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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

Cosserat model for periodic masonry deduced by nonlinear homogenization

Daniela Addessi a, Elio Sacco b,*, Achille Paolone a

aDipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza”, Via Eudossiana, 18, 00184 Roma, ItalybDipartimento di Meccanica, Strutture, Ambiente e Territorio, Università di Cassino Via G. Di Biasio 43, 03043 Cassino, Italy

a r t i c l e i n f o

Article history:Received 7 August 2009Accepted 4 March 2010Available online xxx

Keywords:MasonryNonlinear homogenizationDamage-frictionCosserat continuum

* Corresponding author.E-mail address: sacco@unicas.it (E. Sacco).

0997-7538/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.euromechsol.2010.03.001

Please cite this article in press as: Addessi,Journal of Mechanics A/Solids (2010), doi:10

a b s t r a c t

The paper deals with the problem of the determination of the in-plane behavior of periodic masonrymaterial. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolutesize of the constituents, is derived by a rational homogenization procedure based on the TransformationField Analysis. The micromechanical analysis is developed considering a Cauchy model for masonrycomponents. In particular, a linear elastic constitutive relationship is considered for the blocks, whilea nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and frictionphenomena occurring during the loading history. Some numerical applications are performed ona Representative Volume Element characterized by a selected commonly used texture, without per-forming at this stage structural analyses. A comparison between the results obtained adopting theproposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover,the substantial differences in the nonlinear behavior of the homogenized Cosserat material model withrespect to the classical Cauchy one, are illustrated.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Masonry is a composite material obtained joining natural orartificial bricks by means of mortar layers. Due to the complexnature of heterogeneous masonry materials, the mechanicalbehavior of masonry structures is one of the most challengingmatters of the structural engineering, both from a scientific andprofessional point of view. In fact, the formulation of models forreproducing the complex nonlinear mechanical behavior of themasonry is an active research field.

In the recent literature and in the case of regular masonrycharacterized by periodic texture of bricks and mortar, multi-scalemodeling is presented as a very promising approach, since itmatches the requirement of accurately reproducing microstruc-tural mechanical and geometrical properties with the need toreduce the computational burden with respect to a fully micro-mechanical analysis. In fact, it has been satisfactorily employed inthe structural analysis of periodic masonry (Massart (2003);Massart et al. (2007)), where microscopic and macroscopic scalesare intrinsically coupled. The main idea is to analyze at themicromechanical level a Representative Volume Element (RVE),containing all the information about the microstructure. Then, twodifferent Boundary Value Problems (BVP) are solved: one at the

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D., et al., Cosserat model for.1016/j.euromechsol.2010.03

micro-scale level and the other at the macro-scale level. At themacro-scale, an equivalent homogeneous continuum is considered,whose formulation is completely stated except for the constitutivelaw, which is derived by solving the BVP formulated at the micro-scale. In fact, at the micro-scale all the constituents are modeled indetail, taking into account the geometrical arrangement, the sizeand the specific constitutive laws of bricks and mortar. In order toconnect the two scales, proper bridging conditions are formulated.

Although the multi-scale modeling can be considered a veryinteresting and effective procedure, it can lead to significantcomputational efforts since the nonlinear micromechanicalproblem could not be analytically solved for complex geometriesand nonlinear behavior of the components, as it occurs formasonry; as a consequence, if the finite element solution procedureis adopted, the multi-scale approach requires to solve a nonlinearmicromechanical finite element scheme at each Gauss point of thestructural mesh. In order to reduce the computational costs, varioussimplified approaches have been proposed in literature resulting indifferent homogenization procedures (Pande et al. (1989);Pietruszczak and Niu (1992); Gambarotta and Lagomarsino(1997); Luciano and Sacco (1997); Uva and Salerno (2006)).

It could be remarked that most of the multi-scale models pre-sented in literature are based on the so-called first order homog-enization technique, making use at both macro- and micro-scale ofthe classical Cauchy continuum. Two major disadvantages of thefirst order multi-scale computational approaches, which signifi-cantly limit their applicability especially to the masonry modeling,

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can be mentioned. In spite of the fact that these techniques doaccount for the volume fraction, the distribution and themorphology of the constituents, they do not incorporate theabsolute size of the microstructure, thus making it impossible toaddress geometrical size effects (Kouznetsova et al. (2002)).Another difficulty arises from the intrinsic assumption of unifor-mity of the macroscopic fields attributed to each microstructuralRVE. This uniformity assumption relies on the concept of separationof scales and it is not appropriate in critical regions characterized byhigh deformation gradients, where the macroscopic fields canconsiderably vary. Moreover, when softening behavior is consid-ered for the constituents, the first order homogenization modelslack in correctly describing both global and local structuralresponse, affected by the well-known localization and mesh-dependency problems.

To overcome the mentioned problems, several techniques havebeen proposed in literature. In particular, in Massart et al. (2007)the localization at the macroscopic scale is modeled by the intro-duction of an embedded fracture band. Moreover, generalizedcontinua (Cosserat, higher order) are employed to describe thebehavior of either the microstructural constituents or the homog-enized macrostructure. This has the effect of introducing a materiallength scale into the constitutive description, providing a naturalway to obtain a dependence of the overall response of thecomposite material on the absolute size of the constituents and toachieve a realistic description of a micro-structurally triggeredmacroscopic localization. For instance, higher order homogeniza-tion methods have been proposed (Kouznetsova et al. (2004)),coupling enhanced higher order continua at the macro-scale withclassical Cauchy media at the micro-scale. Following a similarapproach, Forest and Sab (1998) and van der Sluis et al. (1999)proposed to consider a Cosserat continuum at the macro-scale,containing additional kinematical rotation fields with respect toclassical Cauchy model and an enriched description of stresses andstrains. On the other hand, it could be remarked that, if localizationappears with a band which is below the RVE size, continuummodels (even higher order or Cosserat models) are not able toreproduce correctly the kinematics of the problem and strongdiscontinuities could be introduced.

In literature, some reference can be found employing the micro-polar Cosserat continuum at the macro-level to model masonrymaterial in the field of linear elastic behavior. Masiani et al. (1995)developed an identification procedure to evaluate the Cosseratlinear elastic constitutive tensor as function of the shape, size andarrangement of the masonry texture based on a lattice model.Recently, Salerno and de Felice (2009) investigated on the capabilityof the Cauchy and Cosserat continuum models in representing thelinear behavior of periodic brickwork viewedas a rigid blocky system.

Masiani and Trovalusci (1996) and Trovalusci and Masiani(2003) presented multi-scale linear and nonlinear models formasonry, where the microstructural level is described as an in-plane discrete system of rigid bricks interacting with one anotherthrough the mortar joints, modeled as springs in the normal andtangential directions. Recently, in Brasile et al. (2007a) and Brasileet al. (2007b), the mortar joints have been described as nonlinearsprings characterized by a damage model. Indeed, the use ofa discrete model at the micro-scale, where the deformability of theblocks representing the widest part of the masonry volume iscompletely neglected, appears as a simplification, which couldinfluence the accuracy of the models in correctly reproducing theexperimental behavior of masonry. Furthermore, the identificationof the nonlinear constitutive model for the springs connecting therigid blocks, with the aim of reproducing the elasticity of theconstituents and the complex damaging and friction mechanismscharacterizing the mortar joints, could be a difficult task.

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Cerrolaza et al. (1999) presented a homogenization techniquefor the explicit determination of the elastic and plastic materialparameters of a Cosserat equivalent continuum as functions of thedimensions of the blocks and the interface properties and devel-oped a finite element code for nonlinear analysis of Cosseratcontinuum for blocky structures. Moreover, in Stefanou et al. (2008)the Cosserat continuum model for masonry walls is extended inthree-dimensions to describe the in-plane and out-of-planedeformationmodes, starting from a lattice model at themicro-scaleof the masonry wall.

A continuum based model has been proposed by Casolo (2006),who presented a computational homogenization procedure toderive an orthotropic Cosserat equivalent model for regularmasonry made of rectangular blocks, limiting the study to the caseof linear elasticity. In particular, at the micro-scale, bricks andmortar are modeled as Cauchy continua and the elastic response isevaluated by means of a finite element procedure, applying propercombinations of natural and essential boundary conditions.

It is worthwhile noting that there are very few references inliterature, where Cosserat continuum is employed to modelnonlinear behavior of masonry material. Thus, the formulation ofa two-scale nonlinear homogenization procedure for masonry,where at the micro-scale a deformable (“Cauchy”) continuum isconsidered and the nonlinear behavior of the constituents ismodeled in detail, appears as an essential step towards the devel-opment of accurate stress analyses of masonry structures.

