example

4.4

215

Experimental Verification Example

Now that the Thermodynamic State Index (TSI) is defined, we will use two cases to calculate TSI using the experimental data. First, a fully reversed uniaxial cyclic loading on a steel sample is considered, and then a monotonic loading on a steel sample. The experiments were performed in the same conditions and on identical specimens (Fig. 4.1) for the two cases. Now that the Thermodynamic State Index (TSI) is defined, we will use two different experiments to calculate TSI using the measured data. First, a fully reversed uniaxial cyclic loading on a steel sample is considered, and then a monotonic loading case on a steel sample. The experiments were performed in the same conditions and on identical specimens (Fig. 4.9) for the two cases. To evaluate the Thermodynamic State Index (TSI) evolution in both cases, only the effects of plastic deformation are taken into account, for the sake of simplicity. We assume that heat generated during cycling loading is insignificant, because the frequency of loading is very small, and the temperature of the specimen remains constant. Furthermore, because the load is uniaxial, the plasticity is reduced to one dimension. The internal entropy generation is then reduced to Z Δs = t0

t

σ : ε_ p dt ρT

ð4:234Þ

For numerical computation, Eq. (4.234) is simplified to Δsi =

4.4.1

X ðσ  Δεp Þ ρT

:

ð4:235Þ

Tension-Compression Cyclic Loading

The experiment consists of a displacement-controlled test conducted in a material characterization unit, which applies uniaxial tension and compression in repeated and fully reversed cycles to a sample. The experimental data obtained from the digital output consists of force (F) versus displacements (u). Here the fatigue failure

Fig. 4.9 Specimen dimensions and properties

216

4

Unified Mechanics Theory

Fig. 4.10 Testing obtained stress–strain diagram of cyclic loading

Fig. 4.11 Thermodynamic state index evolution for cyclic loading

occurs after 80 cycles. Since we assume a small strain in the model, plastic strain is calculated as follows: εp = ε - εe ,

εp = ε -

σ : E

ð4:236Þ

Figure 4.10 shows the entire engineering stress and strain diagram obtained from the experiment, which is a fully reversed uniaxial cyclic loading. Figure 4.11 depicts the TSI evolution as a function of number cycles. As expected, TSI is initially zero and finally approaches the value of one. Because TSI is an exponential function, it never reaches exactly one. Readers can find more fatigue examples in the papers cited in the references section.

4.5

Dynamic Equilibrium Equations in Unified Mechanics Theory

217

Fig. 4.12 Stress–strain diagram of cyclic loading

4.4.2

Monotonic Loading Test

For this case, the sample is identical to the one used for the cyclic loading. The only difference is that this time it is monotonically loaded until failure [separation of the sample into two pieces] occurs. The loading is uniaxial tension; the experimental data is obtained similarly and treated in the same way as before. Figure 4.12 shows the stress–strain diagram obtained from the monotonic tension test. Figure 4.13 presents the evolution of the TSI for the specimen. TSI starts from zero and approaches one (asymptotically) at the end. We can see that the TSI remains very small for a while before it starts to increase. This domain corresponds to the elastic range of the material. Because we assumed that only plastic deformation could cause degradation in the material, there is no entropy generated in the elastic range according to our assumption. Of course, there is entropy generation due to elastic response; however, it is smaller by orders of magnitude.

4.5

Dynamic Equilibrium Equations in Unified Mechanics Theory

In this section, we will derive the dynamic equilibrium equations using the unified mechanics theory. For further details, readers are referred to Bin Jamal et al. (2021). Newtonian mechanics does not include a term for energy loss; an empirical damping term “C” is used in the dynamic equilibrium equation. However, in unified

218

4

Unified Mechanics Theory

1

Damage Parameter

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

steps

Fig. 4.13 Thermodynamic state index evolution for monotonic loading

mechanics theory energy loss is automatically included because they are part of the universal laws. One-dimensional free vibration analysis, with frictional dissipation, is used to compare the unified mechanics theory results with that of the Newtonian mechanics solution.

4.5.1

Derivation of the Dynamic Equilibrium Equation

The action of a material point, over a time increment, Δt can be defined by the following integral: Z I=

t2

L dt

ð4:237Þ

t1

where L represents the Lagrangian of the system. After we introduce TSI, the Lagrangian of the system can be given by _ Φi Þ L = L ðt, u, u,

ð4:238Þ

where u and u_ represent displacement and velocity of a material point, respectively. The variable Φi is used to represent n number of different independent dissipation [irreversible entropy generating] mechanisms within a given thermodynamic system at a material point. If the integral function in Eq. (4.237) attains a local minimum at a point, q, and let ζ(t) be an arbitrary function that has at least one derivative and vanishes at the initial time, t1, and final time, t2, then for any real number E, closer to zero, the following condition is valid:

4.5

Dynamic Equilibrium Equations in Unified Mechanics Theory

I ðqÞ ≤ J ðEÞ

219

ð4:239Þ

where the real-valued function, J, is represented as follows: J ð E Þ = I ðq þ E ζ Þ

ð4:240Þ

Let’s assume the functional “I” has a local minimum at u = q, and the function J has a local minimum at E = 0. Hence the following equation must hold:

dJ dE



Z E=0

= t1

t2

dL dE

 E=0

dt = 0

ð4:241Þ

Taking the total derivative of Lagrangian, L, yields a new additional term when compared to Newtonian mechanics formulation, dL ∂L du ∂L du_ X ∂L dΦi = þ þ dE ∂u dE ∂u_ dE i = 1 ∂Φi dE n

ð4:242Þ

In Eq. (4.242) that the derivative of Lagrangian with respect to the new axis, Thermodynamic State Index, is non-zero. The displacement is represented as follows: u=q þ E ζ

ð4:243Þ

u_ = q_ þ E ζ_

ð4:244Þ

Hence velocity can be given by

Using Eqs. (4.243) and (4.244), we can write Eq. (4.242) as follows: n dL ∂L ∂L _ X ∂L dΦi = ζþ ζþ dE ∂u ∂u_ ∂Φi dE i=1

ð4:245Þ

For example, a single degree of freedom mass-spring system, subjected to Coulomb friction, is chosen. Hence it is assumed that friction is the only dissipation mechanism. We can write the following total derivative of Φ with respect to E: dΦ ∂Φ du ∂Φ d u_ = þ dE ∂u dE ∂u_ dE Hence, by substituting Eq. (4.246) in Eq. (4.245), we can write

ð4:246Þ

220

4

Unified Mechanics Theory

  dL ∂L ∂L ∂Φ ∂L ∂L ∂Φ _ ζþ þ ζ = þ dE ∂u ∂Φ ∂u ∂u_ ∂Φ ∂u_

ð4:247Þ

Therefore, Eq. (4.241) takes the following form: Z

t2

  Z t2  dL ∂L ∂L ∂Φ ζ dt = dt þ dE E = 0 ∂u ∂Φ ∂u t1 t1 E=0  Z t2  ∂L ∂L ∂Φ _ þ dt = 0 þ ζ ∂u_ ∂Φ ∂u_ t1 E=0

ð4:248Þ

The second term on the right-hand side can be integrated by parts; we get the following equation: Z

t 2 

  ∂L ∂L ∂Φ d ∂L ∂L ∂Φ ζ dt þ þ dt ∂u_ ∂Φ ∂u_ ∂u ∂Φ ∂u t1 E=0  t 2 ∂L ∂L ∂Φ ½ζ E = 0 þ =0 þ ∂u_ ∂Φ ∂u_ t1

ð4:249Þ

By the definition of ζ, the second term in Eq. (4.249) vanishes at the boundary of the integral domain. Hence, using Eqs. (4.243) and (4.244) in Eq. (4.249) and by applying the principle of local state, we can write  ∂L ∂L ∂Φ d ∂L ∂L ∂Φ =0 þ þ dt ∂u_ ∂Φ ∂u_ ∂u ∂Φ ∂u

ð4:250Þ

Compared to the Newtonian mechanics-based Euler-Lagrange equation, Eq. (4.250) has an additional derivative as a function of TSI. Using Lagranged’Alembert’s principle, and variational calculus principles, for an external point load, F, we can write the following equation:  ∂L ∂L ∂Φ d ∂L ∂L ∂Φ þ F=0 þ þ dt ∂u_ ∂Φ ∂u_ ∂u ∂Φ ∂u

ð4:251Þ

Equation (4.251) is the dynamic equilibrium equation in unified mechanics theory. Specific form for a given particular system is derived by writing the thermodynamic fundamental equation for the system and the Lagrangian of the system. In the following section, a mass-spring system with friction as the sole source of dissipation is considered.

4.5

Dynamic Equilibrium Equations in Unified Mechanics Theory

221

Fig. 4.14 Block diagram of a mass-spring system subjected to friction

4.5.2

Vibration of a 1-DOF Mass-Spring System

The governing equation of motion for a mass-spring system, Fig. 4.14, is derived using the unified mechanics theory and compared with that of a Newtonian mechanics equation of motion. Lagrangian of the system is given by L=K -V

ð4:252Þ

where K and V are the total kinetic energy and total potential energy of the system, respectively. In this example, mass is constant as a function of time. For a mass m, excited by an external force, the total kinetic energy of a conservative system (without damping) is given by the following equation: K=

1 _2 m~ u 2

ð4:253Þ

u_ represents the velocity of a mass-spring system in Newtonian mechanics where ~ (with no dissipation). In the unified mechanics theory, we express the net kinetic energy of the mass, with a final velocity of u_ as follows: 1 2 1 _2 u ð1 - ΦÞ mu_ = m~ 2 2

ð4:254Þ

In the same fashion, the potential energy is expressed, by introducing entropy generation due to friction. Hence, the Lagrangian in the unified mechanics theory is given by the following equation: L=

1 1 2 1 1 2 1 _2 1 2 u = mu_ ku m~ u - k~ 2 2 ð1 - Φ Þ 2 ð1 - ΦÞ 2

ð4:255Þ

222

4

Unified Mechanics Theory

In Eq. (4.255), entropy generation due to degradation of the spring stiffness is ignored. Derivatives in Eq. (4.251) can be calculated using Eq. (4.255): ∂L = - ð1 - ΦÞ - 1 ku ∂u

ð4:256Þ

1 1 2 1 2 ∂L 1 mu_ ku = ∂Φ ð1 - ΦÞ2 2 ð1 - ΦÞ2 2

ð4:257Þ

∂ΔS ∂Φ ms ð - ΔSmRs Þ ∂ΔS ms = e ð1 - ΦÞ = R R ∂u ∂u ∂u

ð4:258Þ

Note: Eq. (4.258) is the derivative of entropy with respect to displacement. This , is taken as zero in Newtonian mechanics, even if it is used in a thermodyterm, ∂Φ ∂u namically consistent formulation, because there is no TSI axis. ∂L = ð1 - ΦÞ - 1 mu_ ∂u_   d ∂L d 1 = mu_ ð1 - ΦÞ - 1 þ m€u dt ∂u_ dt ð1 - ΦÞ
dΦ  1 d ∂L dt m€u = mu_ þ 2 dt ∂u_ ð1 - ΦÞ ð1 - ΦÞ dΦ ms dΔS = ð1 - ΦÞ dt R dt    d ∂L ∂Φ d ∂L ∂Φ ∂L d ∂Φ = þ dt ∂Φ ∂u_ dt ∂Φ ∂u_ ∂Φ dt ∂u_ ! !  d 1 d 1 d ∂L 1 2 1 2 = mu_ ku dt ð1 - ΦÞ2 2 dt ∂Φ dt ð1 - ΦÞ2 2 !  1 1 d 1 2 d 1 1 2d _ þ = mu_ - ku2 m u 2 dt ð1 - ΦÞ2 dt 2 ð1 - ΦÞ2 dt 2 !  1 d 1 2 1  ku ð1 - ΦÞ2 dt 2 ð1 - Φ Þ2 !   ms dΔS d ∂L 1 2 1 2 1 1 = mu_ - ku 2 þ 2 2 dt ∂Φ 2 R dt ð1 - ΦÞ2 ð1 - ΦÞ  ðm€ u - kuÞu_

ð4:259Þ ð4:260Þ ð4:261Þ ð4:262Þ ð4:263Þ ð4:264Þ

ð4:265Þ

ð4:266Þ

4.5

Dynamic Equilibrium Equations in Unified Mechanics Theory

∂ΔS ∂Φ ms ð1 - ΦÞ = R ∂u_ ∂u_   ∂ΔS d ∂Φ ms d = ð1 - ΦÞ R dt dt ∂u_ ∂u_  ms d ∂ΔS ms ∂ΔS d = ðð1 - ΦÞÞ þ ð1 - ΦÞ R ∂u_ dt R dt ∂u_

223

ð4:267Þ ð4:268Þ ð4:269Þ

or    d ∂ΔS m 2 d ∂Φ m ∂ΔS dΔS - s ð1 - ΦÞ = s ð1 - Φ Þ ð4:270Þ R dt ∂u_ R dt ∂u_ ∂u_ dt  d ∂L ∂Φ dt ∂Φ ∂u_ ) ( !  ms dΔS 1 1 1 2 1 2 2 ðm€u - kuÞu_ þ = mu_ - ku 2 2 ð1 - ΦÞ2 ð1 - ΦÞ2 R dt ( ) ∂ΔS 1 1 2 1 2 1 ms  ð1 - ΦÞ þ mu_ ku R ∂u_ ð1 - ΦÞ2 2 ð1 - ΦÞ2 2     ∂ΔS dΔS d ∂ΔS m 2 ms ð4:271Þ - s ð1 - Φ Þ  ð1 - ΦÞ R dt ∂u_ R ∂u_ dt Finally, we can write the following equilibrium equation:  1 dΔS m 1 1 1 ms m€ uþ ku þ m u_ þ s R ð1 - ΦÞ dt R ð1 - ΦÞ ð1 - ΦÞ ð1 - ΦÞ  n o ∂ΔS ∂ΔS 1 1 þ F=0  ku2 - mu_ 2 þℊ 2 2 ∂u ∂u_

ð4:272Þ

where the function, ℊ, is the summation of all the terms as a function of ∂ΔS . ∂u_ Newtonian mechanics governing equation of motion for the 1-D sliding friction system shown in Fig. 4.14 is given by m€ u þ μmg sgn ðu_ Þ þ ku = F

ð4:273Þ

where μ is the Coulomb friction coefficient, m is the mass, g is the gravitational constant, and sgn() is the signum function. The damping term “C” in Eq. (4.273) is represented by the Coulomb friction force. Equation (4.273) does not include any dissipation other than mechanical work due to friction. However, other dissipative processes are accounted for in the unified mechanics equilibrium Eq. (4.272). These additional entropy generation mechanisms are defined in terms of change in entropy,

224

4

Unified Mechanics Theory

ΔS, and represented along the thermodynamic state index axis, Φ. The entropy generation function during the sliding friction is derived in the following section.

4.5.3

Entropy Generation in Sliding Friction Contact: Thermodynamic Fundamental Equation

Mechanical work dissipated during friction sliding is given by the following equation: Z Wm =

t2

μmg j du j

ð4:274Þ

t1

The coefficient of friction, μ, is obtained empirically. It is defined by the critical force that initiates the onset of sliding. However, it does not account for heat [continuous friction leads to heat generation] and acoustic losses. The heat generation during sliding friction is dependent on the velocity. A large increment in temperature leads to diffusion of the material, and such process is applied in friction welding applications. This example, for simplicity, does not account for diffusion mechanism entropy generation. Because the purpose of the example is just to derive the dynamic equilibrium equation in unified mechanics theory for a dynamic event. Nevertheless, we simply assume that dissipation due to all other ignored entropy generation mechanisms is proportional to dissipation due to mechanical sliding, Hence, we can write the following relation for total dissipation work: W Td = ð1 þ κ d ÞW m

ð4:275Þ

where W Td is the total dissipation work and κ d is the fraction of work dissipated as heat, acoustic losses, diffusion, etc. The initial potential energy of the spring under free vibration will be dissipated as work of friction, heat, acoustic losses, diffusion, etc. However, this conversion cannot be perfect, and there will be energy lost to unintended work. Hence, the specific entropy generated is given by the following equation: 1 ð1 þ κd Þψ s ΔS = mT

Z

t2

mμg j du j

ð4:276Þ

t1

where ψ s represents the inefficiency factor, as the ratio between the maximum entropy at which TSI, Φ tends to unity (say, at Φ = 0.99) and the maximum entropy that the mass-spring system could generate, for initial potential energy in the spring, if it had zero inefficiencies. Hence, by substituting Eq. (4.276) in Eq. (4.272), we get the following equation:

4.5

Dynamic Equilibrium Equations in Unified Mechanics Theory

n o 1 1 2 1 2 1 1 m m€ uþ ku þ s ku þ mu_ R ð1 - ΦÞ 2 2 ð1 - ΦÞ ð1 - ΦÞ h i 1 ð1 þ κd Þψ s mμg sgn ðu_ Þ = F  mT

225

ð4:277Þ

κ d is required to be able to solve Eq. (4.277). To calculate κ d, we assume the following condition. The rate of heat generation, acoustic losses, and diffusion are dependent on the state of contact surfaces and the impulse force generated due to friction. Since friction is proportional to normal reaction, it will be impulsive when a normal force is impulsive. Due to unevenness (asperity) of the contact surface, a sudden change in the direction of the normal contact force generates an impulsive friction force in addition to the sliding friction force. Impulse force due to interface surface asperities increases the dissipation. It is shown in the literature that the rate of dissipation is maximum in the initial state and is dependent on the TSI, Φ. Similarly, if we represent the dependence of κd with Φ and propose the following relation for the determination of parameter, κ d.  λmg Δt κd = Φ _ mjuj

ð4:278Þ

where Δt represents the duration for which the momentum is calculated and λ is the thermodynamic friction coefficient. The inefficiency factor, ψ s, represents the dissipation of initial strain energy during a free vibration. Once the TSI, Φ, is close to unity (say 0.99), the maximum entropy for a value of, assuming Φc = 1, can be calculated from TSI as follows: h i ms Φ = Φcritical 1 - e - ΔS R

ð4:279Þ

When ΔS0u = 4:605 mRs, Φcritical = 0.99. Keep in mind that Eq. (4.279) is an exponential function; hence, Φcritical = 1.0 only when ΔS = 1. ΔS0u is the maximum entropy, for which TSI is close to unity. For any initial potential energy, the maximum entropy generation cannot be more than the initial potential energy; hence it is given by the following equation: ΔSu =

1 1 2 ku mT 2 0

ð4:280Þ

where u0 is the initial excitation displacement. Hence the inefficiency factor is given by the following equation: ψs =

ΔS ′ u ΔSu

ð4:281Þ

226

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Unified Mechanics Theory

Of course, if we calculate until ΔS = 1 ψ s = 1 will be equal to one. It is less than one ψ s < 1 because we are stopping calculations at Φ = 0.99 Equation (4.277) is solved together with Eqs. (4.276) and (4.281), and the results are presented in the following section.

4.5.4

Comparison with Newtonian Mechanics Results

Free vibration analysis is done for a single degree of freedom system under frictional damping. Equations (4.273) and (4.277) are solved by using Newmark’s method in MATLAB. The model parameters used in the present study are listed in Table 4.2. Simulation results are plotted in Figs. 4.15, 4.16, 4.17 and 4.18 for the input parameters given in Table 4.2. The solution of the equation of motion in unified mechanics theory and Newtonian mechanics for damped oscillation of the mass is shown in Fig. 4.15 for displacement time history and Fig. 4.16 for velocity time history, respectively. It can be readily noticed that both models capture damped oscillation. However, the decay in amplitude is linear for the Newtonian mechanics model, and in unified mechanics, the response is nonlinear. Experimental results on similar models show that the amplitude decay is nonlinear. Since the response of the mass-spring system in the unified mechanics equilibrium equation is dependent on the entropy evolution within the system, a detailed study is required to get the entropy generation function, also known as the thermodynamic fundamental equation. However, this simple example is limited to the derivation of the governing equation of motion using the fundamental principles of the unified mechanic theory. Figure 4.17 shows the response of the mass-spring oscillator subjected to friction, along the Thermodynamic State Index axis. In the present context, the TSI axis represents the thermodynamic state of oscillation. Hence, the interpretation of TSI in this example is limited to the dissipation of initial strain energy stored in the spring. As shown in Fig. 4.17, the oscillations dampen, and the strain energy and kinetic energy of the mass-spring system decays. Even though the Newtonian equation of motion given for the mass-spring system accounts for damping due to friction utilizing a pseudo linear force, the equation does not directly deal with the Table 4.2 List of parameters used to solve the proposed model and the Newtonian mechanics equation Parameter Spring stiffness, k Mass, m Critical TSI, Φcri Friction coefficient, μ Thermodynamic friction coefficient, λ Molar mass for iron, ms Initial spring displacement at t = 0, u0

Value 6.3 × 105 2 0.99 0.2 0.003 55.84 × 10-3 6.25 × 10-4

Unit N/m kg Unitless Unitless Unitless kg/mol m

4.5

Dynamic Equilibrium Equations in Unified Mechanics Theory

227

Displacement, u (UMT model) Displacement, u (Newton's model)

0.0007 0.0006 0.0005

Displacement, u (m)

0.0004 0.0003 0.0002 0.0001 0.0000 -0.0001 -0.0002 -0.0003 -0.0004 -0.0005 -0.0006 -0.0007 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Time (s) Fig. 4.15 Displacement time history for a damped oscillator in the presence of contact surface friction 0.4

Velocity, v (UMT model) Velocity, v (Newton's model)

Velocity, v (m/s)

0.2

0.0

-0.2

-0.4 0.0

0.1

0.2

0.3

0.4

Time (s) Fig. 4.16 Velocity time history time for a damped oscillator in the presence of contact surface friction

228

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Unified Mechanics Theory

Displacement, u (UMT model) 0.4 Velocity, v (UMT model)

0.0006

0.2 0.0002 0.0

0.0000 -0.0002

Velocity, v (m)

Displacement, u (m)

0.0004

-0.2 -0.0004 -0.4

-0.0006 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

TSI

Fig. 4.17 The response of the mass-spring oscillator along with the Thermodynamic State Index (TSI) axis

Fig. 4.18 Displacement amplitude versus time for various static friction coefficient values

thermodynamics of the energy dissipation. However, the unified mechanics theory accounts for the thermodynamics of the system, in the equation of motion, as derived in Eq. (4.272). The dynamic equation of motion in unified mechanics theory must account for all the dissipative processes in the thermodynamic fundamental equations. Unfortunately, these equations are not readily available and must be derived. Figure 4.18 shows the variations in displacement amplitude decay for different frictional coefficients. The linear amplitude decay of the Newtonian equation of motion for friction is not supported by experiments. Tangents to the curves in Fig

4.5

Dynamic Equilibrium Equations in Unified Mechanics Theory

229

4.18 show that the rate of amplitude decay in the unified mechanics theory-based models is high in the initial stage of oscillation when compared with the final stage. This could be due to the macro-level model approximations for the entropy generation equation assumed in Eq. (4.276). A detailed study is required for the identification of all the entropy generation sources that are active in the contact surfaces of the friction block model. Such a study can help in reproducing the nonlinear response of friction block experiments. The unified mechanics theory captures the time at which oscillation stops, as predicted by using the Newtonian mechanics equation of motion. However, the thermodynamic fundamental equation used in this simple example is very crude with several oversimplifications

4.5.5

Comparison with Experimental Data

Test data published in the literature by Feeny and Liang (1996) is used to compare the predictions using the dynamic equilibrium of the unified mechanics’ theory (UMT) and Newton’s equilibrium equation. Feeny and Liang (1996) experimental setup consists of a mass-spring single degree of freedom system with dry friction (Imanian & Modarres, 2015) subjected to an initial displacement. A linear voltage differential transformer is used to sense the motion of mass. A combined effect of Coulomb friction and viscous damping on a single degree of freedom system is investigated by using a similar experimental setup Liang (2005). A list of the model parameters used in this comparative study is given in Table 4.3. Simulations and test data for displacements and displacement-amplitude of vibration are shown in Figs. 4.19 and 4.20, respectively. Both the unified mechanics theory and Newtonian mechanics capture the damped oscillations under Coulomb friction. The amplitude variations, as shown in Fig. 4.20, show a different trend for the unified mechanics theory when compared with the predictions using Newton’s equation of motion. Newton’s equation of motion predicts results closer to the test data. However, it is to be noted that Newton’s model parameter involves experimental curve-fit data, representing the frictional force. Nevertheless, the unified mechanics theory-based model involves TSI, which evolves with the Table 4.3 Parameters used to compare UMT, Newtonian mechanics, and experimental data Parameter Spring stiffness, k Critical TSI, Φc Thermodynamic friction coefficient, λ Mass, m Sliding friction coefficient, μ Molar mass for iron, ms Displacement at t = 0, u0

Value 54 1 0.003 1.51 4.23 × 10-4 55.84 × 10-3 4.25 × 10-3

Unit N/m – – kg – kg/mol m

230

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Unified Mechanics Theory

0.005

Displacement, u (UMT model) Displacement, u (Newton's model) Displacement, u (Test data)

0.004

Displacement, u (m)

0.003 0.002 0.001 0.000 -0.001 -0.002 -0.003 -0.004 -0.005 0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Time (s) Fig. 4.19 Comparison between simulations predictions and test data for the displacement time history. (After Bin Jamal et al. (2020)) 5

-3

Displacement amplitude (×10 m)

Fig. 4.20 Comparison between the predictions and test data for the amplitude time history. (After Bin Jamal et al. (2020))

Test data Newton's model UMT model

4

3

2

1

0 0

1

2

3

4

5

6

7

8

9

10

11

12

Time (s)

thermodynamics of dissipative mechanisms that are associated with friction along the TSI axis. Hence, UMT tracks dissipated energy, not displacement. Therefore, it is expected that empirical models based on Newtonian mechanics give a better match to the experiment they are obtained from. Spring stiffness is assumed to be isotropic under the compressive and tensile loading. Newton’s equilibrium equation agrees with the initial positive peaks of halfcycles of oscillation but does not predict the negative peaks of initial half-cycle data

References

231

for displacement. While comparing the duration of free vibration, UMT predicts a closer result to the test data. Since the equilibrium equation in UMT is a function of the thermodynamic fundamental equation, the model requires the computation of entropy accurately to get a solution matching the test data. Here we conclude the formulation of the unified mechanics theory. In the following chapter applications of the theory are discussed in greater detail.

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Prigogine, I. (1961). Introduction to thermodynamics of irreversible processes (2nd ed.). Interscience. OCLC 219682909. Prigogine, I. (1968). Introduction to thermodynamics of irreversible processes. Wiley-Interscience. Prigogine, I., & Defay, R. (1954). Chemical thermodynamics. Longmans Green and Co.. Prigogine, I., & Herman, R. (1971). Kinetic theory of vehicular traffic. American Elsevier. ISBN 0-444-00082-8. Prigogine, I., & Nicolis, G. (1977). Self-organization in non-equilibrium systems. Wiley. ISBN 0-471-02401-5. Rice, J. R. (1971). Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity. Journal of the Mechanics and Physics of Solids, 19, 433–455. Rice, J. R. (1977). Thermodynamics of the quasi-static growth of Griffith cracks. Journal of the Mechanics and Physics of Solids, 26, 61–78. Richard Swalin (1972). Thermodynamics of Solids, John Wiley & Sons, New York NY Rivas, Á., & Martin-Delgado, M. A. (2017). Topological heat transport and symmetry-protected boson currents. Scientific Reports, 7(1), 6350. https://doi.org/10.1038/s41598-017-06722-x Rivli, R. S. (1981). Some comments on the endochronic theory of plasticity. International Journal of Solids and Structures, 17(2), 231–248. Searles, D. J., & Evans, D. J. (2004). Fluctuations relations for nonequilibrium systems. Australian Journal of Chemistry, 57(12), 1119–1123. https://doi.org/10.1071/ch04115 Sharp, K.; Matschinsky, F. Translation of Ludwig Boltzmann’s Paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI 1877, pp 373–435 (Wien. Ber. 1877, 76:373–435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, p. 164–223, Barth, Leipzig, 1909. Entropy 2015, 17, 1971–2009. https://doi. org/10.3390/e17041971 Sherbakov, S. S., & Sosnovskiy, L. A. (2010). Mechanics of tribo-fatigue systems (p. 407). BSU. Sosnovskiy, L. A. (1987). Statistical mechanics of fatigue damage (p. 288). Nauka i Tekhnika. (In Russian). Sosnovskiy, L. A. (1999). Tribo-fatigue: The dialectics of life (2nd ed., p. 116). BelSUT Press. (In Russian). Sosnovskiy, L. A. (2004). L-risk (Mechanothermodynamics of irreversible damages) (p. 317). BelSUT Press. (In Russian). Sosnovskiy, L. A. (2005). Tribo-fatigue: Wear-fatigue damage and its prediction (Foundations of engineering mechanics) (p. 424). Springer. Sosnovskiy, L. A. (2007). Mechanics of wear-fatigue damage (p. 434). BelSUT Press. Sosnovskiy, L. A. (2009). Life field and golden proportions. Nauka i Innovatsii, 79, 26–33. (In Russian). Sosnovskiy, L. A., & Sherbakov, S. S. (2012). Mechanothermodynamical system and its behavior. Continuum Mechanics and Thermodynamics, 24, 239–256. Sosnovskiy, L. A., & Sherbakov, S. S. (2015). Mechanothermodynamics. Springer. Sosnovskiy, L. A., & Sherbakov, S. S. (2016). Mechanothermodynamic entropy and analysis of damage state of complex systems. Entropy, 18(7), 268. Sosnovskiy, L. A., & Sherbakov, S. S. (2017). A model of mechanothermodynamic entropy in tribology. Entropy, 19, 115. Sosnovskiy, L. A., & Sherbakov, S. S. (2019). On the development of mechanothermodynamics as a new branch of physics. Entropy, 21(12), 1188. Suhir, E. (2019). Failure oriented accelerated testing (FOAT) Boltzmann Arrnhenius Zhurkov equation (BAZ) and their application in aerospace microelectronics and photonics reliability engineering. International Journal of Aeronautical Science and Aerospace Research, 6(3), 185–191. Swendsen, R. H. (2006). Statistical mechanics of colloids and Boltzmann’s definition of the entropy. American Journal of Physics, 74, 187–190.

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Tang, H., & Basaran, C. (2003). A damage mechanics based fatigue life prediction model. Transactions of the ASME: Journal of Electronic Packaging, 125(1), 120–125. Temfack, T., & Basaran, C. (2015). Experimental verification of a thermodynamic fatigue life prediction model. Materials Science and Technology, 31(13), 1627–1632. Turing, A. M. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London B, 237(641), 37–72. Bibcode:1952RSPTB.237...37T. https://doi.org/ 10.1098/rstb.1952.0012.%20JSTOR%2092463 Valanis, K. C. (1971). Irreversibility and existence of entropy. International Journal of Non-Linear Mechanicsm, 6(3), 337–360. Van Kampen, N. G. (1984). The Gibbs paradox. In W. E. Parry (Ed.), Essays in theoretical physics in honour of Dirk ter Haar (pp. 303–312). Pergamon. Valanis and Komkov (1980). Irreversible thermodynamics from the point of view of internal variable theory /A Lagrangian formulation, Archiwum Mechaniki Stosowanej, vol. 32, no. 1, 1980, p. 33–58. Wang, G. M., Sevick, E. M., Mittag, E., Searles, D.J. & Evans, D.J. Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales. Physical Review Letters 89, 050601, (2002) Wang, J., & Yao, Y. (2017). An entropy based low-cycle fatigue life prediction model for solder materials. Entropy, 19, 503. Wang, J., & Yao, Y. (2019). An entropy-based failure prediction model for the creep and fatigue of metallic materials. Entropy, 21(11), 1104. Wang, T., Samal, S. K., Lim, S. K., & Shi, Y. (2019). Entropy production based full-chip fatigue analysis: From theory to mobile applications. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 38(1), 84–95. Whaley, P. W. (1983). A thermodynamic approach to metal fatigue. In Proceedings of ASME. International conference on advances in life prediction methods, Albany, NY, pp. 18–21. Yao, W., & Basaran, C. (2012). Electromigration analysis of solder joints under ac load: A mean time to failure model. Journal of Applied Physics, 111(6), 063703. Yao, W., & Basaran, C. (2013a). Computational damage mechanics of electromigration and thermomigration. Journal of Applied Physics, 114, 103708. Yao, W., & Basaran, C. (2013b). Electrical pulse induced impedance and material degradation in IC chip packaging. Electronic Materials Letters, 9(5), 565–568. Yao, W., & Basaran, C. (2013c). Electromigration damage mechanics of lead-free solder joints under pulsed DC loading: A computational model. Computational Materials Science, 71, 76–88. Young, C., & Subbarayan, G. (2019a). Maximum entropy models for fatigue damage in metals with application to low-cycle fatigue of aluminum 2024-T351. Entropy, 21(10), 967. Young, C., & Subbarayan, G. (2019b). Maximum entropy models for fatigue damage aluminum 2024-T351. Entropy, 21, xx. Yun, H., & Modarres, M. (2019). Measures of entropy to characterize fatigue damage in metallic materials. Entropy, 21(8), 804. Zhang, M.-H., Shen, X.-H., He, L., & Zhang, K.-S. (2018). Application of differential entropy in characterizing the deformation inhomogeneity and life prediction of low-cycle fatigue of metals. Materials, 11, 1917.

Chapter 5

Unified Mechanics of Thermomechanical Analysis

5.1

Introduction

This chapter describes the implementation of the unified mechanics theory (UMT) for inelastic thermomechanical analysis.

5.2 5.2.1

Unified Mechanics Theory-Based Constitutive Modeling Flow Theory and Yield Criteria

Newtonian Mechanics: Elastic Stress-Strain Relationship (Hooke’s Law) For classical von Mises rate-independent plasticity, the elastic constitutive relationship is given by Hooke’s law in the rate form as follows: σ_ = C : ε_ e

ð5:1aÞ

Assuming a small strain case, we can write the following equation:
 ε_ e = ε_ - ε_ p - ε_ θ

ð5:1bÞ

where ε_ ,_εp , and ε_ θ are total strain rate, inelastic strain rate, and thermal strain rate vectors, respectively, and C is the elastic constitutive tensor for the virgin material. In Eq. (5.1) “:” represents the inner product between the fourth-order constitutive tensor C and the elastic strain rate vector, ε_ e .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_5

237

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There is no uniformity in terminology in the plasticity literature; “yield function,” “yield surface,” “plastic potential function,” and “plastic flow function” are all used interchangeably in different publications, essentially referring to the same thing, a boundary between elastic and inelastic behavior of a material, but with different formulations. Von Mises yield criterion is the most basic function originally developed for metals. According to this yield criterion, plastic deformation initiates when the second invariant of the deviatoric stress tensor reaches a critical value. It is surmised that yielding at a stress point begins when the distortional energy reaches a value that is equal to distortional energy at yield in simple uniaxial tension. J2D < k2 if the material is in the elastic state J2D = k2 if the material is yielding where k is a material constant proportional to the initial yield stress in uniaxial tension. In a three-dimensional state of stress, the von Mises criterion can be given by h i 2
2 1
σ xx - σ yy þ σ yy - σ zz þ ðσ zz - σ xx Þ2 þ σ 2xy þ σ 2yz þ σ 2xz = k2 6 In the case of uniaxial tension, this equation becomes 1 2  2σ xx = k2 6 In a simple uniaxial tension case, yielding starts when the applied stress, σ xx, is σ equal to the yield stress of the material, σ xx = σ y. Hence, pyffiffi3 = k: Yield stress of most metals during loading and unloading cycles increases, called isotropic hardening, and the origin of the yield surface moves in the stress space, called kinematic hardening. Considering the general hardening case, for most metals, an elastic-plastic domain can be defined according to the following yield function of the von Mises type: rffiffiffi 2 K ðαÞ  kS - Xk - RðαÞ F ðσ, αÞ = kS - Xk 3

ð5:2Þ

where F(σ, α) is a yield surface separating the elastic from the inelastic domain, σ is the second-order stress tensor, α is a hardening parameter that specifies the evolution of the radius of the yield surface, X is the deviatoric component of the back-stress tensor describing the position of the center of the yield surface in stress space, S is the deviatoric component of the stress tensor given by S = σ - 13 pI whereqpffiffi is the hydrostatic pressure and I is the second-order identity tensor, and RðαÞ 

2 3

K ðαÞ

is the radius of the yield surface in stress space and k k represents the norm operator. A schematic representation of the yield surface is described in Fig. 5.1.

5.2

Unified Mechanics Theory-Based Constitutive Modeling

qffiffiffi Fig. 5.1 Yield surface in the principal stress space.

2 3

239

σ y is the yield stress coordinate on the π

plane

Flow Rule The evolution of the plastic strain is represented by a general associative flow rule of the form ε_ p = γ

∂F  γb n ∂σ

ð5:3Þ

with b n = ∂F being a vector normal to the yield surface in the stress space and ∂σ specifying the direction of plastic flow [this is also referred to as the normality rule], ε_ p are the plastic strain rate vector, and γ is a nonnegative consistency parameter, which is derived later in the chapter. Isotropic Hardening Isotropic hardening describes the increasing yield stress of the material, which is represented by the radius of the yield surface defined in Eq. (5.2). Chaboche (1989) proposed the following evolution empirical function for isotropic hardening in metals: rffiffiffi 2 σ þ R1 ð1 - e - cα Þ K ðαÞ = 3 y0

ð5:4aÞ

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Unified Mechanics of Thermomechanical Analysis

where α is the plastic strain trajectory evolving according to Eq. (5.4b), σ y0 is the initial yield stress in uniaxial tension, R1 is an isotropic hardening saturation value, and c is an isotropic hardening rate material parameter: rffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p p α_ = ε_ ε_ 3

ð5:4bÞ

Using Eqs. (5.3) and (5.4b), we can write the standard definition of equivalent plastic strain trajectory as follows: Z

t1

α= t0

rffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p p ε_ ε_ dt 3

ð5:4cÞ

The Nonlinear Kinematic Hardening (NLK) Rule For implementation, the NLK rule describing the movement of the center of the yield surface in stress space proposed by Chaboche (1989) is used. The model is based on the work of Armstrong and Frederick (1966). Nonlinearities are introduced as a recall term to the Prager (1955) linear hardening rule given in Eq. (5.5) and where c1 and c2 are material parameters: X_ = c1 ε_ p - c2 Xα_

ð5:5Þ

In Eq. (5.5) the first term represents the linear kinematic hardening rule as defined by Prager (1955). The second term is a recall term, often called a dynamic recovery term, which introduces the nonlinearity between the back-stress X and the actual plastic strain. When c2 = 0, Eq. (5.5) reduces to the Prager (1955) linear kinematic hardening rule. The NLK equation describes the rapid changes in the yield stress due to the plastic flow during cyclic loadings and plays an important role even under stabilized conditions (after saturation of cyclic hardening). The NLK rule given by Eq. (5.5) considers the transient hardening effects in each stress-strain hysteresis loop during cycling loadings, and after unloading, dislocation remobilization is implicitly described due to the back-stress effect and the larger plastic modulus at the beginning of the reverse plastic flow. Consistency Parameter, γ In Eqs. (5.3) and (5.4b), γ is a nonnegative consistency parameter representing the irreversible character of plastic flow. The consistency parameter must satisfy the following requirements: 1. For a rate-independent plasticity material model, γ obeys the so-called loading/ unloading and consistency condition: γ ≥ 0 and

F ðσ, αÞ ≤ 0

γ F_ ðσ, αÞ = 0

ð5:6Þ ð5:7Þ

5.2

Unified Mechanics Theory-Based Constitutive Modeling

241

2. For a rate-dependent plasticity, viscoplasticity, material model conditions specified by Eqs. (5.6) and (5.7) are replaced by a constitutive equation of the form γ=

h∅ðF Þi η

ð5:8Þ

where η represents a viscosity material parameter, h i is Macaulay bracket, and ∅(F) is a material-specific function defining the character of the viscoplastic flow. When η → 0 the constitutive model approaches the rateindependent case (Simo & Hughes, 1997). In the case of a rate-independent plasticity, material model F satisfies conditions specified by Eqs. (5.6) and (5.7), and additionally stress states such as F(σ, α) > 0 are ruled out. On the other hand, in the case of a rate-dependent plasticity material model, the magnitude of the viscoplastic flow is proportional to the distance of the stress state to the surface defined by F(σ, α) = 0. Viscoplastic Strain Rate The relation between γ and η expressed in Eq. (5.8) is a general constitutive equation, and different forms of this constitutive relationship describing the material-specific viscoplastic strain rate are used. There are many creep models, and mainly they can be separated into two categories: diffusional creep models and dislocation creep models. In the example here, the creep law proposed by Kashyap and Murty (1981) and extended to the multiaxial case by Basaran et al. (2005) is used in this chapter for the implementation:  n  p   AD0 Eb hF i b - Q=Rθ ∂F e ε_ = kθ E d ∂σ vp

ð5:9Þ

where A is a dimensionless material parameter, which is temperature- and ratedependent, Di = D0e-Q/Rθ is a diffusion coefficient with D0 representing a frequency factor, Q is the creep activation energy, R is the universal gas constant, θ is the absolute temperature in Kelvin, E(θ) is a temperature-dependent Young’s modulus, b is the characteristic length of crystal dislocation (magnitude of Burger’s vector), k is Boltzmann’s constant, d is the average grain size, p is a grain size exponent, and n is a stress exponent for viscoplastic deformation rate. In Eq. (5.9) we can identify h∅ðFÞi = hFin

and η =

 p d kθ eQ=Rθ AD0 En - 1 b b

ð5:10Þ

For a more in-depth study of different viscoplastic models and a comparison of most creep rate models published in the literature, readers are referred to Lee and Basaran (2011).

242

5.2.2

5

Unified Mechanics of Thermomechanical Analysis

Effective Stress Concept and Strain Equivalence Principle

Rabotnov (1969) introduced the effective stress concept, and in the following, we introduce the formulation of Lemaitre and Chaboche (1990) and Lemaitre (1996). To introduce the effective stress concept, it is useful to consider a representative volume element (RVE) of a material loaded by force F. At a point M oriented by a → → plane defined by its normal direction n and its abscissa x along the direction n → (Fig. 5.2), the nominal uniaxial stress is σ = F/δS, where F = n F; δS is the area of the intersection of the plane with the RVE. The effective area of all micro-cracks and micro-cavities that lie in δS is represented by δSDx. It is assumed that no forces are carried by the micro-cracks and micro-cavities. It is convenient to introduce an effective stress concept related to the surface that effectively carries the load, (δS - δSDx), namely, for the n axis, the equation can be given by Rabotnov (1969): σ~x =

F δS - δSDn

ð5:11Þ

The effective stress σ~ is higher than the nominal stress, because stresses are carried by the undamaged [intact] material only. Thermodynamics enables us to define the state at any point by a set of continuous state variables. This postulate means that the constitutive equations written for the surface of (δS - δSDn) are not modified by the damage in the material. Hence, the true stress is the effective stress and not the nominal stress. As a result, we can postulate the following strain equivalence principle by Lemaitre and Chaboche (1990): “Any strain constitutive equation for a damaged material may be derived in the same way as for an intact material except that the usual stress is replaced by the effective stress.” Fig. 5.2 Representative volume element. (After Lemaitre and Chaboche (1990))

5.3

Return Mapping Algorithms

5.2.3

243

Unified Mechanics Theory Implementation

Based on the third law of the unified mechanics theory, the strain energy density rate can be given by 1 1 U_ = σ_ ε_e = ð1 - ΦÞC : ½ε_ e 2 2 2

ð5:12Þ

where Φ is the thermodynamic state index, C is the tangential constitutive tensor, and ε_ e is the elastic strain rate vector. Taking the derivative with respect to strain rate and ignoring the derivative of strain energy rate with respect to thermodynamic state index, Φ, for the simplified version of formulation, yields σ_ = ð1 - ΦÞ C : ε_ e

ð5:13Þ

Of course, ignoring the derivative of Φ with respect to strain rate is not accurate. However, for any incremental procedure, the ignored portion is smaller by orders of magnitude because of the square over strain rate. Moreover, our objective here is to introduce the simplest implementation of unified mechanics theory. For small strain problems, the elastic strain rate can be calculated from the total strain rate vector by subtracting the inelastic and thermal components:
 σ_ = ð1 - ΦÞC : ε_ - ε_ vp - ε_ θ rffiffiffi 2 Φ K ðαÞ  S - X Φ - ð1 - ΦÞRðαÞ F = S - X - ð 1 - Φ Þ 3

ð5:14Þ ð5:15Þ

The evolution of the back-stress, including degradation effects, is given by Φ X_ = ð1 - ΦÞðc1 ε_ vp - c2 Xα_ Þ

ð5:16Þ

It is important to point out that in unified mechanics theory, we only care about degradation along the thermodynamics state index axis. The kinematic hardening function given in Eq. (5.16) is empirical and hence can be defined in many other forms.

5.3

Return Mapping Algorithms

Material nonlinear finite element method implementation requires more sophisticated solution methods compared to linear elastic analysis. A return mapping algorithm is a particular form of a solution algorithm based on a combination between an explicit method and an implicit method. The explicit method makes an

244

5

Unified Mechanics of Thermomechanical Analysis

initial approximation to the solution. The approximated solution is then used in the implicit method to improve the prediction. The following trial (elastic predictor) state can be written as Strnþ1 = Sn þ ð1 - ΦÞ 2GΔenþ1

ð5:17Þ

where G is the shear modulus and Δen+1 is the deviatoric strain increment vector. The increment of the back-stress can then be computed using Eq. (5.16):

 Φ  vp Φ dX Φ nþ1 = ð1 - ΦÞ c1 dεnþ1 - c2 Δγ βX n þ ð1 - βÞX nþ1 where c02 =

ð5:18Þ

qffiffi

2 3 c2

and a generalized midpoint rule for the recall term with the

extreme values of β = 0 and β = 1 corresponds to the backward and forward Euler methods, respectively. Utilizing Eqs. (5.3) and (5.14), incremental form of the viscoplastic strain can be written as dεvp nþ1 = Δγ

Snþ1 - X Φ nþ1 Snþ1 - XΦ nþ1

ð5:19Þ

Substitution of Eq. (5.19) into Eq. (5.18) yields any value of the integration parameter β between 0 and 1: "

S - XΦ c02 Φ nþ1 nþ1 dX Φ nþ1 = anþ1 Δγ c1 X n Snþ1 - X Φ nþ1 with anþ1 =

# ð5:20Þ

c1 ð1 - ΦÞ 1þc02 ð1 - ΦÞð1 - βÞΔγ .

Using the flow rule expressed in Eq. (5.3) allows us to express Eq. (5.17) as Snþ1 - X Φ nþ1 Snþ1 = Strnþ1 - Δγ ð1 - ΦÞ2G Snþ1 - X Φ nþ1

ð5:21Þ

Φ And introducing the relative stress tensor in ξD nþ1 = Snþ1 - X nþ1 yields

Φ tr ξΦ nþ1 = Snþ1 - X nþ1  Snþ1 - Δγ ð1 - ΦÞ2G

Snþ1 - X Φ nþ1 Φ - XΦ n - dX nþ1 ð5:22Þ Snþ1 - X Φ nþ1

Substituting Eq. (5.20) into Eq. (5.22) yields

5.3

Return Mapping Algorithms

245

Snþ1 - XΦ nþ1 Snþ1 - X Φ þ Δγ ½ ð 1 Φ Þ2G þ a  nþ1 nþ1 Snþ1 - XΦ = Bn nþ1

ð5:23Þ

c0

2 where Bn = Strnþ1 - X Φ n þ bnþ1 ΔγX n and bnþ1 = c1 anþ1 . The normal to the yield surface can be expressed in terms of the initial values of the stress; the state variables and the strain increment at each step can be defined as follows:

Snþ1 - X Φ Bn nþ1 n_ nþ1  Snþ1 - X Φ = kBn k nþ1

ð5:24Þ

Taking the trace product of Eq. (5.22) yields Snþ1 - X Φ þ Δγ ½ð1 - ΦÞ2G þ anþ1  nþ1 n Sn - X Φ 2 þ ð1 - ΦÞ2GΔenþ1 þ bnþ1 ΔγX Φ 2 n n =
   Φ 1=2 þ 2 Sn - X Φ n : ð1 - ΦÞ2GΔenþ1 þ bnþ1 ΔγX n g

ð5:25Þ

Using Eq. (5.7) for the rate-independent case or Eq. (5.8) for the rate-dependent case results in the following nonlinear scalar equation for the consistency parameter γ which can be solved by a local Newton method: gðΔγ Þ n Sn - X Φ 2 þ ð1 - ΦÞ2GΔenþ1 þ bnþ1 ΔγX Φ 2


n

n

   1=2 : ð1 - ΦÞ2GΔenþ1 þ bnþ1 ΔγX Φ n g = þ2 rffiffiffi rffiffiffi !

2 2 Δγη K αn þ Δγ - Δγ ½ð1 - ΦÞ2G þ anþ1  - Θ - ð1 - ΦÞ 3 3 Δt Sn - X Φ n

ð5:26Þ After solving Eq. (5.26) for Δγ, the following updating schemes are needed: αnþ1 = αn þ

pffiffiffiffiffi 2=3Δγ

ð5:27Þ

Bn kBn k

ð5:28Þ

vp εvp nþ1 = εn þ Δγ

"

S - XΦ c02 Φ nþ1 Φ nþ1 XΦ nþ1 = X n þ anþ1 Δγ Φ - c Xn Snþ1 - Xnþ1 1 ξΦ nþ1 = ð1 - ΦÞK ðαnþ1 Þ

Bn kBn k

# ð5:29Þ ð5:30Þ

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Unified Mechanics of Thermomechanical Analysis

Φ Snþ1 = ξΦ nþ1 þ X nþ1

ð5:31Þ

  Bn θ ð5:32Þ γ σ nþ1 = ҡð1 - ΦÞtrðεnþ1 Þ I þ 2Gð1 - ΦÞ enþ1 - εvp e nþ1 n nþ1 kB n k

5.3.1

Linearization (Consistent Jacobian)

Differentiating Eq. (5.32) with respect to the total strain but not entropy at the end of the step yields  O ∂Δγ O ∂b nnþ1 : dεnþ1 nnþ1 - 2GΔγ dσ nþ1 = ð1 - ΦÞ C - 2G b ∂εnþ1 ∂εnþ1 

ð5:33Þ

Ignoring differentiation with respect to entropy is not correct. If the strain increments are small, then the ignored part of the differentiation is smaller by an order of magnitude. However, our objective here is to present an earlier and simpler version of the UMT N formulation. nnþ1 is the The operator represents the ordered (dyadic) outer product and b normal vector to the yield surface as d4efined in Eq. (5.3) evaluated at the end of the ∂Δγ increment, and ∂ε can be calculated from Eq. (5.25): nþ 1 b n ∂Δγ = nþ1 ∂εnþ1 K3

ð5:34Þ

Using the following definitions: K3 = K1 þ K2 K1 = 1 þ K2 =

anþ1 K0 þ 3G 2Gð1 - ΦÞ

a0nþ1 Δγ b n bnþ1 1 ∂Θ þ nþ1 ½b ð1 - βÞΔγ - 1 : XΦ n þ 2Gð1 - ΦÞ ∂Δγ 2Gð1 - ΦÞ 2Gð1 - ΦÞ nþ1

In addition,

∂b nnþ1 ∂εnþ1

can be obtained from Eq. (5.24) as



O ∂b nnþ1 ∂b nnþ1 ∂Bn ∂Bn 1 b nnþ1 nnþ1 : = →  I -b B k k ∂εnþ1 ∂ε ∂ε nþ1 nþ1 nþ1 ∂Bn

ð5:35aÞ

5.3

Return Mapping Algorithms

247

and O
 O ∂Δγ ∂Bn b - 1I = 2μð1 - ΦÞ Π I þ b0nþ1 Δγ þ bnþ1 Xn 3 ∂εnþ1 ∂εnþ1 b is fourth-order unit tensor mapped into the unit matrix and I second-order where Π identity tensors mapped into unit column vector and K′ denotes the derivative with respect to the argument in K(α).  4 = b0nþ1 Δγ þ bnþ1 and substituting ∂Bn in Eq. (5.35a) yields Defining K ∂εnþ1

O ∂b nnþ1 ∂b nnþ1 ∂Bn 2Gð1 - ΦÞ b 1 O b =  Π-b nnþ1 nnþ1 - I I 3 kBn k ∂εnþ1 ∂Bn ∂εnþ1 1 þ kBn k

 O O  K4 b b XΦ ð5:35bÞ  I-b nnþ1 nnþ1 nnþ1 : n K3 Using Eqs. (5.34) and (5.35b) into Eq. (5.33) results in the following algorithmic version of the material Jacobian coupling the effects of damage and rate dependency: O b - 1I I þ 2Gð1 - ΦÞδnþ1 Π I 3

O O 2Gð1 - ΦÞ b b b nnþ1 Δγ Π - b nnþ1 nnþ1 nnþ1 - 2Gð1 - ΦÞθnþ1b kBn k   O 4 K b ð5:35cÞ : nnþ1 XΦ n K3

CEVPD nþ1 = ð1 - ΦÞҡI

O

where ҡ is the bulk modulus and δnþ1 = 1 -

Δγ2Gð1 - ΦÞ kB n k

and

θnþ1 =

Δγ2Gð1 - ΦÞ 1 kBn k K3

The pseudocode corresponding to the above algorithm is detailed in Table 5.1. Moreover, Table 5.2 shows the local Newton-Raphson algorithm used to solve for the consistency parameter, which preserves the quadratic rate of convergence of the Newton scheme.

248

5

Unified Mechanics of Thermomechanical Analysis

Table 5.1 Return mapping algorithm-classical theory rate-dependent model εn + 1 = εn +— Δμ Str nþ1 = Sn þ ð1 - Φn Þ2GΔenþ1 qffiffi tr D 2 Ftr nþ1 = Snþ1 - X n - ð1 - ΦÞ 3K ðαn Þ

Update strain Compute trial state Compute trial yield function

IF Ftr nþ1 > 0 THEN Call Newton g(Δγ) = 0 for Δγ n local and solve 2
 2 gðΔγ Þ = Sn - X Φ þ ð1 - ΦÞ2GΔenþ1 þ bnþ1 ΔγX Φ þ 2 Sn - X Φ : n



bnþ1 ΔγXΦ n

n

1=2

n

ð1 - ΦÞ2GΔenþ1 þ qffiffi qffiffi
 - ð1 - ΦÞ 23K αn þ 23Δγ - Δγ ½ð1 - ΦÞ2G þ anþ1  - Θ Δγη Δt where anþ1 =

c1 ð1 - ΦÞ 1þc02 ð1 - ΦÞð1 - βÞΔγ ,

bnþ1 =

c2 c1

anþ1

Update nnþ1 

Snþ1 - X Φ nþ1

XΦ kSnþ1 qffiffinþ1 k

αnþ1 = αn þ εvp nþ1

= εvp n

þ

=

2 3Δγ Δγ kBBnn k

Φ XΦ nþ1 = X n þ anþ1 Δγ

Bn kBn k



Φ Φ where Bn = Str nþ1 - X n þ bnþ1 ΔγX n

Snþ1 - X Φ nþ1

k Bn ξΦ ð 1 Φ ÞK ð α Þ nþ1 nþ1 kBn k

Snþ1 - X Φ nþ1

k

-

c02 c1

 XΦ n

Φ Snþ1 = ξΦ nþ1 þ X nþ1



Bn θ σ nþ1 = ҡð1 - ΦÞtrðεnþ1 ÞbI þ 2Gð1 - ΦÞ enþ1 - εvp n - γ nþ1 kBn k - enþ1 Compute consistent Jacobian   N Q 1 Ob b bI þ 2Gð1 - ΦÞδnþ1 I CEVPD - bI nþ1 = ð1 - ΦÞҡI 3 N - 2Gð1 - ΦÞθnþ1 b nnþ1 nnþ1 b Q N N Φ 4 ΦÞ K b b b - 2Gkð1BΔγ ð n Þ : n Xn n nþ1 nþ1 nþ1 k n K Where δnþ1 = 1 -

Δγ2Gð1 - ΦÞ kBn k

3

and θnþ1 =

1 K3

-

Δγ2Gð1 - ΦÞ kBn k

with

K3 = K1 þ K2 K0 K 1 = 1 þ 3G þ 2Gða1nþ1 - ΦÞ 0 anþ1 Δγ bnnþ1 bnþ1 1 K 2 = 2Gð1 - ΦÞ þ 2Gð1 - ΦÞ ½bnþ1 ð1 - βÞΔγ - 1 : X Φ n þ 2Gð1 - ΦÞ 0  4 = b Δγ þ bnþ1 K nþ1 ELSE Elastic step ð Þnþ1 = ð Þtr nþ1 (Exit) END IF EXIT

∂Θ ∂Δγ

5.4

Thermodynamic Fundamental Equation in Thermomechanical Problems

249

Table 5.2 Local Newton iteration for the consistency parameter-classical theory Let Δγ (0) ← 0 αn + 1(0) ← αn Start Iterations DO_UNTIL |g(Δγ)| < tol k←k+1 Compute nΔγ (k + 1) 2 2
 gðΔγ Þ = Sn - X Φ þ ð1 - ΦÞ2GΔenþ1 þ bnþ1 ΔγX Φ þ 2 Sn - X Φ : n



bnþ1 ΔγXΦ n

1=2

n

n

ð1 - ΦÞ2GΔenþ1 þ qffiffi qffiffi
 - ð1 - ΦÞ 23K αn þ 23Δγ - Δγ ½ð1 - ΦÞ2G þ anþ1  - Θ Δγη Δt
 dg Δγ ðkÞ =

∂gðΔγ ðkÞ Þ ∂Δγ ðkÞ gðΔγ ðkÞ Þ ðkþ1Þ ðkÞ Δγ ← Δγ - dgðΔγðkÞ Þ

END DO_UNITL

5.4

Thermodynamic Fundamental Equation in Thermomechanical Problems

We have defined thermodynamic fundamental relations [equation] in the earlier chapter on thermodynamics. However, we feel it is necessary to restate some basics again to make it easier to understand the derivation for the readers. We find it helpful to quote these basics directly from DeHoff (1993). In irreversible thermodynamics, the so-called balance equation for the entropy generation plays a central role. This equation expresses the fact that the entropy of a volume element changes with time for two reasons. First, it changes because entropy flows into the volume element, and second it changes because there is an entropy generation due to irreversible phenomena inside the matter. The internal entropy generation is always a nonnegative quantity, since entropy can only be created, but never destroyed. This contrasts with energy that cannot be created. We find it helpful to quote DeHoff (1993): Thermodynamic entropy is the measure of how much energy is unavailable for work. Imagine an isolated and closed system with some hot objects and some cold objects. Work can be done as heat is transferred from the hot to the cooler objects; however, once this transfer has occurred, it is impossible to extract additional work from them alone. Energy is always conserved, but when all objects have the same temperature, the energy is no longer available for conversion into work. The entropy [energy unavailable for work or disorder] of the universe increases or remains constant in all-natural processes. It is possible to find a system for which entropy decreases but only due to a net increase in a related system. For example, the originally hot objects and cooler objects reaching thermal equilibrium in an isolated system may be separated, and some of them put in a refrigerator. The objects would again have different temperatures after some time, but now the system of the refrigerator would have to be included in the analysis

250

5

Unified Mechanics of Thermomechanical Analysis

of the complete system. No net decrease in entropy of all the related systems occurs. This is yet another way of stating the second law of thermodynamics. (DeHoff, 1993) The concept of entropy has far-reaching implications that tie the order/disorder of our universe to probability and statistics. Imagine a new deck of cards in order by suits, with each unit in numerical order. As the deck is shuffled, no one would expect the original order to return. There is a probability that the randomized order of the shuffled deck would return to the original format, but it is exceedingly small. An ice cube melts, and the molecules in the liquid form have less order than in the frozen form. An infinitesimally small probability exists that all of the slower-moving molecules will aggregate in one space so that the ice cube will reform from the pool of water. The entropy, or disorder, of the universe increases as hot bodies cool and cold bodies warm. Eventually, the entire universe will be at the same temperature so the energy will be no longer usable. (DeHoff, 1993)

To relate the entropy generation directly to irreversible processes that occur in a system, one needs the macroscopic conservation laws of mass, momentum, and energy in local form, i.e., differential form. The conservation laws are quantified utilizing information from processes such as diffusion, heat flow, and velocity, which are related to the transport of mass, exchange of energy, and exchange of momentum, respectively. Then, the entropy generation can be calculated by using the thermodynamics Gibbs relation, which connects the rate of the change in entropy in the medium to the rate of the change in energy and work. Entropy generation rate has a relatively simple formula: it is a sum of all entropy generating micromechanism terms, each being a product of a flux characterizing an irreversible process, and a quantity called thermodynamic force, which is related to the gradient [non-uniformity] in the system (Mazur & De Groot, 1962). We discussed this topic earlier in Chap. 3 in detail in Onsager relations, as well. The complete entropy generation rate equation can thus serve as a basis for the systematic description of the irreversible processes occurring in a system. We refer to this equation as the thermodynamic fundamental equation. “As yet, the set of conservation laws, together with the entropy balance equation and the equations of state are to a certain extent empty, since this set of equations contains the irreversible fluxes as unknown parameters and can therefore not be solved with the given initial and boundary conditions for the state of the system” (Mazur & De Groot, 1962). At this point, we must therefore supplement the equations with an additional set of phenomenological relationships [Onsager reciprocal relations], which relate the irreversible fluxes and the thermodynamic forces appearing in the entropy source strength. Irreversible thermodynamics, in its present form, is mainly restricted to the study of the linear relationship between the fluxes and the thermodynamic forces as well as possible cross-effects between various phenomena. This is not a very serious restriction, however, since even rather extreme physical situations are still described by linear laws. Together with the phenomenological equations, the original set of conservation laws may be said to be complete in the sense that one now has a consistent set of partial differential equations for the state parameters of a material system, which may be solved with the proper initial and boundary conditions. (Mazur & De Groot, 1962)

5.5

Conservation Laws

5.5

251

Conservation Laws

Thermodynamics is based on two fundamental laws: the first law of thermodynamics or the law of conservation of energy and the second law of thermodynamics or the entropy law. A systematic macroscopic scheme for the description of irreversible processes must also be built upon these two laws. However, it is necessary to formulate these laws in a suitable way for use in continuum mechanics. Since we wish to develop a theory applicable to systems in which the properties are continuous functions of space coordinates and time, we need a local formulation of the law of conservation of energy. As the local momentum and mass densities may change in time, we will also need local formulations of the laws of conservation of momentum and conservation of mass. In solid mechanics, the thermodynamic system is usually chosen as a collection of continuous matter, i.e., the system is closed and does not interchange matter with its surroundings, and the bounding surface of the system in general moves with the flow of matter.

5.5.1

Conservation of Mass

Consider an arbitrary volume V fixed in space, bounded by surface Ω. The rate of change of the mass within the volume V is given by, using the notation used by Malvern (1969) d dt

Z V

Z ρdV =

V

∂ρ dV ∂t

ð5:36Þ

where ρ is the density (mass per unit volume). If no mass is created or destroyed inside V, this quantity must be equal to the rate of the matter flow into the volume V through its surface Ω: Z V

∂ρ dV = ∂t

Z Ω

ρ ~v  dΩ

ð5:37Þ

where ~v is the velocity and dΩ is a vector with magnitude |dΩ| normal to the surface and defined as positive from the inside toward the outside. The quantities ρ and v are all functions of time and space coordinates. Applying Gauss’s theorem to the surface integral in equation (5.37), we obtain ∂ρ = - ρ div ~ν ∂t

ð5:38Þ

Equation (5.38) is valid for any arbitrary volume V, which expresses the fact that the total mass is conserved, i.e., the total mass in any volume element of the system can only change if matter flows into (or out of) the volume element. This equation

252

5

Unified Mechanics of Thermomechanical Analysis

has the form of a so-called balanced equation: the local change of the density is equal to the negative divergence of the flow of mass. The continuity equation in the vector form of equation (5.38) is independent of any choice of coordinate system. The conservation of mass equation can also be written in an alternative form by introducing the substantial time derivative (Mazur & De Groot, 1962): d ∂ = þ ~ν  grad dt ∂t

ð5:39Þ

The substantial time derivative is the rate of change of a quantity (i.e., mass) as ∂ measured by an observer moving along with the flow of matter. The first term ∂t accounts for the time rate of change according to a stationary observer, and the second term ð~ν  gradÞ accounts for the movement of the observer. With the help of Eq. (5.39), we can rewrite Eq. (5.38) as follows: dρ = - ρ div ~v dt

ð5:40Þ

With the specific volume ~v = ρ - 1 , formula (5.40) may also be written as ρ

d~v = div ~v dt

ð5:41Þ

Finally, the following substantial time derivative relation is valid for an arbitrary local property a that may be a scalar or a component of a vector or tensor: ρ

da ∂aρ = þ div aρ~v dt ∂t

ð5:42Þ

which is a consequence of Eqs. (5.38) and (5.39). We can verify Eq. (5.42) directly. According to Eq. (5.39), the left-hand side of Eq. (5.42) is ρ

da ∂a =ρ þ ρ~v  grad a dt ∂t

ð5:43Þ

According to Eq. (5.38), the right side of Eq. (5.42) is ∂aρ ∂ρ ∂a þ div aρev = a þρ þ a div ρev þ ρev  grad a ∂t ∂t ∂t ∂a = að - div ρevÞ þ ρ þ a div ρev þ ρev  grad a ∂t ∂a =ρ þ ρev  grad a ∂t Therefore, Eq. (5.42) is true.

