Failure-Oriented Multi-scawith nucleation and evolu

P.J. Snchez a,b,, P.J. Blanco c,d, A.EaCIMEC-INTEC-UNL-CONICET, Gemes 3450, CP 3000 SabGIMNI-UTN-FRSF, Lavaisse 610, CP 3000 Santa Fe, Argc LNCC/MCTI Laboratrio Nacional de Computao Cientd INCT-MACC Instituto Nacional de Cincia e Tecnologia

a r t i c l e i n f o

Article history:Received 18 January 2012Received in revised form 20 November 2012Accepted 23 November 2012Available online 5 December 2012

Keywords:

ary Conditions is that they guarantee the existence of a physically admissible RVE-size, a concept that we

ion, as well as, the

All rights re

1. Introduction

Materials, in general, have an intrinsic heterogeneous nature.From a micro-structural point of view, they can be composed of

a number of constituents and heterogeneities that govern themechanical response of the material at the macro-structural level.For example, the evolution of inelastic/dissipative mechanisms, aswell as the nucleation process of macroscopic failure phenomena,usually depends on the complex interaction between defects andheterogeneities at smaller length scales. Thus, suitable theoreticalframeworks based on Multi-scale formulations arise as a powerfulmodeling tool to describe more general heterogeneous material

Corresponding author at: CIMEC-INTEC-UNL-CONICET, Gemes 3450, CP 3000Santa Fe, Argentina.

Comput. Methods Appl. Mech. Engrg. 257 (2013) 221247

Contents lists available at

A

w.eE-mail address: psanchez@intec.unl.edu.ar (P.J. Snchez).call through the paper objectivity of the homogenized constitutive response.Several numerical examples are presented showing the objectivity of the formulat

capabilities of the new multi-scale approach to model material failure problems. 2012 Elsevier B.V.0045-7825/$ - see front matter 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cma.2012.11.016served.in the rule that denes how the increments of generalized macro-strains are inserted into the micro-scaleand (ii) the Kinematical Admissibility concept, from where proper Strain Homogenization Procedures areobtained. Then, as a consequence of the HillMandel Variational Principle and the proposed kinematicalassumptions, the FOMF provides an adequate homogenization formula for the stresses in the continuumpart of the body, as well as, for the traction acting on the macro-discontinuity surface.The assumed macro-to-micro mechanism of kinematical coupling denes a specic admissible RVE-dis-

placement space, which is obtained by incorporating additional boundary conditions, Non-StandardBoundary Conditions (NSBC), in the new model. A consequence of introducing these Non-Standard Bound-Multi-scale variational formulationsStrain localization bandsHeterogeneous material failureHomogenized cohesive modelsle Variational Formulation: Micro-structurestion of softening bands

. Huespe a, R.A. Feijo c,d

nta Fe, Argentinaentinaca, Getlio Vargas 333, Petrpolis, Ro de Janeiro, CEP: 25651-075, Brazilem, Medicina Assistida por Computao Cientca, Brazil

a b s t r a c t

This contribution presents the theoretical foundations of a Failure-Oriented Multi-scale variational Formu-lation (FOMF) for modeling heterogeneous softening-based materials undergoing strain localization phe-nomena. The multi-scale model considers two coupled mechanical problems at different physicallength scales, denoted as macro and micro scales, respectively. Every point, at the macro scale, is linkedto a Representative Volume Element (RVE), and its constitutive response emerges from a consistent homog-enization of the micro-mechanical problem.At the macroscopic level, the initially continuummedium admits the nucleation and evolution of cohe-

sive cracks due to progressive strain localization phenomena taking place at the microscopic level andcaused by shear bands, damage or any other possible failure mechanism. A cohesive crack is introducedin the macro model once a specic macroscopic failure criterion is fullled.The novelty of the present Failure-Oriented Multi-scale Formulation is based on a proper kinematical

information transference from the macro-to-micro scales during the complete loading history, even inthose points where macro cracks evolve. In fact, the proposed FOMF includes two multi-scale sub-modelsconsistently coupled:

(i) a Classical Multi-scale Model (ClaMM) valid for the stable macro-scale constitutive response.(ii) A novel Cohesive Multi-scale Model (CohMM) valid, once a macro-discontinuity surface is nucleated,

for modeling the macro-crack evolution.

When a macro-crack is activated, two important kinematical assumptions are introduced: (i) a changeComput. Methods

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Appresponses or, according to the particular interest of the presentcontribution, to understand more precisely the phenomenologiesleading to material failure.

During the last decades, a great diversity of multi-scale method-ologies for material modeling has been proposed and formulatedfrom different perspectives, such as the approaches based on: theEffective Medium [8,24], the Self-Consistent Models [14,5], the Varia-tional Boundary Methods [13,43], the Homogenization Theory ofAsymptotic Expansions [35,10], among others. All these strategieshave rigorous conceptual foundations and have been deducedusing analytical or semi-analytical procedures. Nevertheless, theyare restricted to micro-structures with simple topology of hetero-geneities or adopt specic assumptions, limiting the scope of theirapplicability, specially if material failure is addressed.

In order to circumvent these limitations, multi-scale techniquesbased on the existence of a statistically Representative Volume Ele-ment (RVE), which is associated to the scale where the material het-erogeneities are observable, have gained popularity [2123]. Aparticular class of this multi-scale methodology uses two differentscales: the macro and the micro scale, although generalizationsfor including additional length scales could be considered. In suchapproach, it is assumed that the constitutive behavior at themacro-scale is derived by means of two basic steps: (i) a specicinsertion process of the macro strains into the RVE domain, whichis interpreted as the prescribed action stimulating the micromechanical problem; this step is sometimes called localizationby some authors [9] and (ii) a variationally consistent homogeniza-tion procedure, or volumetric average, of the constitutive responseat the micro-level. Specic kinematical constraints, as well as ener-getic equivalence principles, are additionally included in the modelto obtain a consistent mechanical framework.

In this kind of approach, the notion of constitutive equation is aconcept that is only dened at the micro-structural level (RVE-le-vel). Meanwhile, at the macro-level, the material response is a con-sequence of the micro mechanical behavior and the adopted modelto transfer the information between scales. From this perspective,the constitutive model dened in the micro-mechanical problemturns out to be a particularly important ingredient in the multi-scale formulation.

Regarding the following two general local continuum constitu-tive relation categories that can be adopted to describe the micro-mechanical response:

Constitutive relations derived from associative ow rulesand hardening-based inelastic evolutions models, herecalled for simplicity Standard or Generalized AssociativeMaterials.

Constitutive relations based on non-associative ow rules orstrain softening models, generally adopted for describingdegradation mechanisms and failure,

and inspired in the issues addressed in this paper; the fundamentaldifference between them is attributed to the intrinsic ability of theSoftening-based material models to induce strain localization intonarrow bands. These strain localization bands are the precursor tothe development of strong discontinuities, which can be associatedwith cracks, shear bands, etc. On the contrary, the Standard orGeneralized materials completely inhibit the development of strainlocalization bands.

