Multi-scale limit load analysis for discontinuous rock mass based on the homogenization method

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS

Int. J. Numer. Anal. Meth. Geomech., 23, 995}1019 (1999)

MULTI-SCALE LIMIT LOAD ANALYSIS FORDISCONTINUOUS ROCK MASS BASED ON THE

HOMOGENIZATION METHOD

TAKASHI KYOYA*,1, KENJIRO TERADA1 AND TOSHIKAZU KAWAMOTO2

1Department of Civil Engineering, Tohoku University, Aoba, Sendai 980-8579, Japan2Department of Civil Engineering, Aichi Institute of Technology, Toyota, Japan

SUMMARY

A new method to evaluate the strength of rock mass structures is proposed and examined. The method isbased on the collapse load analysis of elasto-perfectly plastic material along with the homogenizationmethod, which enables the multi-scale analyses for heterogeneous media. The homogenization processreplaces a rock mass with cracks by an equivalent continuum medium with macroscopic sti!ness while thefailure criterion for the rock mass is estimated in the localization process. It is shown that both the averagedsti!ness and the macroscopic failure criterion of the discontinuous rock mass are numerically obtained viathe "nite element analyses. Thus, the failure strength of a rock mass structure is evaluated by the collapseload analysis in the form of Linear Programming with the macroscopic failure criterion. This is the "rstattempt to apply the homogenization method to the strength analysis of rock mass. Copyright ( 1999 JohnWiley & Sons, Ltd.

KEY WORDS: discontinuous rock mass; homogenization method; collapse load analysis

1. INTRODUCTION

The identi"cation of materials is a primal concern for designers of rock mass structures and thecharacterization of their properties is one of the most important problems in the "eld of rockmechanics and rock engineering. Foremost are the deformability and the strength of the rockmass which by itself has a microstructure consisting of intact rocks and spatially distributedcracks. In intuition, if both the shapes and the positions of all the cracks in rock mass were knownvia in situ surveys and if all of their properties were obtained by laboratory tests, a precise analysismight be possible by appropriately modelling the complicated mechanical behaviours. However,it is impossible to know a priori such complex con"gurations and also impractical to take intoaccount the highly non-linear behaviours in the analysis.

Nonetheless, many researchers have tried to involve the e!ect of cracks in a rock mass andthere have been various models with the distributions of discontinuities, see, for example,References 1}8. However, the geometrical information is limited in actual situations; the cracksare measured in some &averaged' sense, such as averaged length, averaged spacing, and dominantdirections, etc. Therefore, in designing rock mass structures, a rock mass is often regarded as anequivalent continuum along with the e!ective material properties, in which the heterogeneities

*Correspondence to: T. Kyoya, Department of Civil Engineering, Tohoku University, Aoba, Sendai 980-8579, Japan

CCC 0363}9061/99/100995}25$17.50 Received 4 February 1998Copyright ( 1999 John Wiley & Sons, Ltd. Revised 30 October 1998

are excluded by some macroscopic approaches. The approaches seem reasonable if the mechan-ical interaction between the rock mass with cracks and the overall structure can be appropriatelymodelled in formulating the analysis.

Within the framework of equivalent continuum approaches, the cracks are considered to playa role of reducing both the sti!ness and the strength of a rock mass. If the reductions of both thesti!ness and the strength of a rock mass are appropriately evaluated, the collapse load of a rockmass structure can also be calculated by solving the boundary value problem de"ned for theequivalent continuum. In this respect, various models have been proposed ever before, in whichthe e!ects of cracks on the sti!ness of rock mass are included in the methods in di!erent forms,see, for example, References 6}8. However, most of these developments do not seem to berigorous since little attention is paid to the strength. It should be noted that, in designing andanalysing rock mass structures, the strength of the constituents is used to judge whether thestructure is safe or not.

On the other hand, the homogenization method, which is based on the singular perturbationtheory with averaging schemes, can provide a reasonable estimate on the sti!ness as well as thestrength of rock masses. The method was originally developed for the analysis of heterogeneousmedia based on the method of two-scale asymptotic expansions and has been recognized asa rigorous mathematical tool for analysing the di!erential equations with rapidly oscillatingcoe$cients, see References 9}11. Although the homogenization theory has been developed withpurely mathematical interest, its systematic way of deriving macro and microscopic governingequations for a heterogeneous body has been in"ltrated into the engineering "elds with the helpof the Finite Element Method (FEM), especially for the analysis of industrially producedcomposites with periodic microstructures.12~14 However, most of geotechnical engineeringproblems and, in particular, the problems arising from rock mechanics must deal not only withthe deformation but also the stability or the collapse load of an overall structure. Therefore, thecharacterization should be based upon the structural analysis of the local structures withsu$cient amount of geometrical information such as cracks and their distributions. Althoughsome applications of the mathematical homogenization are found in the literatures,15~16 most ofthem pay attention of deformation only.

While characterizing the rock mass as a constituent structure, a collapse load analysis used indesigning geotechnical structures can be carried out. The collapse load of an elasto-perfectlyplastic continuum has been successfully analysed in a form of Linear Programming (LP) with the"nite element method, see, for example, References 20 and 21. In the collapse load analysis, it isessential to evaluate the elastic stress distribution within the continuum induced by an appliedload of unit magnitude and to give the failure criterion of the elasto-perfectly plastic materialconstituting the body. Therefore, we can systematically evaluate a limit load of rock massstructures through the collapse load analysis if the sti!ness and the macroscopic failure criterionare successfully determined for discontinuous rock mass by appropriate averaging schemes.Idealizing a discontinuous rock mass as an equivalent continuum in which every material point init includes the Representative Elementary Volume (REV) composed of intact rocks and cracks,the homogenization approach will provide not only its averaged sti!ness but also the macro-scopic failure criterion of the REV of the rock mass by its localization capability. Thus, thecollapse load analysis can readily be combined with the homogenization method which enablesthe multi-scale analysis of highly heterogeneous bodies.

