Fundamentals of Continuum Mechanics of Soils - Home - …978-1-4471-16… · · 2017-08-28Fundamentals of Continuum Mechanics of Soils ... * The interval between the second and

Fundamentals of Continuum Mechanics of Soils

Yehuda Klausner

Fundamentals of Continuum Mechanics of Soils With 210 Figures

Springer-Verlag London Berlin Heidelberg New York Paris Tokyo Hong Kong

Yehuda Klausner, PhD Engineering Consultant, 19 Yarboa Lane, Beer-Sheva 84736, Israel

British Library Cataloguing in Publication Data

Klausner, Yehuda 1926-Fundamentals of continuum mechanics of soils. 1. Soils. Mechanics I. Title 624.15136

Library of Congress Cataloging-in-Publication Data

Klausner, Yehuda, 1926-Fundamentals of continuum mechanics of soils/Yehuda Klausner. p. cm.

ISBN-I3: 978-1-4471-1679-0

DOl: 10.1007/978-1-4471-1677-6

1. Soil mechanics. I. Title. TA710.K549 1991

e-ISBN-I3: 978-1-4471-1677-6

624.1' 5136-dc20 90-20705 ClP

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.

© Springer-Verlag London Limited 1991

Softcover reprint of the hardcover I st edition 1991

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Preface

The world is a poor affair if it does not contain matter for investigation to everyone in every age. Nature does not reveal all her secrets at once. We imagine we are initiated in her mysteries: we are as yet, but hanging around her outer court.

Seneca

Between the Second International Conference on Soil Mechanics and Foundation Engineering (INCOSOMEFE) in 1948 and the 12th International Conference in 1989 the conferences were held at regular intervals of four years. * Over 2500 papers, not including discussions, were presented in the Proceedings. Around 10 000 additional articles were published during these 40 years, in the proceedings of regional and specialty conferences and symposia, as well as in scientific and professional journals. This is an enormous amount of information, and of this 75% or more was added after 1960. The question one must ask himself is: what has come of all these efforts?

In 1960 the ASCE Research Conference on Shear Strength of Cohesive Soils was held in Boulder, Colorado; the special significance of this conference was that it presented an up to date summary of the developments in soil mechanics. We can now say with certainty that, except for minor refinements, the body of knowledge in soil mechanics has not changed much since then, and the topics considered today had already been thoroughly studied by that time: consolidation; pore-water, pore-air, and negative pore pressures; effective pressures; critical void ratio; isotropic and anisotropic properties of soils; electrokinetic properties of soils; drained, consolidated-undrained and undrained tests; disturbed and undisturbed samples; phenomena of saturated, unsaturated and overconsolidated

* The interval between the second and third international conferences was five years. The first conference was held in 1936 in Cambridge, Massachusetts, with the establishment of the International Society of Soil Mechanics and Foundation Engineering.

vi Preface

soils, expansive and collapsible soils, cohesive and cohesionless soils; the critical state theory and the Coulomb-Mohr failure theory. All these topics were understood and discussed at the 1960 conference in great detail.

Nevertheless, several problems remained unresolved and continue to engage our interest today. Uniaxial versus triaxial consolidation, constitutive modeling, coaxiality of stresses and strains, conditions of anisotropy, elastic, viscous and plastic behavior, hysteretic behavior, the role of continuum mechanics, discontinuities, general failure conditions, workhardening and dilatancy - all are topics brought repeatedly to the surface but not resolved, apparently owing to shortcomings in the analytical and conceptual tools applied to the studies.

About the time of the Boulder conference, a small group of graduate students at Princeton University led by Professor W. E. Schmid, aware of these shortcomings, came to the conclusion that although the knowledge of the above topics was substantial, their formulation was deficient and resulted in inconsistencies and misinterpretations, thus obstructing further scientific advances. They began looking at things differently, and revised the formulation of the problems. Unfortunately they did not present their findings efficiently and appealingly and their work went unnoticed.

When Professor A. W. Skempton, in his presidential address at the 5th International Conference in 1961 in Paris, warned of two dangers: "the danger of what might be called handbook engineering", and "the second danger which can be foreseen and which we must strive to avoid can be expressed in the simple word complacency", he actually foresaw what was about to happen. The intellectual challenge posed by the discipline was replaced by computer techniques, numerical methods and data accumulating on job sites. The majority of publications mentioned earlier attest to that aspect of the discipline.

I had the good fortune to have some of the best teachers in my engineering studies. Among them the late M. Reiner of the Technion lIT, who introduced me to rheology, a discipline of which he was a cofounder; the late I. Haber-Schaim of the Technion lIT, an original thinker with a vast practical experience, my tutor and superior in my early engineering practice; the late G. P. Tschebotarioff of Princeton University, a man of integrity and a scientific thinker and leader; Professor J. G. Zeitlen, Professor Emeritus of the Technion lIT, who in 1954 brought the experience and tradition of the Corps of Engineers and the message of Classical Soil Mechanics to the Technion, and ushered me and a number of young faculty members into this field of knowledge, and Professor W. E. Schmid, my graduate study advisor at Princeton University, who led me into

Preface vii

the realm of theoretical studies. This work is a product of their teaching effort.

An early version of the material presented here was given as a one-semester graduate course under the same name at Wayne State University, in 1961 and 1962 (Klausner 1962). Much scientific progress has been made in mechanics of materials and related disciplines since then and an abundance of experimental data on soils has appeared in the scientific and technical literature.

This work intends to review the present state of knowledge in mechanics of soils and place it abreast with the related disciplines. More specifically, it aims to close the gap between soil mechanics, a discipline based greatly on empirical impressions, and continuum mechanics and its many ramifications, and to present a long-needed general scheme, based on the laws of physics, for that important yet intricate material, soils.

The book is intended for soil scientists and scholars of soil engineering, engineering mechanics and material sciences, as well as for graduate students familar with the fundamentals of soil mechanics. It is not conceived as an undergraduate textbook, and is by no means intended to present practical applications of soil engineering.

The organization of the subjects is different from that of the existing textbooks. The phenomena constituting the behavior of soils are presented here not as separate topics like permeability, capillarity, consolidation, shear strength, etc. incidentally related, but as phenomena logically inferring from one another and depending upon one another, and governed by the laws of physics.

The book has two parts, although it is not formally so subdivided: the first part comprises Chapters 1-7 and the second Chapters 9-13, with Chapter 8 connecting the two. Each chapter is based on the previous one, however the experienced reader basically acquainted with the subject will have no problem in studying any individual chapter.

