Computation of stresses and strains in saturated soil

COMPUTATION OF STRESSES AND STRAINS

IN SATURATED SOIL

COMPUTATION OF STRESSES AND STRAINS

IN SATURATED SOIL

PROEFSCHRIFT

ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool Delft,

op gezag van de Rector Magnificus, prof. dr. .1. M. Dirken, in hel openbaar ie verdedigen ten overstaan van hel College van Dekanen op dinsdag 17 september 1985

te 14.00 uur"

KAREL LUDWIG MEIJER

geboren ie Amsterdam, werktuigkundig ingenieur

TR diss

1450

Dit proefschrift is goedgekeurd door de promotor prof. dr. ir. A. Verruijt

Aan: Hennie Willemïen Lodewiek

VOORWOORD

Een groot deel van deze studie vond plaats in het kader van het "Toegepast Onderzoek Waterstaat", een door de Rij kswaterstaat gefinancierd speurwerk-programma.

Het onderzoek is uitgevoerd bij 'net Waterloopkundig Laboratorium. Ik wil beide genoemde instellingen bedanken voor de grote medewerking die is verleend bij de totstandkoming van dit werk.

CONTENTS

page

1 INTRODUCTION 1 1.1 Saturated soil 1 1.2 Background of the present research 1

1.3 Main limitations 3 1.4 Contents of this thesis 3

2 BIBLIOGRAPHY ON CONSOLIDATION 4

2.1 Introduction 4 2.2 From Terzaghi to Biot 4 2.3 Analytical solutions 6 2.4 Recent developments 8 3 MATHEMATICAL FORMULATION 11 3.1 Introduction 11 3.2 Continuum theory 12 3.3 Conventions of notation 14 3.4 Theory of mixtures 15 3.5 Scaling quantities 19 3.6 Definition of kInematical and dynamical quantities 23

3-7 Constitutive equations 28 3.8 Motion of pore water 47 3.9 Motion of the soil matrix 50

3.10 Continuity 52 3.11 Permeability-porosity relations 56

3.12 Infinitesimal strain 57 3.13 Initial and boundary conditions 59

3.14 Plane strain 60 3-15 Weak formulation 61 3.16 Translating coordinate system 65

CONTENTS (continued)

NUMERICAL APPROXIMATIONS 69 4.1 Introduction 69 4.2 Basic equations in vector notation (plane strain) 69

4.3 Galerkin method 75 4.1 Interpolation functions 78 4.5 Interpolation and accuracy 82 4.6 Implementation of boundary conditions 84

4.7 Time integration 86 4.8 Integration of constitutive equations 91

4.9 Stability of global time integration 9 2

4.10 "t = 0" problem 93 4.11 Linear equations 100 4.12 Integrations in the spatial domain 103

COMPUTATIONS WITH WAZAN 108 5.1 Introduction 108 5.2 Numerical experiments on " t * 0 " problem 109

5-3 Consolidation in a cylinder 112 5.4 Comparison of finite and infinitesimal strain

consolidation 11 6

5.5 One-dimensional consolidation 128 5.6 Pore pressure generation beneath a moving load 135

5.7 Moving slope problem 137

LIST OF GLOBALLY USED SYMBOLS ^

REFERENCES 146

SAMENVATTING 159

CURRICULUM VITAE 160

1. INTRODUCTION

1.1 Saturated soil

The earth beneath our feet, the upper part of the earth's crust, the lower limit of our visual field, is an entity with which each individual is con­ fronted in a more or less painful way soon after his first steps. As we get older and stay on our feet more frequently the notion "ground" will have more meanings. Probably the idea of the earth as the mother of all that lives and grows plays a role in all civil i zat ions. The earth as one of the four elements has dominated the natural philosophical world-picture since Empedocles in about 460 B.C.

A few abstractions that are related to the notion "ground" are for ex­ ample: the place of origin, the producer of food, ground as a foundation or as a mineral, the ground as scenic beauty, ground as material that has to be transported by dredging and ground as a depository for matter that we do not want to see anymore but which nevertheless exists.

In this thesis ground is considered in a narrower sense, that is as a porous mass of soil wherein the pores are (nearly) completely filled with water. Although applications of the theory for rock, clay, peat, etc. may easily be indicated, it is primarily saturated sand of which one has to think when soil is mentioned in the subsequent text.

1.2 Background of the present research

The work reported here stems from research done at the Delft Hydraulics Laboratory for the solution of some dredging problems. This work was con­ tinued in a project "Fast deformations in saturated soil", initiated in the framework of "Toegepast Onderzoek Waterstaat ( T O W ) " , a co-operative engagement between the Ministry of Public Works, the Delft Soil Mechanics Laboratory and the Delft Hydraulics Laboratory.

must I do so that the soil will not collapse. In the dredging business, the conditions under which the soil looses its coherence has to be known. An important factor herewith is the rate of loading and the rate of defor­ mation. Apart from the acceleration forces, which often have a negligible magnitude and intrinsic deformation rate dependent soil properties, the main effect of the loading rate is caused by the flow of pore water through the porous soil matrix.

The TOW project was initiated because knowledge of the deformation of saturated soil has not only significance for the dredging world but also for those who are involved with building constructions on the sea bottom. The goal of the TOVJ-project was formulated as: to increase the knowledge of the deformation behaviour of saturated soil.

The following strategy was chosen to achieve this:

- Knowledge must be gained about the elementary stre33-deformation rela­ tions of soil.

- An instrument must be made to translate these elementary relations into macroscopic observable phenomena. Such an instrument is for example a computer programme that couples stress-strain relations to equilibrium equations and to boundary and initial conditions.

When concretely elaborated this boils down to:

- Developing a model for the elementary stress-strain behaviour of soil.

- Developing a computer programme in which these relations are used to compute concrete (non-homogeneous) cases.

The first point of investigation mainly took place at the Delft Soil Me­ chanics Laboratory, the second point was concentrated at the Delft Hy­ draulics Laboratory in the person of the author. Part of this thesis has appeared In a TOVJ report (Meijer, 1981).

3

-1.3 Main limitations

The ultimate goal of the research was the construction of a tool to ana­ lyse foundation and dredging problems in practise. This means the develop­ ment of a computer model and so we are confronted with the limitations of the momentary level of hardware technique. He may say that, although it is technically possible at this moment to analyse numerically the full three-dimensional problem, it is commercially attainable only in a very few cases. Frontiers nowadays are moving quickly, however. We may mention the expanding supply of supercomputers and the coming fifth generation com­ puters. In the underlying research we restricted ourselves to the numeri­ cal analysis of two-dimensional, viz. plane strain proolems.

The second limitation is our ignorance of acceleration forces. This means that transients caused by, for example, seismic events are excluded from the field of application. On the other hand, subjects like the initiation of land slides, stability analysis or suction of sand, where dynamical phenomena are of minor importance, are covered by this approach.

The influence of time is only expressed via the boundary conditions and by the viscous flow of the pore fluid.

1.4. Contents of this thesis

In Chapter 2 a resume is given of highlights in the literature related to the underlying research,

Chapter 3 contains a description of the theoretical foundations and the derivation of the equations which govern the problem mathematically. The numerical method, that is concretized in a computer programme called WAZAN, is worked oat in Chapter 4.

In Chapter 5 a number of problems is presented which are analysed with the WAZAN programme. Compar isons are made w ith analytical solutions and meas­ urements .

A list of globally used symbols and a bibliography are present at the end of the text.

t

-2. BIBLIOGRAPHY ON CONSOLIDATION

2.1. Introduction

In this chapter a global historical resumé will be given

of the develop­

ment of the mathematical theory of consolidation from Terzaghi until now.

