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Micromechanical Basis of Concept of Effective StressAnjani Kumar Didwania1
Abstract: Key developments in the concept of effective stress are surveyed with an emphasis on the ensemble averagimicromechanical approach reported by Didwania and de Boer in 1999. Various assumptions underlying the effective stress fcases like rigid/poroelastic solid material saturated by incompressible fluid under homogeneous conditions are elucidated. It is sfor inhomogeneous saturated porous material the concept of effective stress needs to be generalized by introducing higher gporosity, pore pressure, etc.
DOI: 10.1061/~ASCE!0733-9399~2002!128:8~864!
CE Database keywords: Effective stress; Porous media; Saturation; Micromechanics.
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Introduction
The concept of effective stress originated from the early expmental investigations of the effect of fluid pressure on the demation and strength of porous solid materials. The earliest dissions date back to English geologist Sir Charles Lyell in 18Boussinesq in 1876, and to Osborne Reynolds in 1886 incontext of his now famous dilatancy experiments~Skempton1960!. The major experimental contributions leading to the devopment of this concept came from the works of Fillunger aTerzaghi.~For a detailed historical account, see de Boer 199!Skempton~1960! attributes the formulation of the concept of efective stress to the following quote from a paper of Terza~1936!.
‘‘The stresses in any point of a section through a mass of ecan be computed from thetotal principal stresses nI8 , nII8 , andnIII8 which act in this point. If the voids of the earth are filled wiwater under a stressNW , the total principal stresses consist of twparts. One partnW acts in the waterand in the solid in everydirection with equal intensity. It is called theneutral stress. Thebalance,nI5nI82nW , nII 5nII8 2nW , represents an excess ovthe neutral stressnW and it has its seat exclusively in the solphase of the earth. This fraction of the total principal stressesbe called theeffective principal stresses. Each of the porous materials mentioned~sand, clay, andconcrete, the author! was foundto react on a change ofnW as if it were incompressible andas ifits internal friction were equal to zero. All the measurable effeof a change of stress, such as compression, distortion achange of the shearing resistance are exclusively due to chain the effective stresses,nI , nII , andnIII . Hence, every investi-gation of the stability of a saturated body of earth requiresknowledge of both the total and the neutral stresses.’’
1Dept. of Mechanical and Aerospace Engineering, Univ. of CalifornSan Diego, La Jolla, CA 92093-0411. E-mail: [email protected]
Note. Associate Editor: Franz-Josef Ulm. Discussion open until Jaary 1, 2003. Separate discussions must be submitted for individuapers. To extend the closing date by one month, a written request mufiled with the ASCE Managing Editor. The manuscript for this paper wsubmitted for review and possible publication on March 25, 2002;proved on. This paper is part of theJournal of Engineering Mechanics,Vol. 128, No. 8, August 1, 2002. ©ASCE, ISSN 0733-9399/2002864–868/$8.001$.50 per page.
864 / JOURNAL OF ENGINEERING MECHANICS / AUGUST 2002
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as
This original concept of effective stress thus can be summup as
TSE5T1pFI (1)
where TSE, T, and pF5effective stress, total stress, and popressure, respectively, andI5 identity tensor. The effective stresobtained from this expression, however, is limited to a homoneous isotropic porous matrix constituted of rigid porous matesaturated with an incompressible fluid under quasistatic cotions. Under more general conditions, the knowledge of tostress and pore pressure alone is not sufficient to define the etive stress tensor. The effective stress is thus recognized toconstitutive quantity and cannot be defined independently ofrheological properties of particular porous material and saturafluid under consideration even for homogeneous isotropic cotions. Subsequent investigations, as discussed below, have galized the effective stress expression for incompressible saturafluid under homogeneous conditions to a more general expresof the type
TSE5T1pFB (2)
where the tensorB accounts for the characteristics of the somaterial.