A number of well-established two-scale RVE-based approachesusing Generalized/Associative Materials has been proposed in the lit-erature, see for example [21,23,40,41,29] and references citedtherein. We call this family of approaches: the Classical Multi-scale

222 P.J. Snchez et al. / Comput. MethodsModel (ClaMM). However, in their present form, they cannot beapplied to problems involving material failure because they giveinconsistent results, as will be shown in the following.Adoption of materials with softening, for modeling failure at themicro-mechanical level as a mechanism inducing degradation atthe macro-level, introduces some theoretical issues which shouldbe particularly considered in the development of multi-scaleformulations. Aside from the well-known fact that continuumonstitutive equations with softening have to be regularized, wemention that:

Material softening induces dissimilar mechanical regimes inthe RVE domain, such as loading into strain localizationbands and unloading in the remaining domain. Under favor-able kinematical conditions, this strain-localization phe-nomenon can evolve throughout the RVE and, eventually,nucleate a bifurcation mode at the macro scale. Hence, thetraditional assumptions of Classical Multi-scale Models bywhich a macro-strain is uniformly distributed into themicro-cell, as well as, the standard stress homogenizationprocedure extended over the entire RVE domain, have adebatable physical meaning and need to be reformulated.

Strain localization phenomena induce a size-effect in themicro-cell response, a well-known behavior in the contextof structural analysis [2]. A consequence of this, in the Clas-sical Multi-scale setting, is that the stress homogenizedresponse becomes sensitive with respect to the micro-cellsize [11]. Then, some action must be adopted to remove thisdecient behavior.

From these observations, an alternative class of multi-scalemodels should be developed whenever strain localization phenom-ena are considered. Pursuing this objective, recent contributionshave been proposed. For instance, Belytschko and co-workers[4,39] presented a specic stress homogenization procedureexcluding, from the integration domain where the stresses areaveraged, the zones where strain localization is detected. Theycalled this methodology the Multi-scale Aggregating Discontinu-ities model. By construction, the so obtained homogenizedmechanical response keeps the material stability. A connectionbetween the micro-cell domain measure and the nite elementsize at the macro-level is introduced.

A second-order (gradient-based) computational homogeniza-tion scheme has been proposed by Kouznetsova and co-authors[1618]. Recently, this model has been adapted for the analysisof strain localization problems introducing a generalized periodic-ity boundary condition, that is aligned with the direction of thelocalization band developed in the micro scale domain [6].

An alternative technique for modeling the constitutive responseof a pre-existent macro-adhesive discontinuity is due to Matouand co-workers [20,19]. They postulated a multi-scale model basedon the existence of a volumetric heterogeneous RVE, linked to themacroscopic adhesive-interface. A limitation of their approach isthat the thickness of the assumed RVE is associated with the phys-ical thickness of the macro-adhesive surface. They do not give aprocedure to determine the RVE cell size in more general problems.An additional limitation of this model is that it cannot be triviallyextended to nucleate discontinuities from an initially continuummedium. Thus, according to the authors, the discontinuity surfacesmust be predened in the body.

Verhoosel et al. [42] developed a method for determining thehomogenized macro cohesive model derived from micro-struc-tures with possible nucleation of micro-cohesive cracks and adhe-sive micro-interfaces. Nguyen et al. [26] have extended thisproposal to include micro-cells with a damaged material modelregularized with an enhanced gradient approach. They also intro-

l. Mech. Engrg. 257 (2013) 221247duced the notion of failure zone averaging technique, where thehomogenized procedure for micro-stresses is only extended overthe RVE active damaged sub-domain. Although the so-called

failure zone averaging technique seems to have a consistentphysical meaning, this concept is presented as an a priorimechanical denition and there is no clear relation establishedbetween the imposed macro and micro kinematics. The authorsshow, through numerical experimentation, the existence of objec-tive macro responses, with respect to the RVE-size.

Up to the authors knowledge, a unied framework basedexclusively in physical/modeling arguments, and not only onnumerical ones, where the adopted hypotheses and derived conse-quences are well established and fully justied, has not yet beencompletely developed in the context of material failure multi-scalemodeling. This subject remains an open issue at present and thispaper addresses some contributions in this direction.

In this contribution we present a Failure-Oriented Multi-scaleFormulation (FOMF) that uses a specic insertion of macro-strainsinto the micro scale domain [34]. The Insertion Operator isincorporated in the kinematical description of the RVE and itis used to introduce the concept of Kinematical Admissibilitybetween the strains at macro and micro levels. Then, following apurely variational framework, a number of associated mechanical

(V) The proposed formulation is able to nucleate a macro cohe-sive crack, and model its evolution, from an initially classicalmacro-continuum medium. Therefore, the classical or stan-dard multi-scale model represents a particular case of theproposed multi-scale formulation.

(VI) The micro-mechanical problem is solved considering newkinematical constraints, imposed through additional bound-ary conditions that naturally emerge in the present formula-tion. By construction, they assure the objectivity of the modelwith respect to the RVE-size.

Since our interest in this paper is to highlight the fundamentalaspects of the formulation, additional topics such as: the constitu-tive softening regularization model adopted at the RVE-level, theclassical displacement boundary conditions dened in the RVEdescribing the Taylor, Linear or Periodic models among others,are not discussed in detail in this manuscript.

The remainder of this document is structured as follows. In Sec-tion 2, the governing equations for a macro-mechanical problem,in which failure mechanisms in the form of cohesive cracks can

P.J. Snchez et al. / Comput. Methods Appl. Mech. Engrg. 257 (2013) 221247 223consequences are derived, as for example the micro-equilibriumproblem and the stress homogenization procedure. TheFOMF comprehends two multi-scale sub-models sequentiallycoupled in a consistent manner: (i) a Classical Multi-scale Model(ClaMM) and (ii) a new Cohesive Multi-scale Model (CohMM). Anumber of distinguishing features of the FOMF are shown in detailin the body of the paper and they are summarized as follows:

(I) We systematically follow a well established variationalmethodology.

(II) All developments are based on kinematical hypotheses.(III) The generalized stress homogenization procedures arise as a

consequence of the formulation, as a dual concept from theadopted kinematical hypotheses (ruled by the InsertionOperator) and consistent with the HillMandel principle.Thus, they are not an a priori denition. Furthermore, thisgeneralized stress homogenization procedure is character-ized by an integration over specic RVE sub-domains, andnot over the entire micro-cell.

(IV) The topology of the RVE sub-domain, where the strain local-ization takes place, emerges from the evolution of themechanical problem. It depends on the complexities of theunderlying micro-structure and the loading history.Fig. 1. Multi-scale (macromicro) modnucleate and evolve, are discussed. The need for two homogeniza-tion rules is shown: a classical model for the stressstrain responseduring the macroscopic pre-bifurcation stage and a new one, givinga traction-separation cohesive model for the evolution of macro-scopic displacement discontinuities. Such discontinuities are acti-vated after detecting a macroscopic bifurcation point, and this isdiscussed in Section 2.2. Section 3 is devoted to reviewing thefoundations of Classical Multi-scale Models (ClaMM). Some generaloperators, useful for the subsequent developments, are introduced.In Section 4, a new variational formulation is discussed in detail.We show that, following the same basic theoretical steps used inthe context of ClaMM and introducing additional ingredients, awell-posed model, called Cohesive Multi-scale Model (CohMM)which captures objective cohesive responses from a degraded/localized RVE, is obtained. Section 5 presents several numericalexamples showing the mechanical consistency of the completemulti-scale formulation here proposed (FOMF). In Section 6 theimportant role played by the kinematical restriction imposed onthe boundary of the strain localization domain, named in the paperas NSBC, is highlighted through several examples. Conclusions areexposed in Section 7. For the sake of completeness, Appendix A isdevoted to a brief description concerning the numerical techniquesused for the FOMF implementation into a nite element code.el idealization for failure analysis.

