computational homogenization of uncoupled consolidation in micro-heterogeneous porous media

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2010; 34:1431–1458Published online 8 December 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.862

Computational homogenization of uncoupled consolidation inmicro-heterogeneous porous media

Fredrik Larsson, Kenneth Runesson and Fang Su∗,†

Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden

SUMMARY

Variationally consistent homogenization is exploited for the analysis of transient uncoupled consolidationin micro-heterogeneous porous solids, whereby the classical approach of first-order homogenization forstationary problems is extended to transient problems. Homogenization is then carried out in the spatialdomain on representative volume elements (RVE), which are introduced in quadrature points in standardfashion. Along with the classical averages, a higher-order conservation quantity is obtained. An iterativeFE2-algorithm is devised for the case of nonlinear permeability and storage coefficients, and it is appliedto pore pressure changes in asphalt-concrete (particle composite). Various parametric studies are carriedout, in particular, with respect to the influence of the ‘substructure length scale’ that is represented by thesize of the RVE’s. Copyright q 2009 John Wiley & Sons, Ltd.

Received 3 April 2009; Accepted 1 September 2009

KEY WORDS: computational homogenization; uncoupled consolidation; FE2

1. INTRODUCTION

In order to account for the effect of the material substructure in continuum modeling of solids,one may carry out analytical or computational homogenization on a representative volume element(RVE). This notion presumes complete scale separation such that the subscale solutions interactwith the macroscale only via their homogenized results, typically via a suitably defined conservationlaw involving the balance of macroscale fluxes, cf. Zohdi and Wriggers [1, 2], Miehe et al.[3–5], Fish et al. [6–10]. It is important to note that the subscale modeling, e.g. size, boundaryconditions and finite element discretization of the RVE, represents model errors in the correspondingmacroscale ‘constitutive relations’. In practice, computational homogenization is carried out such

∗Correspondence to: Fang Su, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96Goteborg, Sweden.

†E-mail: [email protected]

Contract/grant sponsor: The Swedish Research Council for Environment, Agricultural Sciences and Spatial Planning;contract/grant number: 245-2006-1223

Copyright q 2009 John Wiley & Sons, Ltd.

1432 F. LARSSON, K. RUNESSON AND F. SU

Figure 1. Illustration of a meso-macro-scale representation of seepage in a micro-heterogeneousporous solid. On the meso-scale, the RVE occupying �� consists of the porous sand/bitumen

matrix with the ballast as inert inclusions.

that the material response is obtained as part of macro-subscale transitions in an ‘nested’ (iterative)procedure for a component subjected to the actual loading.

The aim of this paper is to introduce a variational framework for first-order homogenizationin the spatial domain as applied to the uncoupled pore pressure equation.‡ An application ofsignificant engineering importance is asphalt-concrete, cf. Figure 1. The mesostructure is composedof ballast particles (stones) embedded in a sand/bitumen matrix that is treated as a fluid-saturatedporous medium with nonlinear material properties. In particular, we focus on the size-effect on themacroscale transient response that the micro-structural length scale introduces. An investigation ofthe importance of the size-effect for the similar (but somewhat simpler) problem of transient heatconduction in micro-heterogeneous material was presented by Larsson et al. [11]. This effect maybe ignored a priori, cf. Ozdemir et al. [12], which is often (but not always) a good approximation.

The paper is organized as follows: The pore pressure equation pertinent to ‘uncoupled consol-idation’ is derived in Section 2, and the corresponding space–time variational formulations ofthe multiscale problem are established in Section 3. In Section 4 we introduce the notion ofmacro/mesoscale decoupling upon identifying a macrohomogeneity condition that generalizes theclassical condition of Hill–Mandel to the present class of time-dependent problems. The FE-discretized problem in space–time is considered in Section 5, whereby time integration is carriedout on both the subscale and macroscale using the dG(0)-method. Particular attention is given to thederivation of the macroscale algorithmic tangent forms used in the macroscale Newton iterationsas part of a nested (two-level) macro-subscale computational/iterative scheme. In Section 6, wepresent results from ‘nested’ macro-subscale, i.e full-fledged FE2, computations for the seepage

‡By ‘first-order homogenization’ we here denote the standard assumption that the (scalar) primary variable varies atthe most in a linear fashion within the RVE, i.e. the local gradient field is constant within the RVE.

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2010; 34:1431–1458DOI: 10.1002/nag

COMPUTATIONAL HOMOGENIZATION OF UNCOUPLED CONSOLIDATION 1433

of fluid in asphalt-concrete. The paper is concluded in Section 7 by some final remarks and anoutlook to future work.

2. UNCOUPLED PORE PRESSURE EQUATION—A MODEL PROBLEMOF TRANSIENT SEEPAGE

2.1. Basic relations for binary porous solid

We give a brief summary of the relevant porous media theory (PMT), as proposed by e.g. Ehlerset al. [13], and the simplifying approximations leading to the so-called ‘uncoupled pore pressureequation’. A binary medium consisting of a porous solid with fluid-saturated pores is considered.Superscript S and F are used to denote variables associated with the solid skeleton (S) and thesaturating pore fluid (F), respectively. Small deformations are assumed. The solid phase is assumedto be intrinsically incompressible, while (a small) compressibility of the fluid is assumed.§ Asa result, the porosity (here denoted �) is taken as constant, and the bulk densities are given as�S=[1−�]�S and �F=��F. The relative velocity of the fluid phase (relative to the solid skeleton)

is denoted by wdef= vF−vS, while the seepage velocity (flux) that represents the rate of volume

transport of the fluid mixture is wdef= �w. Finally, we introduce the notation u

def= uF for the porefluid intrinsic pressure.

The pertinent balance equations for the quasi-static motion of the two-phase porous medium are

−r·∇− �g= 0 (1)

dt�+w·∇ = 0 (2)

where (1) represents equilibrium of the two-phase medium, whereas (2) represents the massconservation of the two-phase mixture. Equation (2) is obtained upon combining the continuityequations for the two phases (S,F). It can then be shown, e.g. [13] that the storage function, �, isgiven as

�=�

[1+ u−u0

K F

]+�vol (3)

The total bulk density of the two-phase medium (mixture) is �= �S+ �F. In (1) we introduced thetotal (equilibrium) bulk stress, r, which can be additively split as follows: r=r′−uI, where r′is the classical effective stress according to von Terzaghi and I is the identity tensor. Moreover,we introduced K F as the constant intrinsic compression modulus of the pore fluid (pertinent to a

linearized constitutive assumption for the compression property), and �voldef= u·∇ is the volumetric

strain of the bulk (solid skeleton) in terms of the displacement u. Further, ∇ is the spatial gradientwith respect to coordinates x in �, whereas dt denotes time derivative. Subscript ‘0’ representsthe initial value at t=0, e.g. u0(x) is the initial pore pressure distribution before application ofany external loading.

§Since the fluctuation of �F is small, it is ignored in the momentum equation (1).