For homogeneous isotropic linear poroelastic materials srated with incompressible fluid, Biot and Willis~1957! proposed,B5@12(K/KS)#I , by including the effect of the bulk moduli odry porous solid and fully compacted nonporous solid, denotedK andKS , respectively. Carroll and Katsube~1983! obtained ananisotropic expression forB allowing for the complex kinematicsassociated with anisotropic elastic materials. Assuming matrixcompressibility, de Boer and Kowalski~1983! used the originalform of Terzaghi’s effective stress expression~1! to model plasticporous saturated material. Coussy~1995! adopted expression othe type~2! to describe poroplastic material saturated with incopressible fluids~see also Zienkiewicz et al. 1977!.
Micromechanical investigations of effective stress have brather limited. Applying homogenization method de Buhan aDormieux ~1996, 1999! showed that Terzaghi’s original effectivstress expression is valid for the failure of a purely cohesive mtrix material. For porous material like sandstone, they showedthe effective stress expression is modified with a nonzeroB givenas a function of porosity of the sandstone,B5 f (f)I resultingfrom significant contact area between the grains. Didwania an
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Boer ~1999! and de Boer and Didwania~2001! analyzed the con-cept of effective stress from both macro- and micro-mechanapproaches and showed how both approaches can lead tTerzaghi’s original form~1! and the modified form~2!. Chateauand Dormieux~2001! and Didwania~2001! analyze effectivestress in an unsaturated media using homogenization andsemble averaging approach, respectively, while Gray and Scfler ~2001! focus on a thermodynamic approach.
In what follows, we review the ensemble-averaging basedcromechanical formulation of effective stress of a fluid-saturaporous media~Didwania and de Boer 1999!. We then elucidatethe various assumptions involved in the derivation of special cexpressions like that for rigid/linear poroelastic solid materialcompressible/incompressible fluids, etc., under homogeneconditions. For inhomogeneous conditions, the need for includhigher gradients of porosity, pore pressure, etc., is demonstr
Micromechanical ModelFollowing Didwania and de Boer~1999!, a model of the fluid-saturated porous media in which the solid skeleton is assumeconsist ofN grains of a specific configurationCN, specified by aset of position vectorsra for a51,2, . . . ,N. The grains in gen-eral can be of arbitrary shape and size. However, to keepanalysis tractable, the grains are considered identical and spcal. The spherical grains are immersed in a fluid phase. It isther assumed there is no gravitational field, no relative velobetween the two phases and both phases are isothermal.gravitational terms can be easily included in a manner outliearlier by de Boer and Didwania~1997!.
We recall some definitions and manipulations commonly eployed in the ensemble-averaging approach~Sangani and Didwa-nia 1993; Didwania and de Boer 1999!. We consider an ensemblof realizations and denote byf (CN) the probability of a specificconfigurationCN. In view of the identity of the grains, the appropriate normalization is
E dCN f ~CN!5N! (3)
A reduced probability distributionf (CM) in which the positionsof M grains are specified, can be obtained fromf (CN) by inte-gration
f ~CM !51
~N2M !! E dCN2M f ~CN!
with
E dCM f ~CM !5N!
~N2M !!(4)
Furthermore, a conditional probability can also be defined as
f ~CN2M !5f ~CN!
f ~CM !
with
E dCN2M f ~CN2M !5~N2M !! (5)
We define an indicator functionxS(x;CN), such thatxS(x;CN)51 if the point x is in the solid phase~i.e., inside a grain!,otherwisexS(x;CN)50. For spherical grains of radiiR, the indi-cator function is of the form
xS~x;CN!5 (a51
N
H~R2ux2rau! (6)
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with H5Heaviside distribution. The gradient of the indicatfunction is then
¹xS~x;CN!52 (a51
N
d~R2ux2rau!x2ra
ux2rau (7)
whered5Dirac delta function. Since the boundary of the graihas a measure of zero, the indicator function of the fluid phxF, is 12xS in case of fluid-saturated porous solids. The fluand solid phase volume fractions are then defined by
nF,S~x!51
N! E dCN f ~CN!xF,S~x;CN! (8)
from which, by Eq.~3! we obtain the saturation condition
nF1nS51 (9)
By substituting expression~6! for xS in Eq. ~8! one readily finds
nS~x!5Eux2r1u<R
d3r 1 f ~C15r1! (10)
where d3r 15differential volume element within the grain centered atr1. We defineL as the characteristic length scale fvariation of macroscopic quantities. For slow variations off (C1
5r1) at this macroscopic scaleL, Eq. ~10! can be expanded as
nS~x!5Eux2r1u<R
d3r 1 f ~C15r1!