App224 P.J. Snchez et al. / Comput. Methods2. Macro-model with cohesive crack for modeling materialfailure

In the subsequent paragraphs, the main features of the macro-mechanical equilibrium problem, along with the associated set ofgoverning equations, are presented. The purpose of this Section isto provide the reader a comprehensive presentation of the coupledmacromicro multi-scale formulation. The model is fully deter-

Fig. 3. Schematic representation of the whole constitutive response provi

Fig. 2. Basic ingredients and nomenclature dening a mechanical probll. Mech. Engrg. 257 (2013) 221247mined with the developments addressed in the forthcoming Sec-tions 3 and 4. A summary of the multi-scale formulationdeveloped in this Section is presented in Box 1.

2.1. Preliminaries

The following premises dene the basic assumptions of themodel:

ded by the proposed Failure-Oriented Multi-scale Formulation (FOMF).

em exhibiting strong discontinuities across the material surface S.

s _et _c n 2

App(i) Quasi-static problems are considered. A monotonicallyincreasing pseudo-time variable t 2 0; T, which belongs tothe pre-established interval of analysis 0; T, is used toaccount for the evolution of the non-linear materialresponse.

(ii) Innitesimal strain theory is assumed. Then, X denotes thereference conguration of the body B dened as an openbounded set in Rndim, ndim refers to the spatial dimensionof the problem. A material point of B, at the macro-scale, isdenoted: x, see Fig. 1. In the following, we use the prexmacro- to denote the elds depending on x and omit-ting, for the sake of simplicity, this functional dependence.

(iii) B undergoes a loading process given by body forces per unitof volume, b, and external tractions per unit of surface area,t. As usual, C symbolizes the piece-wise smooth boundary ofX, which can be split into two sub-sets CD and CN , wherepredened values of displacements, uD, and traction, t, areprescribed, respectively. Besides we have: CD [ CN C,CD \ CN ;, and m is the outward unit vector normal to C,as shown in Fig. 1.

(iv) Initially, the macro-scale is idealized as a homogeneous con-tinuum body where displacements, u, macro-strains, e, andmacro-stresses, r, characterize the mechanical state of thecontinuum medium. However, since softening-based mate-rials are considered for the denition of the micro-scaleproblem, an alternative macro-mechanical scenario canarise. A strain localization mode can nucleate and developin x after a critical condition is fullled. This critical condi-tion is governed by the progressive accumulation of micro-failure mechanisms. The strain localized macro-modes canrepresent shear bands, fractures/cracks, slip surfaces, etc.,depending on the considered problem. A fundamentalassumption of the present formulation is that the mechani-cal phenomenology associated to these failure modes areadequately represented, at the macro-scale, by the inclusionof a Cohesive Crack [7,1,15].

(v) Every point x 2 X is linked to the micro-scale through a het-erogeneous RVE. The same concept exists for those points xthat belong to the macro-crack. Thus, it arises a mechanicalrepresentation of the displacement jumps and cohesiveforces characterizing the discontinuity interface througha volumetric micro-cell domain, see Fig. 1.

(vi) For simplicity, it is assumed that the macro-cracks do notintersect neither the Dirichlet boundary CD nor the non-homogeneous Neumann boundary CN . In addition, themacro-cracks do not intersect each other.

The Cohesive Crack concept introduces, in the macro mechanicalformulation, two features which cannot be reproduced bymeans ofthe standard (stressstrain) mechanical description: (i) the activa-tion of a new independent kinematical eld, accounting for dis-placement discontinuities through the faces of the crack(commonly known as Strong Discontinuity Kinematics), and (ii) anadditional generalized stress variable T , called the Cohesive Trac-tion vector or the Cohesive Generalized Force, conjugated to the dis-placement discontinuity eld. Once a macro cohesive crack isactivated at point x, it is not possible to return to the classical con-tinuum description. The constitutive response of the crack is gov-erned, exclusively, by the so-called cohesive model (traction-jump model) which, in turn, is determined from a specic stresshomogenization procedure performed in the micro scale.

Remark 2.1. In the context of softening material modeling, we

P.J. Snchez et al. / Comput. Methodspostulate that a physically admissible RVE does exist. We adopt theclassical denition of RVE given in the literature (see for examplewhere _c is an arbitrary (pseudo-) velocity vector and the symbol sdenotes the external symmetric dyadic product.

We admit the existence of a strain-localization band insidewhich the strain rate, _eB, experiences a jump with respect to thestrain rate in the surrounding continuum medium, _eC , and suchthat: _eB _eC s _et. Additionally, an incrementally linear strainrate-stress rate behavior at the macro-scale is assumed to existas a result of the homogenization procedure:

_r Cet _e 3where the symbol et denotes the history of the strains up to the in-stant t and C represents the homogenized tangent constitutiveoperator which, for general non-linear models, depends on et

(see Section 3.4). Then, the traction continuity condition across Simplies:[25]) in which, a micro-cell is a RVE whenever it has the propertyof material statistical representativeness with respect to micro-structural patterns.

In the present context of analysis, the essential key point todetermine the RVE, and thus its existence, lies on the fact that awell-posed multi-scale model does exist, in the sense that asequence of increasing micro-cell sizes furnishes a convergentmacroscopic homogenized response. Standard multi-scale tech-niques fail to reach this goal, obscuring the notion of theexistence of a physically admissible RVE for softening-basedmaterials [11].

From this point of view, in the following development of themodel, speaking of RVE, Micro-Cell or Unit-Cells is immaterial.These terms are here used only to keep in mind the existence of anunderlying heterogeneous material micro-structure.

2.2. Failure criterion at the macro scale

A criterion to decide when and how a cohesive cracks has tobe inserted, or activated, in the initially continuum medium, is aningredient to be dened in the present formulation. The failure cri-terion here adopted is derived from the RVE Homogenized mechan-ical state. It species the following two attributes of the macro-crack:

- the nucleation pseudo-time tN 2 0; T, dened as the instant atwhich an admissible mechanical condition for the developmentof a bifurcation mode, at the macro-level, is reached for the rsttime during the load history;

the orientation of the discontinuity surface given by the unitvector n orthogonal to the crack direction.

For this purpose, the classical bifurcation analysis based on thespectral properties of the Localization Tensor, is considered[32,30,33]. This criterion determines the mechanical conditionsunder which a strain rate discontinuity, across a given materialinterface, is admissible.

Dening the jump, or discontinuity, operator across a materialsurface S with normal vector n as follows:st jxdx jxdx 8x 2 S;

dx parallel to the n-direction and jjdxjj ! 0 1then, the Maxwells kinematical compatibility condition establishesthat a discontinuity in the strain rate s _et has the followingstructure:

s

l. Mech. Engrg. 257 (2013) 221247 225s _rtn s _Tt _TB _TC Cets _et

n 0 4

HSxnuitytrary fFromwith X

For the macro-points located in S, we introduce the notion of

226 P.J. Snchez et al. / Comput. Methods Appl. Mech. Engrg. 257 (2013) 221247where the term s _rtn _TB _TC represents the jump betweenthe (rate of) traction inside the strain-localization band, _TB,and the (rate of) traction in the continuum part of the body,_TC . Note that (4) has been derived by assuming the same incre-mental constitutive behavior, loading-regime, on both sides ofthe surface S.