In order to reduce the system (1), (2) to an ‘uncoupled pore pressure equation’ based on (2),we first introduce the tacit assumption that the solid skeleton responds as an isotropic solid, whichis characterized by the compression modulus, K , and the shear modulus, G. Hence, we have therelation

r′ =r′0+2Gεdev+K �volI, (4)

where εdef= (u⊗∇)sym is the strain and the subscript ‘dev’ indicates deviator. This means that we

may express the total (equilibrium) stress as

r= r0+2Gεdev+[K �vol−[u−u0]]I (5)

where r0def= r′0−u0I. In particular, we may express the mean stress �m

def= 13 r : I as

�m = �m,0+K �vol−[u−u0] (6)

Using the relation (6), we may eliminate �vol in (3) to obtain the alternative expression

�=�+[1

K+ �

K F

][u−u0]+ 1

K[�m− �m,0] (7)

where it is noted that �m is a (state) function of u and u.The pertinent initial condition is �=�0=� when t=0, and we may adopt the assumption

(without loss of generality) that u0=0, which gives ε0=0.

2.2. Uncoupled pore pressure equation

We shall next introduce the crucial assumption that it is possible to compute the equilibrium stressfield in space/time, �m(x, t), approximately via a suitable assumption: One such viable assumptionis that �m(x, t) can be computed from the undrained condition, i.e. w=0, prevailing throughoutthe entire duration of the loading process.¶ This condition means that � is conserved, �=�, andwe obtain from (3) that the corresponding pore pressure field, u∗, can be expressed in terms ofthe volumetric strain, �∗vol, as follows:

u∗ =u0− K F

��∗vol (8)

Upon introducing this relation into (5), we obtain the constitutive relation that is pertinent to aso-called ‘total stress analysis’ for a single-phase solid, i. e. by assigning elastic properties to theequilibrium stress r∗, as follows:

r∗ = r0+2G∗εdev+K ∗�∗volI (9)

where we introduced the ‘fictitious (undrained) elastic moduli

G∗ =G, K ∗ =K + K F

�(10)

Replacing r by r∗ in (1), while adopting (10), we may compute (in particular) the field �∗m(x, t).

¶This approach is classical in the extreme situation of loading that is applied instantaneously at t=0 and keptstationary, cf. Christian and Boehmer [14].



It is now possible to consider (2) as the ‘uncoupled pore pressure equation’ with

�(u)def= �+

[1

K+ �

K F

][u−u0]+ 1

K[�∗

m− �m,0] (11)

where �(u) is the relevant (bulk)volume-specific storage function. It is noted that we have theinitial condition �∗

m(x,0)= �m,0(x).As to the constitutive (permeability) relation for the seepage velocity w, a quite general expression

for isotropic seepage is as follows:

w=−k(u,∇u)[∇u−�Fg] (12)

Finally, it may conveniently be assumed that the seepage velocity vanishes at t=0 before anyadditional loading has been applied. It then follows from (12) that the initial value of the porepressure, u0, must satisfy the condition ∇u0=�Fg. To simplify the formulation, without losingany generality, we shall set u0≡0 in this paper corresponding to the assumption �F≈0. In otherwords, we may interpret u as the excess pore pressure and w(u,∇u)=−k(u,∇u)∇u. If we, inaddition, assume that �S≈0, then a possible solution for the initial stress field is r0=0. As aresult, �(u) in (11) is simplified as

�(u)=�+[1

K+ �

K F

]u+ 1

K�∗m (13)

3. THE VARIATIONAL MULTISCALE METHOD FOR TRANSIENT SEEPAGE

3.1. Space–time variational format

We consider the spatial (macroscale) domain � with boundary �. Our (ultimate) aim is to resolvethe pore pressure field u(x, t) for (x, t)∈�×[0,T ] while accounting for the subscale effects.As a step toward establishing the full space/time variational format of (11)1, we introduce thetime-dependent spatial variational forms

a(u;�u)def= −

∫�w·∇[�u]dV, l(�u)

def= −∫

�N

qp�u dS (14)

where (14)2 represents the prescribed mass flux, qdef= w·n=qp, out from the Neumann part of

the boundary, �N (n is the outward unit normal to �). The prescribed pore pressure, u=u p, onthe Dirichlet part of the boundary, �D, is satisfied by the appropriate choice of solution space instandard fashion; see below. We also introduce the scalar product between two scalar-valued fieldsu and v, as well as between two vector-valued fields u and v, as follows:

(u,v)def=

∫�uv dV, (u,v)

def=∫

�u·vdV (15)



We are now in the position to establish the relevant space/time variational forms

A(u;�u)def=

∫ T

0(dt�,�u)dt+

∫ T

0a(u;�u)dt+(�(u|t=0+),�u|t=0+) (16)

L(�u)def=

∫ T

0l(�u)dt+(�0,�u|t=0+) (17)

The initial value �0 in (17) is included for completeness only; in reality �0=0.As to the solution space U and the corresponding test space U0, which contain the subscale

fluctuation features, they are defined as U={u∈H(�×[0,T ]) :u=u p on �D} and U0={u∈U :u=0 on �D}, where H denotes the appropriate (Hilbert) space of sufficiently regular functionsin the whole space/time domain �×[0,T ]. We may now formulate the space/time variationalproblem corresponding to the strong format (11)1 as follows: Find u∈U such that

A(u;�u)= L(�u) ∀�u∈U0 (18)

For later use, we also introduce the residual for any given function u′ ∈U as

R(u′;�u)= L(�u)−A(u′;�u) ∀�u∈U0 (19)

and it follows from (18) that R(u;�u)=0 ∀�u∈U0, where u is the exact solution.

3.2. Model-based homogenization—preliminaries

In order to reduce the excessive effort in resolving the fine-scale representation embedded in U, itis not feasible to solve (18) directly. Rather, it is convenient to introduce separation of the macro-and subscales. The classical approach (which is adopted in this paper in principle) is to introduce‘model-based homogenization’, which gives rise to a continuous macroscale problem in a globallydefined ‘smooth’ field u =u.

RemarkA different approach, recently proposed by Larsson and Runesson [15], admits seamless scale-bridging and is based on ‘discretization-based homogenization’. With that approach, the modelerror introduced by homogenization can, in fact, be estimated as part of a ‘model-adaptive’ strategy.

In the model-based homogenization paradigm, the local field is replaced (as a smoothing approx-imation) by the spatially homogenized field:‖

y(x, t) �→〈y〉�(x, t), x∈�, t�0 (20)

whereby it is assumed that a RVE occupies the subscale region �� (with boundary ��).∗∗ TheRVE is centered at the macroscale position x, i.e. 〈x− x〉� =0 for any given x∈�. In standardfashion, the RVE is assumed to be large enough to be statistically representative for the fine-scale

‖The subscale volume average on �� of a function y is denoted as

〈y〉�(x)def= 1

|��(x)|∫��

y dV, x∈�.

∗∗Henceforth, the argument x is suppressed unless there is a risk of confusion.



features, yet small compared with the smallest dimension of the macroscale domain � in orderto allow x to be considered as a continuous coordinate. From (20) now follows that any spatialvolume integral can be represented as∫

�y dV ≈

∫�〈y〉� dV (21)

Similarly, it is convenient to introduce homogenization on the boundary � as††

y(x, t) �→〈〈y〉〉�(x, t), x∈�, t�0 (22)

where the representative surface element (RSE) occupies the subscale region ��. Consequently,∫�y dS≈

∫�〈〈y〉〉� dS (23)

In practice, numerical quadrature is employed at the evaluation of integrals in the spatial domain.Hence, homogenization on the RVE’s is carried out (only) in these macroscale quadrature points.