ð5:44Þ

5.5

Conservation Laws

5.5.2

253

Momentum Principle in Newtonian Mechanics

The momentum principle for a collection of particles states that the time rate of the change in the total momentum for a given set of particles equals the vector sum of all the external forces acting on the particles of the set provided Newton’s Third Law of action and reaction governs the initial forces (Malvern, 1969). Consider a given mass of the medium, instantaneously occupying a volume V bounded by surface A and acted upon by external surface tractions σ (n) and body force b. Then the momentum principle can be expressed as follows: Z A

σ ðnÞ dAþ V

Z ρ b dV =

d dt

Z V

ρ ~v dV

ð5:45Þ

or in rectangular coordinates: Z A

Z σ i dAþ V

ρbi dV =

d dt

Z V

ρ v~i dV

ð5:46Þ

Substituting σ i = σ jinj and transforming the surface integral to a volume integral by using the divergence theorem, we obtain Z  V

 ∂σ ji dv~i dV = 0 þ ρbi - ρ dt ∂xj

ð5:47Þ

Equation (5.47) must be true for any arbitrary volume V. At each point, we have ρ

d~vi ∂σ ji þ ρbi = dt ∂xj

ð5:48Þ

where nj is the component of the normal unit vector n, v~i ði = 1, 2, 3Þ is a Cartesian component of velocity ~v , and xj ( j = 1, 2, 3) is the Cartesian coordinates. The quantities σ ji (i, j = 1, 2, 3) and bi (i = 1, 2, 3) are the Cartesian components of the stress tensor σ and body force b, respectively. For a nonpolar case, the stress tensor σ is symmetric, namely, σ ij = σ ji

ði, j = 1, 2, 3Þ

ð5:49Þ

In tensor notation, equation (5.48) is written as ρ

d~v = div σ þ ρb dt

ð5:50Þ

From a microscopic point of view, the stress tensor σ results from the short-range interactions, [strain] between the particles of the system, whereas b contains the external forces as well as a possible contribution from long-range interactions in the system.

254

5

Unified Mechanics of Thermomechanical Analysis

Using Eq. (5.46), the equation of motion Eq. (5.50) can also be written as ∂ρ~v = - div ðρ~v~v - σÞ þ ρb ∂t

ð5:51Þ

N where ~v~v = ~v ~v is an ordered (dyadic) product. This equation also has the form of a balance equation for the momentum density ρ~v . One can interpret the quantity ðρ~v~v - σÞ as a momentum flow with a convective part ρ~v~v , and the quantity ρb as a source of momentum, but there is no entropy generation part. It is also possible to derive from Eq. (5.48) a balance equation for the kinetic energy of the center of gravity motion by multiplying both members with the components of ~v and then summing over i: ρ

 X3 d 12 ~v2 X3 ∂
∂ ~ σ σ = v v~ þ ρbi v~i ji i i,j = 1 ∂xj i,j = 1 ji ∂xj i dt

ði = 1, 2, 3Þ

ð5:52Þ

or in tensorial notation: ρ

d 12 ~v2 = divðσ  ~vÞ - σ : L þ ρb  ~v dt

ð5:53Þ

where L = grad~v is the spatial gradient of the velocity. L can be written as the sum of a symmetric tensor D, called the rate of deformation tensor and a skew-symmetric tensor W called the spin tensor (or the vorticity tensor) as follows: L=D þ W

ð5:54Þ



 where D = 12 L þ LT , and W = 12 L - LT . Since W is skew-symmetric, while σ is symmetric, it follows that σ : grad ~v = σ ij Lij = σ ij Dij = σ : D

ð5:55Þ

We can also establish the relationship between the strain rate dε/dt and the rate of the deformation tensor D: dε = FT  D  F dt

ð5:56Þ

where F is the deformation gradient tensor referring to the undeformed configuration. When the displacement gradient components are small compared to unity, equation (5.56) is reduced to dε ≈D dt

ð5:57Þ

5.5

Conservation Laws

255

With the help of equations (5.42) and (5.53) become

∂ 12 ρ~v2 1 = - div ρ~v2  ~v - σ  ~v - σ : D þ ρb  ~v 2 ∂t

ð5:58Þ

For the conservative body forces that can be derived from a potential Ψ independent of time, we can write (Mazur & De Groot, 1962) b = - gradΨ,

∂Ψ =0 ∂t

ð5:59Þ

We can now establish an equation for the rate of change of the potential energy density ρΨ. It follows from Eqs. (5.38) and (5.59) that ∂ρΨ ∂ρ ∂Ψ =Ψ þρ = Ψð - div ρevÞ ∂t ∂t ∂t = - div ρΨev þ ρev  gradΨ = - div ρΨev - ρb  ev

ð5:60Þ

Adding Eqs. (5.59) and (5.60) for the rate of change of the kinetic energy 12 ρ~v2 and the potential energy ρΨ:
 n

o ∂ρ 12 ~v2 þ Ψ 1 = - div ρ ~v2 þ Ψ ~v - σ  ~v - σ : D 2 ∂t

ð5:61Þ

This equation shows that the sum of kinetic and potential energy is not conserved, since an entropy source term appears on the right-hand side. However, this entropy source term is one of many. There are far more entropy source terms in thermomechanical loading. It is covered in more detail in the following chapter. Moreover, assuming the potential to be time-independent significantly simplified the formulation, albeit unrealistically.

5.5.3

Conservation of Energy

The first law of thermodynamics relates the work done on the system and the heat energy flowing into the system to the change in total energy of the system. Suppose that the only energy transferred to the system is by mechanical work done on the system by surface tractions and body forces, by heat exchange through the boundary surface, and the heat generated within the system by external agencies (e.g., inductive heating). According to the principle of conservation of energy [the first law of thermodynamics], the total energy within an arbitrary volume V in the system can only change if energy flows into (or out of) the volume considered through its boundary A, which can be expressed as

256

5

d dt

Z

V

Z ρ e dV =

V

Unified Mechanics of Thermomechanical Analysis

∂ρe dV = ∂t

Z

A

Z Je  dA þ

V

ρ r dV

ð5:62Þ

where e is the energy per unit mass, Je is the energy flux per unit surface and unit time, and r is the distributed internal heat source of strength per unit mass. We will refer to e as the total specific energy because it includes all forms of energy in the system. Similarly, we will call Je the total energy flux. With the help of Gauss’s theorem, we can obtain the differential [local form] of the law of conservation of energy as follows: ∂ρe = - div Je þ ρr ∂t

ð5:63Þ

To relate this equation to the previously obtained Eq. (5.61) for the kinetic energy and potential energy, we must specify what are the various contributions to the total specific energy e and the total energy flux Je. The total specific energy e includes the specific kinetic energy 12 v2 , the specific potential energy Ψ, and the specific internal energy u (Mazur & De Groot, 1962): 1 e = ~v2 þ Ψ þ u 2

ð5:64Þ

From a macroscopic point of view, this relation can be considered as the definition of internal energy, u. From a microscopic point of view, u represents the energy of thermal agitation [atomic vibrations] as well as the energy due to the short-range atomic interactions, such as the strain energy. Similarly, the total energy flux includes a convective term ρe~v , an energy flux ðσ  ~vÞ due to the mechanical work performed on the system, and finally a heat flux Jq (Mazur & De Groot, 1962): Je = ρe~v - σ  ~v þ Jq

ð5:65Þ

This equation may be also considered as defining the heat flux Jq. Then the heat flow rate per unit mass is given by ρ

dq = - div Jq dt

ð5:66Þ

where q is the heat flowing into the system per unit mass. If we subtract Eq. (5.61) from Eq. (5.63) and use Eqs. (5.64) and (5.65), we get the balance equation for the internal energy u:

 ∂ρu = - div ρu~v þ Jq þ σ : D þ ρr ∂t

ð5:67Þ

5.6

Entropy Law: Second Law of Thermodynamics

257

It is apparent from equation (5.67) that the internal energy u is not conserved. A dissipation source term appears which is equal but of opposite sign to the source term of the balance equation (5.61) for kinetic and potential energy. With the help of Eqs. (5.42), (5.67) may be written in an alternative form as follows: ρ

du = - div Jq þ σ : D þ ρr dt

ð5:68Þ

The total stress tensor σ can be split into a scalar hydrostatic pressure part p and a deviatoric stress tensor S: σ = S - pI

ð5:69Þ

where I is the unit matrix with element δij(δij = 1, if i = j; δij = 0, if i ≠ j) p = - 13 σ kk. With the help of Eqs. (5.69), (5.68) becomes ρ

du = - div Jq - p div ~v þ S : D þ ρr dt

ð5:70Þ

where the following equality is utilized: I : D = I : grad ~v =

X3

δ i:j = 1 ij

X3 ∂ ∂ ~vi = v~ = div ~v i = 1 ∂xi i ∂xj

ð5:71Þ

Utilizing Eq. (5.41), the time rate of change of internal energy can be written in the following form: du 1 1 = σ : D þ r - div Jq dt ρ ρ

5.6

ð5:72Þ

Entropy Law: Second Law of Thermodynamics

Historically, thermodynamics in the traditional sense was concerned with the study of reversible transformations [which are imaginary]. For an irreversible process in which the thermodynamic state of a matter changes from some initial state to a current state, it can be assumed that such a process can occur along an imaginary reversible isothermal path. The processes defined in this way will be thermodynamically admissible if, at any instant of evolution, the Clausius-Duhem inequality is satisfied. According to the principles of thermodynamics, two more new variables, temperature T and entropy S, are introduced for any macroscopic system. The entropy [disorder/energy unavailable for work] of the universe, taken as a system

258

5

Unified Mechanics of Thermomechanical Analysis

plus whatever surroundings are involved in producing the change within the system, can only increase. Everything that happens in the real world is always an irreversible process, which results in the production of positive entropy and thus a permanent/ irreversible change in the universe (DeHoff, 1993). The variation of the entropy dS may be written as the sum of two and only two terms for a closed system (Mazur & De Groot, 1962): dS = dSe þ dSi

ð5:73Þ

where dSe is the entropy generated by the transfer of heat from external sources across the boundary of the system and dSi is the entropy produced inside the system and by the matter. The second law of thermodynamics states that dSi must be zero for any reversible process and positive for the irreversible transformation of the system from one state to any other state (Mazur & De Groot, 1962): dSi ≥ 0

ð5:74Þ

The entropy supplied externally, dSe, on the other hand, may be positive, zero, or negative, depending on the interaction of the system with its surroundings. For an irreversible process in which the thermodynamic state of a matter changes from some initial state to a different state, it is assumed that such a process can occur along an imaginary reversible isothermal path which consists of a two-step sequence (Krajcinovic, 1996). This is the so-called local equilibrium assumption, which postulates that the thermodynamic state of a material at a given point and instant is completely defined by the knowledge of the values of a certain number of state variables at that instant. The method of local state implies that the laws which are valid for the macroscopic system remain valid for infinitesimally small parts of it, which agrees with the point of view currently adopted in the macroscopic description of a continuous system. However, we need to clarify what we mean by “infinitesimally small,” which is relative. We mean that small enough for laws of continuum mechanics to be applicable but not smaller. This method also implies, on a microscopic model, that the local macroscopic measurements performed on the system are measurements of the properties of small parts of the system, which still contain many of the constituting particles. In general, it is assumed that representative volume element (RVE) is the smallest unit of a continuum. This hypothesis of “local equilibrium” can, from a macroscopic point of view, only be justified by the validity of the conclusions derived from it. Ultrarapid phenomena for which the time scale of the evolutions is in the same order as the atomic relaxation time for a return to thermodynamic equilibrium are excluded from this theory’s field of application (Lemaitre & Chaboche, 1990). As we discussed in Chap. 3, all physical processes can be described with precision by utilizing the proper number of thermodynamic state variables. The processes defined in this way will be thermodynamically admissible if, at any instant of evolution, the Clausius-Duhem inequality is satisfied.

5.6

Entropy Law: Second Law of Thermodynamics

259

In irreversible thermodynamics, one of the important objectives is to relate the dSi, the internal entropy generation, to the various irreversible phenomena which occur inside the matter. Before calculating the entropy production in terms of quantities that characterize the irreversible phenomena, we can rewrite Eqs. (5.73) and (5.74) in a form that is more suitable for the mathematical description of the systems in which the densities of the extensive properties (such as mass and energy, which are needed in conservation laws) are continuous functions of spatial coordinates (Mazur & De Groot, 1962): Z

V

S= dSe =dt

Z

dSi = dt

Z

ρsdV

A

JS,tot  dA

V

γ dV

ð5:75Þ ð5:76Þ ð5:77Þ

where s is the specific entropy per unit mass, JS, tot is the total entropy flux which is a vector that coincides with the direction of entropy flow and has a magnitude equal to the entropy crossing unit area perpendicular to the direction of flow per unit time, and γ is the entropy generation rate per unit volume and per unit time. Utilizing Eqs. (5.75)–(5.77), and Gauss’s theorem, Eq. (5.73) may be written in the following form (Mazur & De Groot, 1962): Z

V

 ∂ρs þ div JS,tot - γ dV = 0 ∂t

ð5:78Þ

where the divergence of JS, tot simply represents the net entropy leaving unit volume per unit time. Equation (5.78) must be true for any arbitrary volume V; hence, we can write ∂ρs = - div JS,tot þ γ ∂t

ð5:79Þ

γ≥0

ð5:80Þ

These last two equations are the local forms of Eqs. (5.73) and (5.74), i.e., the local mathematical expressions for the second law of thermodynamics. Equation (5.79) is formally a balance equation for the entropy density s with an entropy generation rate γ which satisfies the important inequality (5.80). With the help of equation (5.42), equation (5.79) can be rewritten in a slightly different form as follows: ρ

ds = - div JS þ γ dt

ð5:81Þ

260

5

Unified Mechanics of Thermomechanical Analysis

where the entropy flux JS is the difference between the total entropy flux JS, tot and a convective term ρs~v: JS = JS,tot - ρs~v

ð5:82Þ

For applications in continuum mechanics, we must relate the changes in the system to the entropy generation rate. This requires us to obtain explicit expressions for the entropy flux JS and the entropy generation rate γ that appears in equation (5.81). This explicit equation is called the thermodynamic fundamental equation. We postulate the existence of a thermodynamic potential from which the state laws can be derived. Let us assume that a function, with a scalar value, concave with respect to temperature and convex with respect to other variables, allows us to satisfy a priori the conditions of thermodynamic stability imposed by Clausius-Duhem inequality. The specific Helmholtz free energy, φ, is defined as the difference between the specific internal energy density u and the product between the absolute temperature T and specific entropy s: φ = u - Ts

ð5:83aÞ

Differentiating this and with the help of the law of conservation of energy, we have the following relations: dφ = du - Tds - sdT

ð5:83bÞ

= ðδq þ δwÞ - Tds - sdT
 = δq þ δwd þ δwe - Tds - sdT
 = δq þ δwd - Tds þ ðδwe - sdT Þ

ð5:83cÞ ð5:83dÞ ð5:83eÞ

where q is the total heat flowing into the system per unit mass, including the conduction through the surface and the distributed internal heat source, w is the total work done on the system per unit mass by external loads and body forces, wd is the dissipated energy associated with the total work, which may be dissipated in the form of heat, and we is the elastic energy associated with the total work. For the quantitative treatment of entropy for irreversible processes, let us introduce the definition of entropy for irreversible processes: ds =

δq þ δwd T

ð5:84Þ

With the help of Eq. (5.84), we can write the following relation: Tds = du - dwe

ð5:85Þ

5.6

Entropy Law: Second Law of Thermodynamics

261

This is the Gibbs relation which combines the first and second laws of thermodynamics. From the definition of entropy, we also have dwe = dφ þ sdT

or

dφ = dwe - sdT

ð5:86Þ

The Helmholtz free energy, φ, is the isothermal recoverable elastic energy available for work. The specific elastic energy we, namely, the work stored in the system per unit mass during a process, is not a function of the process path [path independent]. It depends only on the end state of a process for a given temperature. The elastic energy is frequently also referred to as the available energy of the process. The elastic energy is the maximum amount of work that could be produced by a system between any given two states if the entropy generation is zero, which is not possible in any process. To find the explicit form of the entropy balance equation, Eq. (5.81), we insert Eq. (5.72) for du dt into Eq. (5.85) with the time derivatives given by Eq. (5.39): ρ

div Jq 1 ρ dwe ρr ds þ σ : Dþ =T T T dt T dt

ð5:87aÞ

Noting that Jq div Jq 1 = div þ 2 Jq  gradT T T T

ð5:87bÞ

it is easy to cast equation (5.87) into the form of a balance equation, Eq. (5.81): ρ

Jq 1 ρ dwe ρr 1 ds þ = - div - 2 Jq  gradT þ σ : D T T dt T T dt T

ð5:88Þ

From comparison with equation (5.81), it follows that the expressions for the entropy flux and the entropy production rate are given by Jq T

ð5:89Þ

1 ρr ρ dwe 1 - 2 Jq  gradT þ σ : DT T T dt T

ð5:90Þ

JS = γ=

Equation (5.90) represents the entropy generation by the internal dissipations. The sum of the first two terms is called intrinsic dissipation or mechanical dissipation. It consists of plastic dissipation plus the dissipation associated with the evolution of other internal variables; it is generally dissipated in the form of heat, but not always. The last two terms are thermal dissipation due to the conduction of heat and internal heat generation. The structure of the expression for γ is that of a bilinear form: it consists of a sum of products of two factors. One of these factors in each term is a flux quantity (heat flow Jq, σ stress tensor) already introduced in the conservation laws. The other factor in each term is related to a gradient of an

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intensive state variable (gradients of temperature and velocity). These quantities which multiply the fluxes in the expression for the entropy production are called thermodynamic forces. As we discussed earlier, actually assignment of flux and thermodynamic force is rather arbitrary. However, their multiplication always must yield the entropy generation rate. According to the second law of thermodynamics, the entropy generation rate γ must be zero if the thermodynamic equilibrium conditions are satisfied within the system. Another requirement that Eq. (5.90) must satisfy is that it is invariant under the transformation of different reference frames since the notions of reversible and irreversible behavior must be invariant under a reference frame transformation. Equation (5.90) satisfies these requirements. Finally, it may be noted that equation (5.90) also satisfies the Clausius-Duhem inequality: σ : D-ρ

  dT dφ gradT - Jq  þs ≥0 dt dt T

ð5:91Þ

Between two particles that are at different temperatures, heat is transferred by conduction, a process that takes place at the molecular and atomic levels. The law of heat conduction for isotropic bodies is given by Jq = - k gradT

ð5:92Þ

where k is the thermal conductivity with units of Btu/fthr°F and Jq is the heat flux. This law of heat conduction was stated first by Fourier who based it on experimental observation. Equation (5.92), also known as Fourier’s law, expresses a linear relation between the heat flux vector Jq and its dual variable gradT. Since solid, opaque bodies are of primary interest in this chapter, heat is transferred from point to point within this body solely by conduction. The field equation of the boundary value problem will, therefore, always be some form of the Fourier heat conduction equation. Of course, heat may be transferred to the surface of the body by other modes of heat transfer. Then the expression for the internal entropy generation rate for thermomechanical problems can be simplified as γ=

k 1 ρ dwe ρr þ 2 jgradT j2 þ σ : DT T dt T T

ð5:93aÞ

Total entropy generation is obtained by time integration of Eq. (5.93a): s=

1 ρ

Z

t2

γ dt

ð5:93bÞ

t1

The specific entropy production rate for small deformation and small strain thermomechanical problems in metals may be simplified as dsi γ σ : ε_ p k r = = þ 2 jgradT j2 þ dt ρ Tρ T T ρ

ð5:93cÞ

5.7

Fully Coupled Thermomechanical Equations

263

However, this simplification assumes that entropy generation due to elastic deformation and all other mechanisms are negligible. Of course, this would be not true for high strain rate loading, elastic fatigue, and most composite materials where significant entropy is generated by internal relative elastic deformations of constituents.

5.7

Fully Coupled Thermomechanical Equations

The formalism of continuum mechanics and thermodynamics requires the existence of a certain number of state variables. For thermomechanical problems in metals at low strain rates of loading, there are two observable variables: the temperature T and the total strain ε. For dissipative phenomena, the current state also depends on the history (loading trajectory) that is represented, in the local state, by the values of internal variables at each instant. Plasticity and viscoplasticity require the introduction of the plastic (or viscoplastic) strain εp (or density of dislocations) as a state variable. Other phenomena, such as softening, hardening, degradation, and fracture, require the introduction of other internal variables. These variables represent the internal state of matter (density of dislocations, the crystal structure of lattice, material phase, polycrystalline grain size, the configuration of micro-cracks and cavities, etc.) Lemaitre and Chaboche (1990) state that “There is no objective way to choose the internal state variables best suited to the study of a phenomenon.” However, this is only true if Newtonian mechanics is used in conjunction with an empirical damage potential, curve fit to a test data. In the unified mechanics theory, internal state variables are defined by the entropy-generating mechanism, which all must be considered. Thermodynamic fundamental equation dictates which state variables must be included. For the general case, state variables will be denoted by Vk(k = 1, 2, . . .) representing either a scalar or a tensorial variable. For small strain formulation, total strain can be written as a summation of elastic and plastic components: ε = ε e þ εp

ð5:94Þ

The relations between the energy, stress tensor, and strain tensor can be obtained using the formalism of thermodynamics. Here we choose the specific Helmholtz free energy, φ, which depends on observable variables and internal state variables: φ = ðε, T, εe , εp , V k Þ

ð5:95Þ

For small strain formulation, the strain appears only in the form of their additive decomposition, so that φððε - εp Þ, T, V k Þ = φðεe , T, V k Þ which shows that (Lemaitre & Chaboche, 1990)

ð5:96Þ

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∂φ ∂φ ∂φ = =- p ∂εe ∂ε ∂ε

ð5:97Þ

and the following expressions define the relation between Helmholtz free energy and thermodynamic state variables: σ=ρ

∂φ ∂εe

ð5:98Þ

s= -

∂φ ∂T

ð5:99Þ

Ak = ρ

∂φ ∂V k

ð5:100Þ

where Ak is a thermodynamic force associated with the internal variables Vk. The vector formed by the variables is the gradient of the function φ in the space of the variables T, εe, and Vk. This vector is normal to the surface φ = constant. Let us return to the equation of the conservation of energy for small strains (utilizing Eqs. (5.57) and (5.68)): ρu_ = - div Jq þ σ : ε_ þ ρr

ð5:101Þ

and replace ρu_ by the expression derived from Eqs. (5.83a–5.83e): ρu_ = ρφ_ þ ρ_sT þ ρsT_

ð5:102Þ

And utilizing φ_ and s_ by their expression as a function of the state variables with the help of equations (5.98–5.100): φ_ =

∂φ e ∂φ _ ∂φ _ 1 V k = σ : ε_ e - sT_ þ Ak V_ k : ε_ þ Tþ ρ ∂εe ∂T ∂V k

2

s_ = -

2

ð5:103Þ

2

∂ φ _ ∂ φ ∂ φ_ 1 ∂σ e ∂s _ 1 ∂Ak _ T: ε_ e Vk = V : ε_ þ Tρ ∂T ρ ∂T k ∂εe ∂T ∂V k ∂T ∂T ∂T 2 ð5:104Þ

We obtain - div Jq = ρT

  ∂s _ ∂σ e ∂Ak _ V k ð5:105Þ T - σ : ðε_ - ε_ e Þ þ Ak V k - ρr - T : ε_ þ ∂T ∂T ∂T

By introducing the specific heat defined by

5.7

Fully Coupled Thermomechanical Equations

C=T

265

∂s ∂T

ð5:106Þ

and considering Fourier’s law for isotropic materials div Jq = - k div ðgrad T Þ = - k∇2 T

ð5:107Þ

where ∇2 denotes the Laplacian operator. Using ε_ p = ε_ - ε_ e : we obtain   ∂σ e ∂Ak _ p _ Vk k∇ T = ρCT - σ : ε_ þ Ak V k - ρr - T : ε_ þ ∂T ∂T 2

ð5:108Þ

This is the fully coupled thermomechanical equation, which can simulate the evolution of temperature due to mechanical work with properly imposed boundary conditions. Ak V_ k represents the non-recoverable energy in the matter corresponding to other dissipation mechanisms. If we simplify the problem to thermomechanical loading at small strain and small strain rates on pure metals, we may be able to ignore other dissipation terms, and then we can write Ak V_ k ≈ 0

ð5:109Þ

which results in the fully coupled elastic-plastic thermomechanical equation: k∇2 T = ρC T_ - σ : ε_ p - ρr - T

∂σ e : ε_ ∂T

ð5:110Þ

Equation (5.110) also allows us to calculate heat flux Jq generated due to elastic and/or inelastic work in a solid body. This will be covered in more detail later in the chapter on fatigue. For the isotropic linear thermoelastic materials, the stress-strain relationship in Newtonian mechanics is given by σ ij = λδij εkk þ 2Gεij - ð3λ þ 2GÞδij αðT - T 0 Þ

ð5:111Þ

where T0 is the reference temperature, α is the isotropic thermal expansion coefficient, λ and G are given by λ=

vE ; ð1 þ vÞð1 - 2vÞ

G=

E 2ð 1 þ v Þ

ð5:112Þ

If the internal heat generation is neglected, equation (5.108) defines the response of isotropic linear thermoelastic materials:

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Unified Mechanics of Thermomechanical Analysis

k∇2 T = ρC T_ þ ð3λ þ 2GÞαT ε_ kk

ð5:113Þ

The last term represents the interconvertibility of thermal and mechanical energy. With the help of Eqs. (5.87) and (5.108), the total specific entropy generation rate for small strains is given by   ds c ∂T 1 ∂σ e ∂Ak : Vk = : ε_ þ dt T ∂t ρ ∂T ∂T

ð5:114Þ

where it is assumed σ : ε_ - ρw_ e = σ : ε_ p - Ak V_ k holds, which represents the total mechanical dissipation rate and Ak V_ k is assumed to be zero. Of course this assumption is not true for elastic fatigue loading or for any problem where plastic work dissipation is not the dominant entropy generation mechanism. In the presence of plastic dissipation, elastic dissipation is small for quasi-static loading. With the help of Eqs. (5.93a), (5.93b), and (5.93c), the specific entropy production rate for small strain and quasi-static loading becomes γ σ : ε_ p r k dsi = = þ - 2 jgrad T j2 dt ρ Tρ T T ρ

ð5:115Þ

The equations governing the temperature, stress, deformation, and entropy production rate in a continuum have been derived in the previous chapters. However, these quantities are all interrelated and must be determined simultaneously in incremental form. In most quasi-static problems, the effect of stresses and deformations on the temperature distribution is quite small and may be neglected. This procedure allows the determination of the temperature distribution resulting from prescribed thermal boundary conditions to become the first step of a thermal stress analysis. The second step is then the determination of the stresses, deformations, and thermodynamic stress index field due to this temperature distribution. Of course, performing thermomechanical analysis in one step is possible, but it is far more computationally intensive. Entropy change caused by the heat transfer between the system and its surroundings does not influence the degradation of the material if the temperature field in the entire body is uniform because of this exchange. If this heat exchange leads to a nonuniform temperature field, then heat coming from outside can lead to irreversible entropy generation in the material as in bending, thermomigration, and other mechanisms. Only the internal entropy generation, namely, the entropy created in the system, should be used as a basis for the systematic description of the irreversible processes, which can be given by

5.8

Numerical Validation of the Thermomechanical Constitutive Model

Z Δs = Δsi = t0

t

σ : ε_ p dt Tρ

Z t t0

 Z t k r 2 dt jgrad T j dt þ 2 T T ρ t0

267

ð5:116Þ

Equation (5.116) shows that the entropy generation is a function not only of the loading or straining process but also of the temperature. However, a uniform increase in temperature in a stress-free field does not cause any change in TSI. While this fundamental equation accounts for strain rate, it does not account for all mechanisms under thermomechanical loading. If the strain rate is very high, there are far more entropy generation mechanisms such as phase change, elastic dissipation, melting, and other mechanisms that can be significant entropy generation sources that this equation does not account for. Thermodynamic state index earlier was given by the following equation: h i ms ðΔsÞ Φ = 1 - e- R

5.8

ð5:117Þ

Numerical Validation of the Thermomechanical Constitutive Model

Two problems are selected to validate the model described in this chapter. First, the model results are compared with testing performed on specimens of thin layer solder joints of Pb37/Sn63 under monotonic and fatigue shear testing at different strain rates and temperatures.

5.8.1

Thin Layer Solder Joint: Monotonic and Fatigue Shear Simulations

Table 5.3 lists the strain rate and temperature ranges for testing and simulations. Monotonic and cycling shear tests were performed on a thin layer of the solder joint (Pb37/Sn63) in pure shearing mode. Figure 5.3 shows the specimen attached to copper plates. Material parameters used for the numerical simulation are shown in Table 5.4. Figure 5.4 shows a comparison between numerical simulation results and test data under monotonic shear testing for different temperatures at a strain rate of 1.67 × 10-3/s. Figure 5.5 shows monotonic shear testing under strain rate of 1.67 × 10-3/s, at 22 °C and inelastic strain range (ISR) = 0.005. Figures 5.6, 5.7, 5.8, and 5.9 show a comparison between the cyclic shear simulation results and test data at 22 °C and strain rate of 1.67 × 10-3/s. Figure 5.10 shows simulation results for isothermal fatigue testing at strain rate 1.67 × 10-4/s at 22 °C with an inelastic strain range of 0.022.

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Table 5.3 Loading scheme used in the analysis Case I 1

Monotonic shear loading Strain rate

2 3 4 5 Case II 1

Strain rate Strain rate Strain rate Strain rate Cyclic shear loading Temp = 22 °C Strain rate

1.67e-3/s a b c d 1.67e-1/s 1.67e-2/s 1.67e-3/s 1.67e-4/s

Temperature (°C) -40 22 60 100 22 22 22 22

1.67e-3/s a

ISR 0.005

Fig. 5.3 Thin layer solder joint attached to copper plates

Table 5.4 Material parameters

Elastic (θ temper. in °K) Young’s modulus (GPa) Shear modulus (GPa) Isotropic hardening R00 (MPa) c (Dimensionless) σ y (MPa) Kinematic hardening c1 (MPa) c2 (Dimensionless) Creep strain rate A (Dimensionless) D0 (mm/s2) b (mm) d (mm) n p Q (mJ/mol) Φcritical

52.10-0.1059θ 19.44-0.0395θ 37.47-0.0748θ 383.3 60.069-0.140θ 2040 180 7.60E+09 48.8 3.18E-07 1.06E-02 1.67 3.34 4.47E+07 1

5.8

Numerical Validation of the Thermomechanical Constitutive Model

Fig. 5.4 Monotonic shear testing under strain rate 1.67 × 10-3/s at different temperature Fig. 5.5 Monotonic shear testing under strain rate 1.67 × 10-3/s at 22 °C

269

270 Fig. 5.6 Cyclic shear simulation vs. test data at 22 °C, strain rate 1.67 × 10-3/s, and ISR = 0.005

Fig. 5.7 Cyclic shear simulation vs. test data at 22 °C, strain rate 1.67 × 10-3/s, and ISR = 0.012

5

Unified Mechanics of Thermomechanical Analysis

5.8

Numerical Validation of the Thermomechanical Constitutive Model

Fig. 5.8 Cyclic shear simulation vs. test data at 22 °C, strain rate 1.67 × 10-3/s, and ISR = 0.02

Fig. 5.9 Cyclic shear simulation vs. test data at 22 °C, strain rate 1.67 × 10-3/s, and different inelastic strain range

271

272

5

Unified Mechanics of Thermomechanical Analysis

Fig. 5.10 Isothermal fatigue at strain rate 1.67 × 10-4/s at 22 °C with inelastic strain range 0.022

Figure 5.11 shows the evolution of the thermodynamic state index under cycling fatigue loading with an inelastic strain range of 0.022. The computational simulations are in good agreement with the experimental results. The differences are due to the imperfect microstructure (such as voids) of the solder joint, while the numerical model assumes a perfect continuum with no defects. Moreover, entropy generation due to internal heat generation was ignored for the sake of simplicity in this example. More examples of unified mechanics theory for thermomechanical problems are provided in detail in the references listed at the end of the chapter.

5.9

Thermomechanical Analysis of Cosserat Continuum: Length Scale Effects

Classical continuum mechanics formulation does not include a term for the size effect. We should clarify here what is meant by “size.” It is not the size of the system, such as 150 stories high building versus a very small bunker. “Size” refers to the dimensions of the microstructure of the material. When the size of the microstructure becomes a dominant factor in the response of the material, in addition to strains, gradients of strains are also included in the continuum mechanics formulation. The size effect is known to be a big factor, especially in the plastic zone of the material response. The incremental theory of plasticity is called strain gradient plasticity theory when strain gradients are also included in the yield surface formulation in addition to strains. Strain gradient plasticity is especially needed when traditional

5.9

Thermomechanical Analysis of Cosserat Continuum: Length Scale Effects

273

Fig. 5.11 Evolution of the damage parameter per Eq. (4.37) for Φcr = 1.0 under fatigue loading with ISR = 0.022

continuum mechanics formulation is unable to represent the material stress-strain behavior due to microstructure size and deformation gradient effects, because traditionally, continuum mechanics formulations are independent of size effect. However, in some mechanics problems, this is not true, and material response is size-dependent, especially low melting point metals such as solder alloys. Class of plasticity theories where material size effect is considered is usually referred to as strain gradient plasticity theories. In strain gradient plasticity theories, it is assumed that classic plastic behavior is due to the slip of statistically stored dislocations, and length [size] scale effects are due to the slip of geometrically necessary dislocations. Geometrically necessary dislocations are due to having a very large strain gradient in the material, like in a thin film subjected to bending. As a result, the dislocation term in Taylor’s flow stress equation is modified to add the geometrically necessary dislocation density: τ=α G b

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρSSD þ ρGND

ð5:118Þ

where α is a geometrical factor that depends on the type and arrangement of the interacting dislocations, G is the shear modulus, b is Burger’s vector, ρSSD is the density of statistically stored dislocations, and ρGND is the density of geometrically necessary dislocations. There are different methods to account for the size effect in continuum mechanics. One of these methods is the use of the Cosserat continuum. In this section, we will discuss (Cosserat continuum) couple stress-based strain gradient theory without

274

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Unified Mechanics of Thermomechanical Analysis

restrictions to any constitutive model. Next, we describe a constitutive model incorporating coupling between degradation and strain rate dependency and considering size effects utilizing a couple of stress theory material length scales. We provide two distinct theories within the same framework, namely, a general couple stress theory where translational and rotational degrees of freedom are treated as independent kinematic variables and a more restrictive reduced couple stress theory, where translational and rotational degrees of freedom are constrained to satisfy a given kinematic relation. The kinematic constraint allows the implementation of the reduced theory into a displacement-based finite element method formulation. The use of this approach relaxes the strong C1 shape function (approximation function) continuity requirements in finite element method and allows the use of C0 approximation functions, thus keeping the convergence features of the element. Here the kinematic constraint is enforced via a penalty function/reduced integration scheme that allows treatment of the reduced theory based on ideas motivated by the general theory. The current description progressively arrives at this formulation after exploring different possibilities within the context of the variational calculus. Throughout all the discussion, the objective should be clear that the final goal is to implement the reduced couple stress theory framework in the form of a user element subroutine within the finite element code ABAQUS which imposes additional restrictions.

5.9.1

Cosserat Couple Stress Theory

The motivation behind Cosserat’s couple stress theory also referred to as the Cosserat continuum, Fig. 5.12, stems from the following arguments. In classical continuum mechanics, it is assumed that a material point occupies no space (has zero volume) and then the size of the material microstructure is not part of the formulation. To include the effects of the finite size of a material’s microstructure while retaining a mathematical continuum, classical continuum mechanics have been extended or generalized. In the Cosserat continuum, each material point is enriched with higher-order kinematic degrees of freedom. In particular, the specification of the position requires also the definition of a rotation. In the more general Cosserat theory, this new kinematic quantity, the micro rotation, is independent of the classical theory rotation vector θk = 12 eijk uj,k where eijk is the alternating pseudotensor and a coma represents a derivative with respect to rectangular Cartesian coordinates. This type of theory has been termed as “indeterminate couple stress theory” by Eringen (1968) and “generalized couple stress theory” by Fleck and Hutchinson (1997). The following discussion elaborates further upon this theory. There are different interpretations of the Cosserat continuum. In Cosserat and Cosserat (1909) couple stress theory, a differential material element is subjected to not only normal and shear stresses but also couple stress components as shown in Fig. 5.12. For linear elastic behavior, the traditional stress components are functions of the strains and the couple stresses are functions of the strain gradients. Two distinct theories are identified in the work of Cosserat and Cosserat (1909). First,

5.9

Thermomechanical Analysis of Cosserat Continuum: Length Scale Effects

275

Fig. 5.12 Cosserat continuum stress point

there is a reduced couple stress theory with the kinematic degrees of freedom being the displacement ui and an associated material rotation θi tied to the displacements by the kinematic constraint in Eq. (5.119) and with the definitions of strain and curvatures expressed in Eqs. (5.120) and (5.121): 1 e u 2 ijk k,j  1
εij = ui,j þ uj,i 2

ð5:119Þ

xij = θi,j

ð5:121Þ

θi =

ð5:120Þ

Second, there is a general couple stress theory in terms of a micro rotation, ωi, which is regarded as an independent kinematic degree of freedom. The traditional continuum mechanics rotation and Cosserat continuum micro rotation are related by a relative rotation tensor αij = eijkωk - eijkθk. For the choice of ωk = θk, the general couple stress theory reduces to the reduced couple stress theory. Equilibrium of the differential element shown in Fig. 5.12, after neglecting body forces and body couples [in Newtonian mechanics], yields σ ji,j þ τji,j = 0

ð5:122Þ

1 τjk þ ejik mpi,p = 0 2

ð5:123Þ

where σ ij and τ ij are the symmetric and antisymmetric components of the Cauchy stress tensor, respectively, and mij is the couple’s stress tensor. Surface stress tractions and surface couple stress tractions are given by

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Unified Mechanics of Thermomechanical Analysis


 ðnÞ σ i = σ ij þ τij nj

ð5:124Þ

qi = mij nj

ð5:125Þ

ð nÞ

where σ i is the traction vector, qi is the couple traction vector, and ni is a surface outward normal vector. For a linear elastic solid, the strain energy density is given by Eq. (5.126) where l is a length scale associated with elastic behavior and may be regarded to be of the order of inter-atomic distances (Koiter, 1964). W=

E v εii εjj þ εij εij þ l2 xij xij 2ð1 þ vÞ 1 - 2v

ð5:126Þ

A constitutive relationship can be directly derived out of Eq. (5.126) as follows: σ ij =

∂W  C ijkl εkl ∂εij

ð5:127Þ

mij =

∂W  Dijkl xkl ∂xij

ð5:128Þ

where Cijkl and Dijkl are elastic constitutive tensors relating stress and couple stress components to elastic strains and elastic curvatures, respectively. The principle of virtual work for the reduced Cosserat theory can be given by [in Newtonian mechanics] ZZZ

ZZZ σ ij δεij dV þ V

ZZ mij δxij dV =

V

ðnÞ

ZZ

σ i δui dA þ A

qi δθi dA

ð5:129Þ

A

The concept of “reduced Cosserat” will be discussed later in this chapter.

5.9.2

Toupin-Mindlin Higher-Order Stress Theory

The Cosserat couple stress theory introduces curvatures as gradients of rotation. Toupin (1962) and Mindlin and Tiersten (1962) and Mindlin (1964, 1965) extended the original Cosserat formulation by considering also gradients of the normal strain components, therefore including all the components of the second gradient of displacement. The additional gradients of strain as variables give rise to new work conjugates in the form of force couples per unit area or higher-order stresses. The physical interpretation of these force couples and higher-order stresses is explained in Fig. 5.12. In this type of solid, Newtonian mechanics equilibrium (neglecting body forces) is given by σ ik,i - τijk,ij = 0

ð5:130Þ

5.9

Thermomechanical Analysis of Cosserat Continuum: Length Scale Effects

277

Fig. 5.13 Schematic of a generally solid body

where σ ij is the symmetric Cauchy stress tensor and τijk is the higher-order stress tensor for a generally solid body (Fig. 5.13). Surface tractions and surface couple tractions (moments) are given in Eqs. (5.124) and (5.125), respectively. The higherorder stresses correspond to force couples per unit area:


 ðnÞ σ i = σ ik - τijk:j ni þ ni nj τijk Dp np - Dj ni τijk r k = ni nj τijk

ð5:131aÞ ð5:131bÞ

where the operators Dj and D are defined as Dj = (δjk - njnk)∂k and D = nk∂k. Generalized strain-displacement and strain gradient-displacement relations can be given by εij =

 1
ui:j þ uj,i 2

ηijk = uk,ij

ð5:132aÞ ð5:132bÞ

A strain energy density function can now be given by W=

1 λε ε þ μεij εij þ a1 ηijj ηikk þ a2 ηiik ηkjj þ a3 ηiik ηjjk þ a4 ηijk ηijk þ a5 ηijk ηkji 2 ii jj ð5:133Þ

where λ and μ are Lamè constants and ai are material constants with units of force. A constitutive relationship can be directly derived out of Eq. (5.133) such that ∂W  C ijkl εkl ∂εij

ð5:134aÞ

∂W  Dijklmn ηlmn ∂ηijk

ð5:134bÞ

σ ij = τijk =

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Unified Mechanics of Thermomechanical Analysis

where Cijkl and Dijklmn are elastic constitutive tensors relating stress and higher-order stress components to elastic strains and elastic strain gradients, respectively. The principle of virtual work can be written by equating an external work increment and an internal increment of strain energy and can be given by [in Newtonian mechanics] ZZZ

ZZZ σ ij δεij dV þ V

5.9.3

ZZ τijk δηijk dV =

V

ðnÞ

ZZ

σ i δui dA þ A

r i ðDδui ÞdA

ð5:135Þ

A

Equilibrium Equations

In this section, we derive the equilibrium equations for two different classes of the Cosserat continuum. Traditionally, the notation used to define a solid body occupying a volume Ω and bounded by a surface ∂Ω defined by an outward normal vector n as schematically is shown in Fig. 5.13. In our formulation, we will use dV for volume differentials and dA for surface area differentials. There are several different interpretations of the Cosserat continuum. In the Cosserat and Cosserat (1909) couple stress theory, a differential material element has not only normal and shear stresses but also couple stress (moment) components as shown in Fig. 5.12. For linear elastic behavior, the usual stress components are functions of the strains and the couple stresses are functions of the strain gradients. Two distinct theories are identified in the Cosserat and Cosserat (1909) as described by Aero and Kuvshinsky (1961), Mindlin (1964), de Borst and Muhlhaus (1992), de Borst (1993), and Shu and Fleck (1999). First, there is a reduced Cosserat couple stress theory with the kinematic degrees of freedom being the displacement ui and an associated material rotation θi tied to the displacements by a kinematic constraint given in Eq. (5.119) and with the definitions of strain and curvatures expressed in Eqs. (5.120) and (5.121), respectively. According to the reduced Cosserat continuum theory, the continuum is assumed to possess bending stiffness allowing for the introduction of additional stress measures in the form of moments per unit area. The presence of the couple stresses (moments) renders the Cauchy stress tensor asymmetric; however, only the symmetric component generates work upon deformation. Second, there is a general Cosserat continuum couple stress theory in terms of a micro rotation ωi which is regarded as an independent kinematic variable. The rotation and micro rotation are related by a relative rotation tensor αij = eijkωk - eijkθk. For the case of ωk = θk, the general Cosserat continuum couple stress theory reduces to the reduced Cosserat continuum couple stress theory. This general theory assumes that at a material point, there is also an embedded micro-volume giving rise to the micro rotation ωi. Different interpretations have been postulated depending on the deformation properties assumed for the embedded micro-volume (see Mindlin and Tiersten (1962), Toupin (1962), and Mindlin (1964, 1965) for a review of the different approaches).

5.9

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Fig. 5.14 Relative deformation in the couple stress theory by Mindlin (1964)

For instance, Fig. 5.14 shows the deformation state in a material point including the micro-volume for the case of pure shear in the solid proposed by Mindlin (1964). Figure 5.15 shows the case of reduced Cosserat continuum couple stress theory where the micro-volume is assumed rigid. Newtonian mechanics equilibrium of the differential element shown in Fig. 5.12 (which is valid for both theories of the Cosserat continuum), after neglecting body forces and body couples, yields σ ji,j þ τji,j = 0

ð5:136Þ

1 τjk þ eijk mpi,p = 0 2

ð5:137Þ

where σ ij and τ ij are the symmetric and antisymmetric components of the Cauchy stress tensor, respectively, and mij is the couple’s stress tensor. Surface stress tractions and surface couple stress (moment) tractions are given by
 σ ðnÞ i = σ ij þ τij nj

ð5:138Þ

qi = mij nj

ð5:139Þ

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Unified Mechanics of Thermomechanical Analysis

Fig. 5.15 Reduced Cosserat continuum stress point

where σ (n)i is the surface traction vector, qi is the surface couple tractions vector, and ni is a surface normal vector. To establish a general framework for elastic material behavior, Cosserat couple stress continuum can be obtained after postulating a strain energy density function dependent on strains and curvatures (rotation gradients).

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Toupin (1962) and Mindlin (1965) extended this theory to include also stretch gradients. In particular, they considered invariants of the strain and strain gradients in the strain energy density function in terms of the following generalized von Mises strain invariant which is given by Fleck and Hutchinson (1997): E2 =

2 0 0 ε ε þ c1 η0iik η0jjk þ c2 η0ijk η0ijk þ c3 η0ijk η0kij 3 ij ij

ð5:140Þ

where prime superscript denotes deviatoric component, ηijk = uk, ij are the strain gradients and the c1,c2,c3 are additional material constant with dimensions of length squared (L2). Equation (5.140) is the basis for the most general strain gradient plasticity theory, and it essentially reveals the phenomenological coupling between the densities of statistically and geometrically stored dislocations. The reduced couple stress theory is just a special case of Eq. (5.140) where only rotation gradients are considered.

5.9.4

Finite Element Method Implementation

The following elastic constitutive relationships can be written for the symmetric Cauchy stress tensor, the asymmetric stress tensor, and the couple stress tensor:

l

σ ij = C ijkl εkl

ð5:141Þ

τij = Dijkl αkl

ð5:142Þ

-1

mij = Dijkl l xkl

ð5:143Þ

where Cijkl is a tangential constitutive tensor relating strains to Cauchy stresses, Dijkl is a constitutive tensor relating curvatures to couple stresses, and Dijkl is a constitutive tensor relating relative rotations to the antisymmetric component of the Cauchy stress tensor, and l is characteristic length. The total potential energy functional ∏ for the general elastic Cosserat continuum can be written by considering separately the contributions from the symmetric and antisymmetric stress tensors and the couple stress tensor as follows: Q

ðui , ωi Þ =

Z 1 C ijkl εkl ðui Þεij ðui ÞdV þ Dijkl xij ðωi Þxkl ðωi ÞdV 2 VZ V Z Z ð5:144aÞ 1 ðnÞ þ Dijkl αij ðui , ωi Þαkl ðui , ωi ÞdV - σ i ui dA - qi ωi dA 2 A A 1 2

Z

V

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where translational and rotational degrees of freedom are independent degrees of freedom. For the discussion that follows, it is convenient to write in the equivalent form of Y ðu, ωÞ = F ðuÞ þ GðωÞ þ H ðu, ωÞ where 1 aðu, uÞ - f ðuÞ 2 1 GðωÞ = bðω, ωÞ - gðωÞ 2 Z 1 Dijkl αij ðui , ωi Þαkl ðui , ωi Þ dV H ðu, ωÞ = 2 F ð uÞ =

ð5:144bÞ

V

R R with aðu, uÞ  C ijkl εij εkl dV and bðω, ωÞ  Dijkl xij xkl dV being symmetric bilinear Ω

V

forms and f(u), g(ω) corresponding to the boundary terms in Eq. (5.144a). For the particular case of the reduced couple stress theory where the stress tensor is symmetric and satisfies the constraint in Eq. (5.119), αij (or equivalently H(u, ω)) vanishes and Eq. (5.144b) reduces to Eq. (5.145a): 1 Π ð ui Þ = 2 Z A

Z

1 C ijkl εkl ðui Þ εij ðui Þ dV þ 2

V ðnÞ σ i ui dA -

Z Dijkl xij ðui Þ xkl ðui Þ dV V

Z qi ωi dA

ð5:145aÞ

A

where now the strain energy contribution from the curvatures becomes a function of the translational degrees of freedom only. Note that Eq. (5.145a) is analogous to the total potential energy functional for the particular case of the so-called Timoshenko beam theory. Using the alternative notation, (5.145a) can be written as ΠðuÞ = F ðuÞ þ G0 ðuÞ

ð5:145bÞ

R where G′(u) = b′(u, u) - g′(u) with b0 ðu, uÞ  Dijkl xij ðui Þxkl ðui ÞdV and g′(u) V

corresponding to the boundary condition associated with the rotation. The prime superscript notation ()′ has been introduced to clarify the fact that in the reduced theory, the curvatures are kinematically constrained to the displacements. Equations (5.144a) and (5.145a) are the basis for the finite element implementation. The reduced theory continuum can also be formulated in terms of independent rotational degrees of freedom with the constraint to the translational degrees of freedom considered as the limit when the term H(u, ω) → 0 in the general Cosserat

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continuum. This implies that Eq. (5.145a) is written in terms of additional independent rotational degrees of freedom but with the additional requirement imposed by the constraint between translational and rotational degrees of freedom. In this case, b ðu, ωÞ that follows: the reduced theory can also be described by the functional Π b ð ui , ω i Þ = 1 Π 2

Z C ijkl εkl ðui Þεij ðui ÞdV þ V

Z

σ i ui dA A

Z Dijkl xij ðωi Þxkl ðωi ÞdV V

Z

ðnÞ

-

1 2

qi ωi dA

ð5:145cÞ

A

b ðu, ωÞ A comparison of Eqs. (5.144b) and (5.145c) reveals that Πðu, ωÞ → Π when H(u, ω) → 0. This means that the general couple stress theory approaches the reduced couple stress theory in the limit of vanishing relative rotation αij.

5.9.5

General Couple Stress Theory: Variational Formulation

Consider the case of the general couple stress continuum where there is no constraint set but instead, there is a space of admissible functions with translational and rotational degrees of freedom. The rotational degrees of freedom are linearly inde→ pendent and belong to the space Q alternatively, and we can define displacements → → → → and rotations as elements of the single space V × Q . The product V × Q results in a → → → → third space with elements being all the ordered pairs of the form u , ω 2 V × Q because translational and rotational degrees of freedom are linearly independent. In terms of the introduced this is equivalent variational

notation, to the

following → → → → → → → → problem; find u , ω 2 V × Q such that for any v , φ 2 V × Q , Π assumes

→ → → → its minimum value at u , ω where V × Q is now the corresponding space of admissible functions. The generalized form of the principle of virtual displacements for the general couple stress theory can be given by Z

Z σ ij εij ðvi ÞdV þ V

Z mij xij ðφi ÞdV þ

V

Z -

qi φi dA = = 0

Z τij αij ðvi , φi ÞdV -

V

ðnÞ

σ i vi dA A

ð5:146Þ

A

where vi and φ i denote

virtual displacement and rotation, respectively. Displacement → → and rotation u , ω are regarded as linearly independent kinematic degrees of freedom. This is consistent with the general couple stress theory and the resulting variational problem is thus unconstrained.

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5.9.6

Unified Mechanics of Thermomechanical Analysis

Reduced Couple Stress Theory: Variational Formulation

The principle of virtual displacements for a reduced couple stress theory can be given by Z

Z σ ij εij ðvi ÞdV þ

V

Z mij χ ij ðvi ÞdV -

ðnÞ

Z

σ i vi dA -

V

A

qi φi ðvi ÞdA = 0

ð5:147Þ

A

In contrast to the case defined in Eq. (5.146), the present principle of virtual → displacements shows that the only kinematic variable is displacement u which defines the strains εij and the curvatures χ ij. This is the definition of the reduced couple stress theory. There are no linearly independent rotational degrees of freedom and there is no constraint to be enforced. However, the displacement approximation (shape) function needs to be C1 continuous in the finite element implementation. However, when there is no kinematic constraint between displacements and rotations, mesh-dependent results are obtained in finite element analysis. This is a major obstacle.

5.9.7

Reduced Couple Stress Theory: Mixed Variational Principle

The principle of virtual displacements with the kinematic constraints imposed in a weak sense leads to Z

Z σ ij εij ðvi ÞdV þ

V

mij χ ji ðφi ÞdV þ V

Z þ

Z

Z τij αij ðvi , φi ÞdV =

V

qi φi dA

ðnÞ

σ i i vi dA A

ð5:148aÞ

A

The kinematic constraint is given by Z ρij αij ðui , ωi ÞdV = 0

ð5:148bÞ

V

The first two terms on the right-hand side of Eq. (5.148a) correspond to virtual work done by the stresses and couple stresses, respectively. The third term corresponds to the virtual work done by the asymmetric component of the stress tensor. Equation (5.148b) is the weak enforcement of the constraint of vanishing relative

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285

rotation. This formulation was proposed by Xia and Hutchinson (1996) to solve the mesh dependency problem that happens in the displacement-based finite element method when Eq. (5.147) is used. Finite Element Method Implementation Details This section discusses the discretization of the equations derived above starting from their corresponding variational equations. In each case, we will schematically show a typical finite element with its associated vector of nodal point degrees of freedom. The nodal degrees of freedom may be translational only, translational and rotational degrees of freedom, or translational and rotational degrees of freedom with additional Lagrange multipliers. To distinguish between the values of the degrees of freedom at any point within the element and its nodal point value, we will use the following notation. For instance, if displacement is being considered, the value at any point within the element will be denoted by the vector u, and its corresponding nodal point value will be denoted by b u. The value of any degree of freedom at any point within the element is obtained via interpolation functions from the known nodal point values. Strains and curvatures are usually calculated at Gauss integration points from the nodal point displacements and rotations values using interpolation functions. We will denote such a function by a subscripted symbol where the subscript indicates the variable that is being interpolated. For instance, we will denote the displacement-curvature transformation matrix by Bχ where the curvature at any point within a given element is obtained out of the nodal point displacement vector as χ = Bχ b u where b u may have translational or translational and rotational degrees of freedom.

5.9.8

General Couple Stress Theory Implementation

In the case of the general couple stress theory, the continuum is free of constraints as → → a result, and u , ω are linearly independent degrees of freedom. In the finite element formulation, C0 continuity of shape functions is enough to satisfy the displacement compatibility requirement. The nodal point displacement degrees of freedom for an n-nodded two-dimensional quadric-lateral element have the following general form: b uTe = ½u1 v1 ω1 . . . un vn ωn  as shown in Fig. 5.16. To describe the finite element discretization, we recall that for a general couple stress theory, principle of virtual work is given by Z

Z σ ij εij ðvi ÞdV þ V

Z -

mij χ ij ðφi ÞdV þ V

qi φi dA = 0 A

Z

Z τij αij ðvi , φi ÞdV -

V

ðnÞ

σ i vi dA A

ð5:149Þ

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Unified Mechanics of Thermomechanical Analysis

Fig. 5.16 2-D finite element for the general couple stress theory

u

ðnÞ bT b u = Nu i,, ε = Bεui,, χ = Bχ ui,, α = Bαui, and σ i = N t where t =  uLetting  t 1 t v1 t ω1 . . . t un t vn t ωn , where subscript n represents node number. Using the constitutive relationships given by Eqs. (5.141)–(5.143), we can write the following discretized form after eliminating the virtual variables:

2 4

Z

Z BTε CBε dV

þ

V

Z BTχ DBχ dV

þ

V

3 BTα DBα dV 5ui

V

Z =

N Tbt dA

ð5:150Þ

A

or using the stiffness matrices, the equilibrium equation can be given by  ε  K e þ K χe þ K αe ui = F

ð5:151Þ

Reduced Couple Stress Theory Implementation: Displacement Formulation → ω are related through the In the case of the reduced couple stress theory, u , ! constraint relationship given in Eqs. (5.119)–(5.121) and are not linearly independent of each other. In terms of a formulation based on translational degrees of freedom only, this constraint implies that approximation functions with C1 continuity are needed to satisfy displacement compatibility requirements. The nodal point displacement degrees of freedom for an n-nodded quadric-lateral element have the following general form: b uTe = ½u1 v1 . . . un vn , where subscript n is the node number as shown in Fig. 5.17. To describe the finite element discretization for reduced couple stress theory, we start with the variational formulation: Z

Z σ ij εij ðvi ÞdV þ V

Z mij χ ij ðvi ÞdV -

V

ðnÞ

Z

σi vi dA A

qi φi ðvi Þ dA = 0 A

ð5:152Þ

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287

Fig. 5.17 Typical finite element for the case of reduced couple stress theory using translational degrees of freedom only

u Letting u = Nui, εe = Bεui, χ e = Bχ ui, and q = NTq bt and introducing the constitutive relationships introduced in Eqs. (5.141)–(5.143), we can write 2 4

Z

Z BTε CBε dV þ

V

3 BTχ DBχ dV 5ui =

V

Z

NT þ N Tq bt dA

ð5:153Þ

A

or equivalently using stiffness matrices, we can write the following equilibrium equation:  ε  K e þ K χe u = F

ð5:154Þ

Reduced Couple Stress Theory: Lagrange Multiplier Formulation → In the case of the reduced couple stress theory, translational u and rotational degrees → of freedom ω are related through a Lagrange multiplier constraint relationship given in Eqs. (5.119)–(5.121) and are not independent of each other. The elements have nodal degrees of freedom shown in Fig. 5.18 [for a 2-D quadric-lateral element]. The nodal displacement degrees of freedom vector uTi = ½u1 v1 ω1 . . . un vn ωn  and an additional nodal point vector of Lagrange mult \ipliers b τ Te = ½τ1 . . . τn  are shown in Fig. 5.18. To describe the finite element discretization, we recall that reduced couple stress theory with the kinematic constraints defined via Lagrange multipliers is given by Z

Z σ ij εij ðvi ÞdV þ V

V

Z þ

qi φi dA A

Z mij χ ji ðφi ÞdV þ

Z τij αij ðvi , φi ÞdV =

V

ðnÞ

σ i vi dA A

ð5:155aÞ

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5

Unified Mechanics of Thermomechanical Analysis

Fig. 5.18 Typical finite element for the case of reduced couple stress theory using Lagrange multipliers

u

Z ρij αij ðui , ωi ÞdA = 0

ð5:155bÞ

A

Using u = Nuui, τ e = Nτbτe ε = Bεui, χ = Bχ ui, and α = Bαui t = Nbt where bt = t u t v t ω . . . t u t v t ω  and after introducing the constitutive relationships, we can 1 1 1 n n n write in the following matrix form: T

Z

2Z 6 6 6 6 4

BTε CBε dVþ V

Z

Z BTχ DBχ dV

V

3 BTα Nτ dV

V

N Tτ Bα dV

0

2Z 7h i 7 bu 4 7 7 bτ = A 5

3 T

N tdA

5

ð5:155cÞ

0

V

or using stiffness matrices h

5.10

K uu K Tuτ

K uτ 0

ih i h i bu = F bτ 0

ð5:155dÞ

Cosserat Continuum Implementation in Unified Mechanics Theory

Up to this point, all our derivations have been based on Newtonian mechanics. In the Cosserat continuum, the side effects are introduced by enhancing the definition of equivalent plastic strain with the addition of an equivalent plastic curvature, which is a gradient of strain. This approach leads to a straightforward extension of the classic flow theory (also known as the theory of incremental plasticity) which allows the treatment of cyclic loading and monotonic loading. First, we will start with the formulation that presents the flow theory equations for the undamaged rateindependent case. A key feature is the coupling between the Cauchy stress and a

5.10

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289

couple of stress components. This coupling is not present in the initial elastic material but progressively appears with the accumulation of plastic curvatures. In the case of degradation, additional terms appear in the expression for the Helmholtz free energy determination. On the other hand, consideration of rate-dependent effects assumes the validity of the same rules for the plastic curvatures as for the plastic strains. This assumption allows for using a single creep law for both components (strain and curvature) and is justified by the kinematic constraint present in the reduced couple stress continuum.