What in soil mechanics is called consolidation, is the time dependent

process of settlement of founding soil after a change in the external

loading. This process is mainly governed by the flow of pore water though

the pores of the soil matrix. The time scales of consolidation vary from

decades for clays with very low permeability to seconds for coarse-grained

sandy soil.

A mathematical approach of the phenomenon was given by Terzaghi about

sixty years ago. Terzaghi considered the simplified one-dimensional pro­

blem.

Biot in 1941 initiated the development to a more general treatment of the

quasi-stationary behaviour of saturated soil. Biot also analysed transient

pore-water interactions (Biot, 1956a and 1956b).

In this chapter we will consider the literature on consolidation. The era

from Terzaghi until now may be roughly divided into three periods: f^om

Terzaghi to Biot (19-41 ), the period before the introduction of the com­

puter and the period from 1970 until now, where the application field of

numerical techniques has increased tremendously.

2.2. From Terzaghi to Biot

A starting point in the development of the present theory is the paper

"Die Berechnung der Durchlassigkeitsziffer des Tones aus dem Verlauf der

hydrodynamischen Spannungserscheinungen" by Karl Terzaghi (1923).

In this publication the differential equation for the pore pressure is

given for a one-dimensional situation.

As it may be seen from the title, the theory was actually set up to deter­

mine the permeability of clay from a simple test. The differential equa­

tion obtained was identical to the heat diffusion equation for which se­

veral solution techniques were available. In the book "Theorie der Setzung

von Tonschichten" (Terzaghi and Fröhlich, 1936) solutions are presented

for the one-dimensional consolidation equation.

One of the Implicit assumptions of the theory is that the porosity change

of the clay is proportional to the change of stress, diminished by the

change of pore pressure. This effective stress principle was explicitly

posed in 19 36 (Terzaghi, 1936) and declared also applicable to shear de­

formations. The effective stress principle states that in a homogeneous

stress situation a change of deformation is independent of the pore pres­

sure. This assumption holds for most kinds of soil by a large degree of

approximation.

A three-dimensional generalization of the Terzaghi theory is given by

Rendulic (1936) . This generalization leads to the two-dimensional diffu­

sion equation. The basic assumption is that the total stress (that is the

sum of the effective stress and the pore pressure) remains constant during

consolidation (see Section 3-12). In 1941 Biot formulated a consistent

three-dimensional theory which may still be seen as the basis of later

developments (Biot, 1941). The assumptions in the original theory were:

a. Material isotropy,

b. linear elastic grain skeleton,

c. small deformations,

d. pore water flow according to Darcy's law,

e. no acceleration forces.

It is noteworhty that Biot does not use the concept of effective stress in

this paper

(nor in subsequent papers). Implicitly, however, a kind of

effective stress is to be recognized, in which the intrinsic compressibi­

lity of the grains Is accounted for (see Section 3-7.2).

Analytical solutions of Biot's equations are known for a number of simple

initial-boundary value problems. These problems will be elaborated in

Section 2.3.

In order to obtain solutions for practical situations with a general geo­ metry , numerical methods are Indispensable. Apart from this, the often very complicated analytical expresssions cannot be evaluated without ad-vaneed numerical techniques.

The numerical method mostly used is a variant of the finite element m e ­ thod. Other methods, for example, finite difference methods or boundary element methods are used much less.

Although different theoretical conceptions can be used to construct finite element models, like variational principles or weighted residual methods, the final numerical formulation is often very similar.

Methods that we may call hybrid methods, are for example used by Booker ('973) and Booker and Small (1975). There the time coordinate in the con­ solidation equations is eliminated by means of Laplace transformations. The resulting time independent partial differential equations are disere-tized with the finite element method, whereafter a numer ical back trans­ formation is performed to the time domain. Despite some advantages of such methods, direct numerical treatment of the governing continuum equations gives more possibilities to abandon the restrictions, mentioned above, Only in one-dimensional cases, where the theories of Biot and Terzaghi coincide, is it possible to construct solutions for some distinct non­ linear problems.

2.3. Analytical solutions

Several techniques have been applied to construct analytical solutions of Blot's equations (Fourier transforms, displacement and stress functions, operator calculus).

Solutions have been obtained for consolidation in a cylinder (axially and radially), a sphere and a half-space (both plane-strain and axial symme­ tric) and some other simple configurations. The following authors can be listed: Biot (1941a), Biot and Clingan (1941), Biot and Clingan (1942), De Josselin de Jong (1957), McNamee and Gibson (I960), Heinric'n and Desoyer (1961), De Leeuw (1964), De Leeuw (1965) and Sanyal (1972).

7

-C o n s o l i d a t i o n and the g e n e r a t i o n of pore p r e s s u r e beneath moving l o a d s a r e

t r e a t e d By Schiffman and Fungaroli (1973) (moving r e c t a n g u l a r load d i s ­

t r i b u t i o n ) and by Madsen (1978) and Yamamoto, Koning, S e l l m e i j e r and Van

Hijum (1978) (pore p r e s s u r e s due to wave a c t i o n ) .

Schiffman, Chen and Jordan (1969) compare a n a l y t i c a l s o l u t i o n s of the

e q u a t i o n s of T e r z a g h i , Rendulic and Biot r e s p e c t i v e l y . The f o l l o w i n g pu­

b l i c a t i o n s on a n a l y t i c a l s o l u t i o n s of B i o t ' s e q u a t i o n s d e s e r v e e x t r a a t ­

t e n t i o n .

" C o n s o l i d a t i o n des s o l s " (Mandel, 1953)

I t i s observed h e r e t h a t a f t e r l o a d i n g , the pore p r e s s u r e may undergo a

l o c a l i n c r e a s e before i t d i f f u s e s . This is in c o n t r a s t to s o l u t i o n s of the

o r d i n a r y heat d i f f u s i o n e q u a t i o n .

"A comparison of the t h r e e - d i m e n s i o n a l c o n s o l i d a t i o n t h e o r i e s of Biot and

T e r z a g h i " (Cryer, 19 63)

Here the phenomenon of the temporary r i s e of pore p r e s s u r e i s r e d i s ­

c o v e r e d . In a d d i t i o n i t is e s t a b l i s h e d t h a t the Biot e q u a t i o n s c o i n c i d e

with the T e r z a g h i R e n d u l i c e q u a t i o n s if P o i s s o n ' s r a t i o i s 0 . 5

The e f f e c t of nonmonotonic behaviour of the pore p r e s s u r e a f t e r a s t e p

wise l o a d i n g , which i s p r e d i c t e d by B i o t ' s theory but not by the T e r z a g h i

-Rendulic t h e o r y , i s u s u a l l y c a l l e d the Mandel-Cryer e f f e c t . This e f f e c t i s

a l s o e x p e r i m e n t a l l y demonstrated by Gibson, Knight and Taylor (1963) and

by V e r r u i j t ( 1 9 6 5 ) . I t i s a l s o observed in pumping experiments in a q u i f e r s

( V e r r u i j t , 1969).

In " C o n s o l i d a t i e in d r i e d i m e n s i e s " (De J o s s e l i n de Jong, 1963) t h e

d i f f e r e n c e s between the t h e o r i e s of Biot and T e r z a g h i - R e n d u l i c a r e f u r t h e r

a n a l y s e d .