With the introduced homogenization, the original problem formulation remains formallyunchanged if integrands are replaced by the homogenized quantities in all space-variational forms.Hence, (14) and (15) are replaced by

a(u;�u)def= −

∫�〈w ·∇[�u]〉� dV, l(�u)

def= −∫

�N

〈〈qp�u〉〉� dS (24)

(u,�u)def=

∫�〈u�u〉� dV, (u,�u)

def=∫

�〈u·�u〉� dV (25)

The key assumption in model-based homogenization is that of separation of scales. That is, forany point x∈�, there exists an independent RVE occupying the domain ��. A consequence ofthis fact is that, for finite RVE-size, the RVEs for two sufficiently close macroscale points are, infact, ‘overlapping’.

Inside each RVE, the subscale solution:‡‡ u(x;x, t)=uM (x;x, t)+us(x;x, t) is split into onesmooth part, uM , and subscale fluctuations, us. The scales are linked by setting uM ≈ u inside eachRVE, where u is the smooth (macroscale) solution defined on �. The steps of connecting uM ≈ u

and computing§§ usdef= us{uM } for given uM define the prolongation and allow for computing the

homogenized quantities in (24) and (25).With the redefinition of the pertinent space-variational forms and the prolongation discussed

above, we now seek to find the macroscale (smooth) solution u∈U that solves (18). Obviously,the solution u∈U (which is still continuous but approximate) will be smoother than the exactsolution u∈U pertaining to the original problem. However, the formal regularity requirementson U will depend on the special model assumption for the homogenization (i.e. prolongation) asdefined later.

††The subscale surface average on �� of a function y is denoted 〈〈y〉〉�(x)def= 1

|��(x)|∫��

y dS, x∈�.‡‡Double arguments, e.g. u(x,x, t), are used to explicitly point out the underlying scale separation. Hence, for anymacroscale coordinate x, there exists an independent function u(x,•, t) defined on ��.

§§Curly brackets {(•)} indicate implicit functional dependence on (•).



4. VARIATIONALLY CONSISTENT FIRST-ORDER HOMOGENIZATION

4.1. Preliminaries

The homogenization properties are defined explicitly via a suitable assumption about the smooth-ness of the macroscale solution uM .¶¶ First-order homogenization will be assumed, i.e. it is assumedthat uM varies only linearly (derivatives up to first-order are included) within the RVE, as shownschematically in Figure 2, and can thus be expressed as

uM (x;x, t)= u(x, t)+ g(x, t) ·[x− x] for x∈�� (26)

where u∈U is the (global) macroscale temperature field and where gdef= ∇u is the macroscale

temperature gradient. From (26) we may directly compute the identities

u(x, t)=uM (x; x, t), g(x, t)def= (∇u)(x, t)=(∇uM )(x; x, t) (27)

whereby it is noted that the gradient ∇uM is constant in ��, and where we introduced themacroscale gradient ∇ with respect to the coordinate x.

RemarkIt holds that

u(x, t)=〈uM (x;x, t)〉�, g(x, t)=〈(∇uM )(x;x, t)〉� (28)

which is a direct consequence of the definition in (27) and not an extra requirement or assump-tion. However, u =〈u〉� and g =〈∇u〉� =〈g〉� since 〈us〉� =0 and 〈∇us〉� =0 in general (beforespecifying boundary conditions on the RVE-problem).

As to the corresponding variations, we obtain directly from (26)

�uM (x;x, t)=�u(x, t)+�g(x, t) ·[x− x] for x∈�� (29)

with the obvious identity �g(x, t)def= (∇[�u])(x, t)=(∇[�uM ])(x; x, t).

Next, we introduce the spaces UM and UM,0 as follows:

UM ={uM :uM (x,x, t)= u(x, t)+ g(x, t) ·[x− x], u∈U} (30)

RemarkThe task of finding uM ∈UM is equivalent to that of finding u∈U.

We may then replace (18) with the (approximate) problem of finding uM ∈UM that solves

A(u{uM };�u)= L(�u) ∀�uM ∈UM,0 (31)

where it was tacitly used that the variation‖‖ �u can be defined implicitly as the linear functionalin terms of �uM and can be expressed as

u{uM }=uM +us{uM }⇒�u=�uM +(us)′{uM ;�uM } (32)

¶¶Although such an evaluation is carried out only in the spatial domain in this paper, it is possible (although notentirely straightforward) to extend the strategy to space–time homogenization (future work).

‖‖We introduced the variation as the Gateaux-derivative (us)′{uM ;�uM }def= (�/��)us{uM +��uM }|�=0.



Figure 2. Illustration in 1D of assumed linear variation of uM (x; x, t) for x ∈�� due to strong subscaleheterogeneity (illustrated by p.w. constant k).

In this manner it is concluded that dim(U)=dim(UM ), since it is possible to determine the field�us=(us)′{uM ;�uM } for each possible choice of uM . Hence, we conclude that the ‘constrained’problem (31) has the same size as the original ‘macroscale problem’ (18). It is noteworthy thatthe ‘Galerkin-type’ formulation in (31) preserves any underlying symmetry of the problem, cf. thediscussion in Larsson and Runesson [15].

We shall simplify the subsequent analysis by henceforth assuming that u is sufficiently smoothon � to allow the approximation us=0 (and �us=0) on �. Moreover, we restrict to the situation thatthe boundary values u p and qp vary sufficiently slowly to represent the macroscopic (homogenized)values in the sense that

〈u〉� = u p on �D and 〈〈qp�u〉〉� = qp�u on �N (33)

where u p and qp are prescribed boundary values. In particular, the condition (33)2 infers that〈〈qp〉〉� = qp and 〈〈qp[x− x]〉〉� =0. In this way certain technical difficulties associated with thehomogenization on the boundary are avoided.

RemarkIt is noted that the weaker condition, as compared with (33)2, 〈〈qp�u〉〉� =〈〈qp�uM 〉〉� in theexpression for l(�u) in (24)2, will be used below at the discussion of the Hill–Mandel macroho-mogeneity condition.



4.2. Variationally consistent homogenization based on a generalized macro-homogeneitycondition

The conventional strategy is to solve for uM from (31) upon using the homogenizing propertyof �uM , which requires a condition known in the literature, e.g. Hori and Nemat-Nasser [16],as the Hill–Mandel macro-homogeneity condition. It assures that the virtual work on the macro-and subscales coincides. As a result, homogenized quantities can be derived for the flux andconservation quantities. In order to obtain the pertinent generalization of the Hill–Mandel conditionto a quite general class of problems (including the presently considered nonlinear time-dependentproblems of the diffusion type), we may expand (31) while using the relations in (32) to obtainthe homogenized macroscale problem: Find uM ∈UM that is the solution of

A(u{uM };�uM )−R(u{uM };(us)′{uM ;�uM })= L(�uM ) ∀�uM ∈UM,0 (34)

where we used the implicit relations us{uM }. Hence, the residual in (34) is with respect to aconsistent variation of the subscale fluctuations us{uM }.