In this paper, we will propose a numerical scheme to analyse the collapse load of rockmass structures. The method extensively utilizes the multi-scale capability of the asymptotic

996 T. KYOYA, K. TERADA AND T. KAWAMOTO

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 23, 995}1019 (1999)

homogenization method and the techniques in the collapse load analysis. The homogenizationprocess enables us to evaluate the averaged sti!ness of the rock mass using the mechanicalproperties of intact rock and the pattern of crack distribution. With the macroscopic (orhomogenized) sti!ness, the averaged elastic stress distribution induced by the external load ofunit magnitude, which is needed in the collapse load analysis, can be obtained through theconventional elastic analysis for the equivalent (or homogenized) continuum which possiblyreveals anisotropy. A numerical example in this paper shows that the macroscopic failurecriterion of the idealized continuum is obtained by applying the localization process in theasymptotic homogenization theory to the REV of the rock mass. Then, the apparent failurestrength or the limit load of a rock mass structure can be evaluated by the collapse load analysisin the form of LP using the homogenization and the localization results. To verify the proposedmethod, we shall present numerical simulations of the direct shear tests of specimens includingarranged crack-like voids and show that the predictions agree with the experimental results fairlywell. It is worthwhile to note that there have been no equivalent techniques that can predict thecollapse load of rock mass structures by estimating the sti!ness and the strength of rock mass.

2. APPARENT STRENGTH OF CRACKED BODY

2.1. Strength of discontinuous rock mass

A material body reveals a strength under some mechanical conditions. This strength appears asthe solution of boundary value problem de"ned by the mechanical condition forced on thematerial body. The apparent strength can be considered as an inherent one when the body isregarded as an element of the material. In general, however, the apparent strength is not theinherent strength of the body that includes microstructure such as various constituents or cracks,but the strength a!ected by strengths of the materials constituents, the size and the pattern ofcracks, the shape and the size of the body and the boundary conditions forced on the body. Therehave been many experimental attempts to investigate the relationship between &strength of rockmass' and various factors, e.g. the properties of constituent materials and the pattern or theorientation of cracks included. Although the complicated interaction between the various factorsare the main sources in their arguments, the defects and the heterogeneity of the medium arelikely replaced by a convenient terminology, such as &scale e!ect of mechanical responses'. Thus,the predictions seem irrelevant especially when estimating the strength of a rock mass. Thediscrepancy in their characterization appears to arise from the assumption that the e!ects of suchfactors can be estimated and that the inherent strength can then be obtained by experiments. Thatis, a specimen of appropriate size including various constituents and cracks is treated asa representative element of rock mass without recalling that an experimental work provides anapparent strength as a solution of the boundary value problem de"ned for this particularspecimen. In this section, observing the general tendency in mechanical tests for crackedmaterials, we will study what is needed to predict the apparent strength of discontinuous orfractured rock masses.

2.2. Direct shear test of cracked specimen

We present here the direct shear tests carried out by Kawamoto et al.22 The specimens aremade of gypsum mortar and has rectangular shapes, as shown in Figure 1, and the size of the

MULTI-SCALE LIMIT LOAD ANALYSIS FOR ROCK MASS 997


Figure 1. Direct shearing test of rectangular gypsum mortar specimen with arranged cracks and with cut throughcracks22

Table I. Material properties of gypsum material

Cohesion c (MPa) 3)33Internal friction angle / (degree) 56)3Young's modulus E (MPa) 3,730Poisson's ratio < 0)16

specimens are "xed to 15 cm in height, 15 cm in width and 4 cm in thickness. In place of realisticcracks, two di!erent types of crack-like voids are arti"cially made by inserting and withdrawingsteel plates of 0)2 mm in thickness and 9 mm in width into the gypsum mortar so that they arearranged periodically in each specimen. One type of crack-emulating voids has the length lessthan that of the period and the other type assumes cut-through cracks (see Figure 1). The materialproperties of the gypsum mortar used in the tests are given in Table I, which had been obtained bytriaxial tests on cylindrical specimens.

The experimental results are provided by relating the apparent shear strengths of the defectedgypsum mortar specimens and the distributions of crack-emulating voids. Figure 2 shows therelationship between the apparent shear strength and the angle of crack-like defects. As can beseen from the results, the apparent shear strength is strongly a!ected by both the orientation ofthe cracks and the con"ning lateral load. The specimen with cracks of plus sign angles would tendto fail in the same directions with the cracks due to the shearing force, P

t, but would not with

those of minus sign angles. It is thus found that the apparent shear strength has the maximumvalue around the angle of !22)53 and this trend appears more clear as the con"ning lateral loadbecomes larger. It is therefore concluded that the apparent strength does not only re#ect thematerial properties of the gypsum mortar but also the consequence of complicated interactionbetween various geometrical characteristics of the specimen: the size and the shape, the distribu-tion patterns of crack-emulating voids, and the &shearing' boundary conditions applied.

The observation we made are quite natural and, in some sense, trivial. It is almost impossible totake into account all the factors in estimating the strength of the defected specimen by treating it



Figure 2. Shear strength of cracked specimen22

as a material element. That is, the factor such as distribution of cracks should be regarded asa structural characteristics of the specimen and be distinguished with the inherent properties ofthe material. If the structural characteristics have simple features, they can be treated in "niteelement analysis by modelling them directly. Such direct modelling, however, is almost impossiblefor cracks distributing in rock mass, and some appropriate averaging scheme to estimate themacroscopic properties of REV of the rock mass is needed. In summary, to predict the apparentstrength of cracked bodies as well as discontinuous rock mass, it is essential that the macroscopicmechanical properties of the REV including the structural characteristics are estimated by somereasonable averaging scheme with the properties of intact rocks and the pattern of crackdistribution, and then the apparent strength must be evaluated by solving an appropriateboundary value problem with given boundary conditions.

3. COLLAPSE LOAD ANALYSIS

The collapse load analysis is an e!ective tool to estimate the stability of structures composed ofplastic-like material, such as steel structures and soil foundations. Although a rock mass hasbrittle nature and does not behave like a plastic material, it is expected that the collapse loadanalysis provides us a limit state for the rock mass structure to sustain the applied load asa solution of a boundary value problem, and that the value of the limit load obtained can besu$ciently utilized in design. In addition, it has a great advantage that the collapse load analysisis very simple to work with. The purpose of this section is to investigate the applicability of thecollapse load analysis to cracked body like a rock mass. In the following subsections, we describethe method of the collapse load analysis and present the numerical example which illustrates itsapplicability to rock-like materials.