2.4. Recent developments

Since the beginning of the use of computers for solving consolidation pro­ blems, at the end of the sixties, some trends can be recognized in the development of programmes. Leaving one-dimensional consolidation out of consideration, the development started with finite element models for linear-elastic plane-strain consolidation based on Biot'3 theory. Christian (1968) analyses the time independent undrained problem with the finite element method. An extension with the consolidation phase is pu­ blished by Christian and Boehmer (1970). Furthermore the following authors all assume linear-elastic soil properties: Sandhu and Wilson (1969), Hwang, Morgenstern and Murray (1970), Yokoo, Yamagata and Nagaoka (1971), Verruijt (1972), Ghaboussi and Wilson (1973) Valiappan, Lee and Boon-lualohr (i 97^ >, Bugrov (1975) and Krause (1978).

A next step was the introduction of non-linear consitutive relations for the soil matrix, such as non-linear elasticity and elasto-plasticity. Small, Booker and Davis (1976) derived incremental euqations governing the consolidation, by differentiation of the Biot equations. This formulation makes it possible to handle non-linear stress-strain relations. Some ex­ amples were given of computations of elasto-plastic soil made with a fi­ nite element programme. Kenter and Vermeer (1978) reported non-linear consolidation calculations, made with the CASCO and ELPLAST programmes. Osaimi and Clough (1979) used Duncan and Chang's (1970) non-linear stress-strain relations. Meijer (1979) uses non-linear constitutive equations in the computation of a moving slope.

The latest evolution in computer programmes is the relaxation of the clas­ sical assumption of infinitesimal deformation gradients. Carter, Small and Booker (1977) extend Biot's consolidation theory so that it is valid for situations with finite strain. They assume a linear relationship between the Jaumann derivative of the effective stress and the deformation rate. A generalization to elasto-plastic soils is given by Carter, Booker and Small (1979).

Prévost (1980, 1981) and Meijer (1981) give similar analyses concerning

9

-the finite strain consolidation of non-linear porous materials.

A number of other authors also gave formulations for the motion of two-phase materials like saturated soil, without going into numerical solution techniqes (Biot, 1963 and 1972, Kenyon 1976, Becker and Mclvor, 1 978 , Ahmadi, 1980, Bowen, 1 9 8 2 ) .

Most of them use the framework of mixture theory in deriving the equa­ tions. At this point it is useful to notice which differences exists b e ­ tween the approaches of Carter et al. , Prévost and Meijer.

- Meijer and Prévost formulate the continuum equations in Lagrangian coordinates, whereas Carter et al. use Eulerian coordinates. The use of Eulerian coordinates leads to simpler forms of the continuum equations. The numerical treatment, however, is seriously hampered by the fact that the computational domain is dependent on the solution being approxi­ mated. Either one uses the "old" configuration when going from one time level to the next and so treating the geometrical non-linearity "expli­ citly", or the element discretization has to be adapted Lteratively within one time step. (These matters will be made more clear in the next chapters).

- Prévost and Carter et al. assume a constant permeability. In the undet— lying thesis a permeability coupled to the porosity is assumed. For cases of anisotropic permeability we pose (as do Carter et al.) that the principal axes of the permeability tensor must rotate according to the rotation of the local deformation gradient.

- Carter et a l . and Prévost assume incompressible grains and an incompres­ sible pore fluid. In our thesis, the consequences of compressible grains are investigated. In addition the finite strain consolidation theory is extended for situations where air in the form of small bubbles is pre­ sent. On the other hand we may say that the infinitesimal consolidation theory of Verruijt for pore water containing air is extended for the finite strain case.

The numerical integration of the ultimate system of ordinary differen­ tial equations is solved in a different way by Carter et al., Prévost and Meijer.

Finally, Meijer also sets up consolidation equations in a moving coor­ dinate system and the computer programme is provided with the possi­ bility of using moving coordinates.

3. MATHEMATICAL FORMULATION

3.1. Introduction

In this chapter the basic principles and assumptions are given that lead to a mathematical model for the dynamic and kinematic behaviour of satu­ rated soil.

The principal step in constructing the model is the idealization of the real soil as a continuum. In this approach the microstructure of matter is negotiated and only imaginary entities like displacements of points, strains and stresses in points are considered as (almost) everywhere con­ tinuous functions of the space and time coordinates. In Section 3.2 some deliberations are given concerning the continuum theory.

In Section 3A attention is given to the continuum theory of mixtures, the theory that seems to be adequate in describing a two (soil-water) of three (soil-water-air) phase material,

The fundamental quantities which play a role in the deformation of a soil-water mixture are:

- Displacements of points of the soil skeleton - Pore water velocity

- Effective stress - Pore pressure

In Sections 3.7, 3-8, 3.9 and 3.10 the equations which couple these quan­ tities are elaborated, i.e.:

- Constitutive equations - Momentum equations - Continuity equations

I t is assumed t h a t a l l p ^ o e s s e s are i s o t h e r m a l and thermodynamical ef­

f e c t s a r e not c o n s i d e r e d .

To help judge the relative importance of the various terms in the equa­ tions some relevant dimensionleas quantities are defined in Section 3-5. By making some assumptions the equations can be simplified to equations of which solutions, or at least the character of the solutions, are known (Section 3 . 1 2 ) .

Boundary and initial conditions are dealt with in Section 3-13. In Section 3.14 the assumptions are given to reduce the general three-dimensional formulation to the plane strain situation. The numerical method applied requires a formulation of the field equations in the form of a functional, therefore the plane strain equations are written in a weak form in Section 3.15.

For some applications where situations in the vicinity of a moving bounda­ ry load are studied, the option of moving coordinates described in Section 3.16 can be useful.

Finally the basic equations are summarized in Section 3.17.

3.2 Continuum theory

Although the adoption of a continuum model as an idealization of matter is more or less self-evident in some branches of applLed mechanics, this is less obvious in soil mechanics.

Soil generally consists of a heterogeneous mixture of several granular components having aggregate dimensions that come up or even exceed the spatial scale of external load variations. (As an extreme example we may think of a weir consisting of basalt b l o c k s ) . In the latter case, of cour­ se there is no question of adequate modelling by a continuum model. A continuum is a mathematical notion whereby points, lines and surfaces are respectively defined with no dimensions at all, no dimensions in two directions and no dimension in one direction. In addition, mathematically defined infinitesimal quantities are used. No physical reality can be described in any aspect by these abstractions.

The justification of a continuum approach lies in the fact that the

conti 13 conti

-nuously distributed field quantities can be identified, after integration in space and time, with macroscopically measurable entities. The volume and the time interval of integration must be such that a consistent iden­ tification is possible. This means that a sufficient number of particles must be present in the volume element to ensure reproduceable spatial averages. The time interval must be so large that accidental variations will be averaged out. Bear (1972) gives some theoretical background on the applicability of continuum theories for porous media.

The advantage of a continuum model for soil is the simplicity of the for­ mulation of the balance equations (of momentum and m a s s ) ; the drawback is the complexity of the constitutive equations which describe the material behaviour. It will b e clear that even for sand consisting of grains of one single material, the response to a change of the external load is strongly dependent on the mutual position of the particles and on the actual dis­ tribution of forces between them.

At present the tendency of a growing number of articles concerning conti­ nuum modelling of soils at a lower level can be observed.

In these theories the particles as such are regarded as a continuum with simple constitutive relations like linear elasticity or infinite stiff­ ness. The global conservation relations couple contact forces and the motions of grains as a whole.

As the main interest generally lies in the global properties and not in the motion of each individual grain, it is attempted via this lower level modelling by using statistical techniques, to obtain insight and to disco­ ver regularities in constructing constitutive equations that can be used in the higher level continuum theories.