In this paper, a nonlinear enhanced homogenization proce-dure is proposed. With the aim of correctly reproduce micro-structural interaction effects, typical of masonry materials,a Cosserat medium is adopted at the macro-scale. On the otherhand, at the micro-scale the standard Cauchy continuum isconsidered to model the masonry RVE constituents. The workcan be considered the natural evolution and continuation of theresearch developed by Sacco (2009), where Cauchy continuummodel is adopted both at micro- and macro-scale. It can beremarked that the extension of the nonlinear homogenizationprocedure to the Cosserat model is not straightforward. Thedisplacement field in the RVE is expressed as the superpositionof a prescribed field, defined as a function of the macroscopicCosserat strains, and a periodic part due to the heterogeneousnature of the RVE. As for the prescribed field, a suitable kine-matical map bridging the macro-Cosserat and the micro-Cauchyscales is formulated for a rectangular RVE. In particular, thirdorder polynomial expansions are selected for the prescribedpart. A linear elastic constitutive law is assumed for bricks,while a coupled damage-friction cohesive model is used formortar joints. The nonlinear homogenization, adopted to derivethe overall Cosserat behavior on the basis of the micro-mechanics, is formulated on the basis of the TransformationField Analysis (TFA), initially proposed, in the framework of theCauchy continuum, by Dvorak (1992) and recently modified andadopted by Sacco (2009) to model masonry material. Accordingto TFA procedure the microscopic inelastic strain field is repre-sented as a finite set of uniform transformation fields. Inparticular, for each mortar joint a uniform inelastic strain isassumed and, as a consequence, a uniform transformation fieldis considered. Herein, a non trivial extension of the TFA proce-dure to Cosserat continuum is performed. A numerical proce-dure, based on a backward-Euler time integration and ona classical predictor-corrector technique, is implemented to solvethe evolutive nonlinear problem. Some numerical applicationsare performed on a RVE characterized by a selected commonlyused masonry texture. Moreover, a comparison between theresults obtained adopting the proposed Cosserat TFA procedureand a nonlinear micromechanical Finite Element Analysis (FEA)

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is presented, developing, for the latter case, a nonlocal approachin order to avoid strain and damage localization.

The paper is organized as follows: Section 2 introduces the Cos-serat strain and stress measures in 2D framework and describes theadopted constitutivemodel for eachmasonry component; Section 3deals with the proposed nonlinear homogenization procedure forCosserat continuum; in Section 4 the solution algorithm is pre-sented; finally, numerical results are presented in Section 5.

2. Micropolar masonry model with periodic microstructure

Heterogeneous masonry material with periodic microstructurecan be replaced by an equivalent homogeneous medium at themacro-level, to analyze the masonry wall behavior.

Herein, a Cosserat continuum is adopted at the macro-level inorder to take into account the microstructural interaction effects,related to the size and shape of the constituents. On the other hand,a classical Cauchy continuum is assumed for the constituents of theRVE at the micro-level, suitably modeling their complex nonlinearconstitutive behavior.

2.1. Macro-level Cosserat model

The 2D Cosserat continuum involves an additional rotationalkinematical degree of freedomwith respect to the classical Cauchymodel. In fact, the displacement vector U ¼ {U1 U2 F}T containsthree independent kinematic fields, representing the translationsU1 and U2 and the rotation F, respectively, at each point X ¼ (X1,X2)T of the body volume U. In the following the analysis is devel-oped in the framework of small strain and displacementassumption.

Adopting the Voigt’s notation, the macroscopic Cosserat strain Eis expressed as:

E ¼ fE1 E2 G12 Q K1 K2gT (1)

where E1 and E2 are the extensional strains, G12 is the in-planeCauchy shear strain, K1 and K2 the micro-curvatures and Q ¼ 2(W � F) is the rotational deformation, with W representing therigid rotation. The dual quantities in the virtual work senseassociated with the deformation components (1) are collected inthe stress vector S, which may be expressed as:

S ¼ fS1 S2 S12 Q M1 M2gT (2)

Fig. 1. (a) Selected RVE for periodic masonry characterized by run

Please cite this article in press as: Addessi, D., et al., Cosserat model forJournal of Mechanics A/Solids (2010), doi:10.1016/j.euromechsol.2010.03

2.2. Micromechanics of the masonry

Hereafter, the analysis is limited to the case of regular periodicmasonry. The micromechanical study of the RVE requires thedefinition of the constitutive laws for bricks and mortar. It can beemphasized that masonry is characterized by a strong nonlinearstress-strain behavior, due to the nonlinear behavior of theconstituents; in particular, the nonlinear response of the mortarstrongly affects the overall behavior of the masonry.

In the following a 2D analysis is developed. To model satisfac-torily the in-plane behavior of the RVE masonry, the generalizedplane state should be adopted, as suggested by Pegon and Anthoine(1997), mainly when compressive failure mechanisms occur. Sincethe compressive failure is not investigated herein and in order tosimplify the formulation, the classical plane strain assumption isadopted.

It is worth noting that, for the determination of the effectiveCauchy continuum, several different RVEs can be chosen to describethe samemasonry geometrical texture; i.e. the definition of the RVEis not unique, but different equivalent RVEs can be consideredleading all to the samemicromechanical response, as demonstratedin literature (Anthoine (1995)). On the contrary, the determinationof the effective Cosserat continuum for a given geometrical texturedepends on the specific chosen RVE, influencing the overall struc-tural response. As a consequence, the choice of the RVE in theframework of the Cosserat homogenized medium is not trivial. Inthe following, the RVE is chosen as the simplest repetitive cell.

Introducing a Cartesian coordinate system (O;x1, x2) in the RVE,the displacement vector is denoted as u¼ {u1, u2}T , while the strainvector, according to the Voigt’s notation, is given by 3¼ {31, 32, g12}T,with the classical meaning of the notation. The chosen RVE ischaracterized by rectangular shape with dimensions 2a1 and 2a2,parallel to the coordinate axes x1 and x2, as shown in Fig. 1a. TheRVE accounts for all the geometrical and constitutive properties ofthe masonry components; in Fig. 1a, the mortar thickness isdenoted by s and the brick sizes by b and h.

The linear elastic stress-strain relationship is adopted for thebrick:

sB ¼ CB3 (3)

where CB is the elastic matrix and sB ¼ fsB1; sB1; sB12gT is the stressvector in the brick.

A damage-friction constitutive law for the mortar material isadopted; it is based on the original proposal by Ragueneau et al.(2000) and properly modified to model the mortar behavior by

ning bond texture; (b) Unit Cell obtained as half of the RVE.

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Sacco (2009). To make the present paper self-consistent the funda-mental equations governing the mortar constitutive behavior arebriefly reported hereafter.

A local coordinate system is introduced in the typical mortarjoint, with T and N denoting directions parallel and orthogonal tothe mortar joint, respectively.

The stress sM, occurring at the typical point of the mortar, isobtained as a suitable combination of two stresses, su and sd,according to the formula:

sM ¼ ð1� DÞsu þ Dsd (4)

where D is the damage parameter. The two stress vectors su and sd

are related to the strain vector in the mortar, 3, by the constitutiveequations:

su ¼ CM3; sd ¼ CMð3� 3pÞ (5)

where:

CM ¼

264 CMTT CM

NT 0CMNT CM

NN 00 0 GM

375 (6)

represents the elasticity matrix of the mortar and 3p is the vector ofthe plastic strain due to the possible unilateral opening effect and tothe friction sliding.

Taking into account the constitutive Eqs. (5), Eq. (4) becomes:

sM ¼ CMð3� pÞ (7)

wherep is the vector of the inelastic strain, conveniently written inthe form:

p ¼8<:

pTpNpNT

9=; ¼ D

8<:hð3NÞ3Thð3NÞ3NgpNT

9=; (8)

accounting for the damage, the unilateral contact by means of thecomponenth (3N) 3Nand the slipbymeansof the componentgp

NT ;h (3N)is theHeaviside function,which assumes the following values: h (3N)¼0 if 3N� 0 and h (3N)¼ 1 if 3N> 0. Because of the simplified form of theinelastic strain (8), the constitutive law (5) is able to provide zeronormalstress in transversaldirection,sdN ¼ 0,aswell as in longitudinaldirection, sdT ¼ 0, when mode I opening of the mortar joint occurs.