Chapters 1-7 contain the basic concepts of mechanics and serve as a foundation for the second part, concerned with the mechanical behavior of soils. Chapters 1-5 expound the concept of strains and stresses and the balance equations, Chapter 6 deals with the application of the balance equations to multiphase mixtures, a rather new branch of mechanics, and Chapter 7 discusses the constitutive equations, a topic generally not elaborated in textbooks.

Chapter 8, The Soil, is a self-contained chapter presenting the sub-structural approach to soils, and is not an absolutely required part of the book. It discusses the constituents of the soil, their properties and their interaction, and is instrumental in

viii Preface

understanding the mechanical behavior of soils as a whole and the boundary conditions at the base of our treatment.

Chapters 9-13 deal with the wide range of phenomena concerning the mechanical behavior of soils, including flow, volumetric behavior, shear stress-strain behavior and failure. In Chapter 9 the balance equations of multiphase mixtures, discussed in Chapter 6 in a general manner, are applied to soils. Chapters 10-13 present the mechanical behavior of soils as derived from the dual constitutive equations, based also on the balance equations of multiphase mixtures. Chapters 10 and 11 are concerned with volumetric phenomena of soils, Chapter 10 looking at the motion of the phases within the voids, that is, the problem of flow in soils, and Chapter 11 at the motion of the solids phase and the free energies involved in these motions; this chapter relates to the topic of consolidation in its widest aspect. Chapter 12 discusses the effect of deviatoric stresses and strains and the respective free energies involved. Finally, Chapter 13 formulates a criterion for failure, based on the free energy of the soil.

Three appendices are found at the end of the book. Appendix A outlines the main rules of tensor calculus and can serve as a reference for the derivations in the main text. Appendix B discu~ses cylindrical coordinates and the transformation of several equations of mechanics into these coordinates. Appendix C is a discourse on rheological modeling. It is hoped that the material included in the appendices, which is beyond that required to master the subject, will serve further studies and help develop their application to soils and other materials.

Should the book raise interest, or controversy, by its stand, I would consider this my reward, as I believe it could stimulate scientific progress. I would like to think that the conceptual errors in the text are minimal. Much effort was invested, within my ability, in order to minimize mistakes, faulty mathematical expositions, and typing and printing errors. I would be grateful to readers who bring any remaining errors to my attention and welcome their constructive criticism.

As a practicing engineer I was fortunate to observe the behavior of matter very closely. On the other hand, I missed the day by day contact and exchange of views with fellow scientists, and could not benefit from their criticism and review of my work.

I have enjoyed a massive and unrepayable support, intellectually and practically, from my close family. My wife Yocheved did all the editing. Her composed judgment, sensible counsel and calm attitude more than once balanced and simmered down my own emotional and bold-tempered statements. My sons, David, Aviel, Meir and Moshe assisted me with their mathemat-

Preface ix

ical expertise and also helped solve many technical problems. To my family I would like to extend my love and deep thanks. May God bless them all.

I extend my thanks to the authors and publishers credited in the illustrations and tables of this book for releasing the publication rights of material in which their intelectual effort was invested.

I am indebted to Springer-Verlag and specifically to SpringerVerlag London Limited, who took the risk of publishing this book and who so patiently waited for the manuscript and encouraged me along the way. I hope their patience pays off.

Thanks to Him who created man in the image of His likeness and bestowed upon him His wisdom to understand, to learn and to teach.

Beer-Sheva March,1990

Y. Klausner

Contents

Preface ........................................................................ v List of Symbols . . . .. . . . . .. .. . . . . .. .. . . . . .. . . . . .. . . . . .. .. . . . .. . . . . .. .. . . ... xix 1 Introduction

1.1 Scope ............................................................ 1 1.2 Historical Notes .............................................. 2 1.3 Classical Soil Mechanics versus Mechanics

of Soils .......................................................... 4 1.4 Theory versus Experiment .... .. .... .. .... .. .... .. .. .. .... 6 1.5 Levels of Investigation ..................................... 8 1.6 The Continuum .............................................. 10 1. 7 Homogeneity and Isotropy ............................... 11 1.8 Soils as Multi-phase Mixtures ........................... 12 1.9 The Methodology of Continuum

Mechanics ..................................................... 13

2 Deformation and Strain 2.1 Deformation and Displacement ......................... 15 2.2 Strain ........................................................... 18 2.3 Strain Measures ............................................. 19 2.4 Invariants of the Deformation Tensor ................ 22 2.5 Small Deformations and Infinitesimal

Strains .......................................................... 24 2.6 The Strain Invariants ...................................... 25 2.7 The Hencky Measure of Strain ......................... 27 2.8 The Properties of the Hencky Measure ............... 30 2.9 Compatibility Equations .................................. 32

3 Kinematics 3.1 Material Derivatives ....................................... 35 3.2 Velocity and Speed ......................................... 35 3.3 Acceleration .................................................. 37 3.4 Material Derivatives of Displacement

Gradients ...................................................... 37 3.5 Strain Rates .................................................. 39

xii Contents

3.6 The Fundamental Theorem of Deformations ................................................ 40

3.7 Rigid Deformation and Motion ......................... 41 3.8 Homogeneous Strain ....................................... 42 3.9 Pure Strain .................................................... 42 3.10 Isochoric Deformation and Motion .................... 42 3.11 Irrotational Motion ......................................... 43 3.12 Laminar Motion ............................................. 44 3.13 Spherical Deformation .................................... 44 3.14 Simple Straining ............................................. 45 3.15 Uniaxial Straining ........................................... 46 3.16 Plane Strain .................................................. 47 3.17 Simple Shear ................................................. 47 3.18 Simple Torsion of a Circular Cylinder ................ 50 3.19 Telescoping Deformation ................................. 52 3.20 Rotational Deformation ................................... 55 3.21 Steady Motion ............................................... 57

4 Balance Equations for Homogeneous Media 4.1 Mass ............................................................ 59 4.2 The General Balance Equation ......................... 61 4.3 Density Balance ............................................. 63 4.4 Forces Acting on Deformable Bodies ................. 63 4.5 Tractions and Body Forces ............................... 64 4.6 Balance of Linear Momentum .......................... 68 4.7 Balance of Moment of Momentum .................... 69 4.8 The Pressure Tensor ....................................... 70 4.9 The Stress Tensor ........................................... 71 4.10 The Stress Invariants ....................................... 72