Next, we make the important assumption that it is possible to choose the prolongation conditionsin such a fashion that the residual in (34) vanishes identically, i.e.

R(u{uM };(us)′{uM ;�uM })=0 ∀�uM ∈UM,0 (35)

As a result, (34) can be replaced by the simpler (homogenized) problem: Find uM ∈UM that isthe solution of

A(u{uM };�uM )= L(�uM ) ∀�uM ∈UM,0 (36)

In practice, (36) and the pertinent subscale problem, whose solution is us{uM }, must be solved ina ‘nested’ fashion via some sort of iteration.

RemarkAlthough not usually written in this fashion, the condition (35) represents a generalization of theHill–Mandel macro-homogeneity condition to the present class of boundary-initial value problems.This may be shown as follows: With the decomposition in (32)1, we reformulate (35) as

A(u{uM };�u)= A(u{uM };�uM )+L((us)′{uM ;�uM }) ∀�uM ∈UM,0 (37)

The standard case is that l(�u)=0 and �0=0, whereby it is concluded that L((us)′{uM ;�uM})=0for any choice of test function. Hence, (37) reduces to an identity between the ‘internal virtual work’in space–time for the original problem involving the ‘fine’ scale (without any scale separation)and the homogenized problem, and this identity is obviously a consequence of the condition (35).In fact, this is the classical format of the Hill–Mandel macro-homogeneity condition in the caseof stationary (static) problems.

As to the explicit formulation of the macro-homogeneity condition in terms of the vanishingresidual in (35), we first note that the residual for any test function �us can be expressed in termsof contributions from each RVE as follows:

R(u;�us)=∫

�R�(u;�us)dV (38)

where it was used that (i) 〈〈qp�us〉〉� =0 (since �us=0) on �N and (ii) R�(u;�us) is the space–time residual localized to each RVE. Upon integration by parts on the RVE and using the strong



format of the mass balance, as given in (2), we may show that the space–time residual on a givenRVE can be expressed as

R�(u;�us) def=∫ T

0[〈dt��us〉�−〈w·∇[�us]〉�]dt+〈[�(u|t=0+)−�0]�us|t=0+〉�

= −∫ T

0

[1

|��|∫

��qn�u

s dS

]dt (39)

where qn=w·n, with wdef= w(u,g), is the outward mass flux on ��. A sufficient (strong) condition

for the Hill–Mandel mcrohomogeneity condition (35) to hold is thus that R�(u{uM };(us)′{uM ;�uM })=0 on each RVE for each possible �uM ∈UM,0.

RemarkAn even stronger condition is that R�(u{uM };�us)=0 for any field �us in a given space offunctions defined locally without imposing any implicit (or explicit) coupling to the actual uM .In such a case, the condition (35) can be ascertained without actually computing the test function�us=(us)′{uM ;�uM } from the pertinent tangent problem.

4.3. Explicit homogenization results

Upon inserting the expression in (29) into (36), while observing the definitions in (14) to (15) andcollecting terms, we may define the macroscale problem in terms of the space-variational forms

(dt�(u),�uM ) = (dt �,�u)+(dt¯U,�g) (40)

a(u;�uM ) = −( ¯w,�g)def= −

∫�

¯w·�gdV (41)

l(�uM ) = l(�u)def= −

∫�N

qp�u dS (42)

The pertinent homogenized quantities of first and second order are given as the (implicit)relations in terms of the macroscale variables u and g:

¯w{u, g} = 〈w〉�, �{u, g}=〈�〉� (43)

¯U{u, g} def= 〈�[x− x]〉� =〈�x〉�−�x (44)

It is noted that � is scalar-valued, whereas ¯U is a vector-valued quantity.

RemarkAlthough the macroscopic quantities u and g are related by the standard kinematic relation g

def= ∇u,they are treated as independent in the implicit macroscopic relations (43) and (44).

RemarkFor the present class of problems, the linear approximation does indeed introduce a size effect inanalogy to the case of classical second-order homogenization, cf. Kouznetsova et al. [17]. This



fact becomes evident in the appearance of the ‘second-order’ term ¯U, and this quantity will vanishonly when the size of the RVE becomes infinitely small.

We are now in the position to establish the relevant macroscale space/time variational forms

A{u;�u} def=∫ T

0(dt �,�u)dt−

∫ T

0( ¯w,∇[�u])dt+(�|t=0+,�u|t=0+)

+∫ T

0(dt

¯U,∇[�u])dt+( ¯U|t=0+,∇[�u]|t=0+) (45)

L{�u} def=∫ T

0l(�u)dt+(�0,�u|t=0+)+( ¯U0,∇[�u]|t=0+) (46)

Finally, we may reformulate (36) as the macroscale balance equation: Find u(x, t)∈U that solves

R{u;�u}def= L{�u}− A{u;�u}=0 ∀�u∈U0

(47)

4.4. Dirichlet boundary conditions on fluctuation field

The most straightforward choice of prolongation condition is defined by homogeneous Dirichletboundary conditions on the fluctuation field us on each RVE, i.e. �us=0 on ��. We may thenintroduce the pertinent spaces for the subscale (local) solution as follows on any given RVE:U� =U�[u, g]={u∈H(��×[0,T ]) :u= u+ g·[x− x] on ��}, U0

� =U�[0, 0]={u∈U� :u=0on ��. As to the macro-homogeneity condition, we note that �us=0 on �� in (39), hence theresidual vanishes and the required constraint is satisfied.

We are now in the position to formulate the (local) subscale space–time variational problem onthe RVE associated with each macroscopic position x∈�: For given macroscale field uM , find thesubscale field us∈U0

� that solves

R�(uM +us;�us)def= L�(�us)−A�(uM +us;�us)=0 ∀�us∈U0� (48)

where we introduced the local space/time variational forms

A�(u;�u)def=

∫ T

0〈dt��u〉� dt+

∫ T

0a�(u;�u)dt+〈�(u|t=0+)�u|t=0+〉� (49)

L�(�u)def=

∫ T

0l�(�u)dt+〈�0�u|t=0+〉� (50)

The relevant time-dependent local spatial (within each RVE) variational forms are

a�(u;�u)def= −〈w·∇[�u]〉�, l�(�u)=0 (51)



5. FE-SOLUTION IN SPACE/TIME—NESTED APPROACH

5.1. FE-discretized format of the subscale problem

Upon employing the dG(0)-method (generalized Backward Euler method) for integrating thesubscale problem (48) in time, we introduce the finite element spaces∗∗∗ (in standard fashion)

U�,hdef= nU�,h associated with the time interval In =(tn−1, tn) whose length is �t= tn− tn−1. The

resulting FE-problem to be solved on the RVE for each interval In then reads: For n=1,2, . . . ,N ,find the solution nus∈U0

�,h , that is the solution of

R�(nuM +nus;�us) =∫Inl�(�us)dt−�ta�(nuM +nus;�us)−〈[n�−n−1�]�u〉�

= 0 ∀�us∈U0�,h (52)

where n�def= �(nu)=�(nuM +nus).

