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Further titles in this series: 1. G.SANGLERAT-THE PENETROMETER ANDSOIL EXPLORATION 2. Q. ZÄRUBA AND V. MENCL - LANDSLIDES AND THEIR CONTROL 3. E.E. WAHLSTROM - TUNNELING IN ROCK 4. R. SILVESTER - COASTAL ENGINEERING, 1 and 2 5. R.N. YONG AND B.P. WARKENTIN - SOI L PROPERTIES AND BEHAVIOUR 6. E.E. WAHLSTROM - DAMS, DAM FOUNDATIONS, AND RESERVOIR SITES 7. W.F. CHEN - LIMIT ANALYSIS AND SOIL PLASTICITY 8. L.N. PERSEN - ROCK DYNAMICS AND GEOPHYSICAL EXPLORATION Introduction to Stress Waves in Rocks 9. M.D. GIDIGASU - LATERITE SOIL ENGINEERING 10. Q. ZÄRUBA AND V. MENCL - ENGINEERING GEOLOGY 11. H.K. GUPTA AND B.K. RASTOGI - DAMS AND EARTHQUAKES 12. F.H. CHEN - FOUNDATIONS ON EXPANSIVE SOILS 13. L. HOBST AND J. ZAJIC - ANCHORING IN ROCK 14. B. VOIGHT (Editor) - ROCKSLIDES AND AVALANCHES, 1 and 2 15. C. LOMNITZ AND E. ROSENBLUETH (Editors) - SEISMIC RISK AND ENGINEERING DECISIONS 16. CA. BAAR - APPLIED SALT-ROCK MECHANICS, 1 The In-Situ Behavior of Salt Rocks 17. A.P.S. SELVADURAI - ELASTIC ANALYSIS OF SOIL-FOUNDATION INTERACTION 18. J. FEDA - STRESS IN SUBSOIL AND METHODS OF FINAL SETTLEMENT CALCULATION 19. A. KEZDI -STABILIZED EARTH ROADS 20. E.W. BRAND AND R.P. BRENNER (Editors) - SOFT-CLAY ENGINEERING 21. A. MYSLIVEC AND Z. KYSELA -THE BEARING CAPACITY OF BUILDING FOUNDATIONS 22. R.N. CHOWDHURY - SLOPE ANALYSIS 23. P. BRUUN - STABILITY OF TIDAL INLETS Theory and Engineering 24. Z. BAZANT - METHODS OF FOUNDATION ENGINEERING 25. A. KEZDI -SOIL PHYSICS Selected Topics 26. H.L. JESSBERGER (Editor) -GROUND FREEZING 27. D. STEPHENSON - ROCKFILL IN HYDRAULIC ENGINEERING 28. P.E. FRIVIK, N. JANBU, R. SAETERSDAL AND L.I. FINBORUD (Editors) - GROUND FREEZING 1980 29. P. PETER-CANAL AND RIVER LEVEES 30. J. FEDA - MECHANICS OF PARTICULATE MATERIALS The Principles 31. Q. ZÄRUBA AND V. MENCL- LANDSLIDES AND THEIR CONTROL Second completely revised edition 32. I.W. FARMER (Editor) - STRATA MECHANICS 33. L. HOBST AND J. ZAJIC - ANCHORING IN ROCK AND SOIL Second completely revised edition 34. G. SANGLERAT, G. OLIVARI AND B, CAMBOU - PRACTICAL PROBLEMS IN SOIL' MECHANICS AND FOUNDATION ENGINEERING, 1 and 2 35. L. RETHÄTI - GROUNDWATER IN CIVIL ENGINEERING 36. S.S. VYALOV - RHEOLOGICAL FUNDAMENTALS OF SOIL MECHANICS 37. P. BRUUN (Editor) - DESIGN AND CONSTRUCTION OF MOUNDS FOR BREAKWATERS AND COASTAL PROTECTION 39. ET. HANRAHAN THE GEOTECTONICSOF REAL MATERIALS: THE e g, €k METHOD DEVELOPMENTS IN GEOTECHNICAL ENGINEERING 38 SOIL PLASTICITY Theory and Implementation W.F.CHEN School of Civil Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. and G.Y. BALADI U.S. Army Engineering Waterways Experimental Station, Vicksburg, Miss., U.S.A. ELSEVIER Amsterdam — Oxford — New York — Tokyo 1985 ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, NY 10017 U.S.A. Library of Congress Cataloging-in-Publication Data Chen, Wai-Fah, 1936- Soil plasticity. (Deveolpments in geotechnical engineering ; 38) Bibliography: p. Includes index. 1. Soils—Plastic properties. I. Baladi, George Y. II. Title. III. Series. TA710.5.Cl*76 1985 624.1'5136 85-I632U ISBN 0-UU+-1+21+55-5 (U.S.) ISBN 0-444-42455-5 (Vol. 38) ISBN 044441662-5 (Series) © Elsevier Science Publishers B.V., 1985 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or other- wise, without the prior written permission of the publisher, Elsevier Science Publishers B.V ./Science & Technology Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands. Special regulations for readers in the USA — This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. Printed in The Netherlands PREFACE This book is based on a series of lectures that the authors gave at Purdue University and elsewhere. In this book, we have attempted to present a simple, concise and reasonably comprehensive introduction of the theory of soil plasticity and its numeri- cal implementation into computer programs. The theory and method of computer implementation presented in this book are appropriate for both solving nonlinear static and dynamic problems in soil mechanics and are applicable for both finite- difference and finite-element computer codes. The book is intended primarily for civil engineers familiar with such traditional topics as strength of materials, soil mechanics, and theory of elasticity and structures, which are primarily concerned with the elastic behavior, but less familiar with the modern development of the mathematical theory of soil plasticity that is required for any engineer working under the general heading of "Nonlinear Analysis of Soil- Structure System". This book attempts to satisfy such a need in the case of soil medium. The scope of the book is