5 Energetics 5.1 Energy Considerations .................................... 75 5.2 Kinetic Energy ............................................... 76 5.3 Potential Energy ............................................ 78 5.4 Internal Energy .............................................. 79 5.5 The Total Energy Balance ............................... 80 5.6 Historical Notes on Irreversible Processes

of the Continuum ........................................... 82 5.7 The Thermodynamic State ............................... 83 5.8 Thermodyp.amic Tensions ................................ 84 5.9 Entropy and Temperature ................................ 85 5.10 The ThermodynamicFunctions ......................... 88 5.11 The Production of Entropy .............................. 89 5.12 Particular Cases of the Thermodynamic

State ............................................................ 92

Contents xiii

6 Multi-phase Mixtures 6.1 Extensive and Intensive Variables ..................... 97 6.2 Density, Volume, Mass and Weight

of Constituents .............................................. 97 6.3 Diffusion Velocity and Barycentric Velocity ........ 99 6.4 The General Balance of Multi-Phase

Mixtures ..................................................... 100 6.5 Multi-phase Density Balance .......................... 101 6.6 Multi-phase Balance of Linear Momentum ........ 103 6.7 Multi-phase Balance of Moment of Momentum .. 105 6.8 Multi-phase Balance of Internal Energy ............ 106 6.9 The Caloric Equations................................... 108 6.10 The Production of Entropy ............................ 110

7 Constitutive Equations 7.1 Scope ......................................................... 113 7.2 Principles of Formulating Constitutive

Equations ................................................... 115 7.3 The Rheological Equation .............................. 116 7.4 Linearity and Non-linearity of Constitutive

Equations ................................................... 117 7.5 The Dual Rheological Equation ...................... 119 7.6 Viscoelastic Models ...................................... 121 7.7

7.8

7.9 7.10 7.11

8 The Soil 8.1

8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13

Dual Volumetric Stress-Strain versus Shear Stress-Strain Relationship .................... . Energy Considerations in View of the Dual Equations ........................................... . The Isotropic Stress-Strain Relationship .......... . Dilatancy ................................................... . Isotropic Non-linear versus Linear Viscoelasticity ............................................. .

Single-phase versus Multi-phase Considerations ............................................ . Soil Constituents ......................................... . The Water ................................................. . Water Solutions .......................................... . Vapor Pressure ........................................... . The Air ..................................................... . Compressibility of Gases ............................... . Air-containing Pores .................................... . The Solid Particles ....................................... . Specific Surface ........................................... . The Mineralogical Structure of Clays .............. . Electric Charges and Exchange Capacity .......... . The Diffuse Double Layer ............................ .

126

129 131 133

135

137 138 139 143 145 148 150 152 155 156 157 162 163

xiv Contents

8.14 The Gouy-Chapman Double-layer Theory of Planar Surfaces ........................................ 164

8.15 Limitations of the Gouy-Chapman Theory ....... 168 8.16 Two Interacting Surfaces in Electrolyte

Solution ...................................................... 170 8.17 The Work of Interacting Surfaces .................... 172 8.18 Osmotic Pressure and Consolidation ................ 178 8.19 Pore Water Pressure ..................................... 180 8.20 Swelling Pressure ......................................... 182 8.21 Factors Affecting the Behavior of Clays in

Consolidation .............................................. 184 8.22 Properties of Clays as Predicted by the

Diffuse Double Layer ................................... 188 8.23 The Structure of Clays .................................. 189 8.24 Interfacial Forces ......................................... 193 8.25 Air-Water Interface ..................................... 196 8.26 Capillarity ................................................... 202 8.27 Suction and Shrinkage ................................... 203

9 Soil as a Multi-phase Mixture 9.1 Introduction ................................................ 211 9.2 Volume and Weight Relations in Soils .............. 211 9.3 Density Balance ........................................... 215 9.4 Balance of Linear Momentum ........................ 218 9.5 Balance of the Internal Energy ....................... 223

10 Flow in Soils 10.1 Introduction ................................................ 227 10.2 Force Fields ................................................ 228 lO.3 Flow Potentials ............................................ 232 10.4 Review of Linear and Non-linear, Saturated

10.5 10.6 lO.7 10.8 lO.9 lO.10 10.11 10.12 10.13 10.14

and Unsaturated Flow ................................... 236 Darcy's Law ................................................ 238 Flow in Saturated Soils .................................. 240 Modes of Saturated Flow ............................... 242 The Coefficient of Permeability ....................... 244 Seepage in Saturated Soils ............................. 248 Unsaturated Flow in Multi-phase Fluids ............ 250 Flow in Unsaturated Soils .............................. 251 Flow in Unsaturated Non-swelling Soils ............ 252 The Boltzmann Transformation Solution ........... 257 Flow in Unsaturated Swelling Soils .................. 259

11 Volumetric Stress-Strain Phenomena 11.1 The Volumetric Stress-Strain

Relationship ................................................ 263 11.2 Volume Changes in Soils ............................... 264

Contents xv

11.3 Consolidation of Saturated Soils ........... . .......... 265 11.4 Terzaghi's Theory of Consolidation .................. 266 11.5 Discussion of Terzaghi's Theory of

Consolidation .............................................. 270 11.6 The Consolidometer ..................................... 274 11. 7 The Consolidation Test ................................. 275 11.8 The Void Ratio-Pressure Dependence ............. 278 11.9 The Pressure and Strain Tensors in Uniaxial

Consolidation .............................................. 281 11.10 The Triaxial Testing Device ........................... 285 11.11 The Spherical Consolidation .. ...... .. .. .. ........ .. ... 289 11.12 The Pressure and Strain Tensors in the

Triaxial Test ................................................ 292 11.13 The Void Ratio-Pressure Curve ..................... 299 11.14 Normally Consolidated and Overconsolidated

Soils . ........... . ....................... . ........ . ............ 305 11.15 Consolidation of Unsaturated Soils .................. 307 11.16 Hysteresis ................................................... 312 11.17 Phenomenological Linear Volumetric

Stress-Strain Relationship ............................. 315 11.18 Modeling the Linear Volumetric Stress-Strain

Relationship ................... ............................. 316 11.19 Constant Spherical Pressure ........................... 318 11.20 General Spherical Pressure ............................. 320 11.21 The Volumetric Plastic Restraint .... .. .. .. .... .. ..... 321 11.22 Effective Pressure ......................................... 322 11.23 Total Pressure ................ .. ................ .. ......... 327 11.24 The Internal Energy and Energy Rate of