3.1. Formulation of collapse load analysis

Let us consider a homogeneous body, which is made of an elasto-perfectly plastic material, asshown in Figure 3. The body is subjected to a body force over ) and/or to a surface traction onL)

talong with the outward unit normal n. We assume that the body is loaded proportionally

according to a load factor a. As the factor a increased, a part of the body would yield and becomeperfectly plastic so that it reveals elasto-plastic behaviours. Then the stress states in the body arerepresented as follows:

pij"ap%

ij#p3

ij(1)

where p%ij

is the elastic stress induced by the unit load, f and/or, t, and p3ij

is the residual stress bythe yielding of parts of the body. Here, the elastic stress satis"es the equilibrium equation:

p%ij,j

#fi"0 in )

p%ijnj"t

ion L)

t(2)

while the residual stress satis"es the following self-equilibrium condition:20

p3ij,j

"0 in )

p3ijnj"0 on L)

t(3)

so as to keep the static equilibrium during the elasto-plastic behaviour.When the yield criterion of the material is given by f (p

ij)"0 and the fundamental elastic stress

distribution p%ij

is known, the lower-bound theorem in plasticity theory states that for the body tobe stable under a certain external load, there must exist a residual stress distribution whichsatis"es the following conditions:

f (pij)"f (ap%

ij#p3

ij)"0 in )

p3ij,j

"0 in )

p3ijnj"0 on Lp

t(4)

Note here that the yield surface should be convex in the stress space.Applying the "nite element discretization to equations (2) and (3), we obtain the system of

simultaneous equations which is represented in matrix forms:

B3 5r8 %"F

B3 5r8 3"0 (5)

Figure 3. Proportional loading for elasto-perfectly plastic body



where r8 % and r8 3, respectively, denote the total elastic stress and residual stress vectors that consistof all the elemental stresses. The matrix B3 is the integrated strain}displacement matrix corre-sponding to the total stress vectors, which is obtained from the equilibrium equation by assumingthat each element has constant stress:

F"

N+

%-%.%/5 iCP

Vi

B5p% d<iD"CN+

%-%.%/5 iPVi

B5 d<iD r8 %

"B3 5r8 % (6)

Here, both the stress vectors r8 % and the matrix B3 are of the extended forms so that themultiplication of B3 by the stress vector, r8 %, would be equivalent to the nodal force vector, F,induced from the external load of unit magnitude.

Meanwhile, the failure criterion can be approximated by a set of tangential planes that areassociated with the outward unit normal vectors by n

k(k"1, 2,2) as seen in Figure 4. Thus, the

failure criterion is represented by a set of inequalities

NT (ar8 %#r8 3)!R)0 (7)

where N is a matrix made by ordering the unit normal vectors on each tangential plane so that theinner products between all of the stresses in "nite elements and the unit normal vectors is wellde"ned. Here, R is a column vector whose components are the distances to each tangential planefrom the origin of the stress space. Thus, the convex region within the yield criterion isapproximated by a convex polyhedron (or equivalently convex polyhedral cone) throughequation (7).

By the above setting for numerical analyses, the collapse load analysis can be formulated as theproblem in Linear Programming (LP) such that

maximize asubject to

GN5(ar8 %#r8 3)!R)0

B3 5r8 3"0 (8)

in which variables to be optimized are the load factor, a, and the residual stress vector, r8 3.

Figure 4. Failure criterion and its tangent planes



The stress distribution in/around a rock mass structure induced by some external loads isstrongly a!ected by the existing cracks since the macroscopic sti!ness (or equivalently, compli-ance) of the rock mass is changed by them. Moreover, the strength of the rock mass also dependson the crack distribution. This means that both the stress distribution and the strength of the rockmass need to be appropriately estimated to predict the limit state of the rock mass structure. Theframework of the limit load analysis described above has an advantage in this situation. That is,the limit state of a rock mass structure can be systematically evaluated by the collapse loadanalysis if both the macroscopic failure criterion of REV of the rock mass is given and theaveraged sti!ness which provides the fundamental elastic stress vector, r8 %, is appropriatelyevaluated.

3.2. Numerical analysis

We now try to predict the apparent shear strength of cracked specimen by the collapse loadanalysis formulated in the above and compare them with the experimental results presented inSection 2.2. The fundamental elastic stress distribution caused by unit shearing load, P

t, was

computed under plane stress condition by the structural analysis, while the stress caused by thehorizontal con"ning load, P

n, was calculated and used as an initial stress in the collapse load

analysis carried out later. The models used in both "nite element analyses are shown in Figures5(a) and 5(b), respectively. The important input datum for these analysis is, as stated in theprevious section, the averaged sti!ness of a cracked body. Although a few theoretical methodwould be applicable for this purpose (see for example References 31 and 32), it is preferable to takethe computational homogenization approach whose procedure and applicability will be dis-cussed in the next sections. The averaged sti!ness used in this computation is given there by (19).As will be emphasized there, the homogenization method enables us to evaluate both theaveraged sti!ness of a heterogeneous body with arbitrary microstructural con"guration and theactual stress distribution within the microstructures. After the homogenized sti!ness determinedfor crack-like void horizontally located is calculated by the homogenization analysis, the

Figure 5. Finite element model of the specimen. (a) Model for shearing condition; (b) model for con"ning condition



averaged sti!ness for each type of specimens is simply evaluated by applying the co-ordinatetransformations according to the crack orientation.

Although an appropriate failure criterion must be chosen for the cracked body, the conven-tional ones are enough to investigate the applicability of the collapse load analysis. We hereassume the macroscopic failure criterion for the specimen by superposing the following twocriteria, both of which are well known and commonly used in geomechanics:

f (pNij)"J((1

2) sN

ijsNij)!ap6

kk!K"0 (9)

Dq6sD"c!p6

ntan / (10)

The former is for the failure of intact material (gypsum mortar) and the latter is for the failurealong crack surfaces. The superposition still provides a convex region of failure surface, asillustrated in Figure 6. The constants a and K in equation (9) were so determined that theDrucker}Prager-type criterion go through inner apices of Mohr}Coulomb criterion for the intactmaterial. The values of constants of the Mohr}Coulomb criterion for the intact material are givenin Table I, and then the constants of the corresponding Drucker}Prager-type criterion aredetermined as a"0)752 and K"1)670 MPa. To study the variation in responses, we preparedanother values of constants in equation (10); the cohesion c as 0)5 and 1)0 MPa, and the frictionangle / as 20 and 303.

The predicted strengths for several angles of crack-like defects are given in Figure 7. In the caseof c"1 MPa and /"203, the predictions agree with those obtained from the periodicallyarranged defects of the "rst type except for the case of P

n"25 MPa. Moreover, it is quite

interesting that the predictions with the weakest strength along cracks, which corresponds to thechoice of constants as c"0)5 MPa and /"203, show good agreement with the experimentalresults for the specimens with the cut through cracks in Figure 2.

In summary, the apparent strength of cracked body can be predicted well by the collapse loadanalysis, in which the body is idealized to be elastic}perfectly plastic, if both the fundamentalelastic stress distribution and the macroscopic failure criterion of the body are reasonablydetermined. It should be, however, noted that only the sti!ness and the strength of the intact rockare available from laboratory tests in most of the cases. Therefore, the characterization of themechanical properties as well as the strength of REV of rock mass is in general a formidable taskeven by applying the theory of equivalent continuum approaches or averaging schemes. It will beshown in the next sections that the homogenization method overcomes such problems arisingfrom the heterogeneity of rock mass.