indicated by the table of contents. It is divided into six chapters. Chapter 1 sets out initially to review the basic continuum theory of soil mechanics. This outline of theoretical fundamentals is presented in a form that is keyed directly to the main exposition of the book. Chapter 2 presents a review of the general techniques used in the constitutive modeling of soils based on the mathemat- ical theories of elasticity and plasticity. Chapter 3 is of introductory character and is restricted to the selected models from the theory of perfectly plastic solids. The purpose of this chapter is to introduce the basic concepts of soil plasticity to the reader that leads him to the front of present research of Chapter 4 on the theory of work-hardening plastic solids without much presupposing prior familiarity with the subject. Chapter 4 is the main theme of the book. It starts with a general description and a detailed derivation of the cap-type of constitutive models for soils that is generally used in practical geotechnical engineering. This is followed by a description of the general procedures for fitting the cap model to a given set of material properties. A numerical method for incorporating the cap model into computer programs is then presented. The model subroutine associated with the preceding numerical algorithm is included at the conclusion of this chapter. Numerical studies of some typical soil mechanics problems using the model subroutine are presented in Chapter 5. Recent advances on the cap-type of constitutive models for soils are summarized briefly in Chapter 6. NOTATION Stresses and strains σ,, σ , σ principal stresses, compressive stress positive 2 3 stress tensor sü deviatoric stress tensor σ normal stress shear stress τ jl\ = σο octahedral normal stress — 1 octahedral shear stress = \(σ + σ + σ) mean normal stress or hydrostatic pressure Ρ χ ν : s, s principal stress deviators 2 3 Si, ε , ε principal strains, compressive strain positive 2 3 strain tensor deviatoric strain tensor y engineering shear strain AV skk = ε, + ε2 + ε3 volumetric strain octahedral normal strain ^oct = ^0 7oct = 2X1 octahedral engineering shear strain Invariants I\ = σ\ + σ2 + σ3 = σα = first invariant of stress tensor Λ = jsijsy = *[(** - °? + {o - σ)2 + (σ - σ)2] + τ% + τ] + τ] y y ζ ζ χ : χ = second invariant of deviatoric stress tensor 7 = %SijS s = third invariant of deviatoric stress tensor 3 jk ki 3V3 J cos 30 = 3 = angle of similarity Θ defined in Figs. 2.1 and 2.2 A11 Ji = Jeveu = i[fe - e )2 + (e - ε)2 + (ε - ε)2] + s2 + ε2 + ε2 y y ζ ζ χ xy 7 = second invariant of deviatoric strain tensor Material parameters E Young's modulus v Poisson's ratio E K = — — = bulk modulus £ G = 7--: = shear modulus 2(1 + v) M = K + f G = constrained modulus c, φ cohesion and friction angle in Mohr-Coulomb criterion a, k constants in Drucker-Prager criterion k yield stress in pure shear Miscellaneous { } vector [ ] matrix C material stiffness tensor ijkl D material compliance tensor ijkl /( ) failure criterion or yield function x, y, z or x,, x , *3 cartesian coordinates 2 Sjj Kronecker delta W(£ij) strain energy density function Ω(σ,) complementary energy density function 7 Chapter 1 THE CONTINUUM THEORY OF SOIL MECHANICS 1.1 INTRODUCTION The most common structures which the geotechnical engineer may be required to design in soil and rock may be divided into three types of problems, namely: (1) the foundation and anchorage problems; (2) the slope and excavation problems; and (3) the earth pressure and retaining wall problems (Chen, 1975). Of all these problems, the engineer is required to make a two-stage process in his design operation: firstly, he has to determine the force field acting on the structural material due to the environmental loadings to which it may be subjected, and secondly, the reaction of the material to that force field, so that it is capable of withstanding the environmental conditions. The first stage involves an analysis of the stresses acting within the structural members; the second involves a knowledge of the properties of the struc- tural material, and in particular its mechanical properties which define the charac- teristic reaction of the material to the force field of its environment. These are typical stress analysis and design problems in soil mechanics, which is a branch of the science of solid mechanics. The word mechanics implies a mathematical formulation of the problem and of the basic equations to be used in its solution. All soil is discontinuous to some extent, but it becomes irrelevant at some scale of aggregation and a continuum view is necessary and valid. In the continuum theory of soil