Constitutive Modeling of Soils and Rocks E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 501

Veröffentlichungsjahr: 2013

Beschreibung

This title provides a comprehensive overview of elastoplasticity relating to soil and rocks. Following a general outline of the models of behavior and their internal structure, each chapter develops a different area of this subject relating to the author's particular expertise. The first half of the book concentrates on the elastoplasticity of soft soils and rocks, while the second half examines that of hard soils and rocks.

Leseprobe

Table of Contents

Preface to the English Edition

Preface to the French

Chapter 1. The Main Classes of Constitutive Relations

1.1. Introduction

1.2. The rheological functional

1.3. Incremental formulation of constitutive relations

1.4. Rate-independent materials

1.5. Notion of tensorial zones

1.6. The main classes of rate-independent constitutive relations

1.7. The main constitutive relations for rate-dependent materials

1.8. General conclusions

1.9. References

Chapter 2. Mechanisms of Soil Deformation

2.1. Introduction

2.2. Remolded soil behavior

2.3. Relationships between discontinuous and continuous medium

2.4. Natural soils

2.5. Conclusion

2.6. References

Chapter 3. Elastoplastic Modeling of Soils: Monotonous Loadings

3.1. Introduction

3.2. Elastoplasticity equations

3.3. Constitutive laws and laboratory tests

3.4. Characterization of natural cohesive soil behavior

3.5. Characterization of frictional soil behavior

3.6. Principles for the derivation of elastoplastic models

3.7. Three-dimensional aspect of the models and calculation of geotechnical works

3.8. Examples of perfect elastoplastic models

3.9. Examples of elastoplastic models with hardening

3.10. Conclusions

3.11. Notations

3.12. References

Chapter 4. Elastoplastic Modeling of Soils: Cyclic Loading

4.1. Soil behavior under drained loading

4.2. Isochoric triaxial tests

4.3. Modeling soil cyclic behavior

4.4. Models based on one or several independent yield surfaces

4.5. Models based on nested yield surfaces

4.6. Generalized plasticity models

4.7. Parameter identification for cyclic plasticity models

4.8. Conclusion

4.9. References

Chapter 5. Elastoplastic Behavior of Ductile Porous Rocks

5.1. Introduction

5.2. Review of typical mechanical behavior of porous rocks

5.3. Formulation of the constitutive model

5.4. Examples of numerical simulations

5.5. Influence of water saturation

5.6. Creep deformation

5.7. Conclusion

5.8. References

Chapter 6. Incremental Constitutive Relations for Soils

6.1. Incremental nature of constitutive relations

6.2. Hypoplastic CloE models

6.3. Incrementally non-linear constitutive relations

6.4. General conclusion

6.5. References

Chapter 7. Viscoplastic Behavior of Soils

7.1. Introduction

7.2. Laboratory testing

7.3. Constitutive models

7.4. Numerical integration of viscoplastic models

7.5. Viscoplastic models for clays

7.6. Conclusion

7.7. References

Chapter 8. Damage Modeling of Rock Materials

8.1. Introduction

8.2. Modeling of damage by mesocracks and induced anisotropy

8.3. Taking into account mesocrack closure effects: restitution of moduli and complex hysteretic phenomena

8.4. Numerical integration and application examples – concluding notes

8.5. References

Chapter 9. Multiscale Modeling of Anisotropic Unilateral Damage in Quasibrittle Geomaterials: Formulation and Numerical Applications

9.1. Introduction

9.2. Homogenization of microcracked materials: basic principles and macroscopic energy

9.3. Formulation of the multiscale anisotropic unilateral damage model

9.4. Computational aspects and implementation of the multiscale damage model

9.5. Illustration of the model predictions for shear tests

9.6. Model’s validation for laboratory data including true triaxial tests

9.7. Application on an underground structure: evaluation of the excavation damage zone (EDZ)

9.8. Conclusions

9.9. References

Chapter 10. Poromechanical Behavior of Saturated Cohesive Rocks

10.1. Introduction

10.2. Fundamentals of linear poroelasticity

10.3. Fundamentals of poroplasticity

10.4. Damage modeling of saturated brittle materials

10.5. Conclusion

10.6. References

Chapter 11. Parameter Identification

11.1. Introduction

11.2. Analytical methods

11.3. Correlations applied to parameter identification

11.4. Optimization methods

11.5. Conclusion

11.6. References

List of Authors

Index

First published in France in 2002 by Hermes Science/Lavoisier entitled “Modèles de comportement des sols et des roches Vol.1 et 2”

First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc.

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

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The rights of Pierre-Yves Hicher and Jian-Fu Shao to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

[Modeles de comportement des sols et des roches. English] Constitutive modeling of soils and rocks / Edited by Pierre-Yves Hicher Jian-Fu Shao.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-020-2

1. Engineering geology--Mathematical models. 2. Soil mechanics--Mathematical models. I. Hicher, Pierre-Yves. II. Shao, Jian-Fu. III. Title.

TA705.M6113 2008

624.1'51015118--dc22

2007046228

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-020-2

Preface to the English Edition

The French version of this book appeared in 2002 as part of the “Material Mechanics and Engineering” series. The objective of this book was to create as complete as possible a corpus of knowledge and methods in this field.

In designing this book on the mechanical behavior of soils and rocks, we gathered together a number of internationally known specialists, who each brought a significant contribution to the knowledge of the experimental behavior of these materials, as well as their constitutive modeling. Our goal was to cover as far as possible the theories at the basis of the different approaches of modeling, and also to address the most recent advances in the field.

In translating this book into English, we hope to make available to a wider scientific and engineering public the approaches and school of thought which have dominated the field of geomaterial mechanics in France over the past few decades. We have put together present-day knowledge of mechanical behavior and their theoretical bases in order to construct an original, analytical framework which, we hope, will give readers a useful guide for their own research. Most of the chapters have been updated in order to include the most recent findings on the respective topics.

Finally, we wish to dedicate this book to the memory of Professor Jean Biarez, who not only played a ground-breaking role in the history of soil mechanics in France, but remains a source of inspiration to many of us today.

Pierre-Yves HicherJian-Fu Shao

Preface to the French Edition

Soils and rocks possess a number of similar characteristics: both are highly heterogenous materials formed by natural grains. This alone gives them certain rheological features which distinguish them from other solid materials, such as a strongly non-linear character, a behavior which depends on the mean stress and shearing which induces volume variations, often dilatancy, which leads to unassociated plastic strains.

