Transport in Porous Media 9:49 59, 1992. 49 �9 1992 Kluwer Academic Publishers. Printed in the Netherlands.

An Elastoplasticity Model in Porous Media

Theories

WOLFGANG EHLERS Technische Hochschule Darmstadt, Institut fiir Mechanik, W-6100 Darmstadt, Germany

(Received: 25 June 1991)

Abstract. In this article, porous media theories are referred to as mixture theories extended by the well-known concept of volume fractions. This approach implies the diverse field functions of both the porous solid matrix and the pore fluid to be represented by average functions of the macroscale.

The present investigations are based on a binary model of incompressible constituents, solid skeleton, and pore liquid, where, in the constitutive range, use is made of the second-grade character of general heterogeneous media. Within the framework of geometrically finite theories, the paper offers a set of constitutive equations for the solid matrix, the viscous pore liquid and the different interactions between the constituents. The constitutive model applies to saturated as well as to empty solid materials, taking into account the physical nonlinearities based on elasto-plastic solid deformations. In particular, the constitutive model concentrates on granular materials like soil or concrete, where the elastic deforma- tions are usually small and the plastic range is governed by kinematically hardening properties.

Key words. Porous media, mixture theories, volume fractions, incompressible constituents, granular solid matrix, viscous pore liquid, finite elastoplasticity, kinematical hardening.

1. Introduction

With in the f r a m e w o r k o f con t inuum mechanics , po rous med ia can be descr ibed by

use o f mix ture theories (he te rogeneous ly c o m p o s e d con t inua with in ternal in terac-

t ions) ex tended by the wel l -known concept of vo lume fractions. This a p p r o a c h

employs po rous med ia mode l s as an immiscible mixture o f const i tuents ~0 i

(i = 1 . . . . , k) and was successfully app l ied to incompress ib le and compress ib le

models , e.g., by Bowen (1980, 1982). Assoc ia ted with this p rocedure is the assump-

t ion o f a s tat is t ical d i s t r ibu t ion o f the different phases th rough the con t ro l space,

thus p roceed ing f rom the idea o f super imposed con t inua where the ind iv idua l

funct ions o f the microscale are replaced by average field funct ions o f the

macroscale . The diverse average funct ions are then unde r s tood as the result o f an

averaging process re la t ing funct ions f rom the microscale towards the macroscale ,

compare , e.g., Hassan i zadeh and G r a y (1979a, b), N igma tu l i n (1979), Ba c hma t and

Bear (1986), or Ehlers and Pl ischka (1991).

Based on this approach , the bas ic theory for po rous med ia as well as the

const i tu t ive equat ions and the t he rmodynamica l res t r ic t ions governing these consti-

tut ive equat ions are general ly well known. Proceeding f rom the concept o f second-

50 WOLFGANG EHLERS

grade materials (Miiller, 1968), porous media models are usually assumed to consist of an elastic solid matrix saturated by an arbitrary number of fluids, whereas the different papers on porous media elastoplasticity (e.g., Koji6 and Cheatham, 1974; Pr6vost, 1981; de Boer and Kowalski, 1983; de Boer and Ehlers, 1986) are partly restricted to geometrically linearized theories or proceed from simplified thermodynamical approaches, in some extent combined with other rig- orous simplifications.

On the other hand, it was only recently that the author published a paper on thermodynamics of elasto-plastic porous media (Ehlers, 1989a) taking into account both the second-grade character of the single constituents and the different influ- ences of geometrically finite theories. Following this, it is the objective of the present article to offer a finite elastoplasticity model suitable to describe the material behaviour of saturated and empty matrices. In particular, the investiga- tions concern a binary system of an elasto-plastic solid matrix saturated by an incompressible viscous pore liquid. The present model furthermore concentrates on granular materials like soil or concrete, where the individual particles of the solid material are considered to be incompressible. Hence, the theory is based on an incompressible binary model, where the elastic strains are usually small and the plastic range is governed by kinematically hardening properties. For simplicity, mass exchanges between the constituents are excluded and, in addition, the model is assumed to be governed by a single temperature function. Finally, note in passing that the same type of basic problem was already investigated, within the scope of a purely mechanical theory, in classical papers on soil mechanical problems (e.g., Heinrich and Desoyer, 1955). Throughout this paper, direct tensor notation will be

used.