The friction is modeled as a classical plasticity problem; theevolution of the inelastic slip strain component gp

NT is governed bythe classical Coulomb yield function:

4�sd�

¼ msdN þ���sdNT ��� (9)

where m is the friction parameter. The non-associated flow rule isconsidered as:

_gpNT ¼ _lsdNTjsdNT j

(10)

with the classical loading-unloading Kuhn-Tucker conditions:

_l � 0 4�sd�� 0; _l4

�sd�

¼ 0 (11)

where l is the inelastic multiplier. A model which accounts for thecoupling of mode I and mode II of fracture is considered for thedamage parameter D evolution. The two quantities hN and hNT,which depend on the first cracking strains 3N,0 and gNT,0, on thepeak values of the stresses sN,0 and sNT,0 and on the fracture ener-gies GcI and GcII, respectively, are introduced in the form:

Please cite this article in press as: Addessi, D., et al., Cosserat model forJournal of Mechanics A/Solids (2010), doi:10.1016/j.euromechsol.2010.03

hN ¼ 3N;0sN;02GcI

; hNT ¼ gNT;0sNT ;02GcII

(12)

The equivalent strain measures YN and YNT are defined as:

YN ¼ h3Ni2; YNT ¼ ðgNTÞ2 (13)

where the bracket operator h�i gives the positive part of thequantity �. Then, the strain ratios are determined as:

h ¼ 1� 1a2

ðYNhN þ YNThNT Þ; b ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiYN32N;0

þ YNTg2NT ;0

s� 1 (14)

with a¼ (YN þ YNT)1/2. Finally, the damage is evaluated according tothe following law:

D ¼ maxhistory

�min

�1;

1h

�b

1þ b

�(15)

3. Nonlinear homogenization technique

In order to derive themacroscopic behavior of periodicmasonry,a compatible nonlinear homogenization procedure, based ona proper formulated kinematic map linking the macro- and micro-levels, is presented. The heterogeneous masonry RVE is subjectedto the macroscopic Cosserat strain E applied to the whole RVE andto the inelastic strainpi (Eq. (8)), with i¼ 1,. ,m, applied to each ofthe m mortar joints.

3.1. Kinematic map

The displacement field for the Cauchy micromechanicalmedium at the point x ¼ (x1, x2)T is expressed in the followingrepresentation form:

u ¼ uðxÞ þ ~uðxÞ (16)

where uðxÞ is the assigned displacement field, depending on themacroscopic deformation E and uðxÞ is the periodic field, satisfyingperiodicity conditions at the RVE boundary (Suquet (1987); Lucianoand Sacco (1998)). Consequently, the compatible strain field for theCauchy medium results in the form:

3 ¼ 3ðxÞ þ ~3ðxÞ (17)

where 3 and ~3 are the strain fields compatible with u and ~u,respectively. Herein, in order to derive a proper expansion for theassigned displacement field uðxÞ, which takes into account all themacroscopic Cosserat deformation components, a least-squaresoptimization procedure is employed, where the macroscopickinematic field U is determined as the rigid body motion that bestfits the microscopic displacement field u(x) on the RVE (Forest andSab (1998)). Similarly to the classical first order homogenization,where both at the macro- and micro-levels a Cauchy medium isconsidered and a first order polynomial expansion is adopted for u,here a third order polynomial function is assumed. It is worthwhilenoting that second order polynomial expansions would not allowto take into account all the Cosserat deformation states, in partic-ular the rotational deformation Q would not be described, asremarked in van der Sluis et al. (1999). Then, by imposing the scaleinvariance condition between the first three components bE of thestrain vector E and the Cauchy strain vector 3 evaluated at thecenter of the RVE (see Fig. 1a), i.e. bE ¼ 3ð0Þ, and by requiring theselast be of grade 2 and the rotation F be affine, as well as some usualrestrictions on the form of the microscopic deformation

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components, Forest and Sab (1998), only five terms are retained inthe third order polynomial expansions expressing each of thecomponents of u. Finally, by considering a generic rectangular cell,the following form of the assigned displacement u results incompact notation:

u ¼ AðxÞE (18)

where

A ¼"x1 0 1

2x2 �ax32 � 3r2x21x2

� �x1x2 �12x

22

0 x2 12x1 �r2a

r2x31 � 3x1x22

� 12x

21 x1x2

#(19)

with

a ¼ 54a21 þ a22

a41r ¼ a2

a1(20)

It has to be underlined that Cosserat strain vector E in Eq. (18)differs from the one defined in Eq. (1); in fact, the fourth compo-nent is redefined as:

bQ ¼ Qþ 12r2 � 1r2 þ 1

G12 (21)

In such a way, the Cauchy deformation modes can be activatedindependently from the Cosserat modes. The stress variableconjugated to bQ is denoted with bZ .

The strain vector at the micro-level (Eq. (17)) can now beexpressed as:

3 ¼ BðxÞEþ ~3ðxÞ (22)

where ~3ðxÞ is the periodic strain, satisfying null average conditionin U, and:

B ¼241 0 0 6ar2x1x2 �x2 00 1 0 6ar2x1x2 0 x10 0 1 3a

r2 � 1

�x22 � r2x21

�0 0

35 (23)

The in-plane periodicity and continuity conditions lead to thefollowing boundary conditions:

~uða1; x2Þ ¼ ~uð�a1; x2Þ cx2˛½�a2; a2�~uðx1; a2Þ ¼ ~uðx1;�a2Þ cx1˛½�a1; a1� (24)

Furthermore, additional conditions with respect to the first orderhomogenization are required to ensure that the fluctuation fieldproduce zero average strain:

Za1�a1

~u1ðx1;�a2Þdx1 ¼ 0;Za2

�a2

~u2ð�a1; x2Þdx2 ¼ 0 (25)

3.2. Nonlinear homogenization

In this section a TFA based procedure, up to now exclusivelyadopted for classical Cauchy media, is properly extended to themicro-polar Cosserat continuum. The total assigned Cosseratmacroscopic strain E is additively decomposed into an elastic partEe and an inelastic part P. Consequently, 6 þ 3m micromechanicalBVPs on the RVE, subjected to the prescribed value of the sixcomponents of the overall elastic strain Ee and to the threecomponents of the inelastic strains pi in the m mortar joints, haveto be solved.

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Firstly, the BVP on the RVE subjected to Ee is formulated. Themicromechanical strain field, resulting after numerically solvingthe BVP, can be written in the following representation form:

e ¼ ReðxÞEe (26)

where the localization matrix Re (x) is introduced, which allows toevaluate the Cauchy local strain at any point of the microscopicmedium, corresponding to the application of the Cosserat strain Ee.Consequently, the average strain in each mortar joint Mi, with UMi

denoting its volume, results as:

eMi ¼ 1

UMi

ZMi

eðxÞdU ¼ 1

UMi

ZMi

ReðxÞdUEe ¼ RMi

e Ee;

i ¼ 1;2;.;m ð27Þas well as in the brick, whose total volume is denoted by UB:

eB ¼ 1

UB

ZB

eðxÞdU ¼ 1

UB

ZB

ReðxÞdUEe ¼ RBeEe (28)

The corresponding homogenized Cosserat stress in the wholeRVE volume U is obtained by applying the generalized Hill-Mandelprinciple:

ETeSe ¼ 1U

ZU

eTsdU¼ ETe1U

264ZB

RTeC

BedUþXmj¼1

ZMj

RTeC

MjedU

375¼ ETeCEe ð29Þ

so that:

Se ¼ CEe (30)

where C represents the overall elastic constitutive matrix. Similarly,the average stress in the mortar joint Mi may be evaluated assMi

e ¼ CMeMi ¼ CMR

Mi

e Ee, as well as in the brick sBe ¼

CBeB ¼ CBRBeEe.

It could be noted that the constitutive relation for a 2D periodicmasonry, characterized by orthotropic texture in a reference framealigned with the principal axes of the material, results as:8>>>>>><>>>>>>:

S1S2S12bZM1M2

9>>>>>>=>>>>>>;¼

26666664C11 C12 0 0 0 0C21 C22 0 0 0 00 0 C33 C34 0 00 0 C43 C44 0 00 0 0 0 D11 00 0 0 0 0 D22

37777775

8>>>>>><>>>>>>:

E1E2G12bQK1K2

9>>>>>>=>>>>>>;e

(31)

where the first 3 � 3 sub-matrix can be recognized as the classicalin-plane elastic constitutive operator for Cauchy continuum, C34 ¼C43 and C44 govern the unsymmetric shear behavior and areinfluenced by the shape of the RVE and D11 and D22 govern theflexural behavior affected by the absolute size of the RVE.