In this study the higher level continuum approach is applied and the con­ stitutive equations are assumed to be known. It must be kept in mind how­ ever that variations computed on the basis of this theory at most have a meaning if they extend over a multiple of the aggregate size.

3.3 Conventions of notation

The symbols applied in this thesis will be defined where they appear in the text for 'the first tine. A number of quantities are referenced by the same symbol throughout this work. They are given in the "List of globally used symbols" -at the end of this book.

A few general conventions have been adopted throughout the text. The fol­ lowing comments are made concerning the usage of indices:

a) With the exception of the cases d-e below, indices (sub- and super­ scripts) serve to distinguish between different usages of a physical quantity. Especially the characters a, s and w refer to different mate­ rials v i z . to air-inclusions, soil grains and pore water respectively. An arbitrary material will also be indicated by a or 6 (in Section

3.1).

b) Superscripted asterisks mark scaled quantities (see Section 3 . 5 ) .

c) Primes and bars indicate a different but related physical quantity (for example k' is the permeability and k the specific permeability).

d) In this thesis only Cartesian tensors will be used in the continuum description of physical quantities. Where necessary, covariant (lower) indices refer to the coordinate direction (1 to 3 ) . Only the characters i, j , k, 1, n, m, p and q will be used for this purpose.

e) The use of a superscript as an exponent will be clear from the context.

Material points of the soil matrix are identified by X, which is the coor­ dinate vector at a reference point at time tp. Subsequently a point X has a coordinate vector x on time t. The description of a continuum where quantitits are regarded as a function of X, is called a Lagrangian or

15

-material description. In the Eulerian or spatial description things are supposed to be a function of x.

Derivatives with respect to X* and K, are indicated respectively by the subscripts ,i and ;i.

Material time derivatives, i.e. derivatives with respect to time, holding X constant are denoted by a dot above the symbol. A bar between the symbol and the dot occasionally indicates which symbols are affected by the dot. In tensor Lai expressions the summation convention of Einstein is in ef­ fect. The notation f[g] will be used for a tensor valued function f that is linear in g.

The Greek capital A preceeding a symbol denotes an incremental quantity.

3.1 Theory of mixtures

In this section saturated soil will be viewed from the optics of the theory of mixtures.

The foundations of the modern continuum theory of mixtures is laid down by Truesdell (Truesdell and Toupin, i 9 6 0 ) . Also reference can be made to Truesdell and Noll (1965, p. 130) and Bowen ( 1 9 7 6 ) . The theory is based on the assumption that each spatial element can be simultaneously filled by several components which are all regarded as a continuum. This theory seems to be the natural starting point for the continuum description of saturated soil. In this section only the basic equations are given as far as they are relevant to an isothermal situation in a two-phase medium subject to negligible accelerations.

Later on in Chapter 3 the resultant equations are examined more closely and linked to the classical equations of consolidation.

As in the continuum approach of a single material, the field equations for a mixture are the balance equations of momentum and mass and the balances of thermodynamical and magnetodynamieal quantities. These latter quanti­ ties are irrelevant for saturated soil. In addition the materials are characterized by their constitutive equations.

the appearance of terms in the partial balance equations which describe the influence of the other constituents.

Quantities that are related to constituent a are indicated in this section by an upper index a. Two phases are considered, the soil matrix, occasio­ nally indicated by an upper index s and the pore fluid indicated by w. We define the following quantities s

n , the relative volume occupied by const ituent a. By definition:

p , the partial density of constituent a:

p is the intrinsic density of component p, the density of the mixture,

( 3 . 2 )

a , the partial tensor of contact stresses.

o , the partial tensor of contact stresses averaged over an elementary

volume of mixture.

-a a a , „ , ,

o = n o (3, i\)

o , the total Cauchy stress:

v , the velocity vector of component a,

v -= x

The partial equation of equilibrium for component a reads

y *

A

* e- o (3.?)

where b is the body force per unit of mass, e.g.

t y - g «

3 1 (3-8)

i,a is the force supply, that is the force on the a-component exerted by the mixture.

According to Truesdell and Toupin (1960, Section 2 1 5 ) :

i

e-

o

( 3 . 9 )

The following reasoning will clarify this equation.

We can imagine that c° is built up by all interaction forces of the compo­ nents Ë on component a, so

ca= i

£* (3.10)

e

The force on component a exerted by component B is equal and of opposite sign to the force exerted by B on a, so

Using this relation, summation of Eq. 3-10 results in Eq. 3-9. Summation over a of Eq. 3.7 results in the equilibrium equation of the mixture:

The continuity equation for each component reads

18

-T o describe the material properties, we need a constitutive equation for each constituent b u t in addition also for the interaction forces ? . Constitutive equations are mathematical formulations for macroscopically observable results of processes that occur on a micro-scale. No matter how these equations are obtained, either from theoretical considerations about the micro-scale processes or by postulation inspired by mathematical ele­ gance, the touchstone has to be an empirical one.

That a certain degree of inconsistence in the formulation of the constitu­ tive equations can still lead to empirically acceptable results may be seen in the equations for the pore water and for the interaction forces in the binary water-soil mixture.

The constitutive equation for the pore water will be stated as

oW.= - P ( pW) 5.. (3.1^)

So it is assumed that the pore pressure is a function of the density of the fluid. Viscous effects are neglected i.e. the fluid transfers n o shear stresses. This assumption is naturally wrong for the case of a real fluid in a capillary environment, where viscosity does play an important role, Eq. 3.14 must b e regarded as the constitutive equation of an imaginary material as part of the mixture.

The constitutive equation for the interaction forces is postulated to consist of a buoyancy term which is a function of the partial density gradients and a drag term which is a function of the velocity difference (Proportional to the so-called diffusion velocity) vw - v3.

Prévost (1980) poses

h= 'h?-l' «*""

v3) (3J5)

P

So here it is assumed that shear stresses do occur, as the only way to transfer forces between grains and fluid is via viscous shear stresses. Substitution of Eq. 3.15 into Eq. 3.7 applied to the pore water phase gives

19

-P .+ pWg 6 + S - (Vw- v3) - 0 (3.16)

;i 3i nw i i

Here the generalized law of Darcy appears for a moving soil matrix. This empirical law, that is more closely examined in Section 3-8. appears here as a result of some vaguely motivated assumptions. In this light the cita­ tion: "It is of importance to notice that in contrast to classical porous media theories, Darcy's law here is derived from the general field equa­ tions" (Prévost, 1980) must be viewed as an exaggeration.

By combining the continuity equation and Darcy'a law the consolidation equation according to Biot (1941) is obtained. In Section 3.10 we return to this subject.

In concluding this section some remarks will be made on the equilibrium equation Eq. 3-12, expressed in the total stress o . It appears to be useful to split off from the total stress the so-called effective stress o. The effective stress is the stress that determines the deformation of the soil matrix (see Section 3 . 7 . 2 ) . According to Skempton (I960):

Where T is determined by the compressibility of the grains. In most cases of practical interest It can be assumed that Y = 1.

3.5 Scaling quantities

In what follows the equations derived will be expressed in customary engi­ neering units. When the influence of particular terms must be investiga­ ted, the equations will be made non-dimensional with the aid of scaling quantities with dimensions of length, stress and velocity as is explained in this section.

Although the physical relevance of the scaling quantities introduced in this section will be come more clear in connection with the physical laws in which they appear, for the moment the ideas are illustrated in Figure 3.1.