A generic algorithm for the iterative solution of (52) within the given timestep employs Newton’smethod. To this end, we first introduce the tangent form of the residual for the considered timestepas follows:

(R�)′(•;�us, du) = −〈�′(•; du)�us〉�−�ta′�(•;�us, du)

= −〈�us�′ du〉�+�t〈∇[�us]·Ydu〉�−�t〈∇[�us]·K·∇[du]〉�∀�us∈U0

�,h (53)

where we introduced the tangent forms

〈�′(•; du)�us〉� = 〈�us�′ du〉� with �′ def= d�

du= 1

K+ �

K F (54)

a′�(•;�us, du) = −〈∇[�us]·Ydu〉�+〈∇[�us]·K·∇[du]〉� (55)

with the tangent tensors Y and K defined via the differential relations

dw= Y(u,∇u)du−K(u,∇u) ·∇[du], Ydef= �w

�u, K

def= − �w�(∇u)

(56)

The iterative algorithm then reads: For k=1,2, . . . , compute the spatial fields

nus(k+1)(x)=nus(k)(x)+ dus(x), nus(0)(x)=0 (57)

where††† the iterative updates dus∈U0�,h are solved from the tangent equations

−(R�)′(nuM +nus(k);�us, dus)= R�(nuM +nus(k);�us) ∀�us∈U0�,h (58)

until the residual R�(nuM +nus(k);�us) is sufficiently small.

∗∗∗In this paper we assume the same (fixed) space mesh in each time-interval; hence, superindex n is droppedsubsequently.

†††The quantity uM (x) is defined below in (64).



5.2. FE-discretized format of the macroscale (homogenized) problem

In this paper we consider only the simplified situation when homogenization is carried out is space(but not in time). As a direct consequence, the time integration on the macro- and subscales maybe chosen identical. Applying dG(0) to the macroscale residual Equation (47), we then obtain the

problem: For n=1,2, . . . ,N , find the solution nu∈UHdef= nUH that is the solution of

R{nu;�u} =∫Inl(�u)dt+�t (n ¯w,∇[�u])−([n�−n−1�],�u)−([n ¯U−n−1 ¯U],∇[�u])

= 0 ∀�u∈U0H (59)

where n ¯wdef= ¯w{nu,∇[nu]}, n�def= �{nu,∇[nu]}, n ¯Udef= ¯U{nu,∇[nu]}.Again, Newton’s method gives the corresponding algorithm for the macroscale iterations: For

iterations K =1,2, . . . , compute the macroscale spatial fields

nu(K+1)(x)=nu(K )(x)+ du(x) (60)

where the iterative updates du∈U0H are solved from the tangent equations

−R′{nu(K );�u, du}= R{nu(K );�u} ∀�u∈U0H (61)

Upon introducing the following algorithmic relations (whose derivation is discussed in the nextsubsection):

d ¯w = Ydu−K· dgd� = C du+B· dgd ¯U = ¯Cdu+ ¯B · dg

(62)

we may compute the tangent form R′{•;�u, du} explicitly as follows:

R′{•;�u, du} = −∫

��uC du dV −

∫�

�uB·∇[du]dV +∫

�∇[�u]·[�tY− ¯C]du dV

−∫

�∇[�u]·[�tK+ ¯B]·∇[du]dV (63)

Note that C is a scalar quantity, Y, B, and ¯C are vectors, whereas K and ¯B are second-ordertensors. In addition, note that all these macroscale quantities are generally state-dependent, e.g.nC

def= C{nu,∇[nu]}.

5.3. Macroscale algorithmic tangent operators

5.3.1. General. Our aim is to establish the algorithmic operators in (62) for variations of themacroscale solution uM (expressed in terms of u and g). We shall then need to compute ‘unitfluctuation fields’ or, rather, sensitivity fields, corresponding to a unit variation of uM , i.e. of



u and g. Hence, we shall be interested in computing the differential du= duM + dus in terms ofdu and dg. In standard fashion, we introduce the representation (while suppressing the argumentsx and t)

uM (x)= u+ g·[x− x]= u+NDIM∑i=1

uM(i)(x)gi (64)

where the ‘unit fields’ uM(i) are given as

uM(i)(x)def= ei ·[x− x]= xi − xi (65)

Upon differentiating the relation for uM in (64) and introducing the following ansatz for dus:

dus(x)= usu(x)du+NDIM∑i=1

us(i)g (x)dgi (66)

we obtain the following representation for the total differential du in terms of the sensitivity fieldsusu∈U0

�,h and us(i)g ∈U0�,h , i=1, ,2, . . . ,NDIM:

du(x)=[1+ usu(x)]du+NDIM∑i=1

[uM(i)(x)+ us(i)g (x)]dgi (67)

We are now in the position to evaluate linearizations of ¯w, � and ¯U in turn. First, from theidentities in (43)1 and (51), the components of d ¯w can be expressed as

d ¯wi = d ¯w·ei = d[ ¯w·ei ]= d[〈w·ei 〉�]=−d[a�(•; uM(i))]=−a′�(•; uM(i), du) (68)

Upon inserting the representation of du from (67) into (68) and using the expansion of a′� in (55),

we obtain the desired algorithmic operators

(Y)i = −a′�(•; uM(i),1+ usu)=〈∇uM(i) ·Y[1+ usu]〉�−〈∇uM(i) ·K·∇usu〉�

= 〈(Y)i 〉�+〈(Y)i usu〉�−〈(K·∇usu)i 〉� (69)

(K)ij = a′�(•; uM(i), uM( j)+ us( j)g )=〈∇uM(i) ·K·∇us( j)g 〉�−〈∇uM(i) ·Yus( j)g 〉�

= 〈(K)ij〉�+〈(K·∇us( j)g )i 〉�−〈(Y)i [x j − x j ]〉�−〈(Y)i us( j)g 〉� (70)

Next, we consider the linearization of � and U(2). From (43)2 we obtain

d�= d〈�〉� =〈�′ du〉� (71)

Upon inserting the representation of du from (67) into (71), we identify the tangent operators

C = 〈�′[1+ usu]〉� =〈�′〉�+〈�′usu〉� (72)

(B)i = 〈�′[uM(i)+ us(i)g ]〉� =〈�′[xi − xi ]〉�+〈�′us(i)g 〉� (73)

Finally, from (44) we obtain

(d ¯U)i = d〈�[xi − xi ]〉� = d〈�uM(i)〉� =〈�′uM(i) du〉� =〈�′[xi − xi ]du〉� (74)



and upon inserting the representation of du from (67) into (74), we identify the tangent operators

( ¯C)i = 〈�′uM(i)[1+ usu]〉� =〈�′[xi − xi ]〉�+〈�′[xi − xi ]usu〉� (75)

( ¯B)ij = 〈�′uM(i)[uM( j)+ us( j)g ]〉� =〈�′[xi − xi ][x j − x j ]〉�+〈�′[xi − xi ]us( j)g 〉� (76)

RemarkAlthough � is assumed not to depend on ∇u, i.e. B

def= ��/�(∇u)=0, it appears that a ‘nonlocal’contribution is obtained upon carrying out the homogenization in the sense that, still, B =0.Likewise, even if we had employed a simplified constitutive law such that w does not depend on u,i.e. Y=0, there would still be a ‘nonlocal’ contribution to Y =0 from the fluctuation field, see below.