5.10.1

Rate-Independent Plasticity Without Degradation

The relationship between the symmetric part of the Cauchy stress tensor and the elastic strains and the couple stress tensor and the elastic curvatures can be written in rate form as follows: σ_ ij = C ijkl ε_ elkl

ð5:156Þ

ℓ - 1 m_ ij = Dijkl ℓ χ_ elkl

ð5:157Þ

where Dijkl = Gδikδjl, where G is the shear modulus. The strains and curvatures are decoupled into elastic and inelastic components which results in ε_ ij = ε_ elij þ ε_ pl ij

ð5:158Þ

ℓ χ_ ij = ℓ χ_ elij þ ℓ χ_ pl ij

ð5:159Þ

Considering the following definition of generalized deviatoric stress norm kΣk:  1=2 0 Σ = Sij Sij þ ℓ - 1 mij ℓ - 1 mij

ð5:160Þ

where Sij is the deviatoric stress tensor and mij is the couple stress tensor, which is deviatoric (it does not have a spherical component). Similarly, the generalized strain tensor norm can be given by  1=2 Ε = εij εij þ ℓ χ ij ℓ χ ij

ð5:161Þ

Next, we need to introduce a yield surface separating the elastic and inelastic domains per classical plasticity theory. To account for the Bauschinger effect observed in metals, a back-stress tensor βij and a couple of back-stress tensors ℓ -1ηij defining the movement of the yield surface in stress space need to be defined. The difference fij = Sij - βij between the back-stress and the deviatoric component of

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the symmetric part of the Cauchy stress tensor is the relative stress tensor fij. In an analogous form for the couple back-stress, there follows that the relative couple b ij which is defined by back-stress ℓ - 1 C b ij = ℓ - 1 mij - ℓ - 1 ηij : ℓ - 1C The generalized relative stress kξk can be described in terms of the relative stresses and given by → h i1=2 b ij C b ij ξ = f ij f ij þ ℓ - 2 C

ð5:162Þ

We can now define a yield surface to distinguish between elastic domain and inelastic domain. In the classical theory of incremental plasticity, the yield surface is defined in terms of a hardening parameter that can be shown to be proportional to the equivalent plastic strain or plastic work. In the couple stress-based strain gradient plasticity theory introduced here, the hardening parameter also incorporates the size effects via the equivalent plastic curvatures. The generalized yield criteria can therefore be given by


F σ, ℓ

-1

rffiffiffi 2 m, α = ξ K ðαÞ 3 

ð5:163Þ

where α is the generalized isotropic hardening parameter and K(α) represents the radius of the yield surface. To complete the constitutive model, it is necessary to define the rate-independent plastic flow rules (i.e., evolution equations for the plastic strains and curvatures) and hardening laws (i.e., the evolution of the hardening parameter and back-stress components). Here it is assumed that the flow rule obeys associative plasticity. Considering a more general stress space with normal stresses and couple stresses, the yield surface can be considered as a hypersphere in a b to the yield surface, F, defined as stress space with normal N b = ∂F  ½b N n, bv → ∂Σ where b n=

∂F ∂σ

f ij  and bv = ξij

∂F ∂l - 1 m

ð5:164Þ

C ij b  ℓ-1 . ξij

And the plastic flow rules are given by ε_ pl ij = γ

f ij ∂F  γ n  γb ∂σ ij ξij

ð5:165aÞ

5.10

Cosserat Continuum Implementation in Unified Mechanics Theory

ℓ χ_ pl ij = γ

b ij C ∂F  γb  γℓ - 1 ν -1 ∂ℓ mij ξij

291

ð5:165bÞ

γ is the consistency parameter which is defined from the loading/unloading conditions and is related to the evolution of the generalized equivalent plastic strain trajectory defined by rffiffiffi 2 γ α_ = 3

ð5:166Þ

The constitutive model is completed with the evolution equations for the backstresses: 2 nij β_ ij = H 0 γb 3 2 νij ℓ η_ ij = H 0 γb 3

ð5:167aÞ ð5:167bÞ

where H′ represents a kinematic hardening modulus which may be a linear or a nonlinear function of the hardening parameter α. For instance, the assumption of a constant kinematic hardening modulus leads to the so-called Prager-Ziegler rule (Fung & Tong, 2001). Introducing Eqs. (5.165a) and (5.165b) into the generalized strain norm for the plastic quantities yields h i1=2 h i1=2 γ _ pl pl pl pl pl -2b b  f þ ℓ C C f Ε = ε_ ij ε_ ij þ ℓ2 χ_ ij χ_ ij ij ij ij ij ξij

ð5:168Þ

pl which implies Ε_ = γ Using this result in Eq. (5.166) and integrating yields Z t rffiffiffi 2 _ pl α ðt Þ = Ε ðτÞ dτ 3

ð5:169Þ

0

This is the generalized version of equivalent plastic strain (trajectory) including the gradients of plastic strains. The evolution equations are complemented by the loading-unloading conditions which allow the determination of the consistency parameter. In terms of the yield function, the following loading/unloading condition must be satisfied: γ≥0 γ≥0


 F σ, ℓ - 1 m, α ≤ 0
 γF σ, ℓ - 1 m, α = 0

ð5:170aÞ ð5:170bÞ

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5

Unified Mechanics of Thermomechanical Analysis

And the consistent condition
 γ F_ σ, ℓ - 1 m, α = 0

ð5:171Þ

Determination of Consistency Parameter Using the definition of the yield function, we can write ∂F ∂F ∂F ∂F _ þ p : ε_ pl þ F_ = : χ_ pl : σ_ þ - 1 : ℓ - 1 m pl ∂ℓχ ∂ε ∂σ ∂ℓ m

ð5:172Þ

Using Hooke’s law together with the plastic flow rule yields F_ = ½b n : C : ε_ þ bv : D : ℓχ_     ∂F ∂F b b b b b :nþ : bv - γ ½n : C : n þ v : D : v ∂εpl ∂ℓχ pl

ð5:173Þ

And imposing the consistency condition yields the consistency parameter γ in the following form: γ=

b n : C : ε_ þ bv : D : ℓ χ_

∂F b ∂F b n:C:b n þ bv : D : bv - ∂ε v pl : n þ ∂ℓχ pl : b

ð5:174Þ

Using the following relations ∂F =∂εpl

rffiffiffi 2 0 2 εpl K 3 3 α

and

∂F =∂ℓχpl

rffiffiffi 2 0 2 ℓχpl K 3 3 α

ð5:175Þ

yields ∂F :b n= ∂εpl

rffiffiffi 2 0 2 εpl S K : 3 3 α Σ

and

∂F : bv = ∂ℓχ pl

rffiffiffi 2 0 2 ℓχ pl ℓ - 1 m K : 3 3 α Σ

ð5:176Þ and then the term within brackets in the denominator reduces to

5.10

Cosserat Continuum Implementation in Unified Mechanics Theory

∂F ∂F :b nþ ∂εpl ∂ℓχ pl rffiffiffi  2 0 2 1
pl 2 pl -1 : bv = : S þ ℓχ : ℓ m = - K0 K ε 3 3 α 3 Σ

293

ð5:177Þ

after introducing εpl : S + ℓχ pl : ℓ -1m = Kα. The consistency parameter γ reduces to γ=

b n : C : ε_ þ bv : D : ℓ χ_ b n:C:b n þ bv : D : bv þ 23 K 0

ð5:178Þ

Using b n : C = 2Gb n

and

b n:C:b n = 2G

S:S k Σk 2

and bv : D = 2Gbv and bv : C : bv = 2G

ℓ - 1m : ℓ - 1m kΣ k2

where G is shear modulus yields -1 1 S : ε_ þ ℓ - 1 : ℓ χ_ 1 S : ε_ þ ℓ : ℓ χ_
γ= = 0 K b 1 þ 3G K Σ Σ

ð5:179Þ

Introducing this result into Hooke’s law yields



2G O 2G O b b b bv : ℓχ_ n n : ε_ n b b K K O

2G 2G O _ = D : ℓ χ_ b bv : ε_ bv bv : ℓ χ_ ℓ - 1m n b b K K σ_ = C : ε_ -

ð5:180aÞ ð5:180bÞ

Using C = κI

O

a  c 1 O - I I þ 2G I 3

and after simplifying yields

and D = 2G

a c

ð5:181Þ

294

5

2 6 κI M ep = 6 4

N

 Iþ2G

Unified Mechanics of Thermomechanical Analysis

N  O ‘ b b n n b - 1I I3 N b K b n bv - 2G b K

b n

N

bv

3

- 2G bN  7 K 7  ‘ b v v 5 b 2G b K

ð5:182Þ

Alternatively, it can be written as 2

b n

3 D : bv 7 b 7 NK bv D : bv 5 D - D: b K

N

C:b n b ep K N M =6 4 bv C : b n - D: b K 6 C - C:

- C:

b n

N

ð5:183Þ

The plastic flow theory just described can be written in the following alternative form, which is useful in the numerical treatment of the problem. Using the generalized vector and matrix notation, for the simple case of isotropic hardening, we make the following definitions: Σ=

h ℓ

σ

-1

i , m

Ε=

h

ε ℓχ

i

, M=

h

C 0

0 D

i

ð5:184Þ

and then the constitutive model equations can be written as Σ = M : Εel. The yield criteria and flow rule follow: rffiffiffi
 2 0 -1 K ðαÞ F σ, ℓ m, α = Σ 3

ð5:185Þ

pl b E_ = γ N

ð5:186Þ

∂F _ pl ∂F _ F_ = Ε Σþ ∂Σ ∂Εpl

ð5:187Þ

then

Using Hooke’s law and the flow rule results in b : M : Ε_ - γN : M : N b þ ∂F  γ N b F_ = N ∂Εpl

ð5:188Þ

Utilizing the consistency conditions γ F_ = 0 gives us the following relations: N : M : Ε_ b b - ∂Fpl  N b N:M:N ∂Ε pl

γ= we have

or after using

∂F b 2  N = - K0 pl 3 ∂Ε

ð5:189Þ

5.10

Cosserat Continuum Implementation in Unified Mechanics Theory

γ=

b : M : Ε_ pl N b :M:N b þ 2 K0 N 3

295

ð5:190Þ

Using b :M:N b = 2G, b : M = 2GN bT, N N

b : M : Ε = 2GN b Ε N

yields γ=

b  Ε_ N b K

ð5:191Þ

Using this result in the generalized Hooke’s law yields  O  _Σ = M - 2G N b b : Ε_ N b K

ð5:192Þ

The constitutive tensor given by Eq. (5.192) is equivalent to Eq. (5.183) where the coupling between strains and curvatures becomes evident in the off-diagonal terms in the generalized constitutive tensor.

5.10.2

Rate-Dependent Plasticity (Viscoplasticity) Without Degradation

In classical continuum mechanics, in the case of a rate-dependent material, the conditions established by Eqs. (5.170a), (5.170b), and (5.171) are replaced by a constitutive equation of the form: γ=

hφðF Þi η

ð5:193Þ

where η represents viscosity. A rate-independent material F satisfies conditions given by Eqs. (5.170a), (5.170b), and (5.171), and additionally stress states such F(σ, ℓ-1m, α) > 0 are ruled out. In other words, the state of stress cannot be outside the yield surface. In the case of a rate-dependent material, on the other hand, the intensity of the viscoplastic flow is proportional to the distance of the state of stress to the yield surface defined by F(σ, ℓ -1m, α) = 0. Therefore, the yield condition for a rate-dependent material can be given by F = ΘðγηÞ whereas Θ(γη) = φ-1(γη)

ð5:194Þ

296

5.10.3

5

Unified Mechanics of Thermomechanical Analysis

Introducing the Thermodynamic State Index

In the following formulation, for the sake of simplicity in presentation, we will ignore the derivatives with respect to entropy. In the Cosserat continuum, TSI is introduced into the formulation using the same approach as in the classical continuum; stress-strain and couple stress-curvature relations can be given by the third law of the unified mechanics theory as follows: σ_ ij = ð1 - ΦÞC ijkl ε_ ekl

ð5:195aÞ

_ ij = ð1 - ΦÞDijkl ℓχ_ ekl ℓ - 1m

ð5:195bÞ

And the yield function now becomes rffiffiffi
 2 Φ -1 F σ, ℓ m, α, Φ = ξ K ðαÞ 3

ð5:196Þ

where superscript Φ indicates modification due to entropy generation at each loading step. The thermodynamic state index can be anisotropic, Φ. The rate-dependent constitutive model equations are summarized in Table 5.5. The constitutive model can also be written in the equivalent alternative form given in Table 5.6, which is more convenient for the numerical implementation of the algorithm using a return mapping scheme. Using the Cosserat continuum generalized stress and strain definitions, for small strain problems, can be given by

vp θ Σ_ = ð1 - ΦÞM e : Ε_ - Ε_ - Ε_

Table 5.5 UMT rate-dependent-strain gradient formulation 1 Third Law of UMT σ_ ij = ð1 - ΦÞCijkl ε_ ekl _ ij = ð1 - ΦÞDijkl ℓχ_ ekl ℓ - 1m Yield function qffiffi
 F σ, ℓ - 1 m, α = ξΦ - 23K ðαÞ Normality rule - 1 C ij ∂F ∂F f ij ε_ pl n and ℓχ_ pl ν ij = γ ∂ℓ - 1 mij  γℓ ij = γ ∂σ ij  γ  γb  γb ξij ξij Isotropic and kinematic hardening laws qffiffi α_ = 23γ Φ 2 0 β_ ij = ð1 - ΦÞ 32 H 0 γb nij and ℓη_ Φ νij . ij = ð1 - ΦÞ 3 H γb

Consistency parameter γ = hφðηFÞi

ð5:197Þ

5.10

Cosserat Continuum Implementation in Unified Mechanics Theory

Table 5.6 UMT rate dependent model-strain gradient formulation 2

297

Third law of UMT _ ij = ð1 - ΦÞDijkl ℓχ_ ekl σ_ ij = ð1 - ΦÞCijkl ε_ ekl ℓ - 1m Yield function qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffi T FðΣ, αÞ = ξΦ PξΦ - 23K ðαÞ Flow rule vp Ε_ = γ ð1 Pξ - ΦÞ Hardening laws qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffi T α_ = 23γ ð1 - ΦÞ ξΦ PξΦ

Φ b X_ = γ 32 H 0 ðαÞð1 - ΦÞN

Consistency parameter γ = hφðηFÞi vp θ where E_ is the total strain rate, Ε_ is the viscoplastic strain rate, and Ε_ is the thermal strain rate. The yield function can be given by

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi T 2 K ðαÞ F ðΣ, αÞ = ξΦ P ξΦ 3

ð5:198Þ

where ξΦ = ΣΦ - XΦ is the generalized relative h i stress previously introduced with Φ -1 Φ Φ the generalized back-stress X  βij , ℓ ηij , where the following definitions are used: Φ 2 nij β_ ij = ð1 - ΦÞ H 0 γ b 3 2 0 ℓη_ Φ νij ij = ð1 - ΦÞ 3 H γ b

ð5:199Þ ð5:200Þ

In Eq. (5.198) P is a constant matrix; hence, PξΦ gives the deviatoric component qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

of ξΦ and 23 ξΦ P ξΦ is the von Misses equivalent stress for the generalized relative stress vector. K(α) represents the radius of the yield surface. The direction of plastic flow and the hardening laws are given for the associative plasticity case as follows: T

Pξ ð1 - ΦÞ rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 α_ = γ ð1 - ΦÞ ξΦ PξΦ 3

ð5:201Þ

Φ 2 b X_ = γ H 0 ðαÞð1 - ΦÞN 3

ð5:203Þ

vp Ε_ = γ

ð5:202Þ

In Eq. (5.203) the hardening modulus H′ can be a nonlinear function of the hardening parameter α. In this formulation, Eq. (5.203) assumes that the couple

298

5

Unified Mechanics of Thermomechanical Analysis

stress continuum back-stresses evolve in the same manner as the symmetric stress tensor back-stress. However, this is not essential for the formulation, because they can be different. The couple back-stress is needed to allow for the uniform movement of the yield surface in the enhanced (Cosserat continuum) stress space. Viscoplastic Creep Law The evolution of the generalized viscoplastic strain rate is specified by a function representative of the viscoplastic flow micro-mechanisms in the material. In the following formulation, the same creep law is assumed to hold for both the strains and the strain gradients. The following viscoplastic strain rate function is one of the most generalized models for low melting point solder alloys. It accounts for multiple creep micro-mechanisms in the polycrystalline alloys as well as accounting for coarsening of the crystals:  n  p AD0 Eb hF i b ∂F e - Q=Rθ kθ E d ∂σ ij  n  p b ∂F AD0 Eb hF i = e - Q=Rθ - 1 ℓ χ_ vp ij kθ E d ∂ℓ mij  n  p b ∂F _Ε vp = AD0 Eb hF i e - Q=Rθ kθ E d ∂Σ ε_ vp ij =

ð5:204Þ ð5:205Þ ð5:206Þ

where A, n, p are material parameters, D0 is the temperature-independent diffusivity constant, k is Boltzmann’s constant, b is Burger’s vector, R is the gas constant, E is the elastic modulus, d is the initial grain size, Q is the creep activation energy, and θ is the temperature in Kelvin. The fluidity parameter η can be summarized as η=

5.10.4

 p kθ d eQ=Rθ AD0 E n - 1 b b

ð5:207Þ

Entropy Generation Rate in Cosserat Continuum

In the Cosserat continuum couple stress theory, the internal energy can be given by ρ

de = σ Sij Dij þ mji χ ij þ ρr - qi,i dt

ð5:208Þ

where σ Sij is the symmetric part of the Cauchy stress tensor, Dij is the rate of deformation tensor, mji is the couple stress tensor, and χ ij are the corresponding

5.10

Cosserat Continuum Implementation in Unified Mechanics Theory

299

curvatures. The rate of change of the Helmholtz free energy φ is written in terms of the symmetric part of the Cauchy stress tensor; thus, ρ

dφ = σ Sij Deij dt

ð5:209Þ

Combining Eqs. (5.208) and (5.209), the difference between the changes in the internal energy and the Helmholtz free energy with respect to a reference state in the presence of couple stresses can be given by 1 Δe - Δφ = ρ

Zt2 σ Sij Dpij t1

5.10.5

1 dt þ ρ

Zt2

Zt2 mji χ_ pl ij

t1

dt þ

1 r dt ρ

t1

Zt2 qi,i dt

ð5:210Þ

t1

Integration Algorithms

In the material nonlinear finite element method, the solution is achieved in small increments with at every increment Newton-Raphson iteration method is used to find the equilibrium. At every increment, the problem may be regarded as analogous to a strain-controlled test. At the beginning of the time step, the total and viscoplastic strain fields and the internal state variables are considered to be known. Assuming that displacement increment is known, the basic problem is to update the field variables to their new values at the end of the time step in a manner consistent with the constitutive model. Below we present the integration algorithm. First, the case of a rate-independent model without degradation using both a radial return and return mapping scheme is presented. Then the rate-dependent model with degradation case with the return mapping algorithm is presented in Table 5.6. Radial Return Algorithm: Rate-Independent Model with Linear Isotropic Hardening Using Eqs. (5.148a) and (5.148b) and the backward Euler finite difference integration scheme, we can write b nþ1 Εpnþ1 = Εpn þ Δγ N rffiffiffi 2 αnþ1 = αn þ Δγ 3

ð5:211Þ

Now consider the following trial state obtained after freezing the plastic flow: Σtrnþ1 = Σn þ 2GΔΕnþ1 b nþ1 Σnþ1 = Σtr - 2GΔγ N nþ1

ð5:212Þ

300

5

Unified Mechanics of Thermomechanical Analysis

from which it is concluded that Σnþ1 þ 2GΔγ = Σtrnþ1

ð5:213aÞ

and tr b nþ1 = Σnþ1 N tr Σnþ1

ð5:213bÞ

Assuming linear isotropic hardening yields, rffiffiffi 2 2 tr K ðαn Þ - K 0 Δγ - 2GΔγ = 0 Σnþ1 3 3

ð5:214Þ

which is solved for the consistency parameter Δγ Linearization of the Incremental Solution Process Consider b nþ1 Σ_ nþ1 = M : Ε_ nþ1 - 2GΔγ N

ð5:215Þ

and then it follows that " # O dΔγ b nþ1 d N dΣnþ1 b nþ1 = M - 2G N þ Δγ dΕnþ1 dΕnþ1 dΕnþ1

ð5:216Þ

using dΔγ 1b = N dΕnþ1 K b nþ1

ð5:217Þ

b nþ1 dN 2G = tr H nþ1 dΕnþ1 Σnþ1

ð5:218Þ

and

where 2‘ O O b - 1I b I-b n n   nþ1 3 4 nþ1 H nþ1 = 0

3 0

‘ N d bv -b v  nþ1

thus,

nþ1

5

ð5:219Þ

5.10

Cosserat Continuum Implementation in Unified Mechanics Theory

2

301

3

N

b nþ1 b N dΣnþ1 Δγ2G 6N 7 þ = M - 2G4 nþ1 H nþ1 5 dΕnþ1 b tr K Σnþ1

ð5:220Þ

Expanding Eq. (5.220) yields M ep nþ1 2 6 κI =4

N

‘ O N b - 1I Iþ2Gδnþ1 I - 2Gθnþ1bnnþ1 bnnþ1 3 O 2G bvnþ1 b nnþ1 b K

3 O 2G b bvnþ1 7 n b nþ1 K 5 N ‘ b bv 2Gδnþ1 - 2Gθnþ1 v nþ1

nþ1

ð5:221Þ When Δt → 0 Eq. (5.221) approaches (5.183). The coupling between the curvature and strain in the tangent stiffness matrix is apparent Return Mapping Algorithm: Rate-Dependent Model-Combined Isotropic/Kinematic Hardening with Degradation. The rate-dependent model, Table 5.6, is integrated using a return mapping algorithm as presented in Simo and Hughes (1997). A backward Euler integration scheme yields the following set of algorithmic equations: Εnþ1 = Εn þ ΔΕnþ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 T αnþ1 = αn þ Δγ ð1 - ΦÞ ξ Pξ 3 nþ1 nþ1 2 Xnþ1 = Xn þ Δγ H 0 ð1 - ΦÞξnþ1 3

ð5:222Þ ð5:223Þ ð5:224Þ

The standard operator split technique defines the following trial state: Σtr nþ1 = Σn þ ð1 - ΦÞMΔΕ nþ1

ð5:225Þ

Using Eq. (5.225), we can write the following relations: Σnþ1 = Σtr nþ1 - MΔγPξnþ1

ð5:226Þ

tr ξtr nþ1 = Σnþ1 - X n

ð5:227Þ

From (5.226) and (5.227), updated relative stress can be obtained in terms of the algorithmic consistency parameter Δγ: ξnþ1 = ΞðΔγ Þ

1 M - 1 ξtr nþ1 1 þ 23 H 0 Δγ ð1 - ΦÞ

ð5:228Þ

302

5

Unified Mechanics of Thermomechanical Analysis

where "

ΔγP ΞðΔγ Þ = M - 1 þ 2 0 1 þ 3 H ð1 - ΦÞΔγ

#-1 ð5:229Þ

To complete the above algorithm, it is still necessary to compute the consistency parameter Δγ, which can be obtained from the yield condition and the constitutive model. Thus rffiffiffi



2 ηΔγ ηΔγ K ðαnþ1 Þ - Θ F ðΔγ Þ - Θ =0  bf nþ1 3 Δt Δt

ð5:230Þ

 1=2

 - 1 ηΔγ as defined in the previous and Θ ηΔγ where b f nþ1 = ξTnþ1 Pξnþ1 Δt Δt = φ section. Equation (5.230) is a scalar nonlinear equation in the consistency parameter Δγ, which can be solved by a local iteration. In the Newton-Raphson iteration scheme, the elastoplastic tangent modulus consistent with the integration scheme is needed to preserve the convergence. Linearizing the above set of equations yields ! O 1 dΣnþ1 Ν = ð1 - ΦÞ ΞðΔγ Þ -
Ν dΕnþ1 1 þ β~

ð5:231Þ

ΞðΔγÞPξnþ1 ffi Ν = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξTnþ1 PΞðΔγÞPξnþ1

ð5:232Þ

where

  θ2 dΘ b 2θ 2 1 ~ ð5:233Þ β = ð1 - Φ Þ 1 b f nþ1 ðK 0 θ1 þ H 0 θ2 Þ þ 1 f nþ1 T 3θ2 θ2 dΔγ ξnþ1 PΞðΔγ ÞPξnþ1 2 θ1 = 1 þ H0 ð1 - ΦÞΔγ 3 θ2 = 1 -

2 0 K ð1 - ΦÞΔγ 3

Finding the Consistency Parameter Recalling Eq. (5.198) and updating the formula given by Eq. (5.228) yields F ðΔγ Þ = bf nþ1 -

rffiffiffi 2 K ðαnþ1 Þ = 0 3

ð5:234Þ

5.10

Cosserat Continuum Implementation in Unified Mechanics Theory

303

1 M - 1 ξtr nþ1 1 þ 23 H 0 Δγ ð1 - ΦÞ

ð5:235Þ

ξnþ1 = ΞðΔγ Þ

In Newton-Raphson iteration to find the solution x of a nonlinear equation of the general form F(x) = 0, the ith correction to the solution is computed as Δxi =
 FðΔxi - 1 Þ ∂F Δxi - 1 - JðΔxi - 1 Þ where J Δxi - 1 = ∂Δx i - 1 represents the “Jacobian” of F(x). Solution of Eq. (5.230) with a local iteration requires the “Jacobian” J(Δγ) to be defined. In the Newton-Raphson iteration, the Jacobian represents the derivative of the function; hence, dbf dF dΘ J ðΔγ Þ =  nþ1 dΔγ dΔγ dΔγ

rffiffiffi 2 dK dΘ 3 dΔγ dΔγ

ð5:236Þ

Equation (5.235) can be given by i-1 h 2 ξtr ξnþ1 = I þ H 0 Δγ ð1 - ΦÞI þ ΔγPM nþ1 3

ð5:237Þ

db f nþ1 , the symmetric matrices P and M are diagonalized. Since both To find dΔγ matrices share the same characteristic space, the following relationships can be written:

P = QΛP QT

ð5:238Þ

M = QΛM QT

ð5:239Þ

and then PM = QΛP ΛM QT where QT = Q-1. The matrices for a plane strain idealization are Q= " Q1 =

h

P1 =

2=3 - 1=3 - 1=3

- 1=3 2=3 - 1=3

0 I3 × 3

0ffiffi p -pffiffi 2=2 2=2

P= "

Q1 0

- 1=3 - 1=3 2=3

h

#

P1 0

i with

pffiffi 2= p 6 ffiffi - 1=p6ffiffi - 1= 6 0 P2

pffiffi # 1=p3ffiffi 1=p3ffiffi 1= 3

i with

and P2 = DIAG½2, 1, 1

ð5:240aÞ ð5:240bÞ ð5:241aÞ ð5:241bÞ

304

5

ΛP = DIAG½1, 1, 0, 2, 1, 1 and

Unified Mechanics of Thermomechanical Analysis

ΛM = DIAG½2G, 2G, 3λ þ 2G, G, 2G, 2G

Now the update formula Eq. (5.237) can be written as ξnþ1 =

h

i-1 2 1 þ ΔγH0 I þ ΔγQΛP ΛM QT ξtr nþ1 3

ð5:242Þ

or after using the following definition:  ΓðΔγÞ =

-1  2 1 þ ΔγH0ð1 - ΦÞ I þ ΔγΛP ΛM 3

ð5:243Þ

As a result, this expression becomes ξnþ1 = QΓðΔγ ÞQT ξtr nþ1

ð5:244Þ

where ΓðΔγ Þ

2

6 6 = DIAG6 6 4



1

,



1

,

1

3 ,

2 2 2 7 ð1 - ΦÞH 0 þ 2G Δγ 1 þ ð1 - ΦÞH 0 þ 2G Δγ 1 þ ð1 - ΦÞH 0 Δγ 7 3 3 3 7 7 1 1 1

5

,

, 2 2 2 1 þ ð1 - ΦÞH 0 þ 2G Δγ 1 þ ð1 - ΦÞH 0 þ 2G Δγ 1 þ ð1 - ΦÞH 0 þ 2G Δγ 3 3 3 1þ

ð5:245Þ Making use of Eq. (5.214), bf nþ1 becomes h i1=2 bf nþ1 = ξtrT QT ΓðΔγ ÞQPQT ΓðΔγ Þξtr nþ1 nþ1 Using Eqs. (5.244), (5.245), and (5.246),

db f nþ1 dΔγ

ð5:246Þ

can be defined as

g dbf nþ1 b = nþ1 dΔγ b f nþ1

ð5:247Þ

T T tr b gnþ1 = ξtr nþ1 QGðΔγ ÞQ PQΓðΔγ ÞQ ξnþ1

ð5:248Þ

where T

5.10

Cosserat Continuum Implementation in Unified Mechanics Theory

305

dΓðΔγ Þ = DIAG½G11 , G22, G33 , G44 , G55 , G66  with dΔγ 2  0 3 ð1 - ΦÞH þ 2G G11 = G22 = G44 = G55 = G66 = -

 2  1 þ 23 ð1 - ΦÞH0 þ 2G Δγ GðΔγ Þ =

G33 = - 

1

0 2 3 ð1 - ΦÞH 2 þ 23 ð1 - ΦÞH0Δγ

Using the definition of K(α), Eq. (5.247) and θ2 from Eq. (5.231) yield the Jacobian needed for the local Newton iteration: b g dF ðΔγ Þ 2 = θ2 nþ1 - K 0bf nþ1 3 dΔγ bf

ð5:249Þ

nþ1

The complete Newton-Raphson integration algorithm is shown in Table 5.7. The local Newton iteration for the determination of the consistency parameter is shown in Table 5.8.

Table 5.7 Return mapping integration algorithm Update strain Compute trial state Compute trial yield function IF Ftr nþ1 > 0 THEN Call Newton local and solve f(Δγ) = 0 for Δγ Compute

Εn + 1 = Εn +— Δu
 vp θ Σtr nþ1 = ð1 - ΦÞM Ε nþ1 - Ε n - Εnþ1 tr ξtr nþ1 = Σnþ1 - X n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffi tr 2 Ftr ξtrT nþ1 Pξ nþ1 nþ1 = 3K ðαn Þ

h i-1 ΞðΔγ Þ = M - 1 þ 1þ2H0 ðΔγP 1 - ΦÞΔγ 3

Update

ξnþ1 = ΞðΔγ Þ 1þ2H0 ð11- ΦÞΔγ M - 1 ξtr nþ1 3

X nþ1 = X n þ Δγ 23 H0 ð1 - ΦÞξnþ1 Σn + 1 = ξ n + 1 + Xn + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αnþ1 = αn þ Δγ ð1 - ΦÞ 23 ξTnþ1 Pξnþ1 Pξ

vp nþ1 Εvp nþ1 = Ε n þ Δγ ð1 - ΦÞ

= Εnþ1 - Εvp nþ1

dΣnþ1 dΕnþ1

 = ð1 - ΦÞ ΞðΔγ Þ -

Compute consistent Jacobian

ELSE Elastic step (EXIT) END IF EXIT

- Εθnþ1

Εenþ1

1 Ν ð1þ~βÞ

N

 Ν

306

5

Unified Mechanics of Thermomechanical Analysis

Table 5.8 Local Newton iteration to determine the consistency parameter Let Δγ (0) ← 0 α(0)n + 1 ← αn Start Iterations DO_UNTIL jF(Δγ j < tol k←k+1 Compute Δγ (k + 1) ðkÞ

ðkÞ  pffiffiffiffiffi - 2=3K ðαnþ1 Þ - Θ ηΔγ F Δγ ðkÞ = bf Δγ nþ1




 ðkÞ

Δt


 bg ðΔγðkÞ Þ = θ2 nþ1 - 23 K 0 ð1 - ΦÞbf nþ1 Δγ ðkÞ J Δγ bf nþ1 ðΔγðkÞ Þ FðΔγ ðkÞ Þ Δγ ðkþ1Þ ← Δγ ðkÞ - JðΔγðkÞ Þ pffiffiffiffiffiffiffiffi
 αnþ1 ðkþ1Þ ← αnþ1 ðkÞ þ 2=3bf nþ1 ð1 - ΦÞ Δγ ðkÞ END DO_UNITL

dΘ dΔγ

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References

307

Lemaitre, J. (1996). A course on damage mechanics. Springer-Verlag. Lemaitre, J., & Chaboche, J. L. (1990). Mechanics of solid materials. Cambridge University Press. Malvern, L. E. (1969). Introduction to the mechanics of continuous medium. Prentice-Hall. Mazur, P., & De Groot, S. (1962). Non-equilibrium thermodynamics. Dover Publications, Inc.. Mindlin, R. (1964). Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, 51–78. Mindlin, R. (1965). Second gradient of strain and surface tension in linear elasticity. International Journal of Solids and Structures, 1, 417–438. Mindlin, R., & Tiersten, H. F. (1962). Effects of couple-stresses in linear elasticity. Communicated by C. Truesdell. Archive for Rational Mechanics and Analysis, 11, 415–448. Prager, W. (1955). The theory of plasticity: A survey of recent achievements (James Clayton Lecture). Proceedings of the Institution of Mechanical Engineers, 169, 41. Rabotnov, Y.N. (1969) Creep Problems in Structural Members, Wiley Shu, J. Y., & Fleck, N. A. (1999). Strain gradient plasticity: Size-dependent deformation of bicrystals. Journal of the Mechanics and Physics of Solids, 47, 297–324. Simo, J., & Hughes, T. (1997). Computational inelasticity. Interdisciplinary applied mathematics. Springer. Toupin, R. (1962). Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis, 11, 385–414. Xia, Z., & Hutchinson, J. (1996). Crack tip fields in strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 44(10), 162–1648.

Chapter 6

Thermomechanical Analysis of Particle-Filled Composites

6.1

Introduction

Modeling the macroscopic constitutive response of composite materials starting from the local description of the microstructure is necessary. Composite materials are heterogeneous. However, continuum mechanics formulation requires homogenization of the medium. The homogenization method used to go from the local level to the macro-level and the localization from macro-level quantities to the corresponding local micromechanics variables must be well defined. Using the Eshelby method (Eshelby, 1957; Ju & Chen, 1994a, b) formulated the effective mechanical properties of elastic multiphase composites containing many randomly dispersed ellipsoidal inhomogeneities with perfect bonding. Within the context of the representative volume element (RVE), four governing micromechanical ensemble-volume averaged field equations utilized to relate ensemble-volume averaged stresses, strains, volume fractions, Eigen strains, particle shapes, and orientations, and elastic properties of constituent phases of the particulate composites. Ju and Tseng (1996) formulated combining a micromechanical interaction approach and the continuum plasticity to predict effective elastoplastic behavior of two-phase particulate composite containing many randomly dispersed elastic spherical inhomogeneities. Explicit pairwise interparticle interactions are considered in both the elastic and plastic responses. Furthermore, the ensemblevolume averaging procedure is employed and the formulation is of complete second order. Let us consider a perfectly bonded two-phase composite consisting of an elastic matrix (named “phase 0”) with bulk modulus k0 and shear modulus μ0, and randomly dispersed elastic spherical particles (named “phase 1”) with bulk modulus k1 and shear modulus μ1. For the effective bulk modulus k
and effective shear modulus μ
, this two-phase composite for the noninteracting solution, neglecting the interparticle interaction effects, was derived by Ju and Chen (1994a) as follows:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_6

309

310

6


k
= k0
μ
= μ0

Thermomechanical Analysis of Particle-Filled Composites

3ð1 - v0 Þðk 1 - k 0 Þφ 1þ 3ð1 - v0 Þk0 þ ð1 - φÞð1 þ v0 Þðk 1 - k 0 Þ



15ð1 - v0 Þðμ1 - μ0 Þφ 1þ 15ð1 - v0 Þμ0 þ ð1 - φÞð8 - 10v0 Þðμ1 - μ0 Þ

ð6:1Þ  ð6:2Þ

where φ is the particle volume fraction and v0 is Poisson’s ratio of the matrix. If the effects due to interparticle interactions are included for the two-phase elastic composites with randomly located spherical particles, the solutions for the effective bulk modulus k
and shear modulus μ
are provided by Ju and Chen (1994b):
k
= k0

 30ð1 - v0 Þφð3γ 1 þ 2γ 2 Þ 1þ 3α þ 2β - 10ð1 þ v0 Þφð3γ 1 þ 2γ 2 Þ
 30ð1 - v0 Þφγ 2 μ
= μ0 1 þ β - 4ð4 - 5v0 Þφγ 2

ð6:3Þ ð6:4Þ

with  α = 2ð5v0 - 1Þ þ 10ð1 - v0 Þ

μ0 k0 k1 - k0 μ1 - μ0

β = 2ð4 - 5v0 Þ þ 15ð1 - v0 Þ



μ0 μ1 - μ0

ð6:5Þ ð6:6Þ

And 
5φ 8α ð 13 14v Þv ð Þ ð 1 þ v Þ 1 2v 0 0 0 0 3α þ 2β 8β2 
 1 5φ  6α 2 25 34v þ 22v ð Þ ð 1 þ v Þ γ2 = þ 1 2v 0 0 0 0 2 16β2 3α þ 2β γ1 =

ð6:7Þ ð6:8Þ

Alternatively, effective Young’s modulus E
and Poisson’s ratio v
of particulate composites are easily obtained through the following relationships: 9k
μ
3k
þ μ


ð6:9Þ

3k
- 2μ
6k
þ 2μ


ð6:10Þ

E
= v
=

The experimental validation of the Ju and Chen model was provided by Nie and Basaran (2005), by comparing the analytical predictions with the experimental data on the particulate composite prepared using lightly cross-linked poly-methyl methacrylate (PMMA) filled with alumina trihydrate (ATH). All filler particles are assumed spherical and both the matrix and the filler are isotropic elastic. The material properties involved are as follows:

6.1

Introduction

311

12

Young’s Modulus (GPa)

Composite [test data] 10 Pairwise interacting solution 8

6 Noninteracting solution 4 PMMA [test data] 2

0

0

0.1

0.2

0.3

0.4

0.5

0.6

Particles Volume Fraction

Fig. 6.1 Effective Young’s modulus as a function of particle volume fraction. (After Nie (2005))

ATH: E1 = 70 GPa, v1 = 0.24. PMMA: E0 = 3.5 GPa, v0 = 0.31 Particle volume fraction: φ = 0.48 The experimental mean value of Young’s modulus for this composite is about 10.2 GPa at room temperature. Figure 6.1 shows the comparisons among the analytical solution (including pairwise interacting solution and noninteracting solution) and experimental results. It is observed that agreement between the pairwise interacting prediction and experimental data is very good for effective Young’s modulus. Based on the foregoing preliminary analytical and experimental comparisons, it appears that Ju and Chen’s analytical micromechanical approach offers a simple, approximate, yet sufficiently accurate framework for the prediction of effective elastic moduli of two-phase particle-filled composites. Particle-filled composites consist of the bulk matrix, filler particles, and interfacial transition zone around particles, which often have very different properties such as coefficient of thermal expansion (CTE) and stiffness. Within the particulate composite microstructure, there are micro-stresses due to the coefficient of the thermal expansion (CTE) mismatch between the matrix and the filler particles. For the composite prepared using lightly cross-linked poly-methyl methacrylate (PMMA) filled with alumina trihydrate (ATH), these micro-stresses can be imaged using the fact that PMMA is stress optically birefringent. The micro-stresses imaged using this technique are indeed due to CTE mismatch as they dissipate at temperatures close to the glass transition temperature Tg of the PMMA. These stresses can be quantified by calculation or by direct measurement and are of the order between 15%

312

6

Thermomechanical Analysis of Particle-Filled Composites

and 75% of the tensile strength of the composite. Therefore, the thermal stress associated with the CTE mismatch between the matrix and particles is an important factor in the failure of particulate composites subjected to thermomechanical loads.

6.2

Ensemble-Volume Averaged Micromechanical Field Equations

To obtain effective constitutive equations and properties of random heterogeneous composites, one typically performs the ensemble-volume averaging process within a mesoscopic representative volume element (RVE). The volume-averaged stress tensor is defined as

σ=

1 V

2

Z σðxÞdx =

16 4 V

V

Z σðxÞdx þ

Z

Xn q=1

V0

3 7 σðxÞdx5

ð6:11Þ

Vq

where V is the volume of an RVE, V0 is the volume of the matrix, Vq is the volume of the qth-phase particles, and n denotes the number of particulate phases [fillers] of different materials (excluding the matrix). Similarly, the volume-averaged strain tensor is defined as

ε=

1 V

2

Z εðxÞdx = V

16 4 V

Z εðxÞdx þ V0

Xn

Z

q=1

3

h i Xn 71 εðxÞdx V 0 ε0 þ V ε 5 q q q=1 V

Vq

ð6:12Þ According to Eshelby’s equivalence principle, the perturbed strain field ε′(x) induced by inhomogeneity can be related to the specified Eigen strain ε
(x) by replacing the inhomogeneities with the matrix material. Sometimes the inhomogeneities may also involve their Eigen strain caused by, i.e., phase transformation, precipitation, plastic deformation, or CTE mismatch between different constituents of the composites. However, it is not necessary to attribute the Eigen strain to any specific source. Then the total perturbed stress is the sum of the two parts, one caused by the inhomogeneity, and the other one is the Eigen stress caused by the Eigen strain. For the domain of the qth-phase particles with elastic stiffness tensor Cq, Mura (1987) provided the following stress-strain relation: i h i h Cq : ε0 þ ε0 ðxÞ - εTq ðxÞ = C0 : ε0 þ ε0 ðxÞ - εTq ðxÞ - ε
q ðxÞ

ð6:13Þ

where C0 is the stiffness tensor of the matrix and ε0 is the uniform elastic strain field induced by far-field loads for a homogeneous matrix material only. εTq is its Eigen

6.2

Ensemble-Volume Averaged Micromechanical Field Equations

313

strain associated with the qth particle, and ε
q is the fictitious equivalent Eigen strain by replacing the qth particles with the matrix material. ε′(x) is the perturbed strain due to distributed Eigen strain εT and ε
is associated with all the particles in the RVE. The stress tensor for the matrix is given in Newtonian mechanics by σ0 = C 0 : ε 0

ð6:14Þ

The strain at any point within an RVE is decomposed into two parts, the uniform strain and the perturbed strain due to the distributed Eigen strain. It is emphasized that the Eigen strain ε
and εT are nonzero in the particle domain and zero in the matrix domain, respectively. The perturbed strain field induced by all the distributed Eigen strains ε
and εT can be expressed by (Mura, 1987) ε0 ðxÞ =

Z

 Gðx - x0 Þ : εT ðx0 Þ þ ε
ðx0 Þ dx0

ð6:15Þ

V

where x, x′ 2 V, and G is Green’s function in a linear elastic homogeneous matrix. For a linear elastic isotropic matrix, the fourth-rank tensor Green’s function can be given by (Ju & Chen, 1994a) Gijkl ðx - x0 Þ =

1 F ð - 15, 3v0 , 3, 3 - 6v0 , - 1 þ 2v0 , 1 - 2v0 Þ 8π ð1 - v0 Þr 3 ijkl

ð6:16Þ

where r  x - x′, r  j rj, and v0 is Poisson’s ratio of the matrix. The components of the fourth-rank tensor F are defined by   F ijkl ðBm Þ = B1 ni nj nk nl þ B2 δik nj nl þ δil nj nk þ δjk ni nl þ δjl ni nk þ B3 δij nk nl   þ B4 δkl ni nj þ B5 δij δkl þ B6 δik δjl þ δil δjk

ð6:17Þ

with the unit normal vector n  r/r and index m = 1–6. From Eqs. (6.13) and (6.15), we arrive at - Aq :

ε
q ðxÞ = ε0

Z - εTq ðxÞ

þ

 Gðx - x0 Þ : εT ðx0 Þ þ ε
ðx0 Þ dx0 x0 2 V ð6:18Þ

V

where  -1  C0 Aq = Cq - C0 Furthermore, the total local strain field ε(x) can be expressed as

ð6:19Þ

314

6

Thermomechanical Analysis of Particle-Filled Composites

ε ð xÞ = ε 0 þ ε 0 ð xÞ = ε0 þ

Z

 Gðx - x0 Þ : εT ðx0 Þ þ ε
ðx0 Þ dx0

ð6:20Þ

V

Using the renormalization procedure of the volume-averaged strain tensor given by Ju and Chen (1994a), as ε = ε0 þ

1 V

Z Z V

 Gðx - x0 Þ : εT ðx0 Þ þ ε
ðx0 Þ dx0 dx = ε0 þ s

V

:

h Xn

i T
φ ε þ ε q q q=1 q

ð6:21Þ

where s is a constant tensor for unidirectionally aligned and similar ellipsoidal filler particles. If the linear elastic matrix material is isotropic and all inclusions are spherical, then the s takes the form of the Eshelby tensor S: Sijkl =

   1 ð5v0 - 1Þδij δkl þ ð4 - 5v0 Þ δik δjl þ δil δjk 15ð1 - v0 Þ

ð6:22Þ

where δij signifies the Kronecker delta. Similarly, using Eqs. (6.11)–(6.13), the ensemble-volume averaged stress field can be given by 2 σ= =

16 4 V

Z C0 : εðxÞdx þ V0

Xn

Z

q=1 Vq

3 h i 7 C0 : εðxÞ - εTq - ε
q dx5

h ii 1 V q C0 : εq - εTq - ε
q V 0 C0 : ε0 þ q = 1 V h

i Xn T
= C0 : ε φ ε þ ε q q q q=1 h

Xn

ð6:23Þ

The effective elastic properties can be obtained, in principle, from Eqs. (6.18), (6.21), and (6.23) since the variables are σ,ε,ε0 ,ε
q. Essentially, one needs to solve the relationship between ε and ε
q, which involves the solution of the integral Eq. (6.18). ε
q depends on interparticle interactions, particle-matrix interactions, and microstructure (i.e., particle sizes, orientation, shapes, volume fractions, locations, configurations, and probability functions) of a composite system. For randomly dispersed particles, one needs to obtain the ensemble-volume averaged relation between ε and ε
q by averaging all possible solutions of the integral Eq. (6.18) for any particle configurations generated according to the specified probability function. Taking the ensemble-volume average of Eq. (6.18) overall qth-phase particles, we obtain

6.2

Ensemble-Volume Averaged Micromechanical Field Equations

315

- Aq : ε
q = ε0 - εTq þ εq 0

ð6:24Þ

where ε0q =

1 Vq

Z Z

 Gðx - x0 Þ : εT ðx0 Þ þ ε
ðx0 Þ dx0 dx

ð6:25Þ

Vq V

If all particles in the qth phase have the same ellipsoidal shape and orientation, then εq 0 can be given by

 T
þ S : ε þ ε ε0q = ε0p q q q q

ð6:26Þ

with

ε0p q =

1 XN q i=1 Vq

8 Z > < Z Ωiq

> :

V

- Ωiq

9 > =

 Gðx - x0 Þ : εT ðx0 Þ þ ε
ðx0 Þ dx0 dx > ;

ð6:27Þ

representing the interparticle interaction effects, where Ωiq is the domain of the qth filler particle in the qth-phase domain Vq and Nq is the number of the [filler] phase q particles dispersed in V. Sq is the Eshelby tensor associated with the qth particle. From Eqs. (6.24) and (6.26), we arrive at 

 - Aq - Sq : ε
q = ε0 - εT þ Sq : εT þ ε0p q

ð6:28Þ

In summary, the three basic governing micromechanical ensemble-volume averaged field equations can be given as follows: h

i Xn
T σ = C0 : ε φ ε þ ε q q q q=1

 Xn T
ε = ε0 þ φ s : ε þ ε q q q q=1

  - Aq þ Sq : ε
q = ε0 - εT þ Sq : εTq þ ε0p q

ð6:29Þ ð6:30Þ ð6:31Þ

To solve Eqs. (6.29)–(6.31) and obtain the effective elastic properties of the composite, it is essential to express the qth-phase average Eigen strain ε
q in terms of the volume average strain ε . Namely, one must solve the integral Eq. (6.18) exactly for each phase, which involves details of random microstructure.

316

6.3

6

Thermomechanical Analysis of Particle-Filled Composites

Noninteracting Solution for Two-Phase Composites

Let us consider a perfectly bonded two-phase composite consisting of a viscoplastic matrix (phase 0) with elastic bulk modulus k0 and elastic shear modulus μ0, and randomly dispersed elastic spherical particles (phase 1) with bulk modulus k1 and shear modulus μ1. For the sake of simplicity, the von Mises yield criterion is used for the matrix. The extension of the present framework to a general yield criterion and a general hardening law, however, is straightforward. Accordingly, at any matrix material point, the deviatoric stress σ and the equivalent plastic strain e p must satisfy the following yield function:   pffiffiffiffiffiffiffiffiffiffiffiffiffi F σD , e p = H ð σ D Þ - K ð e p Þ

ð6:32Þ

where K(e p) is the isotropic hardening function of the matrix-only material. Furthermore, H(σD) signifies the square of the deviatoric stress norm in the matrix only:   H σD = σ : I D : σ

ð6:33Þ

where ID denotes the deviatoric part of the fourth-rank identity tensor:  D  1 1 I ijkl = - δij δkl þ δik δjl þ δil δjk 3 2

ð6:34Þ

To solve the elastoplastic response exactly, the stress at any local point must be known and used to determine the plastic response through the local yield criterion for all possible configurations. This approach is in general infeasible due to the complexity of the statistical and microstructural information. Therefore, a framework in which an ensemble-averaged yield criterion is constructed for the entire composite.

6.3.1

Average Stress Norm in Matrix

Total stress σ(x) at any point x in the matrix can be given by the superposition of the far-field stress σ0 and the perturbed stress σ′ induced by the inclusion particles: σ ð xÞ = σ 0 þ σ 0 ð xÞ

ð6:35Þ

According to the Eshelby theory, the perturbed stress σ ′ at any point in the matrix due to the presence of the particles can be given by 0

Z

σ ðxÞ = C0 : V

 Gðx - x0 Þ : εT þ ε
ðx0 Þ dx0

ð6:36Þ

6.3

Noninteracting Solution for Two-Phase Composites

317

where ε
(x′) denotes the fictitious elastic Eigen strain in the particle-induced by replacing the inhomogeneity [filler] with the matrix material. εT is the Eigen strain of the inhomogeneity. Assuming εT is uniform in the particles, G(x - x′) is the fourthrank tensor of Green’s function defined by Eq. (6.16). According to the Eshelby theory, the Eigen strain ε
(x′) due to a single ellipsoidal inclusion is uniform for the interior points of an isolated (noninteracting) inclusion. Therefore, the perturbed stress for any matrix point x due to a typically isolated inhomogeneity centered at x1 takes the form    σ0 ðxjx1 Þ = C0  Gðx - x1 Þ : εT þ ε
0

ð6:37Þ

where Z Gðx - x1 Þ =

Gðx - x0 Þdx0 For x= 2 Ω1

ð6:38Þ

Ω1

Here Ω1 is the particle domain centered at x1. Alternatively, we can derive G ð x - x1 Þ =

 1  ρ3 H þ ρ2 H2 30ð1 - v0 Þ

ð6:39Þ

where the components of H1 and H2 are given by H 1ijkl ðrÞ = 5F ijkl ð - 15, 3v0 , 3, 3 - 6v0 , - 1 þ 2v0 , 1 - 2v0 Þ H 2ijkl ðrÞ = 3F ijkl ð35, - 5, - 5, - 5, 1, 1Þ

ð6:40Þ ð6:41Þ

where r  x - x1, r  j rj, ρ = a/r, and a is the radius of a spherical particle. The components of the fourth-rank tensor F are given by Eq. (6.17). In Eq. (6.37), ε
0 denotes the solution of the Eigen strain ε
for the single inclusion problem, which is given (from Eq. (6.31) when ε0p q is dropped): ε
0 = - ðA þ SÞ - 1 : ε0 þ ðA þ SÞ - 1  ðI - SÞ : εT

ð6:42Þ

A = ðC1 - C0 Þ - 1  C0

ð6:43Þ

where

We define H(x| ℘) as the square of the current deviatoric stress norm at the local point x, which determines the plastic strain in a particulate composite for a given phase configuration ℘. Since there is no plastic strain in the elastic particles or voids, H(x| ℘) can be written as (Ju & Tseng, 1997)

318

6

( H ðxj℘Þ =

Thermomechanical Analysis of Particle-Filled Composites

σ ðxj℘Þ : ID : σðxj℘Þ 0

x in the matrix otherwise

ð6:44Þ

In addition, hHim(x) is defined as the ensemble average of H(x| ℘) over all possible realizations where point x is in the matrix phase. Matrix point receives the perturbations from particles. Therefore, the ensemble-average stress norm for any matrix point x can be evaluated by collecting and summing up all the current stress norm perturbations produced by any typical particle centered at x1 in the particle domain and averaging over all possible locations of x1, namely, for point x in the matrix: Z

hH im ðxÞ = H 0 þ

 H ðxjx1 Þ - H 0 Pðx1 Þdx1

ð6:45Þ

r>a

Here a is the radius of the particle, P(x1) denotes the probability density functions for finding a particle centered at x1, and H0 corresponds to the far-field deviatoric stress norm in the matrix: H 0 = σ0 : I D : σ0

ð6:46Þ

where ID signifies the deviatoric part of the fourth-rank identity tensor. Assuming that P(x1) is statistically homogeneous, isotropic, and uniform, and P(x1) takes the form Pðx1 Þ =

N V

ð6:47Þ

where N is the total number of filler particles dispersed in volume V. According to the assumption of statistical isotropy and uniformity, Eq. (6.45) can be recast into a more convenient form: h H i m ð xÞ ffi H 0 þ

N V

Z

Z dr

r>a

 H ðrÞ - H 0 dA

ð6:48Þ

AðrÞ

with the help of Eqs. (6.35), (6.37), (6.44), and (6.46), plus using the following identities: Z ni nj dA =

4πr2 δ 3 ij

ð6:49Þ

Aðr Þ

Z ni nj nk nl dA = Aðr Þ

 4πr2  δij δkl þ δik δjl þ δil δjk 15

ð6:50Þ

6.3

Noninteracting Solution for Two-Phase Composites

Z

319

Z dr

r>a

H 1 ðrÞdA = 0

ð6:51Þ

H 2 ðrÞdA = 0

ð6:52Þ

GðrÞdA = 0

ð6:53Þ

Aðr Þ

Z

Z

dr r>a

Aðr Þ

Z

Z

dr r>a

Aðr Þ

Equation (6.48) can be recast as N h H i m ð xÞ = H þ V

Z

0

Z dr

r>a



σ0 ðxÞ : ID : σ0 ðxÞ dA

ð6:54Þ

Aðr Þ

After some lengthy yet straightforward derivations, one can obtain the ensembleaveraged current stress norm at any matrix point x as: hH im ðxÞ = σ0 : T : σ0 þ σT : T
: σT - 2σ0 : T
: σT

ð6:55Þ

σT = AC1 : εT

ð6:56Þ

where

The components of the positive definite fourth-rank tensors T and T
are   T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk   T
ijkl = T
1 δij δkl þ T
2 δik δjl þ δil δjk

ð6:57Þ ð6:58Þ

with 3T 1 þ 2T 2 = 200ð1 - 2v0 Þ2 T2 =

φ ð3α þ 2βÞ2

φ 1  þ 23 - 50v0 þ 35v20 2 2 β

φ ð3α þ 2βÞ2  φ T
2 = 23 - 50v0 þ 35v20 2 β

3T
1 þ 2T
2 = 200ð1 - 2v0 Þ2

ð6:59Þ ð6:60Þ ð6:61Þ ð6:62Þ

α and β are given by Eqs. (6.5) and (6.6), and φ is the filler particle volume fraction.

320

6

Thermomechanical Analysis of Particle-Filled Composites

The ensemble-averaged current stress norm at a matrix point must be established in terms of the macroscopic stress σ to express the effective loading function in terms of the macroscopic stress. In the special case of uniform dispersions of identical elastic spheres in a homogeneous matrix, the macroscopic stress and the far-field stress take the form    σ = C0 : ε - φ ε
0 þ εT    σ0 = C0 : ε - φS : ε
0 þ εT

ð6:63Þ ð6:64Þ

Using Eqs. (6.63), (6.64), and (6.42), the relation between the far-field stress σ0 tensor and the macroscopic stress σ takes the form σ = P : σ0 - Q : σT

ð6:65Þ

P = I þ φðI - SÞðA þ SÞ - 1

ð6:66Þ

Q = φðI - SÞðA þ SÞ - 1

ð6:67Þ

where

with the components of P and Q given by   Pijkl = P1 δij δkl þ P2 δik δjl þ δil δjk   Qijkl = Q1 δij δkl þ Q2 δik δjl þ δil δjk

ð6:69Þ

3P1 þ 2P2 = aφ þ 1

ð6:70Þ

ð6:68Þ

where

1 ðbφ þ 1Þ 2

ð6:71Þ

3Q1 þ 2Q2 = aφ

ð6:72Þ

P2 =

Q2 = a=

1 bφ 2

ð6:73Þ

20ð1 - 2v0 Þ 3α þ 2β

ð6:74Þ

ð7 - 5v0 Þ β

ð6:75Þ

b=

With the help of Eq. (6.65), we arrive at the alternative expression for the ensemble-averaged current stress norm at a matrix point x as:


hH im ðxÞ = σ : T : σ þ 2σ : T : σT þ σT : T





: σT

ð6:76Þ

6.3

Noninteracting Solution for Two-Phase Composites

321






where the positive definite fourth-rank tensors T, T , and T are defined as follows:  T T = P-1  T  P-1  T
T = P - 1  T  P - 1  Q - T
 P - 1  T    

T = P - 1  Q  T  P - 1  Q þ T
- 2T
 P - 1  Q


where the components of T,T , and T





ð6:77Þ ð6:78Þ ð6:79Þ

are

  T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk 


 Tijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk 





 T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk

ð6:80Þ ð6:81Þ ð6:82Þ

with 3T 1 þ 2T 2 = T2 =

3T 1 þ 2T 2 ðaφ þ 1Þ2

T2 ðbφ þ 1Þ2







3T 1 þ 2T2 =


T2 = 3T 1







T2 =

3T 1 þ 2T 2 ðaφ þ 1Þ2

T2 1 þ 2 2 ð bφ þ 1Þ ðbφ þ 1Þ



þ 2T 2 =

3T 1 þ 2T 2 ðaφ þ 1Þ2

T2 bφ - 1 þ ðbφ þ 1Þ2 2ðbφ þ 1Þ

ð6:83Þ ð6:84Þ ð6:85Þ ð6:86Þ ð6:87Þ ð6:88Þ

Because σT = AC1 : εT is spherical stress, Eq. (6.76) of the ensemble-averaged loading function can be simplified as     hH im ðxÞ = σ - σT : T : σ - σT

ð6:89Þ

It should be noted that the effective loading function is pressure-dependent now and not of the von Mises type anymore, because spherical stress influences plastic flow resistance of polymers. In other words, the yield stress is a function of hydrostatic stress. Therefore, the particles have significant effects on the viscoplastic behavior of the matrix materials. Plastic yielding and plastic flow occur only in the matrix because the filler particles are assumed elastic. Since the two-phase

322

6

Thermomechanical Analysis of Particle-Filled Composites

composite is in the plastic deformation range when the ensemble-volume averaged current stress norm in the matrix reaches a critical level. The magnitude of the current equivalent stress norm is utilized to determine the possible viscoplastic strain for any point in the composite.

6.3.2

Average Stress in Filler Particles

If the particle interaction for the two-phase composite is ignored, Eq. (6.30) becomes   ε = ε0 þ φS : εT þ ε
0

ð6:90Þ

With the noninteracting solution ε
0 of the Eigen strain given by Eq. (6.42), we arrive at h i ε = I - φSðA þ SÞ - 1 : ε0 þ φSðA þ SÞ - 1 ðA þ IÞ : εT

ð6:91Þ

The volume-averaged stress tensor for the particles is defined by 1 σ1 = V1

Z

1 σ1 ðxÞdx = V1

V1

Z

  C1 : ε0 þ ε0 ðxÞ - εT ðxÞ dx = C1 : ε0 þ ε01 - εT1

V1

ð6:92Þ where ε01 can be recast as ε01 = S :

8

: 2 N6 - 4 V

Z

Z1

2a

ρ

6

Z

Z1 ρ3 2a



H1 dAdr þ 2 A

Z

Z1 3

ρ5 2a

A

39 > =   2 27 H dAdrH 5 : εT þ ε
0 > ;

    7  L  H1 dAdr5 : εT þ ε
0 þ 0 ρ8

A

ð6:120Þ where A denotes the spherical surface of a particle with radius r. After a lengthy but straightforward mathematical manipulation, the final expression takes the following form: hε
i = Γ : ε
0 þ Γ
: εT

ð6:121Þ

The components of the positive definite fourth-rank tensor Γ and Γ
read   Γijkl = γ 1 δij δkl þ γ 2 δik δjl þ δil δjk   Γ
ijkl = γ 1
δij δkl þ γ 2
δik δjl þ δil δjk

ð6:122Þ ð6:123Þ

where 
5φ 8α ð 13 14v Þv ð Þ ð 1 þ v Þ 1 2v 0 0 0 0 3α þ 2β 8β2 
 5φ  6α
2 25 - 34v0 þ 22v0 ð1 - 2v0 Þð1 þ v0 Þ γ2 = 3α þ 2β 16β2 γ1
=

with γ 1 and γ 2 given in Eqs. (6.7) and (6.8).