Spherical Phenomena .................................... 330 11.25 The Excess Stored Specific Free Energy ...... .. .. . 332 11.26 The Excess Stored Specific Free Energy of

Solids in Linear Viscoelastic Media .......... . ....... 334 11.27 Isotropic Functional Relationship Applied to

the Excess Stored Specific Free Energy .. . ......... 338 11.28 The Excess Stored Specific Free Energy

(Particular Cases) ................ . ........................ 340

12 Shear Stress-Strain Phenomena 12.1 Introduction. ..... .... ........ ... .. ... ........... . ... .. ..... 343 12.2 Density Effects on Shear Stresses ......... .. ......... 344 12.3 The Shear Stress-Strain Relationship ............... 345 12.4 Shear versus Volumetric Stress-Strain

Relationship ........ . ..... . .................. ..... . . ........ 346 12.5 Deviatoric Tests ...................... .... ................. 355 12.6 The Conventional Triaxial Shear Test . . ............ 355 12.7 The Unconfined Compression Test ........ .. ........ 359

xvi Contents

12.8 The Simple Shear Test .................................. 362 12.9 The Direct Shear Test ... .. ... .. ... ... .... . .... .. .. .. .. .. 366 12.10 The Torsion Test and Testing Device ............. .. 375 12.11 Torsion of Solid Cylindrical Samples .. .. .. .. ...... .. 379 12.12 Torsion of Hollow Cylindrical Samples ............. 381 12.13 Introduction to the Pure Deviatoric Test ........... 389 12.14 The Pure Deviatoric Test and its

Equipment ..... ............ ............. . ..... .............. 393 12.15 Stresses and Strains in the Pure Deviatoric

Test .. ............ . .... .. .... . ...... ............ ............... 398 12.16 Linear Deviatoric Stress-Strain

Relationship .................... ....... .. .... .. ... . .. . ... .. . 404 12.17 The Linear Deviatoric Constitutive Equation

for'Soils ............ ........ . ... . .. . ... . .. . ... . . .. .. .. . .. ... . . 405 12.18 Isotropic Strain Functions of Shear

Constitutive Equations .... . . . .. .... .. ... .. .. .. ... . ... .. . 409 12.19 Isotropic Stress Functions of Shear Constitutive

Equations ..... .. ................. .. ........... . ... .. ..... .. . 412 12.20 Spherical Components in the Pure Deviatoric

Test .................... ....................... ... .... .. ..... . . 413 12.21 Pore Pressures in the Pure Deviatoric

Test ........................................................... 415 12.22 The Effect of the Rate of Loading in the

Pure Deviatoric Test .... .. .... ... .. .. ..... .. ............. 417 12.23 Critical Void Ratio and Pure Deviatoric

Loading .. .. ... . . ....... .. .... ... . ... ................... . ..... 417 12.24 The Free Energy and Energy Rate of

Deviatoric Phenomena ..... . .. ... .. .. . .. .... .. ..... . .. .. 423 12.25 The Disbursed Specific Free Energy ................ 425 12.26 The Disbursed Free Energy Applied to Linear

Stress-Strain Relations ...... .............. .............. 425 12.27 The Disbursed Free Energy in the Pure

12.28

12.29

12.30

13 Failure 13.1 13.2 13.3 13.4

Deviatoric Test of Linear Viscoelastic Soils .......................................................... 426 The Disbursed Free Energy in Pure Deviatoric Tests for Isotropic Stress-Strain Relations of Soils ... ... .............. .......................... ... .. .. . . 429 Note on General Non-linear Stress-Strain Relations ..... .. ............................... ....... .. ..... 433 Closing Remarks on Shear Stresses .... . .. .. .. . .... . . 434

Brief Review of Failure Theories .... . . ... .. .. . ..... .. 437 Failure Criteria ............................. .. ............ . 438 The Dual Specific Internal Energy .. ... .. .. .. ... . .. .. 442 Stored and Disbursed Specific Free Energy ........ ........ .. ..... .. ..... . ... . ... .. . . . .. . ;.. . ..... 444

Contents xvii

13.5 Specific Free Energy Balance of the Elastic Medium (A Particular Case) ........................... 446

13.6 Specific Free Energy Balance of Linear Viscoelastic Media ........................................ 448

13.7 The Pure Deviatoric Test of a Linear Viscoelastic Medium ..................................... 449

13.8 Free Energy Balance with Non-linear Stress-Strain Relations ................................... 453

13.9 The Pure Deviatoric Test with Isotropic Relations .................................................... 454

13.10 Appraisal of the Presented Failure Criteria ...................................................... 460

13.11 The Study of Failure through the Pure Deviatoric Test .......................................... .. 462

13.12 Drained Pure Deviatoric Shear Tests ............... 474 13.13 Consolidated Undrained Pure Deviatoric

Shear Tests ..... ......... ................................... 475 13.14 Undrained Pure Deviatoric Shear Tests ............ 477 13.15 Slip Surfaces .................... ............................ 480 13.16 Lateral Earth Pressure .................................. 485 13.17 The Coefficient of Lateral Earth Pressure 487

Appendix A Tensor Mathematics A.1 Introduction .................................... .... ...... .. 491 A.2 The Indicial Notation .................................... 492 A.3 Transformation of Coordinates ....................... 492 A.4 The Summation Convention .................. ......... 493 A.5 The Kronecker Delta .................................... 494 A.6 Contravariant and Covariant Tensors ............... 495 A.7 Symmetric and Skew-symmetric Tensors ........... 496 A8 Addition, Subtraction and Multiplication .......... 497 A.9 Contraction ................................................. 498 A .10 The Line Element ........................................ 498 A.11 The Angle between Vectors ........................... 500 A.12 Lowering and Raising Indices ......................... 501 A.13 The Christoffel Symbols ................................ 501 A .14 Covariant Differentiation of Tensors ................ 503 A .15 Principal Directions of Second-order

Tensors ..... ................................................. 505 A16 Differential Operators ................................... 507 A17 Orthogonal and Cartesian Coordinates ....... .. ... . 508 A.18 Invariants ................................................... 511 A.19 Integrals of Tensor Fields .............................. 513 A.20 Geometrical Representation of Second-order

Tensors .......... .... .... .. ...... .. .......................... 516 A.21 Axially Symmetric Second-order

Tensors .... ....... ......... ...... .. ... ..... .. ...... ..... . .... 523

xviii Contents

Appendix B Cylindrical Coordinates B.1 Introduction ............ .. .................................. 529 B.2 Definition of the Cylindrical Coordinate