Scales of length

At least three length measures may be relevant. The first one is the scale of the micro structure i.e. the size of the grains. In our continuum ap­ proach this scale is not explicitly of Interest except for the considera­ tions mentioned in Section 3 - 2 , which means that this size must be far less than the second one: the spatial scale of the external stress changes. The scope of this study is to gain insight in the kinematic res­ ponse of the soil in consequence of a time-varying external load. We take a characteristic dimension L of the area of application of this load as the scale of length.

i 1 _;

i

L.illllllln,.!

Figure 3-1 Characteristic scaling quantities

A third scale is the measure of displacements of the soil particles fol­ lowing a load change. If we call this measure 1, then 1/L is a logical measure for the strains in the soil. It will be apparent that this small length scale can b e eliminated by posing two stress scales.

Scales of stress

The two stresses which will be used are:

- Scale of external stress variation.

As many problems in saturated soil are related to movement of soil mas­ ses, either excavations or dumpings, we will take the stress related to the weight of the soil:

pW gL (3.18)

Where pw is the density of pure water at the temperature in the situa­

tion considered and g is the acceleration of gravity.

- The second relevant stress measure is a stiffness modulus of the soil

skeleton.

Every constitutive equation of a material that couples dynamical quanti­

t i e s (stresses) to kinematical ones (strains) must contain a so-called

dimensional modulus with the dimension of stress (Truesdell and Toupin,

1960, Section 28). This modulus indicated by C can be, for example, the

bulk modulus, the shear modulus, the yield s t r e s s , etc.

It is clear that 1/L is of the same order as p gL/C.

Time scale

As acceleration effects will be neglected in this study the remaining internal velocity measure is that of the seepage velocity of the pore fluid.

The usual quantity in soil mechanics is the permeability k' with the dimension of velocity, k' is a combined material property of the soil and the pore fluid. We will take a characteristic permeability k' as the scale of velocity.

In Section 3.10 we will see that it is adequate to take the ratio 1/k' for the time scale or

re

(3

-

19)

k

o

=

- r

k

o

( 3

-

2 0 )

p g

(y i s the dynamic v i s c o s i t y ) .

In the case of a l i n e a r e l a s t i c s o i l s k e l e t o n we w i l l d i s t i n g u i s h t h r e e

d i m e n s i o n l e s s time v a r i a b l e s according to Eq. 3-19 with the f o l l o w i n g

d e f i n i t i o n s for the modulus C r e s p e c t i v e l y .

one-dimensional modulus

p l a n e s t r a i n modulus

ps 2(1+v) *>

where K is the three-dimensional bulk modulus and v Poisson's ratio. The corresponding characteristic time parameters are

t_

VMD

UL

tI I k K

t = Hk_

III k K

and t j j j - . r e s p e c t i v e l y T , , t and T, . The l e n g t h L w i l l depend on t h e

s p e c i f i c problem under c o n s i d e r a t i o n .

The diraensionless q u a n t i t i e s provided with an a s t e r i s k a p p e a r i n g in the

next s e c t i o n s a r e s c a l e d with L, p g L or k ' , those l a b e l l e d with a double

a s t e r i s k are s c a l e d with C or with 1 = p

W

gL

2

/C.

3.6 D e f i n i t i o n of k i n e m a t l c a l and dynamical q u a n t i t i e s

A fixed c a r t e s i a n r e f e r e n c e system i s used to d e s c r i b e the motion of mate­

r i a l p o i n t s ( F i g u r e 3 - 2 ) .

Figure 3-2 Coordinate system

In t h i s study d e f o r m a t i o n s and d i s p l a c e m e n t s of the s o i l - w a t e r mixture are

d e s c r i b e d in terms of motion of the g r a i n s k e l e t o n . The pore water motion

i s regarded r e l a t i v e l y with r e s p e c t to the s o i l . A Lagrangian approach i s

a p p l i e d , so a l l q u a n t i t i e s a r e formulated u l t i m a t e l y as a f u n c t i o n of X,

which i s t h e c o o r d i n a t e v e c t o r a t the r e f e r e n c e time t

n

.

In t h i s s e c t i o n a number of k i n e m a t i c a l and dynamical q u a n t i t i e s a r e d e ­

fined t h a t w i l l be used in t h e d e s c r i p t i o n of t h e motion and t h e deforma­

t i o n of t h e s o i l . Only t h o s e e x p r e s s i o n s a r e given t h a t w i l l be used i n

t h e s e q u e l .

For a more e l a b o r a t e t r e a t m e n t of t h i s m a t t e r we r e f e r t o t h e work of

T r u e s d e l i and Toupin (1960) or t h a t of Eringen ( 1 9 6 2 ) .

The p a r t i c l e v e l o c i t y i s :

V = X = U

Material time derivatives of an arbitrary function <t can be converted to spatial time derivatives by use of

In the subsequent text the deformation gradient tensor 3 and its inverse s will play an important role:

(6 is the Kronecker unity tensor)

Spatial derivatives and material derivatives of an arbitrary function are related by

;i , k ki

The deformation rate tensor of Euler (also called stretching) d is defined as the symmetric part of the velocity gradient tensor y

From Eqs. 3.24, 3.26 and 3.28 it follows:

i ij d, . = fc(v . . + v° . ) = %($,. s, . + S., s, . ) u 2 I ; J J ; L 2 Ik kj jk ki (3.30) The s p i n t e n s o r i s t h e skew-symmetric p a r t of v. _ . ■. u . = % tvf . - v° . ) = US,, s, . - S., s . j i j 2 l j j j i i Ik kj jk ki Of t h e v a r i o u s measures of s t r a i n we w i l l use t h e L a g r a n g i a n s t r a i n t e n s o r and t h e l e f t Cauchy-Green t e n s o r , r e s p e c t i v e l y d e f i n e d by i j 2 Ki kj i j ( 3 - 3 3 )

The Lagrangian strain rate tensor is the material derivative of e. From Eqs. 3-30 and 3-32 we derive

The third principal invariant of 3, denoted by .J, is the determinant of S:

J = det (S) (3-35)

The volumetric strain E with respect to the reference situation at time t is

6 - J - 1 (3.36)

In conclusion we give here some useful expressions related to the kinema­ tic quantities defined above.

The first and second principal invariants of the deformation gradient are

£ * a ij ji

The inverse s can explicitly be written as

After some manipulation we can derive from this equation the identity:

Partial derivatives of J with respect to the components of 3 are propor­

tional to the components of s:

| — ■ Js., (3.41)

3 S. . ij

With this relation we can derive for example

II

In the following sections deformat ion gradients defined with respect to

different reference situations will be mentioned.

The reference system that is related to the actual material coordinates

i.e. to the configuration at time t = t when

x = X, is the system to

which the above-mentioned deformation gradients (without superscript) are

defined. Expressed in the material coordinates, the time rate of another

deformation gradient S

1

is to be obtained from

S

h

= S s 3

h

(3.13)

Related to the forces that cause motion and deferma tion, a number of

stress quant it ies i.e. forces per unit of area

are relevant. Extensive

discussions about the notion of stresses can be found in Truesdeli and

Toupin (I960) for example. Apart form the scaling stresses mentioned in

27

-Section 3-5, the total Cauchy stress tensor or true tensor

o already

introduced in Eq. 3-5 is of importance.

Regard an elementary volume on time t (Figure 3-3).

SITUATION AT TIME I

SITUATION AT REFERENCE TIME

Figure 3-3 Cauchy stress

The resultant of all contact and internal forces, decomposed in the cooi—

dinate directions and divided by the unit of area, form the components

of o (tension positive).