Figure 6. Superposed failure criteria



Figure 7. Predicted strength using simple superposed failure criterion

4. THE METHOD OF HOMOGENIZATION AND LOCALIZATION

4.1. The homogenization theory

We are concerned with the mechanical behaviours of a general heterogeneous medium whosemicrostructure occupies a "xed region with characteristic length e. When the medium is subject to



a certain load and given boundary conditions, our primal interest is what would happen as e goesto zero. The mathematical theory of the homogenization method asserts that, as e goes to zero,the heterogeneous body behaves as if it were occupied by some equivalent homogeneous medium.More speci"cally, in linear elasticity, the actual displacement ue tends to the averaged displace-ment "eld u0, which is a solution of the governing equations whose coe$cients have beenhomogenized. The mathematical structure of this problem has explicitly been known to involvethe concepts of generalized convergence such as !-convergence of a functional and G-conver-gence associated with Green function, see e.g., Reference 23. In this mathematical framework,however, neither the microstructural con"guration nor the in situ loading condition at theboundaries for microstructures are speci"ed.

To simplify the mathematical treatment, most of the homogenization theories have beendeveloped in particular, for, periodic microstructures. By assuming the periodic distribution ofa basic microstructural element (called the unit cell), the method successfully relates micro- andmacroscopic variables and, together with the method of two-scale asymptotic expansions, clearlydescribes the asymptotic behaviours of composite materials, see e.g. References 9 and 24}26. Thisversion of the homogenization which is sometimes called the asymptotic homogenizationmethod, allows a systematic derivation of the micro- and macroscopic problems and, therefore, isuseful in numerical analyses of practical engineering problems. Examples of such applications toengineering problems are found in References 27, 12, 14 and others, in which the "nite elementmethod is a basic tool for computations.

In this section, we simply follow these theoretical developments and provides some homogeniz-ation formulae that are necessary for the following discussions. However, it is di$cult to identifythe periodic microstructure, namely a unit cell, for an actual rock mass whose heterogeneity hasmore or less random nature. Nonetheless, assuming that a periodic microstructure which consistsof intact rock and cracks, we shall show that the computational homogenization procedure givesreasonable estimates on the macroscopic sti!ness of a rock mass. Indeed, we can choosea Representative Elementary Volume (REV) over which an averaging process is meaningful if therock mass satis"es the condition of statistical homogeneity, see References 28}30 for furtherdiscussion.

4.2. Variational formulation of the homogenization method

Let us consider a general heterogeneous media of domain ) whose microstructure is composedof linearly elastic solids and is highly heterogeneous in the selected RVE domain >. Denoting bye the order of the heterogeneity and therefore the size of RVE, we introduce the two di!erentscales: one of these is a macroscopic scale denoted by x, in the domain at which the heterogenei-ties are invisible and the other one is a scale for heterogeneities and is referred to as themicroscopic scale, y"x/e. Then the superscripts introduced in variables will indicate their ordersas well as the dependency on both x and/or y in the following descriptions.

If the structure is subjected to a surface traction on L)t

and a prescribed displacementboundary conditions L)

u, respectively, the displacement ue(x)"u (x, y) is the exact solution of the

following boundary value problem de"ned in the domain ):

P) Eeijkl

(x)Lue

k(x)

Lxl

Lvei(x)

Lxj

d)"PL)t

ti(x)ve

i(x) dS#P) f e

i(x)ve

i(x) d) ∀ve

i(x)3< (11)



where ve is a virtual displacement within the admissible space<. We assume that the solution canbe expressed in the following form of two-scale asymptotic expansion:

ui(x, y)"u0

i(x)#eu1

i(x, y)#e2u2

i(x, y)#2 (12)

Then the averaging procedure along with the method of asymptotic expansions provides thefollowing macroscopic problem of the homogenized structure to be solved for the averageddeformation, u0 (x):

P) EHijkl

(x)Lu0

k(x)

Lxl

Lv0i(x)

Lxj

d)"P) f Hi

(x)v0i(x) d)#PL)

ti(x)v0

i(x) dS ∀v0

i(x)3< (13)

where v0 is assumed to be selected from the same space as <. Here, the homogenized elasticitytensor and the averaged body force are, respectively, de"ned as follows:

EHijkl

(x)"1

D>D PYAEijkl

(x, y)!Eijpq

(x, y)Lskl

p(x, y)

LyqB d> (14)

f Hi

(x)"1

D> D PY

fi(x, y) d> (15)

where D> D indicates the volume of the unit cell >. The >-periodic functions vkl have beenintroduced to characterize the mechanical behaviour of the unit cell and are related to the sixfundamental strain distributions as

u1i(x, y)"!skl

i(x, y)

Lu0k(x)

Lxl

#ti(x) (16)

where ti(x) is an arbitrary vector depending only on the macroscopic scale variable x. The indices

k and l take values between 1 and 3 in three-dimensional setting and each of vkl contributes to the#uctuation of the solution within a unit cell domain >. The functions vkl represent the sixdeformation patterns corresponding to the six fundamental strains applied over the cell andtherefore called the characteristic deformations. The asymptotic homogenization method pro-vides the following governing equations for the characteristic deformations:

PY

Eijpq

(x, y)Lskl

p(x, y)

Lyq

Lvi(x, y)

Lyj

d>"PY

Eijkl

(x, y)Lv

i(x, y)

Lyj

d> ∀vi(x, y)3<

1%3(17)

where v(x, y) can be arbitrary functions in the >-periodic admissible space <1%3

. Note that theseparation of the macro- and microscopic variables in equation (16) enables the evaluation ofactual microscopic stress/strain distribution at any point x, in which the actual strain distributionhave been computed by solving equation (13):

reij(x, y)+E

ijkl(x, y)Adkpdlq!

Lspqk

(x, y)

LylB

Lu0p(x)

Lxq

(18)

This capability is called the localization process and makes the method very rigorous.



In the formulation, we have skipped the several steps and omitted some explanations. For thedetailed derivation of the homogenization formulae, one can refer to the appendix or thepublished literature, e.g. References 9 and 12.

4.3. Computation of the averaged stiwness

In this subsection, we compute the averaged sti!ness of the cracked specimen presented inSection 2. The result shows the capability of the homogenization method in characterizing themacroscopic mechanical properties of such cracked body. First we prepare the geometry modelfor the unit cell. Applying the periodicity to the degrees of freedom on the cell boundaries, thespatial discretization gives the "nite element model for this unit cell. By solving (17) in thethree-dimensional setting, the characteristic deformation were obtained as six nodal displacementvectors. Then the homogenized sti!ness is calculated by equation (14) by using a numericalintegration scheme over the unit cell domain >.