Soils and rocks can be studied at different scales. At the scale of one or several grains (from μm to cm), we can examine the discrete phenomena which govern the interactions between grains. They can be described using micro-mechanical models or analyzed in order to better understand the material behavior at a larger scale, typically the size of the material specimen: this approach corresponds to passing from a discontinuous to an equivalent continuous medium. Even though the size of the latter can vary, it has to be “sufficiently large” (typically from 1 cm to 1 dm) compared to the size of the material discontinuities in order to be representative of the equivalent continuous medium, whose behavior can be modeled by using certain concepts of continuous medium mechanics which ignore the notion of scaling in its basic equations.

However, some phenomena, such as the development of defects or cracks within the material specimen, are located at an intermediary scale, called the “meso” scale. It is thus necessary, in a constitutive model for continuous medium, to use scaling techniques in order to take into account these intermediary scales. This approach, still recent but potentially strong, can also be adapted to change the scale from the material specimen to the in situ soil or rock masses in geotechnical work modeling.

The constitutive models developed to describe the mechanical behaviors at the macroscopic scale can be roughly classified into two categories: those adapted to the behavior of “ductile” materials and those adapted to the behavior of “fragile” materials. The first category corresponds mainly to sandy or clayey soils, but also to soft rocks subjected to high confining stresses. The second category corresponds mainly to hard rocks, but also to certain soft rocks and highly overconsolidated clays subjected to small confining stresses. In ductile materials, the non-linear behavior is essentially due to irreversible grain displacements, which leads to a more or less significant hardening and to a pore volume change which induces volume changes at the scale of the specimen. In fragile materials, the non-linear behavior is due to the development of cracks, whose size may vary and whose direction depends on the principal stress directions.

In order to model ductile behaviors, plasticity (elastoplasticity or viscoplasticity) has shown to be an operational framework and the large majority of the constitutive models for soils and certain soft rocks belong to this category. However, for noncohesive soils in particular, the difficulty of characterizing an elastic domain, determining the plastic mechanisms (potential and yield surface) experimentally, has led to the development of specific constitutive models, whose structure can be defined as incrementally non-linear.

In order to model fragile behaviors, the damage mechanics framework has been used to propose constitutive models adapted to describing irreversible phenomena linked to the deterioration of certain physical properties. In particular, they can take into account a large amount of rock properties: irreversible strains, dilatancy, induced anisotropy, hysteresis loop during loading-unloading due to opening and closing of mesocracks and frictional mechanisms along closed mesocracks.

In intermediary materials, the non-linear behavior can be due to microstructural changes, associating damage and hardening phenomena. Models coupling plasticity and damage have been developed to take into account this type of behavior.

After a general presentation of the constitutive models and their internal structures, each chapter will give a brief description of the different approaches mentioned above by focusing on a given class of materials. The first three chapters are devoted to the elastoplasticity theory applied to soils and soft sedimentary rocks. An alternative approach is then presented by means of the so-called incrementally non-linear models. The time-effect in clayey soils is analyzed in the framework of viscoplasticity. The behavior of hard rocks is then studied in Chapters 8 and 9, through the use of the damage theory at different scales. The modeling of the poromechanical behavior is also introduced in order to take into account the hydromechanical coupling in saturated porous rocks.

As the validity of any given model lies in its capacity to reproduce the observed material characteristics, the authors have placed the experimental data, obtained mainly from laboratory testing on intact soil and rock samples, under special consideration. The final chapter is devoted to parameter identification procedures. This is an important topic when dealing with natural materials because, each site being different from another, accurate parameter identification is essential for the quality of geotechnical work calculations, which is the final goal of this modeling approach.

Pierre-Yves HicherJian-Fu Shao

Chapter 1

The Main Classes of Constitutive Relations1

1.1. Introduction

The study of the mechanical behavior of solid materials and its description by constitutive relations was for many years developed within the framework of isotropic linear elasticity characterized by Hooke’s law, plasticity characterized by the Von Mises, Tresca and Mohr-Coulomb criteria, and viscosity characterized in the linear case by Newton’s law. However, since the end of the 1960s, the development of more powerful numerical methods such as the finite element method and the use of high-performance computers has revived the study of material behavior, as it became possible to take into account a more realistic viscoelastoplastic modeling, albeit at the expense of much more complex formalisms.

Inside the three sets of equations defining a continuous medium mechanics problem, i.e. general equations (conservation equations), constitutive laws and boundary conditions, constitutive laws correspond to the more difficult part, particularly since the general framework in which the constitutive equations are inscribed remains often numerically imprecise. It is the comprehension of the absence of “physical laws” in this domain which gradually changed the designation of “constitutive laws” to “constitutive models”. The latter corresponds better to the objective of giving a mathematical form to the mechanical properties of materials, whose complexity has been demonstrated by the diversity of the experimental results.

During the last 30 years, a large variety of constitutive models have been developed and many workshops organized all over the world have shown that it is important for developers as well as users of models to be able to obtain guiding ideas and a general framework of analysis. The objective of this chapter is to try to formulate both of these.

This general framework will be more useable if it can be unified, and we intend to show that it can be applied to elastoplasticity as well to viscoplasticity or damage theory. We thus invite the reader to a wide presentation of constitutive relations for solid materials.

Two preliminary comments need to be made. First, we should explain why the chapter covers rheology in an incremental form. Two main reasons have made such an incremental presentation indispensable. The first is physical and is linked to the fact that, as soon as some plastic irreversibility is mobilized within the material, the global constitutive functional, which relates the stress state σ(t) at a given time t to the strain state ε(t) history up to this time, is in principle very difficult to formulate explicitly as this functional is singular at all stress-strain states (or more precisely non-differentiable, as will be shown). An incremental formulation enables us to avoid this fundamental difficulty. The second reason is numerical and stems from the fact that material behavior, and usually also the modeling of engineering works, exhibits many non-linearity sources which imply that the associated boundary value problem must be solved by successive steps linked to increments of loading at the boundary. Therefore, such finite element codes need to express the constitutive relations incrementally.