Taking into account (2) and considering the minor symmetriesof C, expression (4) yields:

C et

n

_c

n Qet;n _c 0 5where the denition of the so-called Localization Tensor Qet;n isintroduced. Non trivial solutions for the system (5), with _c 0, canbe obtained only if Q is singular:

det Qet;n 0 6Eq. (6) is a necessary condition for the existence of discontinu-

ous strain rate modes in the body, and can also be interpreted as anecessary condition for the activation of velocity discontinuitiesacross a material surface S with normal n [27,3].

The failure criterion here adopted postulates that the nucle-ation of a macro-crack is determined once the instability condi-tion (6) is veried for the rst time (t tN) during the loadinghistory. The criterion also determines the normal vector to thelocalization direction n. In particular, when C possesses majorsymmetries, the solution of (6) has two conjugate eigenvectors:n and the initial opening (pseudo-) velocity direction _ci [28].The solution of the nucleation problem (6) is denoted as:SN ftN;n; _cig.

2.3. Macroscopic kinematical description with strong discontinuities

When a non-trivial solution SN ftN;n; _cig of the problem (6) isdetected at point x, a cohesive crack along the surface denoted S, isthere inserted. Once the crack is active, its orientation remainsxed for all t > tN . The nucleation and evolution of a crack requiresa specic kinematical description which is presented in thisSection.

Let us consider the mechanical problem of a body B, in its con-guration X, exhibiting a discontinuity in the displacement eldacross the surface S (S X) with a unit normal vector n, seeFig. 2(a). In two-dimensional cases, as depicted in the same gure,for each point x 2 S it is possible to dene a local cartesian systemfn; sg. S divides X in two sub-domains: X and X, according tothe direction of n (with n pointing to X).

For subsequent developments, we dene Xu as an arbitrarysub-domain of X (Xu X) including the interface discontinuityS (S Xu), with piece-wise smooth boundary Cu whichcan be split as: Cu S [ S [ CuI [ CuII , see Fig. 2(b). Also, notethat S divides Xu in two sub-domains Xu and Xu (n pointingtowards Xu). The (outward) unit vectors n, n; nI and nIIare normal to the boundaries S; S; CuI and CuII of Xu,respectively.

After the insertion of the macro-crack, a strong discontinuitydevelops on S. Let us consider the total macro-displacement eldu and the arbitrary smooth vectorial function b, dened inX tN; T ! Rndim, which is such that:bx; t : ux; t 8x 2 S and rxbn 0 8x 2 X 7Thus, b represents the displacement jump in S. Following to [38],the discontinuous macro-displacement eld can be written as theaddition of two terms:

z}|{continuous z}|{discontinuous

ux; t ux; t MSxbx; t; 8x 2 X 8generalized strain which, in this case, is characterized by: (i) thevector b in S with normal n and (ii) the regular counterpart eR,see Eq. (10). Then, generalized strains in S are characterizedthrough the list of variables: feR; b;ng.

Kinematically admissible discontinuous macro-displacementsu, are described by the elements u; b of the linear manifold U ,which is dened as:

U u;b ; u 2 H1X; b 2 H1X and ujCD uDn o

11

where H1 is the vectorial space of functions whose rst deriva-tive is square-integrable. In the present context, the vector spaceV characterizing the virtual kinematically admissible actions(velocities), denoted by u^ u^MS b^, reads:

V u^; b^; u^ 2 H1X; b^ 2 H1X and u^jCD 0n o

12

2.4. Variational equilibrium problem at the macro-level

In problems with nucleation and evolution of cohesive cracks,the equilibrium at the macro level is described by means of thePrinciple of Virtual Power (PVP), as follows:

Given prescribed values of body forces b and predened trac-tions t; nd the kinematically admissible displacement eldu; b 2 U , such that:Z

XnSru; b e^R dX

ZSTu;b b^ dS

ZXnS

b u^MS b^ dXZCN

t u^ dC 0; 8u^; b^ 2 V13

where the rst integral term is the internal virtual power in thecontinuous part of the solid, and the second one is the internal vir-tual power of the cohesive crack, noting that this term is present inthe formulation whenever the failure criterion presented in Sec-tion 2.2 has been fullled. In both terms, e^R eRu^; b^ and b^ repre-sent the virtual kinematically admissible generalized-strains. On theother hand, the third and fourth terms in (13) are the external vir-In view of the above denitions, the strain tensor e, compatiblewith Eq. (8), is expressed as:

e rsxu rsxuMSrsxb bsrxu eRu;b;8x 2 X n S 10

wherersx is the symmetric gradient operator. Note that the expres-sion eRu;b is a regular bounded term. Considering the compactsupport of b; MS and u, from expression (10), we have:e rsxu; 8x 2 X nXu.tual p9

: X! R is the Heaviside step function shifted to the disconti-surface S and ux : X! R is a sufciently smooth and arbi-unction with the two requirements showed in Eq. (9)-right.equations (9) MSx has compact support and it coincidesu (MSx 0 8x 2 X nXu).where ux; t is a smooth function in X and is called the continuousdisplacement term, and MSx is a prescribed function, constant intime, usually called the Unit Jump function and dened as follows,see Fig. 2(b):

MSx : HSx ux; HSx 0 8x 2 X1 8x 2 X

; ux 0 8x 2 X nX

u

1 8x 2 X nXu

(ower.

etR x; t; 8x 2 X n S and 8 t 2 0; T 14

tdt tdt

imal increments at t. The last terms in the right side of both

eR, in the argument of function (17), works as a parameter when

2.5. The whole constitutive response via a Failure-Oriented Multi-scale

in the present context of analysis, is only characterized by the reg-

P.J. Snchez et al. / Comput. Methods Appl. Mech. Engrg. 257 (2013) 221247 227Formulation (FOMF)

From the perspective of the variational formulation, the solu-tion of the equilibrium problem at the macro-level requires thedenition of two different constitutive multi-scale models. Eachone is enunciated as follows:

A Classical Multi-scale Model (ClaMM) for the regular domain inthe macro-scale (refer to Section 3 for more details): 8x 2 X n S and 8 t 2 0; T, given the history of regular strains etdtR and the strain incre-

ment deR, obtain the associated increment of macroscopicstress dr.the virtual power of cohesive forces is evaluated. In Section 4 weexploit this feature.produces virtual power through the kinematically admissiblevariations b^, i.e. T is conjugate to b^. Thus, the regular componentwhere the sub-indices identify the pseudo-time at which the vari-ables are evaluated. From now on, and in order to simplify the nota-tion, we remove the sub-indices when variables are evaluated at thepresent time t.