Similarly, after solving the micromechanical problem of the RVEsubjected to an inelastic strain pi prescribed in the mortar joint Mi,the resulting local strain field is expressed in the form:

pi ¼ Rpi ðxÞpi (32)

being Rpi ðxÞ the associated localization matrix which relates theCauchy inelastic strain pi to the Cauchy local strain pi. The elasticstrain in the generic mortar joint Mj is obtained as the differencebetween its total deformation pi;Mj

and the inelastic strain pi as:

hi;Mj ¼ pi;Mj � dijpi ¼

�RMj

pi � dijI�pi ðno sumÞ (33)

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where pi;Mjand RMj

pi are the restriction to the mortarMj of the fieldspi and Rpi , respectively. As for the elastic strain in the brick, itcoincides with the total strain, so resulting:

hi;B ¼ pi;B ¼ RBpipi (34)

with evident meaning of the symbols. Then, the correspondingoverall Cosserat stress, derived by applying again the generalizedHill-Mandel principle, is obtained in the form:

Spi ¼ 1U

264ZB

�RBpi

�TCBhi;BdUþ

Xmj¼1

ZMj

�RMj

pi

�TCM

jhi;Mj

dU

375¼ 1U

264ZB

�RBpi

�TCBRB

pidUþXmj¼1

ZMj

�RMj

pi

�TCM

j�RMj

pi �dijI�dU

375pi

ð35ÞFrom the computational point of view, the evaluation of thelocalization matrix Rpi ðxÞ and, as a consequence, of the strain fieldpi (x) is performed by enforcing the vanishing of the average stress(35), corresponding to the homogenization of a self-equilibratedmicromechanical stress state. To this end the procedure describedin what follows is employed.

The elastostatic problem corresponding to the RVE subjected tothe inelastic strain pi in the mortar Mi under the continuity andperiodicity conditions (24) and (25), leads to the determination of thestrainfieldqi ¼ Qpi ðxÞpi, beingQpi ðxÞ the localizationmatrix. It canbe remarked that the strainfieldqi (x) is characterized bynull averageand by the non zero macroscopic Cosserat stress Sqi ¼ Sipi, where:

Si ¼ 1U

264 ZB

�QB

pi

�TCBQB

pidU

þXmj¼1

ZMj

�QMj

pi

�TCM�QMj

pi � dijI�dU

375 ð36Þ

In order to enforce the vanishing of the Cosserat stress, a macro-scopic Cosserat strain Pi is assigned to the RVE, which produces themicroscopic strain state p

_ iðxÞ ¼ RePi. By the superposition of thetwo strain fields p

_ iðxÞ and qi(x), the total strain is obtained as:

pi ¼ p_ iðxÞ þ qiðxÞ ¼ RePi þ Qpipi (37)

The macroscopic Cosserat stress Spi associated to the field piðxÞ isobtained as the sum of the average stress S

p_i due to p

_ iðxÞ and theaverage stress Sqi due to qiðxÞ, namely:

Spi ¼ Sp_i þSqi ¼ CPi þ Sipi (38)

By enforcing the condition Spi ¼ 0, Eq. (38) gives:

Pi ¼ �C�1Sipi (39)

The linear operator, which allows to evaluate the macroscopicstrain associated with the inelastic strain pi in the RVE, is deter-mined as:

Pi ¼ �C�1Si (40)

so that Pi ¼ Pipi:

Finally, because of Eqs. (39) and (40), Eq. (37) takes the form:

pi ¼�ReP

i þ Qpi

�pi (41)

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and the localization matrix Rpi ðxÞ, introduced in Eq. (32), can beevaluated as:

Rpi ¼ RePi þ Qpi (42)

The average stresses in the i-th mortar joint and in the brick aresMj

pi ¼ CMhi;Mjand sB

pi ¼ CBhi;B, with hi;Mj ¼ ðRMj

pi � dijIÞpi and

hi;B ¼ RBpipi, where R

Mj

pi and RBpi are the average of the matrices

RMj

pi and RBpi in the mortar joint Mj and in the brick, respectively:

RMj

pi ¼ �RMj

e C�1Si þQMj

pi ; RBpi ¼ �R

BeC

�1Si þ QBpi (43)

When the RVE is subjected to the overall elastic strain Ee and to theinelastic strains pi, i ¼ 1, 2, . , m, the superposition of the effectscan be performed. In such a way, it is possible to compute:

� The total overall strain

E ¼ Ee þP1p1 þ/þPmpm (44)

� The overall stress S ¼ Se

� The total average strain in the m mortar joints and in the brickas:

3Mj ¼ R

Mj

e EþXmi¼1

LMj

pi pi; (45.1)

3B ¼ RBeEþ

Xmi¼1

LBpipi (45.2)

where:

LMj

pi ¼ RMj

pi � RMj

e RMj

pi ; LBpi ¼ R

Bpi � R

BeR

Bpi (46)

� The average stress in the m mortar joints and in the brick:

sMj ¼ CM�eM

j þ h1;Mj þ/þ hm;Mj�

¼ CM�3M

j � pj�

sB ¼ CBeB þ h1;B þ/þ hm;B� ¼ CB3B

(47)

Herein, it is assumed that:

� The inelastic strain is constant in each mortar joint;� The nonlinear behavior of the RVE depends on the averagestresses and strains evaluated in each of the m mortar joints.

4. Computational procedure

The proposed nonlinear homogenization procedure initiallyrequires the evaluation of the overall constitutive matrix C, the

matrix Pi and the localization matrices RMj

e and RMj

pi . They arecomputed by solving 6 þ 3m FE linear elastic micromechanicalproblems, by using 4-node displacement-based quadrilateralelements with two translational degrees of freedom for each node.

The damage and plasticity evolution problem in the RVE sub-jected to the macroscopic Cosserat total strain E is solved by a stepby step integration procedure, based on a classical backward-Euleralgorithm (Simo and Hughes, 1998).

In the following, the subscript ‘n’ denotes variables evaluated atthe previous time step tn; for ease of notation, no subscript is usedto indicate the corresponding current quantities, i.e. computed atthe step tnþ1. Moreover, D denotes the increment of the variableinto the time step Dt ¼ tnþ1�tn.

The nonlinear problem at each time step is solved by adoptinga return-mapping algorithm in order to evaluate the overall stressS. To this end, the elastic strain Ee and the inelastic strains pj (j ¼ 1,

periodic masonry deduced by nonlinear homogenization, European.001

Table 1Damage-plastic solution procedure in the mortar joint Mj.

Iteration ‘k þ 1’

Damage evaluationEquivalent strain measures

YkN ¼ h3kNi2; Yk

NT ¼ ðgkNT Þ2 (Eq. (13))Strain ratios

hk; bk (Eq. (14))Damage

Dkþ1 (Eq. (15))

Unilateral effect evaluationif 3kN � 0 then hkþ1ð3NÞ ¼ 0 else hkþ1ð3NÞ ¼ 1if Dkþ1 > 0

Friction plasticity evaluationPrediction phase

pikþ1;tr ¼ pinði ¼ 1;.;mÞ

Trial stressessdkþ1;tr ¼ ½sdkþ1;tr

T ; sdkþ1;tr

N ; sdkþ1;tr

NT �T (Eq. (5))

Trial yield function4kþ1;tr ¼ msd

kþ1;tr

N þ jsdkþ1;tr

NT jCheck plasticityif 4kþ1;tr < 00Dgp;kþ1

N ¼ 0, else

Correction phase

Dgp;kþ1NT ¼ Dl

�����sdkþ1;tr

NT

sdkþ1;tr

NT

����� with

Dlkþ1 � 0 4kþ1 � 0; Dlkþ14kþ1 ¼ 0

Dlkþ1 ¼ 1GM

4kþ1;tr

Update inelastic and elastic average strainspikþ1 ði ¼ 1;.;mÞ; 3kþ1

Compute residualrkþ1 ¼ ReEþPm

i¼1 Lpipi;kþ1 � 3kþ1

D. Addessi et al. / European Journal of Mechanics A/Solids xxx (2010) 1e14 7

ARTICLE IN PRESS

2,.,m), related to the damage and to the unilateral contact-frictioneffects occurring in the mortar joints, have to be calculated.

It is worthwhile noting that, as it is clear from Eq. (45), theaveragestrain3M

jin themortar jointMjdependson theoverall strain

E and on all them inelastic strains pi (i ¼ 1, 2,. ,m). Consequently,the damage and unilateral contact-friction evolution problems inthe m mortar joints result all coupled, giving rise to a complexnonlinear problem. Then, in the spirit of the splitting procedure, theiterative procedure described below is employed, solving a set ofmuncoupled evolutionary problems, one for each mortar joint,considering as frozen the damage and plasticity evolution into theother m � 1. A strain-driven procedure is adopted to evaluate thedamage and plasticity evolution into the mortar joints, when themacroscopic Cosserat strain vector E is assigned.