Our second comment concerns the use of incremental stress and strain rather than the stress and strain rates. Here also, it is the physical nature of the phenomena which determines our choice: in elastoplasticity, and more generally for all non-viscous behaviors, physical time does not play any role and, as a consequence, the derivatives with the physical time have no real meaning. Therefore, the incremental form appears to be intrinsically significant and can in fact be attached straightforwardly to the rate: the incremental strain is the product of the strain rate with the time increment, while the incremental stress is the product of Jaumann’s derivative of the stress tensor with the time increment. It is, however, incorrect to speak of stress and strain increments, since the incremental strain (for example) corresponds to a small strain variation only in the case of a sufficiently small strain.

This chapter begins with a traditional presentation of the rheological functional. We will show the limits of the functional expression and overcome this limitation by establishing the incremental rheological formalism. First, we will cover the case of non-viscous materials. The notion of “tensorial zones” will allow us to present the different classes of non-viscous models. Then, we will come back to the general case by considering models which take into account any kind of irreversibility.

1.2. The rheological functional

The basic concepts of continuous medium mechanics are taken for granted. The tangent linear transformation, characterized by the matrix of the gradient of the material particle positions, is assumed to describe correctly the material geometric deformation, even if some theories, called “second gradient theories”, consider that this first order approximation by the tangent linear transformation from the positions at a given time to the actual positions is not sufficient, and subsequently introduce second order terms [MUH 91]. We also assume that the constitutive law of a material element does not depend on the neighboring elements (some theories called “non-local theories” consider that the behavior of a basic material particle depends on a finite deformation field around that particle [PIJ 87]). These two hypotheses define a specific class of materials called “simple media” [TRU 74] for which we will develop a theoretical analysis.

The starting point of rheology is thus based upon a principle of determinism, which can be expressed as follows: if a given loading path is applied to a material sample, the material response is determined and unique, i.e., the principle of determinism applies only in conditions where there is uniqueness of the rheological response. Passing through a bifurcation point gives several possible responses. The choice of one of these responses is guided by existing imperfections which are not taken into account in the description of the material mechanical state or in the mode of loading application (control in force or in displacement, for example).

The first expression of the principle of determinism is obtained by stating that stress state σ(t) at a given time t is a functional of the history of the tangent linear transformation up to this time t. This implies that it is necessary to know the entire loading path in order to deduce the associated response path.

From a mathematical point of view, this is stated by the existence of a rheological functional F:

(1.1)

where E(t) is the strain part of the tangent linear transformation E at time t, also called the deformation gradient. Deformation gradient E is the Jacobian matrix of position f(X,t) of material point X at time t. The existence of such a functional, and not a function, is related to an essential physical characteristic: for irreversible behaviors, knowing strain ε(t) at time t does not enable us to determine the stress, and vice versa. For example, we can think of viscous or plastic materials where a given level of stress can be related to an infinite number of different strain states.

Since this chapter is devoted to the study of fundamental properties of constitutive relations, the general properties of rheological functional F need to be examined:

– isotropy of F: due to the principle of isotropy of space, F has to be an isotropic function of ε (if σ, ε and the internal variables are subjected to an equal rotation, F remains identical);

– non-linearity of F: the hypothesis of linearity for F is expressed by:

In such a case, the material response to a sum of histories will simply be equal to the sum of the responses to each history. This constitutes Boltzmann’s principle and is the basis of linear viscoelasticity theory, but it is not at all valid in the general case, as in elastoplasticity, for example, where, when we double the strain for example, the stress is obviously not doubled, due to the non-linear behavior.

In the general case, F must be studied in the framework of non-linear functionals:

  F is furthermore non-differentiable as soon as there is some plastic irreversibility. Owen and Williams (1969) showed in fact that the assumptions of non-viscosity and differentiability of stress functional F imply that there is no internal dissipation.

In other words, a non-viscous material whose constitutive functional is differentiable is necessarily elastic. Basically, this is due to the fact that, in plasticity, the tangent loading modulus is not equal to the tangent unloading modulus. Therefore, if we want to describe the behavior of anelastic materials by using a stress-strain relationship, this relation must be formulated using a non-linear and non-differentiable functional:

– degeneration of F: the only case of degeneration of functional F into a function corresponds to elasticity (possibly non-linear and anisotropic), where there is a one-to-one mapping between stress and strain.

Finally, if we want to describe irreversible behavior, we have to consider a nonlinear, non-differentiable functional, which, mathematically, is very difficult to use. We therefore need to study constitutive relations using an incremental formulation rather than a global one.

1.3. Incremental formulation of constitutive relations

We shall now introduce an incremental formulation using a second statement from the principle of determinism. The second principle of determinism, which can be called “in the small” to be distinguished from the first principle “in the large”, is obtained by stating that a small load applied during a time increment dt induces a small uniquely determined response.

As stated previously, this principle applies only if the uniqueness of the incremental constitutive relation is maintained. For bifurcation cases, the principle is no longer valid and the choice of the bifurcated branch will depend on the boundary conditions and material imperfections. In addition, the principle assumes implicitly that the loading rate is kept constant during the time increment (even if it can vary from one increment to another), which excludes dynamic loads due to shocks.

(1.2)

What are the properties of this tensorial function Fh?

The first comment concerns the fact that Fh depends on the previous stress-strain history. This history is generally characterized by some scalar and tensorial variables denoted by h which will appear as parameters in the previous relation. These parameters describe, as far as possible, the actual deformed state of the solid. According to various constitutive theories, they are sometimes called “memory variables”, “hardening parameters”, “internal variables”, etc.

Secondly, Fh must satisfy the objectivity principle, which means that Fh must be independent of any observer movement relative to the solid. Thus, Fh is an isotropic function of all its arguments: dε, dσ and also the state tensorial variables, which characterize its presently deformed state. However, if the material is anisotropic insofar as its mechanical properties are concerned, then Fh is an anisotropic function of dε and dσ.

Finally, Fh is essentially a non-linear function as long as there is some plastic irreversibility. If Fh is linear, we can write:

which is the general form of viscoelastic laws where M is the fourth-order elastic tensor and C the second-order creep rate tensor of the material.

This property of non-linearity for Fh is directly linked to the non-differentiability of rheological functional F, the property of differentiability of F being equivalent to the linearity of Fh.

In conclusion, relation (1.2) corresponds to the general incremental form of the constitutive relations. We will now distinguish between viscous and non-viscous materials in order to represent this incremental form more precisely.