The total stress can be written as the sum of the so-called effective

stress and a term proportional to the pore pressure (see Eq. 3.17). The

effective stress o is the stress that governs the deformation of

particu-late materials in the presence of a pore fluid. In Section 3.7.3 the ef­

fective stress is considered in more detail.

The pore pressure is the hydrostatic part of the fluid stress averaged

over a characteristic volume of pores.

28

-3-7- Constitutive equations

3.7.1 . Introduction

The mechanical behaviour of each of the components of the soil mixture is described by constitutive relations. These are relations between dynamical quant it ies (stresses, stress rates) and kinematieal ones (strains, strain r a t e s ) , Generally they ar ise as the result of theoretical considerat ions and experimental research, mostly on test samples subject to homogeneous stress and strain conditions.

For components that act together in a mixture, one can pose in an analoge-ous way relations between the partial stresses (see Section 3.1) and the partial movement of the components. However, in this situation interac­ tions can also take place between the constituents. For example a change in pore pressure may cause a volume change of the grains. We must keep in mind in this regard that merely macroscopic quantities are related in the sense of Section 3-2 which means in the "higher level" cont inuum approach. It is not hard to imagine that effects which on a lower level are account­ ed for in the equations of motion, on the higher level are implemented in the consti tut ive relat ions. In the next sections the constitutive relations of three const ituents of satura ted soil are examined viz. grains, pore water and air in the form of bubbles.

3.7.2. Effective stress

The so-called "Effective stress principle", formulated by Terzaghi (1923, 1 9 3 6 ) , states that the deformation of saturated soil depends on the inter-granular contact stress averaged over a charac terist ic area. If the con­ tact surfaces are assumed to be points and the material of the grains is incompressible, this leads to the conclusion that a change in pore pres­ sure has no effect on the motion of the soil mass. In this view the con­ tact stress governs the deformation process and is called therefore effec­ tive stress.

29

-Although this principle generally is found to be right in geomechanics, at least sufficiently accurate, a number of researchers did examine the no­ tion of effective stress more closely (Skempton, 1960, Bishop and Skinner, 1977, Carrol, 1980, Verruijt, 1932).

With respect to the deformation of a particulate material two effects may be noted:

a. Deformation of the grains separately.

b. Change of the relative location of the grains by sliding and rolling.

Effect a as well as effect b may be influenced by the pore pressure and the contact stress.

According to Bishop and Skinner (1977), Terzaghi's effective stress, i.e. the intergranular contact stress, determines the shear strength of soil. For the deformation of the soil skeleton however, one may look for another effective stress, which is a combination of contact stress and pore pres­ sure in a way that the deformation is only determined by this effective stress.

In this section we will consider an effective stress mentioned by Skempton (1960) which determines the volumetric strain of the soil. Carrol (1980), Verruijt (1982) and Meijer (1984) treat this matter also. The analysis is partly borrowed from Verruijt (1932). For simplICity we assume in this section infinitesimal deformat ion, incremental li near stress-strain rela-tions for the grains as well as for the skeleton and finally an isotropic stress situation. We consi der an elementary cube (Figure 3.4) with volume V and edge L which is filled with grains with a total volume V and a porosity:

V - V

n = — ^ (3.44)

Figure 3'*1 Elementary volume V

Although not really necessary, the analysis is simplified by assuming that

the grains touch each other and the boundary surface only in points.

The force vector on grain no. m, tangent to surface i is called K

m

.. The

intergranular stress is defined by

1

(3.45)

The total force at surface A in the presence of pore water is

rat

_{- I K}

.m

_{P 6 .}

"

1J

i

lJ

"

iJ

so that the total stress is

(3.46)

U

- P S .

(3.^7)

The stresses o and o are macroscopic quantities with respect to the

grains. In order to look at the deformation of the grains themselves we

define the microscopic stress

0 k j

i.e. the stress inside the grains. The

exact distribution of o is unknown and not interesting in the present

context.

The average grain stress o

K

is again a macroscopic quantity and can be

related to o and tJ. The average stress is

1

ij V.

f ak. dV

C3.48)

The Cauchy equation for equilibrium reads for the increments of stress

(see also Section 3-9)

(3.49)

Together with the trivial relation X<. i = 6.. expression 3.50 can be de­

rived.

a . . = (x . o ., ) x,

lj J lk ;m k;m

(3.50)

After s u b s t i t u t i o n in Eq. 3 . 4 8 , G r e e n ' s formula (Courant and H i l b e r t ,

1937, p . 231) g i v e s

( 3 . 5 1 )

where the surface integral extends over the surface of the grains and K is

the surface traction vector. As is stated by Love (1923, p. 174) this

expression may also be found by application of Betti's reciprocal theorem.

If there are point contacts exclus ively, Eq. 3-51 reduces to

a

g

. - i- £ , K

m

. - P 6..

U ¥ i U U

So for tile cube in Figure 3 - ^ we get

o

e

. - i - o. . - P 6 . - -.— a. . - P 6. .

The average stress determines the total volumetric strain increment e of the grains:

T - h - w

;

kk

8

" - r <T^ ° -

f ) <3

•

5

'

,,

s s s

I

3 If one would be especially interested in the change of grain volume then an effective stress defined by the expression between parentheses is ade­ quate. For example Bow en (1976) uses this stress in his treatise on the theory of mixtures.

The strain increment of the grains relative to the total volume V is cal­ led è s V ^ = è * (1-n) Ê - Ir- fa - (1-n) P] = ~ (a* + n P | (3.55) V s g K K s s A definition that is directed towards this quantity was introduced by Blot

(1956). So this effective stress reads 0 - (1-n) P. So far it is assumed that the relation between stress and strain increments in the grains is a linear one; an assumption which is rather realistic as long as no crushing occurs. However this assumption is far less reliable concerning the volu­ metric strain of the skeleton because of the aforementioned effect b. Nevertheless in this section it is assumed that the volumetric strain increment can be written as (in absence of pore water)

Pore water pressure gives an additional strain increment, so:

K K

(3-56)

(3.57)

Only when no sliding and rolling occurs, is the deformation of grains and pores similar and then Kw = K_. The effective stress 0 is now defined by posing

(3.58)

P = 0 + (1 C3.59)

(3.60)

and Y' - 1 if K„ = ». Eq. 3.57 can be written as

(3.61)

From Eq. 3-55 and 3.61 it follows for the relative change of pore volume 9

(3.62)

wherein Y is defined by

(3-63)

Biot (19^1) and Ceertsma (1957) derive the same expression (in a different notation and by different reasoning) for K = K so Y' = Y.

J ° w s

Eqs. 3-61 and 3.62 can be written in matrix form

Y Y' - n + n Y . t a P ■ M

.t

0 P (3.61)

The energy that is supplied to the grain skeleton by the intergranular forces and the pore pressure is

m

3^

-Ü M È - P ê « o ê + P (§ (3.65)

3

A n e c e s s a r y c o n d i t i o n in o r d e r t h a t a f u n c t i o n U(e,Q) e x i s t s , with U d t as

t o t a l d i f f e r e n t i a l , i s

t 'd\l , _ 9U ,_ ,,-.

o = — and P = -rrr (3.66)

ÓE d t )

So the inverse of matrix M and M itself must be symmetric and consequent­ ly Y' =» Y and K = K . In this case no dissipation occurs and U is the

J w s

increment of strain energy of the grains. Biot (19*11) postulates a priori the function U(e,G) and conclues in this way that Y' = Ï.