Remark 2.2. From a thermodynamical point of view, T onlyequations explicitly introduce, in the arguments of the stress func-tionals, the kinematically admissible strain increments deR and db,at t. Also, the generalized stresses can be written in incrementalform:

r rtdt dr; T T tdt dT 18where the symbols eR and b stand for the history of the gen-eralized strains up to time t dt and d represents the innites-fetR x; t;btx; tg; 8x 2 S and 8 t 2 tN ; T 15

the generalized stress functionals can be written as follows:

r F etR F etdtR ;deR; 8x 2 X n S; 8 t 2 0; T 16

T T etR ; bt T etdtR ;btdt;deR;db; 8x 2 S ;8 t 2 tN ; T 17In order to completely dene the mechanical problem at themacro-scale, constitutive relations for the stress eld, r, and forthe cohesive traction, T , need to be prescribed. In this contribu-tion it is assumed that each one of this constitutive relations isobtained using specic multi-scale formulations that are denedin the following Sections. In general, we consider non-linearmaterial responses, where the functionals of the stress-likemacro-quantities r and T , at time t, depend on the history ofthe strain-like macro-variables up to the time t, here denotedas: etR and b

t.Dening the history of the generalized strains in the spatial do-

mains and time intervals by means of:ular strain component eR.As it is well known, material models with softening charac-

terizing the micro-structural constituents, require a regulariza-tion of the constitutive theories to keep the mechanicalproblem well-posed. Depending on the specic technique usedto perform this regularization, some particular aspects of the fol-lowing ClaMM can be described through different approaches. Toavoid unnecessary details that would obscure the following pre-sentation, we adopt a regularization strategy based on theSmeared Crack Approach to fracture (SCA) [31], which uses astandard continuum mechanical description at the micro-scalelevel and regularizes the softening parameter according to thefracture energy and a predened characteristic length. However,the foundations of the proposed multi-scale formulation workswell with different regularization strategies, such as: non-localschemes, gradient/higher-order models as well as strong discon-tinuity techniques with cohesive laws. In other word, the meth-odology is exible for adopting different regularization strategiesin the RVE. A Cohesive Multi-scale Model (CohMM) for the points in themacro-crack (refer to Section 4 for additional details): 8x 2 S and 8 t 2 tN; T, given the history of generalized strains etdtR ; b

tdtn o

and the generalized strain increment fdeR; dbg, obtainthe associated increment of macroscopic cohesive tractiondT .

The FOMF approach proposed in this contribution is basedon a consistent mechanical coupling between the twosub-models: ClaMM and CohMM. In order to clarify ideas,Fig. 3 sketches the whole homogenized constitutive responseof a macro-point x which, at t tN , nucleates a cohesivecrack.

The basic ingredients dening the macro-mechanical problemwith nucleation and evolution of cohesive cracks are summarizedin Box 1, while both sub-models, the ClaMM and CohMM, are de-scribed in the two following sections.

3. Classical Multi-scale variational Model (ClaMM)

Following the similar axiomatic framework described in[40,41,29], the foundations of the so-called Classical Multi-scaleModel (ClaMM) are here summarized. In addition, we introducethe denitions and nomenclature of new operators, that are use-ful in the subsequent developments of Section 4. We remark, asa particular aspect in this Section, the effect that the kinematicalassumptions and the HillMandel Principle have on the resultingstress homogenization rule and the equilibrium problem at mi-cro-level.

The ClaMM results an important ingredient of the complete pro-posed multi-scale methodology. It is used to obtain the macrohomogenized constitutive response in the following two situa-tions, see the item 4.1 in Box 1 and also Fig. 3:

(i) in those points where, according to the criterion discussed inSection 2.2, no crack formation is detected during the entireload history;

(ii) during the period of stable homogenized material responsein the points where, eventually at a posterior time, a crackis inserted.

From the previous explanations, the kinematics at macro-level,

sic

AppBox 1: Macro-mechanical problem with strong discontinuities. Ba

1.Kinematics:

u uMS b ; 8x 2 Xe eR rsxuMSrsxbrxusb ; 8x 2 X=Su;b 2 U ; U u;b; u 2 H1X; b 2 H1X and ujCD uD

n o1. Virtual kinematically admissible actions (velocities):

u^ u^MS b^u^; b^ 2 V; V u^; b^; u^ 2 H1X; b^ 2 H1X and u^jCD 0

n o2.Variational Equilibrium Problem:

Given b and t; find u; b 2 U such that :ZXnS

r rsxu^MS b^ dXZST b^ dS

ZXnS

b u^MS b^ dXZC

t u^ dC 0; 8u^; b^ 2 V

228 P.J. Snchez et al. / Comput. Methods3.1. Kinematical description at the micro-cell level

Let us consider a reference conguration Xl (Xl Rndim) of theheterogeneous micro-cell Bl having a piece-wise smooth boundaryCl, see Fig. 1. A specic reference system is introduced to describethe mechanical problem at the micro-scale. Points in Bl are de-noted y (y 2 Xl).

Macro-elds depend on x, whereas those dened at the mi-cro-scale depend on y. For the sake of simplicity, all functionaldependencies on y are omitted, while the sub-index l denotesthe variables dened in Bl. Additionally, we use the prex micro- for the elds dened at the micro-scale.

Let eR rsxu eRu; b be the regular component of the strain atthe point x 2 X n S, see Eq. (10). We dene the total micro-straineld, el, as the addition of two contributions:

el IyeR eel 19where the rst term represents the (macro-to-micro) insertion of theregular macro-strain into the RVE, and the second term is the micro-strain uctuation eld given by eel rsyeul, with eul being the uc-tuation of the micro-displacement eld. From now on, the supra-symbol e denotes uctuations in Bl.

The Insertion Operator Iy maps a symmetric second ordertensor, dened at the macro-scale, into a symmetric second order

N

3.Constitutive response via the Failure-Oriented Multi-scale Formu

4:1 8x 2 X=S and 8 t 2 0; T : Given etdtR and deR; find dr;via a Classical Multi-scale Model (ClaM

4:2 8x 2 S and 8 t 2 tN; T : Given fetdtR ;btdtg and fdeR;dbg; finvia the Cohesive Multi-scale Model (Coh

4.Cohesive crack nucleation criterion:

Find SN ftN;n; _cig; verifying the singularity of the Localization Tedet Qet;n 0; 8x 2 X=S and 8n 2 Rndimwhere Qet;n _c Cetn _cn; 8 _c 2 Rndimand Cet is the Homogenized Tangent Constitutive Tensor; obtconcepts and ingredients

l. Mech. Engrg. 257 (2013) 221247tensor eld in the micro-cell domain. In addition, Iy is a linearapplication, thus it can be expressed in the form:

Iy Iy 20with Iy being a fourth order tensor micro-eld, that must be prop-erly dened.

Using the incremental format introduced in the previous Sec-tion, and the property (20), we can alternatively write Eq. (19) asfollows:

del IydeR rsydeul 21We say that del is Kinematically Admissible in Xl, for a givenIy, if it satises:ZXl

del dXl ZXl

IydeR dXl 22

In the Classical Multi-scale Formulation, the Insertion Operator is de-ned as the identity operator, such that for each point y 2 Xl, it canbe characterized by the fourth order identity tensor: I. Then, deR ishomogeneously inserted into Xl, and Eq. (21), results:

del deR rsydeul 23Under this condition, the Eq. (22) yields:

lation (FOMF):

M), refer to Section 3 and Box 2. Update stresses: r rtdt drd dT;

MM), refer to Section 4 and Box 3. Update tractions: T T tdt dT

nsor Qet;n :

ained via a Classical Multi scale Model ClaMM

strained Vector Space of kinematically admissible incremental displace-~ e

Vl u^l; u^l 2 H Xl and u^lsml dCl 0 27

mechaFro

xing l

AppCl

Summarizing, we have: deul 2 ~Ul and u^l 2 Vl ~Ul.Taking into account (21) and (27), it is also possible to introduce

the concept of virtual kinematically admissible (rate of) micro-strain e^l:

e^l Iy e^R rsyu^l; 8 e^R and 8 u^l 2 Vl 28e^R being a virtual kinematically admissible (rate of) macro-regularstrain.