1.4. Rate-independent materials

For non-viscous materials, the loading rate (characterized by time gradation on a loading path) has no influence on material constitutive behavior: a given loading path, followed at any given rate, gives the same response path. In other words, the behavior class considered is rate-independent. This restriction of the constitutive law implies that constitutive function Fh, which relates dε and dσ, is independent of time increment dt, during which the incremental loading is applied. Therefore, Fh is independent of dt and we can write:

(1.3)

or

(1.4)

The possibility of inversing G or H is linked to the uniqueness of the constitutive relations. This question will not be studied here; for more details see [DAR 94, DAR 95a].

From a mathematical point of view this independence of non-viscous behaviors on loading rates implies the following identity:

thus

(1.5)

which states that if the stress rate is multiplied by any positive scalar, the strain rate response is also multiplied by the same scalar.

This is the first property of G: G is a homogenous function of degree 1 in dσ with respect to the positive values of the multiplying parameter. This homogenity property must not be confused with that of “positively homogenous” functions, which is given by;

In addition to this property of homogenity of degree 1, functions G and H, as we have seen in general for function Fh, are non-linear and anisotropic in dσ (or in dε).

1.4.1. Non-linearity of G and H

If, in relation (1.3), dε is the response to an incremental loading dσ, the response to an incremental loading – dσ, following dσ, is not equal to – dε, because plastic irreversibility or damage takes place in the material. Therefore, G and H are necessarily non-linear functions of dσ and dε respectively, which implies that the principle of incremental superposition cannot be rigorously verified, except within the elastic domain, or more generally within a domain of incremental linearity of the constitutive model. Calculus shows, however, that the principle of incremental superposition can be roughly verified along “step-wise” paths, approaching a given loading path [DAR 95b].

1.4.2. Anisotropy of G and H

Following the same reasoning as for function Fh, we can deduce that G and H are anisotropic functions of dσ and dε respectively.

This anisotropy is directly linked to the geometrical meso-structure of the material, which is gradually modified by the strain (particularly irreversible) history. We have seen that this history can be characterized by scalar and tensorial state parameters.

In simple cases, this anisotropy is directly imposed by the choice of these state parameters. If we consider, for example, only scalar memory parameters (such as void ratio), defined independently of any frame, based on the objectivity principle functions G and H will be isotropic functions, which is not supported by experiments.

If we add one single tensor variable (such as the stress tensor) to these scalar memory parameters, G and H are orthotropic functions of dσ and dε respectively, the orthotropy axes being identified with the principal stress or strain axes. In this case, it means that G and H are invariant by symmetry with respect to any plane containing two principal stress or strain directions.

In the more general case of state variables with at least two second-order non-commutating tensorial variables, anisotropy is not defined. Orthotropy thus becomes a constitutive assumption, which must be considered as an approximation of the real behavior of the material for classes of loading in which stress and strain principal axes rotate.

1.4.3. Homogenity of degree 1 of G and H

Having described the three main properties of G, we will now focus on the first (homogenity of degree 1) to see the mathematical consequences of such a property. Let us for this purpose recall Euler’s identity for homogenous regular functions of degree 1, by writing it for a function of two variables:

(1.6)

where partial derivatives δf/δx and δf/δy are homogenous functions of degree 0.

In formulating constitutive relations, it is often more convenient to replace stress tensor σ and strain tensor ε, which are second order, by two vectors of IR6 defined in a six-dimensional related space. In this space, vectorial function G is written:

with summation on index β.

The partial derivatives of a homogenous function of degree 1 are homogenous functions of degree 0. Therefore, functions ∂Gα/∂(dσβ) depend only on the direction of dσ, characterized by the unit vector:

with:

Finally, we obtain:

(1.7)

or:

(1.8)

Equations (1.7) and (1.8) are the general expressions for all rate-independent constitutive relations. Constitutive tensors M and N also depend on state variables and memory parameters, which characterize the loading history. These two matrices are the gradient matrices of non-linear functions G and H, respectively. In that sense, they can be considered as tangent constitutive tensors and are therefore uniquely defined. However, it is possible to construct from them an infinite number of secant constitutive tensors by adding to the M or N lines the components of any unit vector perpendicular to dσ or dε, respectively.

Relations (1.7) and (1.8) will now allow us to propose a classification of all the existing rate-independent constitutive relations with respect to their intrinsic structure.

1.5. Notion of tensorial zones

First of all, we need to define the notion of a “tensorial zone” [DAR 82]. We will call a tensorial zone any domain in the incremental loading space on which the restriction of G or H is a linear function. In other words, the relationship between dε and dσ in a given tensorial zone is incrementally linear. If we denote the tensorial zone being considered as Z, the following definition implies:

In zone Z, the constitutive relation is characterized by a unique tensor Mz. If u belongs to Z, any vector collinear to u also belongs to Z for all real positive values. Therefore, a zone is defined by a set of half-infinite straight lines, whose apex is the same and is at the origin of the incremental loading space. Tensorial zones thus comprise adjacent hypercones, whose common apex is this origin. What does the constitutive relation become on the common boundary of two (or several) adjacent tensorial zones? If Mz1 and Mz2 are constitutive tensors attached, respectively, to tensorial zones Z1 and Z2, they must obviously satisfy the condition of continuity of the response to any loading direction u which belongs to the boundary between Z1 and Z2:

(1.9)

Relation (1.9) can be called a “continuity condition” for zone change. This condition prohibits, in particular, an arbitrary choice of the constitutive tensors in two adjacent tensorial zones.

Furthermore, we will see that conventional elastoplastic relations satisfy this condition by means of the “consistency condition”. This is also the case for damage models when they are built in a rigorous manner. On the other hand, hypoelastic models do not necessarily fulfil this condition, which has to be verified a posteriori. It has been proven that this is not the case for some of these models [GUD 79].

The “response-envelopes”, as proposed by Gudehus [GUD 79], constitute geometrical diagrams which completely characterize a constitutive relation at a given stress strain state after a given strain history. At this state, all the incremental loadings, having the same norm but oriented in all directions, are considered and all the incremental responses are plotted. The extremities of the response vectors form a hypersurface which is called the “response-envelope”. Figure 1.1 gives an example of an elastoplastic model in axisymmetric condition: the continuity of the response at the boundary of two tensorial zones appears well fulfilled. Figure 1.2 gives an example of a model with discontinuities, whereas Figure 1.3 corresponds to a continuous non-elastoplastic model.