System ....................................................... 529 B.3 The Fundamental Tensor ............................... 532 B.4 The Christoffel Symbols ................................ 533 B.5 Covariant Derivatives .. .................................. 534 B.6 Basic Operations of First-order Tensors in

Cylindrical Coordinates ................................. 535 B.7 Elements of Differential Geometry .................. 538 B.8 Equations of Kinematics ................................ 540 B.9 The Strain Tensor ...................... .. .............. .. 541 B.10 The Balance Equations .................................. 543

Appendix C Rheological Modeling C.1 Introduction......... ......... ...... .. ..... ....... .. ..... ... 547 C.2 The Hookean Elastic Element ...... .... .............. 549 C.3 The Newtonian Viscous Element ..................... 549 C.4 Coupling of Rheological Elements ................... 550 C.5 St Venant's Element of Plastic Restraint ........... 551 C.6 The Prandtl Body ......................................... 552 C. 7 The Maxwell Body .................. .. .. .. ............... 553 C.8 The Kelvin Body .......................................... 555 C.9 The Burgers Body ................ .. ...................... 557 C.lO The Relations Between Excitation and

Response ............................ .. ................ .. .... 561 C.ll The Relaxation and Creep Functions ............ .. . 563 C.12 The General Rheological Models .... ................ 565 C.13 Elastic and Dissipative Excitations ............. .. .. .. 569 C.14 The Plastic Restraint ..................................... 570

References . ..... . ..................... . .... . ...... . .... ... .... . ...... . ... . .. 575

Subject Index .. .. ............ . .. .. ................ .... . . ................ . . 597

Symbols

Section of Notation Name definition

8 0 Coefficient of compressibility 11.4

ai Acceleration 3.3, B.7

ai Body force acceleration 10.3

an Constant coefficients C.1

an ae, az Acceleration components B.7

A Constant C.1, C.13 Velocity 3.19 Reciprocal of time factor C.14

A(t) Retardation spectrum C.ll

Ai ith coefficient of creep function C.12

A Dipole dependent function 8.21

Ai Vector, tensor of order one A.4

A,s Tensor of order two A.4

AKM Metric tensor of the Euclidean space 2.9

bn Constant coefficients C.1

B Coefficient of transverse stretching 11.12 Twist per unit length 12.11

xx Symbols


Constant C.1 Relaxation spectrum C.1l Reciprocal of time factor C.14

B(t) Relaxation spectrum C.1l

Bj ith coefficient of relaxation function C.12

B; Body force 4.8

B* I Body force acting on stress tetrahedron 4.5

Bg; Body force on air 9.4

Bn; Body force acting on the nth constituent 6.6

BM Body force on solids 9.4

Bm; Body force on water 9.4

BLM Transformation tensor A.20

CO Coefficient of consolidation 11.4

c Cohesive intercept 12.9

c. Effective cohesive intercept 12.9

clm , Clm , C Cauchy's deformation tensor 2.1

C Mean curvature of air-water interface 8.25 Constant spherical pressure 11.26 Closed contour of integration A.I8 Constant C.13 Reciprocal of time factor C.14

C, Compression index 11.7

C j , C2 Curvatures of air-water interface 8.25

CEC Cation exchange capacity 8.12

CLM , CLM ' C Green's deformation tensor 2.1

Symbols xxi


Cg Excess density supply rate of air 9.3

Cn Excess density supply rate of the nth constituent 6.5

C. Excess density supply rate of solids 9.3

Cm Excess density supply rate of water 9.3

e Constant rate of vertical load 12.15

d Dial reading 10.7 Thickness of cylindrical tube 12.10

d(x) Linear extension in Xi 2.3

d(X) Linear extension in XL 2.3

do Initial dial reading 10.7

d Half distance between particles 8.16

De Effective diameter of soil particles 10.8

D Diffusivity 6.5 Dielectric constant of water solute 8.14

De,D;, D" • Void ratio dependent diffusion coefficient 11.15

D(w) Water content diffusivity 10.12

D( f)ij Moisture diffusivity 10.12

D(fJ) Moisture ratio diffusivity 10.14

~ Dissipative energy 5.5

~(t") Lumped dispersed free energy 13.7

e Electronic charge 4.803 x 1010 e.s.u./electron 8.4 Void ratio 9.2 2.71828, constant base of natural logarithm C.7

e(t) Deviatoric strain factor 13.9

xxii Symbols


en eo, ez Base vectors B.2

eij Traceless strain tensor 2.3 Distortion 7.5

eijk> e ijk Permutation symbol in Xi A.14

eLMN, eLMN Permutation symbol in XL A.14

ee Void ratio corresponding to pressure Pezz 11.7

E Isotropic strain matrix 7.9 Rheological response C.1 Response function C.1

E1, E2 , E3 Rheological responses C.7

E", Response at infinite time C.S

EK Response of the Kelvin model C.14

Es Response of the St Venant element C.14

E Isotropic strain-rate matrix 7.9

EC Exchangeable cations S.12

En Excess internal energy of the nth constituent 6.S

t Distortion 11.27 Strain 12.17 Constant rate of response C.7 Constant response C.S

~ Energy efflux 5.4

~o Complex static elastic modulus C.lO

~(w) Dynamic modulus C.lO

~(wh Dynamic friction C.lO

~(iw) Complex dynamic modulus C.lO

Symbols xxiii


f Force field 10.2

fi Force field, flux vector 10.2

Ii Flux through cross-section a 10.2

f(x) Unit force per volume S.17

[;j Flow or stretching tensor 3.4

f9ij Flow tensor of air 9.5

f.ij Flow tensor of solids 9.5

froij Flow tensor of water 9.5

Counter of immiscible fluids 10.10

F Rheological excitation C.1

r= Excitation function C.1 Activating excitation C.6

FK Excitation acting in the Kelvin model C.14

Fs Excitation acting in the St Venant element C.14

Fl , F2 , F3 Rheological excitations C.S

F(iE) Complex potential 10.9

Fi Force vector 4.4

Fn Unbalanced supply of the nth constituent 6.4

Fni Excess linear momentum supply of the nth constituent 6.6

Function, functional relationship 3.9 Viscoelastic differential operator 11.9 Material coefficient function 11.26 Constant excitation C.7 Constant rate of excitation C.7