ð6:124Þ ð6:125Þ

6.4

Pairwise Interacting Solution for Two-Phase Composites

6.4.2

327

Ensemble-Average Stress Norm in the Matrix

The total stress at any point x in the matrix is given by the superposition of the far-field stress σ0 and the perturbed stress σ′ induced by the filler particles. Assuming the elastic Eigen strain ε
in the filler particle is uniform, the perturbed stress at matrix point x takes the form (Ju & Chen, 1994a)    σ0 ðxjx1 Þ = C0  Gðx - x1 Þ : εT þ ε


ð6:126Þ

where εT is the Eigen strain caused by the CTE mismatch between the matrix and the filler particle. Using the ensemble-volume averaged Eigen strain given in Eq. (6.121), the stress perturbation can be given by the following relation:    σ0 ðxjx1 Þ = C0  Gðx - x1 Þ  Γ : εT þ ε
0

ð6:127Þ

where ε
0 denotes the solution of the Eigen strain for the single inclusion problem, which is given by Eq. (6.42). Therefore, h i σ0 ðxjx1 Þ = - C0  G  ΓðA þ SÞ - 1 i h : ε0 þ C0  G  ΓðA þ SÞ - 1 ðA þ IÞ : εT

ð6:128Þ

Following the same procedure used for the noninteracting solution, we obtain the ensemble-averaged current stress norm at any matrix point x as: hH im ðxÞ = σ0 : T : σ0 þ σT : T
: σT - 2σ0 : T
: σT

ð6:129Þ

where σT is given by Eq. (6.56), and the components of the positive definite fourthrank tensors T and T
read:   T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk   T
ijkl = T
1 δij δkl þ T
2 δik δjl þ δil δjk

ð6:130Þ ð6:131Þ

with 3T 1 þ 2T 2 = 200ð1 - 2v0 Þ2 T2 =

ð3γ 1 þ 2γ 2 Þ2 φ ð3α þ 2βÞ2

 4γ 2 1  þ 23 - 50v0 þ 35v20 22 φ 2 β

ð6:132Þ ð6:133Þ

328

6

Thermomechanical Analysis of Particle-Filled Composites

3T
1 þ 2T
2 = 200ð1 - 2v0 Þ2

ð3γ 1 þ 2γ 2 Þ2 φ ð3α þ 2βÞ2

ð6:134Þ

  4γ 2 T
2 = 23 - 50v0 þ 35v20 22 φ β

ð6:135Þ

In addition, α and β are given in Eqs. (6.5) and (6.6). The ensemble-averaged current stress norm at a matrix point can also be expressed in terms of the macroscopic stress σ. Following the same procedure as in the former section, the relation between the far-field stress σ0 and the macroscopic stress σ takes the form σ = P : σ0 - Q : σT

ð6:136Þ

P = I þ φðI - SÞΓðA þ SÞ - 1

ð6:137Þ

Q = φðI - SÞΓðA þ SÞ - 1

ð6:138Þ

where

where the components of P and Q are given:   Pijkl = P1 δij δkl þ P2 δik δjl þ δil δjk   Qijkl = Q1 δij δkl þ Q2 δik δjl þ δil δjk

ð6:139Þ ð6:140Þ

with 3P1 þ 2P2 = aφ þ 1

ð6:141Þ

1 ðbφ þ 1Þ 2

ð6:142Þ

3Q1 þ 2Q2 = aφ

ð6:143Þ

1 bφ 2 3γ þ 2γ 2 a = 20ð1 - 2v0 Þ 1 3α þ 2β

ð6:144Þ

P2 =

Q2 =

b = ð7 - 5v0 Þ

ð6:145Þ

2γ 2 β

ð6:146Þ

Using Eq. (6.136), we arrive at the alternative expression for the ensembleaveraged current stress norm in a matrix point x:


hH im ðxÞ = σ : T : σ þ 2σ : T : σT þ σT : T





: σT

ð6:147Þ

6.4

Pairwise Interacting Solution for Two-Phase Composites


where the positive definite fourth-rank tensors T, T , and T

329



are defined as

 T T = P-1  T  P-1  T
T = P - 1  T  P - 1  Q - T
 P - 1  T    

T = P - 1  Q  T  P - 1  Q þ T
- 2T
 P - 1  Q


where the components of T, T , and T





ð6:148Þ ð6:149Þ ð6:150Þ

are

  T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk 


 T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk 





 T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk

ð6:151Þ ð6:152Þ ð6:153Þ

with 3T 1 þ 2T 2 = T2 =


3T 1 þ 2T 2 ðaφ þ 1Þ2

T2 ðbφ þ 1Þ2




3T 1 þ 2T 2 =


T2 =



T2 1 þ ðbφ þ 1Þ2 2ðbφ þ 1Þ



3T 1 þ 2T 2 =



T2 =

3T 1 þ 2T 2 ðaφ þ 1Þ2

3T 1 þ 2T 2 ðaφ þ 1Þ2

T2 bφ - 1 þ 2 2ðbφ þ 1Þ ðbφ þ 1Þ

ð6:154Þ ð6:155Þ ð6:156Þ ð6:157Þ ð6:158Þ ð6:159Þ

σT = AC1 : εT is spherical stress in nature and caused by CTE mismatch; hence, Eq. (6.147) of the ensemble-averaged loading function can be simplified as follows:     h H i m ð xÞ = σ - σ T : T : σ - σ T

6.4.3

ð6:160Þ

Ensemble-Average Stress in the Filler Particles

The ensemble-volume averaged strain for two-phase composites can be given by

330

6

Thermomechanical Analysis of Particle-Filled Composites

  ε = ε0 þ φS : εT þ ε


ð6:161Þ

  ε = ε0 þ φSΓ : εT þ ε
0

ð6:162Þ

Substituting Eq. (6.121),

With the noninteracting solution Eigen strain ε
0 given by Eq. (6.42), we arrive at i h h i ε = I - φSΓðA þ SÞ - 1 : ε0 þ φ SΓðA þ SÞ - 1 ðA þ IÞ : εT

ð6:163Þ

The volume-averaged stress tensor for the particles is defined as σ1 =

1 V1

Z σ1 ðxÞdx =

1 V1

V1

Z

  C1 : ε0 þ ε0 ðxÞ - εT ðxÞ dx = C1 : ε0 þ ε0 - εT

V1

ð6:164Þ where ε0 can now be given by 8
a

Z

þ

 H ðxjx2 Þ - H 0 Pðx2 Þdx2

ð6:187Þ

r2 > a

where r1 = j x - x1j, r2 = j x - x2j P(x1) and P(x2) denote the probability density functions for finding a particle centered at x1 and a void centered at x2, respectively. isotroFor simplicity, P(x1) and P(x2) are assumed to be statistically homogeneous,   pic, and uniform. Using the properties of the fourth-order tensor G x - xq , we obtain the ensemble-averaged current stress norm at any matrix point x as: h H i m ð xÞ = H 0 þ þ

N2 V

N1 V Z

Z

Z dr 1

r1 > a

Z

Aðr1 Þ

dr 1 r2 > a

ðσ0 ðxjx1 Þ : Id : σ0 ðxjx1 ÞÞdA

Aðr2 Þ

ðσ0 ðxjx2 Þ : Id : σ0 ðxjx2 ÞÞdA

ð6:188Þ

334

6

Thermomechanical Analysis of Particle-Filled Composites

By carrying out the lengthy but straightforward algebra, we arrive at hH im ðxÞ = σ0 : T : σ0 þ σT : T
: σT - 2σ0 : T
: σT

ð6:189Þ

where σT = A1 C1 : εT

ð6:190Þ

The components of the positive definite fourth-rank tensor T and T
read   T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk   T
ijkl = T
1 δij δkl þ T
2 δik δjl þ δil δjk

ð6:191Þ ð6:192Þ

with 3T 1 þ 2T 2 = 200ð1 - 2v0 Þ2 T2 =

X2 q=1

1  þ 23 - 50v0 þ 35v20 2



φq

3αq þ 2βq  X 2 φq q=1

β2q

φ1 ð3α1 þ 2β1 Þ2  φ T
2 = 23 - 50v0 þ 35v20 2 β1

3T
1 þ 2T
2 = 200ð1 - 2v0 Þ2

2

ð6:193Þ ð6:194Þ ð6:195Þ ð6:196Þ

The ensemble-averaged current stress norm at a matrix point must be established in terms of the macroscopic stress σ to express the effective loading function in terms of the macroscopic stress. In the special case of uniform dispersions of identical elastic spheres in a homogeneous matrix, the macroscopic stress and the far-field stress take the following forms: h

i X2
0 T σ = C0 : ε φ ε þ ε q q q=1 q h

i X2
0 T φ S ε þ ε σ0 = C 0 : ε q q q q=1

ð6:197Þ ð6:198Þ

Using Eqs. (6.197), (6.198), and (6.185), the relation between the far-field stress σ0 and the macroscopic stress σ is given by σ = P : σ0 - Q : σT where

ð6:199Þ

6.5

Noninteracting Solution for Three-Phase Composites

P=I þ

X2

φ ðI - SÞ q=1 q



335

-1 Aq þ S

ð6:200Þ

Q = φ1 ðI - SÞðA1 þ SÞ - 1

ð6:201Þ

The components of P and Q are given by   Pijkl = P1 δij δkl þ P2 δik δjl þ δil δjk   Qijkl = Q1 δij δkl þ Q2 δik δjl þ δil δjk

ð6:202Þ ð6:203Þ

where 3P1 þ 2P2 = P2 =

X2

aφ q=1 q q

þ1

ð6:204Þ

 1 X2 b φ þ 1 q q q=1 2

ð6:205Þ

3Q1 þ 2Q2 = a1 φ1 1 b φ 2 1 1

ð6:207Þ

20ð1 - 2v0 Þ 3αq þ 2βq

ð6:208Þ

ð7 - 5v0 Þ βq

ð6:209Þ

Q2 = aq =

ð6:206Þ

bq =

Using Eqs. (6.199) and (6.189), we arrive at the alternative expression for the ensemble-averaged current stress norm in a matrix point x as:


hH im ðxÞ = σ : T : σ þ 2σ : T : σT þ σT : T


where the positive definite fourth-rank tensors T, T , and T









: σT

are defined as

 T T = P-1  T  P-1  T
T = P - 1  T  P - 1  Q - T
 P - 1  T    

T = P - 1  Q  T  P - 1  Q þ T
- 2T
 P - 1  Q






where the components of T ,T , and T

are

ð6:210Þ

ð6:211Þ ð6:212Þ ð6:213Þ

336

6

Thermomechanical Analysis of Particle-Filled Composites

  T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk 


 T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk 





 T ijkl = T 1 δij δkl þ T 2 δik δjl þ δil δjk

ð6:214Þ ð6:215Þ ð6:216Þ

with 3T þ 2T 2 3T 1 þ 2T 2 = P 1 2 2 a φ þ 1 q=1 q q T 2 = P 2

T2

q = 1 bq φq þ 1

2

3T þ 2T 2

3T 1 þ 2T 2 = - P 1 2 2 q = 1 a q φq þ 1


1  2 þ P2 b φ þ 1 2 q q q=1 q = 1 b q φq þ 1

T 2 = - P 2

T2

3T þ 2T 2



3T 1 þ 2T 2 = P 1 2 2 q = 1 a q φq þ 1

T2

P2 q = 1 bq φ q - 1  = P 2 þ P2 2 2 b φ þ 1 q b φ þ 1 q q=1 q=1 q q T2

ð6:217Þ

ð6:218Þ

ð6:219Þ

ð6:220Þ

ð6:221Þ

ð6:222Þ

Because σT = AC1 : εT is spherical stress caused by the CTE mismatch, Eq. (6.147) of the ensemble-averaged loading function can be simplified as     h H i m ð xÞ = σ - σ T : T : σ - σ T

6.5.3

ð6:223Þ

Ensemble-Average Stress in Filler Particles

The ensemble-volume averaged strain for three-phase composites takes the following form:

6.5

Noninteracting Solution for Three-Phase Composites

ε = ε0 þ


0 T φ S ε þ ε q q q=1 q

X2

337

ð6:224Þ

With the noninteracting solution Eigen strain ε
0 given by Eq. (6.185), we can write h X2   - 1i 0 φ S A þ S ε= I: ε þ φ1 SðA1 þ SÞ - 1 ðA1 þ IÞ : εT1 q q q=1

ð6:225Þ

The volume-averaged stress tensor for the filler particles is defined as σ1 =

1 V1

Z σ1 ðxÞdx = V1

1 V1

Z

 C1 : ε0 þ ε0 ðxÞ - εT ðxÞ dx = C1

V1

 : ε0 þ ε01 - εT1

ð6:226Þ

where ε1 0 can be recast as ε′1 =S :

8
>
> =

2ð1 - 2v0 Þ 10ð1 þ v0 Þφ1 k 1   3k1 þ φ P 3ð 1 - v 0 Þ > > q > > ; :ð3α1 þ 2β1 Þ 1 - 10ð1 þ v0 Þ 2q = 1 ðk 1 - k 0 Þ 3αq þ 2βq ð6:236Þ V2 = 2

β1 - 2ð4 - 5v0 Þ β1

3

2ð4 - 5v0 Þφ1 μ1 7 - 5v0 5 þ 4 h μ P2 φ q i 15 ð 1 - v0 Þ 1 β1 1 - 2ð4 - 5v0 Þ q = 1 β ðμ1 - μ0 Þ

ð6:237Þ

q

6.6

Effective Thermomechanical Properties

In particle-filled composites, there is a thin layer of interfacial layer (interphase) between a filler particle and the matrix. The imperfect interface bond may be due to a very compliant thin interfacial layer that is assumed to have perfect boundary conditions with the matrix and the filler particle. This defines a three-phase composite that includes the filler particle, the thin interphase, and the matrix as shown in Fig. 6.2. Once the effective mechanical and thermal properties of the inner composite sphere assemblage (CSA), which consists of the filler particle and interphase layer of thickness δ, are computed, the composite models that are used for the perfect interface composite model can be applied to this case of composites with imperfect interface conditions.

6.6

Effective Thermomechanical Properties

339

Fig. 6.2 Three-phase composite system. Note: filler particle is polycrystalline

Fig. 6.3 Schematic illustration of composite spherical assemblage (CSA)

The composite sphere assemblage (CSA) model was first proposed by Kerner (1956) and Van der Poel (1958) as shown in Fig. 6.3. Smith (1974, 1975), Christensen and Lo (1979), and Hashin and his co-workers (1962, 1963, 1968, 1990, 1991a, b) improved on the CSA model. The CSA model assumes that the filler particles are spherical and that the action on the particle is transmitted through a spherical interphase shell. The overall macro-behavior is assumed isotropic and is thus characterized by two effective moduli: the bulk modulus k and the shear modulus μ. In this section, we summarize the theoretical solution for effective thermomechanical properties of the CSA model consisting of an elastic spherical filler particle and an elastic interphase layer with a thickness of δ. In the following formulas, k represents the bulk modulus, μ represents the shear modulus, and α represents the coefficient of thermal expansion. The subscripts i, f, and m refer to the interphase, filler particle, and matrix, respectively.

6.6.1

Effective Bulk Modulus

The effective bulk modulus k
for the CSA model as obtained by Hashin (1962) is given below: k
= ki þ

where

1 kf - ki

φ - φÞ þ 33kð1i þ4μ i

ð6:238Þ

340

6

Thermomechanical Analysis of Particle-Filled Composites

φ=



r rþδ

3

ð6:239Þ

r is the radius of the filler particle δ is the thickness of interphase Hashin and Shtrikman’s formulas (Hashin & Shtrikman, 1963) and Walpole’s formulas (1966) provide upper and lower bounds for the bulk modulus of the CSA model. Equation (6.238) is the same as the highest lower bound value.

6.6.2

Effective Coefficient of Thermal Expansion (ECTE)

The effective coefficient of thermal expansion α
for the CSA model as obtained by Levin (1967) is given as   αf - αi 1 1 α = αi þ 1 1 k
ki kf - ki


6.6.3

ð6:240Þ

Effective Shear Modulus

Based on the generalized self-consistent scheme (GSCS) model proposed by Hashin (1962), Christensen and Lo (1979) have given the condition for determining the effective shear modulus of the CSA model as follows: A


2 
 μ μ þ D=0 þB μi μi

ð6:241Þ

where     A = 8 μf =μi - 1 ð4 - 5vi Þη1 φ10=3 - 2 63 μf =μi - 1 η2 þ 2η1 η3 φ7=3    2 þ252½μf =μi - 1η2 φ5=3 - 50 μf =μi - 1 7 - 12vi þ 8vi η2 φ þ 4ð7 - 10vi Þη2 η3 ð6:242Þ B = - 4 μf =μi - 1 ð1 - 5vi Þη1 φ10=3 þ 4 63 μf =μi - 1 η2 þ 2η1 η3 φ7=3 ð6:243Þ  - 504½μf =μi - 1η2 φ5=3 þ150 μf =μi - 1 ð3 - vi Þvi η2 φ þ 3ð15vi - 7Þη2 η3 









6.6

Effective Thermomechanical Properties

    D = 4 μf =μi - 1 ð5vi - 7Þη1 φ10=3 - 2 63 μf =μi - 1 η2 þ 2η1 η3 φ7=3   2  þ252½μf =μi - 1η2 φ5=3 þ25 μf =μi - 1 vi - 7 η2 φ - ð7 þ 5vi Þη2 η3

341

ð6:244Þ

with         η1 = μf =μi - 1 49 - 50vf vi þ 35 μf =μi vf - 2vi þ 35 2vf - vi    η2 = 5vf μf =μi - 8 þ 7 μf =μi þ 4   η3 = μf =μi ð8 - 10vi Þ þ ð7 - 5vi Þ

ð6:245Þ ð6:246Þ ð6:247Þ

φ is given in Eq. (6.239). Based on Van der Poel’s formula for the shear modulus of a particulate composite, Smith (1974, 1975) provided the condition for determining the effective shear modulus of the CSA model as follows: 
2 
 μ μ -1 þ β -1 þ γ=0 α μi μi

ð6:248Þ

where h i α = 4Pð7 - 10vi Þ - Sφ7=3 ½Q - ð8 - 10vi ÞðM - 1Þφ

2 - 126PðM - 1Þφ 1 - φ2=3

ð6:249Þ

β = 35ð1 - vi ÞP½Q - ð8 - 10vi ÞðM - 1Þφ - 15ð1 - vi Þ h i  4Pð7 - 10vi Þ - Sφ7=3 ðM - 1Þφ

ð6:250Þ

γ = - 525Pð1 - vi Þ2 ðM - 1Þφ

ð6:251Þ

M = μf =μi    P = 7 þ 5vf M þ 4 7 - 10vf

ð6:252Þ

with 

Q = ð8 - 10vi ÞM þ ð7 - 5vi Þ   S = 35 7 þ 5vf M ð1 - vi Þ - Pð7 þ 5vi Þ

ð6:253Þ ð6:254Þ ð6:255Þ

φ is given in Eq. (6.239). Equations (6.241) and (6.248) are verified as equivalent in the determination of the effective shear modulus of the CSA model. Solving the above equations, one can determine the exact solution for the effective shear modulus of the CSA model. One of the roots is negative and extraneous. The positive root provides the value of the effective shear modulus.

342

6.6.4

6

Thermomechanical Analysis of Particle-Filled Composites

Effective Young’s Modulus and Effective Poisson’s Ratio

Effective Young’s modulus and Poisson’s ratio for the CSA model can be calculated from the well-known expressions: 9k
μ
3k
þ μ


ð6:256Þ

3k
- 2μ
6k
þ 2μ


ð6:257Þ

E
= v
=

6.6.5

Numerical Examples

To illustrate the effect of imperfect interface and interphase thickness on the overall effective mechanical and thermal properties, we consider a special case of CSA that consists of spherical alumina trihydrate (ATH) particle and interphase with the following properties: For particles: E f = 70 GPa vf = 0:24 αf = 13 × 10 - 6 = ° C For interphase material: vi = 0:31 αi = 70 × 10 - 6 = ° C δ=r = 0:01 and δ=r = 0:001 The nondimensional interface parameter is defined as q = Ef/Ei. A zero value of q implies that there is no abrupt jump in displacement at the interface and the perfectly bonded interface condition exists. At the other extreme, the infinite value of q implies that the interface tractions do not exist, and the filler particle is completely debonded from the matrix. Finite positive values for q define an imperfect interface, which lies between two extreme cases mentioned above (Fig. 6.4). Figures 6.5, 6.6, and 6.7 show the effect of the interphase bond modulus and thickness on the effective shear modulus, Young’s modulus, and bulk modulus of the CSA. The stiffness of the bond has a strong effect on the degradation of the shear

6.6

Effective Thermomechanical Properties

343

15

14.5

δ/ r = 0.01

14 α* 13.5

δ/r = 0.001

13

12.5

12 100

102 104 Interface Parameter q

106

Fig. 6.4 Variation of effective CTE with the interface parameter, q, Nie (2005)

1

0.8 δ/r = 0.01

0.6 m*/mf δ/ r = 0.001 0.4

0.2

0 100

104 102 Interface Parameter q

106

Fig. 6.5 Variation of effective shear modulus with the interface parameter q, Nie (2005)

344

6

Thermomechanical Analysis of Particle-Filled Composites

1

0.8

δ/r = 0.01

0.6 E*/Ef 0.4

δ/ r = 0.001

0.2

0 100

102 104 Interface Parameter q

106

Fig. 6.6 Variation of effective Young’s modulus with the interface parameter q, Nie (2005)

1

0.8 δ/r = 0.01

0.6 k*/kf

δ/ r = 0.001

0.4

0.2

0 100

102 104 Interface Parameter q

106

Fig. 6.7 Variation of effective bulk modulus with the interface parameter q, Nie (2005)

6.7

Micromechanical Constitutive Model of the Particulate Composite

345

0.4 0.35 0.3 ν*

0.25

δ/r = 0.01

0.2 0.15

δ/r = 0.001

0.1 0.05 0 100

102 104 Interface Parameter q

106

Fig. 6.8 Variation of effective Poisson’s ratio with the interface parameter q, Nie (2005)

modulus, Young’s modulus, and bulk modulus. As the interface becomes thinner for the same Ei value, the effective shear modulus increases. On the other hand, for the same interphase thickness, as the elastic modulus value Ei of the interphase decreases, the effective shear modulus decreases. This is also true for both the effective bulk modulus and effective Young’s modulus. Figure 6.8 shows the effect of the interphase bond modulus and thickness on effective Poisson’s ratio. Effective Poisson’s ratio seems to be independent of the interphase stiffness when q ≤ 102 and q ≥ 104. Analytical and numerical evaluation of the effective elastic properties and the thermal expansion coefficients of the composite sphere assemblage show that the bulk and shear moduli are insensitive to the value of Poisson’s ratio of interphase. Numerical results show that the nature of the interphase has a significant effect on the stiffness and the thermal expansion coefficient of the CSA.

6.7

Micromechanical Constitutive Model of the Particulate Composite

The nature of the bond between particles and the matrix has a significant effect on the mechanical behavior of particulate composites. Most analytical and numerical models assume that the bond between the filler and matrix is perfect and can be modeled using the continuity of tractions and displacements across the interface.

346

6

Thermomechanical Analysis of Particle-Filled Composites

However, internal defects and imperfect interfaces are well-known to exist in composites, and the incorporation of such phenomena into the general theory requires modification and relaxation of the continuity of displacements between the constituents. The imperfect interface bond may be due to the compliant interfacial layer or interface damage, which may have been created deliberately by coating the particles with a debonding agent. It may also develop during the manufacturing process due to chemical reactions between the particles and the matrix or due to interface damage from cyclic loading. Most importantly, the strength of the bond at the interface controls the mechanical response and fatigue life of the composite (Basaran & Nie, 2004). By controlling the stiffness of the interphase, it is possible to control the overall behavior of the composite. In the following section, a micromechanical model for particulate composites with the imperfect interface between the filler particles and the matrix and also a CTE mismatch between the filler particle and matrix is presented, based on the work of Ju and Chen (1994a, b) and Ju and Tseng (1996), for perfectly bonded particlematrix interfaces with no CTE mismatch between the filler particles and the matrix. In the micromechanical model, particulate composites are treated as three-phase composites consisting of an agglomerate of filler particles, the bulk matrix, and the interfacial transition zone around the agglomerate as shown in Fig. 6.2. The compliant interfacial transition zone is assumed to have perfect bonding with matrix and filler particles. The inner composite sphere assemblage (CSA), consisting of the particle and the interfacial transition zone, is regarded as an equivalent spherical particle with the effective mechanical and thermal properties derived in the previous section.

6.7.1

Modeling Procedures for Particulate Composites

In this section, we discuss the modeling procedures for a particulate composite, including how the interface properties, CTE mismatch between the particle and the matrix, and an isotropic damage parameter are introduced. The modeling process consists of four steps. The first step is the simplification of the actual particulate composite, which is shown in Fig. 6.9. A is a representation of the actual microstructure for a particulate composite, where filler particles have different sizes and shapes. To get an analytical expression for the effective behavior of the particulate composite, we must make some assumptions. For simplicity, we assume that the particles are uniform in shape and spherical. The interfacial layer is used to model the imperfect interface condition. The displacement jumps at the interface are defined by the deformation of the interphase. The particulate composite is defined as a three-phase system that includes particles, the thin interphase, and the matrix. The simplified microstructure of a particulate composite is shown in Fig. 6.9. The simplified microstructure for the particulate composites includes the particle, the interphase, and the matrix. For the sake of simplicity, the particle and the interphase are regarded as one, namely, a particle-interphase assemblage. The

6.7

Micromechanical Constitutive Model of the Particulate Composite

347

Fig. 6.9 Modeling procedures—step 1: simplification, Nie (2005)

Fig. 6.10 Modeling procedures—step 2: equivalence, Nie (2005)

microstructure of composite is now simplified as C shown in Fig. 6.10. The microstructure of C includes the particle-interphase assemblage and the matrix. The thermomechanical properties are calculated for the new two-phase particleinterphase assemblage. Finding the effective thermomechanical properties for the two-phase (Fig. 6.11) spherical particle-interphase assemblage is essential. The particulate composite is simplified as a two-phase system with perfect interfacial bonding between the particle-interphase assemblage and the matrix. This final modeling step is shown in Fig. 6.12.

348

6

Thermomechanical Analysis of Particle-Filled Composites

Fig. 6.11 Modeling procedures—step 3: thermomechanical properties, Nie (2005) Fig. 6.12 Modeling procedures—step 4: micromechanics, Nie (2005)

Equivalent Continuum Medium

C

6.7.2

Elastic Properties of Particulate Composites

Based on Ju and Chen’s (1994a, b) solution for noninteracting particles, the effective properties of a two-phase composite with imperfect bonding can be obtained by substituting the properties of the particle with the properties of composite sphere assemblage: ) 3ð1 - vm Þðk
- k m Þφf   1þ 3ð1 - vm Þkm þ 1 - φf ð1 þ vm Þðk
- km Þ

( k = km ( μ = μm

15ð1 - vm Þðμ
- μm Þφf   1þ 15ð1 - vm Þμm þ 1 - φf ð8 - 10vm Þðμ
- μm Þ

ð6:258Þ ) ð6:259Þ

Similarly, the pairwise interacting solution for the effective properties of the two-phase composite with the imperfect bond can be given by

6.7

Micromechanical Constitutive Model of the Particulate Composite


k = km

30ð1 - vm Þð3γ 1 þ 2γ 2 Þφf 1þ 3β1 þ 2β2 - 10ð1 þ vm Þð3γ 1 þ 2γ 2 Þφf
 30ð1 - vm Þγ 2 φf μ = μm 1 þ β2 - 4ð4 - 5vm Þγ 2 φf

349

 ð6:260Þ ð6:261Þ

where 

μ k β1 = 2ð5vm - 1Þ þ 10ð1 - vm Þ
m -
m k - k m μ - μm β2 = 2ð4 - 5vm Þ þ 15ð1 - vm Þ



μm μ
- μm

ð6:262Þ ð6:263Þ

and γ1 =

5φf 8β2 2




 8β1 ð13 - 14vm Þvm ð1 - 2vm Þð1 þ vm Þ 3β1 þ 2β2

ð6:264Þ


   5φf 6β1 1 2 25 - 34vm þ 22vm ð1 - 2vm Þð1 þ vm Þ ð6:265Þ γ2 = þ 3β1 þ 2β2 2 16β2 2 where k
and μ
are the effective bulk modulus and shear modulus of the composite spherical-filler assemblage and φf is the filler particle volume fraction. Example Nie (2005) studied lightly cross-linked poly-methyl methacrylate (PMMA) filled with alumina trihydrate (ATH). The following phase properties were used for the numerical analysis: Particle: Ef = 70GPa, vf = 0.24 Matrix: Em = 3.5GPa, vm = 0.31 The particle volume fraction is φf = 0.48 Nondimensional interface parameter is defined as q = Ef/Ei. Three kinds of composites with different interfacial properties were prepared in the lab. Composite A has the strongest interfacial adhesion due to the addition of a special adhesion-promoting additive, where the value of the nondimensional interface parameter q = Ef/Ei can be assumed one. The interfacial adhesion strength of composite C is the weakest due to the addition of a debonding promoting additive. The value of the nondimensional interface parameter q for composite C can be assumed to be a very large number. The interfacial strength of composite B is moderate. The value of nondimensional interface parameter q for composite B can be assumed to be some value between that for composite A and composite C.

350

6

Effective Young’s Modulus (Gpa)

12

Thermomechanical Analysis of Particle-Filled Composites

Composite A [Test Data] Composite B [Test Data]

10

Pairwise Interacting Solution

8

6

4 Noninteracting Solution Composite C [Test Data]

2

0 100

101

102 103 104 Interface Parameter q

105

106

Fig. 6.13 Effective Young’s modulus vs. interface parameter at a volume fraction of 48% (the thickness of interphase is 1/10 of the diameter of particles), Nie (2005)

The effects of the interphase properties on effective Young’s modulus of the two-phase composite are shown in Figs. 6.13, 6.14, and 6.15, where the elastic properties of composites with different interphase properties obtained from test data were also plotted for comparison. These results indicate that the pairwise interacting solution yields a better approximation of the overall elastic modulus. Figures 6.13, 6.14, and 6.15 indicate that as the thickness of the interphase gets larger, the effective elastic modulus value decreases for the same q value.

6.7.3

A Viscoplasticity Model

Viscoplastic flow occurs only in the matrix because filler particles have a very high melting temperature, and they are brittle and deform linear elastic. The magnitude of the current equivalent stress norm of the matrix can be used to determine viscoplastic behavior. When the ensemble-volume average stress norm in the matrix reaches a certain level, the effective yield function for the composite can be given by f ðσ, αÞ =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    σ - σ T : T : σ - σT -

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 þ 2T 2 σ y ðαÞ

ð6:266Þ

6.7

Micromechanical Constitutive Model of the Particulate Composite

351

12 Effective Young’s Modulus (Gpa)

Composite A [Test Data] Composite B [Test Data]

10 Pairwise Interacting Solution

8

6 Noninteracting Solution

4 Composite C [Test Data]

2 0 100

101

102 103 104 Interface Parameter q

106

105

Fig. 6.14 Effective Young’s modulus vs. interface parameter at a volume fraction of 48% (the thickness of interphase is 1/100 of the diameter of particles), Nie (2005) 12

Composite A [Test Data]

Effective Young’s Modulus (Gpa)

Composite B [Test Data]

10 Pairwise Interacting Solution

8 6

Noninteracting Solution

4 Composite C [Test Data]

2 0 100

101

102 103 104 Interface Parameter q

105

106

Fig. 6.15 Effective Young’s modulus vs. interface parameter at a volume fraction of 48% (the thickness of interphase is 1/1000 of the diameter of particles), Nie (2005)

352

6

Thermomechanical Analysis of Particle-Filled Composites

where σ is the average stress in the particulate composite; σT is the stress caused by the CTE mismatch between the matrix and the particle which was derived earlier in Eq. (6.190); T is the fourth-order tensor which is given by Eq. (6.211); T1 and T2 are given by Eqs. (6.217) and (6.218), respectively; σ y ðαÞ is the isotropic hardening function of the composite materials; and α is the equivalent viscoplastic strain that defines the isotropic hardening of the composite. Its rate is given by α_ =

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vp vp -1 T 1 þ 2T 2 ε_ : T : ε_

ð6:267Þ

The factors in the effective yield and the effective plastic strain increment equations are chosen so that the effective stress and effective plastic strain increments are equal to the uniaxial stress and uniaxial plastic strain increment in a uniaxial monotonic tensile test. It should be noted that the effective yield function is spherical pressure-dependent now and not of the von Mises type anymore. So the particles and hydrostatic pressure have a significant effect on the viscoplastic behavior of the matrix. Unified mechanics theory provides a basic framework to introduce degradation evolution intrinsically without an empirical evolution function. According to the strain equivalence principle, the effective yield stress function can be given by r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ σ - σT : T : - σT - T 1 þ 2T 2 σ y ðαÞ f ðσ, αÞ = 1-Φ 1-Φ

ð6:268Þ

The degradation increases the equivalent stress norm of the composite, which tends to amplify the viscoplastic flow. If the kinematic hardening behavior is included, let β define the center of the yield surface  of the composite in the stress space, and the relative stress can be defined as σ - β . Assume the filler particles have the same effect on the stress norm of the matrix as on the kinematic behavior of the matrix; the stress norm defining the viscoplastic behavior of the matrix can be updated as     hH im ðxÞ = σ - β - σT : T : σ - β - σT

ð6:269Þ

Therefore, the effective yield function for the composite can be given as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



ffi σ σ T T -β-σ : T : f ðσ, qÞ = -β-σ 1-Φ 1-Φ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  - T 1 þ 2T 2 σ y þ K ðαÞ

 where q = α, β is chosen as the internal viscoplastic variable.

ð6:270Þ

6.7

Micromechanical Constitutive Model of the Particulate Composite

353

For simplicity, the Perzyna-type viscoplasticity model is employed to characterize the rate (viscosity) in the matrix. Therefore, the effective ensemble-volume average viscoplastic strain rate for the composite can be expressed as ε_ vp = γ

∂f 1 = γn ∂σ 1 - Φ

ð6:271Þ

where   T : 1 -σ Φ - β - σT n = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    T σ : T : 1 -σ Φ - β - σT 1-Φ - β - σ

ð6:272Þ

and γ denotes the plastic consistency parameter γ=

hf i hf i = η 2μτ

ð6:273Þ

where η is the viscosity coefficient and τ is called the relaxation time. And the effective equivalent viscoplastic strain rate is defined as α_ =

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ  T 1 þ 2T 2 1-Φ

ð6:274Þ

The evolution of back-stress, β , depends on the plastic strain or plastic work history. The earliest model for the evolution of back-stress is due to Prager (1956) and was subsequently modified by Ziegler (1959). The Prager kinematic hardening assumption is that during yielding the back-stress increment dβ is equal to the component of dσ in the direction normal to the yield surface. Since the plastic strain increment is also normal to the yield surface, the increment dβ can be written as   dεp dεp dβ = dσ  p jdε j jdεp j

ð6:275Þ

In the case of linear kinematic hardening, we have 

1:5 T 1 þ 2T 2 H 0α_ n dβ =  2 2 T 1 þ 2T 2 þ 2T 1

ð6:276Þ

where H′ is the kinematic hardening modulus. The factor in Eq. (6.276) is chosen so that the effective stress and effective viscoplastic strain increments are equal to the uniaxial stress and uniaxial plastic strain increment in a uniaxial monotonic ensile test.

354

6.7.4

6

Thermomechanical Analysis of Particle-Filled Composites

Thermodynamic State Index

The thermodynamic state index (TSI) is given by

ms Φ = Φcr 1 - e - R

Δs



ð6:277Þ

where Φcr is the critical value of the TSI; however, this term can also be used as a function to map elastic modulus degradation, onto the TSI axis [however, using Φcr is not necessary, and can be taken as one, especially if all entropy generation mechanisms are included in the fundamental equation. In the other examples in this book, it is taken as 1], and R is gas constant, ms is molar mass, and Δs is the change in entropy and must be calculated from the thermodynamic fundamental equation at each Gauss integration point. For thermomechanical analysis of particulate composites, the following thermodynamic fundamental equation represents the majority, but not all, of the entropy generation mechanisms: Z Δs = t0

t

σ : ε_ dt þ Tρ vp

t0 

Z t

 Z t k r 2 jgradT j dt þ dt 2 T T ρ t0

ð6:278Þ

where ρ is the mass density, T is temperature, k is the thermal conductivity of the composite, and r is the distributed internal heat generation rate. Unfortunately, Eq. (6.278) ignores entropy generation due to aging in PMMA and other longterm chemical reactions in the PMMA molecular chains, which are not trivial.

6.7.5

Solution Algorithm

A general return mapping algorithm is used to solve Eqs. (6.270)–(6.278). The general return mapping algorithm was proposed by Simo and Taylor (1985) and summarized in detail by Simo and Hughes (1998). To minimize confusion, the symbol Δ is used to denote an increment over a time step or an increment between successive iterations. For the rate-of-slip γ, we adhere to the following conventions: Δγ = γΔt denotes the increment of γ over a time step Δt, and Δ2γ denotes the increment of Δγ between iterations. Let C be the elastic consistent tangent moduli of the particulate composite; ignoring the derivative of TSI with respect to displacement for the sake of simplicity, the overall stress-strain relationship for small strain formulation can be given by   th σnþ1 = ð1 - ΦÞC : εnþ1 - εvp nþ1 - εnþ1

ð6:279Þ

6.7

Micromechanical Constitutive Model of the Particulate Composite

355

Since εn+1 and εth nþ1 are fixed during the return mapping stage, it follows that Δεvp = -

1 C - 1 Δσ 1-Φ

ð6:280Þ

From Eq. (6.277), we have

 _ = Φcr ms exp - ms Δs Δ_s Φ R R

ð6:281Þ

Assuming that irreversible entropy generation is only due to plastic work and heat generation, from Eq. (6.278) for an isothermal process at each increment, we can write r σ : ε_ þ Tρ T vp

Δ_s =

ð6:282Þ

If the heat generated within the system is negligible, the distributed internal heat source of strength per unit mass is zero [which is not true, especially if the rate of loading is high]. Using Eqs. (6.280) and (6.282), Eq. (6.281) becomes ΔΦ = -

 1 Φcr ms m exp - s Δs σC - 1 Δσ 1 - Φ TρR R

ð6:283Þ

We should point out that in Eq. (6.282) we ignored many prominent irreversible entropy generation mechanisms, such as the relative motion between the filler particles and the matrix, the entropy generation in the filler polycrystals, and aging in PMMA molecular chains. Moreover, if the strain rate of the loading is very high, there will be additional entropy generation mechanisms. However, here the goal is to demonstrate the process as simple as possible. From Eqs. (6.271), (6.274), and (6.276), we also have 1 Δγ n 1-Φ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 þ 2T 2 Δγ αnþ1 = αn þ 1-Φ  1:5 T 1 þ 2T 2 βnþ1 = βn þ  ΔHn 2 2 T 1 þ 2T 2 þ 2T 1 vp εvp nþ1 = εn þ

ð6:284Þ ð6:285Þ ð6:286Þ

Let pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 = T 1 þ 2T 2  1:5 T 1 þ 2T 2 m2 =  2 2 T 1 þ 2T 2 þ 2T 1

ð6:287Þ ð6:288Þ

356

6

Thermomechanical Analysis of Particle-Filled Composites

m3 = -

 m Φcr ms exp - s Δs TρR R

ð6:289Þ

∂f 1 = n ∂σ 1 - Φ

ð6:290Þ

Also, we have

2

∂ f 1 = N 2 ∂σ ð1 - ΦÞ2

ð6:291Þ

∂f = -n ∂β

ð6:292Þ

2

∂ f ∂β

where K 0 =

∂σ y ∂α

2

=N

ð6:293Þ

∂f n:σ = ∂D ð1 - ΦÞ2

ð6:294Þ

∂f = - m1 K 0 ∂α

ð6:295Þ

∂n = -N ∂β

ð6:296Þ

∂n 1 = N ∂σ 1 - Φ

ð6:297Þ

∂n 1 N:σ = ∂D ð1 - ΦÞ2

ð6:298Þ

is the isotropic hardening modulus. 

N  T-n n N = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   σ ffi T σ : T 1 - Φ - β - σT 1-Φ - β - σ

ð6:299Þ

We define the following residual functions as follows: Rα = - αnþ1 þ αn þ

m1 Δγ 1-Φ

2μτΔγ - hf i Δt 1 vp Rε = - εvp nþ1 þ εn þ 1 - Φ Δγn Rγ =

Rβ = - βnþ1 þ βn þ m2 ΔHn

ð6:300Þ ð6:301Þ ð6:302Þ ð6:303Þ

6.7

Micromechanical Constitutive Model of the Particulate Composite

357

Linearizing these residual functions yields m1 m1 m3 ΔγσC - 1 Δσ = 0 Δ2 γ þ 3 1-Φ ð1 - ΦÞ  2 0  m1 K τ 0 0Rα Δ2 γ n Δσ - nΔβ = Rf þ m1 K þ þ 2μ 1-Φ Δt

Rα - Δαnþ1 þ

P : Δσ - Q : Δβ þ R : Δσ - S : Δβ þ

Δ2 γ n þ Rε = 0 1-Φ

ð6:304Þ ð6:305Þ ð6:306Þ

m1 m2 0Δ2 H γn þ m2 H 0 Rα n þ Rβ = 0 1-Φ

ð6:307Þ

where # "

 O  Δγ Δγm3 Nσ 1 -1 -1 Nþ þn σC P= C þ 1-Φ 1-Φ ð1 - ΦÞ2 1 - Φ Q=

Δγ N 1-Φ

ð6:309Þ

" # O  m2 m3 -1 0 ðΔHNσ þ m1 H ΔγnÞ σC R= ΔHN þ 1-Φ ð1 - ΦÞ2 S = I þ m2 ΔHN

"

n0 =

ð6:308Þ

  1 m3  nσ - m1 2 K 0 Δγ σC - 1 nþ 2 1-Φ ð1 - ΦÞ

#

ð6:310Þ ð6:311Þ ð6:312Þ

Combining Eqs. (6.306) and (6.307) yields h

P R

Q S

in

Δσ - Δβ

o

þ

n o n o Δ2 γ n m m1 H 0 þ m H 0 RRεnþR = 0 1 2 2 α β 1-Φ

ð6:313Þ

From Eqs. (6.305) and (6.313), we have

Δ2 γ =

- Rf - m1 K 0Rα - ½ n0 m1 2 K 0 1-Φ

n A

þ 2μ Δtτ þ 1 -1 Φ ½ n0

n

Rε m2 H Rα nþRβ

n A

n

0

n m1 m2 H 0 n

o o

ð6:314Þ

where A-1 = A=

h

h~

P ~ R

P R

Q S ~ Q ~ S

i

i

ð6:315Þ ð6:316Þ

358

6

Thermomechanical Analysis of Particle-Filled Composites

  ~ = P - Q  S-1  R -1 P     ~ = - P - Q  S-1  R -1  Q  S-1 Q     ~ = - S-1  R  P - Q  S-1  R -1 R       ~ = S-1 þ S-1  R  P - Q  S-1  R -1  Q  S-1 S

ð6:317Þ ð6:318Þ ð6:319Þ ð6:320Þ

So we have n

Δσ - Δβ

o

9 8 Δ2 γ > > = < nþ Rε   1 Φ 2 = -A m1 Δ γ > ; : þ Rα m2 H 0 nþRβ > 1-Φ Δεvp = -

1 C - 1 Δσ 1-Φ

Δ2 s =

σΔεvp Tρ

ð6:321Þ

ð6:322Þ ð6:323Þ

Δsnþ1 = Δsn þ Δ2 s

ð6:324Þ

ΔΦnþ1 = - m3 ðσ nþ1 Δε Þ m1 m1 m3 Δαnþ1 = Rα þ ΔγσΔεvp Δ2 γ 1-Φ ð1 - ΦÞ2 vp

ð6:325Þ ð6:326Þ

Then, the viscoplastic strain, consistency parameter, stress, TSI and entropy production rate, and effective equivalent viscoplastic strain are updated at each Gauss integration point and iterated until the norms of residual functions (Eqs. 6.300–6.303) are smaller than a predefined tolerance. Normally, the convergence is achieved when the norms of these residual functions are all equal to or less than 1 × 10-5. The value of the tolerance is problem-dependent. The procedure summarized above is simply a systematic application of Newton’s method to the system of Eqs. (6.270)–(6.278) that results in the computation of the closest point projection from the trial state onto the yield surface. It should be noted that the general return mapping algorithm is unconditionally stable, and the convergence of the algorithms toward the final value of the state variable is obtained at a quadratic rate. Further information on the general return mapping algorithm is given in Simo and Hughes (1998) and Ortiz and Martin (1989). It is necessary to point out that the formulation presented above calculates the TSI at Gauss integration points. In the finite element formulation, it is possible to define TSI at nodal points and solve for TSI and other nodal unknowns. However, this would significantly increase the computational cost. It is much more cost-effective to calculate TSI at Gauss integration points from thermodynamic variables obtained in the previous step. Because the solution process is incremental, this simplification introduces very little error.

6.7

Micromechanical Constitutive Model of the Particulate Composite

6.7.6

359

Consistent Elastic-Viscoplastic Tangent Modulus

An important advantage of the algorithm lies in the fact that it can be exactly linearized in closed form. This leads to the notion of consistent elastic-viscoplastic tangent moduli. Let C be the consistent elastic-viscoplastic tangent moduli, and then the incremental stress-strain relationship based on the third law of the unified mechanics theory can be given by dσ = ð1 - ΦÞC : ðdε - dεvp Þ

ð6:327Þ

Differentiating Eqs. (6.277) and (6.278), we have dΦ = - m3 σdεvp

ð6:328Þ

where m3 is given by Eq. (6.289). Differentiating Eq. (6.284), we have dεvp =

Δγ Δγ dΔγ WNdσ WNdβ þ Wn 1-Φ 1-Φ ð1 - Φ Þ2

ð6:329Þ

where W-1 = I þ

O Δγm3 Nσ σ þn 2 1-Φ ð1 - ΦÞ

ð6:330Þ

Differentiating Eq. (6.285), we have dα =

m1 m3 m1 Δγσdεvp dΔγ 1-Φ ð1 - ΦÞ2

ð6:331Þ

Differentiating Eq. (6.286), we have dβ = -

O  m ΔH m2 m3 ΔH Nσ σ dεvp þ 2 Ndσ - m2 ΔHNdβ 2 1-Φ ð1 - ΦÞ

þ m2 nH 0 dα Differentiating Eq. (6.273) yields

ð6:332Þ

360

6

Thermomechanical Analysis of Particle-Filled Composites

 1 m3  nσ - m1 2 K 0 Δγ σdεvp ndσ - ndβ 2 1-Φ ð1 - ΦÞ   τ m 2K0 dΔγ = 2μ þ 1 Δt 1 - Φ

ð6:333Þ

From Eqs. (6.327) and (6.329) p : dσ - q : dβ þ

Wn dΔγ = dε 1-Φ

ð6:334Þ

where p=

 1 Δγ C-1 þ WN 1-Φ 1-Φ

ð6:335Þ

Δγ WN 1-Φ

ð6:336Þ

q=

From Eqs. (6.329), (6.331), and (6.332), we have m1 m2 H 0 WndΔγ = 0 1-Φ

ð6:337Þ

 1 Δγ m2 ΔHN þ VWN 1-Φ 1-Φ

ð6:338Þ

r : dσ - s : dβ þ where r=

Δγ VWN 1-Φ h

O i Nσ m mm þn σ V = - 1 2 32 H 0 Δγ 1-Φ ð1 - ΦÞ s = I þ m2 ΔHN þ

ð6:339Þ ð6:340Þ

Combining Eqs. (6.334) and (6.337) h

p r

q s

in

dσ - dβ

o

þ

Wn 1-Φ

n

1 m1 m2 H 0

o

dΔγ =

n

dε 0

o

ð6:341Þ

Combining Eqs. (6.329) and (6.333) yields n
dσ - n
dβ = m4 dΔγ 1-Φ where

ð6:342Þ

6.8

Verification Examples

361

n
= n m4 =

 m3 Δγ  nσ - m1 2 K 0 Δγ ðσWNÞ 3 ð1 - ΦÞ

 m3  τ m 2K0 nσ - m1 2 K 0 Δγ ðσWnÞ þ 2μ þ 1 3 Δt 1 - Φ ð1 - ΦÞ

ð6:343Þ ð6:344Þ

From Eqs. (6.341) and (6.342), we have dΔγ =

m4 þ 1 -1 Φ

n
h



1 ~ 1-Φ p


n 1-Φ

 þ ~r dε i n n
a m

Wn 0 1 m2 H Wn

o

ð6:345Þ

where a-1 = a=

h

h

p r

~ p ~r

q s ~ q ~s

i

i

ð6:346Þ ð6:347Þ

From Eqs. (6.334) and (6.337), we have ~dε dσ = p



 1 ~ þ ~r WndΔγ p 1-Φ

ð6:348Þ

From Eqs. (6.345) and (6.348), we have  1  N
 1  ~ ~ ~ þ ~r n 1-Φ p dσ 1 - Φ p þ r Wn ~ h
i n o C= =pn dε Wn m4 þ 1 -1 Φ n
a 0 m m H Wn 1 2 1-Φ

6.8 6.8.1

ð6:349Þ

Verification Examples Material Properties of ATH

ATH (particle filler) is modeled as an isotropic elastic material. The thermomechanical properties of ATH as provided by the manufacturer are as follows: Poisson’s ratio of ATH: νf = 0.24, CTE: αf = 1.47 × 10-6/ ° C Elastic modulus of ATH: Ef = 70 GPa The average diameter of a filler particle is 35 μm.

6

Fig. 6.16 Young’s modulus of PMMA as a function of temperature (Celsius) (Cheng et al., 1990a, b)

Thermomechanical Analysis of Particle-Filled Composites

Young's Modulus (GPa)

362

5 4 3 2 1 0 -50

6.8.2

0

50 100 Temperature

150

200

Properties of PMMA

The matrix poly-methyl methacrylate (PMMA) is a very common polymer, which has been extensively studied. Young’s modulus of PMMA as a function of temperature is as shown in Fig. 6.16. (Cheng et al., 1990a, b, c): Em = - 0:0234 T þ 4:124

ðGPAÞ

where T is the temperature in Celsius. Poisson’s ratio of PMMA is νm = 0.31.

6.8.3

Properties of Matrix-Filler Interphase

The interphase around the particle is also regarded as an isotropic elastic material. In addition, it is reasonable to assume that Poisson’s ratio and CTE of the interphase are the same as that of the matrix (PMMA); however, Young’s modulus of the interphase is less than that of PMMA. For simulations, the thickness of the interphase is taken as 1% of the diameter of filler particles; Young’s modulus is half of that of PMMA, except where specified differently.

6.8.4

Properties of Particulate Composites

The following thermomechanical properties of the particulate composite A were provided by the material manufacturer (Basaran & Nie, 2007). The average specific mass for the composite is ms = 85 g/mole, and the density of the composite is ρ = 1750 kg/m3 .The volume fraction of particle in the composite is φ = 0.48

6.8

Verification Examples

363

Fig. 6.17 Poisson’s ratio of composite A as a function of temperature

Poisson’s ratio

0.4 0.35 0.3 0.25 0.2 0

Fig. 6.18 Coefficient of thermal expansion (CTE) of composite A as a function of temperature

20

40 60 Temperature

80

100

120

140

1.E-04

CTE

8.E-05 6.E-05 4.E-05 2.E-05 0.E+00 0

20

40 60 80 Temperature

100

Poisson’s ratio of the composite A [strong interfacial adhesion] is a function of temperature as shown in Fig. 6.17: ν = 0:008 T þ 0:334 The coefficient of thermal expansion of composite A is given as a function of temperature as shown in Fig. 6.18. When T ≤ 90 ° C: α = 3:035 × 10 - 7 T þ 2:347 × 10 - 5 When T ≥ 90 ° C: α = 1:0992 × 10 - 6 T - 5:012 × 10 - 5

6

Fig. 6.19 Relaxation time of composite A as a function of temperature

Thermomechanical Analysis of Particle-Filled Composites

1.6 Relaxtion Time (s)

364

1.2 0.8 0.4 0 0

20

40

60

80

100

120

Temperature

Viscosity η is the ratio of the loss modulus to the angular frequency, which is determined during the forced harmonic oscillation test (Nielsen & Landel, 1994). The viscosity relaxation time τ is defined by Simo and Hughes (1998): τ=

η 2μ

where μ is the shear modulus. It is important to realize that the controlling factor in the relaxation process is the relative time t/τ. The absolute clock time t is regarded as short or long only when compared with τ. The concept of relaxation time is explained in greater detail by Simo and Hughes (1998). From dynamic mechanical testing, the viscosity relaxation time τ is determined as shown in Fig. 6.19. When T ≤ 90 ° C: τ = 1:12406 × 10 - 6 T 3 - 1:67823 × 10 - 4 T 2 þ 7:91134 × 10 - 3 T þ 7:35 × 10 - 3 When T ≥ 90 ° C: τ = - 2:6348 × 10 - 6 T 4 1:08452 × 10 - 3 T 3 - 1:64629 × 10 - 1 T 2 þ 10:9673T - 271:11 The elastic modulus of composite A can be determined from the properties of the filler particle, the matrix, and the interphase according to the micromechanical model presented earlier in this chapter. The elastic modulus of composite A is also determined by uniaxial tensile tests for strain rates and temperatures as shown in Fig. 6.20: Em = - 0:0005T 2 - 0:021T þ 13:33 þ 0:6 log ðε_ Þ The gas constant is R = 8.3145 Joule/mole/K.

ðGPaÞ

Verification Examples

Fig. 6.20 Elastic modulus of composite A as a function of temperature, Nie (2005)

365

Elastic Modulus (GPa)

6.8

12

Strain rate=1E-3

10

Strain rate=1E-4 Strain rate=1E-5

8 6 4 2 0 0

Fig. 6.21 Critical TSI of composite A as a function of temperature

50

100 Temperature

150

Parameter Φcr

80 60 40 20 0 0

20

40 60 Temperature

80

100

The critical TSI is a function of temperature for composite A is determined from the uniaxial tensile tests at various temperatures as seen in Fig. 6.21: Φcr = 0:1972 T þ 50:22

T ≤ 100 °

where T is the temperature in Celsius. Φcr is not an essential parameter. As long as all entropy generation mechanisms are included in the thermodynamic fundamental equation and no simplifications are made in derivation, if not used, nothing in the formulation changes. In this specific example [some entropy generation mechanisms, such as polymer aging and heat generation, are not included], an Φcr is determined using the statistical methods by matching the elastic modulus degradation determined from test data with the thermodynamic state index (TSI) evolution calculated from the thermodynamic fundamental equation during uniaxial tensile tests. This process allows us to map elastic modulus degradation onto TSI axis. If Φcr = 1 is used, nothing changes in the formulation or the results. In the unified mechanics theory, we only care about the dissipation along the thermodynamic state index axis, which provides the entropy generation. The system will always have many entropy generation mechanisms and degrade along the TSI axis, and the entropy generation rate eventually always goes to zero.

366

6.8.5

6

Thermomechanical Analysis of Particle-Filled Composites

Finite Element Simulation Results

The stress-strain response obtained from the simulations was compared with experimental data as shown in Figs. 6.22 and 6.23 at 24 and 75 °C, respectively. These simulation results are obtained based on the assumption that the thickness of interphase is 1% of the diameter of the particle and the elastic modulus of interphase is 50% of that of PMMA. Furthermore, the CTE mismatch effects are also included, where the temperature when residual stresses begin to build up is assumed to be 100 °C. It is seen that the degradation effects must be accounted for at relatively large strains. The comparison of the TSI from the simulations with elastic modulus degradation that was measured during experiments is given in Fig. 6.22.

6.8.6

Cyclic Stress-Strain Response

Mapping the elastic modulus degradation onto the TSI axis is not necessary if the user is not seeking the exact stress-strain data cycle to cycle from simulations. However, if it is needed to map the elastic modulus degradation onto the TSI axis, the following process can be used. However, according to UMT, only the degradation along the TSI axis is what matters not the degradation along the elastic modulus or any other material property that is a human construct.

40 Newtonian mechanics viscoplasticity model 30

Stress (Mpa)

Fig. 6.22 Comparison of stress-strain relationship among viscoplastic model based on Newtonian mechanics, viscoplastic model based on UMT, and experiments at 750 °C

UMT viscoplasticity model 20 Experimental data

10

0 0.00

0.005

0.01

Strain

0.015

0.02

6.8

Verification Examples

367

Fig. 6.23 Comparison of TSI obtained from simulations versus that measured in experiments in terms of elastic modulus degradation at 24 °C

Experimental Measurement Simulation

0.24

T.S.I

0.20 0.16 0.12 0.08 0.04 0.00 0.000

0.004 0.006 Strain

0.008

1

Displacement(mm)

Fig. 6.24 Applied displacement profile for tension-compression fatigue test with a strain amplitude of 0.6%

0.002

0.5

0 0

10

20

30

40

50

Time (s) -0.5

-1

The critical TSI Φcr is temperature-dependent and can be determined from the test data as follows: First, determine the elastic modulus degradation in cyclic tests. Second, calculate the dissipated plastic strain energy during cyclic testing, (Fig. 6.24) using the stress-strain hysteresis loop, (Fig. 6.25) and then determine the TSI. This process is also equivalent to calculating the fatigue fracture entropy of the material. From cyclic test data, Φcr can be determined as a function of temperature as follows:

6

Thermomechanical Analysis of Particle-Filled Composites

Fig. 6.25 Uniaxial stressstrain hysteresis loop from simulation vs. experimental data for cycle 5 at room temperature with a strain amplitude of 0.6%

60 Test Simulation

Stress (MPa)

368

40 20 0

-0.009

-0.006

-0.003

0.000 -20

0.003 0.006 Strain

0.009

0.003

0.009

-40 -60 -80

Fig. 6.26 Uniaxial stressstrain hysteresis loop from simulation vs. experimental data for cycle 50 at room temperature with a strain amplitude of 0.6%

Test Simulation

Stress (MPa)

60 40 20 0 -0.009

-0.006

-0.003

0.000 -20

0.006 Strain

-40 -60 -80

Φcr = 0:00545T þ 0:659 T ≤ 100 ° C Φcr (T) is not necessary, if all material properties are defined as a function of temperature and all entropy generation mechanisms are formulated properly in the thermodynamic fundamental equation, because the entropy generation is always according to the second law of thermodynamics and defined by the TSI coordinate. Other examples of thermomechanical loading applications where Φcr = 1 are given by Bin Jamal et al. (2020, 2021a, b, c, 2022), Lee and Basaran (2021a, b), and Lee et al. (2022, b, c, d) (Figs. 6.26, 6.27, 6.28, 6.29, 6.30, 6.31, and 6.32).

Verification Examples

369

Fig. 6.27 Uniaxial stressstrain hysteresis loop from simulation vs. experimental data for cycle 104 at room temperature with a strain amplitude of 0.6%

60 Test Simulation

Stress (MPa)

6.8

40 20 0

-0.010

-0.005

0.000 -20

0.005 Strain

0.010

0.003 0.006 Strain

0.009

0.003 0.006 Strain

0.009

-40 -60 -80 Fig. 6.28 Uniaxial stressstrain hysteresis loop from simulation vs. experimental data for cycle 5 at 75 °C with a strain amplitude of 0.6%

Test Simulation

Stress (MPa)

30 20 10 0 -0.009 -0.006

-0.003

0.000 -10 -20 -30 -40

Fig. 6.29 Uniaxial stressstrain hysteresis loop from simulation vs. experimental data for cycle 100 at 75 °C with a strain amplitude of 0.6%

Test Simulation

Stress (MPa)

30 20 10 0 -0.009

-0.006 -0.003

0.000 -10 -20 -30 -40

6

Thermomechanical Analysis of Particle-Filled Composites

Fig. 6.30 Uniaxial stressstrain hysteresis loop from simulation vs. experimental data for cycle 495 at 75 °C with a strain amplitude of 0.6%

30 Stress (MPa)

370

Test Simulation

20 10 0

-0.009

-0.006

-0.003

0.000 -10

0.003 0.006 Strain

0.009

-20 -30 -40 Fig. 6.31 Comparison of TSI at room temperature with a strain amplitude of 0.6%

0.3 0.25

T.S.I

0.2 0.15 0.1

Simulation Experimental Measurement

0.05 0 0

30

60

90

120

Number of Cycles

Fig. 6.32 Comparison of TSI at 75 °C with strain amplitude of 0.6%

0.4

T.S.I

0.3

0.2 Simulation

0.1

Experimental Measurement

0 0

100

200

300

400

Number of Cycles

500

600

References

371

Here, we conclude the presentation of modeling of particle-filled composites with the unified mechanics theory for thermomechanical analysis.

References Basaran, C., & Nie, S. (2004, July). An irreversible thermodynamic theory for damage mechanics of solids. International Journal of Damage Mechanics, 13(3), 205–224. #41. Basaran, C., & Nie, S. (2007). A thermodynamics based damage mechanics model for particulate composites. International Journal of Solids and Structures, 44, 1099–1114. #63. Bin Jamal, M. N., Kumar, A., Lakshmana Rao, C., & Basaran, C. (2020). Low cycle fatigue life prediction using unified mechanics theory in Ti-6Al-4V alloys. Entropy, 22, 24. Bin Jamal, N., Rao, L., & Basaran, C. (2021a). A semi-infinite edge dislocation model for the proportionality limit stress of metals under high strain rate. Computational Mechanics, 68(3), 545–565. https://doi.org/10.1007/s00466-020-01959-2 Bin Jamal, N., Rao, L., & Basaran, C. (2021b). A unified mechanics theory-based model for temperature and strain rate dependent proportionality limit stress of mild steel. Mechanics of Materials, 155, 103762. Bin Jamal, N., Lee, H. W., Rao, L., & Basaran, C. (2021c). Dynamic equilibrium equations in unified mechanics theory. Applied Mechanics, 2(1), 63–80. Bin Jamal, N. M., Rao, L., & Basaran, C. (2022). Unified mechanics theory based flow stress model for the rate-dependent behavior of bcc metals. Materials Today Communications, 31, 103707. Cheng, W.-M., Miller, G. A., Manson, J. A., Hertzberg, R. W., & Sperling, L. H. (1990a). Mechanical behavior of poly(methyl methacrylate)–Part 1: Tensile strength and fracture toughness. Journal of Materials Science, 25, 1917–1923. Cheng, W.-M., Miller, G. A., Manson, J. A., Hertzberg, R. W., & Sperling, L. H. (1990b). Mechanical behavior of poly(methyl methacrylate)–Part 2: Temperature and frequency effects on the fatigue crack propagation behavior. Journal of Materials Science, 25, 1924–1930. Cheng, W.-M., Miller, G. A., Manson, J. A., Hertzberg, R. W., & Sperling, L. H. (1990c). Mechanical behavior of poly(methyl methacrylate)–Part 3: Activation processes for fracture mechanism. Journal of Materials Science, 25, 1931–1938. Christensen, R. M., & Lo, K. H. (1979). Solutions for effective shear properties in three-phase sphere and cylinder models. Journal of the Mechanics and Physics of Solids, 27, 315–330. Eshelby, J. D. (1957). The deformation of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London, Series A, Mathematical Science, 241(1226), 376–396. Hashin, Z. (1962). The elastic moduli of heterogeneous materials. Transaction of the ASME, Journal of Applied Mechanics, 29, 143–150. Hashin, Z. (1968). Assessment of the self consistent scheme approximation: Conductivity of particulate composites. Journal of Composite Materials, 2(3), 284–301. Hashin, Z. (1990). Thermoelastic properties and conductivity of carbon/carbon fiber composites. Mechanics of Materials, 8, 293–308. Hashin, Z. (1991a). The spherical inclusion with imperfect interface. Transaction of the ASME, Journal of Applied Mechanics, 58, 444–449. Hashin, Z. (1991b). Thermoelastic properties of particulate composites with imperfect interface. Journal of the Mechanics and Physics of Solids, 39(6), 745–762. Hashin, Z., & Shtrikman, S. (1963). A variational approach to the theory of the elastic behavior of multiphase materials. Journal of the Mechanics and Physics of Solids, 11, 127–140. Ju, J. W., & Chen, T. M. (1994a). Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities. Acta Mechanica, 103, 103–121.

372

6

Thermomechanical Analysis of Particle-Filled Composites

Ju, J. W., & Chen, T. M. (1994b). Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. Acta Mechanica, 103, 123–144. Ju, J. W., & Tseng, K. H. (1996). Effective elastoplastic behavior of two-phase ductile matrix composites: A micromechanical framework. International Journal of Solids & Structures, 33(29), 4267–4291. Ju, J. W., & Tseng, K. H. (1997). Effective elastoplastic algorithms for ductile matrix composites. Journal of Engineering Mechanics, 123(3), 260–266. Kerner, E. H. (1956). The elastic and thermoelastic properties of composite media. The Proceedings of Physical Society, 69B, 808–813. Lee, H. W., & Basaran, C. (2021a). A review of damage, void evolution and fatigue life prediction models. Metals, 11, 609. Lee, H. W., & Basaran, C. (2021b). Predicting high cycle fatigue life with unified mechanics theory. Mechanics of Materials, 104, 116, ISSN 0167-6636. Lee, H. W., Basaran, C., Egner, H., Lipski, A., Piotrowski, M., Mroziński, S., Noushad Bin Jamal, M., & Rao, C. L. (2022a). Modeling ultrasonic vibration fatigue with unified mechanics theory. International Journal of Solids and Structures, 236–237, 111313. Lee, H. W., & Basaran, C. (2022). Predicting high cycle fatigue life with unified mechanics theory. Mechanics of Materials, 164, 104116. Lee, H. W., Fakhri, H., Ranade, R., Basaran, C., Egner, H., Lipski, A., Piotrowski, M., & Mroziński, S. (2022c). Modeling corrosion fatigue with unified mechanics theory. Material & Design, 224, 111383. Lee, H. W., Fakhri, H., Ranade, R., Basaran, C., Egner, H., Lipski, A., Piotrowski, M., & Mroziński, S. (2022d). Modeling fatigue of pre-corroded bcc metals with unified mechanics theory. Materials & Design, 111383. https://doi.org/10.1016/j.matdes.2022.111383 Levin, V. M. (1967). Thermal expansion coefficients of heterogeneous materials. Mechanics of Solids, 2(1), 58–94. Mura, T. (1987). Mechanics of elastic and inelastic solids: Micromechanics of defects in solids (2nd ed.). Martinus Nijhoff Publishers. Nie, S., & Basaran, C. (2005). A micromechanical model for effective elastic properties of particulate composites with imperfect interfacial bonds. International Journal of Solids & Structures., 42(14), 4179–4191. #62. Nie, S., Basaran, C., Hutchins, S., & Ergun, H. (2006). Failure mechanisms in PMMA/ATH acrylic casting dispersion. Journal of Mechanical Behavior of Materials, 17(2), 79–95. #39. Nielsen, L. E., & Landel, R. F. (1994). Mechanical properties of polymers and composites (2nd ed.). Marcel Dekker, Inc. Ortiz, M., & Martin, J. E. (1989). Symmetry-preserving return-mapping algorithms and incrementally extremal paths: A unification of concepts. International Journal for Numerical Methods in Engineering, 28, 1839–1853. Prager, W. (1956) A New Method of Analyzing Stresses and Strains in Work-Hardening Plastic Solids. Journal of Applied Mechanics, 23, 493–496. Simo, J. C., & Hughes, T. J. R. (1998). Interdisciplinary applied mathematics, mechanics and materials, computational inelasticity. Springer. Simo, J. C., & Taylor, R. L. (1985). Consistent tangent operators for rate-independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering, 48, 101–119. Smith, J. C. (1974). Correction and extension of van der Poel’s method for calculating the shear modulus of a particulate composite. Journal of Research of the National Bureau of Standards-A. Physics and Chemistry, 78A(3), 355–361.

References

373

Smith, J. C. (1975). Simplification of van der Poel’s formula for the shear modulus of a particulate composite. Journal of Research of the National Bureau of Standards-A. Physics and Chemistry, 79A(2), 419–423. Van der Poel, C. (1958). On the rheology of concentrated suspensions. Rheologica Acta, 1, 198. Walpole, L. J. (1966). On bounds for the overall elastic moduli of inhomogeneous systems-I. Journal of the Mechanics and Physics of Solids, 14, 151. Ziegler, H. (1959). A Modification of Prager’s Hardening Rule. Quarterly of Applied Mathematics, 17, 55–65.