5yF f ~x!1R2
5¹2f ~x!1O~R4!G (11)
wherey5volume of a single grain. Neglecting terms of the ordo(R/L)2 and higher, we obtain
nS~x!5y f ~x! (12)
We further define a conditional volume fractionnMF,S for a speci-
fied configuration ofM grains
nMF,S~x!5nF,S~x;CM !
51
~N2M !! f ~CM ! E dCN2M f ~CN!xF,S~x!
51
~N2M !! E dCN2M f ~CN2M !xF,S~x! (13)
We note that the conditional volume fraction also satisfiessaturation conditionnM
F 1nMS 51.
Let uF,S(x;CN) be any flow quantity pertaining to the fluid osolid phase at positionx in the presence of the configurationCN
of the grains. The phase-ensemble average ofuF,S is defined byaveraging over all the configurations such that the pointx is in theappropriate phase
^uF,S&~x!51
nF,SN! E dCN uF,S~x;CN!xF,S~x! f ~CN! (14)
The conditional average ofuF,S(x;CN) on the specified configu-rationCM of M grains can be similarly defined~Didwania and deBoer 1999!.
It can be easily shown that
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¹^uF&5^¹uF&2¹nF
nF ^uF&
11
nF Eux2r1u5R
dSr 1 nr 1f ~C1!^uF&1~x! (15)
where^uF&15average ofuF conditional to the presence of a gracentered atr1; dSr 15differential surface element of this grainandnr 15unit normal directed out of the grain. The integrationover all the grains touching the pointx. Similar expressions canbe obtained for gradient of conditionally averaged quantit¹^uF&M as derived in Didwania and de Boer~1999!.
Rearranging Eq.~15!
¹^uF&5^¹uF&11
nF Eux2r1u5R
dSr 1 nr 1f ~C15r1!
3@^uF&1~x;C1!2^uF&~x!# (16)
The average ofuS(x;CN) pertaining to the solid phase is giveby Eq. ~14! and by virtue of the identity of the grains reduces
^uS&~x!51
nS~x! Eux2qu<Rd3q f~C15x!^uS&1~q;x! (17)
whereq5point within the grain. We note thatf (C15x) is alsothe local number density of particles by Eq.~12!.
Ensemble-Averaged Fluid-Phase Momentum Balance
Next we derive continuum momentum balance equations forfluid phase using these concepts that we have developed. Focase of quasistatic limit with no relative velocity between the tphases, as considered here, we ignore the fluid viscosity anertia contributions. The fluid pressure, at any pointq in the fluid,is then given as
¹pF~q;CN!50 (18)
By conditionally ensemble averaging over Eq.~18!, we have byEq. ~16! a hierarchy of equations withN members given as~Did-wania and de Boer 1999!
¹^pF&21
nF Eux2r1u5R
dSr 1 nr 1f ~C15r1!
3@^pF&1~x;C1!2^pF&~x!#50 (19)
¹^pF&M21
nMF E
ux2rM11u5RdSr M11 nr M11f ~CM11!
3@^pF&M11~x!2^pF&M~x!#50 (20)
with M51, . . . ,N22.To obtain the continuum level description, in principle, w
have to solve this hierarchy sequentially to obtain^pF&1 . Once^pF&1 is obtained, we have an ensemble-averaged balance ofmentum, Eq.~19!, for the fluid phase.
Ensemble-Averaged Solid-Phase Momentum Balance
Next we direct our attention to the solid phase. We allow the smaterial to have arbitrary constitutive behavior and proceed wthe force balance on a single grain located at positionx is
2Eux2qu5R
dSq nqpF~q;CN!1Fc~x;CN!50 (21)
866 / JOURNAL OF ENGINEERING MECHANICS / AUGUST 2002
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wherepF(q;CN)5fluid pressure at a pointq in the presence of aspecific grain configurationCN with one of the grains located atxandFc5sum of all contact forces due to other grains in contwith this particular grain.