xxiv Symbols


~ Number of immiscible liquids 10.10 Compliance C.10

%'0 Proportionality constant of plastic restraint C.S

%,(iw) Complex compliance C.lO

%,(iw)* Complex elastic compliance C.lO

g(w) Dynamic viscosity C.1O

gi Gravity acceleration 9.2

glm, glm' g~ Metric tensor in Xi 2.1

qn' q2, q3 General viscosity coefficients C.3

9 Gravity acceleration 10.3

G Elastic shear modulus 7.6

Gi Elastic shear modulus of the ith element 7.6

G. Specific gravity of solids 9.2

Gil) Specific gravity of water 9.2

Gr, G3 Elastic shear modulus of different elements 7.6

GLM , GLM ,

Gkt Metric tensor in XL 2.1

() Function, functional relationship 7.2 Viscoelastic differential operator 11.9

()(T) Measure of solubility of air in water 8.6

he Head, capillary 8.26

hi Head, hydrostatic 8.26

h Molar specific heat 8.S

hi Specific energy efflux S.4

Symbols xxv


h gi Specific energy efflux of air 9.5

hni Specific energy flux of the nth-constituent 6.8

h nib Deviatoric specific energy flux of the nth constituent 11.24

hOi Specific energy efflux of solids 9.5

hroi Specific energy efflux of water 9.5

H Total entropy 5.9 Piezometric head 10.8 Length of consolidating path 11.3

Hn Excess entropy supply of the nth constituent 6.9

Imaginary unit C.10

ith viscoelastic element in the model C.10

3 Internal energy 5.4 Number of viscoelastic elements in the model C.1O

3. Internal energy of solids C.10

~.b Disbursed internal energy of solids 11.24

3.i Stored internal energy of solids 11.24

3.0 Residing stored internal energy of solids 11.24

L\3.i Excess stored internal energy of solids 11.24

f} Jacobian A.3, B.2

f}' Inverse Jacobian B.2

k Bolzmann's constant 1.3805 x 10-16 ergfK 8.14 Isotropic coefficient of permeability 10.5 Failure parameter 13.2 Constant C.1

xxvi Symbols


ka Restricted active coefficient of lateral earth pressure 13.16

kp Restricted passive coefficient of lateral earth pressure 13.16

k ij Coefficient of permeability 10.5

ku, k22' k33 Coefficients of permeability in the principal directions 10.5

k1' k3 General elastic moduli C.2

k Coefficient of permeability 10.8

K Coefficient of lateral earth pressure 11.6

Ka Coefficient of lateral earth pressure at rest 11.9

K Temperature in Kelvin 8.5

Ka Active coefficient of lateral earth pressure 13.16

Kp Passive coefficient of lateral earth pressure 13.16

Kij Intrinsic permeability 10.8

5t Kinetic energy 5.2

lij Body moment 4.5

lnij Body moment of the nth constituent 6.7

L Length of flow path 10.8 Line, line element A.18

La Initial length 2.3

Ln Length after straining 2.3

m Mass element 4.1

mg Mass of air 9.2

Symbols xxvii


mn Mass of the nth constituent 6.2

m. Mass of solids 9.2

m", Mass of water 9.2

min Couple traction in direction ni 4.5

mijn Surface couple in direction ni 4.5

mijk Couple stress 4.5

mnijk Couple stress of the nth constituent 6.7

m Summation counter of substates 5.7

M Total mass 4.1

M(s) Frequency function of creep C.ll

Mi Torque, moment 4.4

9Jl Number of sub states 5.7

n Porosity 9.2

ni Unit vector in Xi normal to a surface 2.3, A.20

n+, n- Ionic concentration 8.14

+ -n Xl , n oo , noo Ionic concentration per unit volume in the bulk water 8.4

n Number of constituents in a mixture 6.2

N(s) Frequency function of relaxation C.ll

N Normal vector A.19

N Number of coordinates A.3

N L Unit vector in XL normal to a surface 2.3, A.20

Pb Spherical stress of the second order

xxviii Symbols


91 Number of constituents in a mixture 6.2

Pc Capillary pressure, suction 8.26 Cell pressure 12.13

Pczz Vertical pressure corresponding to void ratio ec 11.7

Pe Effective pressure 8.18

p(d)e Effective pressure at half-distance d between particles 8.19

Pg Pressure of the air constituent or pore air pressure 8.7,9.4

Pgc Critical pressure of air 8.7

Pi Air-water interfacial pressure 8.25

Pm Mean spherical pressure 4.9

Consolidation pressure 8.18

PmO Cohesion 13.5

Po Preconsolidation pressure 11.8 Constant initial pressure 11.18 Osmotic pressure 8.17

Pp Pore pressure 11.9, 11.12

PI Repulsive pressure 8.17

Pu Unconfined compression strength of undisturbed soil 8.23

Pur Unconfined compression strength of remolded soil 8.23

PIO'O Water vapor pressure 8.5

Pro Pore water pressure 8.18

Symbols xxix


Pm Average pore water pressure 11.4

p(d)m Pore water pressure at half-distance d between particles 8.19

pF Measure of water tension 8.27

Pij Pressure tensor 4.8

Pu, P22, P33 Principal stresses 7.5

P: p': p", Equivalent pressure corrections 12.14

Psij Stress in solids or effective pressure 9.4

P lXlX Trace of pressure tensor 4.8

P Isotropic stress matrix 7.9

P Isotropic stress-rate matrix 7.9

Pb Traceless isotropic stress matrix 7.9

Pi Volumetric isotropic stress matrix 7.9

1} Constant spherical pressure 11.9 Differential operator C.1

1}q qth step of constant deviatoric loading 13.7

'P Potential energy 5.3

q Specific energy supply 5.4

qg Energy supply of air 9.5

qn Specific energy supply of the nth constituent 6.8

qnb Deviatoric specific energy supply of the nth constituent 11.24

q. Energy supply of solids 9.5

qm Energy supply of water 9.5

xxx Symbols

Section of

Notation Name definition

qi Specific discharge vector 10.4

qgi Flux of air 10.4

q~i Flux of solids 10.4

qroi Flux of water 10.4

iiroi Diffusion velocity of water with respect to the solids 10.5

q Counter of deviatoric step loadings 12.17

Q Discharge 10.8

Qi Total discharge vector 10.4

Qgi Discharge of air 10.4

Qsi Discharge of solids 10.4

Qroi Discharge of water 10.4

Q Differential polynomial C.1

Q Number of deviatoric step loadings 12.17

r Cylindrical coordinate B.2

(. Region in Xi 2.1

r Counter of spherical stress increments 13.11

R Linear stress-strain relationship function 7.3 Gas constant, 8.314 Jmole-1 deg-1 8.7 Radius of air-water interface 8.25 Radius of cylindrical sample 12.11