Remark 3.1. Expression (24) is widely known as the StrainHomogenization Principle. In the present framework, we remarkthe fact that expression (24) is not a Principle, but a conditionderived from the Kinematical Admissibility concept, introduced in(22). Then, alternative multi-scale models can be derived byadopting different Insertion Operators Iy that lead to StrainHomogenization Procedures not coinciding with (24).

Remark 3.2. From (26), alternative Classical Multi-scale Modelscan be dened by adopting different sub-spaces of the minimallyconstrained vector space ~Ul (and Vl), such as: the Taylor or homo-geneous strain model, the Linear or afne RVE-boundary displace-ment model, the Periodic RVE-boundary displacement uctuationsmodel. Moreover, it can be proved that the formulation with min-imum kinematical restrictions, given by the denition (26), repre-sents the so-called Uniform boundary Traction model [40].

3.2. The HillMandel Variational Principle of macro-homogeneity

In order to guarantee the energetic consistency between themacro and micro scales, the HillMandel Variational Principle isassumed. This principle establishes an equivalence between thevirtual internal power at the macro-scale and the homogenizedvirtual internal power at the micro-scale:

r e^R 1jXljZXl

rl e^l dXl;

8 e^R and 8 e^l kinematically admissible 29where rl is the stress tensor eld in the micro-scale. Alternatively,ment uctuations at the RVE-level: Ul. We establish that dul isadmissible if deul 2 ~Ul, where ~Ul is dened as:~Ul deul; deul 2 H1Xl and Z

Cl

deulsml dCl 0( )

26

from where we can dene the minimally constrained vector spaceVl of virtual kinematically admissible actions (velocities), u^l, asfollows:

1Z( )deR 1jXljZXl

del dXl 24

with jXlj being the measure of Xl.Replacing del, dened in (21), in (24) and using standard math-

ematical manipulation, Eq. (24) results equivalent to the followingkinematical constraint at the micro-cell level:ZCl

deulsml dCl 0 25where ml is the outward unit vector normal to the boundary Cl.Prescription (25) denes what is known as the Minimally Con-

P.J. Snchez et al. / Comput. Methodswriting r and rl in term of stress increments, as shown in expres-sion (18)left, and considering that the variational equation (29) issatised throughout the history, this Variational Principle reads:Homogenization Operator H:

Hdrl dr 1jXljZXl

Iy T drl dXl 32

which, for the particular case of the ClaMM, it results in the conven-tional stress homogenization procedure:

Hdrl dr 1jXljZXl

drl dXl 33

Note that the Insertion Operator, Iy, plays a fundamental rolein the characterization of the Stress Homogenization OperatorH,see Eq. (32). Any change in Iy, i.e. changes of the kinematicaldescription, would introduce modication in the stress homogeni-zation process to hold the energetic consistency, the HillMandelPrinciple in this case. We exploit this concept in Section 4.

3.2.2. Second consequence of the HillMandel Principle: the RVEequilibrium problem

Alternatively, considering arbitrary variations u^l and xing tozero the variations e^R in expression (31), it is derived theRVE equilib-rium problem that is written in a variational format as follows:

Given the history of the regular strains etdtR and a kinemati-cally admissible increment deR; nd the incremental micro-dis-placement uctuation eld deul 2 eUl, such that:Z

Xl

drl rsyu^l dXl 0; 8 u^l 2 Vl 34

Eq. (34) prescribes a self-equilibrated micro-stress eld, whichcan be understood as a reaction to the imposed generalized loadingsystem induced by IydeR.

Remark 3.3. From the topics discussed in the present Section, theconceptual guidelines for the formal treatment of a kinematicaland variationally consistent multi-scale formulation can be sum-marized in three basics steps:

(1) Postulate the kinematical description at the micro-level byintroducing the denition of the strain tensor eld elthrough the selection of the Insertion Operator Iy.

(2) Apply the (incremental) Kinematical Admissibility concept,Eq. (22), from where a Strain Homogenization Procedure(3)nical responsem expression (31), considering arbitrary variations of e^R andto zero u^ , it is derived the so-called micro-to-macro Stress-dr e^R 1jXljZXl

drl e^l dXl;

8 e^R and 8 e^l kinematically admissible 30Inserting (28) into (30), and after some algebra, the HillMandel

Variational Principle is expressed as:

dr 1jXljZXl

Iy T drl dXl" #

e^R 1jXljZXl

drl rsyu^l dXl 0;

8 e^R and 8 u^l 2 Vl 31

where Iy T represents the transpose of the Insertion Tensor Iy. Inthe present context: Iy T Iy I.

3.2.1. First consequence of the HillMandel Principle: the homogenized

l. Mech. Engrg. 257 (2013) 221247 229emerges.Postulate the energetic equivalence between the macro andmicro scales through the HillMandel Variational Principle.

AppOnce dened this theoretical framework and following purelyvariational arguments, both mechanical ingredients: (i) thehomogenized stresses, i.e. the denition of the Stress-Homogeni-zation Operator H, and (ii) the RVE mechanical equilibriumequations, emerge as direct consequences of the formulation.

Assumptions of purely kinematical character, such as those de-scribed in points (1) and (2) above, guide our proposal to general-ize the Classical Multi-scale Formulation for modeling materialfailure problems.

3.3. Constitutive model at the micro-scale

In general, the stress state in the RVE, rl, depends on the historyof the micro-strains through a given constitutive functional of thetype:

rl Fl etl

35

For the problems addressed in this work, the functional Fl de-scribes a generic dissipative inviscid material model with regular-ized softening, not necessarily derived from a convex potential.

3.4. Homogenized tangent constitutive tensor

The linearization of the macro-homogenized constitutive re-sponse denes the so-called Homogenized Tangent Constitutive Ten-sor C. Evaluation of C is necessary for two reasons: (i) numericalprocedures to solve the fully coupled non-linear macromicromulti-scale problem, generally requires its computation and (ii)as it is explained in Section 2.2 and based on this tensor, we denea material stability criterion in order to determine the instantwhen a cohesive crack is introduced in a point x at the macro-level.The last item motivates this Section.

Let us consider the conventional denition of the constitutivetangent operator, which can be expressed as:

Cet DrDeR

eR

DdrDdeR

eR

36

where DDeR denotes the total derivative with respect to eR. Then,regarding the dependence of dr with drl through expression (33),jointly with the denition of rl in (35) and el in (21), the expres-sion (36) can be rewritten as follows:

C DdrDdeR

eR

1jXljZXl

Cl dXl|{z}Taylor Contribution

1jXljZXl

Cl@ deel@ deR

jeR dXl|{z}Fluctuation contribution

CT eC 37where Cl @drl@del

eR

represents the tangent constitutive tensor for

each point y, at the RVE-level, which depends on the functionalFl, see Eq. (35). From expression (37), two terms contribute tothe homogenized tangent constitutive tensor C: the rst one calledthe Taylor term, CT , and the second one called the uctuation con-

tribution eC . The Taylor component CT is determined through directaveraging of the micro-constitutive tangent tensors Cl. The compu-

tation of the uctuation component eC is here summarized for acartesian system with base vectors (e1; e2; e3). It is given by:

eC 1jXljZXl

Clijpq rsyDeuklpq" #

ei ej ek el 38

230 P.J. Snchez et al. / Comput. Methodswhere Deukl is the Tangential Displacement Fluctuation vector deter-mined by solving one linear system of equations for each macro-strain component eksel, with k; l 1;2;3. This set of variationallinear problems is enunciated as follows: for k; l 1;2;3 ndDeukl 2 Vl such that:ZXl

ClrsyDeukl rsyu^l dXl ZXl

Cl eksel rsyu^l dXl" #

;

8 u^l 2 Vl 39Additional details about this derivation can be found in [40,29].