Upon averaging this force balance according to Eq.~14! weobtain
2 f ~C15x!Eux2qu5R
dSq nq^pF&1~q;x!1“•TSC50 (22)
with TSC5contact stress among the grains and can be compexactly from the knowledge ofFc ~see Zhuang, et al. 1995; Goddard and Didwania 1997; and Didwania, et al. 2001 for riggrains!. If the contact area between the grains is not negligiblethe contact area depends on pore pressure in some mannesituation is more complex. The case of significant grain conarea has been treated by de Buhan and Dormieux~1999! recently.If the grain contact area depends on pore pressure, one exTSC also to depend on pore pressure in general. The utility ofconcept of effective stress in this case will then be questionaas none of the expressions~1! or ~2! will be valid. We restrictourselves here to the case of point contact between the grFurthermore, we note that by averaging the single grain fobalance@Eq. ~22!#, the role of fluid pressure is separated from tbeginning, a key aspect of the concept of effective stress.
Expanding¹^pF& by Taylor’s series expansion around thgrain center,x @in o(R/L)#
¹^pF&~q!5¹^pF&~x!1pR2
10¹¹2^pF&~x!1O~R4! (23)
which leads to
Eux2qu5R
dSq nq^pF&~q!
5y¹^pF&~x!1ypR2
10¹¹2^pF&~x!1O~R4! (24)
Homogeneous Porous Media
Neglecting higher-order terms@o(R/L)2# in Eq. ~24!, and usingEq. ~11! we write Eq. ~22! after some manipulation~Didwaniaand de Boer 1999! as
2nS¹^pF&~x!2nSEux2qu5R
dSq nq@^pF&1~q;x!,
2^pF&~q!] 1“•TSC50 (25)
We note that once we knowpF&1 from Eqs. ~19!–~20!, inprinciple, we can evaluate the first integral assuming that it dnot diverge. Evaluation ofpF&1 requires solution of the hierarchof Eqs. ~19!–~20!, which is a very difficult task and can be accomplished numerically at present only for a limited numberparticles. An useful approach has been to find a physically meingful closure to these sets of equations, also known as effecmedium approximation. However, we take an alternativeproach and try to cast this term in a more familiar form befodiscussing specific closure.
The terms under the integral sign in Eqs.~19! and ~25! looksimilar and are of opposite sign but do not cancel each otheEq. ~25!, the integration is carried out over the surface of a pticle centered atx, while in Eq.~19! the integration is over all the
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particles that touch the pointx. The integrand of the term in Eq~19! depends strongly on the distance from the particle centeronly weakly on the positionr1 of the center itself and thereforeTaylor series expansion can be carried out in this variable.brevity, denote the bracketed quantity in the integrand of Eq.~19!by F(x;r1) and let x5r11s. Then we have~Didwania and deBoer 1999!
f ~r1!F~r1s;r1!5 f ~x!F~x1s;x!
2s•“x@ f ~x!F~x1s;x!#1¯ (26)
By changing the integration variablex1s to q we find that thefirst term in the expansion is equal to the integral term in Eq.~25!.The second term in the expansion corresponds to an extra sterm.
Making the appropriate substitution in expression~25! byusing Eq. ~26!, we obtain the two overall momentum balanequations as
Fluid Phase
“•TF1IF50 (27)
Solid Phase
“•TS1IS50 (28)
where
TF52nF^pF&I (29)
IF5^pF&¹nF1Eux2r1u5R
dSr 1 nr 1f ~C15r1!
3@^pF&1~x;C1!2^pF&~x!# (30)
TS52nS^pF&@ I2TSA#1TSC (31)
IS52IF5^pF&¹nS2Eux2r1u5R
dSr 1 nr 1f ~C15r1!