mechanics that includes the mathematical theories of elasticity, plasticity, and viscosity, the basic sets of equations are: (1) equations of equilibrium or motion; (2) conditions of geometry or compatibility of strains and displacements; and (3) material constitutive laws or stress-strain relations. Clearly, both the equations of equilibrium and the equations of compatibility are independent of the characteristics of the material. They are valid for metals, soils as well as rock or concrete materials. The differentiating feature of various material behaviors is accounted for in the material constitutive relationships which idealize the behavior of actual materials. Once the material stress-strain relationship is known, equations of equilibrium and of compatibility are used to determine the state of stress or strain when an idealized body is subjected to prescribed forces. The stress-strain or constitutive relationship for a material depends on many factors, including the homogeneity, isotropy and continuity of the body material, its reaction to loading over a period and the rate and magnitude of loading. Under general environmental and loading conditions, such a relationship can be highly nonlinear, anisotropic and irreversible, and the difficulties involved in stress analysis can be virtually insuperable, even with the present developments in computational 2 techniques like the finite-element method. If, however, the unit under load is large enough and the environmental and loading conditions are defined with certain limits, it is possible to assume that soils under load can be idealized and treated as linear elastic or nonlinear elastic, and perfectly plastic or work-hardening plastic materials for the purpose of stress and strain analysis, thus providing the necessary stress-strain relationship for the solution of an idealized soil mechanics problem. These elastic/ plastic stress-strain relationships and their applications to soils under the general heading of "Soil Plasticity: Theory and Implementation" are discussed in the follow- ing chapters. This chapter will present a simple, concise and reasonably comprehensive introduc- tion to the mechanics of soil that will set the stage for the subsequent chapters. The reader is assumed to be familiar with the more elementary aspects of stress analysis and some basic concepts of elasticity, viscosity and plasticity. A recent two-volume comprehensive book entitled "Constitutive Equations for Engineering Materials" by Chen and Saleeb (1982, 1986) may prove helpful in this respect as an introduction to constitutive modeling of engineering materials. Although the basic concepts of stress analysis and strain analysis can be found in a number of standard books, but for completeness, some of the developments involving stress and strain transformations in three dimensions are collected here in a form which is keyed directly to the main exposition of the present chapter and the chapters that follow. Since the mathematical theories of elasticity, viscosity, and plasticity all follow the same course, we therefore present the material in the same sequence. Firstly, the notions of stress and strain are developed (Sections 1.3 and 1.4); secondly, strain equations describing the geometry of deformation of a continuum, and stress equa- tions expressing the basic physical principles of equilibrium or of motion are set up (Sections 1.5.1 and 1.5.2) but in order to arrive at a system of equations which enable the state of stress and strain to be calculated, stress-strain relations must be obtained, which idealize the behavior of actual materials (Section 1.5.3). The form of such relations are not entirely arbitrarily. They must satisfy the basic principles of con- tinuum mechanics including the restrictions from thermodynamic laws. It is, of course, the simple mathematical expressions on soils that simulate the actual rela- tions between stress and strain for practical use (Section 1.6). 1.2 NOTATIONS For purposes of generalization, symbolic forms of the equations, using index notation and summation convention, have been used. The notations used in the text are those conventionally used in continuum mechanics. For computer programming purposes, matrix notations are most convenient. Thus, for specific material models, the numerical procedures for a solution have been illustrated in matrix notations in a cartesian coordinate system. In this book, we shall restrict ourselves to a right-handed cartesian coordinate system with a set of three, mutually orthogonal, x, y, and z axes. For future convenience, the 3 axes are more conveniently named as x x , and x for a general discussion, rather