2. Fundamentals

The present section offers a brief review of kinematics, balance laws and the volume fraction concept for incompressible porous media together with the basic constitu- tive set-ups. Concerning the different notions and several further information, the reader is referred to Bowen (1980) or Ehlers (1989a, b). In what follows, all introduced functions are assumed to be sufficiently smooth in spacetime.

Consider a liquid-saturated porous solid as an immiscible mixture of constituents ~o i with particles X i (i = S: solid phase; i = F: liquid phase), then, the macroscopic formulation implies each constituent to be assigned its own motion function, viz.:

x = z i ( x i , t). (1)

Therein, X~ are the reference position vectors of ~0 i at time t = to. The volume fractions hi(x, t) are defined as the local ratios of the constituent volumes v i with respect to the bulk volume v. Associated with each q~i is an effective density QiR which is defined as the average mass of q)i per unit of v i and a partial or bulk density r defined as the average mass of q~i per unit of v. The density functions are

AN ELASTOPLASTICITY MODEL IN POROUS MEDIA THEORIES 51

coupled by the volume fractions, i.e.

~i = ni~iR. (2)

Thus, it is easily seen from (2) that constituent incompressibility, implying the effective density functions to be constant during deformation (~eR= const.), does not cause macroscopic incompressibility since the bulk densities can still change through changes in volume fractions.

Excluding mass exchanges between the solid and the liquid phases, the concept of volume fractions and the governing set of balance equations are given with respect to Ehlers (1989a, b):

Concept of volume fractions:

n s + n v = 1. (3)

Balance of mass:

(ng~ + nediv xi = 0. (4)

Balance of momentum:

div T i+ 9 i ( b - ~e) + pi= o, pS+ pF = O. (5)

Balance of moment of momentum:

T i = T Te - M i, M s + M v = 0. (6)

Balance of energy:

pi(ei)~ = - p i ' x e + T e ' L e + p e r i - d i v q i + e e , eS+ eF = 0. (7)

In the above equations, "l "i, d, r i, and qi are the partial Cauchy stress tensors, the internal energy densities, the external beat supplies, and the heat influx vectors of cp e. The quantities pi, M i and e i are defined as the supply terms of momentum, moment of momentum and energy representing the transfers to cp e caused by the respective other constituent that occupies x at time t. For the present binary model, the momentum supples may also be interpreted as the interaction forces or the drag forces, respectively, taken per unit of bulk volume. Furthermore, in using the body force density b instead of b i, it is understood that b = b s = b F. The material time derivatives (...)~ are defined by

0(. . . ) + grad(. . . ) �9 xe, (8) ('" ')~ = at

xi characterizing the constituent velocities of ~oe. In addition, ~e are the correspond- ing constituent accelerations. The symbol grad means partial differentiation with respect to the spatial position x; div is the divergence operator corresponding to grad. Furthermore,

L e = grad xe (9)

52 W O L F G A N G EHLERS

is the spatial velocity gradient of 99 ( Concerning the moment of momentum balance equations, it must be noted that the skew-symmetric supplies M e cause nonsymmet- ric partial stresses T( On the other hand, it was shown by Hassanizadeh and Gray (1979b), for mixtures with immiscible constituents, that T ~ must be symmetric as far as the individual q~ are microscopically nonpolar. Thus, for a liquid-saturated porous medium with nonpolar constituents, it is sufficient to conclude that

T i : T Ti and M i = 0. (10)

Finally, it is seen from (4) that, for the incompressible model under discussion, the usual balance of mass equations reduce to balance equations for the volume fractions.