At the previous iteration k, the macroscopic Cosserat elasticstrain Ee is evaluated on the basis of the total strain E and of themacroscopic strains Pi associated to the inelastic strains pi in themortar joints. Then, the total and elastic average strains in themortar joints, 3M

jand hi;Mj

, are computed.At the current iteration and for all the mortar joints Mj, the

damage is updated on the basis of the 3Mjevaluated at the previous

iteration. Then, the unilateral contact problem is solved, by evalu-ating the Heaviside function h(3N). Finally, if damage is active in themortar joint Mj the friction problem is solved adopting a predic-tion-correction technique. A trial prediction of the inelastic strainspi is computed by assuming them equal to the ones evaluated at theprevious time step tn. The normal and trial shear stress are thenevaluated, on the basis of which the trial yield function f iscalculated. The correction phase is performed if f > 0. Once thedamage, the unilateral contact and the friction problems are solvedin all the mortar jointMj, the values of the inelastic strain vectors pj

are updated, then the new values of the total average strains 3Mjare

determined. Thus, a further iteration is performed where damage,unilateral contact and friction problems have to be solved again inall the mortar joints, until a convergence test is satisfied. Thescheme of the described solution algorithm is reported in Table 1,where the apex ‘Mj’ is omitted for ease notation.

Writing Eq. (45.1) in residual form, the residual strain can beevaluated in each mortar joint as:

rMj;kþ1 ¼ RMj

e EþXmi¼1

LMj

pi pi;kþ1 � 3Mj;kþ1 (48)

The norm of the residual vectors, representing the residual error atthe end of the kþ 1-th iteration in eachmortar joint, is computed asrkþ1 ¼ Pm

i¼1 krMi;kþ1k. If the error is lower than a given tolerancethe iterative procedure is stopped.

5. Numerical results

Hereafter, some numerical applications are reported. In partic-ular, a classical texture (running bond) is selected for the RVE,representing a class of widely spread masonry structures. In sucha case, the number of mortar joints which is considered in theanalysis is set equal to 8, as shown in Fig. 1, where also the numberassociated to each joint is reported. The geometrical parameters ofbricks and mortar are assumed as follows: size of the brick b ¼ 240mm, h ¼ 120 mm; thickness of the mortar joints s ¼ 10 mm.Furthermore, the material mechanical parameters are:

E (MPa) n 3N,0 gNT,0 GcI (MPa) GcII (MPa) m

Brick 18, 000 0.15Mortar 1000 0.15 0.0005 0.001 0.00125 0.00217 0.5

where E and n are Young and Poisson’s moduli, respectively.

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Firstly, the validation of the nonlinear numerical homogeniza-tion is performed comparing the results obtained by the proposedprocedure with the ones determined by micromechanical FiniteElement Analyses (FEA). In particular, four tests are performed,applying to theRVE tensile, symmetric shear, rotational deformationandmicro-curvature macroscopic loading histories. Concerning thefirst two analyses, the results obtained by the TFA procedure havealready been shown in Sacco (2009). Herein, a comparisonwith themicromechanical responses is presented to validate the proposednonlinear homogenization. Moreover, the latter two applicationsallow to investigate on the performance of the RVE, when it is sub-jected to Cosserat typical macroscopic strain components.

As for the micromechanical analyses, a 2D displacement-based4-node finite element is formulated considering different consti-tutive laws for brick, head joints and bed joints. In particular, thelinear elastic relationship, Eq. (3), is assumed for bricks, while thedamage-plastic constitutive law described in Eqs. (4)e(15) isconsidered for mortar joints. In order to avoid strain and damagelocalization in the mortar joints, a nonlocal integral model isadopted, defining the nonlocal equivalent strain measures as:

YNðxÞ ¼

ZUM

YNjðx�yÞdUZUM

jðyÞdU; YNTðxÞ ¼

ZUM

YNTjðx�yÞdUZUM

jðyÞdU(49)

where y is a typical point of the head joints or bed joints and j is thestandard Gaussian weighting function, namely:

j ¼ exp

� jx � yj2

r2

!(50)

periodic masonry deduced by nonlinear homogenization, European.001

-0.001 0 0.001 0.002 0.003-6.0

-3.0

0.0

3.0

[MPa

]

Σ1

E1

FEATFA

O

AB

C

DE

F

Fig. 2. Tensile test: comparison of the results obtained by micromechanical FEA and bythe proposed procedure.

D. Addessi et al. / European Journal of Mechanics A/Solids xxx (2010) 1e148

ARTICLE IN PRESS

with r ¼ 15 mm. Indeed, the internal length for the mortar shouldbe set even smaller; in such a case it could be possible to reproducea non uniform damage distribution along the thickness of themortar. Indeed, the continuum description of the mortar already

A

DamageDeformed RVE

B

C

Fig. 3. Tensile test: deformed configurations (first column), maps of damage (second columnin Fig. 2.

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represents an approximation, as the failure is due more often to thedevelopment of cracks at the interface between joints and bricksrather than to the mortar damage. Nevertheless, aiming to capturean overall damage in the thickness direction and to adopt a quitereasonable FE discretization, the assumed value of the internallength can be considered as satisfactory, allowing to recoverdamage variation along the mortar length. The above nonlocalequivalent strain measures are then used to evaluate the strainratios h and b by means of Eq. (14).

Initially, the tensile test is performed on the RVE subjected toa compression in the vertical direction kept constant during thewholehistory, E2¼�4.0�10�4, and a strainhistory in thehorizontaldirection with E1 varying from 0 to 30.0 � 10�4, to �5.0 � 10�4.

Due to the double symmetry of the geometrical scheme andloading condition, micromechanical FEA is performed by consid-ering only a quarter of the RVE under suitable boundary conditions.A regular mesh of 169 elements and 256 nodes is used for thecomputations.

In Fig. 2 the overall response curve is plotted in terms of themacroscopic stress component S1 versus the macroscopic strain E1,where the continuous curve is referred to the proposed procedureresults, while the diamond symbols concerns the FEA solution. Itcan be noted that a very satisfactory agreement is obtained. The

1.75E-04 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00

-9.98E-02

1.10E+00

1.75E-04 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00

-9.98E-02

1.10E+00

1.75E-04 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00

-9.98E-02

1.10E+00

-1.99E-03-1.76E-03-1.54E-03-1.32E-03-1.09E-03-8.70E-04-6.47E-04-4.24E-04-2.01E-04 2.20E-05 2.45E-04

-2.21E-03

4.68E-04

-2.62E-03-2.13E-03-1.64E-03-1.14E-03-6.47E-04-1.53E-04 3.41E-04 8.35E-04 1.33E-03 1.82E-03 2.32E-03

-3.12E-03

2.81E-03

-8.58E-03-4.68E-03-7.78E-04 3.13E-03 7.03E-03 1.09E-02 1.48E-02 1.87E-02 2.26E-02 2.65E-02 3.04E-02

-1.25E-02

3.44E-02

Plastic Shear Strain

) and plasticity (third column) evaluated for different values of the strain E1 as reported

periodic masonry deduced by nonlinear homogenization, European.001

-0.002 -0.001 0 0.001 0.002-2.0

-1.0

0.0

1.0

2.0

[MPa

]

Σ12

Γ12

FEATFA

O

AB C

Fig. 4. Shear test: comparison of the results obtained by micromechanical FEA and bythe proposed procedure.

D. Addessi et al. / European Journal of Mechanics A/Solids xxx (2010) 1e14 9

ARTICLE IN PRESS

mechanical behavior of the RVE is characterized by an initial linearelastic response, followed by the damaging of the head joints andthe subsequent damaging and frictional slip of the bed joints.During the unloading and reverse loading paths, an elastic and

DamageDeformed RVE

A

B

C

Fig. 5. Shear test: deformed configurations (first column),maps of damage (second column) and

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perfect plastic response is observed, followed by the unilateralcontact effect of the head joints. Note that at this point only thefrictional mechanism is activated, corresponding to the ultimatestrength of the RVE associated to the imposed vertical compression.It can be noted that only in the path DE the TFA analysis is not inperfect agreement with the FEA solution.

In Fig. 3 the RVE deformed configurations, the damage and shearplastic strainmaps at three steps of the analysis, namely points A, B,C in Fig. 2, are reported. It can be observed that both the damageand plasticity are almost uniformly distributed along the longitu-dinal direction of the head and bed joints, since the beginning ofthe analysis, becoming uniform after few steps. This evidencevalidates the assumption on which the nonlinear homogenizationprocedure is essentially based.

Then, the symmetric shear test is performed consideringaconstant compression in thevertical direction,E2¼�3.0�10�4, anda symmetric shear history with G12 varying from 0 to 20.0� 10�4, to�20.0� 10�4.