In recapitulation it is seen that the condition K = K can be derived along two lines:

a. Via tho assumption that there exists a function U ( e , 0 ) .

b. Via the statement that a pore pressure increment causes a similar volu­ me change of grains and pores.

Assumption b is less stringent than a, it implies that no sliding and rolling occurs due to pore pressure increments, whereas assumption a re­ quires moreover the same for the intergranular stress.

As it will be seen in Section 3*7.^.j a value of Y unequal to one has unpleasant consequences for the formulation of the field equations in the non-linear finite strain case. We look therefore to the actual values that may occur in practice.

According to Skempton (I960) Y usually exceeds 0.999 for sand and 0.99 for clay. However, for rock much smaller values occur. An evaluation of Y for an artificial configuration is given below. We cons ider a cubic array of equal spheres with radius R. It suffices to analyse one cell formed by a cube with edge 2R which contains one sphere.

So here is

Yong and Warkentin (1975, p.308) give for the volumetric strain

E -

3(-2/3

)

1 - 2v K s is Poisson's ratio, so 1/3 „2/3 ,1-2v. 2/3 (3.67)

v

. 4 , R 3

s 3 (3.68) 1 - n - r (3.69)

1

(3.71) Y

. , - I ( i - )1 / 3 (I=2g) 2/3 (3.73)

s 1 -v

In practice the ratio c/K will not exceed 1 0 , so that for v = 0 Y 3

will be larger than 0.99.

In the sequel it is assumed that Y = 1, except for Section 3-7 --U. where some generalisations are given of the findings of this section for general stress-strain relations and large deformations.

3.7.3. Grain skeleton

The question as to which constitutive equations should be applied in a computer programme for the simulation of granular soil, could be answered easily if there was at hand a generally accepted model, as there is for example for steel. This is however by no means the case. Developments in the field of constitutive models are in full swing.

It is possible to take a particular model and to bui ld around it the pro­ gramme as it were, perhaps with the advantage of having the possiblity to make use of specific features of the model in order to get an optimal efficiency. The drawback however is the low flexibility, so that adaption of the programme to changed views may be difficult.

Another approach is to leave as much room as possible for constitutive relations and to arrange the programme In such a way that a variety of models can be implemented. The advantage of such is that for problems for which a less complicated model suff ices, computer time can be saved by taking a simple model instead of fixed build-in relations.

In the development of the programme WAZAN the second approach has been chosen, so as much freedom as possible has been left for the formulation of constitutive relations.

Despite this premise it is necessary to assume a general form for the equations which after all does curtail generality. The theory of mechani­ cal non-linear constitutive relations is extensively treated in the book of Truesdell and Noll (1965): "The non-linear field theories of mecha­ nics". In this work a number of known and also new constitutive equations for solids and fluids are derived on the basis of the most general rela­ tions. However, the concept of plasticity which is frequently applied in soil mechanics is hardly discussed. For this subject it is better to refer to Hill (1950) for example.

A great number of constitutive models for granular soil have been publish­ ed. Depending on the purpose, these models vary in complexity from simple, for example linear elasticity in the study of propagation of vibrations, to very sophisticated models to fit subtle material tests wherein the soil is subjected to non-monotone stress and strain paths. For partly drained

-

37

-situations with relatively fast stress changes, the behaviour of the poro­ sity is of great importance since the generation of pore pressures depends directly on it.

A class of consti tutive models which seems to comprise a big part of the known soil models is the class of models of the so-called "rate type" (Truesdell and Noll, 1965 p. 9 5 ) . In the following, constitutive equations of the rate type are dealt with and the specific form is given as it must be delivered to the WAZAN programme.

The stress and deformation gradient in a rate-type material have to obey a differential equation of the form

F(o, o, Sh, Sh) = 0 (3.74)

where o is the effective Cauchy stress and 3 is the deformation gradient with respect to some configuration which may be specified independent of the current reference situation.

The general form of Eq. 3.7^ can be reduced by taking into account a few fundamental axiomatic principles valid for arbitrary constitutive rela­ tions (Truesdell and Toupin, 1960, Truesdell and Noll, 1 9 6 5 ) .

One of these is the principle of material invariance with respect to an observer whether or not moving. This means for instance that the response of a tensile loaded bar will not change if the bar is rotated together with the forces.

Application of the fundamental principles for an isotropie material while assuming that Eq. 3-7^ is linear in the stress fluxion leads to relation

o = F(a, Sh, d) (3.75)

where d is the deformation rate tensor and o the eo-rotational effective stress rate or Jaumann derivative defined by

(3.76)

The dependency on S in Eq. 3.75 cannot be arbitrary, but must be in the form ShSh T( Sh Sh ) , i.e. the left Cauchy-Green tensor c (Truesdell and

ik jk Noll, 1965, p. 5 3 ) .

Eq. 3.75 describes material behaviour like (Cauchy) elasticity Co and d are a b s e n t ) , viscosity (o and S are absent) and plasticity (see the re­ marks b e l o w ) .

For application in WAZAN it is assumed in addition that the dependency on d is also linear, so that finally the relation can be written as

0 = D(o, Sh) [ d ] (3.77a)

o = D. „ . d, , (3.77b) ljkl kl

If S is absent in D, Eq. 3.77 defines a so-called hypo-elastic material (Truesdell and Noll, 1965, p. 1 0 4 ) .

For infinitesimal deformation gradients Eq. 3.77 becomes

The last two terms of Eq. 3.78 may be neglected only if the 3tress level is well below the stiffness of the material or if the rotations are very small with respect to the strains. At the end of this subsection we come back to this subject.

Neglecting the last terms of Eq. 3.78 and putting the equation in an in­ cremental form gives

This form is encountered frequently in soil mechanics literature. In the sequel w e will refer occasionally to Eqs. 3.77, 3.73 as well as 3-79 with the term "incremental relations". These relations state a linear relation­ ship between a strain rate or increment and a stress rate or increment. The ability to make the relation dependent not only on the stress but also on the deformation gradient with respect to an arbitrary (however

kinema 39 kinema

-tically possible) reference situation gives the possibility of taking into account changes in fabric. A number of authors (for example Thomas, 1955, Davis and Mullenger, 1979, Mroz, 1980, Nemat-Nasser and Shokooh, 1980, Lee, 1981, Lubarda and Lee, 1981) consider the relation between elasto-plasticity and equations of the kind of Eq. 3.77.

Some aspects concerning the generalization of elasto-plasticity relations, which are derived usually for infinitesimal deformation gradients, to equations of the rate-type according to Eq. 3.75 will be outlined below. Typical for e la3to-plastic models are:

a. The total strain is spli t up into a reversible, elastic part and an irreversible, dissipative plastic part.

b. Often the stress is decomposed into a hydrostatic part and a deviatoric part.

c. The function tha t relates the plast ie strain change in dependence of the stress change is not continuous. For example: for a particular stress change there may be no plasti c strain change, while a plastic strain change docs occur for a stress change that differs an infinite­ simal amount.

This kind of branches in the material behaviour mainly plays a trick if load alterations occur and forms a serious obstacle in the numerical treatment.

The fi rst difficulty is the general isati on of the infinitesimal strains to the elastic and plastic finite strains. Which strain measure must be used and with respect to which reference situation?

As an illustration we mention the following possibilities.

An i ncremental relation like the inverse of Eq. 3.77 seems to be plausible for the plastic deformation-rate tensor, whereby the deformation gradient Sl is for example defined with respect to an imaginary state with a regu­ lar grain packing, so

40

-d

p

= H

F

(a, SM [ o ] ( 3 . 8 0 )

The s t r e s s - s t r a i n r e l a t i o n for an i s o t r o p i c e l a s t i c m a t e r i a l can be w r i t ­

t e n ( T r u e s d e l l and N o l l , 1965, p . 139):

( 3 . 3 1 )

The symmetric strain measure ce again is the left Cauchy-Green tensor. F is an isotropic tensor function of ce.