Chapter 7

Unified Micromechanics of Finite Deformations

7.1

Introduction to Finite Deformations

In this chapter, finite (large) deformation micromechanics is discussed. The formulation is more complicated than the small deformation theory; therefore, it is easier to explain with an example. Micromechanical modeling of a polymer is used as an example. A dual-micro-mechanism rate-dependent constitutive model is used to describe the thermomechanical response of amorphous polymers below and above glass transition temperatures. Material property definitions, the evolution of internal state variables, and plastic flow rules are revisited to provide a smooth and continuous transition in material response around glass transition temperature, θg. To formulate the large deformation mechanics, it is necessary to describe the kinematics of the constitutive model based on finite deformation tensors. Consider a body with an initial volume of Vo in undeformed (original or initial) configuration (Σo) at time to which deforms into a volume V in the current (deformed) configuration (Σ) at time t, as shown in Fig. 7.1. This body can be uniquely defined with a continuous one-to-one mapping (χ) of position vectors r and x in the original configuration and current configuration, respectively. r is the material (local) coordinate of a particle, that is, the coordinates of its location in the reference coordinate system. x is the spatial coordinates, defining location at time t. Both material coordinates and spatial coordinates were discussed earlier in Chap. 2. Deformation gradient tensor F describes the transformation of a line element dr at the position of r in the original configuration to a deformed line element dx at the position of x in the current configuration (Eq. (7.1)). Unique transformation ensures a non-singular, nonnegative determinant of deformation gradient F or Jacobian J (Eq. (7.3)). The spatial gradient of velocity tensor L is the gradient of the velocity _ field ~ v and relates deformation gradient tensor to its material time derivative F:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_7

375

376

7

Unified Micromechanics of Finite Deformations

Fig. 7.1 Deformation from the original configuration to the current configuration

x = χðr, t Þ F = ∇xðχ Þ =

ð7:1Þ ∂x ∂r

ð7:2Þ

J = detðFÞ > 0

ð7:3Þ

~ vðx, t Þ = x_ ðx, t Þ

ð7:4Þ

L = gradð~ vÞ =

∂x_ ∂r

F_ = L  F

ð7:5Þ ð7:6Þ

Deformation gradient tensor F can be decomposed into an elastic part Fe and plastic part Fp, and information related to thermal deformation is included in the elastic part of the deformation gradient tensor. Plastic deformation gradient tensor F p defines a deformation with respect to the initial configuration Σo to an intermediate, “relaxed,” or “natural” configuration Σ′ which is followed by an elastic deformation tensor Fe to the final configuration (Σ). The symmetric part of the spatial gradient of velocity tensor L is defined as stretching rate tensor D, and the asymmetric part of velocity gradient tensor is defined as spin tensor W: F = Fe Fp ; J = J e J p

ð7:7Þ

J e = detðFe Þ > 0

ð7:8Þ

J = detðF Þ > 0

ð7:9Þ

symmetric ðLÞ = D; asymmetricðLÞ = W; L = D þ W

ð7:10Þ

p

p

7.1

Introduction to Finite Deformations

377

Using Eq. (7.7), the elastic and plastic part of the spatial gradient of velocity tensor L can be defined as -1

L = Le þ Fe Lp Fe

ð7:11Þ

where e -1 Le = F_ Fe

ð7:12Þ

_ p p-1

ð7:13Þ

Lp = F F

Stretching rate tensor D and spin tensor W for elastic and plastic parts can be derived similarly from Eq. (7.10), where superscripts “e” and “p” represent elastic and plastic parts of corresponding quantity, respectively: symðLe Þ = De ; asymðLe Þ = We ; Le = De þ We

ð7:14Þ

symðL Þ = D ; asymðL Þ = W ; L = D þ W

ð7:15Þ

p

p

p

p

p

p

p

Right stretch tensor U and left stretch tensors V and rotation tensor R can be found from the right and left polar decompositions of the deformation gradient tensor F as follows: F = RU

ð7:16Þ

F = VR

ð7:17Þ

U and V stretch tensors are positive definite symmetric tensors and R are proper orthogonal tensors. Cauchy tensor C and Almansi tensor B can be formulated as follows: C = FT F = UU

ð7:18Þ

B - 1 = F - TF - 1 = V - 1V - 1

ð7:19Þ

In the same fashion, the following relations can be written for elastic and plastic parts of the deformation gradient tensor F: Fe = Re Ue

ð7:20Þ

F =V R

ð7:21Þ

e

e

e

T

Ce = Fe Fe = Ue Ue Be

-1

-T

= Fe Fe

-1

-1

= Ve Ve

ð7:22Þ -1

ð7:23Þ

378

7

Unified Micromechanics of Finite Deformations

Fp = Rp Up

ð7:24Þ

F =V R

ð7:25Þ

p

p

p

T

ð7:26Þ

Cp = Fp Fp = Up Up Bp

-1

-T

= Fp Fp

-1

-1

= Vp Vp

-1

ð7:27Þ

Multiplicative split of deformation gradient tensor F produces nonunique, locally defined intermediate (relaxed) configurations. One way to solve this problem is to introduce co-rotational rate definitions of elastic and plastic deformation gradients based on (director) orthonormal vectors which was proposed by Mandel (1972). The resulting elastic and plastic parts of the deformation rate tensors are invariant upon any superposed rigid body rotations. However, the arbitrariness of intermediate configuration can be also removed by setting the plastic spin tensor Wp equal to zero (Eq. (7.28)). In this case, elastic and plastic deformation gradients will include rotations, which can be handled with proper selection of stress and rate measures establishing a frame indifferent model: Wp = 0

ð7:28Þ

Jp = 1

ð7:29Þ

Assuming that plastic flow is irrotational (Eq. (7.28)) and incompressible (Eq. (7.29)), Jacobian of total deformation gradient J and material time derivative p of the plastic deformation gradient F_ will be J = Je

ð7:30Þ

F_ = Dp Fp

ð7:31Þ

p

Because of the incompressible plastic flow and irrotational plastic flow assumptions, we have the following relations: Lp = Dp

ð7:32Þ

trðL Þ = trðD Þ = 0

ð7:33Þ

-1

ð7:34Þ

p

p

L = Le þ F e D p F e

The system is assumed to be at rest initially (t = 0) which provides the following initial conditions: Fp ðr, 0Þ = I

ð7:35Þ

F ðr, 0Þ = I

ð7:36Þ

e

7.2

7.2

Frame of Reference Indifference

379

Frame of Reference Indifference

The material constitutive model must be invariant with respect to the frame of reference (coordinate system). Therefore, a quantity or an equation is frame indifferent (or objective) if it is invariant to changes in the frame of reference. Let the rigid body motion of any material point χ(r, t) be defined by a proper orthogonal rotation tensor Ω(t) and a vector ro(t) for any duration t as follows: χ ðr, t Þ = Ωðt Þ½χ ðr, t Þ - O þ ro ðt Þ

ð7:37Þ

where the orthogonal characteristic of rotation tensor is defined by ΩT Ω = I

ð7:38Þ

Here we define a stress tensor as objective if it does not depend on the frame of reference. In Eq. (7.37), Ω(t) represents rigid body rotation and ro(t) represents rigid body translation. In this context, frame indifference of a vector a and a second-order tensor A are defined in terms of transformation rules as follows: a = Ωa

ð7:39Þ

A = ΩAΩT

ð7:40Þ

Transformation of deformation gradient tensor F with respect to a change in the frame of reference can be obtained from Eq. (7.31) as follows: F = ΩF

ð7:41Þ

Therefore, the deformation gradient tensor is not objective (it is not the frame of reference indifferent) according to Eq. (7.40). Using Eq. (7.41) Cauchy tensor C in the new transformed configuration can be given by T

C = F F = FT ΩT ΩF = FT F = C

ð7:42Þ

Since the Cauchy tensor is a Lagrangian tensor referring to the original frame of reference, the Cauchy tensor should not be expected to change with the current coordinate system (frame of reference) according to Eq. (7.40). Therefore, Cauchy tensor C is not objective, but still invariant with respect to changes in the current frame of reference in the deformed configuration. Using Eq. (7.41) and the right polar decomposition rule in Eq. (7.16), we can write F = RU = ΩF = ΩRU

ð7:43Þ

380

7

Unified Micromechanics of Finite Deformations

Since right polar decomposition is unique, Eq. (7.43) implies that R = ΩR

ð7:44Þ

U=U

ð7:45Þ

Therefore, right stretch tensor U and rotation tensor R are not objective. However, similar to the Cauchy tensor, the right stretch tensor refers to the original frame of reference, and hence right stretch tensor is invariant to changes in the current frame of reference. Similarly, using Eq. (7.19), Almansi tensor B in the transformed configuration, we can write T

B = FF = ΩFFT ΩT = ΩBΩT

ð7:46Þ

Therefore, Almansi tensor B is objective. Using Eq. (7.41) and the left polar decomposition rule in Eq. (7.17) yields F = VR = ΩF = ΩVR

ð7:47Þ

According to the transformation of rotation tensor in Eqs. (7.44) and (7.47) can be rewritten as F = VR = ΩVΩT ΩR

ð7:48Þ

Since left polar decomposition is also unique, Eq. (7.48) implies that V = ΩVΩT

ð7:49Þ

Therefore, the left stretch tensor V is objective. Taking the time derivative of Eq. (7.41), it can be shown that the material time derivative of deformation gradient tensor F is not objective: :

_ þ ΩF_ F = ΩF

ð7:50Þ

Using Eq. (7.50) and the definition of velocity gradient tensor given in Eq. (7.6), we can write : 
-1 T _ þ ΩF_ = ΩFF _ _ - 1 ΩT ΩF F = LF = ΩF Ω þ ΩFF

ð7:51Þ

Equation (7.51) implies that the velocity gradient tensor is not objective as shown below: _ T L = ΩLΩT þ ΩΩ

ð7:52Þ

7.2

Frame of Reference Indifference

381

Using definitions of stretch rate tensor and spin tensor given in Eq. (7.10) _ T L = D þ W = ΩðD þ WÞΩT þ ΩΩ

ð7:53Þ

D = ΩDΩT

ð7:54Þ

_ T W = ΩWΩT þ ΩΩ

ð7:55Þ

which implies that

Therefore, stretch rate tensor D is objective, while spin tensor W is not objective. Using the multiplicative decomposition definition of deformation gradient given in Eq. (7.7), we can write F = Fe Fp = ΩFe Fp

ð7:56Þ

Hence, we can write the following relations: Fe = ΩFe

ð7:57Þ

Fp = Fp

ð7:58Þ

Therefore, elastic and plastic deformation gradients are not objective. Since plastic deformation gradient refers to the original and intermediate frame of reference, plastic deformation gradient is not objective but invariant to changes in the current frame reference. Using definitions for elastic and plastic rotation tensors, left stretch tensor, right stretch tensor, Cauchy tensor, and Almansi tensor in Eqs. (7.20), (7.21), (7.22), (7.23), (7.24), (7.25), (7.26), and (7.27), the following relations can be written: Fe = Re Ue = ΩRe Ue

ð7:59Þ

Re = ΩRe

ð7:60Þ

Ue = Ue

ð7:61Þ

Fe = Ve Re = ΩVe ΩT Ω Re

ð7:62Þ

Ve = ΩVe ΩT

ð7:63Þ

Ce = Ue Ue = Ue Ue = Ce

ð7:64Þ

Be = Ve Ve = ΩVe ΩT Ω Ve ΩT = ΩBe ΩT

ð7:65Þ

Fp = Rp Up = ΩRp Up

ð7:66Þ

Rp = ΩR

ð7:67Þ

p

382

7

Unified Micromechanics of Finite Deformations

Up = Up

ð7:68Þ

Fp = Vp Rp = ΩVp ΩT Ω Rp

ð7:69Þ

Vp = ΩVp ΩT

ð7:70Þ

Cp = Up Up = Up Up = Cp

ð7:71Þ

Bp = Vp Vp = ΩVp ΩT ΩVp ΩT = ΩBp ΩT

ð7:72Þ

Therefore, elastic and plastic rotation tensors Re, Rp are not objective. Cauchy tensors, elastic, and plastic right stretch tensors and Ce, Cp, Ue, Up are also not objective, but invariant to changes in the current frame of reference. Almansi tensors, elastic, and plastic left stretch tensors Be, Bp, Ve, Vp are all objective. Using elastic and plastic velocity gradient definitions in Eqs. (7.12) and (7.13) with transformation rules for elastic and plastic deformation gradient in Eqs. (7.57) and (7.58) yields :

Le = Fe Fe

-1


 _ e Fe - 1 ΩT = ΩLe ΩT þ ΩΩ _ T = ΩFe þ ΩF :

Lp = Fp Fp

-1

-1 p = F_ Fp = Lp

ð7:73Þ ð7:74Þ

Equation (7.73) shows that the elastic component of the velocity gradient is not objective. According to (7.74) plastic component of the velocity, the gradient is not objective either, but it is invariant to changes in the current frame of reference, because the plastic velocity gradient refers to original and intermediate reference frames, but does not refer to the current frame of reference. Using definitions of elastic stretch rate tensor and spin tensor given by Eq. (7.14), we can write _ T Le = De þ We = ΩðDe þ We ÞΩT þ ΩΩ

ð7:75Þ

De = ΩDe ΩT

ð7:76Þ

_ T We = ΩWe ΩT þ ΩΩ

ð7:77Þ

which implies that

Therefore, elastic stretch rate tensor De is objective, while elastic spin tensor We is not objective. Similarly, using the plastic stretch rate definition given in Eq. (7.15) and applying the irrotational plastic flow assumption in Eq. (7.28), we can write the following relations: Lp = Dp = Dp which implies that:

ð7:78Þ

7.3

Unified Mechanics Theory Formulation for Finite Strain

Dp = Dp

383

ð7:79Þ

Therefore, plastic stretch rate tensor Dp is not objective but invariant to changes in the current reference frame. To summarize, it has been shown that e e p e _ ,R ,R ,L ,We are not objective; C,U, Fp, Ue, Ce, Up, Cp, Lp, Dp are F,R,F,L,W,F not objective but invariant to changes in the current reference frame; and B,V,D, Ve, Be, Vp, Bp, De are objective.

7.3

Unified Mechanics Theory Formulation for Finite Strain

A thermodynamic potential is concave with respect to temperature (θ) and convex with respect to all other internal state variables. It provides a basis for satisfying conditions of thermodynamic stability imposed by Clausius-Duhem inequality. Specific Helmholtz free energy (ψ) forms such a basis. Helmholtz free energy (ψ) is defined as the difference between specific internal energy (u) and the product of absolute temperature (θ) and specific entropy (s): ψ = u - θs

ð7:80Þ

The first law of thermodynamics (also known as the conservation of energy law) states that energy can be transported, or converted from one form to another, but cannot be destroyed or created. Accordingly, the internal energy of a system can change by heat flow into or out of the system (δq), heat generated within the system due to internal scattering and friction (r), mechanical work done on the system by the external load (δw), or all other kinds of work done on the system (δw′) during any process, including but not limited to electrical, thermal, chemical, etc. Considering _ can be given only the thermomechanical effects, rate of change in internal energy, u, by u_ = Q þ l þ r

ð7:81Þ

where Q is the rate of net heat flow into the system and l is the rate of net work done on the system and r is the heat generated in the system, due to internal friction, scattering, and chemical reactions. The second law of thermodynamics states that there exists a thermodynamic state function, entropy, which increases in the universe for all types of real processes due to irreversible entropy production. Unlike energy, entropy (s) is not only transferred across boundaries of a system, but it may also be created in the system which is called entropy production. Clausius-Duhem inequality describes a condition imposed by the second law of thermodynamics in terms of a

384

7

Unified Micromechanics of Finite Deformations

nonnegative entropy production rate (γ) per unit volume for any kind of irreversible process which is defined as γ = ρ_s þ divðJs Þ > 0

ð7:82Þ

Js is net entropy flux into the system. Heat flows into the body can be defined in terms of heat flux as follows:
 ρQ = - div Jq

ð7:83Þ

Substituting the time derivative of Eq. (7.80), internal energy definition given by Eq. (7.81), and heat flow equation in Eq. (7.83) into Eq. (7.82), internal entropy production rate density can be rewritten as
  div Jq ρ r ρ
_ >0 þ γ = divðJs Þ þ l - ψ_ - θs θ θ θ

ð7:84Þ

r is the internal heat generation source strength and ρ is the mass density.

7.3.1

Thermodynamic Restrictions

The main premise of the unified mechanics theory is the derivation of the thermodynamic fundamental equation, which must contain all entropy generation terms for all active mechanisms. Of course, the amount of entropy generated in each micromechanism will be different. After a comparison of all mechanisms, it is reasonable to ignore mechanisms that generate entropy in smaller amounts by orders of magnitude. For dual-mechanism rate-dependent constitutive modeling of polymers, the material response can be resolved into two components which necessitate multimechanism generalization of multiplicative decomposition in Eq. (7.7) and description of different Helmholtz free energy functions and associated fundamental equation assuming that linear addition is applicable for Eq. (7.80) Accordingly, Eqs. (7.12), (7.13), (7.14), (7.15), (7.20), (7.27), and (7.31) hold for each component of the material’s resistance mechanism. Subscripts “I” and “M” will be used to designate the component of a quantity in the Intermolecular mechanism and Molecular network mechanism, respectively. For a description of dissipation inequality, total Helmholtz free energy density in (original) reference configuration is written as a summation of defect energy (ΨD) and elastic energy stored in an intermolecular structure (ΨI) and molecular network structure(ΨM):


 Ψ CeI , CeM , A, θ = ΨI EeI , θ þ ΨM CeM , θ þ ΨD ðA, θÞ

ð7:85Þ

Defect energy (ΨD) is assumed to depend on a stretch like tensor A and temperature (θ), elastic energy stored in an intermolecular structure (ΨI) is assumed to

7.3

Unified Mechanics Theory Formulation for Finite Strain

385

depend on logarithmic elastic strain in the intermolecular structure EeI and temperature (θ), and elastic energy stored in molecular network structure (ΨM) is assumed to depend on elastic Cauchy tensor in molecular network structure CeM and temperature (θ). If a similar decomposition also holds for specific entropy and specific Helmholtz free energy, we can write the following relation:


 s CeI , CeM , A, θ = sI EeI , θ þ sM CeM , θ þ sD ðA, θÞ

ð7:86Þ




 ψ CeI , CeM , A, θ = ψ I EeI , θ þ ψ M CeM , θ þ ψ D ðA, θÞ

ð7:87Þ

Helmholtz free energy density in reference configuration (Ψ) can be simply related to specific Helmholtz free energy function (ψ) through Eq. (7.88) and in rate form, as follows:

 ρo ψ CeI , CeM , A, θ = Ψ CeI , CeM , A, θ

ð7:88Þ



 _ Ce , Ce , A, θ ρψ_ CeI , CeM , A, θ = J - 1 Ψ I M

ð7:89Þ

where ρo and ρ are mass densities in reference configuration and deformed configuration, respectively. Note that since eigenvalues of elastic Cauchy tensor CeI and logarithmic elastic strain tensor EeI corresponding to the intermolecular structure are related through Eq. (7.90) and eigenvectors of these tensors are identical, it is possible to consider Helmholtz free energy density (ΨI) and specific entropy (sI) associated with intermolecular structure as functions of temperature (θ) and elastic Cauchy tensor CeI ; hence, we can write the following relations:
 1

 eigenvalue EeI = ln eigenvalue CeI 2
e 
 ΨI EI , θ  ΨI CeI , θ

 sI EeI , θ  sI CeI , θ

ð7:90Þ ð7:91Þ ð7:92Þ

The time derivative of Helmholtz free energy can be given by

 3 ∂ΨI EeI , θ ∂ΨM CeM , θ e e _ _ : CI þ : CM þ 7 6 ∂CeI ∂C
eMe  7
e e  6 7 ∂ΨD ðA, θÞ _ ∂ΨI EI , θ _ C , C , A, θ = 6 Ψ 6 _ I M :θþ 7 :Aþ 6 þ 7
 ∂θ ∂A 4 ∂ΨM CeM , θ _ ∂ΨD ðA, θÞ _ 5 þ :θþ :θ ∂θ ∂θ 2

ð7:93Þ

According to the principle of virtual work, rate of work done per unit volume of the deformed body (external work) is balanced with internal work, while total work done on the system is stored as elastic strain energy (represented by the first two

386

7

Unified Micromechanics of Finite Deformations

terms in Eq. (7.95) and dissipated as plastic work (represented with the last two terms in Eq. (7.95)): ρl = w_ int w_ int = ΓeI

:

LeI

þ

ΓeM

:

þ

LeM

ð7:94Þ

J - 1 ΓpI

:

LpI

þ

J - 1 ΓpM

:

LpM

ð7:95Þ

where ΓeI , ΓeM , ΓpI , and ΓpM are stress measures conjugate to the rate of deformation measures LeI, LeM, LpI, and LpM , which were defined in Eqs. (7.12) and (7.13). Noting -1 that J - 1 = J e , the J-1 the multiplier in front of the last two terms in Eq. (7.95) recovers work definitions from intermediate configuration to deformed configuration. The requirement of frame indifference for internal work can be given by the following relation: :

w_ int = w

ð7:96Þ

int

which implies that 

ΓeI :LeI þΓeM :LeM þ þJ - 1 ΓpI :LpI þJ - 1 ΓpM :LpM



=



e e e e ΓI :LI þΓM :LM þ p -1 p p -1 p þJ ΓI :LI þJ ΓM :LM



ð7:97Þ

Using transformation rules for elastic and plastic velocity gradients in Eqs. (7.73) and (7.74), we can write the following relations: 

ΓeI :LeI þJ - 1 ΓpI :LpI þ þΓeM :LeM þJ - 1 ΓpM :LpM



 =

_ T ÞþJ - 1 Γ :Lp þ ΓI :ðΩLeI ΩT þΩΩ I I e _ T ÞþJ - 1 Γp :Lp þΓM :ðΩLeM ΩT þΩΩ M M e

p



ð7:98Þ

or 

ΓeI :LeI þJ - 1 ΓpI :LpI þ þΓeM :LeM þJ - 1 ΓpM :LpM



" =

ðΩT ΓI ΩÞp:LeI þ ðΩT ΓMpΩÞp:LeM þ p e

-1

e

þJ ΓI :LI þJ - 1 ΓM :LM þ e _ T þΓe :ΩΩ _ T þΓI :ΩΩ M

# ð7:99Þ

The first two terms on the right-hand
side of Eq. (7.99) indicate that stress measures corresponding to elastic work ΓeI , ΓeM are objective as shown in
 _ T is a skew-symmetric tensor (Eqs. (7.55) Eqs. (7.100) and (7.101). Since ΩΩ and (7.77)), the last two terms on the right-hand side
 of Eq. (7.99) imply that stress measures corresponding to elastic work ΓeI , ΓeM are symmetric (Eqs. (7.102) and (7.103)). The third and fourth terms on the right-hand side of Eq. (7.99) show that stress measures corresponding to plastic work ðΓpI , ΓpM Þ are not objective but invariant to changes in the current frame of reference as shown in Eqs. (7.104) and (7.105). Hence,

7.3

Unified Mechanics Theory Formulation for Finite Strain

387

e

ΓI = ΩΓeI ΩT

ð7:100Þ

e

ΓM = ΩΓeM ΩT eT

e

ΓI = ΓI ; ΓeI = ΓeI e

e

T

ð7:101Þ T

ð7:102Þ T

ð7:103Þ

ΓM = ΓM ; ΓeM = ΓeM p ΓI

= ΓpI

ð7:104Þ

p ΓM

= ΓpM

ð7:105Þ

Using symmetry property of stress tensor in Eqs. (7.102) and (7.103), and irrotational plastic flow definition given in Eq. (7.28), total internal work over the whole volume of a system can be written as Z

_ int = W

Z w_ int dV =

V



ΓeI : DeI þ ΓeM : DeM þ J - 1 ΓpI : DpI þ J - 1 ΓpM : DpM dV

V

ð7:106Þ Total external work acting on the system can be described in terms of surface tractions on boundaries of the system and body forces acting on the system as follows: _ ext = W

Z

Z ρ l dV =

V

σ

ðnÞ

Z  χ_ dA þ

b  χ_ dV

ð7:107Þ

V

A

where A represents the surface and V represents volume. Consider the principle of virtual work for a special case defined in Eq. (7.34) as L = gradðχ_ Þ = LeI = LeM

ð7:108Þ

DpI = DpM = 0

ð7:109Þ

where it is assumed that

The principle of virtual work can then be rewritten for this special case from Eqs. (7.106) and (7.107) as Z

ðnÞ

σ

Z _  χdA þ

A

_ b  χdV = V

A

Z

Z

_ σðnÞ  χdA þ

ð7:110Þ

_ ½ðΓeI þ ΓeM Þ : gradðχÞdV

ð7:111Þ

V

Z

Z _ b  χdV =

V

½ΓeI : DeI þ ΓeM : DeM dV

V

388

7

Z

_ σðnÞ  χdA þ

A

Z

Z

Unified Micromechanics of Finite Deformations

Z _ b  χdV =

V

div½χ_  ðΓeI þ ΓeM ÞdV V

divðΓeI þ ΓeM Þ  χ_ dV

-

ð7:112Þ

V

Since Eq. (7.112) is true for any choice of V and gradðχ_ Þ, from the first terms on the left- and right-hand side, we can write
 σ ðnÞ = ΓeI þ ΓeM n

ð7:113Þ

which is essentially Cauchy’s stress theorem describing the relation between stress tensor and surface tractions. From the second terms on the left- and right-hand side of Eq. (7.112), we can write
 div ΓeI þ ΓeM þ b = 0

ð7:114Þ

which represents Cauchy’s equation of motion for stationary systems. Therefore, stress measures in Eqs. (7.113) and (7.114) correspond to Cauchy stress T components in the intermolecular mechanism TI and molecular network mechanism TM, respectively. Hence, we can write the following relations: T = ΓeI þ ΓeM

ð7:115Þ

ΓeI = TI = TT I

ð7:116Þ

ΓeM = TM = TM T

ð7:117Þ

Consider the principle of virtual work for a second special case defined from Eq. (7.34) such that L = gradðχ_ Þ = LeI þ FeI DpI FeI

-1

-1

= LeM þ FeM DpM FeM = 0

ð7:118Þ

or LeI = - FeI DpI FeI

-1

ð7:119Þ

-1

LeM = - FeM DpM FeM

ð7:120Þ

Accordingly, the principle of virtual work can be rewritten for this special case using Eqs. (7.106) and (7.197) as _ int = W

Z  V

or




-1



ΓeI :
- FeI DpI FeI þJ  - 1 ΓpI :Dp pI þp -1 þΓeM : - FeM DpM FeM þJ - 1 ΓM :DM

 dV

ð7:121Þ

7.3

Unified Mechanics Theory Formulation for Finite Strain

_ int W =

Z h

J - 1 ΓpI - FeI ΓeI FeI T

-T



389

  i T -T : DpI þ J - 1 ΓpM - FeM ΓeM FeM : DpM dV

V

ð7:122Þ Since the velocity gradient is assumed to be zero for this special case, the velocity field will be also equal to zero. Therefore, external work will be equal to zero
_ ext = 0 . As a result, individual terms inside the parenthesis in Eq. (7.122) should W be equal to zero for arbitrary selection of V, DpI , and DpM : J - 1 ΓpI - FeI ΓeI FeI T

-T

=0

-T

J - 1 ΓpM - FeM ΓeM FeM = 0 T

ð7:123Þ ð7:124Þ

Using definitions of stress measures in Eqs. (7.116) and (7.117), it can be shown that T

ΓpI = JFeI TI FeI

-T

T

-T

ΓpM = JFeM TM FeM

ð7:125Þ ð7:126Þ

which form the definition of elastic symmetric Mandel stress in intermolecular structure and molecular network structure as T

MeI = JFeI TI FeI T

-T

-T

MeM = JFeM TM FeM

ð7:127Þ ð7:128Þ

whereas since the trace of plastic stretch rate is equal to zero due to incompressible plastic flow, assumption stress conjugate to plastic stretch rate should be a deviatoric tensor. Therefore, we can write
 ΓpI = dev MeI
 ΓpM = dev MeM

ð7:129Þ ð7:130Þ

Finally, elastic second Piola-Kirchhoff stress tensor in intermolecular structure and molecular network structure can be defined as follows: -1

SeI = JFeI TI FeI

-T

ð7:131Þ

390

7

Unified Micromechanics of Finite Deformations -1

-T

SeM = JFeM TM FeM

ð7:132Þ

Using definitions of elastic Mandel stress given by Eqs. (7.127) and (7.128) and symmetric second Piola-Kirchhoff stress given by Eq. (7.131) and Eq. (7.132), we can write the following relation: MeI = CeI SeI MeM

= CeM SeM

ð7:133Þ ð7:134Þ


 Therefore, it has been shown that stress measures
ΓeI , ΓeM , ΓpI , ΓpM 
[or ΤI , ΤM , MeI , MeM  with deformation rate conjugates of DeI , DeM , DpI , DpM form a frame indifferent framework for dual-mechanism elastic-viscoplastic constitutive model. The thermodynamic restrictions on constitutive relations can be obtained by substituting Eqs. (7.86), (7.89), and (7.93) into Eq. (7.84) as follows:

 Jq =0 θ

ð7:135Þ

Jq θ

ð7:136Þ

divðJs Þ - div Js =


 1 ρr div Jq  ∇x ðθÞ þ > 0 2 θ θ
e 
e  

  ∂ΨM CM , θ ∂ΨD ðA, θÞ - 1 ∂ΨI EI , θ þ þ þ ρs θ_ = 0 J ∂θ ∂θ ∂θ
e e 
e e  - 1 ∂Ψ EI , CM , A, θ ρs EI , CM , A, θ = - J ∂θ
e e 
 ∂ψ EI , CM , A, θ s EeI , CeM , A, θ = ∂θ
e 

 ∂ΨI EI , θ 1 _e = : C ΓeI : LeI - J - 1 I θ ∂CeI
e  ∂ΨI EI , θ e ΓeI : LeI = TI : LeI = TI : DeI = J - 1 : C_ I ∂CeI
e    e e 1 e-1 e e-T - 1 ∂ΨI EI , θ _ : C_ I TI : DI = : CI = J F TI FI 2 I ∂CeI
  -1  ∂ΨI EeI , θ e e e-T J FI TI FI = SI = 2 ∂CeI γ ther = -

ð7:137Þ ð7:138Þ ð7:139Þ ð7:140Þ ð7:141Þ ð7:142Þ ð7:143Þ ð7:144Þ

7.3

Unified Mechanics Theory Formulation for Finite Strain


e 

 1 e e - 1 ∂ΨM CM , θ _e = : C ΓM : LM - J M θ ∂CeM
e  ∂ΨM CM , θ e ΓeM : LeM = TM : LeM = TM : DeM = J - 1 : C_ M ∂CeM
e   e e 1
e e e - 1 ∂ΨM CM , θ _ TM : DM = FM TM FM : CM = J : C_ M 2 ∂CeM
  -1  ∂ΨM CeM , θ e e e-T J FM TM FM = SM = 2 ∂CeM

 1 p p -1 p -1 p - 1 ∂ΨD ðA, θ Þ _ J ΓI : LI þ J ΓM : LM - J γ mech = : A >0 θ ∂A

391

ð7:145Þ ð7:146Þ ð7:147Þ ð7:148Þ ð7:149Þ

Equation (7.136) relates entropy flux to heat flux. Equation (7.137) defines irreversible entropy production per unit volume in a deformed configuration associated with heat conduction which is always positive according to Fourier’s law (Eq. (7.150)). Equation (7.140) provides the relation between specific entropy and specific Helmholtz energy. Equations (7.144) and (7.148) describe stress measures derived from Helmholtz free energy functions. Equation (7.149) defines irreversible entropy production due to mechanical dissipation per unit volume in a deformed configuration. The reason for the appearance of J-1 term in front of Eq. (7.149) is that all stress measures and their conjugate rate measures refer to intermediate configuration (relaxed configuration), while irreversible entropy production rate density refers to deformed configuration. Heat flux is given by Jq = - k∇x ðθÞ

ð7:150Þ

where k is the temperature-dependent thermal conductivity of the material. Specific heat can be expressed in terms of specific entropy (Eq. (7.151)) and terms of Helmholtz free energy (Eq. (7.153)) by exploiting the relation given by Eq. (7.139). Using linear decomposition assumption in Eqs. (7.85) and (7.86), specific heat can be rewritten in terms of separate components of specific entropy and Helmholtz free energy as shown in Eq. (7.154):
 ∂s EeI , CeM , A, θ ∂θ
e 


e   ∂sM CM , θ ∂sI EI , θ ∂s ðA, θÞ þ þ D c=θ ∂θ ∂θ ∂θ
e   -1 e ∂Ψ EI , CM , A, θ ∂ J c= -θ ∂θ ρ ∂θ c=θ

ð7:151Þ ð7:152Þ ð7:153Þ

392

7

Unified Micromechanics of Finite Deformations


e 
  

∂ΨM CeM , θ ∂ΨD ðA, θÞ ∂ J - 1 ∂ΨI EI , θ þ þ c= -θ ∂θ ∂θ ∂θ ρ ∂θ

7.3.2

ð7:154Þ

Constitutive Relations

In constitutive modeling of amorphous polymers for large deformations, dual decomposition of material response into two parallel working mechanisms of intermolecular structure and molecular network structure is widely used (Gunel and Basaran 2009, 2011a, b). A triple-mechanism constitutive model has also been proposed to include a secondary mechanism for the molecular network structure (Srivastava and Anand 2010). Both the dual- and triple-mechanism models are proven to be successful in describing the large deformation behavior of amorphous polymers at different isothermal conditions. To extend the applicability of such models to non-isothermal conditions, several refinements on material property definitions and viscoplastic flow rule definitions are necessary. Polymers have a very low melting temperature; hence, using time-independent plasticity is not an option. In dual-mechanism constitutive models, a material response is assumed to be controlled by two mechanisms working in parallel (intermolecular structure and molecular network structure), as depicted in Fig. 7.2. Intermolecular (I ) and molecular network mechanisms (M ) work in parallel; deformation in both mechanisms is equal to each other and equal to total deformation (Eq. (7.155)), while total stress is the summation of stresses due to intermolecular interactions (I) and molecular network interactions (M ) (Eq. (7.156)). Subscripts “I” and “M” represent intermolecular and molecular network components of associated quantity, respectively:

Fig. 7.2 Schematic representation of a material model

F = FI = FM

ð7:155Þ

T = TI þ TM

ð7:156Þ

Molecular Network Resistance, M

Intermolecular Resistance, I

7.3

Unified Mechanics Theory Formulation for Finite Strain

393

Intermolecular Resistance (I) In most polymers, the initial elastic response due to intermolecular resistance is governed by van der Waals bonds with surrounding molecules. A Helmholtz free energy per unit volume in reference configuration is considered for constitutive relation describing intermolecular resistance which was developed by Anand and On (1979) and Anand (1986):   o
 n 1 2 
 2 ΨI EeI , θ = G devðEIe Þj2 þ K - G tr EeI - - 3Kαðθ - θo ÞtrðEeI Þ 2 3

ð7:157Þ

where θo is the initial temperature, θo = θ(r, to), and G, K, α are temperaturedependent shear modulus, bulk modulus,
e and coefficient of thermal expansion, respectively. Elastic logarithmic strain EI
is related to the right elastic stretch
 tensor CeI and elastic deformation gradient FeI through the following relations: 1 ln CeI 2  T  1 EeI = ln FeI FeI 2 EeI =

ð7:158Þ ð7:159Þ

Utilizing Eq. (7.144) symmetric second Piola-Kirchhoff stress tensor SeI and Cauchy stress tensor TI can be obtained from Helmholtz free energy density function corresponding to intermolecular resistance as follows: SeI


 ∂ΨI EeI , θ =2 ∂CeI

TI = J - 1 FeI SeI FeI

T

ð7:160Þ ð7:161Þ

Since Helmholtz free energy density corresponds to macroscopic elastic energy stored, (ΨI) is an isotropic function of CeI elastic right Cauchy tensor; hence, CeI and ∂ΨI =∂CeI are coaxial and their product is a symmetric tensor, called elastic Mandel stress tensor MeI which is given by MeI = CeI SeI

ð7:162Þ

Relation between elastic Mandel stress MeI and elastic logarithmic strain EeI can be obtained from Helmholtz free energy function (Eq. (7.157)) and second PiolaKirchhoff stress tensor SeI , and the definition is given in Eq. (7.160); hence, we can write

394

7

Unified Micromechanics of Finite Deformations


 
 MeI = 2Gdev EeI þ K tr EeI - 3αðθ - θo Þ I

ð7:163Þ

Kinematic hardening in intermolecular structure is modeled by a defect energy function per unit volume in intermediate (relaxed) configuration (Anand et al. 2009): ΨD ðA, θÞ =

h i 1 B ln ða1 Þ2 þ ln ða2 Þ2 þ ln ða3 Þ2 4

ð7:164Þ

where ai represents eigenvalues of a stretch-like internal variable A which is a symmetric unimodular tensor, detA(x, t) = 1. Defect energy (ΨD) is an isotropic function of symmetric unimodular stretch-like tensor A. As a result, A and ∂ΨD/∂A are coaxial and their product is a symmetric deviatoric back-stress tensor Mback which is given by Eq. (7.165), and the evolution equation for A is given by Eq. (7.166) with an initial condition defined by Eq. (7.167) as follows: Mback = 2dev

 ∂ΨðA, θÞ A = B ln ðAÞ ∂A

ð7:165Þ

A_ = DpI A þ ADpI - γA ln ðAÞνpI

ð7:166Þ

AðX, 0Þ = I

ð7:167Þ

where γ represents the dynamic recovery, B is the temperature-dependent back-stress modulus, and νpI is the equivalent plastic stretch rate in intermolecular structure. Driving stress for plastic flow in intermolecular structure is defined as
 Meff = dev MeI - Mback

ð7:168Þ

The equivalent plastic stretch rate is given by Eq. (7.169), effective equivalent shear stress is given by Eq. (7.170), and mean normal pressure is defined in Eq. (7.171), and they are all defined in terms of tensorial variables as follows: pffiffiffi νpI = 2 DpI 1 τI = pffiffiffi jMeff j 2 1
 pI = - tr MeI 3

ð7:169Þ ð7:170Þ ð7:171Þ

The evolution of plastic deformation gradient in the intermolecular mechanism can be rewritten from Eq. (7.31) as follows: p F_ I = DpI FpI

ð7:172Þ

7.3

Unified Mechanics Theory Formulation for Finite Strain

FpI ðr, 0Þ = I

395

ð7:173Þ

Only the effective equivalent shear stress drives the plastic flow, and it is the source of plastic dissipation. Some part of plastic work is stored as energy associated with back-stress. In polymers, once the effective shear stress level reaches a critical level so that the energy barrier to molecular chain segment rotation is exceeded, plastic flow takes place. This is in contrast to the dislocation slip mechanism in metal plasticity. It is important to point out that the concept of dislocation and dislocation motion-induced plasticity model does not apply to amorphous polymers because the concept of dislocation cannot be justified in amorphous polymers. According to the cooperative model, viscous flow in a solid amorphous polymer may take place only when several polymer segments move cooperatively which also accounts for the significance of activation volume during the yield process. The flow rule for amorphous polymers is essentially based on the energy distribution statistics of individual segments (Fotheringham and Cherry 1978). In simple terms, the cooperative model flow rule is based on the average probability of simultaneous occurrence of n thermally activated transitions across an energy barrier (activation energy, Q) inducing a macroscopic strain increment of νo (Fotheringham et al. 1976; Fotheringham and Cherry 1978). Plastic flow characteristics of amorphous polymers are strongly temperature- and rate-dependent. According to the strain ratetemperature superposition principle, an increase in temperature will have the same effect on the yield stress as a decrease in strain rate (Francisco et al. 1996). Equivalence of time and temperature describes that the yielding of amorphous polymers at low temperatures is comparable to that at high strain rates. Therefore, Eyring plots (yield stress-temperature ratio vs. plastic strain rate curves) for various temperatures can be shifted vertically and horizontally with respect to a reference temperature (θref) to obtain a master curve describing yield stress behavior over a wide range of temperatures and strain rates. Richeton et al. (2006) proposed that both horizontal shift (ΔHh) and vertical shift (ΔHv) should follow Arrhenius-type temperature dependence. The resulting yield stress definition relates yield behavior of polymer with β mechanical loss peak at temperatures below the glass transition temperature θg through introducing activation energy at β-transition temperature, i.e., yield behavior in polymers is controlled by segmental motions of polymer chains, and the reference state for yielding is chosen as β-transition. An increase in yield stress due to an increase in strain rate is attributed to a decrease in molecular mobility of molecular chains, while a slow deformation rate allows polymer chains to slip past each other, resulting in lower resistance to flow. At low temperatures near secondary transition temperature (θβ), secondary molecular motions are restricted, and chains become stiffer which also increases the yield stress, while the increase in temperature provides more energy to polymer chains facilitating relative motion between polymer molecular chains. For temperatures above the glass transition temperature θg, characteristic plastic strain rate equation was modified by Williams-Landel-Ferry (WLF) parameters (c1, c2). Although characteristic plastic strain rate definitions at temperatures below and above the glass transition temperature, θg, are continuous functions of temperature in separate domains, piece-wise definition with respect to the glass transition temperature, θg, results in an unrealistic change in plastic flow behavior around glass transition temperature, i.e., the derivative of plastic strain rate equation is

396

7

Unified Micromechanics of Finite Deformations

discontinuous at θg. Srivastava and Anand (2010) proposed a modified version of the flow rule in intermolecular structure which incorporates different values of activation energy for the glassy region and rubbery region, but still, the constitutive model makes an abrupt change in the activation energies at the glass transition temperature, θg, which creates a discontinuity problem in computational mechanics simulations. To provide a smoother transition in flow characteristics around glass transition temperature, θg, characteristic plastic strain rate given by Eq. (7.174) and equivalent shear plastic stretch rate given by Eq. (7.175) can be modified in the following forms: ν



= νoI

exp


!# ln ð10Þc1 θ - θg
 1 þ exp c2 þ θ - θ g 

nI τI V p  νI = ν sinh 2kB θ

Q - I kB θ

"

ð7:174Þ ð7:175Þ

where νoI is the pre-exponential factor, QI is the activation energy for plastic flow in intermolecular structure, kB is Boltzmann’s constant, c1 and c2 are WLF parameters, nI is the number of thermally activated transitions necessary for plastic flow, V is the activation volume, and τI is the net effective stress which is defined as τI = τI - SI - αP pI

ð7:176Þ

where αp is hydrostatic pressure sensitivity parameter and SI is plastic flow resistance in intermolecular structure. The evolution of intermolecular resistance to plastic flow can be given by
 S_ I = hI SI - SI νpI

ð7:177Þ

with the initial condition given by SoI = SI ðr, 0Þ

ð7:178Þ

where hI is a parameter characterizing hardening-softening and SI is saturation value for plastic flow resistance in intermolecular structure which can be defined as follows: SI = bðϕ - ϕÞ

ð7:179Þ

where b is a temperature- and rate-dependent parameter that relates the saturation value of plastic flow resistance to order function (ϕ - ϕ). Resistance
to plastic flow (SI) increases with the order in the material and becomes constant SI when order parameter (ϕ) reaches a critical value (ϕ) which is also a temperature- and ratedependent variable. When intermolecular resistance reaches saturation value, steady-

7.3

Unified Mechanics Theory Formulation for Finite Strain

397

state plastic flow occurs and the plastic flow rate becomes equal to the applied strain rate. The evolution equation for order parameter is defined as ϕ_ = gðϕ - ϕÞνpI

ð7:180Þ

ϕo = ϕðr, 0Þ

ð7:181Þ

where g is a temperature-dependent parameter. The evolution equation for plastic deformation gradient is given by Eqs. (7.172) and (7.173) which completes the definition of material behavior in intermolecular structure. Strain hardening becomes insignificant as temperatures approach glass transition temperature, θg, and completely vanishes above the glass transition temperature θg (Richeton 2006). Definition of a vanishing internal resistance right at the glass transition temperature θg causes also discontinuity in yield behavior of polymers. Since annealing at high temperatures well above the glass transition temperature, θg, clears past thermomechanical history of a polymer by providing an alternative stationary molecular configuration at a higher energy level, internal resistance is bound to vanish above or around the glass transition temperature θg. Therefore, the underlying problem is essentially the assumption that glass transition temperature takes place at a single temperature point and internal resistance becomes zero abruptly exactly at the glass transition temperature θg. In the constitutive model presented in this
chapter, similarly, variables b, g, SI , ϕ , hI that characterize hardening-softening behavior in the post-yield region shall also provide a smooth transition from temperatures below the glass transition temperature θg to temperatures above θg. It should be noted that viscoplastic models as of the writing of this book available in the literature are all-phenomenological and serve as a mathematical tool to fit experimentally observed behavior into a curve. These empirical models can provide reasonably accurate predictions for viscoplastic characteristics of amorphous polymers for only isothermal cases. In the case of non-isothermal loading, which includes temperature change in the material concurrently with loading, most material models available in the literature predict unrealistic results. A comparison of viscoplastic models for amorphous polymers from literature and the improved version of the dual-mechanism model is presented in Fig. 7.3. Temperature variations of characteristic viscoplastic shear strain rates in different models are presented by normalizing with respect to characteristic viscoplastic strain rate at the reference glass transition temperature for PMMA (387 K). Material properties are taken from Srivastava and Anand (2010), while WLF parameters in the model discussed in this chapter and Richeton et al. (2006) are taken as their original values. Viscoplastic models presented in this chapter are all applicable for temperatures both above and below the glass transition temperature. In the model derived earlier in this chapter, the temperature dependence of viscoplastic stretch rate is directly employed by utilizing physically motivated Williams-Landel-Ferry parameters in a new format as presented in Eq. (7.174). Temperature-dependent activation energy approach used in Anand’s model does not provide a smooth change in behavior of intermolecular mechanism but an abrupt increase in viscoplastic strain rate over a relatively narrow temperature window. Therefore,

398

7

Unified Micromechanics of Finite Deformations

Fig. 7.3 Comparison of different viscoplastic models in literature in terms of the  temperature dependence of normalized characteristic viscoplastic shear strain rate νp =νpðθ = θg Þ , Gunel (2010)

according to Anand’s model, there is no rubbery region, but the material response is liquid-like as the viscoplastic strain rate increases by six orders of magnitude in a narrow temperature interval (2 °C) around the glass transition temperature. On the other hand, in Richeton’s model, there is also a remarkable change in viscoplastic strain rate due to the piece-wise definition with respect to the glass transition temperature, but it also causes a discontinuous derivative of viscoplastic rate function at the glass transition temperature. Accordingly, the material response predicted by Anand or Richeton viscoplastic models has a significant (abrupt) change in stress value at the glass transition temperature. An improved version of the dualmechanism constitutive model presented in this chapter provides a gradual transition in material response with respect to the temperature around the glass transition. Though Anand’s model has a continuous definition of activation energy in the temperature domain, a remarkable difference between activation energies in glassy and rubbery regions still produces an abrupt change in the polymer’s response. According to Fig. 7.3, Anand’s viscoplastic model is invariant of temperature above glass transition temperature, while Richeton’s model provides a relatively gentler transition in response. As a result of this rapid change in viscoplastic response in Anand’s model, it cannot predict material behavior accurately under non-isothermal conditions. These models were further studied by Gunel and Basaran (2010) in terms of predictions for creep strain rate in a creep test with a stress level of 0.6 MPa which is conducted at different temperatures as depicted in Fig. 7.4. Figure 7.4 supports previous observations just discussed. All models predict the same creep strain rates at temperatures below the glass transition. For assurance of accurate and realistic modeling of material response around the glass transition temperature, every aspect of material property definitions that describe hardeningsoftening behavior and flow function must be continuous in the temperature domain and must have continuous derivatives with respect to temperature.

7.3

Unified Mechanics Theory Formulation for Finite Strain

399

Fig. 7.4 Comparison of different viscoplastic models in literature in terms of creep strain rates at different temperatures in response to applied stress of 0.6 MPa

Molecular Network Resistance (M) Resistance in the molecular network to deformation varies according to molecular orientation and relaxation process. If there is enough stretch in polymer chains, the network resists relaxation, and resistance increases with increasing stretch. In literature, there is a consensus on the modeling of the plastic flow behavior of intermolecular structure based on the Eyring cooperative model, but there is still debate on the modeling of molecular network resistance. Arruda and Boyce (1993) modeled molecular network resistance with a rubber elasticity model based on the eight-chain network of non-Gaussian chains similar to the transient response of elastomers which represents nonlinear rate-dependent deviation from the equilibrium state (Bergstrom and Boyce 1998). Unfortunately, the network resistance predicted by the eight-chain model is not accurate in describing the orientational hardening behavior of amorphous polymers and simulations do not match experimental observations. This is because of temperature dependence of rubbery modulus and the number of rigid links between polymer chain segments, which control the response. Therefore, molecular network description based on the eight-chain model (Richeton et al. 2007; Arruda et al. 1995; Palm et al. 2006; Boyce et al. 2000) is merely a numerical tool to match experimentally observed stress-strain response. Instead, a simpler two-constant constitutive model for rubber networks developed by Gent (1996) was shown to describe strain hardening due to polymer chain stretching better than the statistical-mechanical entropic rubber elasticity model (eight-chain model) while resulting in a similar stress-strain response according to Ames et al. (2009) and Srivastava and Anand (2010). Gent’s (1996) model-free energy per unit volume in reference configuration describes elastic energy stored in molecular network structure in terms of the first invariant of stretch in polymer chains as

400

7




ΨM CeM ,

Unified Micromechanics of Finite Deformations

 1 I1 - 3 θ = - μM I M ln 1 2 Im 

ð7:182Þ

where μM and IM are temperature-dependent rubbery shear modulus and limit the extensibility of polymer chains, respectively. Since volume change in a polymer is considered in the elastic deformation gradient associated with intermolecular structure, it is essential to define Gent free energy
 in terms of distortional elastic deformation gradient in network structure, FeM d , as given by Eq. (7.183) which produces no change in volume.
I1 is the first invariant of elastic distortional Cauchy tensor in network structure, CeM d :


 -1 FeM d = J =3 FeM

  det FeM d = 1
e
T
 CM d = FeM d FeM d 
 I 1 = tr CeM d

ð7:183Þ ð7:184Þ ð7:185Þ ð7:186Þ

Second Piola-Kirchhoff stress tensor SeM and Cauchy stress tensor TM can be derived from Gent free energy as follows: SeM


 ∂ΨM CeM , θ =2 ∂CeM

TM = J - 1 FeM SeM FeM T

ð7:187Þ ð7:188Þ

Using Gent free energy in stress definitions results in

 - 1h i I -3 1

 
 - 1 SeM = J - 2=3 μM 1 - 1 I - tr CeM d CeM d 3 IM

-1

  I1 - 3 -1 dev BeM d TM = J μ M 1 IM

ð7:189Þ ð7:190Þ


 Elastic distortional Almansi tensor, BeM d , is defined in terms of distortional
 elastic deformation gradient FeM d Þ as


BeM

 d



T = FeM d FeM d

ð7:191Þ

7.3

Unified Mechanics Theory Formulation for Finite Strain

401

For elastic Mandel stress tensor MeM and equivalent shear stress τM and equivalent plastic shear strain rate νpM for a molecular relaxation in-network, the structure can be given by the following relations: MeM = CeM SeM

-1

  I1 - 3 e MM = μM 1 dev CeM d IM 1 τM = pffiffiffi MeM 2 p ffiffiffi νpM = 2 DpM

ð7:192Þ ð7:193Þ ð7:194Þ ð7:195Þ

The evolution of plastic deformation gradient in molecular network mechanism can be rewritten from Eq. (7.31) as follows: p F_ M = DpM FpM

ð7:196Þ

FpM ðr,

ð7:197Þ

0Þ = I

A molecular network is responsible for resistance to chain alignment which resists relaxation as stretch in-network increases. A similar observation for elastomers was also found that plastic molecular chain stretch is inversely proportional to the effective creep rate (Bergstrom and Boyce 1998). In laboratory experiments, it was observed that in the post-yield region (at large deformations), the controlling mechanism is the molecular network. The molecular network mechanism has a dominant contribution to stress change in the post-yield region, while the amount of elastic recovery upon unloading is associated with plastic strain in the molecular network. Temperature dependence of molecular relaxation in network structure is characterized by a classical Arrhenius function, and the plastic flow rule is described with a simple power law as follows:

νpM

= νoM

exp

Q - M kB θ



τM SM

nM ð7:198Þ

where νoM is the pre-exponential factor, QM is the activation energy for molecular relaxation in network structure, kB is Boltzmann’s constant, nM is a strain-rate sensitivity parameter, and SM is a stress measure describing the resistance of network structure to relaxation which increases with increasing plastic stretch rate as defined as follows:
 S_ M = hM ðλpM - 1Þ SM - SM νpM with initial condition

ð7:199Þ

402

7

Unified Micromechanics of Finite Deformations

SoM = SM ðr, 0Þ

ð7:200Þ

where hM is a parameter characterizing molecular relaxation in material, SM is temperature- and rate-dependent saturation value of the network resistance, and λpM is a plastic stretch that is related to plastic Almansi tensor in network structure BpM as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffi trðBpM Þ λM = 3 T

BpM = FpM FpM

ð7:201Þ ð7:202Þ

Plastic deformation gradient evolution given by Eqs. (7.196) and (7.197) completes the constitutive modeling of material behavior in molecular network structure. According to Eq. (7.199), network resistance will increase continuously
as plastic stretch ðλpM Þ in polymer chains increases and reaches a constant value SM which depends on temperature and stretch rate. Plastic stretch-dependent evolution of resistance to plastic flow also ensures the correct prediction of elastic recovery in the unloading path. In Anand’s model, net driving stress for plastic flow in intermolecular mechanism includes an additional resistance term accounting for dissipative resistance to plastic flow ðSb or S2 Þ taking place at large deformations. According to Anand’s model, this dissipative resistance evolves with plastic stretch in the intermolecular mechanism or total stretch in the intermolecular mechanism. In the model derived in this chapter, dissipative resistance (SM) at large deformations in molecular network mechanism evolves with plastic stretch in molecular network branch which controls material response at large deformations (post-yield region). In Anand’s model, a third mechanism must be added for the true prediction of elastic recovery during the unloading and cooling of a pre-heated sample. It was argued that the third mechanism for molecular network structure was introduced due to a necessity driven by the experimentally observed complex response. However, the constitutive model presented in this chapter involves non-isothermal conditions without introducing the additional third mechanism. Finally, utilizing the relation given by Eq. (7.196) and specific Helmholtz free energy definitions (Eqs. (7.157), (7.164), (7.182)), the temperature-dependent governing constitutive equation can be given as ρcθ_ = ∇x ðk∇x ðθÞÞ þ rþ    1 þJ - 1 τI þ γBj ln ðAÞj2 νpI þ τM νpM þ 2 !
 -1 ∂S 1 eI _ e 1 ∂SeM _ e 1 ∂ Mback A -1 _ :A : CI þ : CM þ þJ θ 2 ∂θ 2 ∂θ 2 ∂θ

ð7:203Þ

7.4

Thermodynamics State Index

403

where the first two terms are heat conduction representing heat transfer within the material during transient state and heat source due to passive heating (external heating) or active heating (internally generated heat by internal friction or scattering, etc.). The last two terms represent heat induced due to intrinsic dissipation and thermoelastic effect representing conversion between mechanical and thermal energy in the elastic range. In the case of coupled thermomechanical loading, temperature increase due to mechanical work and temperature change due to heat transfer between material and surroundings are all included in Eq. (7.203). Based on descriptions of stress components and strain rate measures, irreversible entropy production due to mechanical dissipation per unit volume in deformed configuration can be rewritten from Eq. (7.149) as follows: 2

3  ∂ΨD ðA, θÞ p devð Þ - 2dev A :DI þ 7 J-1 6  ∂A 6

7>0 γ mech = 4 5 ∂Ψ ð A, θ Þ θ D þγ A : ln ðAÞνpI þdevðMeM Þ:DpM ∂A h   i J-1 1 γ mech = τI þ γBj ln ðAÞj2 νpI þ τM νpM > 0 θ 2

MeI

ð7:204Þ

ð7:205Þ

Due to the associative plasticity rule, irreversible mechanical entropy production (γ mech) is always positive:

7.4

Dp Meff NpI = Ip = M j DI eff j

ð7:206Þ

Me Dp M NpM = M p = e DM MM

ð7:207Þ

Thermodynamics State Index

There are two major irreversible entropy production mechanisms, thermal dissipation (Eq. (7.137)) due to heat exchange between the system and surroundings and mechanical work dissipation formulated by Eq. (7.205). Critical entropy (Scr) [also referred to as fracture entropy in the literature] is a characteristic value of a material. The link between irreversible material degradation and the amount of heat generated due to non-conservative dissipation mechanisms (i.e., plastic work, internal friction, scattering mechanisms, chemical reactions, etc.) or entropy production due to mechanical work is well established. Heat conduction within metals is usually very fast leading to negligible thermal gradients [at low-frequency loads, also referred to as pseudo-static in some books] and leading to quick dissipation of heat due to the high thermal diffusivity of metals. In the case of materials with low thermal diffusivity such as polymers, heat transfer may take place over a prolonged

404

7

Unified Micromechanics of Finite Deformations

period with significant thermal gradients. However, thermal gradients cannot be responsible for the failure of chemical bonds since material degradation (or damage) is a consequence of the formation of small voids or cracks at the microscale by the breakage of chemical bonds between molecules. Thermal dissipation may result in deterioration of material properties which is usually insignificant compared to degradation by mechanical dissipation. Thermodynamics state index (TSI) is given by h i ΔS Φ = Φcr 1 - e - ms R

ð7:208Þ

When TSI reaches a critical value Φcr, corresponding to a critical entropy level (Scr) that can be defined as failure or Φ = 1 can be defined as failure, depending on the application. [Because TSI is an exponential function, it never reaches 1; commonly 0.999 is considered as the final value, depending on the computing resources.] The critical TSI value depends on the critical entropy level (Scr), and it is a characteristic property of material [like toughness] and can be calculated or measured. Hence, we can calculate Φcr from the critical entropy value: h i ½Scr - So  Φcr = 1 - e - ms R

ð7:209Þ

where So is initial internal entropy value, which may be taken as zero. Any irreversible process induces degradation in microstructure, and internal entropy increases according to the second law of thermodynamics. Total entropy production due to mechanical and thermal dissipation can be calculated at any time step as follows: Zt Smech = Smech jt = to þ

γ mech dt

ð7:210Þ

to

Zt  Smech = Smech jt = to þ

J-1 θ

h

 i 1 p p 2 τI þ γBj ln ðAÞj νI þ τ M νM dt 2

ð7:211Þ

to

Zt Sther = Sther jt = to þ

γ ther dt

ð7:212Þ

to

Zt Sther = Sther jt = to þ

 -

to


 1 ρr  ∇ div J ð θ Þ þ dt q x θ θ2

ð7:213Þ

It is common practice to assume that there is no initial damage to the material. Of course, this assumption is unrealistic, but it is for expediency:

7.5

Definition of Material Properties

Smech jt = to = Sther jt = to = 0

405

ð7:214Þ

At failure, internal entropy production reaches a critical value (Scr) which is of course temperature-dependent. Unlike metals, the temperature dependence of critical entropy is essential for the case of stretching of polymers due to changes in the failure mode of amorphous polymer chains. Amorphous polymers display a brittle failure at very low temperatures (θ < < θg) without any significant plastic deformation, while at high temperatures (θ > θg), ductile failure occurs after a significant amount of plastic work. Critical TSI parameter is defined in such a way that at constant temperatures as S → Scr(θ), Φ → 1. Nonnegative entropy production assures that Φ ≥ 0, while for an intact material we assume (S = 0), the damage is assumed to be zero: h  i m ΦðθÞ = 1 - exp - s Scr ðθÞ R

ð7:215Þ

According to the incremental form of TSI evolution, degradation will increase at a much faster rate at low temperatures (sudden brittle failure), while degradation rate will be relatively slower at high temperatures (prolonged ductile failure). TSI evolution merely depends on the thermodynamic fundamental equation which incorporates all micro-mechanisms responsible for entropy generation due to thermomechanical loading. Derivative of TSI with respect to entropy can be given by   m m ΔΦ = Φcr s exp - s Smech R R ΔS

7.5

ð7:216Þ

Definition of Material Properties

Material properties are vital for constitutive models. Material properties must be defined over a large temperature and rate of loading range and should be continuous and smooth over the transition region. Functions for material properties presented herein are mathematical tools to describe the influence of temperature and loading rate in a continuous form. Formulations for these material properties representing temperature dependence are in a form that provides smooth continuity in the temperature domain and has a continuous first derivative with respect to temperature. Some material properties can be obtained by conducting isothermal tests at different temperatures (both above and below the glass transition temperature)  and different loading rates such as E, ν, I m , νoI , QI , nI , Bg , X B , V, αp , γ . Some material parameters are difficult to measure directly such as

406

7

Unified Micromechanics of Finite Deformations



 hI , b, g, νoM , QM , hM , nM , μM , ϕ , SM , yet these properties/parameters can be obtained by statistical methods. According to the free-volume theory of William et al. (1955), the plastic flow rule (yield function) can be constructed for equivalent plastic shear strain rate at temperatures above the glass transition temperature θg using Williams-Landel-Ferry (WLF) equations. Similarly, rate dependence of glass transition temperature can be considered in terms of temperature-time equivalence of glass transition as follows:
 8 g ν > < θref þ c2 log =νref g cg1 - log ðν=νref Þ θg = > : θref g

ν > νref ν≤ν

ð7:217Þ

ref

where cg1 and cg2 are WLF parameters associated with θg, νref is the reference stretch rate, and ν is the equivalent stretch rate which is defined by pffiffiffi ν= 2 j D j

ð7:218Þ

Temperature and rate dependence of elastic modulus (E) is given by 2


 3

 1
 θ - θ g þ θE 1
5 E = 4 2 Eg þ Er - 2 Eg - E r tanh ΔE þX E ½θ - ðθg þθE Þ h  i ν × 1 þ sE log ref ν ( E θ ≤ θg þ θE Xg XE = E Xr θ > θg þ θE

ð7:219Þ ð7:220Þ

where Eg and Er are for glassy and rubbery elastic modulus corresponding to temperatures confining the glass-rubber transition region. X gE and X rE represent the rate of change of elastic modulus with respect to temperature in glassy and rubbery domains, respectively. sE is the rate of loading sensitivity of elastic modulus, while θE and ΔE define origin temperature and width of glass-rubber transition window, respectively. Experimental studies on temperature and rate dependence of storage modulus and elastic modulus of poly-methyl methacrylate (PMMA) indicate that PMMA is highly sensitive to the rate of loading and temperature at constant humidity. Modulus of PMMA continuously decreases with increasing temperature with a remarkable drop around the glass transition temperature θg over a 10 C–20 C temperature window, depending on the frequency of loading (Gunel 2010). Poisson’s ratio ðνÞ is assumed to be only temperature-dependent and defined as

7.5

Definition of Material Properties

407




 1
 θ - θg þ θ E 1
ν = νg þ νr - νg - νr tanh 2 ΔE 2

ð7:221Þ

where νg and νr are for glassy region and rubbery region Poisson’s ratios, respectively. Shear modulus (G) and bulk modulus (K ) are defined as follows: G=

E 2ð1 þ νÞ

ð7:222Þ

K=

E 3ð1 - 2νÞ

ð7:223Þ

Temperature and rate dependence of rubbery modulus (μM) is modeled like the elastic modulus function as follows: 2


 3

 1
g  h  i θ - θg þ θμ 1
g r r ν 5 μM = 4 2 μM þ μM - 2 μM - μM tanh log 1 þ s μ Δμ νref þX μ ½θ - ðθg þθμ Þ

( Xμ =

ð7:224Þ X gμ

θ ≤ θg þ θμ

X rμ

θ > θg þ θμ

ð7:225Þ

Definitions of rubbery shear modulus parameters in Eq. (7.225) are identical to those for elastic modulus. Temperature dependence of the critical value of order (IM), and saturation value of parameter (ϕ), the limit of polymer chain extensibility
 plastic flow resistance of molecular network SM are defined below: ϕ =




    
 θ - θg þ θϕ 1  1  ϕg - ϕr tanh þ X ϕ θ - θg þ θϕ ϕg þ ϕr 2 2 Δϕ

(

ð7:226Þ X gϕ X rϕ

θ ≤ θg þ θϕ ð7:227Þ θ > θg þ θϕ


  1
g  
θ - θg þ θM 1
g r r I M = I M þ I M - I M - I M tanh þ X M θ - θg þ θM 2 2 ΔM Xϕ =

( XM =

ð7:228Þ X gM

θ ≤ θg þ θμ

X rM

θ > θg þ θμ

ð7:229Þ

408

7

Unified Micromechanics of Finite Deformations




 1
g  θ - θg þ θS 1
g r r = SM þ SM - SM - SM tanh 2 ΔS 2 
 þ X S θ - θg þ θS  g X S θ ≤ θg þ θS Xs = X rS θ > θg þ θS

SM

ð7:230Þ ð7:231Þ

Definitions of parameters above equations are to those for elastic
identical  modulus. Saturation value for network resistance SM and critical value of order parameter (ϕ) were assumed to decrease with increasing temperature, while limited chain extensibility (IM) increases with increasing temperature based on observations in experiments. Similar to models of Richeton et al. (2005a, b) and Anand et al. (2009), back-stress is assumed to vanish above the glass transition temperature, θg. However, a decrease in back-stress modulus (B) with increasing temperature asymptotically approaches zero at a temperature around the glass transition temperature θg, which can be defined as follows:




 θ - θg þ X B θg - θ B = Bg 1 - tanh ΔB ( g XB θ ≤ θg XB = 0 θ > θg

ð7:232Þ ð7:233Þ

Parameters b and g characterizing hardening-softening behavior in intermolecular structure can be defined as follows:


    θ - θg þ θg 1 1 g= g - gr tanh g þ gr 2 g 2 g Δg 
 þ X g θ - θg þ θg  g Xg θ ≤ θg þ θg Xg = 0 θ > θg þ θg !b3 νpI b = b1 exp ðb2 θÞ p νref parameters required  for the constitutive  oOther νI , QI , V, αp , nI , hI , γ, νoM , QM , hM , nM are all basic constants.

ð7:234Þ ð7:235Þ ð7:236Þ model

7.6

Applications of Finite Deformation Models

7.6

409

Applications of Finite Deformation Models

For verification of the constitutive model, laboratory experiments of isothermal and non-isothermal stretching of PMMA were performed. Simulation results were compared with test data in terms of stress-strain curves for isothermal tests and temperature-displacement-force histories for non-isothermal tests.