Rb R2 Radii of air-water interface 8.25

RH Relative humidity 8.5

Rb Deviatoric linear stress-strain relationship function 7.3

Symbols xxxi


Re Reynolds number 10.8

Rj Spherical linear stress-strain relationship function 7.5

Rp Hydraulic radius 10.8

R Radius in cylindrical coordinates 3.18

R~, R(x) Rotation tensor in Xi 2.7

RXt, R(X) Rotation tensor in XL 2.7

RKMPQ Riemann -Christoffel curvature tensor 2.9

rQ Region in XL 2.1 Differential polynomial C.1

9\ Number of spherical stress increments 13.11

S Length of line element in Xi A.10

Sij Traceless pressure tensor 4.8 Deviatoric stress 7.5

SOij Shear stress of soil 9.4

S Surface area 2.1 Length of line element in XL A.10

S Degree of saturation 9.2

Si Projection vector of area 10.2

sa I Projection of cross-section a in direction i 10.2

Sf Degree of saturation of the fth immiscible liquid 10.10

SI Sensitivity 8.23

J Constant shear stress 12.27 Constant rate of deviatoric stress 12.17

J r rth step of constant vertical load 12.17

e; Energy supply 5.4

xxxii Symbols


\S(t)q Stored free energy of the qth step loading 13.7

\S(t') Lumped stored free energy 13.7

t Time 3.1

t Traction vector 4.5

ti Traction vector 4.5

tin Traction in direction of vector ni 4.5

tii , tii Stress tensor, component of stress tensor 4.5

t~ Stress vector acting on stress tetrahedron 4.5

tt Stress tensor acting on stress tetrahedron 4.5

t nn , tn Normal traction 4.5

tnt' tt Tangential traction 4.5

tnii Stress tensor acting on the n-th constituent 6.6

T Temperature in Kelvin degrees 5.9 Tangential component in the octahedral plane A.20 Time factor C.14