The unit cell model of the cracked specimen is composed of intact gypsum mortar andcrack-like voids as shown in Figure 8 and the gypsum mortar was assumed to be an isotropicmaterial whose Young's modulus and Poisson's ratio are given in Table I. The unit cell wasdivided into about 10,000 three-dimensional rectangular "nite elements. The void part wasidealized as one layer consisting of "nite elements with very compliant material so that the resultswould not be a!ected by the numerical errors. Carrying out the "nite element analysis, thehomogenized sti!ness of the specimen with horizontal cracks was estimated as

[EH]"

3)875 0)447 0)692 0 0 02)308 0)441 0 0 0

3)874 0 0 01)116 0 0

Sym. 1)591 01)070

]103(MPa) (19)

To con"rm the practical validity of this estimate, we computed the average sti!ness for variousspacing of crack-like voids and compared the results with those obtained by usual representativevolume analyses (see Figure 9) in which either the strain or the stress is applied uniformly over thedomain >. Figure 8 shows the results for various spacings of the crack-like voids. As can be seenfrom this "gure, the macroscopic sti!ness with the computational homogeneization methodappears to be reasonable since it reveals the intermediate estimates.

In addition to the evaluation of the macroscopic material properties, the asymptotic homogen-ization method enables us to compute the actual stress distribution within the unit cell. Forsimplicity, we here assume that the macroscopic strain has been obtained after the overallstructure is analysed with the homogenized quantities. The localization formulae (18) was appliedto the unit cell giving the homogenized elasticity matrix (19).

Remark 1. The specimen used in Section 2.2 has crack-like voids and so does the unit cell model.Therefore, the setting for this numerical analysis is consistent with the actual test condition.However, an actual rock mass has cracks of zero thickness and, then, the unit cell model like theone presented here is expected to provide lower sti!ness values than the actual ones. With



Figure 8. Unit cell of cracked specimen

Figure 9. Comparison of averaged sti!ness

appropriate mathematical and numerical modelling schemes, the numerical results would bemore reliable, but the current results seem allowable from practical points of view.

5. MULTI-SCALE STRENGTH ANALYSIS FOR ROCK MASS STRUCTURE

5.1. Introductory remarks

The purpose of this study is to construct the macroscopic failure criterion of the REV of a rockmass, which is a formidable task in rock mechanics, and eventually propose a method of collapseload analyses for rock mass structures. The former is found by applying the computationalhomogenization method while the latter can be carried out within the framework of LP. In thissubsection, we shall complete our analysis method for collapse load of rock mass structures byextensively utilized the rigorous capabilities of the homogenization and localization processes ofthe homogenization theory.

The asymptotic homogenization method plays two roles in our proposed analysis method forcollapse loads of rock mass structures. One is to evaluate the macroscopic material properties bytaking into account the microstructural geometries such as sti!ness of intact rock and crackdistributions, and the other is to a priori estimate the limit load of the REV for every macroscopicloading pattern. For the latter estimation, the localization capability in the homogenizationmethod is utilized in evaluating the local (or equivalently) microscopic responses of all the



possible pattern of macroscopic stress "eld, whose 2-norm is non-zero and has a unit magnitude.Then the allowable magnitude of macroscopic stress can be estimated for each loading pattern.

The feature of this approach is that we are concerned with both the macro- and the microscopicmechanical behaviours of rock mass. Moreover, these behaviours are related to each other due tothe mathematical structure of the asymptotic homogenization method. Together with suchmultiscale nature, the details of our strategy are described below.

5.2. Macroscopic failure criterion of rock mass by the localization process

When the macroscopic stress at a point x in a rock mass is given by p6mn

(x), the actualmacroscopic strain can be evaluated by the following reciprocal relationship:

eNkl(x)"(EH)~1

klmnp6mn

(20)

Due to the localization capability of the homogenization method, the microscopic stress distribu-tion within the unit cell, p0

ij(y), can be computed by the corresponding microscopic strain which is

obtained by localizing the deformation (see Figure 10). That is, by the macroscopic strain given inequation (20), the microscopic stress "eld measured in y is given by the following linear mappings:

p0ij(y)"AEijkl

(y)!Eijpq

(y)Lskl

pLy

qB eN

kl(x) (21)

which corresponds to equation (18). Therefore, if an appropriate failure criterion is introduced forthe intact material within the microstructure, the microscopic limit load of rock mass can beestimated as a threshold of the microscopic responses.

Let us suppose that the failure criterion of an intact rock has been obtained by laboratory tests.We assume that the failure surface is convex and can be expressed in the following form:

f (p0ij)"0 (22)

where p0ij(y) is the microscopic stress "eld within a unit cell. Based on this criterion, we might be

able to de"ne the failure condition at a macroscopic point x in a rock mass as follows: themicrostructure would be stable if the induced microscopic stress satis"es f (p0

ij)(0 everywhere in

the unit cell, and it would become unstable if f (p0ij)"0 holds at a certain point within the unit

cell. Here, the microscopic stress p0ij(y) is computed via formulae (20) and (21), if the macroscopic

stress p6ij(x) at that point has been appropriately evaluated. That is, the elastic limit case of the

macroscopic stress is recognized as the stress at failure of an element of rock mass. In other words,&failure occurs at a point x in a rock mass if the microstructure located at that point becomes

Figure 10. Cracked rock mass and its unit cell of the microstructure



Figure 11. Assumption of macroscopic failure due to microscopic failure

unstable in the sense that the microscopic stress at some point in the unit cell reaches the limitstate'. This assumption may correspond to the initial yield of an intact rock as shown in Figure 11at a point in the unit cell. Thus, the macroscopic strength of the rock mass element can bedetermined from global}local viewpoints.