From the preceding consideration, the present model is defined by the concept of volume fractions (3), the mixture balance equations (4)-(7) and the Bowen-Trues- dell version of the entropy inequality for mixtures (cf., e.g., Ehlers, 1989a, b) together with the following set of constitutive postulates which, from the principle of equipresence, must be assumed in the first place to depend on a common set of independent state variables 4, viz.

{i/li, ~]i, T i + ,~nil, qi, pV _ 2 grad n F} = ~(~), 3 (11)

= {| grad | Fs, Gs, xv - Xs, DF}.

Therein, O e and r? i are the specific free energy and entropy densities of ~0e, | is the common absolute temperature function, and D F is the symmetric part of LF; the inclusion of the Lagrangian multiplier ,~ into (11)l represents the incompressibility constraint for the model under discussion. Furthermore, the concept of second- grade materials requires the incorporation of the second-grade pair of solid

3 deformation gradients, (Fs, Gs), into (11)2 where

3 F s = Grads x and Gs = Grads Fs. (12)

In (12), Grads means partial differentiation with respect to the solid reference

position Xs; (. 3..) is the symbol for a tensor of third order. However, (11) defines a homogeneous binary medium consisting of an incom-

pressible inviscid elastic skeleton saturated by an incompressible viscous liquid, thus modelling the same type of medium as was investigated, e.g., by Heinrich and Des| (1955).

3. Constitutive Framing

It is known from solid mechanics for single continua that the concept of finite elastoplasticity must be based on the multiplicative decomposition of deformation gradients connected with the suggestion of a stress-free solid intermediate configu- ration where the purely plastic deformation state in stored into the memory of the material. Following this, porous media elastoplasticity must result in a multiplica-

AN ELASTOPLASTICITY MODEL IN POROUS MEDIA THEORIES 53 3

tive decomposition of the second-grade pair (F s, Gs) into elastic and plastic parts, viz.

3 3 3

(Vs, Gs) = (Vse, Gse) ~ (Fsp, Gsp). (13)

The symbol ( . . . , . . . . 3 .) o (. , . 3..), the composition, is the binary operator govern- ing the linear group of second-grade pairs. Since the composition rule is induced by the chain rule (Cross, 1973), it is easily proved that

F s = FseFsp ,

Gs = [(GseFsp)~Fsp] -3~ + (Fsel~Sp) 3 (14)

where ( . . . )3 defines a contraction of the arguments in brackets towards a third-or- der tensor and (. 3 .)2.~ �9 means transposition with respect to the indices 2 and 3. Concerning further information about the concept of second-grade elastoplasticity, the reader is referred to Ehlers (1989a, b). Moreover, it was shown by the author (Ehlers, 1989a, b) that the above concept is compatible with an additive decompo- sition of solid-strain and strain-rate measures. For example, the Lagrangian strain tensor E s and its elastic and plastic parts, IZSe and FSp , a re given by

1 T 1 T Ks = g(FsF s - I), ESp = g(FspFsp - I), Esr = E s - ESp. (15)

Furthermore,

D s = Fs T l[(Ese)~ -t- (Esp)~JFs 1 (16)

is the symmetric part of L s represented by the contravariant push-forward transfor- mation of the elastic and plastic Lagrangian strain rates, (Ese)~ and (Esp)~, from the solid reference towards the actual configuration.

By combination of the basic set of constitutive postulates, (11), and the above concept of second-grade elasto-plasticity,

{0 i, 17 i, T i+ 2nq, qi, pF _ 2 grad n F} = ~ ( E ) , 3 3 (17)

"~ = {| grad | Fse, Gse, Fsp, Gsp, xv - Xs, DF} 3

holds, where (Fs, Gs) has been replaced by the elastic and plastic second-grade pairs, (Fse, Gse) and (Fsp, Gsp).