T3 Volumetric retardation time 11.18

T, Critical temperature of gases 8.7

Tn Temperature of the nth constituent 6.9

T,,( Relaxation time C.7

T,,(j Relaxation time of the ith element C.ll

T"t Retardation time C.8

T"ti Retardation time of the ith element 7.6

T;,ti Shear retardation time of the ith element 7.6

Symbols xxxiii


To Consolidation time factor 11.4

Tni Excess linear momentum supply of the nth constituent 6.6

T z Torque 12.11

T Tangential vector A.19

T Surface tension 8.25

<d Transmittance C.1

~ Total energy 5.5

ua Pore air pressure (Soil Mechanics notation) 11.22

Uro Pore water pressure (Soil Mechanics notation) 11.22

ui , Ui Displacement in Xi 2.1

U(J Tangential deformation 12.11

U Average degree of consolidation 11.4

Ur Degree of consolidation 11.4

U Dimensionless mid-plane potential 8.21

UL , UL Displacement in XL 2.1

V Speed 3.2

Vi Velocity 3.2, B.7

Vi Mean velocity or barycentric velocity 6.3

Vgi Velocity of air 9.3

Vmi Velocity of the reference constituent 6.3

Vni Velocity of the nth-constituent 6.3

VSj Velocity of solids 9.3

xxxiv Symbols


V"'i Velocity of water 9.3

Ugi Diffusion velocity of air 9.4

Uni Diffusion velocity or peculiar velocity of the nth constituent 6.3

U.i Diffusion velocity of solids 9.4

U"'i Diffusion velocity of water 9.4

Vnmi Velocity of the nth constituent relative to the mth constituent 6.3

V Volume 2.1

Vb Volume of dry soil 8.27

V f Volume of the fth immiscible liquid 10.10

Vg Volume of the air constituent 8.7

Vgm Molal volume of air 8.7

Vn Volume of the nth constituent 6.2

V. Volume of solids 9.2

V", Volume of water 9.2

Vo Volume of voids 9.2

W Water content 9.2

WL Liquid limit 8.2

Wp Plastic limit 8.2

Ws Shrinkage limit 8.27

Wi,W Vorticity vector 3.4

wij Spin tensor 3.4

Symbols xxxv


ro. Specific stress-work 5.2

rom Specific mechanical work 5.5

ron Specific non-mechanical work 5.5

W Total weight of soil 6.2

Wg Weight of air 9.2

Wn Partial weight of the nth constituent 6.2

W. Weight of solids 9.2

WID Weight of water 9.2

emm Mechanical power 5.5

emn Non-mechanical power 5.5

em, Work of repulsive force 8.17

em. Stress-power 5.2

x Length of vector Xi 2.2

Xi, Xi Deformed coordinates 2.1

Xi Deformed position vector A.3

Xi ,L Deformation gradient in Xi 2.1

XL, XL Un deformed coordinates 2.1

XL Undeformed position vector A.3

XL ,m Deformation gradient in XL 2.1

y Dimensionless potential at ; 8.16

Z Cylindrical coordinate B.2

+ -Zj , Zi Valency of the ith ion 8.4

xxxvi Symbols


Z Compressibility factor 8.7

Z Dimensionless potential at particle surface 8.14

a Angle of distortion 3.17 Angle A.19 Normalization factor of creep C.11

ag Density concentration of air 9.5

a' 9 Density concentration of the dissolved gas 8.6

an Density concentration of the nth constituent 6.2

a. Density concentration of solids 9.5

an> Density concentration of water 9.5

f3 Angle A. 20 Normalization factor of relaxation C.11

Y Angle, angle of rotation 3.17, A.20 Angle between vectors B.5

Yb Dry unit weight 9.2

Yg Unit weight of air 9.2

Y. Unit weight of solids 9.2

Yn> Unit weight of water 8.26

Yn>o Unit weight of water at 4 °C 8.26

r k,ij Christoffel symbol of the first kind in Xi A.13

rK,LM Christoffel symbol of the first kind in XL A.13

rt Christoffel symbol of the second kind in Xi A. 13

rfM Christoffel symbol of the second kind in XL A.13

<5(t) Dirac delta function C.lO

Symbols xxxvii


c5ij , c5ij , c5} Kronecker unit tensor in Xi A.5

c5 LM , c5LM ,

c5kt Kronecker unit tensor in XL A.5

Ll Volumetric dilatation 2.4 Vertical deformation 10.7 Settlement 11.7

f ij Strain tensor, general 2.3

fU' f22, f33 Principal strains 7.5

inf Infinitesimal strain tensor 2.3 f ij

A Almansi-Hamel strain tensor 2.3 f ij

E Eulerian strain tensor 2.2 f ij

G Green-St Venant strain tensor 2.3 f ij

H Hencky strain tensor 2.3 f ij

L fij Lagrangian strain tensor 2.2

s Swainger strain tensor 2.3 fij

Em Volumetric strain 7.5

f <t<t Trace of strain tensor 2.3 Spherical strain 7.5

EfJfJ Dilatancy 11.27

Ei Specific internal energy 5.4

Eig Specific internal energy of air 9.5

Ein Specific internal energy of the nth constituent 6.8

Eis Specific internal energy of solids 9.5

Eiro Specific internal energy of water 9.5

ESt Specific kinetic energy 5.2

xxxviii Symbols


Ep Specific potential energy 5.3

E.b Disbursed specific internal energy 11.24

E.i Stored specific internal energy 11.24

AEtii Excess stored specific internal energy 11.24

E.iO Residing stored specific internal energy 11.24

Et Specific total energy 5.5 , Specific free enthalpy 5.10

~ Gravity potential 10.3

1] Specific entropy 5.9 Shear viscosity coefficient 7.6 Viscosity 10.8

1]2' 1]3 Shear viscosity of elements 12.16

1]9 Specific entropy of air 9.5

1]i Shear viscosity coefficient of the ith element 7.6

1]. Specific entropy of the nth constituent 6.9

1]. Specific entropy of solids 9.5

1]ro Specific entropy of water 9.5

f} Moisture ratio 9.2, C.5

t'J( t) Coefficient of volumetric plastic restraint 11.5 Plastic restraint C.5

(J Angle of cylindrical coordinates 3.18, B.2 Angle A. 11 Failure angle 13.15

e Angle of cylindrical coordinates 3.18

(J" Critical angle of failure 13.15

Symbols xxxix


Og Volumetric air fraction 9.2

Os Volumetric solids fraction 9.2

Oro Volumetric moisture fraction 9.2

°roi Initial moisture content 10.13

Oros Saturated moisture content 10.13

Oror Moisture content between initial and saturated state 10.13

Elastic bulk modulus 7.6 Reciprocal of double layer thickness 8.14

"1, "3 Elastic bulk moduli of elements 11.18

"m Elastic bulk modulus of the mth element 11.18

". Bulk modulus of solids 11.4

"ro Bulk modulus of water 11.4

A Scalar, proper number A.15

A:z Vertical stretch 11.7

AT Trouton's viscous traction 7.6

A.\i:, km Latent roots of second order tensors A.15

~ Proper number A. 15

A(X) Linear stretch in Xi 2.3

A(X) Linear stretch in XL 2.3

A Volumetric stretch 2.4

J1. Coefficient of friction 13.5

J1.; Volumetric viscosity coefficient of the ith element 7.6

xl Symbols


112, J.l3 Bulk coefficients of viscosity of elements 11.18

v Kinematic viscosity 10.8 Ratio of transverse dilatation 11.12

Vnm The mth thermodynamic subs tate of the nth constituent 6.9

Vnmb Thermodynamic deviatoric substates 11.24

~n Extensive quantity of the nth constituent 6.3

~ Extensive quantity per unit mass 4.2 Specific total energy 5.5 Dimensionless distance in the diffuse double layer 8.14

~ Total extensive quantity of volume 4.2 -BLM , Bij Dimensionless deviatoric tensor A.21

ITA Octahedral invariant of a tensor Ajj A.18

P Density 4.1 Charge density per unit volume of solution 8.10

P g Density of air 9.2

Pm Density of the reference constituent 6.5

Pn Density of the nth constituent 6.2

P~ Density of solids 9.2

Pro Density of water 9.2

a Supply of quantity per unit mass 4.2 Surface charge density 8.14 Total pressure (Soil Mechanics notation) 11.22

a' Effective pressure (Soil Mechanics notation) 11.22

aj Body force 4.5

an Specific supply rate of the n-th constituent 6.4

Symbols xli


T Octahedral shear stress 13.5

Teu Shear stress of consolidated-undrained test 13.13

Tb Shear stress of drained tests 13.12

Tm m-th thermodynamic tension 5.7

Tnm m-th thermodynamic tension of the n-th constituent 6.9

Tnmb Thermodynamic deviatoric tension 11.24

Toct Octahedral shear stress 13.2

Tu Shear stress of undrained test 13.14

cf> Angle of internal friction 12.9 Potential 10.2 Tensorial function A.18

cf>. Effective angle of internal friction 12.9

cf>n Coefficients of polynomials 2.7

<I> Mechanical and non-mechanical energy flux 5.5

<I> (x, y) Real part of complex potential 10.9

<l>c Capillary potential 10.3

<l>b Heat potential 10.3

<1>, Electric potential 8.14 Electrochemical potential 10.3

<l>g Gravitational potential 10.3

<l>n Specific efflux of the n-th constituent 6.4

<l>p Pressure potential 10.3

<l>t Total potential 10.3

xlii Symbols


<Pro Moisture potential 10.3 Pore water potential 11.9

<Py Overburden potential 10.3

«I» Total flow potential 10.3

q; Scalar A.16

X Specific enthalpy 5.10 Bolzmann transformation function 10.13 Pore pressure parameter 11.22

x(t) Relaxation function C.1O

1fJ Specific free energy 5.10 Imaginary part of flow line 10.9

1jJ( t) Creep function C.lO

1fJob Disbursed specific free energy 11.25

1fJ.i Stored specific free energy 11.25

1fJoiO Residing specific free energy 11.25

d 1fJoi Excess specific free energy 11.25

'I' Matric potential 10.3

w Frequency C.lO

Q Angle of twist 12.11

II Overburden potential 10.3

~n nth isotropic stress-strain relationship coefficients 7.9

:In nth volumetric isotropic stress-strain relationship coefficients 7.9, 12.17

'n nth dilatancy coefficients of isotropic stress-strain relationship 7.9,

Notation Name

nth shear isotropic stress-strain relationship coefficients

lA, I1A' IlIA Principal invariants of a tensor Aij

lA' I1A' IlIA Moment invariants of a tensor Aij

grad Gradient of a scalar

div Divergence of a vector

curl Rotation of a vector

Laplacian differential operator

Unit vector

Symbols xliii

Section of definition

7.9, 12.17

A.18

A.18

A.16

A.16

A.16

10.2, A.14

B.2

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