7.6.1

Material Properties

To simulate material response under isothermal and non-isothermal conditions, it is essential to determine appropriate material parameters. Isothermal tests on PMMA are required to obtain material parameters for the dual-mechanism viscoplastic constitutive model presented in earlier sections. PMMA response is highly sensitive to loading rate and temperature. Therefore, experiments must be conducted at different loading rates and different temperatures at constant humidity. In addition, the thermal properties of PMMA are also necessary for a fully coupled temperaturedisplacement finite element analysis. Most ofthe mechanical material properties of
PMMA E, ν, I m , νoI , QI , nI , Bg , X B , V, αp , γ can be obtained by conducting isothermal tests at different temperatures and different strain (displacement) rates. Since some material parameters such as (hI ,b,g,νoM ,QM ,hM ,nM ,μM ,ϕ ,SM ) cannot be directly observed in macroscale experiments or their influence on material response cannot be isolated from others, these parameters can be obtained by statistical methods only. Rate dependence of glass transition temperature of PMMA is
based on  the freevolume theory of Williams et al. (1955), while WLF parameters cg1 , cg2 needed in Eq. (7.224) are provided by Richeton et al. (2005a, b) and reference glass transition temperature is provided by Nie (2005). Variation of glass transition temperature with frequency (rate) can be obtained as shown in Fig. 7.5. Fig. 7.5 Rate-dependent glass transition temperature of PMMA

384

382

380 θ.g(ν) (°K)

378

376

374

20

10 10−5

ν (S-1)

25

410

7

Unified Micromechanics of Finite Deformations

1800

H series

Elasc Modulus (MPa)

1600 1400

M series

1200

L series

1000 800 600 400 200 0 0

20

40

60

80

100

120

140

160

Temperature (°C)

Fig. 7.6 Temperature- and rate-dependent elastic modulus of PMMA, Nie (2005)

5.0

Limited Chain Extensibility

4.8 4.6 4.4 4.2

H series

M90

4.0

M series

3.8 3.6

L series

H90

3.4 3.2 3.0 0

20

40

60

80

100

120

140

160

Temperature (°C)

Fig. 7.7 Temperature- and rate-dependent limited chain extensibility of PMMA

Figure 7.6 depicts the elastic modulus as a function of temperature for different strain rates, H series (loading rate of 0.9 mm/s), M series (0.09 mm/s), and L (0.009 mm/s). Figure 7.6 indicates that the temperature
sensitivity of elastic modulus  becomes smaller at slower loading rates. Parameters E g , E r , θE , ΔE , X gE , X rE , sE as a function of temperature and rate can be obtained from test data by statistical methods. Temperature dependence of limited chain extensibility (IM) can be obtained from fracture strain values in isothermal tests. Figure 7.7 shows the limited chain extensibility as a function of temperature and strain rate.
Two data points (H90 and  M90 values) are outliers. Using test data, parameters I gM , I rM , X gM , X rM , θM , ΔM can be obtained as a function of temperature and loading rate.

7.6

Applications of Finite Deformation Models

411

Fig. 7.8 The ratio of yield stress to temperature vs. plastic strain rate plot of PMMA, Nie (2005)

The temperature variation of Poisson’s ratio in the glass-rubber transition region can be assumed
to be identical to that of elastic modulus (θE, ΔE), while glassy and rubbery values νg , νr can be assumed to be constant. Poisson’s ratio in the glassy regime is given by Nie (2005), and Poisson’s ratio in the rubbery regime is assumed to be a value close to “0.5” to impose nearly incompressible conditions at high temperatures. Material parameters characterizing 
viscoplastic features of deformation associated with an intermolecular structure νoI , QI , nI , Bg , X B , V, αp , γ can be determined by using the ratio of yield stress to temperature (σ y/θ) versus plastic strain rate (νp) plots (Eyring plots) obtained experimentally as presented in Fig. 7.8. It is normally assumed that, when plastic flow occurs, the plastic rate is equal to the total strain rate. Since PMMA displays brittle failure under tension at very low temperatures without pronounced yielding, the data set for Eyring plots are limited. A master curve can be constructed by shifting the data set horizontally and vertically to a reference temperature (θref) as defined as

 1 1 θ θref

 σ  1 1 y Δ = Hv - ref θ θ θ

Δð log ðνpI ÞÞ = H h

ð7:237Þ ð7:238Þ

According to Richeton et al. (2005a, b), if θref is selected as θg, horizontal shift (Hh) and vertical shift (Hv) factors can be related to material properties through the following relation: Hh =

QI kB ln ð10Þ

H v = Bg - X gB θg =

ð7:239Þ γ 3

ð7:240Þ

412

7

Unified Micromechanics of Finite Deformations

where QI is the activation energy for plastic flow in intermolecular structure, kB is Boltzmann’s constant, Bg and X gB are parameters for bulk modulus, and γ is the parameter characterizing dynamic recovery in hardening characteristics of the intermolecular structure. The equation for the master curve can be derived from the plastic flow rule in intermolecular structure (Eq. (7.174)) as follows: σ y 2k B  αp  - 1 = sinh - 1 1θg V 3

p n1 ! νI I ν

ð7:241Þ

where nI is the number of thermally activated transitions necessary for plastic flow, V is the activation volume, and αp is the pressure sensitivity parameter. Temperaturedependent characteristic viscoplastic flow rate (ν) is defined in Eq. (7.175), which is derived from the flow rule of the theory of plasticity defined in Eq. (7.174). Effective stress ðτI Þ at the glass transition temperature, θg, can be approximated from Eq. (7.176) for a one-dimensional case. In this example, it is assumed that backstress (Mback) and plastic flow resistance in intermolecular structure (SI) vanish around glass transition temperature, whereas applied stress at yielding is equal to yield stress and normal pressure is one-third of applied stress: σ1 = σy

ð7:242Þ

Mback ffi 0

ð7:243Þ

SI ffi 0

ð7:244Þ

τI ffi σ 1 = σ y

ð7:245Þ

1 1 σ = σ 3 1 3 y  αp  τI = 1 σ 3 y

ð7:246Þ

pI =

ð7:247Þ

Using regression analysis for fitting a master curve to experimental data with shift factors Hh = 4900 K and Hv = - 40MPa material parameters, (νoI, QI, nI, Bg, XB, V, αp and γ) can be calculated. Activation volume (V ) and activation energy (QI) were assumed to be constant. Back-stress modulus asymptotically approaches zero around the glass transition temperature. The remaining parameter is the back-stress modulus definition ΔB which controls transition temperature range which was selected as 5 C to ensure a smooth change in hardening characteristics of the material in non-isothermal simulations (Fig. 7.9). Implementing a 1-D version of the constitutive model in software, like MATLAB, with isothermal conditions is an expedient way to determine the structure and the remaining parameters (hI, b, g) in intermolecular
 parameters associated with molecular network resistance νoM , QM , hM , nM , μM . These parameters cannot be directly observed in macroscale experiments. Hence, it is necessary to run simple 1-D simulations to estimate values for these parameters. Influence of parameters (hI, b, g) which control hardening-softening characteristics of response on

7.6

Applications of Finite Deformation Models

413

200

Back Stress Modulus (MPa)

180 160 140 120 100 80 60 40 20 0 0

20

40

60

80

100

120

140

160

Temperature (°C) Fig. 7.9 Temperature-dependent back-stress modulus of PMMA


 stress-strain curves and parameters νoM , QM , hM , nM , μM which control post-yield responses is presented in Fig. 7.10a–h. Arrows in Fig. 7.10a–h indicate an increasing trend of the corresponding parameter, while other parameters are held at a constant value. Parameters from intermolecular network mechanism (b, g, hI) control strain hardening-softening behavior in yield region, whereas parameters from molecular network mechanism ðhM , μM , νpM , QM , nM Þ control post-yield behavior at large deformations. 1-D MATLAB® simulations allow us to observe the influence of parameters on different aspects of stress-strain curves and achieve an overall acceptable curve fitting to stress-strain curves from isothermal test data. When parameters (b, g, hI) are determined, b is assumed to be both viscoplastic strain rate and temperature-dependent, g is assumed to be temperature-dependent, and hI is assumed as a constant. Parameters for molecular network ðhM , νpM , QM , nM Þ are assumed to be constant except for rubbery modulus which is taken as both temperature- and rate-dependent. The 
critical  value for the order parameter (ϕ ) and saturation value for network resistance  SM are obtained from test data of Ames et al. (2009). Initial values for a parameter (ϕ) and intermolecular resistance to plastic flow (SI) are usually assumed to be zero (while molecular
network resistance to plastic flow (SM) is assumed to be at 10% of  saturation value SM Þ. Complete list of material parameters included in the constitutive model is presented in Table 7.1: ϕðr, 0Þ = 0,SI ðr, 0Þ = 0,SM ðr, 0Þ = 0:1SM ðθo Þ,θo = θðr, 0Þ

414

7

Unified Micromechanics of Finite Deformations

40

b

30

stress

stress

30

40

20

10

10

0 0

(a)

0.2

0.4

strain

0.6

0.8

1

0 0

(b)

40

hI

stress

stress

0

0.4

strain

0.6

0.2

0.4

strain

0.6

hM

30

20

0.8

1

(d) 00

0.2

0.4

strain

0.6

1

0.8

1

no

m

M

M

30

stress

stress

0.8

40

30

20

10

0 0

1

10

40

(e)

0.8

40

10

(c) 0

0.2

50

30

20

g

20

20

10

0.2

0.4

strain

0.6

0.8

1

0

(f) 0

0.2

0.4

strain

0.6

40

100

n

M

80

30

Q

stress

stress

M

60

40

10

20

(g)

0 0

20

0.2

0.4

strain

0.6

0.8

1

(h)

0 0

0.2

0.4

strain

0.6

0.8

1


 Fig. 7.10 Influence of material parameters b, g, hI , hM , μM , νoM , QM , nM on stress-strain curves of PMMA. Gunel (2010)

7.7

Numerical Implementation of Dual-Mechanism Model

415

Table 7.1 Material parameters for PMMA constitutive model, Gunel (2010) Parameters

0.001

Parameters gr

32.58

ϕg

0

Δθ (K )

5

c1(K )

87.5 9

θϕ(K ) Δϕ(K )

0 10

θg (K )
 X gg K - 1

5 0.07

c2

116

X gϕ ð1=K Þ

-4.5 × 10-6

b1(MPa)

2.72 × 1010

θref g ðK Þ cg1 ðK Þ cg2

Parameters ϕg

9.55

1.73 × 10

X rϕ ð1=K Þ

0

b2(1/K )

-4.58 × 10-2

Eg(MPa) Er (MPa)

500 1.5

I gM I rM

7.04 4.6

b3

4.04 × 10-2 0.001

θE(K ) ΔE(K ) sE X gE ðMPa=KÞ X rE ðMPa=KÞ νg νr μgM ðMPaÞ μrM ðMPaÞ θμ(K ) Δμ(K ) sμ X gμ ðMPa=KÞ

-8 15 0.06 -15.05 0 0.31 0.49 9 0.5 5 14 0.03 -0.4

θM(K ) ΔM(K ) X gM ð1=K Þ X rM ð1=K Þ

X gS ðMPa=KÞ X rS ðMPa=KÞ Bg (MPa) X gB ðMPa=KÞ ΔθB (K )

-20 12 0 0.001 35 0.2 5 5 -0.01 0 10 2.15 5

X rμ ðMPa=KÞ

0

gg

7.97

7.7

-2

g

ν (s ) ref

-1

373

SgM ðMPaÞ SrM ðMPaÞ θS(K ) ΔS(K )

νpref ðs - 1 Þ νoI ðs - 1 Þ QI(J/K) V(m3) αp nI hI γ(MPa) νoM ðs - 1 Þ QM(J/K) hM nM ms(g/mol)

2.43 × 1012 1.56 × 10-19 1.39x10-27 0.21 2.17 40.42 60 4.33 × 109 1.30 × 10-19 14.43 5 100.13

Numerical Implementation of Dual-Mechanism Model

The dual-mechanism constitutive model is implemented numerically based on the staggered method with an isothermal split. In each loading step, the temperature value at the end of the time increment is taken as constant, and after mechanical equilibrium is satisfied, thermal equations are solved under a fixed configuration to update the temperature increment in the following time increment. Since this scheme is only conditionally stable, different time increments are used for isothermal cases at different temperatures, and a suitable time increment is chosen based on convergence study results. Objectivity (frame indifference) was ensured by using stress components and their conjugate rate components in reference configuration (in material coordinates). Final forms of the constitutive equations are obtained after a transformation of evolution equations to reference configuration. The evolution of elastic and plastic deformation gradient tensors is approximated by the exponential operator (Weber and Anand 1990). When an exponential operator is used in combination with the backward Euler scheme, plastic incompressibility, and symmetry of state variable tensors are conserved with sufficient accuracy by only

416

7

Unified Micromechanics of Finite Deformations

including the first two terms of series representation of the exponential function. Rate dependency of some material properties (E, μM, θg) is implemented in a numerical algorithm such that property values are calculated based on strain rates in the previous time increment and strain rates are updated based on initial and final deformation gradients at the end of the time increment. Therefore, equilibrium is satisfied at discrete time increments assuring thermodynamic consistency. Numerical integration of evolution equations of plastic deformation gradients FpI ,FpM and stretch-like internal variable A based on exponential mapping is obtained from Eqs. (7.172), (7.196), and (7.166) as presented below:
 ðFpI Þnþ1 = exp Δt ðDpI Þnþ1 ðFpI Þn
 ðFpM Þnþ1 = exp Δt ðDpM Þnþ1 ðFpM Þn

 Anþ1 = An exp Δt ðDpI Þnþ1 þ exp Δt ðDpI Þnþ1 An - γAn ln ðAn Þ
 exp Δt ðDp Þ I nþ1 pffiffiffi × 2

ð7:248Þ ð7:249Þ

ð7:250Þ

where Δt = tn+1 - tn is the time increment, γ is a constant representing dynamic recovery associated with kinematic hardening, νpI and νpM are equivalent plastic stretch rates in intermolecular and molecular network structure, and DpI and DpM are plastic stretch rate tensors in intermolecular structure and molecular network structure, respectively. Owing to unconditional stability of implicit time integration scheme, the classical backward Euler method is preferred for integration of evaluation equations of internal state variables, ς_ = f ðς, θÞ. For a generalized internal state variable (ς), numerical integration of evolution equation can be performed as ςnþ1 = Δtf ðςnþ1 , θnþ1 Þ þ ςn

ð7:251Þ

Further in-depth details of the implementation of the dual-mechanism viscoplastic model are given by Gunel (2010).

7.7.1

Simulating Isothermal Stretching of PMMA

A fully coupled temperature-displacement analysis is performed to investigate the adiabatic effect in stretching of PMMA. Eight-node linear brick elements (C3D8T) are used as element types in Abaqus finite element code. Convergence studies using different mesh sizes and time steps were conducted. In Fig. 7.11, the influence of time increments on the convergence of different aspects of material response is presented. In simulations, a rectangular prism model was uniaxially stretched at a displacement rate of 1 mm/s for 50 s, while the temperature was kept constant at 90 C. In convergence studies for time increment, maximum axial stress (σ),

Numerical Implementation of Dual-Mechanism Model

7.7

417

12

True Stress (MPa)

10

8

Δt=0.0001

6

Δt=0.001 4

Δt=0.01

2

Δt=0.1

0

(a)

0

0.2

0.4

True Strain

0.6

0.8

1

0.0445 0.0440

Strain Rate (s-1 )

0.0435 0.0430

νpI

νpM

d

0.0425 0.0420 0.0415 0.0410 0.0405

(b)

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

me increment (Δt, sec)

Fig. 7.11 Convergence study results for time increment in the viscoplastic model in terms of (a) true stress-strain curves and (b) strain rates p equivalent viscoplastic strain rate ðνpI , νM Þ, and equivalent stretch rate (d) must be monitored. True stress-strain curves presented in Fig. 7.11a indicate a fast convergence even for the largest time increment of Δt = 0.1 s. Convergence of equivalent plastic strain rates ðνpI , νpM Þ requires a small time increment, which also indicates a slow convergence of internal state variables (SI, SM, ϕ. . .). Another interesting point that can be observed from Fig. 7.11b is that equivalent viscoplastic strain rate in an intermolecular network ðνpI Þ is larger than the equivalent stretch rate (d ) which might seem like an error. However, this is due to the numerical method that is used to

418

7

Unified Micromechanics of Finite Deformations

0.050

12

0.045 0.040 0.035

8

0.030 6

0.025 0.020

4

σ11 νpI νpM d

2

0.015

Strain Rate (s-1)

True Stress (MPa)

10

0.010 0.005

0

0.000 0

10

20

30

40

50

60

me (sec) Fig. 7.12 Comparison of true stress and strain rate histories

capture strain-softening characteristics and is controlled by the parameters b and g in the constitutive model. In Fig. 7.12, true stress, equivalent viscoplastic strain rates ðνpI , νpM Þ, and equivalent stretch rate (d ) histories are presented. At the beginning of a simulation, the equivalent stretch rate (d ) is larger than the equivalent plastic strain rate in both p p mechanisms
e e  ðνI , νM Þ, and continuous increase in elastic strain in both mechanisms F I , F M causes an increase in stress level up to the onset of yielding at which equivalent viscoplastic strain rate in intermolecular mechanism ðνpI Þ becomes larger than the equivalent stretch rate (d ) and remains larger over some period which corresponds to the yielding and strain-softening region. During this period, the negative elastic strain rate results in a decrease in elastic strain in the intermolecular mechanism FeI and hence decrease in stress level TI. After strain-softening, equivalent viscoplastic strain rate in intermolecular mechanism ðνpI Þ remains equal to the equivalent stretch rate (d). In the meantime, the difference between equivalent viscoplastic strain rates in the molecular network ðνpM Þ and equivalent stretch rate (d) starts to increase which leads to an increase in elastic strain in the molecular network FeM . As a result, stress TM level in the molecular network also increases which describes strain-hardening characteristics of material due to chain locking in the molecular network. However, further stretch in the molecular network continuously increases stress TM level in the molecular network causing the increase in driving stress for plastic flow (τM) which in return starts to increase equivalent viscoplastic strain rate in the molecular network ðνpM Þ once again. Therefore, the change in stress level at large deformations gradually becomes smaller. Mesh sensitivity of results is also very important in finite element analysis. In Fig. 7.13, the influence of the size of mesh seeds (in Abaqus) on the convergence of

7.7

Numerical Implementation of Dual-Mechanism Model

419

different aspects of material response is presented. A rectangular prism sample was uniaxially stretched at a displacement rate of 1 mm/s for 50 s, while the temperature was kept constant at 100 C. In mesh sensitivity studies, axial stress (σ) history, equivalent viscoplastic strain rate ðνpI , νpM Þ, and equivalent stretch rate (d ) are all monitored. In Fig. 7.13a, results for different mesh sizes are presented. The seed size encircled in Fig. 7.13a is used in the final simulations for PMMA since sufficient convergence is achieved at this mesh size for true stress value (σ) and equivalent plastic strain rate values ðνpI , νpM Þ as shown in Fig. 7.13b, c, respectively. A coarser mesh than the selected one (seed size = 3) would also be reasonable for convergence of (true) stress values except for the coarsest one (seed size = 10). However, a finer mesh is always desirable for convergence of equivalent strain rate values and associated state variables (Fig. 7.13c). Results shown in Fig. 7.14 indicate that simulation results and experimental measurements are reasonably in good agreement. The constitutive model is especially successful in describing temperature and rate dependence of material response as well as strain-softening and strain-hardening characteristics. Due to the limited size of the data set used for the determination of viscoplastic flow rule for intermolecular mechanism, it is really difficult to determine material parameters involved in the viscoplastic model which resulted in some minor differences in response. Dual-mechanism viscoplastic model for PMMA is verified and validated for more challenging non-isothermal conditions as presented in the following section. Entropy production values are calculated as an average of cumulative entropy production of all Gauss integration points at the cross-section of the center of the narrow section of the test sample. The difference between mean entropy production and maximum entropy production in the cross-section is less than 5% for all isothermal test simulations which indicates that no damage localization is predicted at the cross-section which is similar to the case of brittle failure of materials. As brittle failure takes place in the form of sudden rupture of the specimen with concurrent crack formation and propagation at many locations, damage localization is not observed contrary to ductile failure of materials for which damage localization is an essential part of the crack initiation stage around microscopic defect regions, while crack propagation takes place over a longer time compared to brittle failure. Therefore, damage quantification based on average entropy production in the crosssection is a proper method for brittle failure, while local thermodynamic state index is essential for damage quantification in ductile failure. It is also important to note that damage evolution and material degradation due to mechanical load are assumed to result from only plastic energy dissipation excluding all other entropy generation mechanisms for the sake of simplicity. Therefore, in order to fully account for the evolution of microscopic defects that lead to stiffness and strength degradation in material, it is necessary to include all entropy generation mechanisms associated with energy loss associated with new surface creation at microscopic defect regions, heat generation, chemical reactions, aging, etc. The latter approach would be more accurate. It is possible to account for the difference between failure in tension and

420

7

Unified Micromechanics of Finite Deformations

number of elements (in log scale)

1.E+05

1.E+04

1.E+03

1.E+02

1.E+01 0

(a)

2

4

6

8

12

10

seed size

9 8

True Stress (MPa)

7 6

seed size=10 seed size=5 seed size=2.5 seed size=1.25 seed size=0.625 seed size=3

5 4 3 2 1 0

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

True Strain 0.070 0.065

νp1

Strain Rate (s-1)

0.060

νp2

d

0.055 0.050 0.045 0.040 0.035 0.030

(c)

0

2

4

6

8

10

12

seed size

Fig. 7.13 Mesh sensitivity of the ABAQUS model in terms of (b) true stress-strain curves and (c) strain rates (a) for different mesh densities

7.7

Numerical Implementation of Dual-Mechanism Model

35

ISO-H30 (exp.) ISO-H60 (exp.) ISO-H90 (exp.) ISO-H30 (sim.)

ISO-M30 (exp.)

35

ISO-M60 (exp.)

30

ISO-M90 (exp.)

25

ISO-M30 (sim.)

20

ISO-M60 (sim.)

15

ISO-M90 (sim.)

10 5 0

0.1

0.2

0.3

0.4

0.5

Strain

(c)

30

Stress (MPa)

(b)

40

Stress (MPa)

Stress (MPa)

(a) 50 45 40 35 30 25 20 15 10 5 0

421

0

0

0.2

0.4

0.6

0.8

1

1.2

Strain ISO-L30 (exp.) ISO-L60 (exp.)

25

ISO-L90 (exp.)

20

ISO-L30 (sim.)

15

ISO-L60 (sim.)

10

ISO-L90 (sim.)

5 0

0

0.5

1

Strain

1.5

2

Fig. 7.14 Comparison of Abaqus fea simulation and testing results for isothermal testing of PMMA in (a) H series, (b) M series, and (c) L series tests

failure in compression or predict damage evolution in the case of ductile failure by deriving a more comprehensive thermodynamic fundamental equation. Total irreversible thermal entropy production in ISO-TEST simulations (Sirr, ther, ) test is completely created by the transfer of heat generated due to plastic dissipation during isothermal stretching. Temperature rise due to plastic dissipation in isothermal stretching is very small; hence, thermal gradients within specimen during testing are also small, as a result producing a negligible amount of irreversible entropy production in comparison with Sirr, ther, pre-test and total irreversible mechanical entropy production during isothermal tests (Sirr, mech, test). Mechanical damage in materials can be described as the formation of microcracks and voids at the micro-level with a corresponding degradation of stiffness at the macro-level. On the other hand, thermal damage in materials is the deterioration of material properties which is the weakening of chemical bonds between molecules or change in length of the molecular bond. The evolution of thermal damage is much slower than the evolution of mechanical damage. Remarkable thermal gradients over long periods are necessary to induce permanent cracks and voids in materials, e.g., freezing-thawing cycles, thermal shock, etc. Therefore, mechanical and thermal damage induced in the material has different entropy generation sources, and both must be defined separately in the case of thermomechanical loading. As failure in isothermal stretching of PMMA was completely induced by mechanical entropy generation mechanisms, TSI must be based on the evolution of irreversible mechanical entropy production. In metal fatigue tests, it is found that cumulative-specific

422

7

Unified Micromechanics of Finite Deformations

entropy generation is constant at the time of failure, which is independent of the structural geometry of a sample, load intensity, and loading rate for small frequencies. The critical-specific entropy value is usually referred to as the fatigue fracture entropy (FFE). It is observed by many researchers that the rate dependence of critical entropy density is negligible (at small frequencies), while critical entropy density increases with increasing temperature because of increasing plastic dissipation before failure. Mechanical damage is completely based on irreversible mechanical entropy production; however, due to intrinsic heat dissipation, the temperature variation of plastic dissipation is also observed in critical entropy production values. Critical entropy is also referred to as the fatigue fracture entropy (FFE) in some publications. The logarithm of critical entropy value Sirr, cr for PMMA as a function of temperature is given by Gunel (2010): log ðSirr,cr Þ = 0:0405θ - 2:5653

ð7:252Þ

The selection of critical entropy is based on an assumed failure definition. If an engineer defines a 50% reduction in mechanical stiffness or a 10% increase in electrical resistance as a failure, that will be a different critical entropy value than if the complete fracture is defined as the failure. In Eq. (7.252) complete fracture of the sample is the failure definition. It is much easier to define a thermodynamic state index value as a failure point than using the critical entropy, which must be measured, as a failure.

7.7.2

Simulating Non-isothermal Stretching of PMMA

Dual-mechanism constitutive model (viscoplastic) is also applied to non-isothermal stretching of PMMA to large deformations with specific emphasis on the transition of material response around the glass transition temperature. In non-isothermal mechanical tests on PMMA, the temperature of samples spanned over θg ± 50o C range during which a complete glass transition is bound to take place and any change in material response due to this large temperature difference could be monitored. It is well-known that for some polymers viscoplastic behavior right around the glass transition temperature exhibits significant change. This change is manifested as a significant increase in stiffness during the cooling of a specimen. Contrary to other polymer constitutive models, here it is shown that this PMMA model does not experience any abrupt changes in response during stretching under non-isothermal conditions, and the response is continuous and smooth. It is important to point out that material constitutive models for finite-strain behavior of amorphous polymers can only be verified in non-isothermal loading conditions. Concurrent heat transfer with axial stretching of PMMA requires a fully coupled temperature-displacement analysis.

7.7

Numerical Implementation of Dual-Mechanism Model

423

Fig. 7.15 Axial force histories in non-isothermal experiments and simulations of PMMA for H series (0.9 mm/s forming rate)

Axial force histories of H, M, and L series with respect to normalized time (tn) are presented in Figs. 7.15, 7.16, and 7.17. Simulation and experiment results in terms of axial force histories are reasonably in good agreement. A unified mechanics formulation of plastic flow rule for intermolecular structure and appropriate material property definitions ensures smooth transition of response around the glass transition temperature. Here we conclude our discussion of the large deformation analysis of polymers.

424

7

Unified Micromechanics of Finite Deformations

Fig. 7.16 Axial force histories in non-isothermal experiments and simulations of PMMA for M series (0.09 mm/s forming rate)

References

425

Fig. 7.17 Axial force histories in non-isothermal experiments and simulations of PMMA for L series (0.009 mm/s forming rate)

References Ames, N. M., et al. (2009). A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: Applications. International Journal of Plasticity, 25(8), 1495–1539. Anand, L. (1986). Moderate deformations in extension-torsion of incompressible isotropic elastic materials. Journal of the Mechanics and Physics of Solids, 34, 293–304. Anand, L., & On, H. (1979). Hencky’s approximate strain-energy function for moderate deformations. Journal of Applied Mechanics, 46, 78–82. Anand, L., et al. (2009). A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: Formulation. International Journal of Plasticity, 25(8), 1474–1494. Arruda, E. M., & Boyce, M. C. (1993). Evolution of plastic anisotropy in amorphous polymers during finite straining. International Journal of Plasticity, 9(6), 697–720. Arruda, E. M., Boyce, M. C., & Jayachandran, R. (1995). Effects of strain rate, temperature and thermomechanical coupling on the finite strain deformation of glassy polymers. Mechanics of Materials, 19(2–3), 193–212. Bergström, J. S., & Boyce, M. C. (1998). Constitutive modeling of the large strain time-dependent behavior of elastomers. Journal of the Mechanics and Physics of Solids, 46(5), 931–954. Boyce, M. C., Socrate, S., & Llana, P. G. (2000). Constitutive model for the finite deformation stress-strain behavior of poly(ethylene terephthalate) above the glass transition. Polymer, 41(6), 2183–2201. Fotheringham, D. G., & Cherry, B. W. (1978). The role of recovery forces in the deformation of linear polyethylene. Journal of Materials Science, 13(5), 951–964.

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Fotheringham, D., Cherry, B. W., & Bauwens-Crowet, C. (1976). Comment on “the compression yield behavior of polymethyl methacrylate over a wide range of temperatures and strain rates”. Journal of Materials Science, 11(7), 1368–1371. Francisco, P., Gustavo, S., & Élida, B. H. (1996). Temperature and strain rate dependence of the tensile yield stress of PVC. Journal of Applied Polymer Science, 61(1), 109–117. Gent, A. N. (1996). A new constitutive relation for rubber. Rubber Chemistry and Technology, 69(1), 59–61. Gunel, E. M., & Basaran, C. (2009). Micro-deformation mechanisms in thermoformed alumina trihydrate reinforced poly(methyl methacrylate). Materials Science and Engineering: A, 523(1–2), 160–172. Gunel, E.M., & Basaran, C. (2010). Stress whitening quantification of thermoformed mineral filled acrylics. Journal of Engineering Materials and Technology, 132(3), 031002–031011. Gunel, E. M., & Basaran, C. (2011a). Damage characterization in non-isothermal stretching of acrylics: Part II experimental validation. Mechanics of Materials, 43(12), 992–1012. Gunel, E. M., & Basaran, C. (2011b). Damage characterization in non-isothermal stretching of acrylics: Part I theory. Mechanics of Materials, 43(12), 979–991. Hashin, Z., & Shtrikman, S. (1963). A variational approach to the theory of the elastic behavior of multiphase materials. Journal of the Mechanics and Physics of Solids, 11(2), 127–140. Mustafa Eray Gunel (2010), Large Deformation Micromechanics of Particle Filled Acrylics at Elevated Temperatures, Dissertation submitted to Department of Civil, Structural and Environmental Engineering, University at Buffalo, SUNY Mandel, J. (1972). Plasticite classique et viscoplasticite (Lecture Notes). International Center for Mechanical Sciences. Nie, S. (2005). A micromechanical study of the damage mechanics of acrylic particulate composites under thermomechanical loading. (Ph.D. Dissertation). In Civil, structural, and environmental engineering. State University of New York at Buffalo. Palm, G., Dupaix, R. B., & Castro, J. (2006). Large strain mechanical behavior of poly(methyl methacrylate) (PMMA) near the glass transition temperature. Journal of Engineering Materials and Technology, 128(4), 559–563. Richeton, J., et al. (2005a). A formulation of the cooperative model for the yield stress of amorphous polymers for a wide range of strain rates and temperatures. Polymer, 46(16), 6035–6043. Richeton, J., et al. (2005b). A unified model for stiffness modulus of amorphous polymers across transition temperatures and strain rates. Polymer, 46(19), 8194–8201. Richeton, J., et al. (2006). Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: Characterization and modeling of the compressive yield stress. International Journal of Solids and Structures, 43(7–8), 2318–2335. Richeton, J., et al. (2007). Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates. International Journal of Solids and Structures, 44(24), 7938–7954. Srivastava, V., & Anand, L. (2010). A thermo-mechanically-coupled large-deformation theory for amorphous polymers in a temperature range which spans their glass transition. International Journal of Plasticity, 26(8), 1138–1182. Weber, G. & Anand, L. (1990). Finite Deformation Constitutive Equation and a Time Integration Procedure for Isotropic, Hyperelastic-Viscoplastic Solids. Computer Methods in Applied Mechanics and Engineering, 79, 173–202. Williams, M. L., Landel, R. F., & Ferry, J. D. (1955). The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. Journal of the American Chemical Society, 77(14), 3701–3707.

Chapter 8

Unified Mechanics of Metals Under High Electrical Current Density: Electromigration and Thermomigration

8.1

Introduction

Electromigration and thermomigration are irreversible mass diffusion mechanisms under high current density and high-temperature gradient, respectively. However, thermomigration can take place alone in the absence of electrical current, while electromigration cannot happen without thermomigration due to Joule heating, except for special circumstances where boundary conditions prevent a thermal gradient from happening.

8.2

Physics of Electromigration Process

When a metal is subjected to an electrical potential gradient, the current enters from the anode side and travels to the cathode side, and the electrons travel from the cathode to the anode side. Electromigration is a mass diffusion-controlled phenomenon. When a conductor is subject to a high current density, the so-called electron wind transfer part of its momentum to the atoms (or ions) to make the atoms (or ions) move in the direction of the current. As a result, the degradation of the metal conductor occurs mainly in two forms: in the anode side, the atoms will accumulate and finally form hillocks and the vacancy concentration on the cathode side will form voids. Both hillocks and voids will cause the degradation of the material and eventual failure. The damage evolution due to electromigration can be modeled as an irreversible mass transport process. The purpose of this chapter is to present the formulation of modeling the electromigration and thermomigration-induced material degradation process using the unified mechanics theory

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_8

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428

8

Unified Mechanics of Metals Under High Electrical Current Density. . .

The physics of electromigration has been extensively investigated. It is believed that in 1861 French physicist M. Gerardin was the first to observe electromigration in liquid metal alloys. Skaupy (1914) introduced the concept of “electron wind,” which facilitated the modeling of electromigration using the diffusion theory. Black (1969) established an empirical relationship between the mean time to failure and current density for confined thin films on a substrate. The experiments by Blech (1976) revealed that the stress gradient could act as a counterforce to electromigration under high current density. In addition, for thin films, Blech (1976) proposed a length scale called “Blech’s critical length” below which mass diffusion due to electron wind force is counterbalanced by stress gradient driving force in the opposite direction. To solve an arbitrary boundary value problem with irregular boundaries or an initial value problem with arbitrary conditions and composite material properties involving electromigration, there must be a rational mechanics-based current density-strain constitutive relation and a thermodynamic fundamental equation of the material. Then this formulation can be implemented in a finite element method (or any other computational mechanics procedure) to solve any boundary value problem. The classical definition of electromigration refers to the structural damage caused by ion transport in metals because of high current density. Electromigration is usually insignificant or nonexistent at low current density levels. Quantification of what “high current density” is studied extensively by Ye et al. (2003a, b, c, d, e, f, 2004a, b, c, d, 2006), and Basaran et al. (2003). Electromigration is mass transport in a diffusion-controlled process under certain driving forces. The driving force here is more complicated than what is involved in a pure diffusion process, in which the concentration gradient of the moving species is the only driving force component. The electrical driving forces for electromigration consist of the electron wind force and the direct field force. The electron wind force refers to the effect of momentum exchange between the moving electrons and the ionic atoms when an electrical charge—a direct field force—is applied to a conductor with metallic bonds. When current density, which is proportional to the electron flux density, is high enough, this momentum exchange effect becomes significant, resulting in a noticeable mass transport referred to as electromigration. Low melting point alloys when used at elevated homologous temperature (>0.4Tmelt Kelvin) are prone to have considerable atomic diffusivity. For example, solder joints in microelectronics packaging are prime examples of this category

8.2.1

Driving Forces of Electromigration Process

The Electric Field The electrical current generates two driving mechanisms; one is attributed to the electron wind forces, as originally suggested by Skaupy (1914), which refers to the effect of the momentum exchange between the moving electrons and the ionic atoms. This momentum exchange happens because of the scattering of free valence

8.2

Physics of Electromigration Process

429

electrons. Scattered electrons collide with metal atoms and push them in the direction of electron flow (Seith 1955). On the other hand, atoms move in the opposite direction of the applied electric field when they are ionized, the latter mechanism, the static force of the electric field, is considered the direct field force. The net effect of these two forces is the so-called electrical field driving force of electromigration. Stress Gradient When atoms diffuse from the cathode side to the anode side, they leave behind vacancies on the cathode side and mass accumulation on the anode side. As a result, there is tensile stress on the cathode side and compressive stress on the anode side. This stress differential is responsible for the stress gradient. Of course, the mechanical stress gradient influences the electromigration process. There is an interaction between the stress gradient and the electromigration driving forces. Stress gradient can counteract or enhance the electromigration process, depending upon the interaction among all diffusion driving forces. Essentially, stress gradient is another driving force of mass transport. However, in thermomigration in the absence of external stress gradients, the mass moves from the hot side to the cold side, and compression on the cold side and tension on the hot side develop, which counters the thermal gradient-induced force. Thermal Gradient The Joule heat generated under high current density is highly localized. Hence, there is a thermal gradient in the medium, which leads to thermomigration. The thermal gradient is one of the strongest driving forces of diffusion. Thermomigration cannot be ignored especially when the thermal gradient is large, in which case the thermomigration can be the dominant diffusion driving force. The physical explanation behind thermomigration is not well understood. However, recent literature shows that thermomigration could play a significant role in the electromigration-induced failure. Atomic Vacancy Concentration Gradient It has been estimated that the mass flux due to vacancy concentration gradient is small compared to those induced by electrical field forces, stress gradient, and thermal gradient. The atomic vacancy concentration gradient is usually small at the beginning compared to the other diffusion driving forces. However, as the mass migration progresses, vacancy concentration gradient increases significantly. At a certain point in time, the vacancy concentration gradient becomes large enough to slow the diffusion.

8.2.2

Laws Governing Electromigration and Thermomigration

Vacancy diffusion is governed by the vacancy conservation equation; mechanical deformation is governed by Newtonian mechanics; heat transfer is governed by

430

8

Unified Mechanics of Metals Under High Electrical Current Density. . .

Fourier’s law; an electric field is governed by Maxwell’s equation of conservation of charge; and degradation of the material is governed by the laws of thermodynamics. Figure 8.1 shows a microelectronics solder joint before and after electromigration/ thermomigration failure. In the following section, all these laws are used to model the unified mechanics of electromigration and thermomigration processes. When the only acting load is an electrical potential field, the primary deformation mechanism is due to atomic diffusion, which is a mass transport by atomic motion process. Diffusion in solids occurs only at the atomic or molecular level. However, diffusion in solids may be observed on a macroscopic scale given enough time. Diffusion that results in the net transport of matter over macroscopic distances is considered a nonequilibrium process. It does not stop until the phase eventually achieves fully thermodynamic equilibrium or diffusion driving force is removed or counterbalanced. The laws of diffusion were first developed by Adolf Fick when studying the diffusion of the saltwater system (Fick 1855). Fick’s diffusion laws establish the mathematical relationship between the rate of diffusion and the concentration gradient, which is the driving force of mass transport. Fick’s First Law Fick discovered by direct observation that the magnitude of the mass flux is proportional to the magnitude of the concentration gradient in the isotropic continuum given by

AI

Silicon Die

140 μm Underfill

Solder

100 μm

Ni UBM

Cu PCB

Underfill

Ni UBM

Solder

Cu Plate 40µm 700X

Fig. 8.1 Electronics solder joint profile after electromigration and thermomigration failure

8.2

Physics of Electromigration Process

431

J = - D∇C

ð8:1Þ

where: J is the mass flux, which is defined as a vector quantity, with units of (molm-2s-1).
Q d D is diffusivity, with units of m2/s; it is a function of temperature D = D0 exp - RT where D0 is temperature-independent material constant, Qd is the activation energy of diffusion, R is the gas constant (8.31 J/mole K), and T is the temperature in Kelvin. C is the concentration of the matter in question. Combining the mass conservation equation with Fick’s first law yields Fick’s second law. Fick’s Second Law: Nonsteady State Diffusion Fick’s second law can be given in its most elementary form as ∂C = D∇2 C ∂t

ð8:2Þ

Until the twentieth century, diffusion in materials was addressed at a microscopic scale only. Einstein showed the consistency of the random motion of microscopic particles in the presence of molecules. Einstein derived a relationship for the chaotic motion of small particles as vD =

DF kB T

ð8:3Þ

where vD is the drift velocity, unit m/s; D is the diffusion coefficient; F is the driving force of the diffusion, unit Newton; kB is Boltzmann’s constant; and T is absolute temperature, Kelvin. From Eq. (8.3) we can easily deduce that for self-diffusion, the driving force due to concentration gradient has the following form: F = - k T∇C

ð8:4Þ

In addition, Fick’s second law has a more general form as follows: ∂C D∇F = kT ∂t

ð8:5Þ

where F is the driving force of the diffusion. In the presence of electrical current density, the driving force has four primary components:

432

8

Unified Mechanics of Metals Under High Electrical Current Density. . .

F=

X

! Fi

= F em þ F tm þ F σ þ F s

ð8:6Þ

i

where Fem is the electron wind force, Ftm is the driving force due to temperature gradient, Fσ is the stress gradient driving force, and Fs is the driving force due to chemical potential gradient. Each driving force will be introduced separately in the following sections. These forces also lead to grain coarsening and phase changes in the metals. However, we are primarily interested in the mechanical implications in here.

8.2.3

Electromigration Electron Wind Force

The first-ever observation of electromigration occurred long before it became an engineering problem. In 1861, a French scientist, M. Gerardin, first noticed the phenomenon of electromigration. Nevertheless, at that time, no real scientific explanation was provided for this phenomenon. Although Michael Faraday’s discovery of electrochemistry a few decades earlier may have suggested an explanation, it would have been incorrect because electrons had not been discovered yet. Without electrons, a proper scientific explanation would not have been possible. The first reasonable scientific explanation of electromigration driving force came from Skaupy (1914), who recognized that in metals the moving free conduction band electrons could drag atoms along through a frictional force (scattering), which he called “electron wind.” After that, a series of publications by Huntington and Grone (1961) Bosvieux and Friedel (1962), and Genoni and Huntington (1977) proposed that at high current densities, conduction band electrons collide (scatter) with metal atoms/ions and push them in the direction of the electron flow. In other words, the “electron wind” force drives positive ions (or atoms) in the direction of the electron flow, which is opposite to the direction of the electrostatic force of the electric field (Fig. 8.2). In electromigration studies, it has been customary to distinguish two types of contributions: one from the scattering of the electrons with the ions in question (the electron wind term) and that from the direct interaction of an atomic charge with the applied electric potential field (the direct force) (Landauer and Woo 1974). There→ fore, the electromigration driving force, F em, exerted on an ion can be expressed as a summation of these two forces acting on ions: →

→

→

F em = F wind þ F direct

→

= ð - Z wind þ Z el Þe E →

= Ze E

= Z  eρj

ð8:7Þ

8.2

Physics of Electromigration Process

433

Fig. 8.2 Kinematics of electromigration

→

where e is the electron charge, E is the electric field potential (V/cm), ρ is the resistivity of the metal (ohmmeter (Ωm)), j is the current density (Amp/cm2), and Z* is a dimensionless quantity called the effective charge (also called an effective nuclear charge or the affective valence), which is the combined net attractive positive charge of nuclear protons acting on valence electrons. When a charged particle penetrates through condensed matter, it polarizes the medium, which in turn reacts, back slowing down the penetrating particle. To introduce a measure of the inertia of ions, Brandt and Kitagawa (1982) introduced the concept of effective charge. The effective charge, Z*, relates to how a diffusing atom interacts with the conduction band electrons. The interaction with the conduction band electrons is a complex function of the electronic structure, a quantum mechanical effect. The effective charge is defined in terms of the stopping power of the charge Z(S) and the power of the proton moving in the same medium at the same velocity (Sp) (Barberan and Echenique 1986):
1 Z  = S=Sp 2

ð8:8Þ

Z* is commonly negative in metal conductors due to the influence of the “electron wind” Ye et al. (2004a, b, c, d). For high conductivity metals, i.e., Al or Cu, the value of Z* is about [-1.0] (Lloyd et al. 2004). For pure tin, the Z* value was reported to range from [-80] to [-160] (Kuz’menko and Osirovskii 1962; Lodding 1965; Sun and Ohring 1976). For single-crystalline Sn, the Z* value was reported to range from [-10] to [-16], while for polycrystalline, Sn is [-12] at about 200 °C. These values seem to be favored by Sorbello’s electromigration theory, which predicted the effective charge number for Sn at 185 °C as [-10] (Sorbello 1973). His theory is based on the pseudopotential calculation of the driving force for atomic migration in metals in the presence of electric current. The fact that Z* is temperature-dependent is also reported by Singh and Ohring (1984) (see Fig. 8.3).

434

8

Unified Mechanics of Metals Under High Electrical Current Density. . .

Fig. 8.3 Effective charge number versus temperature relation for Sn (After Singh and Ohring (1984))

Fig. 8.4 Energymomentum relationship that determines the value of effective mass. (After Lloyd (2004))

Energy,E

Ef m* ∝

1 d 2E dk 2

Momentum,k

The effective charge, Z*, is determined by the effective mass, m*, of electrons available for interaction (scattering) in which the energy produced is not exceeding the Fermi energy, Ef, which is the total energy difference between the lowest occupied conduction band and the highest conduction band, of uncharged metal at absolute zero temperature. The effective mass m* is proportional to the second derivative in the energy-versus-momentum, E - k relationship, a quantum mechanical effect near Fermi level energy E = Ef (Lloyd et al. 2004). Above the inflection point, m* is negative; below it m* is positive (Fig. 8.4).

8.2

Physics of Electromigration Process

8.2.4

435

Temperature Gradient Diffusion Driving Force

Thermomigration has been known since Ludwig (1856). As Chen et al. (2012) stated, when an inhomogeneous binary alloy is annealed at an elevated isotropic temperature field with no thermal gradient, it will become homogeneous. However, when a homogeneous binary alloy is annealed under a temperature gradient, the alloy will become inhomogeneous. This de-alloying phenomenon is called Soret effect after Soret (1879). The phenomenon is caused by mass transport due to temperature gradient. The role of thermomigration in the presence of high current density in microelectronics solder joints has been ignored until recently. It has been shown by Ye et al. (2003a, b, c, d, e, f), Abdulhamid et al. (2008), and Basaran et al. (2008) that thermomigration plays a significant role in the presence of high current density. Diffusion driving force due to thermal gradient can be bigger than electron wind forces. If they are in the opposite directions, the thermal gradient can negate the electron wind force and if they are in the same direction can hasten the failure process significantly. The temperature gradient-induced diffusion driving force is usually represented by Huntington (1972): F tm =

Q ∇T T

ð8:9Þ

where Q* is the heat of transport, which is the isothermal heat transmitted by the moving atom in the act of jumping a lattice site less the intrinsic enthalpy, and T is the temperature in Kelvin. The most accepted thermomigration theory is very much based on the same concepts as electromigration. In electromigration, the force acting on the diffusing atom is the momentum exchange due to the collisions (scattering) of diffusing atoms and scattered electrons in the conduction band. However, in thermomigration, in the absence of electrical current, the driving force in the direction of the temperature gradient results from the fact that the energy, therefore the momentum of the atoms (phonons) at higher temperatures is greater than that of the lower temperature ones. This gradient in the momentum produces a driving force for mass transport from the hot side to the cold side. Based on this argument, Q* for metal with a high Z* value should also be high. Similar to Z*, the sign of Q* can either be positive or negative depending on the effective mass of the charge carriers. According to Huntington (1972), thermomigration driving force can be divided into an intrinsic part and a part coupled to carrier electrons and phonons: Q = Qin þ Qel þ Qph

ð8:10Þ

436

8

Unified Mechanics of Metals Under High Electrical Current Density. . .

The intrinsic considerations include the energy transported by the moving atom, the energy required to prepare a place [a lattice site] to receive it, and, for the vacancy diffusion mechanism, the formation energy for the counter-moving vacancy, which is given by Q = βEm - E f

ð8:11Þ

where Em and Ef are the energies of motion and formation for the moving atoms, respectively, and β is a dimensionless fraction less than 1 which gives the fraction of the kinetic energy carried by the moving atom. Q* can be either positive or negative depending on the magnitude of the kinetic energy and the formation energy. According to Campbell and Huntington et al. (1969), β was reported to be about 0.81; hence, Q* is estimated to be about 1ev, which is in accordance with the experimental measurement of 0.23 ev.

8.2.5

Stress Gradient Diffusion Driving Force

Due to mass transport from cathode to anode, and from the high-temperature side to low-temperature side, there are more vacancies on one side and more mass accumulation on the other side. Consequently, on the mass accumulation side, atoms are subjected to compression, and on the vacancy accumulation side, atoms are subjected to tension. Therefore, there is a stress gradient between two sides of the continuum. This stress gradient is also another diffusion driving force. Tensile stress makes diffusion easier and compressive stress makes diffusion more difficult to happen. The role of mechanical stress gradients as a diffusion driving force in conjunction with electromigration was first explored in a series of experiments by Blech (1976) and Blech and Herring (1976). It was shown that a critical current density threshold exists below which mass transport is stopped in a thin-film aluminum sample. This threshold was found to be inversely proportional to the thin-film Al stripe length. It was suggested that electromigration in a short segment tends to induce back-stress when they did an electromigration test on a set of short Al thin-film strips deposited on a substrate as depicted in Fig. 8.5.

Void

Void AI ~

e

Extrusion

Extrusion

Length

AI

AI TIN

e~

Fig. 8.5 Schematic of Al thin-film strips of different lengths showing stress gradient influence due to electromigration

8.2

Physics of Electromigration Process

437

Figure 8.5 shows the effect of stress gradient on Al thin film on a substrate. The Al thin-film strips are deposited on a substrate of TiN. It can be observed that the longer the length, the larger the depletion, and extrusion on the cathode and anode sides, respectively, due to electromigration. When the thin film is below a “critical length,” there is no observable depletion or extrusion. Stress gradient diffusion driving force is neutralizing the electron wind force-driven diffusion. The dependence of mass transport on the strip length was explained by the effect of the stress gradient. When the electromigration process transports Al atoms from cathode to anode, the anode will be in compression, while the cathode will be in tension. Based on the Nabarro-Herring model of equilibrium vacancy concentration in a stressed solid, the tensile region has more, and the compressive region has fewer, vacancies than the unstressed region. As a result, there exists a vacancy concentration gradient between the cathode side and the anode side. The stress gradient induces an atomic flux of mass diffusing from anode to cathode, and it opposes the mass flux driven by electromigration from the cathode side to the anode side. Blech’s critical length comes into effect when a thin film on a substrate is subjected to current density with blocking boundary conditions. A stress gradient provides a mass transport force in opposition to electromigration. If the current density is sufficiently low, a stress gradient will be generated that can stop electron wind force-driven diffusion. The stress gradient depends on the length of the strip; the shorter the thin film strip, the greater the gradient. At a certain length defined as the “critical length,” the stress gradient is large enough to counterbalance electromigration force so no depletion at the cathode or extrusion at the anode occurs. The critical length can be calculated when the net force due to electromigration and stress gradient is zero, as shown in Eq. (8.12). The force equilibrium equation in the x-direction for an atom can be rewritten as F = F e þ F σ = - Z  eρj þ f Ω

∂σ =0 ∂x

ð8:12Þ

where f is the atomic vacancy relaxation ratio, the ratio of atomic volume to the volume of an atomic vacancy, Ω is the atomic volume, and σ is the spherical stress. Solving Eq. (8.12), ∂σ Z  eρj = fΩ ∂x fΩ j∂x =  ∂σ Z eρ

ð8:13Þ

The integration of Eq. (8.13) produces one of the most important characteristics of electromigration. Given maximum stress that can be sustained by the conductor, the product of current density and the length of the conductor determines whether electromigration diffusion can occur or not. This product was discovered by I. A. Blech and is referred to as the “Blech product” (Blech 1976; Blech and Herring 1976):

438

8

Unified Mechanics of Metals Under High Electrical Current Density. . .

jlB =

fΩ ðσ - σ 0 Þ Z  eρ m

ð8:14Þ

where lB is known as Blech’s critical length, σ m is the maximum spherical stress that can be supported by the conductor, while σ 0 is the initial spherical stress in the conductor. If the product of the current density and the length is exceeded, electromigration mass transport can occur; if it is not, no electromigration can happen. If all thin-film conductors could be designed so that jlB stays below the critical value, electromigration can be avoided since mass transport comes to a complete halt. However, this scenario is only true for a pure metal thin film on a substrate, because it assumes that metal is not an alloy and there is no temperature gradient acting on the system.

8.3

Laws of Conservation

Electromigration is an electron flow-assisted diffusion process. The process can be assumed to be controlled by a vacancy diffusion mechanism, in which the diffusion takes place by vacancies switching lattice sites with adjacent atoms. In isothermal conditions, the process is driven by electrical current caused by mass diffusion, stress gradient-induced diffusion, and diffusion due to atomic vacancy concentration. In the presence of electrical current, due to electrical resistance, there is always Joule heat production that leads to thermal gradient, which interacts with other diffusion driving forces. Under the presence of these four forces, the atomic vacancy flux equation can be given by combining Huntington’s (1972) and Kirchheim’s (1992) flux definitions, and then adding the influence of temperature gradient and vacancy concentration yields the following equations:

"

→ → ∂C v = -∇  q þG ∂t

→   → → Cv Cv Z  e C v Q ∇T q = - Dv ∇C v þ ð - f ΩÞ∇σ þ -ρ j kT kT kT T

→

→

ð8:15Þ # ð8:16Þ

Combining these two equations yields    →  →  ∂C v fΩ → Z  eρ → Q → = D v ∇2 C v ∇  Cv j þ ∇  C v ∇σ þ 2 ∇ðC v ∇T Þ þ G kT kT ∂t kT

ð8:17Þ

8.3

Laws of Conservation

8.3.1

439

Vacancy Conservation

Based on mass/vacancy conservation law, the following derivation can be given (Fig. 8.6): 


 qx - qxþΔx ΔyΔz þ GΔxΔyΔz Δt = ðc þ ΔcÞðV þ ΔV Þ - cV ð - ∇q þ GÞVΔt = ΔðcV Þ set Δt → 0 ∂ðCV Þ ∂c ∂V ) ð - ∇q þ GÞV = =V þc ∂t ∂t ∂t ∂V=V ∂c ∂c ∂εV ) - ∇q þ G = þc = þc ∂t ∂t ∂t ∂t ∂c ∂εV ) = - ∇q þ G - c ∂t ∂t

ð8:18Þ

where CV0 is the thermodynamic equilibrium vacancy concentration in the absence of stress field, c is the normalized vacancy concentration given by c = CCV0V , CV is the instantaneous vacancy concentration, and εV is the volumetric strain. It is assumed that a diffusing atom leaves behind a vacancy or takes an interstitial position (both cases lead to volumetric strain only at the lattice site), and t is the time. Substituting Eqs. (8.6), (8.8), and (8.11) into Eq. (8.5), we obtain the vacancy flux, q, in the following form:   cf Ω Ze cQ sp ð∇ϕÞc þ ∇σ þ 2 ∇T q = - Dv C v0 ∇c þ kT kT kT where: Initial State Δx, Δy, Δz, V= ΔxΔyΔz, c After Δt qx

qx+Δ x

z

Δ

y x

Δ x

έxΔtΔ

Fig. 8.6 Illustration of vacancy conservation law

Δ

Δx+ έxΔtΔx Δy+ έyΔtΔy Δz+ έzΔtΔz V+ ΔV c+Δc

ð8:19Þ

440

8

Unified Mechanics of Metals Under High Electrical Current Density. . .

 Q  Dv = vacancy diffusivity, Dv = Do e - kT Z = effective charge number e = electron charge ϕ = electric potential j = current density (vector) f = vacancy relaxation ratio, ratio of atomic volume to the volume of a vacancy Ω = atomic volume k = Boltzmann’s constant T = absolute temperature in Kelvin σ sp = spherical part of the stress tensor, σ sp = trace(σ ij)/3 Q = heat of transport, the isothermal heat transmitted by the moving atom in the process of jumping a lattice site less the intrinsic enthalpy G = vacancy generation/annihilation rate can be expressed by (Sarychev et al. 1999) G= -

Cv - C ve τs

ð8:20aÞ

Cev = thermodynamic equilibrium vacancy concentration given by Cve = Cv0 e

ð1 - f ÞΩσ sp kT

ð8:20bÞ

Cv0 = equilibrium vacancy concentration in the absence of stress τs = characteristic vacancy generation/annihilation time. Hence,

G = - C v0

c - exp



ð1 - f ÞΩσ sp kT



τs

ð8:20cÞ

To solve Eq. (8.15) using the finite element method (method of weighted residuals), we can write the following relationship:

 ∂c δc C V0 þ ∇  q - G dV = 0 ∂t V

Z

ð8:21Þ

By expanding Eq. (8.21), we can write

 ð1 - f ÞΩσ sp Z Z -c exp kT ∂c δc  CV0 dV þ C V0 δc  dVþ τs ∂t V V   Z Ze cQ cf Ω δc∇  DV C V0 ∇c þ  ∇ϕ  c þ ∇σ sp þ þ 2 ∇T dV = 0 kT kT kT V ð8:22Þ

8.3

Laws of Conservation

441

The concentration vector can be represented in terms of nodal values, that is, δc = N T  δci

ð8:23Þ

Using Eq. (8.23) into Eq. (8.22), after simple organization, we can derive the stiffness matrices. However, first, we have to define the terms in Eq. (8.22). If we define C = Cv/Cv0 as the normalized concentration, then the vacancy diffusion equation could be rewritten as ∂C v ∂t

  →   →   → Z  eρ → Q → fΩ → ∇  C∇σ þ 2 ∇ C∇T = Dv ∇2 C þ ∇ C j þ kT kT kT þ

G C vo

ð8:24Þ

where initially, C = 1 (or Cv = Cv0). A vacancy can be considered as a substitutional species at the lattice site with a smaller relaxed volume than the volume of an atom. When a vacancy switches lattice site with an atom or when another vacancy is generated/annihilated, local volumetric strain occurs. As proposed by Sarychev et al. (1999), the vacancy causes volumetric strain at the lattice site, because the volume of the vacancy is different from the volume of the atom. This volumetric strain is composed of two parts, εm ij , the g volumetric strain due to vacancy flux divergence, and εij , the volumetric strain due to vacancy generation: 1 → → f Ω∇  q δij 3 1 ε_ gij = ð1 - f Þ ΩGδij 3 ε_ m ij =

ð8:25Þ ð8:26Þ

where δij is the Kronecker delta. Thus, the total volumetric strain rate due to electrical current is _m _ gij = ε_ elec ij = ε ij þ ε

h i Ω → → f ∇  q þ ð1 - f ÞG δij 3

ð8:27Þ

The total volumetric strain rate due to current stressing is then h → i → ε_ elec = Ω f ∇  q þ ð1 - f ÞG

ð8:28Þ

The volumetric strain caused by the current stressing is superimposed onto the total strain tensor with strains due to other loadings; thus, total strain [for small strain case] can be given by

442

Unified Mechanics of Metals Under High Electrical Current Density. . .

8

= ε_ mech þ ε_ therm þ ε_ elec ε_ total ij ij ij ij

ð8:29Þ

where ε_ total is the total strain rate tensor, ε_ mech is the strain rate due to mechanical ij ij therm is the strain rate due to thermal load, and ε_ elec loading, ε_ ij ij is the volumetric strain rate due to electromigration. Following the standard procedure to obtain the finite element method stiffness matrices leads to  1 K cc = 2

Z NT  V

1  NdV Δt

3 ∂ fΩ Q ∇T nþ1 sp þ 7 6 ∂x þ kT nþ1 ∇σ nþ1 þ kT 2  2 ∂N T nþ1 sp 7  NdV  DV  6 K cc =  4 c f Ω ∂∇σ nþ1 5 Z e V ∂x þ  ∇ϕnþ1 þ nþ1 kT nþ1 kT nþ1 ∂cnþ1  2ð1 - f ÞΩ 3 sp  ð1 - f ÞΩσ nþ1 ∂σ sp nþ1 Z exp 1  3 kT nþ1 kT nþ1 ∂cnþ1 5  NdV K cc = - N T  4 τs V Z

ð8:30Þ

ð8:31Þ

ð8:32Þ

Eventually, we have  cc  1  2  3 K nþ1 = K cc þ K cc þ K cc

ð8:33Þ

By taking derivative with respect to temperature, we can obtain the coupled stiffness matrix of vacancy concentration and temperature as follows:



sp  ð1 - f ÞΩσ sp σ sp ð1 - f ÞΩ ∂σ nþ1 1 nþ1 nþ1 ½K cT  = - N   exp    NdV τs kT nþ1 kT nþ1 ∂T nþ1 T nþ1 V " 



Z ∂∇σ sp ∇σ sp cf Ω cQ ∂N T ∂ ∇T nþ1 nþ1 nþ1 -2  DV  þ  þ 2 kT nþ1 T nþ1 T nþ1 ∂T nþ1 kT nþ1 ∂x V ∂x  Z e 1  ∇ϕnþ1  cnþ1   NdV T nþ1 kT nþ1 Z

T

ð8:34Þ By taking derivative with respect to displacement u, we can write the following relation: 

K 1cu



sp  ∂N T cnþ1 f Ω ∂ ∂σ nþ1  ΒdV = DV   kT ∂x ∂εnþ1 ∂x V Z

ð8:35Þ

8.4

Newtonian Mechanics Force Equilibrium

 2 K cu = -

Z

443 ð1 - f ÞΩσ

sp nþ1

ð1 - f ÞΩ e kT nþ1 NT  kT nþ1 τs V

∂σ sp nþ1  ΒdV ∂εnþ1

ð8:36Þ

M where the strain displacement relation is given by εM nþ1 = Β u . And we can obtain the coupled stiffness matrix of concentration and mechanical displacement by the following:

 cu  1  2 K nþ1 = K cu þ K cu

ð8:37Þ

The proper approach would be also to take a derivative of the potential with respect to entropy and find the related stiffness matrices. However, that will add entropy as a nodal unknown to our equations. For sake of computational simplicity, we take the entropy generation rate values from the previous step and calculate them at the Gauss integration point. Assuming the increments are small, the error should be small. However, this simplification is not mathematically true and does contribute to error.

8.4

Newtonian Mechanics Force Equilibrium

The force equilibrium equation must be written according to the laws of unified mechanics theory. However, here we try to provide a simple implementation, which ignores the derivative of displacement with respect to entropy. The thermodynamic state index is introduced into the formulation later on. In the absence of body forces, the Newtonian mechanics force equilibrium equation has the following form: σ ij,j = 0

ð8:38Þ

For small strains, the elastic strain-stress constitutive model can be established as σ = C  ðε - εvp - εD - εTE Þ

ð8:39Þ

where: εvp εD εTE C

is the viscoplastic strain vector is strain vector due to diffusion is strain vector due to thermal expansion is the tangential constitutive tensor

Using the principle of virtual work, we can write Y u

Z

Z

=

δε  σ dV = V

δε  C  ðε - εvp - εD - εTE ÞdV

ð8:40Þ

V

After the standard derivation process, we end up with the mechanical stressinduced deformation-related stiffness matrices as follows:

444

8

Unified Mechanics of Metals Under High Electrical Current Density. . .



K uu nþ1 =

Z BT  C  B dV

ð8:41Þ

V

The stiffness matrix for unit vacancy concentration change-induced deformation can be obtained by taking the derivative of Eq. (8.40) with respect to concentration c: 

K uc nþ1 =

Z

∂σ sp nþ1  NdV ∂cnþ1

ΒT  V

ð8:42Þ

With a similar operation, we can obtain the stiffness matrix components concerning the deformation induced by unit temperature change as follows:  uT K nþ1 =

8.5

Z ΒT  V

∂σ sp nþ1  NdV ∂T nþ1

ð8:43Þ

Heat Transfer

The transient heat transfer equation has the following form: ρC p

∂T - ∇ðkh ∇T Þ - ρQ = 0 ∂t

ð8:44Þ

where: ρ = mass density of the material Cp = specific heat kh = coefficient of heat transfer Q = heat generated within the body, which can be expressed as Q = QJ þ QP þ QV

ð8:45Þ

where: QJ = Joule heating, which can be written as QJ =

1 Δt

Z Δt

QJ ðt Þdt

Using linear assumption, Eq. (8.46) can be expanded as

ð8:46Þ

8.5

Heat Transfer

445

QJ = E nþ1 

1 1 1 1  E - E nþ1   ΔE nþ1 þ ΔE nþ1   ΔEnþ1 R nþ1 R 3 R

ð8:47Þ

where E is the electrical field intensity defined as E= -

∂ϕ ∂x

ð8:48Þ

Based on Ohm’s law, the flow of electrical current description is given by j=

E 1 ∂ϕ =-  R R ∂x

ð8:49Þ

QP is heat generated due to plastic deformation at step n + 1 which can be described as QP = σ nþ1 : ε_ pl nþ1

ð8:50Þ

where ε_ pl is the plastic strain rate. The entire plastic work is not dissipated as heat. Some part is stored in dislocations which contribute to hardening. Equation (8.50) is an acceptable simplification because plastic work in electromigration is negligible. QV is the heat due to vacancy flux which can be expressed by QV = q : F k

ð8:51Þ

where: q is vacancy flux Fk is the effective driving force which has the following form:

 kT Q F k = - Z  e∇ϕ þ f Ω∇σ sp þ ∇T þ ∇c T c

ð8:52Þ

Equation (8.51) can be expanded at step n + 1 as QV = Dv C V

2

cnþ1 kT nþ1 Q ∇T nþ1 ∇cnþ1 þ Z  e∇ϕnþ1 þ f Ω∇σ sp þ nþ1 kT nþ1 cnþ1 T nþ1 ð8:53Þ

Using the method of weighted residuals, Eq. (8.44) can be rewritten as

446

Unified Mechanics of Metals Under High Electrical Current Density. . .