Following this, the plastic deformation gradients are understood as internal state variables; that means, an additional constitutive equation, the flow rule, must be given for (Esp); or any related plastic strain rate. Furthermore, it is known from the literature (cf., e.g., de Boer and Brauns, 1990) that porous solid materials generally show the effect of kinematical hardening. Thus, to govern the multiaxial Bauschinger effect, one must additionally specify the finite back-stress tensor ys. Therefore, the set (17)1 must be completed by

(Esp)s = ( E s p ) s ( - , . . . ) and = Ys(E , . . . ) . (18)

54 WOLFGANG EHLERS

For the present model, thermodynamical restrictions result from the dissipation principle for mixtures together with the above constitutive assumptions. Using standard arguments (Bowen, 1976; Ehlers, 1989b), it is easily proved that a linear expansion about the so-called mixture equilibrium state,

3 3

= = {| grad | = 0, Fsr Gse, FSp, GSp, XF - - XS = O, I) F = 0}, (19)

yields the model to be governed by

{O i, ~ i} = f( | Fs~, Fsp), qS + qF = _ rio grad | (20)

Ti= --nipl + TiE, pV = p grad nF + pF,

where /?o is the coefficient of thermal conductivity for the whole system. As is known from theories of constrained materials, the solid and liquid partial stresses and the momentum supply, respectively, can be expressed by the effective liquid pressure p (note that the Lagrangian multiplier 2 reduces to p) and so-called 'extra quantities' denoted by the additional subscript (..-)E:

~--~Se + aESe J S,

T F = 2#eDv + v F ( D F �9 ])[, (21)

pF = --e0 grad | -- $~(xv -- Xs) --

_ F ~ - l ( p F ~ V ~ s ~ ) ~ _ F s V , f F ~ F ~ S p ) !

Therein, #F and v F are the macroscopic viscosity parameters of the pore liquid, as is the entropy coupling parameter, and (. . .) 1 defines a contraction of the arguments in parantheses towards a vector. Furthermore, in the case of is| permeability, the general permeability tensor reduces to

(nF) 2? FR Sv - kF I (22)

where 7 FR is the effective specific weight of the pore liquid and k F is the coefficient of permeability. Finally, the dissipation principle yields a restriction for the finite back-stress tensor, viz.

ys =de tFsFs (pS ~?O s V 30V'~ p 0--g2s )F (23) x

In porous media theories, the above suggestion was firstly made, however, in a more simplified form by the author (Ehlers, 1989a), whereas, in the frame of thermodynamics for single solid continua, similar suggestions have been made, e.g., by Phillips (1974) who assumed ys to be the 'thermodynamical reference stress'.

However, concerning the solid 'extra stresses' T s, it should be noted that in soil mechanics, these quantities are usually known, following Terzaghi, as the 'effective

AN ELASTOPLASTICITY MODEL IN POROUS MEDIA THEORIES 55

stresses' of the porous soil material; the reader who is interested in more details about the 'effective stress principle', is referred to de Boer and Ehlers (1990a). Further details, especially with respect to the problem of uplift, friction and capillary forces in porous media, can be taken from de Boer and Ehlers (1990b).

Considering applications to practical engineering problems, it is useful to simplify the above constitutive model with the aid of the 'principle of constituent separation' (Ehlers, 1989a), thus implying the liquid free energy function to be independent of the solid deformation state. The consequences of this approach with respect to the representation of (20)-(23) are obvious and must not be discussed in more detail. However, it is easily seen that the simplified model also applies, without any modification, to empty porous matrices. In this case, i.e., in the absence of p, the solid extra stresses T s and the partial stresses T s coincide.

Finally, note in passing that it was not necessary to specify the energy supply terms e i, because the temperature variation of the whole system must be calculated from the sum of the energy balance equations (7). Following this, the present elastoplasticity model for liquid-saturated porous solids is properly defined. Addi- tional relations concerning the elastic and the plastic ranges of the solid constituent can be taken from the following Section.

4. An Elastoplastieity Model for Granular Porous Solids

Concerning an elastoplasticity model for granular porous solids, further simplifica- tions of the theory may be introduced due to a suggestion originally proposed by Green and Naghdi (1965) who assumed the solid free energy density to be split into elastic and plastic parts,

@s(o ' Ese, Esp) = 0se(| Ese) + OSp(Esp), (24)

0se(O, Eso) = ~Se(o) + ~7~~

where, additionally, (24)= corresponds to the assumption of hyperelasticity. Fur- thermore, it is known from experiment that the elastic domain of granular solids (e.g., soil or concrete) is usually very small, whereas the plastic loading leads to considerable displacements.