In fact, the number of tensorial zones characterizes how a given model describes the irreversibility due to plasticity or damage, and the directional change of behavior, i.e. how constitutive tensor M (or N) evolves with the direction of loading u (or v). More precisely, the number of tensorial zones of a given constitutive model is an intrinsic criterion, which fully represents the model structure. Therefore, we have chosen this criterion to classify, in the next section, the different rate-independent constitutive models.

1.6. The main classes of rate-independent constitutive relations

1.6.1. Constitutive relations with one tensorial zone

The first class of relations that we are going to look at is related to the simplest assumption that there is only one tensorial zone. Therefore:

Therefore:

or:

The behavior is therefore entirely reversible (except possibly in the case of the existence of memory parameters h, but this corresponds rather to an artefact in hypoelasticity). As there is a unique linear relationship between dε and dσ (incremental linearity), we have here in this first class all the elastic laws, isotropic or anisotropic, linear or non-linear (in this last case, M and N depend on the actual state of stress).

The best way to reproduce an elastic behavior (without any internal dissipation) in a rigorous manner is to introduce an elastic potential V defined by:

As V is an exact total differential, we obtain the following expression:

Therefore:

using Schwarz’s identity. As a consequence, matrices M and N are symmetric and tensors C and D, defined by

have “major” symmetries

In the general case of non-linear elasticity, the existence of a potential also implies conditions of “integrability” [LOR 85], which have to be satisfied by the components of M and N. All these laws are called “hyperelastic”, while, in the absence of a potential, they are called “hypoelastic”. The hypoelastic models generate energy dissipation, and should thus not be used in practice, the behavior represented by these models being poorly identified.

1. 6. 2. Constitutive relations with two tensorial zones

In the presence of two tensorial zones, we can call one the “loading zone” and the other the “unloading zone”. We thus define two different behaviors (two different constitutive tensors), one representing the loading condition, and the other the unloading condition. Each matrix is attached to a different tensorial zone, these two tensorial zones being separated by a hyperplane in dσor dε space. A loading-unloading criterion, a linear and homogenous inequation in dσ or dε, allows us to discriminate between the two behaviors. The hyperplane equation corresponds, by construction, to the zero value of the loading-unloading criterion. The continuity condition at the crossing of the hyperplane gives a link to the two constitutive tensors and the hyperplane equation:

(1.10)

Numerous constitutive models follow these general rules and are therefore based on the definition of two tensorial zones. Their formalisms are basically similar, even if, sometimes, the detailed equations do not clearly show their fundamentally bilinear structure. These models are divided into three different families: elastoplastic models with one plastic potential, hypoelastic models with a unique loading-unloading criterion and damage laws. We will now examine them successively.

1.6.2.1. Elastoplastic models with one plastic potential

The first assumption concerns the additive decomposition of the incremental strain into an elastic part (reversible) and a plastic part (irreversible):

(1.11)

The plastic deformations exist only beyond a given limit surface, “the elastic limit”, which depends on the loading history and evolves due to the hardening created by plastic strains, as has been shown experimentally. Its equation is given by:

(1.12)

The loading condition is obtained by writing that the incremental stress is directed outwards from the elastic limit. The unloading condition is obtained if the incremental stress is directed inwards. It follows that:

(1.13)

The equation of the hyperplane, the border between the two zones in the dσ space, is thus given by:

When the elastic limit is reached, the direction of the incremental plastic strain is given by the flow rule which is often specified in terms of a plastic potential g(σ) as:

(1.14)

where dλ is an arbitrary scalar, whose value is determined by the consistency rule which mathematically expresses that, the state of stress reaching the elastic limit and the loading condition being maintained, the elastic limit follows the state of stress by hardening. Therefore, the consistency condition can be written as:

(1.15)

thus:

which gives the value of dλ:

(1.16)

Therefore, continuity condition (1.10) is verified:

∀dσ such that

because:

and:

with dλ proportional to (∂f/∂σ.dσ) from (1.16).

Therefore, the consistency condition in elastoplasticity theory allows us to verify the continuity condition in our general theory. The following equation gives the general mathematical form of the elastoplastic models with one single potential, in which the forms of functions f and g are not developed:

(1.17)

where Me is the elastic tensor and α is a scalar equal to 0 or 1:

or:

The hardening rule can be “isotropic” when f varies in an homothetic manner in the stress space, “kinematic” when f is translated in the stress space, or “rotational” when f can turn around the origin of the stress space (this last hardening has been introduced more recently, for example in [LAD 97]).

1.6.2.2. Hypoelastic models with one single loading-unloading criterion

In this type of model, there is no distinction between elastic and plastic strains and the notion of elastic limit surface is non-existent. The two tensorial zones are separated by a hyperplane, having a linear homogenous equation of the following form:

The loading zone corresponds to the loading condition:

(1.18)

with the associated constitutive tensor M+. The unloading zone is defined by:

(1.19)

with the associated constitutive tensor M-. Here there is a complete similarity with the elastoplasticity theory. Inequations (1.18) and (1.19) correspond to inequations (1.13), whereas tensors M+ and M-can be compared to tensors Mep and Me, respectively.

However, the continuity condition must be fulfilled a posteriori here. It can be written:

(1.20)

Condition (1.20) prevents any arbitrary choice for M+ and M-. It corresponds to the consistency condition in elastoplasticity theory. Tensors M+ and M-have to be dependent on memory parameters, which are linked to the stress tensor and to the plastic deformation, as in elastoplastic models.

Figure 1.2 shows the response envelopes obtained for the Duncan-Chang model, which is a non-linear isotropic hypoelastic model with a specific loading-unloading criterion. It is easy to verify that such a model cannot be continuous at the border between the two zones.

1.6.2.3. Damage models

The main assumption behind these models is that damaged material loses part of its mechanical elastic properties. This is shown by introducing a damage parameter D, which can be a scalar or a tensor. By introducing the damage parameter into an elastic formulation, we obtain a rheological functional which expresses the elastic behavior of a damaged material [MAZ 86]. This approach has proved to be very easy to use in monotonic loading, but not so much elsewhere, due to the nature of the functional.

In the framework of the incremental expression of the constitutive relations, damage models can be constructed by distinguishing a reversible damaged behavior from an irreversible behavior [MAZ 89]. A limit surface where damage appears is introduced:

(1.21)

When the incremental stress is directed towards the outside of this surface, additional damage is created, while the damage remains constant when the incremental stress is directed towards the inside of the surface. The loading-unloading criterion is therefore given by the sign of ∂f/∂σ.dσ:

(1.22)

When the limit surface is reached and the loading criterion is verified, damage is produced in the direction given by the damage evolution law:

(1.23)

thus

(1.24)

where

(1.25)

As in elastoplasticity, we must assure that the consistency equation is always verified.