Due to thesymmetryof thegeometrical scheme,micromechanicalFEA is performed by considering only half of the RVE, which is thesmallest periodic RVE often called as Unit Cell; the following peri-odicity conditions are prescribed on the Unit Cell (Fig. 1b):

1.75E-04 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00

-9.98E-02

1.10E+00

1.28E-04 7.32E-02 1.46E-01 2.19E-01 2.92E-01 3.65E-01 4.38E-01 5.11E-01 5.84E-01 6.58E-01 7.31E-01

-7.29E-02

8.04E-01

1.68E-04 9.60E-02 1.92E-01 2.88E-01 3.83E-01 4.79E-01 5.75E-01 6.71E-01 7.67E-01 8.62E-01 9.58E-01

-9.56E-02

1.05E+00

2.36E-07 1.35E-04 2.70E-04 4.05E-04 5.40E-04 6.75E-04 8.10E-04 9.45E-04 1.08E-03 1.22E-03 1.35E-03

-1.35E-04

1.49E-03

1.17E-06 6.72E-04 1.34E-03 2.01E-03 2.69E-03 3.36E-03 4.03E-03 4.70E-03 5.37E-03 6.04E-03 6.71E-03

-6.70E-04

7.38E-03

3.97E-06 2.27E-03 4.54E-03 6.80E-03 9.07E-03 1.13E-02 1.36E-02 1.59E-02 1.81E-02 2.04E-02 2.27E-02

-2.26E-03

2.49E-02

Plastic Shear Strain

plasticity (third column) evaluated for different values of the strainG12 as reported in Fig. 4.

periodic masonry deduced by nonlinear homogenization, European.001

0 2E-005 4E-005 6E-005 8E-005 0.0001

Θ0.0

10.0

20.0

30.0

40.0

Z

MicromechanicalTFA

A

B

C

[MPa

]

Fig. 6. Unsymmetric shear test: comparison of the results obtained by micro-mechanical FEA and by the proposed procedure.

DamageDeformed RVE

A

B

C

Fig. 7. Unsymmetric shear test: deformed configurations (first column), maps of damage (secas reported in Fig. 6.

D. Addessi et al. / European Journal of Mechanics A/Solids xxx (2010) 1e1410

ARTICLE IN PRESS

Please cite this article in press as: Addessi, D., et al., Cosserat model forJournal of Mechanics A/Solids (2010), doi:10.1016/j.euromechsol.2010.03

up�a1; x02

� ¼ upa1; x02

� � a2 � x02 � 0upx01 � a1;�a2

� ¼ upx01;0

�0 � x01 � a1

upx01;�a2

� ¼ upx01; a1;0

�0 � x01 � a1

(51)

Note that the integral boundary conditions, Eq. (25), are automat-ically satisfied, since a Cauchy deformation mode is analyzed. Aregular mesh of 336 elements and 488 nodes is used for the FEA.

In Fig. 4 the macroscopic shear stress component S12 versus themacroscopic shear strain G12 is shown, where, as in Fig. 2, thecontinuous curve denotes the results obtained with the proposedprocedure and the diamond symbols concerns the FEA solution.Also in this case, a very good agreement is obtained.

As in the previous application, in Fig. 5 the deformed configu-ration of the RVE with the damage and shear plastic strain maps atthree steps of the analysis, namely points A, B, C in Fig. 4, arereported. After the initial linear elastic response, the nonlinear

5.75E-05 3.29E-02 6.57E-02 9.85E-02 1.31E-01 1.64E-01 1.97E-01 2.30E-01 2.63E-01 2.95E-01 3.28E-01

-3.28E-02

3.61E-01

1.75E-04 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00

-9.98E-02

1.10E+00

1.75E-04 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00

-9.98E-02

1.10E+00

-1.15E-04-9.75E-05-8.03E-05-6.32E-05-4.60E-05-2.88E-05-1.17E-05 5.51E-06 2.27E-05 3.99E-05 5.70E-05

-1.32E-04

7.42E-05

-1.77E-02-1.57E-02-1.37E-02-1.17E-02-9.70E-03-7.71E-03-5.71E-03-3.72E-03-1.72E-03 2.76E-04 2.27E-03

-1.97E-02

4.27E-03

-3.25E-02-2.88E-02-2.50E-02-2.12E-02-1.74E-02-1.36E-02-9.81E-03-6.03E-03-2.24E-03 1.55E-03 5.34E-03

-3.63E-02

9.13E-03

Plastic Shear Strain

ond column) and plasticity (third column) evaluated for different values of the strain bQ

periodic masonry deduced by nonlinear homogenization, European.001

Table 2Damage values in the mortar joints resulting from the TFA.

bQ (10�5) M1 M2 M3 M4 M5 M6 M7 M8

2.0 0 0 0 0 0 0 0 04.0 0 0.54 0.54 0 0 0 0 010.0 0 0.98 0.98 0 0.34 0.72 0.72 0.34

0 2E-006 4E-006 6E-006 8E-006 1E-005

Κ20.0

100.0

200.0

300.0

400.0

M2

MicromechanicalTFA

A

B

[N/m

m]

Fig. 8. Flexural test: comparison of the results obtained by micromechanical FEA andby the proposed procedure.

D. Addessi et al. / European Journal of Mechanics A/Solids xxx (2010) 1e14 11

ARTICLE IN PRESS

behavior appears due to the activation of both the damaging andplasticity mechanisms in head and bed joints (Fig. 5A). It has to benoted that, while damage is almost uniformly distributed in themortar joints, shear plastic strains are initially located into the headjoints. After that, damage tends to localize into bed joints up to thecomplete deterioration, while plasticity spreads trough the bedjoints, where a friction mechanism is activated (Fig. 5B and C).During the subsequent unloading and reloading paths the RVEresponse is characterized by the progression of the friction plas-ticity. It could be remarked that staircase crack pattern could beobtained by properly setting both the shear and compressionloading histories and the mechanical parameters of the head joints,which are generally weaker than the bed mortar joints.

After that, the response of the RVE subjected to the shear rota-tional deformation is analyzed by applying the loading history, withbQ varying from 0 to 1.0 � 10�4, to 0. In this case micromechanicalFEA is performed by discretizing all the RVE by 650 elements and930 nodes.

In Fig. 6 the macroscopic unsymmetric shear stress componentbZ versus themacroscopic unsymmetric shear strain bQ is shown, thecontinuous curve referring to the results obtained with theproposed procedure and the diamond symbols to the micro-mechanical FEA solution. Note that, in this case, the two curvesdeparts a little. In particular, after the initial linear elastic branch,the stiffness deterioration due to the damaging process startsbefore in the micromechanical analysis than in the TFA.

In order to investigate on the little different response recoveredby the TFA and the FEA, a comparison of the damage and plasticityevolution is reported. In fact, in Fig. 7 the RVE deformed configu-rations, the maps of the damage and plastic shear strain, evaluatedin themortar joints for three loading steps by the FEA, are reported.Moreover, in Tables 2 and 3 the values of the damage and plasticvariables, respectively, determined in the eight mortar joints bymeans of the proposed TFA procedure, are reported for the samethree loading steps indicated in Fig. 6. Note that, at the step A nodamage and plasticity is activated yet in the joints, whereasmicromechanical analysis shows the initiation of the damage andplasticity in the upper right and bottom left bed joints (Fig. 7A). Atthe step B only the vertical joints M2 and M3 show the presence ofdamage and plasticity, while in the corresponding micro-mechanical maps (Fig. 7B) they involve the upper right and bottomleft bed joints and the middle head joints, where damage is verysevere. Finally, at step C the damage and plasticity spreads overallthe bed joints both in Table 2 and in Fig. 7C.

Hence, the FEA leads to amore severe damage evolution than theTFA. For this reason, the slope of the degraded stiffness in themicromechanical analysis is lower than in the TFA. Such differencesin the damage evolution could be expected andmainly related to thesimplified assumptions onwhich the TFA procedure is based, i.e. the

Table 3Plastic variable values in the mortar joints resulting from the TFA.

bQ(10�5) M1 M2 M3 M4

2.0 0 0 0 04.0 0 �2.76 � 10�3 �2.76 � 10�3 010.0 0 �1.24 � 10�2 �1.24 � 10�2 0

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assumption of uniform distribution of the inelastic strain in eachmortar joint, computed as functionof average strain and stress in thejoint.

The response of the RVE during the unloading is linear elastic,characterized by the damaged stiffness.