Thus, the solid extra stresses or the effective solid stresses, respectively, can be determined by use of an elasticity law of Hookian type, viz.

~s = 2#S[,se + vS(~s . I)l, ~s = det FsF;ITSF~-e - 1, FSe = FSTp 1 EseF~pl. (25)

Therein,

o ~ S e F T s ~lffSe ,~s = p~osVsp ~ sp = po~ ,~f ~ ,

S Se ~ i S ^ ,Oo~sffS~176 = poSff (r~o, F~p)lF~p=oon~t. = #sf~ o. tso + ~' ( r so . I) ~ (26)

56 W O L F G A N G E H L E R S

together with (14)1 and (21)1 have been used. The symbol (..~.) characterizes quantities defined in terms of the plastic intermediate configuration of q)S. Note in passing that a representation of the elastic and plastic responses with respect to this intermediate configuration represents the most natural frame in finite elastoplastic- ity (Ehlers, 1989b). However, ~s is the Kirchhoff extra stress tensor of the intermediate configuration; p oSs = p S det Fs is the partial solid density in the refer- ence position of <o s. The elastic strain tensor f'Se is defined as the contravariant push-forward transformation of the Lagrangian strain Ese from the solid reference towards the intermediate configuration (Ehlers, 1989a, b). Since this transformation is carried out with the aid of the plastic part of the deformation gradient, Fsp, it is easily seen that the stored energy function (26)2 must be defined with frozen plastic variables. Thus, #s and v s are understood as the macroscopic Lain6 parameters of the granular solid measured at constant Fsp. Furthermore, for the incompressible solid material under discussion, the material parameter v s can be understood as a structural parameter representing the macroscopic compressibility of the porous matrix caused by variations in porosity.

The plastic range of granular porous materials is governed by the constitutive relations (18) together with a convenient yield condition, compare, Figure 1, which, in the high pressure regime, may be completed by a convenient cap model.

Following this, a temperature independent yield function for kinematically hardening matrices can be introduced by the extension of de Boer's yield function (de Boer, 1988) towards the finite kinematically hardening range, viz.

P =(I~ID) '/2 1 +7 (IID)3/2 j + B I - - x =0, (27)

compression

/ / ~ ~ simpieshear

//\ 300/-'*k~/~\ extension / / - ~ .~3oo .-- \ \

.

(a)

i•pression rc

(b)

i / r

Fig. 1. Yield funct ion for brittle and granular solid materials. (a) deviatoric plane: gl/$2/$3: principal values of a s - ~s . (b) hydrostat ic plane: a c = = ~,/2rc/[1 - ( 2 / x / ~ ) 7 ] t/3, at = . ~ t c / [ 1 + (2 / . , /~)7] '/3, b = ~/~.

A N E L A S T O P L A S T I C I T Y M O D E L I N P O R O U S M E D I A T H E O R I E S 57

where

fib =I As ~,s)o 5(ZE _ . ({S _ VgS)O, (28)

iilD 1 A S _ _ q S ) D - - q S ) D -~- 3 (gE . (,~S E - - ~ S ) D ( , ~ S .

In (28), the symbol ( . . . )D indicates the deviatorical part of the respective tensorial object. The parameters r , y, and ~c are material functions depending on the angle of internal friction, ~o, and the cohesion, c, compare, de Boer (1988):

fi = �89 sin q~, ? = 4 sin ~o, ~: = c cos qo. (29)