8

 ∂T δT ρC p - ∇ðkh ∇T Þ - ρQ dV = 0 ∂t V

Z

ð8:54Þ

Using integration by parts and divergence theorem, we obtain Z δT  V

∂T k dV - h ρC ∂t p

Z ∇δT  ∇TdV V

1 Cp

Z δT  QdV = 0

ð8:55Þ

V

Substituting δT = NT δT into Eq. (8.54) after simple finite element method operations, we arrive at the following equation:

Z N T Cp V

∂T k dV þ h ρ ∂t

Z

∂N  ∇TdV V ∂x

Z

 N T  QdV δT = 0

ð8:56Þ

V

By taking the derivative of Eq. (8.56) with respect to temperature, T, we can find the heat capacity matrix and heat transfer matrix which can be separately expressed as   

Z

CTT nþ1 = Z

K 1TT =

K 2TT =

Z

Z NT  V

∂N T kh ∂N   dV ρ ∂x V ∂x

NT  V

CP  NdV Δt

ð8:57Þ ð8:58Þ

J ∂Qnþ1  NdV ∂T nþ1

∂ϕnþ1 ∂ϕnþ1 ∂ϕnþ1 ∂Δϕnþ1 ð8:59Þ   ∂x ∂x ∂x ∂x V  1 ∂Δϕnþ1 ∂Δϕnþ1   NdV þ ∂x 3 ∂x Z Z V  3 ∂Qnþ1 DV C V cnþ1 K TT =  NT   NdV = NT  ∂T nþ1 k V V # " sp ∂σ Q ∂N Q ∇T nþ1 k∇c nþ1 nþ1 k k  N þ fΩ Nþ  N F nþ1  N - 2F nþ1 cnþ1 ∂T nþ1 T nþ1 ∂x T 2nþ1 dV T 2nþ1 =

N  T

1 ∂R - 2 R ∂T nþ1

ð8:60Þ Eventually, we can get the temperature “stiffness” matrix by adding them all together:

8.6

Electrical Conduction Equations



447

 TT  1  2  3 K TT nþ1 = Cnþ1 þ K TT þ K TT þ K TT

ð8:61Þ

The term stiffness is normally used for the mechanical force-displacement relationship. Here we are using the term “stiffness” as an analogy. The coupled term concerning concentration field-induced temperature change can be obtained by taking the derivative of the total potential equation with respect to c which yields 

Tc =K nþ1

Z NT  V



Z V ∂Qnþ1 ∂σ sp ∂N DV C V
k 2 k  NdV = F nþ1  N - 2F nþ1 cnþ1 f Ω nþ1  ∂cnþ1 kT nþ1 ∂cnþ1 ∂x

V  ∂N kT nþ1 cnþ1  - ∇c  N þ 2 ∂x cnþ1

ð8:62Þ Similarly, we can get the change of temperature induced by plastic deformation by taking the derivative of Eq. (8.56) with respect to u: 

K Tu nþ1 = -

Z NT  V

∂QPnþ1  NdV = ∂unþ1

Z NT  V



dev ∂σ nþ1 1 ∂ΔεPl nþ1  BdV þ ∂εnþ1 Δt ∂εnþ1 ð8:63Þ

where σ dev represents deviatoric stress tensor. And Joule heating can be considered by adding the following term into the general stiffness matrix: Z Z   h i J 1 ∂N Tϕ T ∂Qnþ1 K nþ1 = - N   NdV = = N T  J - ΔJ  dV 3 ∂x ∂ϕ V V nþ1

8.6

ð8:64Þ

Electrical Conduction Equations

The electrical field in metal is governed by Maxwell’s equation of conservation of charge. Assuming steady-state direct current and no internal volumetric current source, Maxwell’s equation reduces to Z j  n dA = 0 A

where: A is the surface of a control volume n is the outward normal to surface j is the electrical current density defined in Eq. (8.49)

ð8:65Þ

448

8

Unified Mechanics of Metals Under High Electrical Current Density. . .

The divergence theorem can be used to convert the surface integral into a volume integral which yields to Z

∂  jdV = 0 ∂x V

ð8:66Þ

The equivalent weak form is obtained by introducing an arbitrary variation in the electrical potential field, δ ϕ, and integrating it over the volume: Z δϕ  V

∂  jdV = 0 ∂x

ð8:67Þ

Using first the chain rule and then the divergence theorem, this statement can be rewritten as Z

∂δϕ  jdV = 0 V ∂x

ð8:68Þ

By introducing the definition of (8.48) and (8.49), the governing conservation of charge equation becomes Z

∂δϕ 1 ∂ϕ dV = 0   V ∂x R ∂x

ð8:69Þ

Introducing δϕ = NT  δϕN into the above equation yields

Z

 ∂N T 1 ∂ϕ   dV δϕN = 0 R ∂x V ∂x

ð8:70Þ

Utilizing Eq. (8.70), we can obtain the stiffness terms: h

i Z ∂N T 1 ∂N K ϕϕ   dV nþ1 = R ∂x V ∂x

ð8:75Þ

The influence of temperature change on the current field can be taken into account by adding the following term: h

K ϕT nþ1

i

 1 ∂R ∂N T  NdV = E R2 ∂T nþ1 V ∂x Z

Eventually, we obtain the total stiffness matrix for an element:

ð8:76Þ

8.7

Thermodynamic Fundamental Equation for Electromigration and Thermomigration

2

½K uu ½Kucnþ1  nþ1  cu K ½  ½Kccnþ1  ½K nþ1 fug = 4 ½K nþ1 Tu ½KTc nþ1  nþ1  0

0

38 → 9 8 9 0 ½K uT nþ1  < u = < fu = 0 5 cv ½K cT nþ1  = qqc K Tϕ T ; ½K TT ½ : : T; nþ1  nþ1  ϕ j ½K ϕT ½Kϕϕ nþ1  nþ1 

449

ð8:73Þ

It is easy to notice that the finite element nodal point has displacement, vacancy concentration, temperature, and electrical potential as nodal unknowns. An accurate analysis, according to unified mechanics theory, would also include entropy generation rate as a nodal unknown. Of course, it does complicate the computation process. dev ∂σ sp ∂σ sp ∂σ sp ∂ΔεPnþ1 nþ1 nþ1 ∂σ nþ1 Terms ∂cnþ1 , ∂Tnþ1 , , , and in Eq. (8.73) can be derived individually ∂ε ∂ε ∂εnþ1 nþ1 nþ1 nþ1 once we introduce the viscoplastic model.

8.7

Thermodynamic Fundamental Equation for Electromigration and Thermomigration

In this section, the thermodynamic fundamental equation accounting for most but not all entropy production mechanisms is derived in a way appropriate for implementation in the finite element method. Ignored mechanisms included but are not limited to grain coarsening and phase change and are assumed to account for a small portion of the total entropy generation. Of course, including their contribution is more accurate. In the derivation below, for the sake of completeness, and clarity, some obvious thermodynamics relations are restated. According to the second law of thermodynamics for any macroscopic system, the entropy of the system is a state function. The differential of the entropy, dS, may be written as the sum of two terms: dS = dSe þ dSi

ð8:74Þ

where dSe is the entropy supplied to the system and dSi is the entropy produced inside the system. The second law of thermodynamics states that dSi must be equal to zero for reversible and positive for irreversible processes: dSi ≥ 0

ð8:75Þ

We establish the local form of entropy as Z S=

ρsdV V

ð8:76Þ

450

8

Unified Mechanics of Metals Under High Electrical Current Density. . .

dSe =dt

Z Js,tot  dΩ

ð8:77Þ

Ω

dSi = dt

Z γdV

ð8:78Þ

V

where s is the specific entropy per unit mass, Js, tot is the total entropy flow per unit area and unit time, and γ is the entropy source strength or internal entropy production per unit volume per unit time. Using the local form of entropy, we can get the local form of the variation of entropy as follows: ∂ρs = - divJs,tot þ γ ∂t

ð8:79Þ

γ≥0

ð8:80Þ

With the help of the material time derivative of the volume integral, which is given by d ∂ = þ v  grad dt ∂t

ð8:81Þ

Eq. (8.79) can be rewritten in a slightly different form as follows: ρ

ds = - divJs þ γ dt

ð8:82Þ

where the entropy flux, Js, is the difference between the total entropy flux and a convective term: Js = Js,tot - ρsV

ð8:83Þ

In obtaining (8.80) and (8.82), it is assumed that the definition given by Eqs. (8.74) and (8.75) also hold for infinitesimally small parts of the system. This assumption agrees with the Boltzmann (1877) relation where the entropy state is related to the disorder parameter via Boltzmann’s constant using statistical mechanics, discussed earlier in Chap. 3.

8.7.1

Entropy Balance Equations

Based on the above definitions, the degradation of the system is directly related to the rate of change of the entropy (thermodynamic state index). This enables us to

8.7

Thermodynamic Fundamental Equation for Electromigration and Thermomigration

451

obtain more explicit expressions for the entropy flux and internal entropy production rate. With the assumption of a caloric equation of state, Helmholtz free energy, Ψ, in differential form can be written as dΨ = du - Tds - sdT

ð8:84Þ

where u is internal energy and T is the temperature (here we use upper case, Ψ, to distinguish from the electrical potential ϕ). Rearranging terms in Eq. (8.84) leads to Tds = du - dΨ - sdT

ð8:85Þ

To get the expression for free energy, we need to specify a function with a scalar value concave with respect to temperature T and convex with respect to other variables. Here we use Ψ = Ψðε, εe , εp , T, V k Þ

ð8:86Þ

where Vk can be any internal state variable determined by the specific application at hand and physical processes that is leading to generating entropy for our specific definition of failure. In small strain theory, the strain can be written in the form of their additive decomposition as follows: εe = ε - εp

ð8:87Þ

Ψ = Ψðεe , T, V k Þ

ð8:88Þ

so that

The time derivative of the free energy is given by dΨ ∂Ψ dεe ∂Ψ dT ∂Ψ dV k : = e: þ : þ dt ∂ε dt ∂T dt ∂V k dt

ð8:89Þ

From Eq. (8.85), we obtain T

ds du dΨ dT = -s dt dt dt dt

ð8:90Þ

Using the first principle of thermodynamics, the conservation of energy equation can be given by

452

8

ρ

Unified Mechanics of Metals Under High Electrical Current Density. . .

X du = - divJq þ σ : gradð~vÞ þ Jk  Fk dt k

ð8:91Þ

where u is the total internal energy, Jq is the heat flux, σ is the stress tensor, ~v is the rate of deformation, Jk is the diffusional flux of component k, and Fk is the body force acting on the mass of component k. Using the conservation of energy equation in the form given by Eq. (8.91) and Eqs. (8.82) and (8.83), we can write the rate of change of entropy density as follows:

X ds 1  ∂Ψ dεe ∂Ψ dT þ : Jk  Fk - ρ : ρ = - divJq þ σ : gradðvÞ þ dt T ∂T dt ∂εe dt k  dT ∂Ψ dV k -s þ : dt ∂V k dt

ð8:92Þ

where σ : gradð~vÞ = σ : ðD þ WÞ

ð8:93Þ

and D (symmetric) and W (skew-symmetric) are the rate of deformation tensor and spin tensor, respectively. Due to the symmetry of σ, we obtain the following relation: σ : ðD þ W Þ = σ : D

ð8:94Þ

For a small strain case, we can make the following assumption: dε σ : D=σ : =σ : dt



dεe dεp þ dt dt

 ð8:95Þ

After rearranging Eq. (8.92) and comparing it with Eq. (8.82), we can obtain Js = T1 Jq and the following entropy production rate term:

 dεe ∂Ψ dεe 1X 1 dεp 1 1 þ J  grad ð T Þ þ J  F þ : þ σ : ρ σ : q dt T dt T k k k T ∂εe dt T2

 ρ ∂Ψ dT ρ ∂Ψ dV k þ : sþ T T ∂V k dt ∂T t

γ= -

ð8:96Þ In materials with internal friction, all deformations cause a positive entropy production rate γ ≥ 0, which is also referred to as the Clausius-Duhem inequality. Using the following relations:

8.7

Thermodynamic Fundamental Equation for Electromigration and Thermomigration

σ = ρð∂Ψ=∂εe Þ s= -

∂Ψ ∂T

453

ð8:97Þ ð8:98Þ

If we assume that entropy generation due to elastic strain rate is much smaller compared to other mechanisms, we can simplify Eq. (8.96) as follows: γ= -

1 1X 1 ρ ∂Ψ dV k J  grad ð T Þ þ Jk  Fk þ σ : e_ p : q 2 T T T ∂V k dt T k

ð8:99Þ

or if the heat flux term Jq is replaced by Jq =

1 CjGradðT Þj2 T2

ð8:100Þ

where C is the thermal conductivity tensor, the simplified entropy production rate can be given by γ= -

1X 1 ρ ∂Ψ dV k 1 CjGradðT Þj2 þ J  F þ σ : ε_ p : 2 T k k k T T ∂V k dt T

ð8:101Þ

We identify Jk in Eq. (8.101) as the flux, and Fk as the effective driving force term, as defined in the Onsager relation:   → Q → kT → Fk = Z  ejρ þ ð - f ΩÞ∇σ - ∇T ∇C T C

ð8:102Þ

In Eq. (8.101) the irreversible dissipation includes two parts: the first term is called heat dissipation caused by conduction inside the system, while the second, third, and fourth terms account for other irreversible processes in the system, and we will call it intrinsic dissipation. It is important to point out that the simplification we did in the entropy generation rate term is not necessarily accurate. However, it simplifies computation. An accurate derivation should include all entropy-generating micro-mechanisms in the thermodynamic fundamental equation. Ignoring the entropy generation due to elastic deformations as well, using Eqs. (8.101) and (8.102), we can write total entropy as Z Δs = to

t1

γ dt

ð8:103Þ

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Unified Mechanics of Metals Under High Electrical Current Density. . .

Δs →

2 C v Deffective Q∇T kT → 1 2  Z l eρj - f Ω∇σ þ CjgradðT Þj þ þ ∇C T C kT 2 T2 =t0  1 ρ ∂Ψ dV k dt þ σ : εp : T T ∂V k dt

Zt

ð8:104Þ Thermodynamic state index (TSI) was given by

Fig. 8.7 Cross-section of test module and solder joints (Ye et al. 2003a)

Table 8.1 Time to failure Current density (Amp/cm2) 0.6 × 104 0.8 × 104 1.0 × 104

Experiment time to failure (hours) 1058.7 446.6 228.7

h i ms Φ = 1 - e - Δs R

Finite element simulations Time to failure (hours) 1098.2 435.33 222.41

8.7

Thermodynamic Fundamental Equation for Electromigration and Thermomigration

455

Table 8.2 Material properties for SAC405 Property Vacancy relaxation time (Ts) Effective charge number (Z*) Resistivity (R)

Unit seconds N/A μm3s-3A-2Kg

Atomic volume (Ω) Vacancy concentration at stress-free state (Cv0) Young’s modulus (E) Poisson’s ratio Initial yield stress (σ y0)

μm-3 μm-3 GPa N/A MPa

Coefficient of thermal expansion (CTE) Linear kinematic hardening cons (c1) Nonlinear kinematic hardening cons (c2) Saturation isotropic hardening (R1) Isotropic hardening rate (c) Dimensionless constant (A) Burger’s vector (B) Initial average grain size Grain size exponent Stress exponent Creep activation energy

K-1 Kg.s.μm-1 Kg.s.K-1.μm-1 Kg.s.μm-1 N/A N/A μm μm N/A N/A μm2.s-2.Kg. mol-1 Kg.μm2.s-2 Kg.μm-3 μm2.s-2.K-1 Kg.μm.s-3.K-1

RðT Þ = 1:52 × 1011 þ3:50 × 108 T ðK Þ 2.71 × 10-11 1.107E+06 E(T ) = 57.7 - 0.056T(K ) 0.33 pffiffiffi σ y ðT Þ = 79:98 þ 95= d - 0:2133T ðK Þ 18.9 × 10-06 9.63 × 103 7.25 × 102 0 383.3 7.60 × 109 3.18 × 10-04 2.45 3.34 6.65 7.95 × 1016 -3.68 × 10-08 7.39 × 10-15 2.19 × 1014 5.73 × 107

Thermodynamic State Index

Thermodynamic State Index

Heat of transport(Q*) Density Specific heat Heat conductivity

95.5%Sn4%Ag0.5%Cu 1.80 × 10-03 10

Fig. 8.8 TSI evolution (a) in three different solder joints at -20 °C (b) in solder #7 at different temperatures

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Unified Mechanics of Metals Under High Electrical Current Density. . .

80 Experimental 70

Simulational

Time to Failure (hour)

60 50 40 30 20 10 0 -50

-45

-40

-35

-30 -25 -20 Ambient Temperature (ć)

-15

-10

-5

0

Fig. 8.9 Comparison of experiment and simulation result

8.8

Example

The solder joint in the electronic package shown in Fig. 8.7 was studied using the formulation given in this chapter. Ye et al. (2003a, b, c, d, e, f) published laboratory test data and finite element simulation results of time to failure shown in Table 8.1 for solder joints under different current density levels. Material properties used in the analysis are given in Table 8.2, Figs. 8.8 and 8.9. In these example simulations, critical TSI is set for Φcr = 0.094. Here we conclude our discussion on electromigration and thermomigration. More examples on this topic are provided in the publications listed in the references section.

References

457

References Abdulhamid, M., Li, S., & Basaran, C. (2008). Thermomigration in lead-free solder joints. International Journal of Materials and Structural Integrity, 2(1/2), 11–34. Barberan, N., & Echenique, P. M. (1986). Effective charge of slow ions in solids. Journal of Physics B: Atomic and Molecular Physics, 19(3), L81. Basaran, C., Lin, M., & Ye, H. (2003). A thermodynamic model for electrical current induced damage. International Journal of Solids and Structures, 40(26), 7315–7327. Basaran, C., Li, S., & Abdulhamid, M. (2008). Thermomigration induced degradation in solder alloys. Journal of Applied Physics, 103, 123520. Black, J. R. (1969). Electromigration failure modes in aluminum metallization for semiconductor devices. Proceedings of the IEEE, 57(9), 1587–1594. Blech, I. A. (1976). Electromigration in thin aluminum films on titanium nitride. Journal of Applied Physics, 47(4), 1203–1208. Blech, I. A., & Herring, C. (1976). Stress generation by electromigration. Applied Physics Letters, 29(3), 131–133. Boltzmann, L. (1877). Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI, 373–435. Bosvieux, C., & Friedel, J. (1962). Electrolysis of metallic alloys. Physics Chemistry Solids, 23, 123–136. Brandt, W., & Kitagawa, M. (1982). Effective stopping-power charges of swift ions in condensed matter. Physical Review B, 25, 5631. Erratum Phys. Rev. B 26, 3968 (1982). Campbell, D. R. & H.B. Huntington. (1969). Thermomigration and Electromigration in Zirconium. Physical Review, 179(3):601–601. Chen, C., Hsiao, H.-Y., Chang, Y.-W., Ouyang, F., & Tu, K. N. (2012). Thermomigration in solder joints. Materials Science and Engineering: R: Reports, 73(9–10), 85–100. Fick, A. (1855). On liquid diffusion. Philosophical Magazine and Journal of Science, 10, 31–39. Genoni, T. C., & Huntington, H. B. (1977). Transport in nearly-free-electron metals. IV. Electromigration in zinc. Physical Review B, 16, 1344. Huntington, H. B., & Grone, A. R. (1961). Current-induced marker motion in gold wires. Journal of Physics and Chemistry of Solids, 20(1–2), 76–87. Huntington, H. B. (1972). Electro-and thermomigration in metals. Diffusion, papers presented at a seminar of the American Society of Metals. Kirchheim, R. (1992). Stress, and electromigration in Al-lines of integrated circuits. Acta Metallurgica et Materialia, 40(2), 309–323. Kuz’menko, P. F., & Osirovskii, L. F. (1962). Electric transfer of silver in copper. Izvestiya Vysshikh Uchebnykh Zavedenii, Chernaya Metallurgiya, 11, 146–149. Landauer, R., & Woo, J. W. F. (1974). Driving force in electromigration. Physical Review B, 10, 1266. Lloyd, J. R., Tu, K. N., & Jasvir Jaspal, J. (2004). The physics and materials science of electromigration and thermomigration in solders, chapter 20. In K. J. Puttlitz & K. A. Stalter (Eds.), Handbook of lead-free solder technology for microelectronic assemblies. Taylor & Francis. Lodding, A. (1965). Current induced motion of lattice defects in indium metal. Journal of Physics and Chemistry of Solids, 26(1), 143–151. Ludwig, C. (1856). Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, 20, 539. Sarychev, M. E., et al. (1999). General model for mechanical stress evolution during electromigration. Journal of Applied Physics, 86(6), 3068–3075. Seith, W. (1955). Diffusion in Metallen, Platzwechselreaktionen. Springer. Singh, P., & Ohring, M. (1984). Tracer study of diffusion and electromigration in thin tin films. Journal of Applied Physics, 56(4), 899–907.

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Skaupy, F. (1914). Electrical conduction in metals. Verband Deutscher Physikalischer Gesellschaften, 16, 156–157. Sorbello, R. S. (1973). A pseudopotential based theory of the driving forces for electromigration in metals. Journal of Physics and Chemistry of Solids, 34(6), 937–950. Soret, C. (1879). Archives des Sciences Physiques et Naturelles. Geneve, 3, 48–61. Sun, P. H., & Ohring, M. (1976). Tracer self-diffusion and electromigration in thin tin films. Journal of Applied Physics, 47, 478. Wever, H., Seith, W. & Elektrochem, Z. (1955). Huntington, H.B. Electro- and thermomigration in metals, proceedings of the seminar of the American Society of Metals. 59, 942. Ye, H., Basaran, C., & Hopkins, D. (2003a). Thermomigration in Pb-Sn solder joints under joule heating during electric current stressing. Applied Physics Letters, 82(8), 1045. Ye, H., Hopkins, D., & Basaran, C. (2003b). Measuring joint reliability: applying the moire interferometry technique. Advanced Packaging, 1, 17–20. Review article. Ye, H., Basaran, C., & Hopkins, D. (2003c). Damage mechanics of microelectronics solder joints under high current densities. International Journal of Solids and Structures, 40(15), 4021–4032. #34. Ye, H., Basaran, C., & Hopkins, D. (2003d). Numerical simulation of stress evolution during electromigration in IC interconnect lines. IEEE Transactions on Components and Packaging Technologies, 26(3), 673–681. Ye, H., Basaran, C., & Hopkins, D. (2003e). Mechanical degradation of microelectronics solder joints under current stressing. International Journal of Solids and Structures, 40(26), 7269–7284. Ye, H., Basaran, C., & Hopkins, D. (2003f). Measurement of high electrical current density effects in solder joints. Microelectronics Reliability, 43(12), 2021–2029. Ye, H., Basaran, C., & Hopkins, D. (2004a). Pb Phase coarsening in eutectic Pb/Sn flip-chip solder joint under electrical current stressing. International Journal of Solids and Structures, 41, 2743–2755. Ye, H., Basaran, C., & Hopkins, D. (2004b). Deformation of microelectronic solder joints under current stressing and numerical simulation I. International Journal of Solids and Structures, 41, 4939–4958. Ye, H., Basaran, C., & Hopkins, D. (2004c). Deformation of microelectronic solder joints under current stressing and numerical simulation II. International Journal of Solids and Structures, 41, 4959–4973. Ye, H., Basaran, C., & Hopkins, D. (2004d). Mechanical implications of high current densities in flip chip solder joints. International Journal of Damage Mechanics, 13(4), 335–346. Ye, H., Basaran, C., & Hopkins, D. (2006). Experimental damage mechanics of micro/power electronics solder joints under electrical current stresses. International Journal of Damage Mechanics, 15, 41–68.

Chapter 9

Fatigue of Materials

Fatigue life prediction of materials has been widely studied. An extensive literature survey of fatigue life prediction models is available in Lee and Basaran (2021); hence, it is not necessary to repeat a literature survey herein. However, most of the research is based on empirical curve fitting models under the framework of Newtonian mechanics. Fatigue test data is used for curve fitting a degradation evolution function using stress, strain, or energy as a variable. Unified mechanics theory (UMT), on the other hand, unifies the universal laws of motion of Newton by incorporating the second law of thermodynamics directly into Newton’s laws at the ab initio level. UMT introduces an additional linearly independent axis called thermodynamic state index (TSI), which can have values between 0 and 1. Evolution along the TSI axis follows Boltzmann’s entropy formulation and the thermodynamic fundamental equation of the material. As a result, governing differential equations of any system automatically include energy dissipation and degradation evolution. When UMT is used, there is no need for an empirical degradation evolution function obtained by curve fitting to fatigue test data. However, UMT does require deriving the analytical thermodynamic fundamental equation of the material. In this chapter, thermodynamic fundamental equations for several different fatigue problems are presented. The theory of elasticity assumes that there is no irreversible deformation during elastic loading. However, it is well-known that under cyclic loading in a linear elastic regime, any metal will still fatigue. Therefore, the second law of thermodynamics is not violated. There is always irreversible entropy generation in any closed system, which is ignored in the Newtonian mechanics-based theory of elasticity. If we apply a cyclic load on metal within its elastic range [below the yield stress], the corresponding strain would always be the same at every cycle, according to the theory of elasticity, because the second law of thermodynamics is not included in the universal laws of Newton directly. In summary, there is no TSI axis in the Newtonian space-time coordinate system. Moreover, in Newtonian mechanics, the laws of thermodynamics are satisfied separately from than laws of Newton [not in the same

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_9

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differential equation] (such as satisfying the thermodynamic consistency using the Clausius-Duhem inequality). As a result, in Newtonian continuum mechanics, derivative of displacement with respect to entropy is zero.

9.1

Predicting High Cycle Fatigue Life of Metals

Fatigue life prediction with the unified mechanics theory eliminates the need for curve fitting an empirical evolution function to fatigue test data. In unified mechanics theory, the degradation evolution occurs along with the linearly independent thermodynamic state index (TSI) axis which is quantified by the thermodynamic fundamental equation of the material. The coordinate value of TSI starts at 0 and ends at 1. However, because TSI is an exponential function, computations are usually stopped before it reaches 1.00, which is possible only if the power term is infinite. The thermodynamic fundamental equation provides the entropy generation quantity in the material under a given load. Based on this entropy value, a new TSI coordinate is calculated, which is the normalized form of Boltzmann’s entropy formulation of the second law of thermodynamics. In the formulation, the free energy of the system is degraded by the entropy generated.

9.1.1

Thermodynamic Fundamental Equation

In this section, we derive the thermodynamic fundamental equation for high cycle fatigue where the sample is subjected to a nominal stress level that is well below the yield stress of the material and the loading frequency is quasi-static. The following assumptions are made in the derivation: 1. Applied maximum nominal stress is below the yield stress of the metal; hence, no macroscopic plastic deformation is expected to happen. 2. A mechanism called microplasticity is expected to happen at some micro-defect sites at the micro-level (Lemaitre et al., 1999; Doudard et al., 2005;). 3. Point defects, including atomic vacancies, interstitials, and impurities, can be built-in with the original crystal growth or created by energy input during the fatigue process. 4. Input mechanical energy increases atomic vacancies and dislocation densities. However, the increasing dislocation density only causes hardening in the microlevel and never induces macroscopic plastic deformation as the maximum applied nominal stress is below the metal’s yield stress value. The vacancy generation/ diffusion and dislocation motion (e.g., cross-slip) around inclusions induce irreversible deformations at a micro-level in elastic cyclic loads, (Marti et al., 2020; ; Ho et al., 2017; Mughrabi, 2009).

9.1

Predicting High Cycle Fatigue Life of Metals

461

5. Vacancy concentration gradient in the specimen will result in vacancy diffusion, and temperature gradient in the specimen will result in thermomigration. 6. Temperature evolution in the specimen is assumed to happen by atomic-frictiongenerated heat, heat conduction, microplastic work, and thermoelastic damping.

9.1.2

Entropy Generation Mechanisms

Six mechanisms generate entropy during high cycle metal fatigue at quasi-static frequencies. These are: I. Configurational entropy, ΔSc II. Vibrational entropy, ΔSvib III. The entropy generation due to vacancy concentration gradient-driven diffusion, ΔSd IV. The entropy from heat conduction, ΔST V. The entropy generation due to atomic-friction-generated heat, ΔSr VI. The entropy generation due to microplasticity ΔSμp. Because entropy is an additive property, we can write the thermodynamic fundamental equation for the total entropy production as follows: ΔS = ΔSc þ ΔSvib þ ΔSd þ ΔST þ ΔSr þ ΔSμp

ð9:1Þ

I. Configurational Entropy Generation The micro-mechanisms in the crystal include the rearrangement of atoms which results in entropy production during cycling elastic loading. The configurational entropy Sc is a concept from statistical thermodynamics that uses the binomial distribution formula given by N! ðN - nÞ!n!

ð9:2Þ

where N is the number of lattice sites and n is the number of vacancies. Replacing the disorder parameter W in Boltzmann-Planck entropy equation yields the following form of configurational entropy (Kelly et al., 2012; Abbaschian & Reed-Hill, 2009):
Sc = kB ln

N! ðN - nÞ!n!

 ð9:3Þ

Atomic defects are atoms missing from their proper lattice sites. These vacancies are rearranged due to the thermally activated mass transport. The change in configurational entropy ΔSc from the temperature state of 1 to the temperature state of 2 is given by

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 N! N! ΔSc = kB ln - k B ln ðN - n2 Þ!n2 ! ðN - n1 Þ!n1 !

ð9:4Þ

where n1 and n2 are the number of vacancies at the temperature state of 1 and the temperature state of 2, respectively. Equation (9.4) can also be written in the following form: 

 1 - C v2 1 - C v1 - Cv1  ln ΔSc = kB N C v2  ln C v2 C v1

ð9:5Þ

where Cv1 and Cv2 are the thermodynamic equilibrium vacancy concentrations at the temperature state of 1 and the temperature state of 2, respectively. II. Vibrational Entropy Generation The vibrational entropy is also a concept from statistical thermodynamics that replaces disorder parameter W in the Boltzmann-Planck entropy equation with the phase state of the atoms as they vibrate, which is defined by momentum and position coordinates (Fultz, 2010; Atkins et al., 2018; Wollenberger, 1996; Laughlin & Hono, 2014; Kelly et al., 2012; Abbaschian & Reed-Hill, 2009; Fultz, 2010). The vibrational state of the atoms changes when vacancies are created. There are various models proposed to precisely calculate vibrational entropies when a certain number of vacancies are removed (Mishin et al., 2001). We can assume that the change in vibrational entropy is the same when each atomic vacancy is created (Abbaschian & Reed-Hill, 2009; Burton, 1972). Therefore, the total vibrational entropy in the system can be given by Svib = nΔSv

ð9:6Þ

In which n is the number of atomic vacancies created and ΔSv is the change in vibrational entropy when one atomic vacancy is created. Equation (9.6) is valid because of the additive property of entropy. The variation ΔSvib is given by ΔSvib = ðn2 - n1 ÞΔSv

ð9:7Þ

Comparing Configurational Entropy and Vibrational Entropy Magnitudes The parameters necessary to calculate the configurational and vibrational entropy magnitudes are listed in Table 9.1. Entropy calculations for these two mechanisms are based on the number of vacancies [or the vacancy concentration], which is related to the temperature at that state. If we assume that after several cycles of loading, the temperature in the metal specimen increases from T0 = 298 K to Ti = 398 K, then we can calculate the evolution of ΔSc and ΔSvib. Figure 9.1 shows configurational entropy as a function of temperature. Figure 9.2 shows vibrational entropy as a function of temperature. We should point out that during high cycle fatigue at quasi-static frequencies, the

9.1

Predicting High Cycle Fatigue Life of Metals

463

Table 9.1 Parameters to calculate configurational and vibrational entropy Parameters Avogadro’s number Atomic weight The mass density of steel Vacancy formation energy for Fe Boltzmann’s constant

NA AwFe ρsteel Δhf kB

Vibrational entropy for bcc crystals

Δsv

Value 6.023 × 1023 55.85 7.894 1.08 8.62 × 10-5 1.38 × 10-23 2.4

Units atoms=mol gr=mol gr=cm3

eV eV/K J/K kB/atom

Fig. 9.1 Configurational entropy (JK-1 m-3) versus temperature evolution

temperature increase in the sample is about 15 °C. The purpose of this example where ΔT = 100 °C is used is to show the magnitude of entropy generation for a hypothetical case. III. Entropy Generation due to Vacancy Concentration Gradient-Driven Diffusion During the fatigue process concentration of vacancies in the metal are not uniform. As a result, there is a vacancy concentration gradient. Vacancy concentration gradient-driven diffusion and thermomigration are governed by the vacancy conservation equation (or mass conservation) (Basaran & Lin, 2007; Ye et al., 2004; Ye et al., 2006).

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Fig. 9.2 Vibrational entropy (JK-1 m-3) versus temperature evolution

The concentration of atomic vacancies should be higher around the edge and lower around the center for a specimen with a round cross-section (cracks usually develop from outside to inward); hence, they cause the vacancy diffusion. The entropy generation from vacancy gradient-driven diffusion ΔSd is extensively studied by Basaran and Lin (2008), Li et al. (2009), and Yao and Basaran (2013a, b, c). It is given by Z ΔSd = t0

t


2 ! C v Dv Q ∇T kB T dt þ ∇c T c kB T 2

ð9:8Þ

where kB is Boltzmann’s constant, Cv0 is the thermodynamic equilibrium vacancy concentration in the absence of a stress field, Cv is instantaneous atomic vacancy concentration, c is normalized vacancy concentration c = Cv/Cv0, Dv is effective vacancy diffusivity, T is absolute temperature, and Q is the heat of transport, which is the isothermal heat transmitted by the moving atom in the process of jumping a lattice site. To compare the order of magnitude of Eq. (9.8) with other entropy generation mechanisms, the following condition is assumed: the specimen is initially placed at room temperature (298 K). After several cycles, the temperature at the specimen’s gage section uniformly increases to 398 K. We assume the vacancy concentration at the center is ten times smaller than at the edge (Fig. 9.3). The thermodynamic equilibrium vacancy concentration Cv can be calculated by

9.1

Predicting High Cycle Fatigue Life of Metals

465

Fig. 9.3 An illustration of a vacancy concentration gradient


 TΔSv - Δhf Cv = exp kB T

ð9:9Þ

The vacancy concentration at room temperature (298 K) and 398 K can be obtained as follows: C v,398 K = 2:35 × 10 - 13 C v0 = C v,298 K = 6:1 × 10 - 18

ð9:10Þ

The normalized vacancy concentration c and normalized vacancy gradient are therefore given by C 0:1C v,388 K cedge = v,398 K , ccenter = C v0 Cv0   ∇c = cedge - ccenter =1 cm = 3:5 × 106 m - 1

ð9:11Þ

The effective vacancy diffusivity Dv is given by Dv = D0 exp



-Q RT

ð9:12Þ

For BCC iron (α-Fe), D0 = 2:8 × 10 - 4 ms , activation energy Q = 251 kJ/mole,   2 and R = 8:3145 moleJ K [63]. At 398 K, Dv = D0 exp -RTQd ≈ 3 × 10 - 37 ms . We can calculate the value of Eq. (9.8) as follows: 2

m2
2 1:38 × 10 - 23 J=K × 398 K 6 -1 s _ Sd = 3:5 × 10 m 3852 1:38 × 10 - 23 J=K × ð398 KÞ2 2:35 × 10 - 13 × 3 × 10 - 37

= 8 × 10 - 91 JK - 1 s - 1 ð9:13Þ From this calculation, it is clear that the entropy generation due to mass transport is negligible compared to other entropy generation mechanisms.

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IV. Entropy Generation due to Heat Conduction In metal high cycle fatigue, the temperature gradient causes an irreversible heat flow. The entropy generation equation is given by ΔST = -

 Z t
∇T  ∇T dt kh T2 t0

ð9:14Þ

where kh is coefficient of heat conduction and ∇T is the temperature gradient. Again, to calculate the order of magnitude of this entropy generation mechanism, we K assume a simple one-dimensional temperature gradient ΔT Δx = 10 =cm from the gage w section to the grip section (Fig. 9.3). The thermal conductivity of steel is K h = 50 Km . After several cycles, the temperature increases from T0 = 298 K to Ti = 398 K and then remains in a steady-state until near failure at around 1.4 × 105 cycles (operating frequency at 30 Hz). The evolution of entropy generation due to heat conduction ΔST for a specimen is shown in Fig. 9.4. V. Entropy Generation due to Atomic-Friction (Scattering)-Generated Heat The entropy generation due to atomic-friction-generated heat is given by Z ΔSr = t0

t

ρr dt T

ð9:15Þ

Fig. 9.4 Entropy production due to thermal conduction (JK-1 m-3) versus the number of cycles

9.1

Predicting High Cycle Fatigue Life of Metals

467

Temperaure varation for uniaxial compression temperature(K)

600 550 500 450 400 350 300 250 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

time(picosecond) 0.001/ps

0.005/ps

0.01/ps

Fig. 9.5 Temperature variation with time under uniaxial compression under different strain rates

At the atomic scale, the temperature is defined by the intensity of atomic vibrations. The atomic-friction-generated heat per unit mass, r, is due to the increasing intensity of atomic vibrations in the lattice from micro-mechanisms such as the breaking of atomic bonds and phonon-phonon scattering, phonon-electron scattering, and electron scattering (called internal friction) (Gao, 1997; Ragab & Basaran, 2009; Chu et al., 2015). This term is essentially different than the heat generated through the macroscopic observable plastic work or thermoelastic sources. It is possible to estimate the internal heat generation due to internal frictions using molecular dynamics simulations. Figures 9.5 and 9.6 show temperature variation with time under uniaxial compression and tension under different strain rates obtained by molecular-level simulations (Lee & Basaran, 2022). It is shown that the temperature fluctuation during uniaxial compression and tension are not completely reversible at higher strain rates (e.g., for 0.01/picosecond at 20 ps). As a result, temperature increases by 250 K for compression but decreases by 90 K for tension. However, at lower strain rates, the temperature fluctuation is negligible (e.g., for strain rates below 0.001/ps). It is therefore concluded that at low strain rates, the temperature fluctuation is solely the result of thermoelastic coupling. Atomic-friction-generated heat is negligible and can be neglected below 0.001/ picosecond rate of loading. VI. Entropy Generation due to Microplasticity The term microplasticity is used in two-scale models, which was initially developed by Lemaitre et al. (1999) and then reformulated by Doudard et al. (2005). These researchers developed the microplasticity model to calculate the amount of microplastic strain during high cycle fatigue and then used the microplastic strain as a variable in an empirical function to curve fit the test data to predict the fatigue life. In UMT-based model, microplastic strain is needed just to calculate the amount of entropy generation.

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Temperature variation for uniaxial tension temperature (K)

320 300 280 260 240 220 200 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

time(picosecond) 0.001/ps

0.005/ps

0.01/ps

Fig. 9.6 Temperature variation with time under uniaxial tension for different strain rates

In two-scale models, the fatigue in the material is investigated at macroscopic and microscopic scales independently. From a macroscopic point of view, the material deforms elastically during the elastic cyclic fatigue loading, while from the microscopic point of view, some microplasticity is activated, because of high-stress concentration, at defects and localized dislocation slip planes (Fig. 9.7). The two-scale model is based on an RVE (representative volume element) and divided into two parts: the elastic matrix part and the elastic-plastic inclusion part. The law of localization and homogenization is applied to deduce the relationship between macroscopic stress tensors and the microscopic stress and microscopic plastic strain tensors (Lemaitre et al., 1999; Doudard et al., 2005; Fan et al., 2018). Microscopic stress tensor is given by σ μ = Σ - 2μð1 - bÞεμp b=

6ðK þ 2μÞ 5ð3K þ 4μÞ

ð9:16aÞ ð9:16bÞ

where σ is microscopic stress tensor, Σ is macroscopic stress tensor, μ is Lame’s constant, K is bulk modulus, and εp is the microscopic plastic strain tensor. Charkaluk and Constantinescu (2009) proposed a modified version of Eq. (9.16a) using a self-consistent scheme proposed by Kröner (1961). In the case of isotropic elastic behavior with a classically defined deviatoric plasticity, one can write the following relationship: σ μ = Σ - 2μð1 - bÞð1 - f v Þεμp

ð9:17Þ

9.1

Predicting High Cycle Fatigue Life of Metals

469

Fig. 9.7 A representative volume element of the two-scale model, Σ is a macroscopic elastic stress tensor, E is a macroscopic elastic strain tensor, σ μ is microscopic stress tensor, and εμp is the microscopic plastic strain tensor

where fv is the volume fraction of micro-defects experiencing (slip) microplasticity. Assuming that the material experiences the same elastic behavior at the mesoscopic and the macroscopic scale, the previous relation implies the following relation: εμe = E - ½ð1 - bÞð1 - f v Þεμp

ð9:18Þ

where εμe is the microscopic elastic strain tensor. Charkaluk and Constantinescu (2009) report that, although it cannot exactly be correlated to the volume ratio, it is assumed that the relative surface ratio covered by the activated dislocation slip bands can be used to represent fv. For low carbon steel, which has a fatigue limit [meaning no microplasticity happens] of 235 MPa, Cugy and Galtier (2002) reported empirical values of fv = 3% for a nominal stress amplitude of 180 MPa, a value of fv = 10% for a nominal stress amplitude of 250 MPa, and a value of fv = 20% for a nominal stress amplitude of 300 MPa. We believe that fv may be calculated by dislocation dynamics simulations for a given initial defect ratio or measured by SEM. A computational scheme for the microplastic strain increment was proposed by Charkaluk and Constantinescu (2009), as follows:

Δεμp = qffiffi 3 2

qffiffi  μ 3   2 Anþ1 - σ y0

Anþ1       2μð1 - bÞð1 - f v Þ þ 23 h Anþ1

ð9:19Þ

in which h is the hardening modulus and Anþ1 is given by  2 Anþ1 = devðΣn Þ - 2μð1 - bÞð1 - f v Þ þ h εμp,n þ devðΔΣÞ 3 b=

6ðK þ 2μÞ 5ð3K þ 4μÞ

ð9:20aÞ ð9:20bÞ

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The yield criterion is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi 2 μ f = ðSμ - αμd Þ : ðSμ - αμd Þ σ =0 3 y0

ð9:21Þ

and the hardening rule is given by α_ μ = h_eμp

1 μ ðσ - αμ Þ = h σ μy0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 μp μp 1 μ E_ E_ ðσ - α μ Þ 3 ij ij σ μy0

ð9:22Þ

where Sμ is the microscopic deviatoric stress, αμd is the deviatoric part of the microscopic back-stress of kinematic hardening, eμp is the microscopic equivalent plastic strain, σ μy0 is the microscopic yield stress, (σ μ - αμ) is the translational direction of the microscopic yield surface under Ziegler’s rule, and h is the kinematic linear hardening modulus defined by  the slope of the stress-strain curve for a finite μ μ plastic strain value as follows: h = σ - σ y0 =εμp . The entropy generation due to microplasticity is then given by Z ΔSμp =

t

t0

ϕf v

σ μ : ε_ μp dt T

ð9:23Þ

in which σ μ and ε_ μP are micro-stress and strain rate tensors. During the microscopic plastic deformation, some plastic work is stored as dislocation stored energy, which is accounted for as the hardening of the material in Eq. (9.22). fv is a constant that establishes a relation between the macro-stresses and micro-stresses utilizing the law of localization and homogenization. It is the ratio of the activated volume of the micro-defects VD to the elastic matrix Vmatrix (Charkaluk & Constantinescu, 2009): fv =

VD V matrix

ð9:24Þ

Some researchers determined fv value by high-resolution microscopic inspection of the surface area along the central axis of the specimen around the center of the gage part during the test to identify activated slip bands in different stages of fatigue (Cugy & Galtier, 2002). For high cycle fatigue of DP600 steel under conventional low-frequency tension-compression loading (e.g., operating at 30 Hz), the material deformation is expected to happen in an athermal regime, where trans-granular crack initiation that leads to surface failure occurs due to local stress and strain concentration at slip bands in ferrite grains (Torabian et al., 2017). The percentage of the observed slip bands’ area in the microscope inspection surface area can be used as a first approximation to calculate the volume fraction of micro-defects in the matrix, VD/Vmatrix. This value can be determined by using microscopy such as TEM or SEM, as shown in Fig. 9.9.

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Predicting High Cycle Fatigue Life of Metals

471

Fig. 9.8 Visualization of microplasticity entropy calculation Table 9.2 Material parameters for DP600 steel Young’s modulus Hardening coefficients Poisson’s ratio Density

210,000 MPa 1000 MPa

Macroscopic yield stress Microscopic yield stress

440 MPa 260 MPa

0.3

Thermal expansion coefficient coefficientcoefficients Heat capacity

10 × 10/C 460 J kg1 -1 K 80 s

7894 kg m-3

Equivalent thermal exchange duration, τeq,

6 o

Figure 9.8 depicts the visualization of microplasticity, inside the hexagon region, microscopic stress and microplastic strain. In Table 9.2 material constants used for simulation are given. The microscopic yield stress in this microplasticity model is the mean fatigue limit of the specimen. We should clarify that a fatigue limit does not mean that there is no fatigue failure if the stress is below the fatigue limit. Not resulting in fatigue at any stress level would violate the second law of thermodynamics. It just indicates that no microplasticity occurs below the macroscopic fatigue limit stress. However, all other entropy generation mechanisms are still active, regardless of stress level and all materials’ fatigue at stress levels below the fatigue limit. In Eq. (9.23), TSI, (ϕ), is a metric of entropy evolution. It is assumed that the increase in the percentage of activated dislocation slip planes follows the second law of thermodynamics, which is represented by TSI. However, each microplasticity site is considered to be independent and has a constant fv. Microplasticity sites are not expected to coalesce before the onset of final failure (Fig. 9.8).

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Fig. 9.9 Dislocation slip bands in ferrite grains on the surface of the specimen activated at 250 MPa stress amplitude. N = 107 cycles, ΔT < 15 ° C. (After Torabian et al. (2017))

There is no fv data available in the literature obtained by TEM or SEM for DP600 steel at hand. However, Torabian et al. (2017) provided fatigue test data on DP600 steel and the results of fractography studies. The appearance of slip bands in ferrite grains on the surface of the specimen around the gage part under 30 Hz fatigue loading is shown in Fig. 9.9. The arrows indicate the slip bands, which are estimated to be around 10% of the observed surface. We use fv=10% as the initial value for the calculations. Of course, fractography studies for individual cases are necessary, since the defect ratio in materials is the highly variable depending quality of the material and manufacturing methods. The microplasticity calculation in this section is based on the following assumptions: 1. The hardening behavior of the material is approximated by a bilinear kinematic hardening curve. Hardening modulus is defined as the slope between (loweryielding strength) LYS and UTS (ultimate tensile strength) from the stress-strain curve, as a first approximation. 2. The microscopic hysteresis loops are stabilized, and they are the same at each defect site. The evolution of TSI, ϕ, from 0 to 1 only induces microplasticity at more defect sites, without any macroplasticity. 3. The energy dissipation due to kinematic hardening at microplasticity sites is small enough to be ignored (Naderi et al., 2010)

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Predicting High Cycle Fatigue Life of Metals

473

Fig. 9.10 Micro-stress-strain curve for each inclusion (defect) site under 300 MPa nominal stress amplitude

A microstress-microstrain hysteresis loop at a defect location, calculated by Eqs. (9.17)–(9.22), is shown in Fig. 9.10. The entropy production due to microplasticity for a specimen undergoing 1.4 × 105 cycles (at a frequency of 30 Hz) is shown in Fig. 9.11.

9.1.3

Comparison Between Different Entropy Generation Mechanisms

The magnitude of each entropy generation mechanism is summarized in Table 9.3. In Table 9.3, it is observed that the configurational entropy and vibrational entropy generation mechanisms are around the same order of magnitude. They are very small compared to heat conduction and microplasticity. The diffusion mechanism and atomic-friction-generated heat are also very small. Furthermore, between the microplasticity and heat conduction, entropy generation due to microplasticity is

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Fig. 9.11 Entropy generation (MJK-1 m-3) due to microplasticity versus the number of cycles under 300 MPa nominal stress amplitude Table 9.3 Order of magnitude of different entropy generation mechanisms 1 2 3 4 5 6

Mechanisms Configurational entropy Vibrational entropy Vacancy diffusion Thermal conduction Atomic-friction-generated heat Microplasticity

The magnitude of entropy generation ≈10-7 JK-1 m-3 ≈10-8 JK-1 m-3 Negligible due to low diffusivity ≈104 JK-1 m-3 Negligible based on molecular dynamics simulations ≈106 JK-1 m-3

two orders of magnitude bigger. Hence, we conclude that the total entropy production, the thermodynamic fundamental equation, can be simplified as ΔS = ΔSconf þ ΔSvib þ ΔSd þ ΔST þ ΔSr þ ΔSμp ≈ ΔSμp

ð9:25Þ

Therefore, the thermodynamic state index can be given by 0

0

B B B ϕ = ϕcr B @1 - exp @

Z

t

- ms

ϕf v

t0

R

σ μ :ε_μP ρT

11 dt

CC CC AA

ð9:26Þ

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Predicting High Cycle Fatigue Life of Metals

475

Incorporating the TSI into microplastic work accounts for the increased probability of activated micro-defect sites that emerge in the specimen during the fatigue process. TSI is an exponential function, hence never reaches unity. In practice, we determine a critical value ϕcr as a threshold. In this example, the specimen is assumed to fail when ϕ reaches ϕcr = 0.995, since the probability of reaching maximum entropy at this state is 99.5%.

9.1.4

Temperature Evolution due to Mechanical Work

The temperature of the specimen plays an important role in high cycle fatigue life. From the entropy point of view, the cumulative entropy production and TSI calculation are related to the temperature of the specimen. Inspecting Eq. (9.26), if the temperature is controlled (i.e., the specimen is air-cooled, e.g., near room temperature), the resulting entropy production will be larger, hence resulting in shorter fatigue life. However, self-heating during loads at low frequencies (i.e., 30 Hz) does not exceed ΔT = 15 K. This amount of temperature difference will not change the fatigue life significantly. However, as the frequency of the loading increases, ΔT increases. In ultrasonic frequencies, ΔT reach several hundred degrees. The temperature of the specimen is governed by the fully coupled thermomechanical equation derived from classical continuum mechanics as follows: k h ∇2 T = ρC T_ - σ : ε_ p - ρr - T




∂σ e : ε_ ∂T




 ∂Ak _ þ Ak - T Vk ∂T

ð9:27Þ

∂s where ρ is the mass density; C = T ∂T is specific heat; εe and εP are elastic and plastic strain vectors, respectively; σ is the stress tensor; r is the strength per unit mass of the internal distributed heat source; and Ak is a thermodynamic force associated with the internal thermodynamic variables, Vk. Using Eq. (9.27), we can calculate the evolution of temperature in the specimen due to mechanical work. To simplify this equation, we can ignore plastic strain ε_ p , internally distributed heat source r, and other thermodynamic variables, Ak V_ k. These assumptions are justified, because of the following reasons:

1. The sample is subjected to elastic loads only, as such, there is no uniform macroplastic strain. However, the contribution of microplasticity is not neglected. 2. Based on molecular dynamics simulations, the contribution of r is negligible at 30 Hz loading.  3. Ak - T ∂Ak V_ k represents the non-recoverable energy corresponding to the ∂T

internal coupling source (such as grain coarsening, phase transformation, etc.). However, for metals, this non-recoverable energy only represents 5–10% of the mechanical dissipation (plastic work) and is often negligible for high cycle fatigue.

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Therefore, Eq. (9.27) can be simplified to the following expression:  ∂Σ _e ρC T_ - ϕf v σ μ : ε_μP - k h ∇2 T - T : E =0 ∂T

ð9:28Þ

If we ignore the thermal fluctuation due to thermoelastic damping and simplify the conduction term as - k h ∇2 T ffi ρC τθeq, Eq. (9.28) can be written in the following time integration form: ρC

 εμP,nþ1 - εμP,n T - T0 þ ρC n =0 Δt τeq  σ μn : εμP,nþ1 - εμP,n T - T0 T nþ1 = T n þ ϕf v Δt - n ρC τeq

T nþ1 - T n - ϕf v σ μn : Δt




ð9:29Þ

ð9:30Þ

The temperature in the sample is calculated at each cycle using Eq. (9.30). Figure 9.12 shows the temperature evolution for DP600 steel operating at 30 Hz for various stress amplitudes obtained using Eq. (9.30). The entropy generated due to microplasticity is strongly affected by the stress amplitude. A rapid climbing stage followed by a steady state is observed. However, Eq. (9.29) cannot capture the abruptly raising temperature when a specimen is near failure when the temperature spikes.

Fig. 9.12 Temperature evolution versus the number of cycles at various stress amplitudes

9.1

Predicting High Cycle Fatigue Life of Metals

9.1.5

477

Entropy and TSI Calculations

Figure 9.13 shows accumulated entropy at failure for various stress amplitudes. Accumulated total entropy at failure is a constant value and is sometimes also referred to as the fatigue fracture entropy (FFE), which is known to be independent of stress amplitude. Figure 9.14 shows the thermodynamic state index as a function of the number of cycles. TSI, ϕ, reaches 1 around 8.5 × 104 cycles under 325 MPa applied stress. For stress amplitude of 300 MPa, TSI, ϕ, reaches 1 at 1.4 × 105 cycles. For stress amplitude of 275 MPa, TSI, ϕ, reaches 1 at around 3.6 × 105 cycles. If we relate the stress amplitude with the number of cycles to failure, a bilinear curve can be plotted, in Fig. 9.15. In Fig. 9.14, it is easy to observe that the TSI increases slowly in the early stage of cycling and then suddenly begins to evolve rapidly after a certain critical threshold. This trend can be explained by Eq. (9.28), in which TSI is included in the microplasticity entropy production equation. In the early stages, there are too few microplasticity sites. Because microplasticity is the largest contributor to entropy generation, the corresponding TSI calculated by the entropy production is small at the early stages. After a certain number of cycles, enough micro-defects are activated, and more entropy is produced.

Fig. 9.13 Accumulated total entropy production versus the number of cycles for various stress amplitudes [this is also referred to as the fatigue fracture entropy]

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Fig. 9.14 TSI evolution versus the number of cycles for various stress amplitudes

Fig. 9.15 Fatigue life prediction with UMT for stress amplitude versus the number of cycles for various stress amplitudes [when TSI reaches ϕcr = 0.995, the specimen is considered as failed]

9.1

Predicting High Cycle Fatigue Life of Metals

479

Fig. 9.16 Comparison of fatigue data and unified mechanics theory predictions for tensioncompression fatigue tests on dual-phase steel (DP600) without any pre-strain [the experimental SN curve is obtained from Torabian et al., 2017]

In Fig. 9.16, simulation results are compared with test data. It shows that the predicted cycles to failure obtained from unified mechanics theory and the test data match reasonably well. Test data in Fig. 9.16 shows expected scatter, especially at lower stress levels. It is very well-known that fatigue test data is always stochastic. Unified mechanics theory, which is based on Boltzmann’s entropy formulation, also yields stochastic expected life. In this section, unified mechanics theory is used for predicting the high cycle fatigue life of DP600 steel. The thermodynamic fundamental equation of metals under high cycle fatigue is derived. In the fatigue life prediction, no empirical curve fitting function is needed. The entropy generation due to atomic vacancy configuration, atomic vibration, and mass transport is extremely small compared to entropy generation due to microplasticity. It is also assumed that entropy generation due to grain coarsening during high cycle fatigue is negligible. The entropy generation due to microplasticity is the most dominant entropy generation mechanism. This mechanism can be visualized as some microplastic inclusions at localized defect sites inside an elastic matrix which has its microstress and micro-strain based on laws of localization and homogenization. Energy is dissipated through the microplastic work at the locations of these inclusions. The number of inclusions increases as TSI ϕ increases from 0 to ϕcr. Metal high cycle fatigue life prediction results based on the theory of unified mechanics are compared with experimental test data. It is shown that the prediction and test data match very well.

480

9

9.2

Predicting Ultrasonic Vibration Fatigue Life

Fatigue of Materials

Low cycle, high cycle, very high cycle, and ultrahigh cycle fatigue occur on different entropy generation mechanisms. In each one, the magnitude of entropy generation for each mechanism is different. For example, during high cycle fatigue, internal heat generation is usually less than ≈15 ° C temperature change in a test sample, while in very high cycle, fatigue temperature change can go to ≈150 ° C or much higher. On the other hand, while microplasticity is the dominant entropy generation mechanism in a high cycle and very high cycle fatigue, it is assumed that it does not happen in ultrahigh cycle fatigue, which happens when the stress value is below the nominal fatigue limit of a metal. In this section, predicting the very high cycle fatigue life of metals with the unified mechanics theory is discussed. Simulation results are compared with the fatigue life test data. It is shown that the unified mechanics theory can predict very high cycle fatigue life very well, without relying on traditional curves fitting an empirical function to fatigue test data. The objective is to predict the fatigue life of metals subjected to ultrasonic vibration where the maximum nominal stress is below the yield stress of the material. The nomenclature used in this section is given in Table 9.4. Table 9.4 Nomenclature used for ultrasonic vibration fatigue life model Nomenclature ϕ Thermodynamic state index ϕcr Critical thermodynamic state index Δs Change in total specific entropy ms The molar mass of the material R Universal gas constant ST Thermal conduction entropy Sr Internal friction entropy Sμp Microplasticity entropy kh σμ εμe εμP Σ E fv Sμ αμd eμp h σy Ψf

Thermal conductivity Microscopic stress tensor The microscopic elastic strain tensor The microscopic plastic strain tensor Macroscopic stress tensor Macroscopic strain tensor The volume fraction of activated microdefects Microscopic deviatoric stress Deviatoric of microscopic back-stress Microscopic equivalent plastic strain Hardening modulus Microscopic yield stress (fatigue limit) Frequency coefficient

ρ r rdrag rdis Bdrag v μ b ϱ ϱm αH γ_ a E υ

Mass density Internal heat generation [IH] IH due to drag mechanism IH due to dislocation motion Drag coefficient Dislocation velocity Shear modulus The magnitude of the Burger’s vector Total dislocation density Mobile dislocation density Taylor’s constant Shear strain rate Lattice constant Young’s modulus Poisson’s ratio

α cV θ τeq t eq

Thermal expansion coefficient Specific heat capacity Total change in temperature Equivalent conduction time Fraction of heat conducted

9.2

Predicting Ultrasonic Vibration Fatigue Life

9.2.1

481

Thermodynamic Fundamental Equation

During the very high cycle fatigue of metals, six prominent mechanisms generate entropy. These are: I. II. III. IV. V. VI.

Configurational entropy Sc. Vibrational entropy, Svib Entropy generation due to diffusion Sd Entropy generation due to heat conduction ST Entropy generation due to internal heat generation Sr Entropy generation due to microplasticity, Sμp

Among these six mechanisms, the first three are too small compared to other mechanisms (Lee & Basaran, 2021). Therefore, we discuss the entropy generation due to the last three mechanisms. Entropy is an additive property; hence, we can write the following equation for the total entropy to obtain the thermodynamic fundamental equation: ΔS = ΔST þ ΔSr þ ΔSμp

ð9:31Þ

The following derivation is based on these assumptions: 1. Maximum nominal stress during fatigue testing is below the yield stress of the material; hence, no macroscopic plastic deformation is expected. However, microplasticity is expected to happen at defect sites at the micro-level. 2. Input mechanical energy increases atomic vacancies and dislocation density. However, the increasing dislocation density only causes hardening at a microlevel and never induces macroscopic plastic deformation, as the maximum nominal stress is below the metal’s yield stress. The vacancy generation/diffusion and dislocation motions (e.g., cross-slip) around defects induce localized microplasticity at a micro-level. 3. Temperature evolution in the specimen is due to several independent mechanisms, such as internal atomic-friction-generated heat, heat conduction, microplastic work, and thermoelastic damping.

I. Entropy Generation due to Heat Conduction Temperature gradient causes an irreversible heat flow across the specimen during fatigue loading. The entropy change is given by ΔST = -

Z t
kh t0

 ∇T ∙ ∇T dt T2

ð9:32Þ

where kh is coefficient of heat conduction and ∇T is the temperature gradient from the gage section to the grip section of the tested sample, respectively. T are the absolute temperature at the gage center.

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In Eq. (9.32), the amount of entropy generation due to heat conduction is strongly dependent on the test duration. To calculate the order of magnitude of this entropy generation mechanism for ultrasonic vibration fatigue, the following boundary conditions are used: According to measurements during ultrasonic vibration testing, the temperature at the gage section of the steel [EN 1.0577] test specimen raises from 300 K to about 450 K, while the nominal normal stress amplitude is 400 MPa. The thermal conductivity of the material is kh = 50WK-1 m-1. Assume the temperature varies parabolically along the centerline of the sample from the gage section to the grip section, as T = C1x2 + C2x + C3. If the boundary conditions are set to be Tgage(t) = T(x = 0, t), Tgrip(t) = T(x = ± L/2, t) where L2 = 30 mm is the length between gage and grip sections. The one-dimensional thermal gradient that changes  as the number of cycles increases can be obtained as ∇T = L8x2 T grip ðt Þ - T gage ðt Þ. II. Entropy Generation due to Microplasticity During very high cycle fatigue, microplasticity happens at defect sites. The concept of microplasticity was introduced in the previous section of this chapter. Hence, we just present the final equation. The entropy generation equation due to microplasticity is given by Z ΔSμp = t0

t

Ψf

! σ μ : ε_μP ϕf v dt T

ð9:33Þ

Compared to the microplasticity-generated entropy formulation in the previous section, Eq. (9.23), the new Eq. (9.33), has an extra term, the frequency coefficient Ψf, which is necessary to relate the forced vibration frequency with the entropy generation rate. It is well understood that the fatigue life measured for a specimen is affected by the loading frequency. For example, under the same stress amplitude, fatigue life is shorter when loading is at a 10–50 Hz frequency range, compared to the fatigue life when the loading frequency range is 15–30 kHz. For FCC metals, it is well-known that the difference in fatigue life is due to the occurrence of timedependent cross-slip of dislocations and vacancy production/diffusion involved in the persistent slip band formation. Marti et al. (2020) established a correlation between low- and high-frequency fatigue life of pure polycrystalline copper with mechanisms of slip band formation. The failure and early slip marking S-N curves they reported from experiments show that the fatigue life is longer at higher frequencies, and the number of cycles for early slip markings to emerge is also higher at “high” than “low” frequencies, for a given stress amplitude. The similarity of these tendencies suggests a correlation between both failure and early slip marking S-N curves. It is assumed that such a correlation is a reasonable hypothesis because the formation and emergence of slip bands play a key role in crack initiation and fatigue failure in ductile single-phase metals. The initiation of persistent slip bands at the surface of fatigued specimens requires crossslip of screw dislocations to promote slip localization and irreversibility. In addition,

9.2

Predicting Ultrasonic Vibration Fatigue Life

483

the generation of vacancies in persistent slip bands and their diffusion toward the matrix favor their emergence. Marti et al. (2020) concluded that cross-slip activation is the main mechanism that leads to early slip markings. For the ultrasonic vibration of BCC metals, Torabian et al. (2017) reported that there is a transition between material deformation mode from thermally activated regime at stress amplitudes below the fatigue limit to athermal mode at stress amplitudes above the fatigue limit. Since in this example the macroscopic stress amplitude is above the fatigue limit, the athermal regime where the mobilities of screw and edge dislocations are equivalent and screw dislocations can cross-slip is of interest. The cross-slip probability P is given by Marti et al. (2020): P=β


 l δt τ - τcsR exp css V kB T l0 t 0

ð9:34Þ

where β is a normalization coefficient ensuring that 0 < P < 1, l/l0 is the ratio of the length of a screw dislocation segment to a reference length l0 = 1 mm, δt/t0 is the ratio of the considered time to a reference time, V is the activation volume associated with cross-slip, (τcss - τcsR) compares the resolved shear stress τcss on the cross-slip system to a threshold critical stress τcsR required to activate cross-slip, kB is Boltzmann’s constant, and T is the absolute temperature. The duration in which τcss is greater than τcsR in one cycle is given by Marti et al. (2020): 

0 Δt 1cycle = @1 -

2 arcsin π

τcsR τcss

1 A1 f

ð9:35Þ

where f is the frequency of cyclic loading. Assuming all the parameters in Eq. (9.34) are frequency insensitive, Eq. (9.35) shows that the time to activate cross-slip is inversely related to the frequency. For two different loading frequencies, 30 Hz and 20 KHz at the same stress amplitude, the following equation must be satisfied to obtain the same cumulated probability of cross-slip activation: T A = Δt 1cycle,30 Hz × N 30 Hz = Δt 1cycle,20k Hz × N 20 kHz

ð9:36Þ

where TA is the duration when τcss is greater than τcsR and N is the number of cycles. According to Eqs. (9.35) and (9.36), at a given stress amplitude, the material subjected to f1 loading frequency reaches the same cumulative probability of activating initial micro-defects (early slip marking) as the material under f2 loading frequency when N1 f 1 = N2 f 2

ð9:37Þ

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Fatigue of Materials

where N1 is the number of cycles experienced under frequency f1 and N2 is the number of cycles experienced under frequency f2. Test data indicates that the fatigue life in a frequency range of 10–50 Hz is not strongly frequency sensitive. If we set the operating frequency fr = 10~50 Hz as the reference frequency, and fp = 15~30 KHz as the frequency during ultrasonic vibration testing, then Ψf can be defined as Ψf = fr/fp, and Eq. (9.33) can be rewritten in the following conditional form: Z ΔSμp =

t

Ψf

t0

8 < 1, Ψf = 100 : , fp

! σ μ : ε_μP ϕf v dt T

ð9:38Þ

if f p < 100 Hz ðtypical servo - hydraulic machineÞ if f p is in the order of KHz ðultrasonic vibrationÞ

ð9:39Þ

The frequency coefficient Ψf = fr/fp is used to account for the fact that at higher frequencies, more cycles are needed to activate the same volume of dislocation slip bands. It indicates that at a given stress amplitude, under the same number of cycles, the specimen produces lower cumulative microplasticity-generated entropy at high frequency than at low frequency, because under the same number of cycles, the probability of generating microplasticity is smaller at high frequency than at low frequency. The high and low frequency mentioned here refer to the ultrasonic vibration frequency and traditional servo-hydraulic machine frequency, respectively. Equation (9.39) is only applicable to up to the frequencies in the order of kHz, because at higher frequencies (e.g., MHz or GHz [ultrahigh cycle fatigue]), different micro-mechanisms and fatigue failure occur (Marti et al., 2020; Torabian et al., 2017). We should also point out that Eq. (9.39) is also applicable only for the class of materials where dislocation slip band formation dominates the crack initiation process (Mughrabi, 2002). In the literature, many micromechanics models for very high cycle fatigue life prediction utilize cyclic irreversible slip, which is equivalent to fv in the formulation presented above. III. Entropy Generation due to Atomic-Friction (Scattering)-Generated Heat The entropy change due to internal atomic-friction (scattering)-generated heat is given by Z ΔSr = t0

t

ρr dt T

ð9:40Þ

At the atomic level, the temperature is defined as the intensity of atomic vibrations. The internally generated heat per unit mass, r, is the increasing intensity of atomic vibrations in the lattice due to events such as the breaking of atomic bonds, phonon-phonon scattering, phonon-electron scattering, electron scattering (referred to as internal friction here), and dislocation motions (Gao, 1997; Ragab & Basaran, 2009; Chu et al., 2015). We should emphasize that this term is different than the heat generated through the macroscopic observable plastic work or thermoelastic sources.