The incremental strain can then be considered as the sum of two contributions: the first being the “degraded elastic” type, the second, irreversible, is induced by the increase in damage:

(1.26)

with Me(D) the damaged elastic matrix, and α a scalar equal to 0 or 1.

or:

(1.27)

We can see that, by using the notions of tensorial zone and the continuity condition, it is possible to give a unified presentation of elastoplasticity, hypoelasticity with loading-unloading criteria, and damage theory.

1.6.3. Constitutive relations with four tensorial zones

In order to describe more precisely the incrementally non-linear behavior of materials, particularly soils, it appears preferable to take into account two plastic mechanisms in the framework of elastoplasticity and two loading-unloading criteria for the hypoelastic constitutive relations. Each criterion can be associated, independently of the other, with a loading or unloading condition according to the direction of the incremental stress, which leads to the definition of four tensorial zones and four constitutive tensors. If the criteria are not independent, another theory, albeit one which is the same in principle, has to be built.

Figure 1.3.Response envelopes [GUD 79] in axisymmetric conditions for the octo-linear Darve’s model [DAR 82] with four tensorial zones (characterized by fs) for four different stress levels. The continuity of the response envelopes is verified

1.6.3.1. Elastoplastic models with double plastic potential

For each state of stress, following a given loading history, there are two yield surfaces which pass through this state of stress when the first plastic deformations develop:

(1.28)

For each surface, we define a criterion of loading-unloading defined by the following relation:

(1.29)

In the incremental stress six-dimensional space, we can therefore define four tensorial zones separated by two hyperplanes having the following equations:

(1.30)

When the loading condition for criterion i is fulfilled, there is a plastic deformation dεpi, whose direction is given by:

(1.31)

(1.32)

where:

The four tensorial zones can be defined by the values of α1 and α2 and the correspondent constitutive tensors are:

– tensorial zone I: (0, 0), Me

– tensorial zone II: (1, 0), Mp1e2

– tensorial zone III: (0, 1), Me1p2

– tensorial zone IV: (1, 1), Mp1p2

(1.33)

(1.34)

(1.35)

(1.36)

We can easily verify that conditions (1.33) and (1.36) are equivalent, as well as conditions (1.34) and (1.35).

Even if there are four limits, only two continuity conditions have to be satisfied, which correspond to the two consistency conditions. Due to the structure of elastoplastic relations, the two other relations are automatically verified.

1.6.3.2. Hypoelastic models with two loading-unloading criteria

For each state of stress, two loading-unloading criteria are defined in order to obtain a more progressive change of the constitutive tensor with the direction of dσ (with only one criterion, the tensor can have only two expressions, one corresponding to the loading condition, the other one to the unloading condition):

(1.37)

The two hyperplanes having for equation:

and:

define four tensorial zones in the dσ space. If we characterize each zone by the signs of expressions (1.37), we get the four different zones with the attached constitutive tensors:

– zone I: (+, +), M++

– zone II: (+, -), M+-

– zone III: (-, +), M-+

– zone IV: (-, -), M--

The four continuity conditions are given by:

(1.38)

We note that, due to the more general structure of the hypoelastic models, the four conditions (1.38) cannot be reduced to two, as in the elastoplastic models with double plastic potential.

1.6.4. Constitutive relations with n tensorial zones (n > 4)

All these models have an incrementally piecewise linear structure. In the case of elastoplastic models with m plastic potentials, corresponding to m loading-unloading criteria, each state of stress can be located at the same time on m elastic limits, to which we can associate m flow rules. The elastic limit corresponds locally to a vertex, which can be interpreted as the creation of a local singularity by the stress state encountering the elastic limit surface and deforming it locally. Some mechanisms can be associated (yield surface i is therefore identical to plastic potential i) or unassociated.

With each plastic mechanism is associated a loading-unloading criterion and two tensorial zones. The total number of tensorial zones is therefore equal to:

In the case of hypoelasticity, we can find identical structures by the direct introduction of m loading-unloading criteria. Obviously, the model has to be constructed so that it will respect the continuity condition at each change of tensorial zone.

An elastoplastic model has been constructed by using three deviatoric mechanisms and one isotropic mechanism, which correspond to 16 tensorial zones [HIC 85, HUJ 85]. A hypoelastic model with eight tensorial zones, the “octo-linear incremental model” [DAR 82], is presented in Chapter 6.

1.6.5. Constitutive relations with an infinite number of tensorial zones

A constitutive model can be considered to have an infinite number of tensorial zones, if each direction of dσ space is linked to a given tangent constitutive tensor which varies in a continuous manner with this direction. There are three different types of model.

Historically, the first models of this type were developed by Valanis [VAL 71]. They are written in the following form

(1.39)

where ζ plays the role of an “intrinsic time” for the material (which is at the origin of the designation “endochronic models” for this type of constitutive laws), and corresponds to the length of the path followed in the strain space:

The incremental non-linearity is thus given by the scalar ⏐⏐dε⏐⏐ which always remains positive, regardless of the direction of the incremental loading. Therefore, irreversible strains, independent of the strain rate since the relation is homogenous of degree 1, can be obtained without introducing either loading-unloading criteria or an elastic domain.

In this class of models, we find models with a non-linear incremental term of a tensorial nature, called “interpolation type” models since they are based on a nonlinear interpolation between given constitutive responses, the non-linearity being linked to the type of interpolation rule used. Among these models, we can cite the “incremental non-linear of second order” model [DAR 80, DAR 88], which has the following general form:

(1.40)

and some other models with different interpolation forms [CHA 79, DIB 87, ROY 86].

Models with a “bounding surface”, proposed by Dafalias [DAF 80], have led to the development of an incrementally non-linear relation called “hypoplasticity” [DAF 86] by creating a dependency of the incremental response on the direction of incremental stress.

The term “hypoplasticity” was also used by Kolymbas [KOL 77] and Chambon [CHA 94] for endochronic type models:

(1.41)

which are the object of specific developments unlike the initial model developed by Valanis.

1.6.6. Conclusion

Metallic mono-crystals have a finite number of defined directions of sliding planes; therefore, this specific microstructure is the basis of models which have a finite number of plastic potentials. In granular materials, however, plastic sliding can occur in any direction of the space along tangential planes at grain contacts. The mechanical behavior of such materials is thus more likely to be incrementally nonlinear.