5.3.2. Dirichlet boundary conditions—tangent problem defining sensitivity fields within RVE. Whatremains is to establish the ‘tangent relations’ from which the sensitivity fields usu(x) and us(i)g (x)can be solved for any given time based on the FE-discrete format in space/time. To this end, weconsider the residual equation in (52), which must hold for any (variation of the) macroscale fielduM(x) in terms of du and dg. Hence, we have the conditions

R�(u;�us)=0, R�(u+ du;�us)=0 ∀�us∈U0�,h (77)

and upon linearizing, we obtain the pertinent tangent condition

(R�)′(•;�us, du)=0 ∀�us∈U0�,h (78)

Now, inserting the expansion of du from (67) into (78), and utilizing that the resulting expressionmust hold for any differential changes du and dg, we obtain the (two) sets of equations:

(i) Solve for usudef=n

usu∈U0�,h from

〈�′�ususu〉�−�t〈∇[�us]·Yusu〉�+�t〈∇[�us]·K·∇[usu]〉�=−〈�′�us〉�+�t〈∇[�us]·Y〉� ∀�us∈U0

�,h (79)

(ii) Solve for us(i)gdef=n

us(i)g ∈U0�,h, i=1,2, . . . ,NDIM, from

〈�′�usus(i)g 〉�−�t〈∇[�us]·Yus(i)g 〉�+�t〈∇[�us]·K·∇[us(i)g ]〉�=−〈�′�us[xi−xi ]〉�+�t〈∇[�us]·Y[xi−xi ]〉�−�t〈∇[�us]·K·ei ]〉� ∀�us∈U0

�,h

(80)

RemarkUnder stationary conditions (when such conditions prevail), we may simplify (79) and (80) to thefollowing two problems:

−〈∇[�us]·Yusu〉�+〈∇[�us]·K·∇[usu]〉� =〈∇[�us]·Y〉� ∀�us∈U0�,h (81)

−〈∇[�us]·Yus(i)g 〉�+〈∇[�us]·K·∇[us(i)g ]〉� =〈∇[�us]·Y[xi − xi ]〉�−〈∇[�us]·K·ei ]〉�∀�us∈U0

�,h (82)

from which the fields usu(x) and us(i)g (x), i=1, . . .NDIM, can be solved, respectively.



5.3.3. Dirichlet boundary conditions—tangent problem defining sensitivity fields within RVE forgeneralized Darcy’s seepage law. A straightforward generalization of the standard seepage modelby Darcy [18] to include nonlinear response is accomplished by assuming that the permeabilityconstant depends on ∇u, i.e.

w=−k(∇u)∇u (83)

As a consequence, Y=0. The following simplifications are then obtained in (69) and (70):

(Y)i = −〈(K·∇usu)i 〉� (84)

(K)ij = 〈(K)ij〉�+〈(K·∇us( j)g )i 〉� (85)

whereas C , B, ¯C, and ¯B remain formally unchanged as compared with (72), (73), (75), (76) withthe appropriate values of usu(x) and us(i)g (x), i=1,2, . . . ,NDIM.

As to the corresponding sensitivity fields, it is concluded that (79) is simplified as [correspondinga′�(•;�u,1)=0]

〈�′�ususu〉�+�t〈∇[�us]·K·∇[usu]〉� =−〈�′�us〉� ∀�us∈U0�,h (86)

whereas (80) is simplified as

〈�′�usus(i)g 〉�+�t〈∇[�us]·K·∇[us(i)g ]〉�=−〈�′�us[xi − xi ]〉�−�t〈∇[�us]·K·ei ]〉� ∀�us∈U0

�,h (87)

RemarkUnder stationary conditions (86) simplifies to

〈∇[�us]·K·∇[usu]〉� =0 ∀�us∈U0�,h (88)

from which it is concluded that usu=0. It is emphasized that this result is obtained only in the

special case of Darcy’s law and stationary conditions. As a result, we obtain Y=0, C=〈�′〉�, ¯C=〈�′[x− x]〉� such that, e.g. d ¯w=−K· dg. Moreover, (87) simplifies to

〈∇[�us]·K·∇[us(i)g ]〉� =−〈∇[�us]·K·ei ]〉� ∀�us∈U0�,h (89)

In the special case that the permeability tensor K is truly constant, then the sensitivity fields us(i)gare truly time-independent and can be solved (and stored) once and for all for each RVE.

6. COMPUTATIONAL RESULTS FOR ASPHALT-CONCRETE

6.1. Substructure characteristics of simplified particle composite arrangement

6.1.1. Preliminaries. We consider a nearly periodic micro-structure of particles embedded ina matrix, as shown in Figure 3. Without loss of generality but with considerable savings incomputational effort, we restrict the analysis to 2D (plane seepage), and we choose the size (side



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-8

-6

-4

-2

0

×104 Pa

(a)

(b) (c)

Figure 3. (a) Plane seepage due to symmetrically loaded half-space (considered in the compu-tational example) based on a subscale model of periodic arrangement of particles in matrix;

(b) distribution of mean stress ¯�Rm (as part of the computation of ¯�m(x, t)= ¯�R

m(x)�(t)); and (c)choice of ramp loading function �(t).

length) of the RVE as lRVE=2ls, where ls is the (unperturbed) lattice side length. The particleradius is chosen as R=0.4ls; hence, in the present case of 2D-representation the volume fractionof particles is Vpart/V ≈50%. As to the subscale material properties, it is assumed (for the sakeof simplicity) that both the particles and the matrix are isotropic and homogeneous.

6.1.2. Permeability properties. The nonlinear permeability relation in (83) is adopted. Note thatdim(k)=Ns/m4. A more common form of the permeability relation than (83) is w=−kn, where

n(∇u)def= (1/�Fg)∇u is the hydraulic gradient (g=|g| is the gravitational constant). The explicit

form of the corresponding pore pressure gradient-dependent permeability coefficient k, for whichdim(k)=1, is proposed as follows:

k(|n|)=

⎧⎪⎨⎪⎩k1|n|�−1 if |n|�l

k2

[1− 2

|n|]

if |n|�l(90)

where the (independent) material parameters are k1, ��1 and l . The remaining (dependent)parameters are given as

2def= l

�−1

�, k2

def= �k1�−1l (91)



0 1 2 3 4 5 6 7 80

5

10

15

20

25

=1

=1.2

=1.5

Figure 4. Nonlinear permeability relation expressed as |w|/k1 vs |n| for l =5 and different values of �.

The typical response is shown in Figure 4. The corresponding tangent tensors are

K=k(|n|)I+q(|n|)n⊗n, Y=0 (92)

where the scalar q is defined as

q(|n|)={[�−1]k1|n|�−3 if |n|�l

k22|n|−3 if |n|�l(93)

RemarkThis nonlinear permeability relation was essentially proposed by Hansbo [19] based on experimentaldata for soft clay under uniaxial flow conditions. It follows directly that the classical Darcy’s lawwith constant permeability, k=k1, is obtained for the choice �=1.