A summary of this Section is presented in Box 2, where the basicingredients dening the ClaMM are shown.

4. Cohesive Multi-scale variational Model (CohMM)

The variational multi-scale formulation presented in this Sec-tion is envisaged for the modeling of the macroscopic post-criticalstage of materials whose micro-structure undergoes degradationand failure. According to the methodology proposed in this contri-bution, a cohesive-crack is inserted in a macro-point x when thestressstrain state fullls the criterion discussed in Section 2.2. Inthis case, the Homogenized Cohesive-type Constitutive Relation,developed in this Section, denes the corresponding constitutiveresponse of the point x.

We show that, following the same systematic approach of theprevious Section (see in particular the Remark 3.3), and incorporat-ing some rational kinematical ingredients, a variationally consis-tent framework can be established for modeling material failurein a multi-scale setting. The following developments describe, indetail, the item 4.2 in Box 1.

4.1. Kinematical description of the RVE

Let us focus our attention in the pseudo-time interval t > tN ,with tN being the instant when the macro-crack nucleationcriterion is rst fullled at a certain point x. An increment of the gen-eralized macro-strain is characterized by a kinematically admissibleincrement of the displacement jump db across themacro-crack Swithnormaln, andbyan incrementof the regular counterpartdeR. Thus, thekinematics in x is determined by: fdeR; db;ng.

Similarly to the ClaMM, the incremental micro-strain eld, del,can be described as the addition of two contributions:

del I ydeR;db deel 40involving an adequate denition of the Insertion Operator I y , whichdepends on y, and a uctuation component: deel : rsy deul. Withsimilar arguments to those given in the previous Section, only theregular component, deR, is uniformly inserted in Xl. Then, expres-sion (40) can be rewritten as follows:

del deR I Lydb rsydeul 41and the kinematical description reduces to dene the new operatorI Ly, which establishes how the localized macro-mode is in-serted into the RVE, such that it is consistent with the physical phe-nomenology of material failure at the micro-scale.

We denote XLl (XLl Xl) the sub-domain where the micro-

strain eld, el, localizes. The criterion that denes XLl, at the nucle-

ation time tN , is described in Section 4.3. It is postulated that anincrement of the localized macro-kinematics, characterized bythe elements fdb;ng, is uniformly inserted into XLl, instead of beinguniformly inserted into Xl. Then, the macro-to-micro kinematicalinformation transference is dened through the so-called Failure-Oriented Insertion Operator I Ly:

I Lydb /Llydbsn

8y 2 Xl; /Lly jXl jjXLl j

Ml

8y 2 XLl(

l. Mech. Engrg. 257 (2013) 221247M 0 otherwise42

esiFo

oun

the

Appand fundamental role in the mathematical formulation. Let usconsider the so called Irwins characteristic length of the material:ch EGF=r2u (whereE is theYoungsmodulus,GF the fracture energyand ru represents an ultimate limit stress). As exposed by Bazantand Planas [2], this physical parameter, ch, is related to thewidth ofthe localization band in softening materials. Then, l could beassociated with ch. However, from themodelling point of view, it isconvenient to differentiate both parameters, such that, even for awhere /Lly is a collocation function and l represents the thick-ness where the strain localization process takes place, see Fig. 4.Furthermore, M is a length parameter that, from the denition of/Lly, results:

M l jXljjXLlj43

Remark 4.1. In the present methodology M results immaterial,as we show in the following. Contrarily, l plays a very important

Classical Multi-scaleFormulation

Standard Boundary Condition (SBC)

Coh

StandardB

(a) (b)

Fig. 4. Minimal Kinematical Constraints (boundary conditions) to be imposed on(CohMM).

P.J. Snchez et al. / Comput. Methodsgiven material with xed ch, a localization bandwidth l, can betaken depending on the adoptedmodel for regularizing thematerialsoftening. Typically, for example, considering a smeared crackapproach at the RVE level, l should be associated to the niteelement size, while the ratio l=ch is used to dene an intrinsicsoftening modulus of the continuum that regularizes the materialresponse during the softening regime. Alternatively, by adopting astrong discontinuity model [27], it can be considered that: l ! 0while ch remains xed.

In the present formulation, and from Remark 4.1, the parameterl, is called the localization bandwidth at the RVE nite elementdiscretization level.

Since the Failure-Oriented Insertion Operator I Ly is linear,expression (42) can be rewritten as:

I Lydb ILy db dbsnl

8y 2 XLl0 otherwise

(44

where ILy is a third order tensor eld, that maps vectors dbinto symmetric second order tensors of the form given by (44).Note that, irrespective of the micro-cell size, for a givenvalue of the macro displacement jump increment db; I Lyimposes identical micro-strains into the localized sub-domain XLl.The tensor ILy has the following property: let R be a symmet-ric second order tensor and x an arbitrary vector, then:

R ILyx ILy T

R x 1lRn x 8y 2 XLl

0 otherwise

(8x 2 Rndim

45Hence ILy

Tis such that:

ILy T

R 1lRn 8y 2 XLl

0 otherwise

(46

We dene that del, given by (41) and (44), is KinematicallyAdmissible if it fullls the following two constraints:

ZXl

deR I Lydbh iz}|{IydeR ;db

dXl ZXl

del dXl 47

Z h i Z

Localizedsub-domain

ve Multi-scalermulation

daryCondition(SBC)

Non-Standard(additional)BoundaryCondition(NSBC)

micro-cell: (a) Classical Multi-scale Model (ClaMM), (b) Cohesive Multi-scale Model

l. Mech. Engrg. 257 (2013) 221247 231XLl

deR I Lydb|{z}IydeR ;db

dXLl XLl

del dXLl 48

which, considering (42) and (44), are equivalent to:

deR dbsn

M 1jXlj

ZXl

del dXl 49

deR dbsn

l 1jXLlj

ZXLl

del dXLl 50

In the present formulation, equations (49) and (50) dene the StrainHomogenization Procedures. Notice the similarities between (49) and(50) and the strain homogenization formula (24), of the previousSection.

Considering the denitions (41) and (42), and after simplemathematical manipulation, expression (49) results equivalent tothe following kinematical restriction on the micro-displacementuctuation increments deul:ZCl

deulsml dCl 0 51Eq. (51) represents the conventional kinematical constraint, alreadyanalyzed in the context of ClaMM. Therefore, we call this expression

the Standard Boundary Condition (SBC) for the proposed multi-scale formulation.

Similarly, in view of (41) and (44), the strain homogenizationprocedure (50) imposes an additional kinematical constraint ondeul, given by:ZCLl

deulsmLl dCLl 0 52

where CLl is the boundary of XLl and m

Ll its outward unit normal vec-

tor. Observe that Eq. (52) naturally emerges from the adopted kine-matical assumptions and induces kinematical boundary conditionsto be applied on the new boundary CLl. Hereafter, expression (52) iscalled the Non-Standard Boundary Condition (NSBC). It is interestingto observe here that the kinematical boundary condition NSBC to beprescribed over the new boundaryCLl is of the same type of the kine-matical boundary condition SBC applied over the boundary of theRVE, Cl. Furthermore, from the theoretical point of view they donot need to be identical. See the examples in Section 5.4 and 5.6.