In order to describe this non-linearity, i.e. the non-linear character of the relationship between the incremental strain and stress, we can use a piecewise linear formalism. Elastoplasticity theory guides us in the construction of such models by the definition of notions such as elastic limit, flow rule and hardening variables. However, calibrating these different elements can be a difficult task and it has been shown that their predictions along non-proportional loading could be of poor quality [WOR 84, WOR 88], mainly due to limitations induced by the difficulty of identifying elastic domains, flow rules and hardening variables.

Hypoelastic models with multiple loading-unloading criteria require fewer initial hypotheses and increase the choice of elaboration of the constitutive tensors, which must however verify a posteriori the continuity conditions. We can say that the modeler pays for his increased choice by a lack of constraint, which is certainly a life lesson of broad concern!

The incrementally non-linear formalism increases to an even greater extent the range of choices which is only limited by the homogenity condition. The advantage of the incrementally non-linear constitutive “interpolation type” relations is the reintroduction of some guides strongly linked to the material mechanical properties by proposing a non-linear interpolation between known behaviors in conventional loading conditions. We shall come back to this aspect in Chapter 6.

1.7. The main constitutive relations for rate-dependent materials

Viscosity plays a less central role in solid materials than in fluids. It completes an elastoplastic behavior, either by adding an additional incremental strain, or by influencing the plastic strain. These two hypotheses are the basis of the proposed classification [DAR 90].

1.7.1. First class of incremental strain decomposition

The hypothesis for the models belonging to this first class consists of assuming that the incremental strain can be divided into an instantaneous and a delayed part:

(1.41)

The instantaneous strain can be interpreted as being of an elastoplastic type and the delayed strain of a viscous type. Therefore:

(1.42)

In the previous section we have seen that dεep can be written:

Thus:

(1.44)

(1.45)

where M is the elastoplastic constitutive tensor and C the creep rate, both tensors being dependent on the loading history.

We should note that, for saturated materials, the stresses to be considered are effective stresses and that the considered creep tests are drained.

1.7.2. Second class of incremental strain decomposition

We assume in this case that incremental strain can be decomposed into a reversible and an irreversible part:

(1.46)

The reversible part can be considered as purely elastic and the irreversible part as viscoplastic:

(1.47)

If Me is the elastic tensor, the elastic law being incrementally linear, we have:

(1.48)

The viscoplastic strain is generally determined by using the viscoplastic potential theory developed by Perzyna [PER 63]:

(1.49)

Due to the fact that the actual stress state can be outside the actual elastic limit, scalar dλ is no longer determined by the consistency condition, but directly by the intensity of the viscoplastic strains.

1.8. General conclusions

Having reviewed all the main classes of constitutive relations, we must emphasize two main choices when constructing a constitutive model.

The first choice concerns the structure of the constitutive model, which should be better adapted to the problem to which it is addressed. This first choice has been widely discussed: is viscosity necessary or not? If yes, which type of viscous behavior is more pertinent: adding viscous behavior to elastoplastic strain or taking into account viscoplastic strains? How many tensorial zones are needed to describe the elastoplastic strains properly? Should we stay in the framework of elastoplasticity or damage theory, well defined and yet restrictive, or should we give ourselves more degrees of freedom by working in the framework of hypoelasticity or incrementally non-linear constitutive equations? Without a precise and solid knowledge of the material microstructure [DAR 05a, DAR 05b], the objective criteria for choosing one direction cannot be well defined, which can partially explain that the use of a given model in finite element analyses is rarely rationally justified. For a better understanding of the limitations of main constitutive models, the reader can refer to the results of two international workshops which took place in Grenoble in 1984 and in Cleveland in 1988, which were devoted to comparing model performances along various loading paths [WOR 84, WOR 88].

The second necessary choice is linked to the description of the loading history and leads to the following questions: which state variables? Which hardening variables? Which memory parameters? These questions have not been fundamentally addressed in this chapter, even if they play a central role in the quality of the model prediction. In the general case of an unproportional loading, our ignorance on this topic remains significant. The only insurance that we have is the major role played by induced anisotropy on the stress-strain response and therefore, the necessity of taking it into account in the modeling. Elastoplasticity also seems to require discrete memory parameters, which can characterize the last changes in the loading direction.

Finally, we should note that the topic concerning constitutive models cannot be developed further without taking into account the significant developments made in the field of discrete numerical simulations concerning granular materials. We also have to take into consideration the demand coming from practitioners concerning the need for parameter identification methods based on inverse analyses of in situ test results, and not only from laboratory testing, which are often more difficult and more expensive to perform.

1.9. References

[CHA 79] R. CHAMBON and B. RENOUD-LIAS, “Incremental non-linear stress-strain relationships for soils and integration by F.E.M.”, Int. Conf. Num. Meth. in Geomech., W. Wittke (ed.), publ. Balkema, vol. 1, 405-413, 1979.

[CHA 94] R. CHAMBON, J. DESRUES, W. HAMMAD and R. CHARLIER, “Cloe a new rate-type constitutive model for geomaterials. Theoretical basis and implementation”, Int.J. Num. Anal. Meth. in Geomech., 18(4), 253-278, 1994.

[DAF 86] Y.F. DAFALIAS, “Bounding surface plasticity I. Mathematical foundation and hypoplasticity”, J. Eng. Mech., 112(9), 966-987, 1986.

[DAF 80] Y. DAFALIAS and L. HERRMANN, “A bounding surface soil plasticity model”, Int. Symp. of Soils under Cyclic and Transient Loading, Pande and Zienkiewicz (eds.), publ. Balkema, vol. 1, 335-345, 1980.

[DAR 80] F. DARVE, “Une loi rhéologique incrémentale non-linéaire pour les solides”, Mech Res. Comm., 7(4), 205-212, 1980.

[DAR 82] F. DARVE and S. LABANIEH, “Incremental constitutive law for sands and clays. Simulations of monotonic and cyclic tests”, Int. J. Num. Anal. Meth. in Geomech., 6, 243275, 1982.

[DAR 87] F. DARVE, “L’écriture incrémentale des lois rhéologiques et les grandes classes de lois de comportement”, Manuel de Rhéologie des Géomatériaux, F. Darve (ed.), publ. Presses des Ponts et Chaussées, 129-152, 1987.