The following values were chosen for the matrix material: k1,matr=10−11m/s, l =5 and �matr=1.2. Since the particles are significantly less permeable than the matrix, we choose k1,part=10−15m/s, l =5 and �part=1.2.

6.1.3. Storage function. The storage function, �, for each constituent of the micro-heterogeneoussubscale structure (matrix and particles) is defined in (13). The task is to determine/compute Kand the total mean stress field �∗

m(x, t) for the two constituents within each RVE based on theavailable data in the literature.

One may envision different strategies for computing the appropriate value of the field �∗m(x, t)

in the different constituents in an approximate manner based on the constitutive relation in (12).The approach taken here is to carry out a simplified homogenization based on the assumptions:(i) Adopting the Hill assumption on homogeneous stress �∗

m within the RVE, and (ii) assuming



that the elastic moduli are homogeneous within each constituent. In such a case the homogenizedmacroscale constitutive relation for r∗ thus reads

¯r∗ =2G∗εdev+ K ∗�∗volI (94)

where ε∗ def= (u∗⊗∇)sym is the macroscale strain field, and where G∗ and K ∗ are computed as:

G∗ =[[

1− VpartV

]1

G∗matr

+ VpartV

1

G∗part

]−1

, K ∗ =[[

1− VpartV

]1

K ∗matr

+ VpartV

1

K ∗part

]−1

(95)

With the given values of G∗ and K ∗, it is possible to solve for the macroscale displacementfield u∗, preferably using the same FE-mesh as for the pore pressure, to obtain ¯r∗ from (94).Consequently, the total mean stress field ¯�∗

m is computed for each RVE. Finally, due to the Hillassumption on homogeneous subscale stress state within the RVE, we obtain for the constituents:

�∗m,matr(x, t)= �∗

m,part(x, t)= ¯�∗m(t) ∀x∈�� (96)

It is convenient to parameterize the (surface) loading as f(t)= fR�(t), where f

Rrepresents the

spatial distribution and �(t) is a given function such that �(0)=0. The chosen ramp function �(t)and times in the integration scheme are shown in Figure 3(c). Owing to the assumed linearity, we

obtain ¯r∗(t)= ¯r∗R�(t) and, hence,

�∗Rm,matr(x)= �∗R

m,part(x)= ¯�∗Rm ∀x∈�� (97)

is in complete analogy with (96).In the present case, it appears that the following values relating to the (drained) effective

properties are known from the literature: Kpart=1014 Pa, K =2.2×109 Pa, where the notationKpart=K S

part is used for the intrinsic (as well as for the bulk) compression modulus pertinent to theparticles (stones), whereas the notation K is used for the macroscale compression modulus thatrepresents a homogenization of the binary mixture. If we, once again, adopt the Hill assumption toobtain the effective (homogenized) elastic properties, we may reuse the formula in (95) to computethe compression modulus for the matrix (asphalt/bitumen-sand mixture) as follows:

Kmatr=[1− Vpart

V

][1

K− Vpart

V

1

Kpart

]−1

(98)

As to the effective (drained) shear moduli, Gmatr and Gpart, they can be computed from Kmatr andKpart, respectively, using the assumed values matr=part=0.35.

We are now in the position to compute the pertinent ‘undrained’ moduli (G∗matr,G

∗part) and

(K ∗matr,K

∗part) as follows:

G∗part=Gpart, K ∗

part=Kpart+ K F

�part, G∗

matr=Gmatr, K ∗matr=Kmatr+ K F

�matr(99)

where we choose the values K F=2×109 Pa, �part=10−5 and �matr=0.2. Finally, these valuesare used in (95) to compute the appropriate value of the ‘undrained homogeneous’ moduliG∗ and K ∗.



Computational results of the macroscale mean stress ¯�∗Rm are shown in Figure 3(b). For the

purpose of stress calculation, the right vertical boundary is horizontally restrained and is placedsufficiently remote not to significantly affect the stress distribution for the ‘fictitiously elastic’soil body.

6.2. Single RVE with control of macroscale pore pressure—iteration algorithm

The macroscale pore pressure 〈u〉� is never computed directly as part of the general algorithm.However, it is simple to compute 〈u〉� in a post-processing step when us (and thus u=uM +us)has been computed within a given RVE. We then obtain

〈u〉�{u, g}= u+〈us〉� (100)

for known values of (u, g). Rather than controlling the time history of the ‘primary’ macroscalevariables (u, g) for a given RVE-problem, we may conveniently choose to control the time historyof (〈u〉�, 〈g〉� = g) via the (u, g)-driven algorithm. To this end, we consider the residual in termsof the implicit relation

rdef= 〈u〉�{u, gpres(t)}−〈u〉�,pres(t) (101)

for any given combination (〈u〉�,pres(t), 〈g〉�,pres(t)= gpres(t)) that describes the prescribed (e.gexperimentally observed) time-history. Applying Newton’s method to solve for u(t) at any giventime t , we thus obtain the algorithm: For k=1,2, . . . , compute

u(k+1) = u(k)+ du with du=−[ ¯A(k)]−1r (k) (102)

where ¯A=1+〈usu〉�. Although only the tangent field usu is needed for the computation of ¯A, it isstill necessary to solve the system (usu, u

s(1)g , us(2)g , . . .) in a coupled fashion. Apparently, in each

iteration defined by the values (u(k), g(k)), we compute the local field us and, hence, 〈u〉(k)� which

gives r (k). Finally, it is noted that 〈u〉� = � for the choice �=u, which means that ¯A= C if weset �′ =1 in (72).

RemarkFor the purpose of illustration, it is useful to compute the intrinsic phase volume averages‡‡‡〈u〉matr

� (t) and 〈u〉part� (t).

6.3. Computational two-scale FE2-algorithm

6.3.1. Simplified solution based on stationary subscale response and ignored second-order storagecoefficient. In order to assess the importance of including the ‘non-classical’ second-order storage

coefficient ¯U arising from the variationally consistent homogenization of the subscale transientterm, it is illuminating to carry out the assessment by comparing with the results of a model basedon arguments put forward by Ozdemir et al. [12] in the context of heat conduction. They ignored

‡‡‡〈y〉phase�def= (1/|�phase

� |)∫�phase�

y dV .



0

0.1

0.2

0.1

0.2

0.3

0.3

0

0.1

0.2

0.3

0

0.1

0.2

0

0.3

0

0.5

1

1.5

2

x104

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

(a) (b)

(c) (d)

Figure 5. Macroscale pore pressure (Pa) distribution at selected times: (a) t=0.004s;(b) t=0.025s; (c) t=0.029s; and (d) t=0.05s.