3@^pF&1~x;C1!2^pF&~x!# (32)
and
TSA5R
y^pF& Eux2qu5RdSq nqnq@^pF&~q!2^pF&1~q;x!# (33)
As mentioned earlier, the integral term in Eqs.~30! and~32! isover all the particles that touch the pointx. In an isotropic porousmedia this integral term, that is the contribution@^pF&1(x;C1)2^pF&(x)# averaged over all possible particle orientationspoint x, is identically zero. Hence, the set of Eqs.~29!–~33! sim-plify to
TF52nF^pF&I (34)
IF5^pF&¹nF (35)
TS52nS^pF&@ I2TSA#1TSC (36)
IS52IF5^pF&¹nS (37)
and
TSA5R
y^pF& Eux2qu5RdSq nqnq@^pF&~q!2^pF&1~q;x!# (38)
This leads to the expression for total stress as
J. Eng. Mech. 2002
t
ss
T5TF1TS52^pF&@ I2TSA#1TSC52^pF&I
1R
y Eux2qu5RdSq nqnq@^pF&~q!2^pF&1~q;x!#1TSC
(39)
Special Cases
Rigid Grains and Incompressible Fluid
We assumepF&15^pF& as a closure to Eqs.~19! and ~20!, i.e.,the fluid phase pressure on the surface of any grain is the severywhere and is equal to the pore pressure^pF&. Hence, for thiscase,TSA50, and since the particles are rigid, the Terzaghi’sfective stress expressionTSE5TSC5T1^pF&I is obtained.
Elastic Grains and Incompressible Fluid
We examine the approximations under which Biot and Wil~1957! results can be derived from our expressions. Withinframework of geometrically linear theory, the volumetric chanof the partial solid phaseeS is obtained by addingeSR, the vol-ume change resulting from that of the grains toeSN, the volumechange resulting from rearrangement of the grains,~de Boer andDidwania 2001!
eS5eSN1eSR (40)
The two experiments chosen by Biot and Willis~1957!, jacketedand unjacketed compressibility tests suggest that for a binmodel with compressible skeleton and fluid
TCS52nSKeSNI (41)
and
^pF&52KSeSR (42)
We once again seek a closure to Eqs.~19! and ~20! of the form,^pF&15^pF&, i.e., the fluid phase pressure on the surface of agrain is the same everywhere and is equal to the pore pres^pF& implying TSA50.
Substituting Eqs.~41! and~42! in Eq. ~36! and rearranging weobtain
nSKeSN5pS2nSKSeSR (43)
Thus, with Eq.~43!, Eq. ~40! yields
nSKeS5pS2nSKSeSR1nSKeSR (44)
5pS2nSKSeSRS 12K
KSD (45)
Making use of Eq.~45! after some rearrangement~Lade and deBoer 1997; de Boer Didwania 2001! we have for the total stress
TS1F5TS1TF52ns^pF&S 12KS
KSRD I1TES (46)
Other Cases
In general, the closurepF&15^pF& may not be valid and maydepend on the rheology of the grain. In such a situation, oneto choose the appropriate closure. In addition, the expression~39!is valid for compressible fluids also to the extent that the flu
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phase pore pressure variation can be approximated by onlyleading-order term in Eq.~24!. The set of expressions derivehere provides a closed set of equations providing all the strewhich can be computed directly from microstructural details.
Inhomogeneous Porous Media
A fluid-saturated porous media can be inhomogeneous durapid variation porosity,nS. In this case, the higher-gradient termin Eq. ~11! cannot be neglected. Inhomogeneities can also afrom rapid variation in fluid pressure in the pores even in a pormedia with homogeneous porosity, e.g., in compressible fludue to fluid density or temperature variations. In this particucase, the higher-gradient terms in Eq.~24! cannot be neglected. Inany of these situations, the analysis of the previous section ilonger valid. We have to account for gradient terms in porosand pore pressure and these terms will arise in the expressiototal stress given in Eq.~39!. This emphasizes, as well, as pointo suitable modification of effective stress expression by retainhigher-order terms in the ensemble-averaging framework.
Conclusion
In summary, we have surveyed the key developments in thecept of effective stress with an emphasis on the ensemaveraging based micromechanical approach. Various assumpunderlying the effective stress for special cases like rigporoelastic solid material saturated by incompressible fluid unhomogeneous conditions are elucidated. For inhomogeneousrated porous material the concept of effective stress needs tgeneralized by introducing higher gradients of porosity, pore psure, etc.
Acknowledgments
The writer acknowledges the partial support of U.S. National Sence Foundation Grant Nos. INT-9605036 and CTS-9510121NASA Grant No. NAG 3-1988 toward completion of the presework.
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