Finally, the nonlinear behavior of the RVE, experienced under theapplication of the secondmicro-curvature Cosserat component K2, isinvestigatedbyapplyingK2 varying from0 to4.0�10�6, to 2.0�10�6,to 10.0 � 10�6, to 0. The same discretization adopted in the previousexample is used to perform the micromechanical FEA. In Fig. 8 theCosseratmacroscopicmicro-coupleM2 versus themicro-curvatureK2is reported. As in the previous examples, the continuous curve refersto the results obtainedwith theproposedprocedure and thediamondsymbols to themicromechanical FEA solution. It isworthwhilenotingthat, also in this case, the two curves obtained by means of the twoprocedures are not inperfect agreement, due to the different damageevolution in themortar joints. In fact, the stiffness degrading processappears more severe in the curve obtained by micromechanical FEA,with respect to the one calculated by TFA. In Tables 4 and 5 the valuesof the damage and plastic variables, respectively, in the eight mortarjoints are reported for the two loading steps indicated in Fig. 8. Theinitial elastic response is followed by the initiation of the damage inthe two right bed joints, which are located in the region of the RVEexperiencing a tensile deformation state. While at the step A of theanalysis (Fig. 8) damage and plasticity evaluated by TFA are locatedonly into the bed jointsM6 andM8, at the same stepmicromechanicalanalysis show the presence of damage also in the head joint M3

(Fig. 9A). Then, the damage spreads over the right middle head jointand grows maintaining such mechanism (Fig. 9B), whereas damageand plasticity remains localized in the bed joints M6 and M8 whenevaluated by TFA. Indeed, plastic deformations are located mainly inthe right middle head joint during all the loading process (Fig. 9B).

The unloading branch is linear elastic characterized by thedamaged stiffness.

In order to investigate on the mechanical response obtained bymeans of the TFA procedure under more sophisticated loadinghistories, the RVE is now subjected to different coupled macro-scopic Cosserat strain components. In particular, the aim is to

M5 M6 M7 M8

0 0 0 00 0 0 07.3 � 10�4 2.59 � 10�3 2.59 � 10�3 7.3 � 10�4

periodic masonry deduced by nonlinear homogenization, European.001

Table 4Damage values in the mortar joints resulting from the TFA.

K2 (10�6) M1 M2 M3 M4 M5 M6 M7 M8

4.0 0 0 0 0 0 0.9 0 0.910.0 0 0 0 0 0 1.0 0 1.0

Table 5Plastic variable values in the mortar joints resulting from the TFA.

K2 (10�6) M1 M2 M3 M4 M5 M6 M7 M8

4.0 0 0 0 0 0 �1.58 � 10�4 0 1.58 � 10�4

10.0 0 0 0 0 0 5.63 � 10�4 0 5.63 � 10�4

A

B C

Fig. 10. Scheme of the pre-stressed shear masonry panel.

D. Addessi et al. / European Journal of Mechanics A/Solids xxx (2010) 1e1412

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reproduce loading conditions on the RVE typical of structuralapplications on a shear pre-stressed masonry panel shown inFig. 10, where combination of compression strain E2, correspondingto a pre-stress about of �3.0 MPa, symmetric shear strain G12,unsymmetric rotational deformation bQ and micro-curvature K2 aremainly expected, when RVEs located at the center (A), bottom left(B) and bottom right (C) of the panel are taken into consideration.To this end, the symmetric shear test presented above is nowperformed by simultaneously applying to the RVE histories of theCosserat strains.

Firstly, the loading history shown in Table 6 is applied to the RVE.In Fig. 11 the macroscopic shear stress S12 versus the applied

shear strain G12 is shown, by considering the four cases g ¼ 0, g ¼ 1,g ¼ 2 and g ¼ 4. It has to be noted that, the case g ¼ 0, i.e. bQ ¼ 0,corresponds to a simple Cauchy loading history. Numerical resultsshow that the presence of Q leads to a more rapid damage evolu-tion into the joints, giving rise to a more severe degrading process.

DamageDeformed RVE

A

B

Fig. 9. Fexural test: deformed configurations (first column), maps of damage (second columnin Fig. 8.

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In the following examples, the RVE is simultaneously subjectedto a constant strain component E2 and to a history of the compo-nents G12 and K1 or K2, in order investigate on the influence of thepresence of the Cosserat micro-curvatures on the damage evolutionand on the overall behavior of the RVE. The loading historiesreported in Tables 7 and 8 are adopted.

Firstly the influenceof thevariationofK1 is analyzed. In Fig.12 themacroscopic shear stress S12 versus the applied shear strain G12 isillustrated. Four curves are depicted corresponding to the cases f¼ 0,i.e.K1¼0, and to f¼5, f¼10 and f¼20. It is evident that the presenceof the micro-curvature component K1 does not significantly affectthe shear response of the RVE, except for the case f ¼ 20. It can be

1.72E-04 9.87E-02 1.97E-01 2.96E-01 3.94E-01 4.93E-01 5.91E-01 6.90E-01 7.88E-01 8.86E-01 9.85E-01

-9.83E-02

1.08E+00

1.75E-04 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00

-9.98E-02

1.10E+00

-3.83E-03-3.07E-03-2.30E-03-1.53E-03-7.67E-04 2.09E-07 7.67E-04 1.53E-03 2.30E-03 3.07E-03 3.83E-03

-4.60E-03

4.60E-03

-1.34E-02-1.08E-02-8.07E-03-5.38E-03-2.69E-03 7.34E-07 2.69E-03 5.38E-03 8.07E-03 1.08E-02 1.34E-02

-1.61E-02

1.61E-02

Plastic Shear Strain

) and plasticity (third column) evaluated for different values of the strain K2 as reported

periodic masonry deduced by nonlinear homogenization, European.001

Table 6Loading history hist1.

t E1 E2 (10�4) G12 (10�4) bQ (10�4) K1 K2

0 0 0 0 0 0 01 0 �3.0 20.0 �g 0 02 0 �3.0 �20.0 g 0 03 0 �3.0 0 0 0 0

-0.002 -0.001 0 0.001 0.002-2.0

-1.0

0.0

1.0

2.0

[MPa

]

Σ12

Γ12

g = 0g = 1g = 2g = 4

Fig. 11. Shear test: comparison of the RVE response subjected to hist1 for differentvalue of g.

Table 7Loading history hist2.

t E1 E2 (10�4) G12 (10�4) bQ K1 (10�6) K2

0 0 0 0 0 0 01 0 �3.0 20.0 0 f 02 0 �3.0 �20.0 0 �f 03 0 �3.0 0 0 0 0

-0.002 -0.001 0 0.001 0.002-2.0

-1.0

0.0

1.0

2.0

[MPa

]

Σ12

Γ12

f = 0f = 5f = 10f = 20

Fig. 12. Shear test: comparison of the RVE response subjected to hist2 for differentvalue of f.

-0.002 -0.001 0 0.001 0.002-4.0

-2.0

0.0

2.0

4.0

[MPa

]

Σ12

Γ12

f = 0f = 5f = 10f = 20

Fig. 13. Shear test: comparison of the RVE response subjected to hist3 for differentvalue of f.

D. Addessi et al. / European Journal of Mechanics A/Solids xxx (2010) 1e14 13

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noted that, the nonlinear responseof theRVE ismore complexwhenhigher values of the curvature K1 are considered.

Finally, the shear response in presence of a variation of themicro-curvature K2 is shown. In Fig. 13, the stress S12 versus theshear strain G12 is reported for the cases corresponding to f ¼ 0, i.e.K2 ¼ 0, and to f ¼ 5, f ¼ 10 and f ¼ 20. Differently from the previouscase, when the micro-curvature K1 is applied, the presence of K2strongly influences the RVE response, also for the lower values of f.In particular, in the case f ¼ 0, which can be considered as the RVEresponse under Cauchy strains only, damage tends to localize onlyin the bed joints up to the complete degradation (see Fig. 5B and C);afterthat the response is governed by the friction mechanisms. Thepresence of K2 results in a more severe damaging process (see thedash-dot and dash curves in Fig. 13, where damage starts beforethan in the solid curve and, after the initial linear elastic branch, thestiffness is more degraded), which progresses also in the verticaljoints, especially in the joints M2 and M3, reaching, for high valuesof K2, the complete deterioration as well. Then, the damagemechanism becomes prominent in presence of K2 with respect tothe friction one. It can be emphasized that the effect of the micro-

Table 8Loading history hist3.

t E1 E2 (10�4) G12 (10�4) bQ K1 K2 (10�6)

0 0 0 0 0 0 01 0 �3.0 20.0 0 0 f2 0 �3.0 �20.0 0 0 �f3 0 �3.0 0 0 0 0

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curvature K2 on the nonlinear shear response is much moresignificant with respect to K1. This is due to the specific texture ofthe RVE made of two bed joints which spread along the wholehorizontal size of the RVE. When, the micro-curvature K2 is appliedon the RVE a significant part of the two bed joints is subjected toa tensile state and, hence, to a severe damage. On the contrary,when the micro-curvature K1 acts on the RVE only a small part ofthe head joints is damaged.