It is physically reasonable to define Se in Eq. 3.81 with respect to a situation that arises by relieving all external forces.

Differentiation of Eq. 3-81 and some manipulation yields (Truesdell and Noll, 1965, p. 108)

(3.82)

FQ is the gradient of F with respect to ce, c is the Jaumann derivative of the strain tensor,

Provided that the inverse exists, Eq. 3-82 can alternatively be written as

o = FcC d F ~1( a ) + F~1(o)d] (3.83)

So we have obtained a relation like Eq. 3.77 in the hypo-elastic form (no explicit occurrence of S ) .

The inverse relationship is symbolically denoted by Eq. 3.84 wherein by use of the superscript e is also stressed that the elastic deformation rate is considered.

de = He( o ) [ o ] (3.84)

If the r e l a t i o n Eq. 3.81 i s l i n e a r ( a l s o c a l l e d neo-Hookean) then i t can

be w r i t t e n as

a, = % E. .. . (S, S, - 6. . ) - % E. ., fc, - 6, . ) (3-85)

Ij

z

i j k l kn In kl * I j k l kl kl

where E i s t h e t w o - p a r a m e t r i c f o u r t h - o r d e r t e n s o r :

E

i j R l ^

( K

' I

G ) 6

i j \ l

+ 2 G S

l k

6

j l

( 3

-

8 6 )

K and G are independent of c ,

The i n v e r s e of E i s c a l l e d B i . e .

( 3 . 8 7 )

B

i j ^ " -T>c?h 6

i A i

+

h

6ik 6ji ( 3-8 8 )

Eq. 3.83 then becomes

0 . . = E . .

L 1

( B , n 6 . + B, 0 5. ♦ 5. 6

n

)d ( 3 . 8 9 )

i j i j k l nlpq pq mk kmpq pq In km In mn

So the simple i s o t r o p i c r e l a t i o n

0. « E . , d (3.90)

i j ïjmn mn

does not correspond with I s o t r o p i c l i n e a r e l a s t i c i t y ( i t i s not a

neo-Hookean r e l a t i o n ) in the common s e n s e , u n l e s s a l l s t r e s s e s a r e z e r o .

Summation of Eqs. 3-80 and 3-81 y i e l d s t h e t o t a l deformation r a t e :

d = (H

p

+ H

e

) [ a ] (3.91)

I n v e r s i o n g i v e s

So here, at least formally, the incremental relations according to Eq. 3-77 arise.

The plastic part of Eq. 3.92 may be either present or not and occasionally may be described by different functions Hp depending on the actual loca­ tion of the so-called yield surface. The yield surface is an imaginary surface in a space spanned by the three effective principal Cauchy stres­ ses (see below) .

The yield surface is described by one or more scalar equations:

*k( o , Sp, r) = o (3.93)

£ is a vector with parameters which brings into account the influence of the load history.

If $. < 0 no plastic deformation occurs, if $, = 0 it does and the plastic branch no. k i.e. H^ is activated.

k

It is convenient to make some remarks at this point about the notion of principal values and principal directions of symmetric second-order ten­ sors .

A symmetric second-order tensor can be represented by a symmetr ic matrix. By a suitable rotation of the coordinate system, this matrix can be redu­ ced to a diagonal matrix with the so-called principal values of the tensor on the diagonal. The directions of the axes of the associated coordinate system are the principal directions. The quest ion may arise how in a par­ ticular material model the relations are between the principal directions of strains, strain rates, stresses and stress rates.

In this regard the following theorems are of impor tance:

a. A symmetr ic isotropic tensor function of one argument is coaxial with this argument (has the same principal directions).

b. Every isotropic tensor function f of t can be written as

f. .= a Ó..+ a.t. + a_t.. t, . ij o ij 1 ij 2 ik kj

(3.94)

^3

-where a , a-,, and a2 are functions of the invariants of t.

c. The gradient of a function F of the invariants of t is an isotropic

So we can infer that for an isotropic elastic material a is coaxial with ce (Eq. 3 . 7 8 ) .

A specific choice of the plasticity relation:

d..= h r | L . (3.95)

where h is a function of the invariants of a, o and eventually of c^ leads to coaxiality of d and o.

To conclude this subsection, the dimensionless form of Eq. 3-77 will be considered in the light of Section 3.5.

Firstly Eq. 3.77 is rewritten as

o = D[d] + wo - do) (3.96)

The stresses are scaled with the expression 3-13, the function D is made dimensionless with the dimensional modulus C which is a characteristic stiffness. If the kinematic quantities are scaled with p gL/C, then all dimensionless quantities are of the same order of magnitude. We get

5 = D ' [d ] + Ë ^ Cu o ~ o co ) (3.97)

So the condition of neglecting the rotational terms is

w

figBs << 1 (3.98)

3-7.^ Penalization of the effective stress concept

We start with Eq. 3.77b, the relation between the Jaumann effective stress rate and the stretching. The inverse of D is called R. Analogous to

Sec-tion 3-7.2 we state (with K, = K„

here o is the Jaumann rate of the intergranular stress and o the effective stress. So it follows:

D.

ijkk. • 0..= ff. + (6.. 5 5 — ) P ij iJ iJ 3K (3.100)

This can be written also by

ij ij ijkk (3.101 )

If D is linear isotropic (i.e. E, see Eq. 3.86) we get

( 3 . 1 0 2 )

T

ijkl"

5

l k V 3

(1

"

ï ) Ó

i A l

( ï

w o r d i n g t o Eq. 3.63)

(3-103)

ij (3.104)

The rate of relative pore volume change Q is the difference of the total volume change and the volume change of the grains. In Section 3.10 we shall see that

• s

0 = Jd + J(1 - n ) £ - (3.105)

In Section 3-7.2 the first term was called ê, the second -ê . Eq. 3-51 has been derived with the sole assumptions: point contact of the grains

and equilibrium. It is easy to see that this equation also holds for the finite strain case, so

P_ = 1 _ { 1

(3.106)

K l3(l-n) kk ' V p s

With Eqs. 3-99, 3-105 and 3-106, we get for 0 :

! ■

R

i i k i \ i

m n

% n

+

H

T

t i

k k

-

n ) f ( 3

-

1 0 7 )

If D equals E, Eq. 3-107 reduces to

! - h '< V r

( y

-

n )

"

<3

-

108)

s Similar to Eq. 3.62.

So in the general case of non-linearity and large deformations the assump­ tion of compressible grains leads to an expression for the effective stress which is substantially complex.

Only for a linear relationship like Eq. 3.90, the effective stress rela­ tion and the G - 0 relation involve a scalar quantity Y.

3.7.5 Pore water - air bubble mixture

The compressibility of pure water is small compared with the bulk compres­ sibility of the soil skeleton. For example it equals to about 5. 1 0 ~1 0 m /N whereas a typical compressibility of the soil is in the order of 10*" m /N or higher. In this study the compressibility of water is therefore neglected.

However if the pore water contains air in the form of bubbles then the compressibility of the mixture increases considerably. Schuurman (1966), Verruij t (1969) and Barends (1979 ) study the compressibility of a water-air bubble mixture. A number o f physical phenomena play a role such as compression and dissolving of the air, evaporation of the water, effects of surface tension etc. In addition the fact is important whether the air