Although we have been able to obtain the macroscopic strength of the REV of the rock massfor arbitrary loading pattern, it is preferable to know total strength for every possible stressdistribution in the macroscale prior to the structural analysis. In other words, the macroscopicfailure criterion must be prepared for the collapse load analysis of rock mass structures, whichhave been formulated in the framework of LP. The strategy for a priori obtaining the macroscopicfailure criterion of rock mass is summarized as follows: representing the average stress using thatof unit magnitude as

p6ij"bp6 @

ij(Ep6 @

ijE"Jp6 @

mnp6 @mn"1) (23)

the value of b at the failure is determined by

b"max Mb@'0 D f (b@p@0ij

(y)))0, ∀y3>N (24)

Here, p6 @ij

and p@0ij

(y) are related to each other (that is same as the combination of equations (20)and (21)) such that

p@0ij

(y)"CAEijkl(y)!E

ijpq(y)

Lsklp

LyqB (EH)~1

klmnD p6 @mn

(25)

Using the above equations, if we prepare the su$cient number of average stresses of unitmagnitude, p6 @

ij, so that they are uniformly distributed on the unit sphere in six-dimensional stress

space, we can obtain a set of average stresses, p6ij"bp6 @

ij, which yields the failure surface to be

expressed by the failure criterion fN (p6ij)"0. This process can be recognized as a series of numerical

tests for the strength of the REV including in"nite number of cracks. Figure 12 illustrates thisapproach. Note that, in actual situations, it is impossible to carry out equivalent in situ tests.

Remark 2. A typical example of failure criteria for an intact rock might be a function of stressinvariants. A possible choice could be the Drucker}Prager-type criterion:

f (p0ij)"J(1

2s0ijs0ij)!ap0

kk!K"0 (26)

in which a and K are material constants. Here, s0ij

is the diviatoric stress of the microscopic stressp0ij. Recall that no hardening law is needed in the collapse load analyses.



Figure 12. Stresses of unit magnitude on the unit sphere and stresses at failure

Remark 3. The judgement for the macroscopic failure seems conservative since the conditionf (p0

ij)"0 postulates the onset of microscopic failure at a certain point in the unit cell, but may not

necessarily imply the overall instability of overall rock mass structure. This matter shouldcarefully be studied in another opportunity. Nevertheless, the macroscopic failure criterion for therock mass element thus obtained is supposed to work well as a slightly conservative estimate indesigning of rock mass structures.

Remark 4. Since the failure criterion of intact rock is assumed to form a closed convex cone in themicroscopic stress space, the subset of the domain within the macroscopic stress space that isrestricted by equation (23) is also convex due to the linearity of mapping (24). Consequently, thestress obtained by equation (23) is certainly on the boundary of a closed convex region in themacroscopic stress space. This guarantees not only the analysis with LP but also any kind ofconvex analyses.

Remark 5. Since the geometry of a microstructure including cracks is speci"ed in spatialco-ordinate, the macroscopic failure criterion of a rock mass has to be made as a function of thesix dimensional stress components, and would be approximated by a six-dimensional quadraticsurface. This makes it easy to handle co-ordinate transformation, etc.

5.3. The procedures of the analysis

The procedures of the collapse load analysis with the homogenization method is summarizedas follows:

(1) Data input: For a given rock mass, the sets of sti!nesses and strengths of intact rocks aredetermined by laboratory tests. The photographs or the sketches of outcrops are preparedfor identifying the representative crack geometry and the region for a unit cell is de"ned.

(2) Computing homogenized sti+ness and body force: Based on the geometrical information ofthe representative crack distribution within the unit cell and using the properties of theintact rocks, the characteristic deformation, skl

p(y), is computed by solving equation (17).

Then the homogenized sti!ness, EHijkl

, and body force, f Hi

, are evaluated by using equations(14) and (15).

(3) Evaluating the macroscopic failure criterion: Preparing su$cient number of stresses of unitmagnitude so that uniform distribution of the unit sphere in the six-dimensional stress



space and using the failure criterion of intact rocks, the values of factor b for determiningthe macroscopic stress at failure are estimated by equations (24) and (25). Then, using theleast-squares method, the macroscopic failure criterion for rock mass are obtained asa six-dimensional quadratic surface as shown in the numerical example in the nextsubsection.

(4) Computing the fundamental elastic stress distribution: Within the framework of linearelasticity, the boundary value problem for a rock mass structure, namely equation (13), issolved along with the homogenized sti!ness, EH

ijkl, and body force, f H

i. By applying an unit

external load, the solution is provided as the averaged displacement, uNi(x)"u0

i(x), the

averaged strain, eNij(x)"1

2(uN

i, j#uN

j,i), and then the averaged stress, p6

ij(x)"EH

ijkleNkl(x),

which is the fundamental elastic stress associated with the unit load.(5) Collapse load analysis of rock mass structure: Collapse load of rock mass structure is

computed in the form of LP given in equation (8) using the elastic stress distribution, p6ij(x),

and the macroscopic failure criterion, fN (r6 )"0, which have been evaluated.

5.4. Numerical example

We provide here the numerical example of the collapse load analysis whose procedure has beenjust presented. To validate the analysis method, we consider the same rock mass structures asthose of the strength tests presented in Section 2.2. Therefore, we have already "nished Steps1 and 2 in the above algorithm. The material properties, the geometry and the analysis conditionsare the same as before. The homogenized elasticity constants and the averaged body forcehave also been evaluated via the formulae in equations (14) and (15) in Step 2 as presented inequation (19).

In Step 3, by assuming appropriate failure condition of the microstructures, the macroscopicfailure criterion of rock mass is determined. We here take the Drucker}Prager-type criterion asa condition of limit state for the intact gypsum mortar:

f (p0ij)"J(1

2s0ijs0ij)!ap0

kk!K"0 (27)

in which the same constants are set as those in equation (8) in Section 3.2. Then the procedurepresented in Section 5.2 are carried out to obtain an approximated quadratic surface that isequivalent to the macroscopic failure criterion. That is, 728 patterns of the macroscopic stress ofunit magnitude are needed to approximate a unit sphere in the six-dimensional stress space. Foreach of them the factor b can be determined by

max Mb'0 D f (bp@0ij

(y)))0, ∀y3>N (28)

Finally, the six-dimensional quadratic surface can be approximated as follows:

fN (Mp6 N)"Mp6 Nt[A]MpN N#MbNtMp6 N!1"0 (29)

where the coe$cient matrix [A] and the data vector MbN can be determined by the least squaremethod. Here, M N is a six-dimensional column vectors and [ ] is a six-by-six square matrix. Itshould be noted that the matrix [A] must be non-negative de"nite because the macroscopicfailure surface is convex if that is also in the microscopic one. Also, in-between points areinterpolated by the given points of macroscopic stresses so that the totality expresses a continu-ous surface and therefore arbitrary stress states can be dealt with in the actual collapse loadanalyses.