l9 2 3 than the more familiar x, y, and z for a specific engineering application. Herein, they will be used interchangeably. In the continuum mechanics, it is conventional to use tensile stress as a positive quantity and compressive stress as a negative quantity. Problems of engineering analysis and design in soil, however, are generally concerned with compressive stresses in most cases. For convenience, therefore, the continuum mechanics sign convention will be followed in this book for general discussion but will be reversed in the text, when a specific application is made. Notations and symbols in the text are explained when they first occur. Detailed discussion of index notation and summation convention can be found in the book by Chen and Saleeb (1982). 1.3 STRESSES IN THREE DIMENSIONS In soil mechanics theory, the soils or rocks are regarded as continua as a rule. This permits the use of the notions of stress and strain. The relationship between stress and strain in an idealized material forms the basis of the mathematical theories of elasticity, plasticity and viscosity which can in turn be applied to actual geotechnical materials to estimate stress or strain in a specified force field. An understanding of stress and strain and principles of stress and strain analysis is therefore essential to the engineer modeling the behavior of, and designing struc- tures in, soil and rock. These principles are summarized briefly here and in the following section. Details of this development are given elsewhere (see Chen and Saleeb, 1982). 1.3.1 Definitions and notations The analysis of stress is essentially a branch of statics which is concerned with the detailed description of the way in which the stress at a point of a body varies. In two dimensions, this involves only elementary trigonometry, and the use of Mohr's circle is found to be most convenient. In three dimensions, however, index notation is preferred for the calculation of stresses across any plane at the point. Herein, only the three-dimensional case will be worked out. The stress at a point P in a solid body may be obtained by considering a small plane area bA at random orientation with a unit normal vector n originating at P (Fig. 1.1). { Then, if bF is the resultant of all the forces exerted on bA, the limit of the ratio bFJbA { as bA tends to zero is called the stress vector T- at the point P across the plane whose x normal vector is «,·, that is: Tf = lim |ζ (1.1) δΑ-+ο oA The dimensions of T("] are force per unit area. The state of stress at the point P is completely specified or defined if we know all 4 ~*z x. Fig. 1.1. Stress vector T at point P on an area element with a unit normal vector /i,.. t the values of Γ!Λ) corresponding to various /?,. Since there are an infinite number of n through the point P, we shall have an infinite number of values of Τ(1° which, in t general, differ from each other. Thus, the infinite number of values of T^ are needed in order to characterize the state of stress at the point. It turns out, however, that these stress vectors are related to each other through Newton's law of motion or equi- librium. In fact, the value of T\n) for any «, can be calculated once the stress vectors T\]\ Tf\ and T{p are known for the three mutually perpendicular area elements whose normals are in the direction of the coordinate axes x, y, z or equivalently JC, , JC, and JC, respectively. This is known as Cauchy's formula, it has the simple form: 2 3 T\n) = T(Pn + Τψη + Γί3)/2 (1.2) x 2 3 where the stress vector T^ at the point P with a unit normal «,· = («!, n, n ) = (/, m, n) defined by its directional cosines is expressed as a linear 2 3 combination of the three stress vectors on the plane-area elements perpendicular to the three coordinates at the point. Therefore, it is clear that the three stress vectors Τγ\ Τγ\ Τ\3) define the state of stress at the point completely. Since T is a vector quantity, it can be more conveniently represented by three t components: a normal stress component and two tangential stress components. For example, the stress vector T\2) associated with the coordinate plane area y or x has the 2 three components: normal stress o or σ and shear stresses x and x or σ and σ y 22 yx yz 21 23 in the direction of the three coordinate axes y, x, and z or x , JC, , and x , respectively, 2 3 as shown in Fig. 1.2, or: T{? = (σ , σ , σ ) = (τ , σ, τ ) (1.3) 21 22 23 γχ γ γζ in which the index notation of σ for normal stress and σ , σ for shearing stresses, 22 21 23 and the engineering notation of σ for normal stress and r , x for shearing stresses will ν yx yz

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