0 0.5 1 1.5 2 2.5

x104x104

0

0.05

0.1

0.15

0.2

0.25

0.30 0.5 1 1.5 2 2.5

0

0.05

0.1

0.15

0.2

0.25

0.3

(a) (b)

Figure 6. Macroscale pore pressure profiles (vertical sections) at (a) x1=0m and (b) x1=0.1016m.

the subscale transient such that the subscale problem can be considered as ‘quasi-stationary’ at alltimes. It must be noted that such a ‘quasi-stationary’ solution is still time-dependent in view ofboundary conditions on the RVE (from the macroscale solution) that vary with time. Moreover, inorder to compute this simplified solution, henceforth denoted ustat, it is still necessary in general tosolve for the corresponding sensitivity fields usu(x) from (81) and us(i)g (x), i=1, . . .NDIM from (82)in each new timestep in the case that the subscale problems are intrinsically nonlinear. (In the specialcase when the classical Darcy’s permeability law is adopted, then Y=0 and usu≡0 while, still,

us(i)g (x) =0, i=1, . . .NDIM.) In conclusion, the algorithm for obtaining the macroscale solutionis, in practice, equally cumbersome (or expensive) as in the case of the variationally consistenthomogenization strategy.



0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Figure 7. Results of FE2-computations. Figure shows the influence of the particle size on the transientresponse for an RVE in a selected macroscale Gauss point (located at x=(0.1100,0.1080)m).

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Figure 8. Results of FE2-computations. Figure shows the influence of the particle sizeon the transient response for an RVE in a selected macroscale Gauss point (located atx=(0.1100,0.1080)m). This influence is characterized by the ratio between the actual and

stationary subscale solution, denoted by ustat(x, t).

RemarkIn the case of complete subscale linearity, defined by w=−k ·∇u (with the choice �=1 resultingin constant value k=k1) and �=cu, it is possible to carry out the homogenization a priori to



0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

24


0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

24




10-84000

5000

6000

7000

8000

9000

10000

11000

12000

560

580

600

620

640

660

680

700

720

740

760

10-7 10-6 10-5 10-4 10-3 10-2 10-8 10-7 10-6 10-5 10-4 10-3 10-2


1.135

1.14

1.145

1.15

1.155

1.16

1.165

1.17

1.88

1.89

1.9

1.91

1.92

1.93

1.94

900905910915920925930935940945950

44

(a) (b) (c)

Figure 12. Results of FE2-computations for lRVE=0.01m. Figure shows snapshotsof the resolved field u(x, t) for an RVE in a selected macroscale Gauss point(located at x=(0.1100,0.1080)m) and for selected times: (a) u(x, t=0.004s); (b)

u(x, t=0.025s); and (c) u(x, t=0.05s).

obtain the intrinsically linear macroscale problem formulated in strong form as follows:

Cstat dt ustat−[Kstat ·∇ustat]·∇=0 in �×[0,T ] (103)

subjected to the boundary conditions in Figure 3(a) and the initial condition ustat(x,0)=0 forx∈ �. Here we introduced the time-independent coefficients Cstat= c=〈c〉� and Kstat.

The macroscale problem (103) is solved using dG(0) in time. Note that the macroscale gradientfield g(x, t)=〈g〉�(x, t) can be computed from the solution u(x, t) of (103) simply as g=∇u.Although it is not possible to compute the volume average 〈u〉�(x, t) only with macroscopicinformation, it can be achieved via solution of the RVE-problem and subsequent homogenizationassuming stationary conditions on the subscale in the spirit of Ozdemir et al. [12].



1.2235

1.224

1.2245

1.225

1.2255

1.93551.9361.93651.9371.93751.9381.93851.9391.93951.94

760.5

761

761.5

762

4 4

(a) (b) (c)

Figure 13. Results of FE2-computations for lRVE=0.001m. Figure shows snap-shots of the resolved field u(x, t) for an RVE in a selected macroscale Gauss point(located at x=(0.1100,0.1080)m) and for selected times: (a) u(x, t=0.004s); (b)

u(x, t=0.025s); and (c) u(x, t=0.05s).

1.2231

1.2232

1.2232

1.2232

1.2233

1.9369

1.9369

1.937

1.937

1.9371

1.9371

1.9372

1.9372

1.9373

1.9373

749.82749.84749.86749.88749.9749.92749.94749.96749.98750750.02

4 4

Figure 14. Results of FE2-computations for lRVE=0.0001m. Figure shows snap-shots of the resolved field u(x, t) for an RVE in a selected macroscale Gauss point(located at x=(0.1100,0.1080)m) and for selected times: (a) u(x, t=0.004s); (b)

u(x, t=0.025s); and (c) u(x, t=0.05s).

6.3.2. FE2-algorithm—numerical results. The final set of results are obtained by solving the full-fledged meso-macroscale problem (FE2-problem). A few typical results of the macroscale porepressure development are shown in Figure 5 for selected times, while lRVE is chosen to be 0.001m.The boundary conditions are indicated in Figure 5(a). Typical macroscale pore pressure profilesare shown in Figure 6.

In particular, we are interested in assessing the influence on the subscale transient character ofthe substructure (length) scale, which is represented by RVE-size. In other words, the RVE-size isrepresentative for the particle size. Figure 7 shows how 〈u〉�/u, 〈u〉matr

� /u, and 〈u〉part� /u developwith time for two different sizes of the RVE located in a given macroscale Gauss point. It canbe clearly seen how the subscale solution is ‘delayed’ when the RVE becomes larger (which isexpected). Figures 8–10 present the same type of information in different ways in order to highlightthe comparison between different choices of the RVE-size. Yet another way of presenting theseresults is given in Figure 11.

Finally, Figures 12–14 show snapshots of the resolved field within an RVE in a selectedmacroscale Gauss point for a number of times and for different particle sizes. These calculations are



carried out (as a post-processing step) according to the algorithm that was outlined in Section 6.2,whereby the already computed results of u(t) and g(t) at the selected times in the FE2-algorithm‘drive’ the RVE-computations. In order to obtain enhanced resolution of the local fields, however,a finer subscale mesh was used to give the results in Figures 13 and 14.

7. CONCLUSIONS AND OUTLOOK

In this paper we have outlined a modeling framework for the computational homogenization oftransient uncoupled consolidation in micro-heterogeneous porous media with fluid-filled pores.Owing to the use of a variationally consistent approach, a second-order storage quantity (second-order tensor) occurs along with the classical averages. This quantity will affect the transientresponse for sufficiently large size of the RVE (the subscale dimension), i.e. the effect vanisheswhen the RVE is sufficiently small. In such a case the method by Ozdemir et al. [12] is recovered,which was based on the assumption of ‘quasi-stationary’ subscale response.

As an outlook to future developments, the presented framework for computational homogeniza-tion will be applied to the case of fully coupled consolidation (without the approximation in thispaper leading to an uncoupled pore pressure equation). Hereby, the displacement and pore pressurefields are coupled in a monolithic fashion, and the concept of first-order homogenization becomesless obvious. For example, it is possible to combine first-order homogenization of the subscalepore pressure field with the Taylor assumption for the subscale displacement field.

ACKNOWLEDGEMENTS

The research was supported by The Swedish Research Council for Environment, Agricultural Sciencesand Spatial Planning, contract No. 245-2006-1223, which is gratefully acknowledged.

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computational homogenization of uncoupled consolidation in micro-heterogeneous porous media

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