For our particular model problem, it is preferable to investigate the characteristics of thematrix [A] and the vector MbN. The six eigenvalues of [A] is computed as

j1"3)393]10~3, j

2"2)658]10~3

j3"1)193]10~3, j

4"0)491]10~3

j5"0)129]10~3, j

6"7)650]10~7 (30)

and the constant vector MbN corresponding to the diagonal matrix was

MbN"M0 0 0)0333 0 0)0072 !0)0179N5 (31)

It is found from the order of each eigenvalue that the sixth eigenvalue j6

can be regarded as zero.Therefore, the matrix [A] is non-negative since all the eigenvalues are non-negative. This positivesemi-de"niteness of [A] implies that the quadratic surface is elliptic. Also, this quadratic surfacecan be identi"ed with an elliptic paraboloid in six-dimensional stress space because the sixthcomponent b

6of MbN, which corresponds to the zero eigenvalue j

6, is non-zero (negative).

To see the e!ects of crack-like voids on the macroscopic strength, the investigation of theprincipal direction of the elliptic paraboloid would make the geometrical interpretation easy. Theeigenvector corresponding to j

6is computed as

m6"Mp6

11p622

p633

q623

q631

q612

Nt

"M!0)663 !0)210 !0)719 0 0 0N5

"M!cos 493 !cos 783 !cos 443 0 0 0N5 (32)

whereas the failure surface formed by the Drucker}Prager criterion of the intact rock is open tothe direction of hydrostatic compression:

m)"M!1

3!1

3!1

30 0 0N5

"M!cos 54)73 !cos 54)73 !cos 54)73 0 0 0N5 (33)

Comparing these two directions, it is found that the direction given by (32) is inclined slightlyaway from the direction of hydrostatic compression. This fact must be due to the existence ofcracks being normal to x

2-axis as seen in Figure 10. In conclusion, the proposed approach to the

evaluation of macroscopic failure appears to be quite reasonable.As the "nal step in the collapse load analysis, we again analyse the cracked specimen discussed

in Section 2. Using the macroscopic failure criterion obtained in the above, we modi"ed it toa failure criterion in three-dimensional stress space corresponding to the plane stress conditionand approximated it by 24 tangential planes. The shear strength of the specimen is then estimatedas a collapse load of the solution of LP. The results are shown in Figure 13. As can be seen fromthe "gure, all the predicted strength values are lower than those of the test results presented inFigure 2. This tendency seems natural since our current setting for the failure of the microstruc-ture lies in a conservative side. However, the anisotropic feature of strength is well represented forthe specimen with many crack-like voids.

Remark 6. As above, the lower strength predicted by the proposed method results from thede"nition of failure at the microscopic level and, as a result, the macroscopic failure criterionbecomes conservative. The onset of microscopic failure in the unit cell causes the macroscopic onewhile, in the direct shear tests, the strength was measured by the maximum load at failure of the



Figure 13. Predicted strength by the failure criterion made by homogenization method

cracked specimen. It is, however, di$cult to identify what triggers the actual macroscopic failure.In fact, &the maximum load at failure' in the actual tests is also an ambiguous measure.

Remark 7. If another condition of failure in the microscopic scale is adopted, the results might bere"ned. For example, the coe$cient b can be determined by carrying out the collapse loadanalysis of the unit cell at the microscopic level. Even in such a case, a convexity of themacroscopic failure criterion is expected if the resulting failure surface for the unit cell is convex.Thus, there seems to be various similar settings that could well describe the real phenomena in themicroscopic level.

6. CONCLUSIONS

The method of estimating the strength of rock mass structures are proposed in this paper. Themajor consequence is that our method for the collapse load analysis simulates the anisotropicfeature of the strength, which is caused by the microscopically distributed cracks. The multi-scalenature of the strength of rock mass is re#ected in the analysis by extensively utilizing thecapability of the computational homogenization method. It should be noted that the successfulresults are owing to our strategy; we treated the discontinuous rock mass as a structural elementand not as a material.

We have emphasized that the homogenization method enables not only to evaluate thehomogenized material properties for arbitrary geometrical con"gurations of a REV, but also tocompute the actual stress "eld within a microstructure. More speci"cally, the macroscopicdeformation at point x is localized to the microscopic stress distribution measured in y. Then themicroscopic response is used to judge a certain type of microscopic failure within each unit cell,which is located at a macroscopic material point, x. Then the estimation of the macroscopicstability successfully takes into account the microscopic mechanical behaviour. The proposedmethod can su$ciently work well as a slightly conservative estimation of the limit load indesigning of rock mass structures. Although the numerical example presented in this papershowed the conservative trends, the limit state of the microscopic responses can easily be replacedby other more appropriate ones if necessary.

In conclusion, with the proposed method, it is a relatively easy task to estimate the macro-scopic stability for actual rock mass structures even if their microstructures have arbitrarygeometrical con"gurations. Because the computational homogenization method does not restrict



anything on the microstructural geometry, appropriate numerical modelling techniques wouldenable the proposed collapse load analysis for rock mass structures. This feature must make theproposed method more practical and will be discussed in the near future.

ACKNOWLEDGEMENTS

The authors would like to thank Mr. T. Osada in OHYO Geotech. Co. Ltd. Japan, for hiscontribution to the numerical analyses. We would also like to acknowledge the valuablesuggestions on the collapse load analysis by Professor S. Otsuka at Nagaoka University ofTechnology, Japan. Special thanks are due to Professor Noboru Kikuchi at The University ofMichigan, who motivated us to study the computational homogenization method.

APPENDIX: FORMULATION OF THE HOMOGENIZATION METHOD

Although the homogenization method has been popular in some engineering "elds, there are fewreports based on its utilization in goemechanics. Therefore, we present an almost completeprocess to derive the homogenization formulae.

Let us suppose a heterogeneous medium consisting of periodic microstructure as shown inFigure 14. We denote the domain of the iverall structure by ) and that of the unit cell of theperiodic microstructure by Y. The measure introduced in these domain are related by thecharacteristic dimension of the cell, which is denoted by e. Since the macroscopic structure can beregarded as an assembly of the periodic cells, the "eld variables including the elastic sti!ness andthe body force must be periodic functions, which change their values rapidly in the domain ) withthe small parameter e. To represent the speci"c dependency on the parameter, a superscript e isattached to each variables.

When the static equilibrium problem in linear elasticity are concerned, the displacement vector"eld ue

i(x) satis"es the following variational form:

P) Eeijkl

(x)Lue

k(x)

Lxl

Lvei(x)

Lxj

d)"PL)t

ti(x)ve

i(x) dS#P) f e

i(x)ve

i(x) d) ∀ve

i(x)3< (34)

where Eeijkl

(x) is the elastic sti!ness tensor and f ei(x) is the body force. Here, the surface traction

tigiven on the boundary surface L)

tis independent of microstructure and < is the admissible

space for the variation ve de"ned on L)u. To measure the rapidly changing "eld variables,

Figure 14. Microstructure and its unit cell



a microscopic scale y"x/e is introduced so that they are rewritten as

Eeijkl

(x)"Eijkl Ax,

x

eB"Eijkl

(x, y)

f ei(x)"f

i Ax,x

eB"fi(x, y)

uei(x)"u

i Ax,x

eB"ui(x, y) (35)

These are assumed to be >-periodic with respect to the microscale y, i.e.