[DAR 88] F. DARVE and H. DENDANI, “An incrementally non-linear constitutive relation and its predictions”, Constitutive Equations for Granular Soils, Saada and Bianchini (eds.), publ. Balkema, 237-254, 1988.

[DAR 90] F. DARVE, “The expression of rheological laws in incremental form and the main classes of constitutive equations”, Geomaterials Constitutive Equations and Modeling, F. Darve (ed.), publ. Elsevier, 123-148, 1990.

[DAR 94] F. DARVE, “Stability and uniqueness in geomaterials constitutive modelling”, 3rdInt. Workshop on Localisation and Bifur. Theory for Soils and Rocks, Chambon Desrues Vardoulakis (eds.), publ. Balkema, 73-88, 1994.

[DAR 95a] F. DARVE, E. FLAVIGNY and M. MEGHACHOU, “Constitutive modelling and instabilities of soils behaviour”, Computers and Geotechnics, 17(2), 203-224, 1995.

[DAR 95b] F. DARVE, E. FLAVIGNY and M. MEGHACHOU, “Yield surfaces and principle of superposition revisited by incrementally non-linear constitutive relations”, Int. J. of Plasticity, 11(8), 927-948, 1995.

[DAR 05a] F. DARVE and F. NICOT, “On incremental non-linearity in granular media. Phenomenological and multi-scale views”, Int. J. Num. Analyt. Meth. in Geomech., 29(14), 1387-1410, 2005.

[DAR 05b] F. DARVE and F. NICOT, “On flow rule in granular media. Phenomenological and multi-scale views”, Int. J. Num. Analyt. Meth. in Geomech., 29(14), 1411-1432, 2005.

[DIB 78] H. DI BENEDETTO, “Modélisation du comportement des géomatériaux”, Doctoral thesis, INPG/ENTPE, 1978.

[GUD 79] G GUDEHUS, “A comparison of some constitutive laws for soils under radially symmetric loading and unloading”, Int. Conf. Num. Meth. in Geomech, W. Wittke (ed.), publ. Balkema, vol. 4, 1309-1324, 1979.

[HIC 85] P.Y. HICHER, “Comportement mécanique des argiles saturées sur divers chemins de sollicitation monotones et cycliques. Application à une modélisation élasto-plastique et visco-plastique”, Doctoral thesis, E.C.P., 1985.

[HUJ 85] J.C. HUJEUX, “Une loi de comportement pour les chargements cycliques des sols”, Génie Parasismique, publ. Presses des Ponts et Chaussées, 1985.

[KOL 77] D. KOLYMBAS, “A rate-dependent constitutive equation for soils”, Mech Res. Comm., 4(6), 367-372, 1977.

[LAD 97] P. LADE and S. INEL, “Rotational kinematic hardening model for sand. Part I Conception of rotating yield and plastic potential surfaces”, Comp. and Geotechn., 21(4), 183-216, 1997.

[LOR 85] B. LORET, “On the choice of elastic parameters for sand”, Int. J. Num. Anal. Meth. in Geomech., vol. 9, 285-292, 1985.

[MAZ 86] J. MAZARS, “A description of micro and macro-scale damage of concrete”, Eng. Fract. Mech., 25(5/6), 729-737, 1986.

[MAZ 89] J. MAZARS and G. PIJAUDIER-CABOT, “Continuum damage theory. Application to concrete”, J. Eng. Mech., 115(2), 345-365, 1989.

[MUH 91] B. MUHLHAUS and E.C. AIFANTIS, “A variational principle for gradient plasticity”, Int. J. Solids Struct., 28, 845-857, 1991.

[OWE 69] D.R. OWEN and W.O. WILLIAMS, “On the time derivatives of equilibrated response functions”, ARMA, 33(4), 288-306, 1969.

[PER 63] P. PERZYNA, “The constitutive equations for work-hardening and rate-sensitive plastic materials”, Proc. Vibrational Problems, 4(3), 281-290, 1963.

[PIJ 87] G. PIJAUDIER-CABOT and Z.P. BAZANT, “Non-local damage theory”, ASCE J. Engng. Mech., 113, 1512-1533, 1987.

[ROY 86] P. ROYIS, “Formulation mathématique de lois de comportement. Modélisation numérique de problèmes aux limites en mécanique des solides déformables”, Doctoral thesis, INPG/ENTPE, 1986.

[TRU 74] C. TRUESDELL, Introduction à ua Mécanique Rationnelle des Milieux Continus, Masson, 1974.

[VAL 71] K.C. VALANIS, “A theory of viscoplasticity without a yield surface”, Arch. of Mech., vol. 23, 517-551, 1971.

[WOR 84] GRENOBLE WORKSHOP, “Constitutive Relations for Soils”, Gudehus, Darve and Vardoulakis (eds.), publ. Balkema, 1984.

[WOR 88] CLEVELAND WORKSHOP, “Constitutive Equations for Granular Soils”, Saada and Bianchini (eds.), publ. Balkema, 1988.

1 Chapter written by Félix DARVE.

Chapter 2

Mechanisms of Soil Deformation1

2.1. Introduction

Mechanical soil behavior is generally studied within the framework of continuous medium mechanics, which provides a way of formulating constitutive models adapted to the specific nature of these materials. Given the extremely diverse nature of soils, it is necessary to investigate first of all the possibility of proposing models flexible enough to be adapted to a vast range of natural materials. Secondly, it is necessary to assess the procedure by which the parameters for a given soil are obtained. The field of soil mechanics has always favored the perfect elastic-plastic Mohr-Coulomb model. Numerous elastoplastic models have thus been developed, improving the representation of observable non-linear behavior. However, these models come up against a problem of parameter determination on account of the generally small number of experimental field data. From the outset of our study, we will present a number of experimental results which clearly show the soil’s mechanical behavior by drawing attention to the common aspects that do not rely on the nature of the constituents. This enables us to propose a mode of behavior applicable to a large range of materials. We will then discuss the possibility of relating the representative parameters of equivalent continuous medium to parameters representative of a discontinuous medium as well as the pertinent representation scales of this discontinuous medium. Our investigation is limited to monotonous axisymmetric compression loading on dry or water-saturated materials. Section 2.2 is devoted to remolded laboratory-prepared soils, considered as continuous materials. Section 2.3 studies the relationship between a discontinuous and equivalent continuous medium. Section 2.4