Furthermore, to guarantee the convexity of the yield surface, especially with respect to the failure state, the parameter 7 must not exceed the value ~ = 3x/3 (cf., de Boer

q: o and Dresenkamp, 1989). Hence, cp ~< ~0 = 76,98 . As is usual in purely kinematically hardening plasticity, the material parameters

included into the yield function are assumed to depend on the onset of plastic deformations and are then taken to be constant during the kinematical hardening process. Thus, the angle of internal friction can be determined from the well-known M ohr - C ou lom b yield condition in comparison with the above yield function evaluated at the point of first plastic yielding with respect to the simple shear test. For example, for comparison obtains that

with experimental data of cohesionless soil, one

s - - s ( 3 0 ) ~o = arc sin s + s

AS where s are the principal values of ZE which, in the case of simple shear (O = 0 and thus, III D = 0, cf., Figure 1), are linked to the principal invariants (28) via

1 ^ = 3(s -}" s ) , ( I D = , I I I D = 0 --+ s 3 = ~(s 2 -~- s ) . ( 3 1 )

Note that, at the first onset of plastic deformations, c / s= 0. However, by combination of (19), (24), and (25), the finite back-stress tensor qs

given in terms of the plastic intermediate configuration is assumed to yield

qs = cl (det FSp ) 2/3RsD p q- C 2 det F s p l,

r = F~elySFT e 1,

where

o~/Sp c1 = 4pSs C3~p and

~ S p = R s p Esp RsVp,

~s 0~ sp c2 = CoS ~3(det FSp )

(32)

(33)

58 WOLFGANG EHLERS

are material functions to be expressed by the first and the square root of the third plastic invariants, viz.

ip = Bsp. I and Bsp = (det Fsp)-2/3(2~sp q- I),

IIIp = (det Fsp) 2. (34)

Therein, IIIp is the purely volumetric plastic strain invariant whereas i v characterizes the respective volume-preserving strain. In addition, the functions Cl and c2 can also be determined from convenient measures of the plastic strain path represented by both a volumetric and an isochoric path variable. The latter approach is necessary, when the plastic loading path consists not only of monotonous loading curves but also on loading and reverse loading histories. Concerning the above representation of the back-stress tensor, note that r is defined as the covariant pull-back transformation o f u s (cf. (23)), namely, from the actual towards the solid intermediate configuration. Furthermore, Ksv is the 'plastic Karni-Reiner strain tensor' of the intermediate configuration defined as the push-forward rotation of the plastic Lagrangian strain IZsp by use of the plastic rotation Rsp (note that, from (14)1 together with the polar decomposition rule, Fsp = VspRsp was used). For further information about the concept of Karni-Reiner strains, the reader is referred to Ehlers (1989b).

To complete the set of equations governing the plastic range of porous media, one must apply a convenient form of a flow rule to the purely plastic strain rate of the intermediate configuration of q~s defined by

l~Sp = FsVv l([=Sp);Vspl. (35)

However, as was shown by de Boer (1988), it is useful to introduce a nonassociated flow rule in order to confirm to experimental data (additionally compare, e.g., Lade, 1977). Thus, it is assumed that

liSp = �89 -l/2A[( ~s - q,s)o + ~il], (36)

where the material function ~ depends on the angle of dilatation, vp. The A-factor included into the above flow rule can be either obtained from (27) with the aid of the well-known consistency condition, (F)~ = 0, or it can directly be computed from the yield function itself. However, note in passing that, in the frame of numerical calculations carried out, e.g., by use of the finite element method, the former approach results in an incremental procedure whereas the latter procedure requires the definition of a so-called 'consistent tangent operator' which can be obtained from the above theory by analogy with, e.g., Simo and Taylor ( 1985); although this paper is restricted to usual metal plasticity.

5. Conclusion

In the present article, a unified approach to liquid-saturated incompressible solid materials was presented by use of general porous media theories defined as mixture

AN ELASTOPLASTICITY MODEL IN POROUS MEDIA THEORIES 59

theories extended by the volume fraction concept. The elastoplasticity model for the solid constituent, within the 'simplified constitutive approach', concentrated on granular matrices like, e.g., soil or concrete, thus offering a convenient tool for applications (e.g., soil mechanics, petroleum industries, concrete technologies, etc.).

Finally, note again that the simplified model applies to saturated as well as to empty porous solid materials.

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