9.2

Predicting Ultrasonic Vibration Fatigue Life

485

In the following section, the equation governing internal heat generation is derived analytically. It has been shown that the rate of internal heat generation is composed of two mechanisms: 1. The drag (friction due to internal scattering) process involves phonon drag, electron drag, and radiation drag (Blaschke et al., 2020; De Hosson et al., 2001; Galligan et al., 2000). The internal heat generation equation for this mechanism is given by ρr drag = ϱBdrag v v

ð9:41Þ

2. The dislocation motion during the plastic deformation (Huang et al., 2008; Parvin & Kazeminezhad, 2016). The internal heat generation equation for this mechanism is given by ρrdis =

pffiffiffi 1 2 μb ϱ_ - αH μb ϱγ_ 2

ð9:42Þ

where ρ is the mass density and Bdrag and v represent the effective drag coefficient and velocity of dislocation, respectively. The terms μ, b, ϱ, and αH are the shear modulus, the magnitude of Burger’s vector, total dislocation density, and Taylor’s hardening parameter, respectively. In Eq. (9.41), the velocity of dislocation is related to the applied shear strain rate, γ_ , as follows: v=

γ_ ϱm b

ð9:43Þ

where ϱm is the density of mobile dislocations. Hence, the entropy change due to internal friction can be given by Z ΔSr = t0

t

ρr dt = T

Z t  drag pffiffiffi  ϱB v v þ 12 μb2 ϱ_ - αH μb ϱγ_ dt T t0

ð9:44Þ

Using Eq. (9.44), the evolution of ΔSr for the specimen under ultrasonic vibration with 400 MPa stress amplitude is shown in Fig. 9.20. Thermodynamic Fundamental Equation Finally, the thermodynamic fundamental equation can be given in the following form: !  Z t Z t
σ μ : ε_μP ∇T ∙ ∇T dt þ kh Ψf ϕf v Δs = dt ρT ρT 2 t0 t0 Z t  drag pffiffiffi  ϱB v v þ 12 μb2 ϱ_ - αH μb ϱγ_ dt þ ρT t0

ð9:45Þ

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Fatigue of Materials

To calculate the TSI evolution versus the number of cycles, material constants need to be obtained beforehand. The temperature evolution in the sample is computed using a fully coupled thermal-mechanical equation. To verify the computed temperature from the fully coupled thermal-mechanical equation, a thermographic camera is used to record the temperature profile of the specimen focusing on the gage center during fatigue testing. Modal Analysis An ultrasonic vibration fatigue test is usually carried out in the resonance frequency of the sample in the axial direction. Therefore, modal analysis is necessary to determine the exact natural frequencies of the tested specimen. To compute the natural frequencies of the test specimen, usually finite element analysis is utilized. The tension-compression mode [in the axial direction] resonance frequency of the sample was determined to be 19,864 Hz. Then, for the axial direction resonance frequency, the stress distribution in the sample must be determined for selected levels of the vibration amplitude applied at the support end of the sample. A linear relationship between the vibration amplitude and the average stress in the center of the sample is obtained: σ = 20,510  A

ð9:46Þ

where σ is the average normal stress in the center of the sample in MPa and A is the vibration amplitude in mm. Finite element analysis results indicate that stress is greater on the surface of the specimen and smaller in the center. These stresses are in the range of about ±1% of the average stress in the narrow gage section. The average stress is σ = 380 MPa for vibration amplitude of A = 0.0185 mm (Fig. 9.17). It is shown that a temperature gradient between the specimen’s gage section and grip section increases slowly in phase 2 and then rises rapidly when a macro crack forms in phase 3. Thermodynamic State Index (TSI) Computation Material properties used in the example are given in Table 9.5. The microscopic yield stress is the mean fatigue limit of the specimen; it is assumed that no microplasticity occurs below the macroscopic fatigue limit. It should be noted that the microplasticity model in this subsection is based on the following assumptions: 1. The hardening behavior of the material is approximated by a bilinear kinematic hardening. Hardening modulus is defined as the slope between (lower yield strength) LYS and UTS (ultimate tensile strength) from the test stress-strain curve, as a first approximation. 2. The microscopic hysteresis loops are stabilized and they are the same at each defect site. The evolution of TSI, ϕ, from 0 to 1 only induces microplasticity at more defect sites; however, they never coalesce until the final rupture.

9.2

Predicting Ultrasonic Vibration Fatigue Life

487

Fig. 9.17 (a) Sample dimensions along the axis, (b) normal stress distribution along with the sample, (c) displacement distribution along the specimen axis for average stress of σ = 380 MPa. (After Lee et al. (2022))

Table 9.5 Material parameters for EN 1.0577 steel Young’s modulus Hardening coefficient

197,265 MPa 879 MPa

Poisson’s ratio

0.27

Mass density The volume fraction of active defects fv Critical thermodynamic state index

7820 kg m-3 20%

After Lee et al. (2022)

ϕcr = 0.97

Macroscopic yield stress Microscopic yield stress [fatigue limit] Thermal expansion coefficient Specific heat capacity cV Frequency coefficient Ψf

400 MPa 368 ± 3.3 MPa 12.610-6 K-1 470 J kg-1 K-1 510-3

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Fatigue of Materials

Computing Internal Heat Generation It has been shown that during the deformation of iron and iron alloys, the proportion of the main types of activated dislocations is approximately 60% of 12 ah1 1 1i, 20% of ah1 0 0i, and 20% of ah1 1 0i, which are unaffected by the temperature of deformation, alloy content, amount of strain, dislocation density, or rate of straining (Dingley & Hale, 1966). The magnitude of Burger’s vectors can be calculated by  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a h2 þ k 2 þ l2 For BCC and FCC lattice 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kbk = ðaÞ h2 þ k 2 þ l2 For simple cubic lattice kbk =

ð9:47Þ

where kbk is the magnitude of Burger’s vector, a is the lattice constants in units of an angstrom (Å), and h, k, l are the components of Burger’s vector. The parameters in Eqs. (9.41)–(9.43) can be obtained by discrete dislocation dynamics. In the literature, these values are available for the material used in this example, as follows: 1. Burger’s vector values are provided in Dingley and Hale (1966). The terms related to Burger’s vector are summed according to their proportion. 2. Taylor’s hardening parameter αH is a dimensionless parameter ranging from 0.05 to 2.6. It is used to characterize the relation between the plastic flow stress of a pffiffiffi material and its dislocation density, τ = τ0 þ αμb ρ . Taylor’s parameter from Davoudi and Vlassak (2018) is used, since it has a similar stress-strain curve as the material of interest in this example. 3. The initial dislocation density value for mild steel is given by Bin Jamal et al. (2021). 4. Moving dislocations experience a drag due to their interaction with the crystal structure, and this drag coefficient Bdrag determines the dislocation glide time between obstacles (Blaschke, 2019; Blaschke et al., 2019). In terms of the dominant drag mechanism, it is assumed that the phonon scattering with dislocations [phonon-defect scatter] constitutes the dominant mechanism. Bdrag value is provided by Blaschke et al. (2019). 5. The evolution of dislocation density during plastic deformation has been extensively studied in the literature. It is widely accepted that when nominal macrostress is well below the yield stress, the microplastic deformation that results in stabilized hysteresis micro-stress-strain loops happens only at micro-defect sites. Therefore, a constant dislocation generation rate Δϱ at each defect site is assumed. The material parameters needed in Eq. (9.45) are summarized in Table 9.6. Figure 9.19 shows entropy generation, Eq. (9.44), due to internal heat generation.

9.2

Predicting Ultrasonic Vibration Fatigue Life

489

Table 9.6 Parameters for mild steel EN 1.0577 Parameter Lattice constants The magnitude of Burger’s vector

Symbol a b

Shear modulus Taylor’s hardening parameter Initial dislocation density Drag coefficient due to transverse phonon drag Dislocation density increment in each cycle

μ αH ϱ Bdrag Δϱ

Value 2.856 60% of 0.87a 20% of 1.00a 20% of 1.41a 7.76 × 104 1 0.88 × 1014 0.02 1.523 × 1014

Units 10-10 m Angstrom

MPa – m/m3 mPa-s m/m3

After Lee et al. (2022)

9.2.2

Temperature Evolution due to Mechanical Work

The temperature increase due to mechanical work is governed by the fully coupled thermomechanical equilibrium equation given below:

 ∂σ e ∂Ak _ 2 p _ Vk : ε_ þ ρr þ T Ak ρcV T - k∇ T = σ : ε_ þ T ∂T ∂T

ð9:48Þ

where the first term is the rate of heat storage, the second term is the heat conduction, the third term is the energy dissipation due to macroplastic work, the fourth term is the thermoelastic coupling source, and the fifth term is the heat generation due to internal frictions, and the last term is non-recoverable energy corresponding to the ∂s is defined as specific heat capacity; εe and εP are elastic internal source. cV = T ∂T and plastic strain tensors, respectively; σ is the stress tensor; r is the strength per unit mass of the internal distributed heat generation; and Ak is a thermodynamic force associated with the internal thermodynamic variable, Vk. Equation (9.48) yields the evolution of temperature due to mechanical work with properly imposed boundary conditions. To simplify this equation, we can ignore plastic strain ε_ p and other thermodynamic variables Ak V_ k . These assumptions are valid, because 1. We are investigating mechanical response under elastic loads only, as such, there is no uniform plastic strain at the macro-level. However, the contribution from microplasticity  is not neglected. ∂Ak _ V k represents the non-recoverable energy corresponding to the 2. Ak - T ∂T

internal sources (e.g., sound waves, grain coarsening, phase transformation, etc.). However, for metals, this non-recoverable energy only represents 5% or less of the total dissipation.

490

9

Fatigue of Materials

Hence, Eq. (9.48) can be rewritten into the following form: h  i pffiffiffi 1 ρcV T_ - k∇2 T = Ψf ϕf v σ μ : ε_μP þ ϱBdrag v v þ μb2 ϱ_ - αH μb ϱγ_ 2

ð9:49Þ

where ρ is the mass density and Bdrag and v represent the effective dislocation-drag coefficient and velocity of dislocation, respectively. The terms kh, ϕ, and cV are coefficient of heat conduction, the thermodynamic state index, and the specific heat capacity, respectively. The relative surface ratio covered by activated micro slip bands is represented by fv. The terms μ, b, ϱ, and αH are the shear modulus of the bulk material, the magnitude of Burger’s vector, total dislocation density, and Taylor’s hardening coefficient, respectively. If we ignore the thermal fluctuation due to thermoelastic dissipation and assume that the temperature field is uniform, we can simplify the conduction term as - k h ∇2 T ffi ρcV τθeq (Torabian et al., 2017).

Where the terms, θ, and τeq are the total change in temperature (Tn - T0) and an equivalent conduction time used to characterize the heat transfer between the specimen and the surroundings (Charkaluk & Constantinescu, 2009; Zhang et al., 2013; Chrysochoos et al., 2012). Then, the following equation is obtained:
 h  i ðT - T 0 Þ 1 = Ψf ϕf v σ μ : ε_μP þ ϱBdrag v ∙ v þ μb2 ϱ_ ρcV T_ þ n τeq 2 pffiffiffi - αH μb ϱγ_

ð9:50Þ

The time discretization of the above equation is given by
μ  γ_ p nþ1 2 Δt Δt Δt ðT n - T 0 Þ þ Ψf ½ϕf v ðσ μ : ˙εμP Þ þ ϱBdrag ϱm b τeq ρcV ρcV pffiffiffi Δt 1 2 Δϱnþ1 Δt þ α μb ϱγ_ = 0: μb ρcV 2 Δt ρcV H ð9:51Þ

ΔT nþ1 = -

After rearranging the terms in Eq. (9.51), we get the following equation for calculating the change in temperature:
μ  h  i γ_ p nþ1 2 1 Ψf ϕf v σ μ nþ1 ε_μP Δt þ ϱBdrag Δt ρcV ρcV ϱm b pffiffiffi 1 1 1 2 α μb ϱγ_ μp Δt þ μb Δϱnþ1 ρcV H ρcV 2 ð9:52Þ

ΔT nþ1 = - t eq ðT n - T 0 Þ þ

9.2

Predicting Ultrasonic Vibration Fatigue Life

491

Fig. 9.18 Measured surface temperature at the gage center of the 11 monitored samples versus the number of cycles (in millions) during fatigue tests. (After Lee et al. (2022))

In Eq. (9.52) dimensionless time factor, t eq =

Δt τeq ,

represents the fraction of heat

conducted. Equivalent conduction time, τeq, the value measured and reported by Charkaluk and Constantinescu (2009), is used in this example. Temperature evolution in the material is calculated at each cycle using Eq. (9.52). It should be emphasized that Eq. (9.52) is used to simulate the phase 1 and phase 2 temperature evolution. Equation (9.52) cannot capture the abrupt temperature rise in phase 3 when a macro crack initiates, because the mechanisms of macroplasticity, macro crack formation, and crack propagation are not included in the formulation. Figure 9.18 shows the measured surface temperature at the gage center versus the number of cycles, and Fig. 9.19 shows the simulations for the same based on Eq. (9.52).

492

9

Fatigue of Materials

Fig. 9.19 Surface temperature predictions at gage center versus the number of cycles under various stress amplitudes based on Eq. (9.52). Compared to test data (Fig. 9.18). (After Lee et al. (2022))

9.2.3

Comparison Between Different Entropy Generation Mechanisms

The entropy generation due to heat conduction, microplasticity, and internal frictions during phase 1 and phase 2 is obtained using Eq. (9.45). The measured temperature rise during phase 1 (shown in Fig. 9.18, up to 2.5 × 105 cycles) matches the predictions shown in Fig. 9.19. Some researchers reported a change in temperature in phase 1, ΔTphase1, to be a constant value (Guo et al., 2015), analogous to the elastic modulus of a material. The temperature rise during phase 2 follows a parabolic function. When the simulation results shown in Fig. 9.19 are compared with test data shown in Fig. 9.18 [before the onset of phase 3], we observe a small difference at the end of phase 2. This discrepancy may be due to the uncertainty of the formation of cross-slip mechanisms [which leads to microplasticity], introduced by the frequency effects. However, the difference between the prediction and measured temperature is very small for all practical purposes. Figure 9.20 shows that the entropy generation due to microplasticity is small at the beginning and starts to increase after a critical threshold. This is because the evolution of activated micro-defects follows the TSI. As more micro-defects are activated, more entropy is produced.

9.2

Predicting Ultrasonic Vibration Fatigue Life

493

Fig. 9.20 Total entropy generation (MJK-1 m-3) versus the number of cycles under 400 MPa nominal normal stress, for different mechanisms. (After Lee et al. (2022))

The order of magnitude of each entropy generation mechanism is summarized in Table 9.7 From Table 9.7, it is obvious that during ultrasonic vibration, very high cycle fatigue, the entropy generation due to thermal conduction, atomic-friction-generated heat, and microplasticity are about the same order of magnitude. Still, microplasticity is the main contributor. Therefore, none of these three entropy generation mechanisms should be ignored. The thermodynamic state index can then be given by  13 Z t drag 1 2 Z t
0
Z t  pffiffi ðB v vþ2μb ϱ_ - αH μb ϱγ_ Þ σ μ :ε_μ ms k h ∇TT∙2∇T dt dt - Ψf ϕf v T P dt T B 6 C7 t0 t0 t0 B C7 ϕ = ϕcr 6 41 - exp @ A5 ρR 2

ð9:53Þ

494

9

Fatigue of Materials

Table 9.7 Order of magnitude of different entropy generation mechanisms 1 2 3

Mechanism Thermal conduction Internal friction Microplasticity

The magnitude of entropy generation ≈0.25 MJK-1 m-3 ≈0.85 MJK-1 m-3 ≈3 MJK-1 m-3

After Lee et al. (2022)

Equation (9.53) never reaches one, because it is an exponential function. In practice, we determine a critical value of TSI ϕcr as a threshold that we consider the specimen is failed taking into account computational time. In this example, ϕcr = 0.97 was chosen as a critical value.

9.2.4

Computing Thermodynamic State Index

The computation of TSI requires a summation of the entropy generation for all mechanisms. The material properties necessary for calculating TSI are presented in Tables 9.5 and 9.6. The material properties can be regarded as constant between 25 and 150 °C based on test data for EN 1.0577 steel, which shows a small change in material properties in this temperature window. The temperature in the specimen during the ultrasonic vibration test does not exceed 130 °C (around 400 K) before rapid temperature rise at phase 3 when a macro crack initiates. Therefore, material constants are used without a temperature correction. Using Eq. (9.53), we can compute the TSI evolution for EN 1.0577 steel operating at 20 kHz. The entropy generation and TSI evolution for different stress amplitudes are shown in Figs. 9.21 and 9.22, respectively. The cumulative specific entropy at failure [fatigue fracture entropy] is validated to be a material constant independent of stress amplitude, specimen’s dimensions and geometry, and loading frequency. It is commonly used as a damage indicator to predict the number of cycles to failure. Figure 9.21 shows that fatigue fracture entropy (FFE) is the same (a material constant) regardless of the stress level during testing. This is similar to test results reported widely by Naderi et al. (2010), Yun and Modarres (2019), and Ribeiro et al. (2020). Figure 9.22 shows the TSI evolution between 0 and 1. The evolution of TSI is based on the cumulative specific entropy of the system; therefore, for the same critical TSI (i.e., ϕcr = 0.999), the cumulative specific entropy will be the same regardless of the stress amplitude, frequency of loading, or geometry of the sample (Fig. 9.21, Fig. 9.23). The comparison of the unified mechanics theory-based model simulation S-N curve and the experimental S-N curve is presented in Fig. 9.24. From Fig. 9.21, it is observed that the cumulative entropy generation for 1.0577 steel under various stress amplitudes is the same (4.1 MJK-1 m-3) at failure. This value is a material constant regardless of the test frequency, loading path, loading type, geometry of the specimen, or stress state. However, the time it takes to reach the fatigue fracture entropy will be different in each case.

9.2

Predicting Ultrasonic Vibration Fatigue Life

495

Fig. 9.21 Entropy generation versus cycles for 1.0577 steel under various stress amplitudes. (After Lee et al. (2022))

A unified mechanics theory-based model is formulated for predicting the very high cycle fatigue life of EN 1.0577 steel under ultrasonic vibration operating at 20 kHz. The thermodynamic fundamental equation under ultrasonic fatigue is derived. Contributions of different mechanisms to the total entropy generation are compared. The entropy generations due to microplasticity, thermal conduction, and atomic-friction-generated heat are the dominant entropy generation mechanisms. Microplasticity, which makes the biggest contribution to entropy generation, can be visualized as having some microplastic inclusions in an elastic matrix. Energy is dissipated through the microplastic work at these inclusions. The number of inclusions increases as TSI, ϕ, increases from zero to the predefined ϕcr critical vale of TSI. The evolution trend of numbers of inclusions leads to very little entropy generation in the beginning, because in the beginning, very few inclusion sites are activated. However, after cycles accumulate, the number of inclusions experiencing microplasticity gradually increases and entropy generation due to microplasticity becomes more evident. To obtain the absolute temperature of the specimen needed in the thermodynamic fundamental equation without test data, a fully coupled thermomechanical equation that includes a heat storage term, a heat conduction term, a microplasticity term, and

496

9

Fatigue of Materials

Fig. 9.22 TSI evolution versus cycles for 1.0577 steel under various stress amplitudes. (After Lee et al. (2022))

a self-heating term due to internal frictions is derived. However, the derived thermalmechanical equation cannot capture the abrupt temperature rise in phase 3 when macro cracks initiate because the mechanisms of macroplasticity, crack formation, and crack propagation are not included in the model. Very high cycle fatigue life prediction results based on the unified mechanics theory are compared with experimental test data. The comparison shows that the prediction and test data match very well.

9.3

Predicting Low Cycle Fatigue Life

In low cycle fatigue, where the nominal stress is above the yield limit, the plastic work is the dominant dissipation mechanism. Entropy generation in the plastic dissipation process is given by Δs =

1 ρT

Z

t2 t1

σ : dεp

ð9:54Þ

9.3

Predicting Low Cycle Fatigue Life

497

Fig. 9.23 Unified mechanics theory simulated S-N data points for 1.0577 steel under various stress amplitudes. (After Lee et al. (2022))

where ρ is the mass density of the material and σ and ε p are the stress and plastic strain, respectively. T represents the temperature. Integral limits t1 and t2 represent the time bounds of the mechanical loading. For one-dimensional case, the total plastic strain, ε p(t), is calculated as follows: εp ðt Þ = εtotal ðt Þ -

σ ðt Þ E

ð9:55Þ

where εtotal(t) is the total strain at time t and σ(t) and E are to the stress at time t and Young’s modulus, respectively. The true stress-strain graph published by Carrion et al. (2017) for Ti-6Al-4V alloy, is used for the comparison between experimental data and the numerical predictions by Bin Jamal et al. (2020). Material constants used are given in Table 9.8. Assuming that the gage section of a dog bone AST standard sample experiences uniform strain, 5 mm section was considered in the computational model to reduce the computational cost. The diameter of the specimen is 6.35 mm. In ABAQUS, linear brick elements, C3D8R are used. One end of the sample is defined with a zero displacement (fixed) boundary condition and the other end is

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Fatigue of Materials

Fig. 9.24 UMT simulated S-N data points for 1.0577 steel compared with experimental data. The blue line and squares are simulated results from UMT. (After Lee et al. (2022)) Table 9.8 Material parameters used in the numerical model for tensile loading in Ti-6Al-4V alloy

Material parameter Young’s modulus, E Poisson’s ratio, ν Mass density, ρ Critical TSI, Φc Hardening parameter, K Hardening exponent, r Yield strength, σ y0 Molar mass, ms The reference temperature, T

Value 106 0.31 4540 1 968.00 0.64 992.00 0.047867 298

Unit GPa kg/m3 MPa MPa kg/mol K

Bin Jamal et al. (2020)

subjected to controlled displacement loading in the axial direction. After a mesh convergence analysis, an optimum seed size of 0.9 mm is fixed for all the simulations. Figures 9.25 and 9.26 show a comparison between the test data and numerical simulation for monotonic tension and compression experiments, respectively. Figure 9.27 depicts simulation results for engineering stress-strain hysteresis loops for 1.2% strain amplitude. Figure 9.28 presents TSI evolution for different strain amplitudes during cyclic loading. Figure 9.29 shows a comparison between the test data and computational predictions for low cycle fatigue life at different strain amplitudes.

9.3

Predicting Low Cycle Fatigue Life

499

1.2x109

A 9

True Stress (Pa)

True Stress (Pa)

1.0x10

8.0x108

6.0x108

1.1x109 1.0x109 1.0x109 1.0x109 9.8x108 9.6x108 9.4x108 9.2x108 9.0x108

0.007 0.008 0.009 0.010 0.011 0.012

Strain

4.0x108

Enlarged view of A

Test data Numerical simulation

2.0x108

0.0 0.00

0.01

0.02

0.03

Strain

Fig. 9.25 Comparison between monotonic tensile stress-strain graphs obtained from the test data (Carrion et al., 2017) and numerical simulation (Bin Jamal et al., 2020)

1.5x109

B 1.10x109 1.08x109 1.06x109 1.04x109

6.0x10

0.025

1.02x109 1.00x109

8

0.020

9.0x10

1.12x109

0.015

8

0.010

True Stress (Pa)

True Stress (Pa)

1.2x109

Strain

Enlarged view of B

3.0x108

Test data Numerical (Seed size - 0.64 mm) Numerical (Seed size - 0.85 mm) Numerical (Seed size - 1.0 mm)

0.0

0.0

0.1

0.2

0.3

Strain

Fig. 9.26 Comparison between test data (Carrion et al., 2017) and numerical simulation (Bin Jamal et al., 2020), for monotonic compressive stress-strain

500

9

1.5x109

1.5x109

Hysteresis loops at 1.2% strain amplitude

Hysteresis loops at 1.2% strain amplitude 1.0x109

1.0x109

5.0x108

Stress (Pa)

Stress (Pa)

Load cycle-1 5.0x108

0.0

Load cycle-1

0.0

-5.0x108

-5.0x108

Load cycle-50

-1.0x109

-1.5x109 -0.015

Fatigue of Materials

-0.010

-0.005

0.000

0.005

Load cycle-50

-1.0x109

0.010

0.015

-1.5x109 -0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

Strain

Strain

b

a

Fig. 9.27 Predictions for engineering stress-strain hysteresis loops for 1.2% strain amplitude. (a) Hysteresis loops at 1.2% strain amplitude for 50 cycles; (b) comparative hysteresis plot for the first cycle and 50th cycle (Bin Jamal et al., 2020)

Damage (TSI)

1.0

0.5

1.2% Strain 1% Strain 0.8% Strain 0 50 0 10 00 15 00 20 00 25 00 30 00 35 00 40 00 45 00 50 00 55 00 60 00 65 00 70 00

-50 0

0.0

Number of Cycles

Fig. 9.28 Predicted damage for different strain amplitudes during cyclic loading (Bin Jamal et al., 2020)

Here we conclude our discussion on fatigue life prediction with the unified mechanics theory. Further details may be found in the references cited below.

References

501

0.024

Test data Analytical approach: 1-D Numerical approach: 3-D

Total strain amplitude

0.022 0.020 0.018 0.016 0.014 0.012 0.010 0.008 0

2000

4000

6000

8000

Nf

Fig. 9.29 Comparison between the test data (Carrion et al., 2017) and predictions (Bin Jamal et al., 2020) for low cycle fatigue life (Nf) at different strain amplitudes

References Abbaschian, R., & Reed-Hill, R. E. (2009). Physical metallurgy principles. Van Nostrand, ISBN 9780495438519. Atkins, P., Paula, J., et al. (2018). Physical chemistry (11th ed.). Oxford University Press, ISBN 9780198769866. Basaran, C., & Lin, M. (2007). Electromigration induced strain field simulations for nanoelectronics lead-free solder joints. International Journal of Solids and Structures, 44(14–15), 4909–4924. Basaran, C., & Lin, M. (2008). Damage mechanics of electromigration induced failure. Mechanics of Materials, 40(1–2), 66–79. Bin Jamal, M. N., Kumar, A., Lakshmana Rao, C., & Basaran, C. (2020). Low cycle fatigue life prediction using unified mechanics theory in Ti-6Al-4V alloys. Entropy, 2020(22), 24. Bin Jamal, M. N., Rao, C., & Basaran, C. (2021). A unified mechanics theory-based model for temperature and strain rate dependent proportionality limit the stress of mild steel. Mechanics of Materials, 155, 103762. Blaschke, D. (2019). Properties of dislocation drag from phonon wind at ambient conditions. Materials, 12(6), 948. Blaschke, D., Mottola, E., & Preston, D. (2019). Dislocation drag from phonon wind in an isotropic crystal at large velocities. Philosophical Magazine, 100(5), 571–600. https://doi.org/10.1080/ 14786435.2019.1696484 Blaschke, D. N., Mottola, E., & Preston, D. L. (2020). Dislocation drag from phonon wind in an isotropic crystal at large velocities. Philosophical Magazine, 100, 571–600. Burton, J. J. (1972). Vacancy-formation entropy in cubic metals. Physical Review B, 5, 2948. Charkaluk, E., & Constantinescu, A. (2009). Dissipative aspects in high cycle fatigue. Mechanics of Materials, 41(5), 483–494, ISSN 0167-6636. Chrysochoos, A., Boulanger, T., & Morabito, A. (2012). Dissipation and thermoelastic coupling associated with fatigue of materials. Mechanics, Models, and Methods in Civil Engineering, 147–156. https://doi.org/10.1007/978-3-642-24638-8_7 Chu, Y., Gautreau, P., Ragab, T., & Basaran, C. (2015). Temperature dependence of Joule heating in zigzag graphene nanoribbon. Carbon, 89, 169–175.

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Cugy, P., & Galtier, A. (2002). Microplasticity and temperature increase in low carbon steels. In A. F. Blom (Ed.), Proceedings of the 8th international fatigue congress, 3–7 June 2002 (pp. 549–556). EMAS, Barnsley. Davoudi, K., & Vlassak, J. (2018). Dislocation evolution during plastic deformation: Equations vs. discrete dislocation dynamics study. Journal of Applied Physics, 123(8), 085302. https://doi.org/10.1063/1.5013213 De Hosson, T. M., Roos, A., & Metselaar, E. D. (2001). Temperature rise due to fast-moving dislocations. Philosophical Magazine A, 81(5), 1099–1120. Dingley, D., & Hale, K. (1966). Burgers vectors of dislocations in deformed iron and iron alloys. Proceedings of the Royal Society of London Series A. Mathematical and Physical Sciences, 295(1440), 55–71. https://doi.org/10.1098/rspa.1966.0225 Doudard, C., Calloch, S., Cugy, P., Galtier, A., & Hild, F. (2005). A probabilistic two-scale model for high-cycle fatigue life predictions. Fatigue & Fracture of Engineering Materials & Structures, 28(3), 2005. Fan, J., Zhao, Y., & Guo, X. (2018). A unifying energy approach for high cycle fatigue behavior evaluation. Mechanics of Materials, 120, 15–25, ISSN 0167-6636. Fultz, B. (2010). Vibrational thermodynamics of materials. Progress in Materials Science, 55(4), 247–352. Galligan, J., McKrell, T., & Robson, M. (2000). Dislocation drag processes. Materials Science And Engineering: A, 287(2), 259–264. Gao, S. (1997). Quantum kinetic theory of vibrational heating and bond breaking by hot electrons. Physical Review B, 55(3), 1876. https://doi.org/10.1103/PhysRevB.55.1876 Guo, Q., Guo, X., Fan, J., Syed, R., & Wu, C. (2015). An energy method for rapid evaluation of high-cycle fatigue parameters based on intrinsic dissipation. International Journal of Fatigue, 80, 136–144. https://doi.org/10.1016/j.ijfatigue.2015.04.016 Ho, H. S., Risbet, M., & Feaugas, X. (2017). A cyclic slip irreversibility based model for fatigue crack initiation of nickel base alloys. International Journal of Fatigue, 102, 1–8, ISSN 0142-1123. Huang, M., Rivera-Díaz-Del-Castillo, P. E. J., Bouaziz, O., & Van Der Zwaag, S. (2008). Irreversible thermodynamics modeling of plastic deformation of metals. Materials Science and Technology, 24, 495–500. https://doi.org/10.1179/174328408X294125 Kelly, A., & Knowles, K. (2012). Crystallography and crystal defects. Wiley, ISBN 9780470750148. Kröner, E. (1961). Zur plastischen verformung des vielkristalls. Acta Metallurgica, 9(2), 155–161. Laughlin, D., & Hono, K. (2014). Physical metallurgy (5th ed.). Elsevier, ISBN 9780444537713. Lee, H. W., & Basaran, C. (2021). A review of damage, void evolution, and fatigue life prediction models. Metals, 11(4), 609. Lee, H. W., & Basaran, C. (2022). Predicting high cycle fatigue life with unified mechanics theory. Mechanics of Materials, 164, 104116. Lee, H. W., Basaran, C., Egner, H., Lipski, A., Piotrowski, M., Mroziński, S., Noushad Bin Jamal, M., & Rao, C. L. (2022). Modeling ultrasonic vibration fatigue with unified mechanics theory. International Journal of Solids and Structures, 236–237, 111313. Lemaitre, J., Sermage, J., & Desmorat, R. (1999). A two scale damage concept applied to fatigue. International Journal of Fracture, 97, 67. Li, S., Abdulhamid, M. F., & Basaran, C. (2009). Damage mechanics of low temperature electromigration and thermomigration. IEEE Transactions on Advanced Packaging, 32(2), 478–485. Marti, N., Favier, V., Gregori, F., & Saintier, N. (2020). Correlation of the low and high frequency fatigue responses of pure polycrystalline copper with mechanisms of slip band formation. Materials Science and Engineering: A, 772, 138619, ISSN 0921-5093. Mughrabi, H. (2002). On ‘multi-stage’ fatigue life diagrams and the relevant life-controlling mechanisms in ultrahigh-cycle fatigue. Fatigue & Fracture of Engineering Materials & Structures, 25(8–9), 755–764.

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Mishin, Y., Sørensen, M. R., & Voter, A. F. (2001). Calculation of point-defect entropy in metals. Philosophical Magazine A, 81(11), 2591–2612. Mughrabi, H. (2009). Cyclic slip irreversibilities and the evolution of fatigue damage. Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science, 40(6), 1257–1279. Naderi, M., Amiri, M., & Khonsari, M. M. (2010). On the thermodynamic entropy of fatigue fracture. Proceedings of the Royal Society A, 466, 423. Parvin, H., & Kazeminezhad, M. (2016). Modeling the temperature rise effect through highpressure torsion. Materials Science and Technology (United Kingdom), 32, 1218–1222. Patricio E. Carrion, Nima Shamsaei, Steven R. Daniewicz, Robert D. Moser. (2017). Fatigue behavior of Ti-6Al-4V ELI including mean stress effects, International Journal of Fatigue, 99, 87–100. Ragab, T., & Basaran, C. (2009). Joule heating in single-walled carbon nanotubes. Journal of Applied Physics, 106, 063705. Ribeiro, P., Petit, J., & Gallimard, L. (2020, July). Experimental determination of entropy and exergy in low cycle fatigue. International Journal of Fatigue, 136, 105333. Torabian, N., Favier, V., Dirrenberger, J., Adamski, F., Ziaei-Rad, S., & Ranc, N. (2017). Correlation of the high and very high cycle fatigue response of ferrite based steels with strain ratetemperature conditions. Acta Materialia, 134, 40–52, ISSN 1359-6454. Wollenberger, H. J. (1996). Physical metallurgy (4th ed.). Elsevier, ISBN 9780444898753. Yao, W., & Basaran, C. (2013a). Electromigration damage mechanics of lead-free solder joints under pulsed DC: A computational model. Computational Materials Science., 71, 76–88. Yao, W., & Basaran, C. (2013b). Computational damage mechanics of electromigration and thermomigration. Journal of Applied Physics, 114, 103708. Yao, W., & Basaran, C. (2013c). Damage mechanics of electromigration and thermomigration in lead-free solder alloys under alternating current: An experimental study. International Journal of Damage Mechanics, 23, 203–221. Ye, H., Basaran, C., & Hopkins, D. C. (2004). Deformation of solder joint under current stressing and numerical simulation––I. International Journal of Solids and Structures, 41(18–19), 4939–4958. Ye, H., Basaran, C., & Hopkins, D. C. (2006). Experimental damage mechanics of micro/power electronics solder joints under electric current stresses. International Journal of Damage Mechanics, 15(1), 41–67. Yun, H., & Modarres, M. (2019). Measures of entropy to characterize fatigue damage in metallic materials. Entropy, 21(8), 804. Zhang, L., Liu, X., Wu, S., Ma, Z., & Fang, H. (2013). Rapid determination of fatigue life based on temperature evolution. International Journal of Fatigue, 54, 1–6. https://doi.org/10.1016/j. ijfatigue.2013.04.002

Chapter 10

Corrosion-Fatigue Interaction

10.1

Corrosion

In this section, first, we summarize the basics of the metal corrosion process. The information presented below is a summary of the corrosion chapter in Callister and Rethwisch (2014). Metals experience chemical interaction with their environment. As a result of this interaction, they experience material loss by dissolution and by the formation of nonmetallic scale or film. Corrosion is an electrochemical process, where there is a transfer of electrons from one material to another. Metals have a metallic bond where positive ions float in a sea of free negatively charged valence electrons. During the corrosion process, metals give up electrons in what is called an oxidation reaction. For example, an iron atom (Fe) has two valence electrons in its electronic configuration [1s22s22p63s23p63d64s2]. Fe can experience oxidation by giving up these two valence electrons; as a result, it becomes (2+) a positively charged ion. This can be represented in chemistry notation as follows: yields

Fe


→ Fe2þ þ 2e -

ð10:1Þ

where e- symbolizes an electron. The site where oxidation happens is called the anode. The term anodic reaction is also used to refer to the oxidation process. The electrons given up by a metal must be physically transferred to other chemical species, in what is termed a reduction reaction. Given up electrons become part of the other chemical species. The site where reduction reaction occurs is called the cathode. An electrochemical reaction contains one oxidation reaction (anodic site) and a reduction reaction (cathodic site). The total rate of oxidation equals the total rate of reduction, or, in other words, all electrons released by the oxidation site must be absorbed by the reduction site. As an example, we can formulate rusting of iron, Fe, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9_10

505

506

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in water with dissolved oxygen, which takes place in two separate steps. Electron configurations of hydrogen and oxygen are given by [1s1], [1s22s22p4]. Step 1—Fe is oxidized Fe2+, as Fe(OH)2, in chemistry notation: yields yields 1 Fe þ O2 þ H2 O


→ Fe2þ þ 2OH -


→ FeðOHÞ2 2

ð10:2Þ

Step 2—To Fe3+ as Fe(OH)3 according to yields 1 2FeðOHÞ2 þ O2 þ H2 O


→ 2FeðOHÞ3 2

ð10:3Þ

The compound Fe(OH)3 is called rust. Readers are referred to Callister and Rethwisch (2014) for a more in-depth study of corrosion.

10.2

Thermodynamic Fundamental Equation of Corrosion-Fatigue

Entropy is an additive property. Hence, we need to identify all entropy-generating mechanisms involved.

10.2.1

Entropy Generation Mechanisms During Corrosion

I. Entropy Generation due to Electrochemical Reaction: Activation Overpotential, ΔSact Overpotential is the additional potential (beyond the thermodynamic equilibrium) needed to drive an electrochemical reaction at a certain rate. Accordingly, activation overpotential is the additional electric potential needed for the activation of electrode reaction at the desired rate, which includes the activation of oxidation reaction at anode and cathode, and the activation of reduction reaction at anode and cathode. The thermodynamic fundamental equation [entropy generation mechanisms] for metal corrosion has been extensively studied based on the theory of chemical kinetics and fundamentals of irreversible thermodynamics. Entropy generation can be computed by multiplying the thermodynamic flux (corrosion current density) and thermodynamic force (electrochemical affinity) based on the Onsager reciprocal relations. Entropy generation for activation overpotential can be given in the form of the sum of the product of fluxes, Ji, and forces, Pi (Gutman, 1998): ΔSact =

1 Xm J ∙ Pi i-1 i T

ð10:4Þ

10.2

Thermodynamic Fundamental Equation of Corrosion-Fatigue

  1 ~ þ J c - αc A ~ J a αa A T    ~ α A -1 J a = j0 exp a RT    ~ - αc A -1 J c = - j0 exp RT

ΔSact =

507

ð10:5Þ ð10:6aÞ ð10:6bÞ

where Ja and Jc are the fluxes (current densities) of anodic flux (direct) and cathodic flux (reverse) half-reactions, respectively. Ja and Jc are defined by Butler-Volmer equation, which formulates the relationship between the electrical current passing through an electrode and the voltage difference between the electrode and the electrolyte, j0 is the exchange current in a dilute solution in the equilibrium state ~ = 0 and Ja + Jb = 0), and αa and αc are the charge transfer coefficients of the (A ~ is the electrochemical electrochemical reactions, anodic and cathodic, respectively. A n P ~= vi μi, in which vi are the affinity of the total reaction which is expressed as A i=1

stoichiometric coefficients, μi are the chemical potentials, and n is the total number of components (initial substances and products). For an electrochemical reaction, the ~ is usually determined by replacing μi with electrochemical electrochemical affinity A potential μ ~i : μ ~i = μi þ zi Fη

ð10:7Þ

where zi is the effective charge number of the ith chemical specie, F is Faraday’s constant, η is the overpotential defined by η = (E - Ecorr) in which Ecorr is the equilibrium electrode potential (open circuit potential, corrosion potential), and E is the working electrode potential on the metal specimen. For the simple electrochemical half-reaction Fe $ Fe2þ þ 2e -

ð10:8Þ

~ = zFη. the electrochemical affinity is given by A Butler-Volmer equation is given by h



i α zFη α zFη J = J a þ J c = j0 exp a - exp - c RT RT

ð10:9Þ

The exchange flux, j0, is a function of the concentrations of the reduced and oxidized species: j 0 = F ðk c C O Þα ðk a C R Þ1 - α

ð10:10Þ

508

10

Corrosion-Fatigue Interaction

where ka and kc are the anodic and cathodic reaction rate constants, respectively. CO and CR are the concentration of oxidized and reduced species, respectively. Therefore, Eq. (10.5) can be rewritten as ΔSact =

 1 ~ þ J c αc A ~ J αA T a a

ð10:11Þ

Equation (10.11) is the entropy generation equation for a simple oxidation ~ = z F ðE - E corr Þ. It should be noted reaction considering the chemical affinity of A that for single-step reactions, αa + αc = 1. In this specific case, they are called symmetry factors (Bockris & Nagy, 1973). Imanian and Modarres (2015) extended Eq. (10.11) to a general oxidationreduction (redox) reaction, as follows [using their original notation]: ΔSact =

 1 ~ M þ J M,c ð1 - αM ÞA ~ O þ J O,c ð1 - αO ÞA ~ M þ J O,a αO A ~ O ð10:12Þ J M,a αM A T

where JM, a and JM, c are the irreversible anodic and cathodic activation fluxes (thermodynamic fluxes) for oxidation reaction and JO, a and JO, c are the anodic and cathodic activation fluxes (thermodynamic fluxes) for reduction reaction, respectively. αM and αO are the charge transport coefficients for oxidation reaction and ~ O are the electrochemical affinity (ther~ M and A reduction reaction, respectively. A modynamic force) for oxidation and reduction reactions induced by the electrochemical potential, respectively. Figure 10.1 explains the four different fluxes formulated in Eq. (10.12). The flux can be related to current density by using Faraday’s second law as follows: I M,a = zM FJ M,a

ð10:13Þ

where IM, a is the anodic current density for the metal dissolution, zM is the number of electrons involved in the metal dissolution reaction, JM, a is the irreversible anodic activation flux, and F is Faraday’s constant. Imanian and Modarres (2015) included three additional entropy production mechanisms in addition to the activation overpotential. These three mechanisms are: 1. The entropy generation due to chemical reaction overpotential, ΔSreact. 2. The entropy generation due to diffusional overpotential, ΔSconc 3. The entropy generation due to ohmic loss, ΔSΩ Entropy generations due to the diffusional overpotential and the ohmic loss are negligible when the solution is well mixed (very low chemical concentration gradient) and when the electrolyte has a strong conductivity, respectively (Onsager, 1931). Therefore, these two mechanisms are trivial II. Entropy Generation due to Chemical Reaction Overpotential, ΔSreact Chemical reaction overpotential entropy generation is given by

10.2

Thermodynamic Fundamental Equation of Corrosion-Fatigue

509

Fig. 10.1 Schematic showing the anodic and the cathodic parts of the iron corrosion reaction in a neutral electrolyte. (Adapted from Popov (2015) The red dashed line in the middle constructs the Tafel plot)

ΔSreact =

1 Xr υA j=1 j j T

where r is the number of chemical reactions involved, T is the absolute temperature, υj is the chemical reaction rate in units of mol/sec, and Aj is the chemical reaction affinity in the units of KJ/mol. III. Entropy Generation due to Diffusion Overpotential, ΔSconc The diffusion overpotential happens when the initial concentration of the solution fails to remain constant due to the consumption of reactant at the electrode. This mechanism is predominant at a high rate of cathodic reactions, which leads to the depletion of dissolved species in the adjacent solution. IV. Entropy Generation due to Ohmic Loss, ΔSΩ The ohmic loss is due to the resistance to the flow of electrons in the electrodes and protons in the electrolyte. This loss becomes nontrivial at high current density or when the electrolyte has low conductivity. Hence, the thermodynamic fundamental equation is obtained by summation of all entropy generation mechanisms as follows:

510

10

Corrosion-Fatigue Interaction

ΔScorr = ΔSact þ ΔSreact þ ΔSconc þ ΔSΩ ΔScorr =

ð10:13aÞ

  1 ~ þ J M,c ð1 - αM ÞA ~ O þ J O,c 1 - αO ÞA ~ M þ J O,a αO A ~O J α A T M,a M M  1 1 Xr υ A þ J E þ J E þ ΔSΩ þ j j M,c M O,c O conc,c conc,c j=1 T T ð10:13bÞ

Usually, during a corrosion test conditions are imposed to limit entropy generation due to activation overpotential and chemical reaction overpotential only. Therefore, the thermodynamic fundamental equation can then be simplified to include only these two mechanisms: ΔScorr =

 1 ~ M þ J M,c ð1 - αM ÞA ~ O þ J O,c ð1 - αO ÞA ~ M þ J O,a αO A ~O J M,a αM A T r 1 X v A þ T j=1 j j ð10:14Þ

Calculation of Entropy Production To compute the corrosion entropy production, the irreversible anodic and cathodic activation fluxes JM, a and JM, c for oxidation and the irreversible anodic and cathodic activation fluxes JO, a and JO, c for reduction must be determined. They are given by 

 ~M αM,a A exp -1 RT    ~M - αM,c A -1 J M,c = - j0M exp RT    ~ α A J O,a = j0R exp O,a O - 1 RT    ~O - αO,c A -1 J O,c = - j0R exp RT J M,a = j0M



ð10:15Þ ð10:16Þ ð10:17Þ ð10:18Þ

where j0M and j0R are the exchange fluxes for the oxidation reaction and reduction reaction, respectively. These exchange fluxes can also be expressed in the following form: j0M = j0Fe , j0R = j0O2 . αM,a, αM,c, αO,a, αO,c are the charge transfer ~ M and A ~ O are chemical affinities related to the overpotential, for coefficients. A oxidation and reduction, respectively. It is usually assumed that the redox reaction during the electrochemical corrosion only involves the metal oxidation and the reduction of dissolved oxygen, and they are both single-step reactions that follow the symmetry property. Therefore,

10.2

Thermodynamic Fundamental Equation of Corrosion-Fatigue

αM,a þ αM,c = 1;

αO,a þ αO,c = 1

511

ð10:19Þ

The effective charge coefficients αM, a and αO, C can be calculated based on the slope of the Tafel plot: βa =

2:303RT αa zF

ð10:20aÞ

βc =

2:303RT αc zF

ð10:20bÞ

where coefficients βa and βc are the slopes of the two branches of the Tafel plot (also known as Tafel slope, or Tafel constant). Exchange current density is a function of the concentrations of the reduced and oxidized species. They are usually in the following range depending on the test conditions (Popov, 2015; Chen et al., 2018): j0Fe = 10 - 4  10 - 5 A=cm2 ,

j0O2 = 10 - 6  3 × 10 - 7 A=cm2

ð10:21Þ

In summary, the calculation of corrosion entropy production requires the corrosion potential, corrosion currents, and charge transfer coefficients. The detailed procedure for the determination of these variables is discussed later in the chapter.

10.2.2

Entropy Generation Mechanisms: Fatigue

Entropy is an additive property. Therefore, we can add the entropy generation due to fatigue and entropy generation due to corrosion. The entropy generation during ultrasonic vibration fatigue has six entropy generation mechanisms during very high cycle fatigue. However, only three are dominant entropy generation mechanisms: ΔSmec = ΔST þ ΔSr þ ΔSμp

ð10:22Þ

where ΔST is the entropy generation due to thermal conduction, ΔSr is the entropy generation due to internal heat, and ΔSμp is the entropy generation due to microplastic work. They are given by the following equations: ΔST = -

Z t t0

 ∇T ∙ ∇T dt kh T2

ð10:23Þ

512

ΔSr =

10

Z t t0

Corrosion-Fatigue Interaction

 Z t  drag pffiffiffi  ρr drag þ ρr dis ϱB v v þ 12 μb2 ϱ_ - αH μb ϱγ_ dt = dt T T t0 ! Z t σ μ : ε_μP ΔSμp = Ψf ϕf v dt T t0

ð10:24Þ ð10:25Þ

where kh is coefficient of heat conduction, ∇T is the temperature gradient, ρ is the mass density, r is the internal heat generation due to drag and dislocation motion mechanisms, and Bdrag and v represent the effective drag coefficient and velocity of dislocations, respectively. The terms μ, b, ϱ, and αH are the shear modulus, the magnitude of Burger’s vector, total dislocation density, and Taylor’s hardening parameter, respectively. Ψf is an ultrasonic vibration frequency coefficient, ϕ is the thermodynamic state index, fv is the activated volume fraction of micro-defects (inclusions, vacancies, dislocations), σ μ is microscopic stress tensor, and εμP is the microscopic plastic strain tensor Figure 10.2 depicts the comparison of fatigue life for uncorroded and corroded samples.

Fig. 10.2 Stress vs. the number of cycles to failure test data for the uncorroded and saltwatercorroded ASTM A656 structural steel samples subjected to 20 KHz vibrations and corrosion. (After Lee et al. (2022))

10.4

10.3

Comparing Simulation Results and Test Data

513

Thermodynamic State Index (TSI)

The evolution of TSI is based on the cumulative-specific entropy production Δs during corrosion fatigue. Δs is the summation of all entropy generation mechanisms discussed above. The degradation of the material happens according to the second law of thermodynamics which is the basis for the thermodynamic fundamental equation of the material and progresses along the TSI axis:  1 1 ðΔScorr þ ΔSmec Þ = ΔSact þ ΔSreact þ ΔST þ ΔSr þ ΔSμp ρ ρ   1  ~ M þ J M,c ð1 - αM ÞA ~ O þ J O,c 1 - αO ÞA ~ M þ J O,a αO A ~O Δs = J M,a αM A ρT  Z t  drag Z t ð10:26Þ 1 Xr ∇T ∙ ∇T r þ r dis dt dt þ υ A k þ j j h 2 j = 1 ρT T ρT t0 t0 ! Z t μ μ _ σ : εP þ Ψf ϕf v dt ρT t0 Δs =

10.4

Comparing Simulation Results and Test Data

Using the formulation presented above, the cumulative entropy generation calculated is shown in Fig. 10.3. Fatigue fracture entropy (FFE) is calculated and measured to be 4.1 MJ K-1 m-3. The TSI evolution is presented in Fig. 10.4. TSI value during mechanical loading starts from the endpoint of the TSI value due to corrosion, instead of 0. Because entropy is an additive property, the contributions from corrosion and fatigue are added. Stress versus the number of cycles to failure curves for the uncorroded samples and corroded samples are presented in Fig. 10.5. Corrosion degradation leads to a decrease in fatigue life. At the stress amplitude of 400 MPa, the fatigue life decreases from 2.79∙106 cycles to 1.58∙106 cycles due to the corrosion degradation, which is around a 40% reduction in fatigue life. It is important to point out that in Fig. 10.5, the test data fluctuates around the unified mechanics theory solution, because Boltzmann’s formulation of the second law is for all possible complexions. If we did an infinite number of experiments, the data will always fluctuate around Boltzmann’s solution. A simple explanation can be given by flipping a coin; we will get 50% head and 50% tail if we flip the coin an infinite number of times. Here we conclude our discussion of corrosion-fatigue interactions.

514

10

Corrosion-Fatigue Interaction

Fig. 10.3 Cumulative entropy production in the corroded samples subjected to ultrasonic vibration MK fatigue. The cumulative entropy production starts from 0.25 Km 3 because samples were first corroded and then subjected to fatigue loading. (After Lee et al. (2022))

Fig. 10.4 TSI evolution of the corroded samples subjected to ultrasonic vibration fatigue. (After Lee et al. (2022)). TSI due to corrosion is 0.2; hence, the curves start from 0.2

References

515

Fig. 10.5 Unified mechanics theory-based simulated stress vs. number of cycles to failure curves for the corroded and uncorroded samples, compared with experimental data. (After Lee et al. (2022))

References Bockris, J., & Nagy, Z. (1973). Symmetry factor and transfer coefficient. A source of confusion in electrode kinetics. Journal of Chemical Education, 50(12), 839. https://doi.org/10.1021/ ed050p839 Chen, L., Hu, J., Zhong, X., Zhang, Q., Zheng, Y., Zhang, Z., & Zeng, D. (2018). Corrosion behaviors of Q345R steel at the initial stage in an oxygen-containing aqueous environment: Experiment and modeling. Materials, 11(8), 1462. https://doi.org/10.3390/ma11081462 Gutman, E. (1998). Mechanochemistry of materials. Cambridge International Science Publishing. Imanian, A., & Modarres, M. (2015). A thermodynamic entropy approach to reliability assessment with applications to corrosion fatigue. Entropy, 17(12), 6995–7020. https://doi.org/10.3390/ e17106995 Lee, H. W., Basaran, C., Egner, H., Lipski, A., Piotrowski, M., Mroziński, S., Bin Jamal, N., & Rao, L. (2022). Modeling ultrasonic vibration fatigue with unified mechanics theory. International Journal of Solids and Structures, 236–237, 111313. https://doi.org/10.1016/j.ijsolstr.2021. 111313 Onsager, L. (1931). Reciprocal relations in irreversible processes. I. Physical Review, 37(4), 405–426. https://doi.org/10.1103/physrev.37.405 Popov, B. (2015). Electrochemical kinetics of corrosion. Corrosion Engineering, 93–142. https:// doi.org/10.1016/b978-0-444-62722-3.00003-3 William D. Callister, Jr. and David G. Rethwisch (2014), Materials Science and Engineering: An Introduction, 9E, Wiley, 2014

Index

A Acrylic, 3, 210 Alumina trihydrate (ATH), 310, 311, 342, 349, 361

B Boltzmann, 3, 116, 117, 148, 150, 153–156, 159, 161, 163, 164, 176, 180, 195, 199, 200, 204, 210, 211, 241, 298, 396, 401, 412, 431, 440, 450, 459, 460, 463, 464, 479, 483, 513 Boltzmann entropy, 135, 151–153, 200

C Composite material, 3, 211, 263, 309, 352, 428 Continuum mechanics, 1–3, 5, 13, 73–76, 81, 89, 91, 93, 98–100, 112–114, 116, 118, 119, 123, 125, 130, 133–145, 147–150, 251, 258, 260, 263, 272–275, 295, 309, 460, 475 Corrosion, 3, 130, 147, 505–507, 509–514 Corrosion fatigue, 3, 147, 506, 513 Couple stress, 20, 21, 96, 274–276, 278–290, 297–299

D Damage, 119, 142, 145–150, 242, 247, 263, 346, 404, 405, 419, 421, 422, 427, 428, 494, 500 Deformation gradient tensor, 113, 121, 254, 375–380, 415

Diffusion, 130, 134, 179, 224, 225, 241, 250, 427–431, 435–438, 441, 443, 460, 461, 463, 464, 473, 474, 481–483, 509 Disorder, 111, 114–118, 135, 153, 200, 202, 203, 210–212, 249, 250, 257, 450, 461, 462 Dissipation, 2, 86, 87, 91, 99, 100, 109, 113, 124–130, 133, 134, 142, 143, 149, 218–221, 223–226, 228, 257, 261, 265–267, 365, 384, 391, 395, 403, 404, 419, 421, 422, 453, 459, 472, 475, 489, 490, 496

E Electromigration, 3, 427–456 Electronics packaging, 428 Energy, 2, 3, 12, 86, 88, 91–94, 96–115, 117–124, 126, 127, 133–138, 140–145, 148, 150, 153–156, 158–161, 165–167, 170–177, 179–181, 183–188, 190–193, 196–209, 211, 213, 214, 217, 218, 221, 224–226, 228, 230, 238, 241, 243, 249–251, 254–257, 259–261, 263–266, 276–278, 280–282, 289, 298, 299, 367, 383–385, 391, 393–403, 412, 419, 431, 434–436, 451, 452, 459, 460, 463, 465, 470, 472, 475, 479, 481, 489, 495 Entropy, 3, 12, 91, 98, 100–105, 107–119, 121, 123, 125, 126, 128–130, 134–138, 141–157, 194–196, 199–203, 205, 207, 208, 210–212, 214, 215, 217, 218, 221–226, 229, 231, 246, 249–251, 254, 255, 257–263, 266, 267, 272, 296,

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Basaran, Introduction to Unified Mechanics Theory with Applications, https://doi.org/10.1007/978-3-031-18621-9

517

518 298–299, 354, 355, 358, 365, 367, 368, 383–385, 391, 403–405, 419, 421, 422, 443, 449–454, 459–482, 484, 485, 488, 492–496, 506, 508–511, 513, 514 Eshelby, 309, 312, 314–317, 332 Euler equation, 106, 107 Eulerian descriptions, 38, 49, 55–57, 59

F Failure, 11, 99, 108, 115, 117, 133, 135, 142, 146, 147, 149, 150, 210, 211, 215, 217, 312, 404, 405, 411, 419, 421, 422, 427–430, 435, 451, 454, 456, 466, 470, 471, 476, 477, 479, 482, 484, 494, 512, 513, 515 Fatigue, 3, 99, 108, 117, 125, 130, 133, 137, 141, 145–148, 150, 215, 216, 263, 265–272, 346, 367, 421, 422, 459–501, 511, 513, 514 Fatigue corrosion interaction, vii, 3, 505–515 Field theories, 11 Finite deformations, 375–425 Finite strain, 3, 34, 57–73, 78, 136, 383–403, 422 Flow theory, 237–241, 288, 294 Fluid mechanics, 18, 49 Fracture, 133, 142, 145, 147, 150, 263, 367, 403, 410, 422, 494, 513

G Gibbs–Duhem relation, 104–107

H Homogenizations, 309, 468, 470, 479

L Lagrangian description, 38, 48, 54, 57 Lagrangian mechanics, 86, 87, 91, 133 Large strain, 34, 55, 60, 273, 366 Laws of motion, 2, 5, 11, 12, 50, 81, 91, 119, 133, 142, 152, 154, 459 Life prediction, 146, 150, 459, 460, 478, 479, 484, 496, 500

M Manufacturing, 346, 472 Metal corrosion, 505, 506 Microplasticity, 469–475, 477, 479, 492–495

Index Modeling, 2, 3, 28, 57, 117, 124, 127, 142, 309, 346–348, 371, 375, 384, 392, 398, 399, 402, 427, 428

N Newtonian mechanics, 81, 83, 86, 91, 96, 98, 113, 119, 123, 124, 126, 133, 136, 138, 139, 142, 148, 150, 217, 219–223, 226, 229, 237, 238, 253–255, 263, 265, 275, 276, 278, 279, 288, 313, 366, 429, 443–444, 459

O Onsager reciprocal relations, 128, 129, 506

P Particle filled, 3 Particle filled composite, 3, 311, 338, 371 Piola–Kirchoff stress, 76–78, 389, 390, 393, 400 Plasticity, 31, 81, 124, 126–128, 136, 137, 139, 142, 215, 237, 238, 240, 241, 263, 272, 273, 281, 288–295, 297, 309, 392, 395, 403, 412, 468 Polymer, 3, 148, 210, 321, 362, 365, 375, 384, 392, 393, 395, 397–400, 402, 403, 405, 407, 422, 423 Poly-methyl-meth-acrylate (PMMA), 310, 311, 349, 354, 355, 362, 366, 397, 406, 409–411, 413–425

R Rational mechanics, 428 Reliability, 150

S Solder alloys, 273, 298 Solder joints, 3, 267–272, 428, 430, 435, 455, 456 Solid mechanics, 18, 39, 73, 251 Strain, 1–3, 33–37, 40, 41, 43, 44, 46, 52–57, 60, 62–76, 96, 97, 108, 111, 113, 117, 119, 123, 125–127, 130, 136, 137, 139, 140, 143, 145, 148, 214, 216, 226, 237, 239–246, 253, 254, 256, 262–267, 272–278, 280–282, 284, 285, 288–291, 295–299, 301, 303, 309, 312–317, 322–324, 327, 329, 330, 333, 336, 337,

Index 352–355, 358, 364, 367–370, 385, 393, 395–399, 401, 403, 406, 409–411, 413, 416–420, 439, 441–443, 445, 451–453, 459, 467–471, 475, 480, 485, 488, 489, 497, 498, 500, 501, 512 Stress, 1, 2, 12–15, 18–32, 35, 37, 41, 73, 76–79, 83–85, 95, 96, 99, 108, 111, 117, 119, 121, 123, 126, 130, 137–142, 146, 150, 217, 238–245, 253, 257, 261, 263, 266, 273–282, 284, 288–290, 295–299, 301, 309, 311–314, 316–323, 327–338, 350, 352, 353, 358, 366, 378, 379, 386–396, 398–403, 408, 411–413, 415, 416, 418, 419, 428, 429, 432, 436–440, 447, 452, 455, 459, 460, 464, 468–489, 492–497, 512, 513, 515

T Thermal mechanical analysis, 237–306 Thermodynamic fundamental equation, 3, 102, 107, 113, 134, 139, 147, 153, 210, 212, 214, 220, 224–226, 228, 229, 231, 249–250, 260, 263, 354, 365, 368, 384, 405, 421, 428, 449–456, 459–461, 474, 479, 481–488, 495, 506, 509, 510, 513

519 Thermodynamics, 2, 3, 12, 86, 91–93, 97–102, 104, 105, 107–116, 118–128, 130, 133–138, 140, 142, 144, 145, 148–153, 155–157, 198, 201, 210, 211, 213, 226, 228, 229, 242, 243, 249–251, 255, 257–264, 266, 267, 272, 296–298, 354, 358, 368, 383–392, 403–405, 416, 419, 422, 430, 439, 443, 449–451, 454, 459–462, 464, 471, 474, 475, 477, 480, 485–487, 489, 490, 493–496, 506, 508, 512, 513 Thermomechanical analysis, 272–288 Thermomigration, 3, 117, 266, 427–456, 461, 463

V Very high cycle fatigue, 480–482, 484, 493, 495, 496, 511 Viscoplasticity, 126, 241, 263, 295, 350–353

Y Yield surface, 31, 124, 127, 136–138, 140, 238–240, 245, 246, 272, 289, 290, 295, 297, 298, 352, 353, 358, 470