6. Conclusions

A nonlinear homogenization procedure able to describe theconstitutive response of regular masonry material has been intro-duced. In particular, the 2D Cosserat continuum model has beenadopted for the equivalent medium at the macro-level, while thestandard 2D Cauchy continuum has been employed at the micro-level. Since periodic textures have been considered for masonry,the proposed homogenization technique held on the assumption ofperiodicity conditions, generalized in order to consider Cosseratdeformation modes, imposed on a RVE. The bridging relationsbetween the macro and micro levels have been stated by formu-lating a suitable kinematic map, which expresses the micro-mechanical displacement fields as a function of the macroscopicCosserat deformation components. The higher order polynomialexpansions used for the macro-micro kinematic map has allowedto analyze micromechanics deformation modes richer than in the

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classical first order homogenization framework, as also flexural andunsymmetric shear modes have been included. A damage-unilat-eral contact-friction model has been adopted for the mortar joints.Then, in the spirit of the TFA procedure, the overall elastic consti-tutive matrix and the localization tensors have been evaluated bylinear FEAs of the RVE, on the basis of which the nonlinear damageand plasticity evolutive problems at the typical point of themacroscopic equivalent medium have been solved by a step-by-step analysis. Furthermore, the analysis of the nonlinear micro-mechanical response of the RVE has been carried out by means ofa FEA on the basis of the coupled damage-plastic model adopted forthe mortar joints. In particular, since an integral type regularizationtechnique has been employed within head and bed mortar joints,pathological localization problems have been avoided. Thecomparison between the numerical results obtained by theproposed procedure and the ones evaluated by the nonlinearmicromechanical FEA has shown a very satisfactory agreementwhen macroscopic Cauchy deformation components are applied tothe RVE, then validating the assumption of uniformly distributedinelastic strains along the longitudinal direction of the joints, onwhich TFA procedure is founded. Some discrepancies otherwiseemerge when the response of the RVE under macroscopic Cosserattypical strain components is analyzed, as a consequence of both thesimplified assumptions of the TFA and of the adopted regulariza-tion procedure. Also the RVE response has been analyzed under theapplication of loading histories combining Cauchy and Cosseratdeformation components with the aim of reproducing loadingconditions typical of structural applications on shearing masonrypanels; it has appeared that the nonlinear response of the RVE isdefinitely influenced by the presence of the Cosserat components,which strongly affect its nonlinear behavior by influencing thedamage initiation, the damage evolution rate and the frictionplastic flow. In this cases the symmetric shear response of the RVEhas been investigated and the response in the absence of the Cos-serat components, which can be considered a Cauchy typicalresponse, has been reported for comparison. It has clearly emergedthat the simultaneous presence of the Cosserat strain componentstogether with the Cauchy ones can definitely modify the typicalresponse of the masonry RVE, mostly in the nonlinear range. Suchresults confirm the relevance of the use of the micro-polar Cosseratcontinuum for developing accurate models for masonry.

Note that in the present paper the analyses have beenperformedonly at the RVE scale without developing structural computations.Thus, aim of further developments is to implement the proposednonlinear homogenization technique in a finite element code forstructural uncoupled multi-scale analyses in order to validate theCosserat capability to naturally account for size effects.

References

Anthoine, A., 1995. Derivation of the in-plane elastic characteristics of masonrythrough homogenization theory. International Journal of Solids and Structures32 (2), 137e163.

Please cite this article in press as: Addessi, D., et al., Cosserat model forJournal of Mechanics A/Solids (2010), doi:10.1016/j.euromechsol.2010.03

Brasile, S., Casciaro, R., Formica, G., 2007a. Multilevel approach for brick masonrywalls part i: a numerical strategy for the nonlinear analysis. Computer Methodsin Applied Mechanics and Engineering 196, 4934e4951.

Brasile, S., Casciaro, R., Formica, G., 2007b. Multilevel approach for brick masonrywalls part ii: on the use of equivalent continua. Computer Methods in AppliedMechanics and Engineering 196, 4801e4810.

Casolo, S., 2006. Macroscopic modelling of structured materials: relationshipbetween orthotropic cosserat continuum and rigid elements. InternationalJournal of Solids and Structures 43 (3e4), 475e496.

Cerrolaza, M., Sulem, J., Elbied, A., 1999. A cosserat non-linear finite element analysissoftware for blocky structures. Advances in Engineering Software 30 (1), 69e83.

Dvorak, G., 1992. Transformation field analysis of inelastic composite materials.Proceedings of the Royal Society of London A 437, 311e327.

Forest, S., Sab,K.,1998.Cosseratoverallmodellingofheterogeneousmaterials.MechanicsResearch Communications in Numerical Methods in Engineering 4, 449e454.

Gambarotta, L., Lagomarsino, S., 1997. Damage models for the seismic response ofbrick masonry shear walls part ii: the mortar joint model and its application.Earthquake Engineering and Structural Dynamics 26, 441e462.

Kouznetsova, V.G., Geers, M.G.D., Brekelmans, W.A.M., 2002. Multi-scale constitu-tive modelling of heterogeneous materials with a gradient-enhanced compu-tational homogenization scheme. International Journal for Numerical Methodsin Engineering 54, 1235e1260.

Kouznetsova, V.G., Geers, M.G.D., Brekelmans, W.A.M., 2004. Multi-scale second-order computational homogenization of multi-phase materials: a nested finiteelement solution strategy. Computer Methods in Applied Mechanics andEngineering 193 (48e51), 5525e5550.

Luciano, R., Sacco, E., 1997. Homogenization technique and damage model for oldmasonry material. International Journal of Solids and Structures 32 (24),3191e3208.

Luciano, R., Sacco, E., 1998. Variational methods for the homogenization of periodicheterogeneous media. European Journal of Mechanics e A/Solids 17 (4),599e617.

Masiani, R., Rizzi, R., Trovalusci, P., 1995. Masonry as structured continuum. Mec-canica 30 (6), 673e683.

Masiani, R., Trovalusci, P., 1996. Cauchy and Cosserat materials as continuummodels of brick masonry. Meccanica 31 (4), 421e432.

Massart, T.J. 2003. Multiscale modelling of damage in masonry structures. Ph.D.thesis, Technische Universiteit Eindhoven.

Massart, T.J., Peerlings, R.H.J., Geers, M.G.D., 2007. An enhanced multi-scaleapproach for masonry wall computations with localization of damage. Inter-national Journal for Numerical Methods in Engineering 69 (5), 1022e1059.

Pande, G., Liang, J.X., Middleton, J., 1989. Equivalent elastic moduli for brickmasonry. Computers Geotechnical 8 (5), 243e265.

Pegon, P., Anthoine, A., 1997. Numerical strategies for solving continuum damageproblems with softening: application to the homogenization of masonry.Computers and Structures 64, 623e642.

Pietruszczak, S., Niu, X., 1992. A mathematical description of macroscopic behaviourof brick masonry. International Journal of Solids and Structures 29 (5), 531e546.

Ragueneau, F., La Borderie, C., Mazars, J., 2000. Damage model for concrete-like mate-rials coupling cracking and friction, contribution towards structural damping: firstuniaxial applications. Mechanics of Cohesive-Frictional Materials 5 (8), 607e625.

Sacco, E., 2009. A nonlinear homogenization procedure for periodic masonry.European Journal of Mechanics e A/Solids 28 (2), 209e222.

Simo, J.C., Hughes, T.J.R., 1998. Computational Inelasticity. Springer.Salerno, G., de Felice, G., 2009. Continuum modeling of periodic brickwork. Inter-

national Journal of Solids and Structures 46 (5), 1251e1267.Stefanou, I., Sulem, J., Vardoulakis, I., 2008. Three-dimensional cosserat homoge-

nization of masonry structures: elasticity. Acta Geotechnica 3, 71e83.Suquet, P., 1987. Elements of homogenization for inelastic solid mechanics. In:

Homogenization Techniques for Composite Media. Springer-Verlag, Berlin.Trovalusci, P., Masiani, R., 2003. Non-linear micropolar and classical continua for

anisotropic discontinuous materials. International Journal of Solids and Struc-tures 40 (5), 1281e1297.

Uva, G., Salerno, G., 2006. Towards a multiscale analysis of periodic masonrybrickwork: a fem algorithm with damage and friction. International Journal ofSolids and Structures 43, 3739e3769.

van der Sluis, O., Vosbeek, P.H.J., Schreurs, P.J.G., Meijer, H.E.H., 1999. Homogeni-zation of heterogeneous polymers. International Journal for Solids and Struc-tures 36, 3193e3214.

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