Eijkl

(xp, y

q#>)"E

ijkl(x

p, y

q)

fi(x

p, y

q#>)"f

i(x

p, y

q)

ui(x

p, y

q#>)"u

i(x

p, y

q) (36)

Based on the method of two scale asymptotic expansions within the framework of singularperturbation theories, the displacement "eld can be represented as

ui(x, y)"u0

i(x)#eu1

i(x, y)#e2u2

i(x, y)#2 (37)

where each term in the right-hand side has the same smoothness. Here the term u0i(x) represents

the &averaged' or &macroscopic' displacement "eld while the higher terms such as u1i(x, y) and

u1i(x, y) represent the small perturbations of this "eld and will vanish in the limit eP0. This

means that the e!ects of the microstructure will be invisible when there are in"nitely many cells inthis structure ). Due to the multi-scale repositioned of the primal variable, its derivative is givenby the following expanded form:

Lui(x, y)

Lxj

"ALu0

iLx

j

#

Lu1i

LyjB#e A

Lu1i

Lxj

#

Lu2i

LyjB#O(e2) (38)

in which the chain rule has been utilized. It is worth while to note that the zeroth-order term of thestrain "eld involves the perturbed strain when measured at microscopic scale.

The substitution of these variables into (34) yields the following form:

P) Eijkl

(x, y) ALu0

k(x)

Lxl

#

Lu1k(x, y)

LylB

Lv0i(x)

Lxj

d)

#P) Eijkl

(x, y) ALu0

k(x)

Lxl

#

Lu1k(x, y)

LylB

Lv1i(x, y)

Lyj

d)

!PL)tiv0i(x) dS!P) f

i(x, y)v0

i(x) d)#O(e)"0

∀v0i(x)3<, ∀v1

i(x, y)3<]<

1%3(39)



By taking the limit eP 0, we can separate the boundary value problem given by (34) into themicro- and the macroscopic one as follows:

P) C1

D> D PY

Eijkl

(x, y)Lu1

k(x, y)

Lyl

Lv1i(x, y)

Lyj

d>D d)

"P) C1

D> D PY

LEijkl

(x, y)

Lyj

v1i(x, y) d>D

Lu0k(x)

Lxl

d) ∀v1i(x, y)3<]<

1%3(40)

P) C1

D> D PY

Eijkl

(x, y) ALu0

k(x)

Lxl

#

Lu1k(x, y)

LylB d>D

Lv0i(x)

Lxj

d)

"P) C1

D> D PY

fi(x, y) d>D v0

i(x) d)#PL)

tiv0i(x) dS ∀v1

i(x, y)3<]<

1%3(41)

where <1%3

is the >-periodic admissible function space. Here, the following Lemma 1 has beenused (see, Reference 10 for the proof ):

¸emma 1. If ge(x)"g(x, y) is >-periodic with respct to y, and if g (x, y) is integrable in the sense of¸ebesgue in domain )]>, then

lime?0 P) E

ijkl(x, y) A

Lu0k(x)

Lxl

#

Lu1k(x, y)

LylB

Lv1i(x, y)

Lyj

d)"0 ∀v1i(x, y)3<]<

1%3(42)

where D> D is the measure of the domain > of the unit cell of the microstructure.

While, for a given u1(x, y), equation (41) is the variational form for the solution u0(x), equation(40) can be regarded as the problem to be solved for u1(x, y) by providing u0(x). The solution ofthe former would be

u1i(x, y)"!skl

i(x, y)

Lu0k(x)

Lxl

#ti(x) (43)

if the >-periodic function sklp(x, y) is the following variational problem:

PY

Eijpq

(x, y)Lskl

p(x, y)

Lyq

Lvi(x, y)

)yj

d>"PY

Eijkl

(x, y)Lv

i(x, y)

Lyj

d> ∀vi(x, y)3<

1%3(44)

The existence and uniqueness of this problem can be proved by the following lemma (seeReference 10):

¸emma 2. ¸et us suppose that Eeijkl

(x)"Eijkl

(x, y) is >-periodic, and it is integrable in domain)]>. For a vector function F

i(x, y) is also >-periodic, the equation

PY

Eijkl

(x, y)L/

k(x, y)

Lyl

Lvi(x, y)

Lyj

d>"PY

Fi(x, y)v

i(x, y) d> ∀v

i(x, y)3<

1%3(45)

admits a >-periodic solution /k(x, y) which is de,ned up to an additive constant, if and only if

PY

Fi(x, y) d>"0 (46)

is hold.



The set of vector functions sklp(x, y) has the symmetry with respect to the indices k and l, and they

are called the characteristic functions which characterize the mechanical behaviour of themicrostructure. Thus, substituting equation (43) into equation (41), we obtain the equation for thesolution u0(x) as

P) EHijkl

(x)Lu0

k(x)

Lxl

Lv0i(x)

Lxj

d)"P) f Hi

(x)v0i(x) d)#PL)

ti(x)v0

i(x) dS ∀v0

i(x)3< (47)

where

EHijkl

(x)"1

D>D PYAEijkl

(x, y)!Eijpq

(x, y)Lskl

p(x, y)

LyqB d>

f Hi

(x)"1

D> D PY

fi(x, y) d> (48)

are called the homogenized elasticity tensor and the homogenized body force, respectively.After the characteristic functions and the averaged displacement are known, the actual

displacement "eld can be estimated via

uei(x)"u0

i(x)#eu1

i(x, y)#O(e2)"u0

i(x)!eskl

i(x, y)

Lu0k(x)

Lxl

#O (e2) (49)

and so also the strain and stress as follows:

eeij(x, y)"e0

ij(x, y)#O(e)

"

1

2 CAdipdjq!Lspq

i(x, y)

LyjB#Adjpdiq

!

Lspqj

(x, y)

LyiBD

Lu0p(x)

Lxq

#O(e) (50)

peij(x, y)"p0

ij(x, y)#O(e)

"Eijkl

(x, y) Adkpdlq!

Lspqk

(x, y)

JylB

Lu0p(x)

Lxq

#O(e) (51)

This post-processing is sometimes called the localization since the actual microscopic mechanicalbehaviours are evaluated by giving the macroscopic deformation.

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