Box 2: Classical Multi-scale Model (ClaMM): given etdtR and deR; nd dr and C (Equations governing the stable material response of amacro-point)

1.Kinematics:

del IydeR deel deR rsydeul; 8y 2 XlIy Iy ; 8y 2 Xl1.a Kinematical Admissibility Concept and Strain Homogenization Procedure:Z

Xl

I ydeR dXl ZXl

del dXl ) deR 1jXljZXl

del dXl

deul 2 eUl; eUl deul; deul 2 H1X and ZCl

deulsml dCl 0( )

1.b Virtual kinematically admissible actions (velocities):

u^l 2 Vl; Vl u^l; u^l 2 H1X andZCl

u^lsml dCl 0( )

e^l I ye^R rsyu^l; 8 e^R and 8 u^l 2 Vl2.Hill-Mandel Variational Principle of Macro-Homogeneity:

dr e^R 1jXljZXl

drl e^l dXl ; 8 e^R and 8 e^l kinematically admissible

2.a First consequence of the Hill-Mandel Principle: Stress Homogenization Operator H

Hdrl 1jXljZXl

Iy T drl dXl 1jXljZXl

drl dXl dr

2.b Second consequence of the Hill-Mandel Principle: Micro-Equilibrium problem

Given etdtR and deR; find deul 2 eUl such that :ZXl

drl rsyu^l dXl 0 8 u^l 2 Vl

3.Constitutive Model at the micro-scale:

rl Fletl 4.Homogenized Tangent Constitutive Tensor:

C CT eC4.a Taylor contribution

CT 1jXljZXl

Cl dXl

;

232 P.J. Snchez et al. / Comput. Methods Appl. Mech. Engrg. 257 (2013) 2212474.b Fluctuation contribution

For k; l 1;2;3; find Deukl 2 Vl such that :ZXl

ClrsyDeukl rsyu^l dXl ZXl

Cl eksel rsyu^l dXl" #

eC 1jXljZXClijpq rsyDeuklpq dXl

" #ei ej ek ell8 u^l 2 Vl

expression (46), this Operator reads:Z

AppRemark 4.2. The distinguishing features of the present CohMMapproach are threefold: (i) the nature of the adopted Insertion OperatorI Ly dened in expression (42), (ii) the Kinematical Admissibilityconcept given by expressions (47), (48) and (iii) the additional derivedkinematicalconstraintsgivenbyexpression(52).Thesefeaturesrepresentthe most salient differences between the present model and existingmulti-scale formulationsaccounting for failure and strain localization.

The set of Eqs. (51) and (52) denes what we call the MinimallyConstrained Vector Space of kinematically admissible incremental dis-placement uctuations in the RVE: eU Ll. Formally, the micro-eld deulis kinematically admissible if deul 2 eU Ll:eU Ll deul; deul 2 H1Xl; Z

Cl

deulsml dCl 0 and(

ZCLl

deulsmLl dCLl 0)

53

Then, the minimally constrained vector space VLl of virtual kinemat-ically admissible actions (velocities), u^l, can be dened as:

VLl u^l; u^l 2 H1Xl;ZCl

u^lsml dCl 0 andZCLl

u^lsmLl dCLl 0( )

54from where: eU Ll VLl. Furthermore, it follows that:VLl Vl 55where Vl has been dened in expression (27).

Considering (41), (44) and (54), it is also possible to introducethe concept of virtual kinematically admissible micro-strain e^l:

e^l ILy b^rsyu^l /Llyb^snM

rsyu^l; 8 b^ and 8 u^l 2 VLl56

where b^ is a virtual kinematically admissible macro-displacementjump dened in S. Comparing Eq. (41) with (56), it can be noticedthat, for t > tN; deR does not induce virtual actions in themicro-scale.

In Fig. 4, we compare the Minimal Kinematical Constraints that areimposed on a generic RVE using both, the ClaMM and the CohMM ap-proaches. As we show in Section 4.2, the additional restriction (52)gives mechanical consistency to the formulation, in the sense that thehomogenized response results objective with respect to the RVE-size.

The Standard Boundary Condition (51) is included in themodel fromthe beginning of the analysis. Meanwhile, the Non-Standard BoundaryCondition (52) is introduced once the sub-domain XLl is properly de-ned, i.e. after fullling the macro-crack nucleation criterion att tN . In this context, the adopted incremental framework plays afundamental role because it allows for a time continuous kinematicaltransition, from the ClaMM to the new CohMM, which is adapted forthe treatment of generalizedkinematics (strainandmacro-discontinu-ities). The topology ofXLl has to be established according to themicro-cellmechanical state at tN . In Section4.3,wepropose a kinematical cri-terion to dene the geometry and boundaries of XLl.

Remark 4.3. Alternative Failure-Oriented Multi-scale sub-modelscan be adopted by taking sub-spaces of eU Ll and VLl, such as: (i) theTaylor model, (ii) the Linear or Afne model and (iii) the Periodicmodel. Note also that, any alternative combination between themulti-scale sub-models derived from the Standard BoundaryCondition (51) and those derived from the Non-Standard BoundaryCondition (52), are also possible. From this perspective, expression(52) represents the most exible Kinematical Constraint.

Henceforth, we follow the same conceptual steps and variationalarguments discussed in the context of the ClaMM. Thus, once the kine-matics of the multi-scale model has been dened, the HillMandelPrinciple is postulated and its variational consequences are derived.

P.J. Snchez et al. / Comput. MethodsThis fact evidences thekinematical character of theassumptions incor-porated in thepresentmulti-scale framework, justifying thecommentsin the items (I) and (II) listed in the introductory Section.HLdrl 1jXLlj XLldrl n dX

Ll; for t > tN 61

Observe that the homogenization of the stresses at the micro-level, Eqs. (60) and (61), is not an a priori mechanical denition,but it arises as a consequence of the adopted admissible kinemat-ics, dened through the operator I Ly, and the HillMandelPrinciple.

Theexpression (60) is aNon-Standardhomogenizationprocedurefor stresses. It has two distinctive featureswith respect to the classi-cal stress homogenization approach dened in Eq. (33):

(i) the homogenization is performed considering the contribu-tion from points y 2 XLl, according to the denition of theadopted insertion operator I Ly, and

(ii) the homogenization is performed with the incremental trac-tion drln.

The previous considerations give a full justication for the point(III) enunciated in the introductory Section.

Remark 4.4. The present stress homogenization procedureensures an objective connection between dT and db, irrespectiveof the RVE-size. Observe that the application of the additional Non-Standard Boundary Condition (52) associates the increments of thegeneralized strains, db, with the strains inserted into XLl, since:R L4.2. The HillMandel Variational Principle of macro-homogeneity

In order to guarantee energetic consistency, the HillMandelVariational Principle of Macro-Homogeneity is postulated. For thepresent case we adapt this Principle to deal with the generalizedstresses, T , and generalized virtual displacement, b^, which are de-ned at the macro-level. Thus, the macro internal virtual power perunit of crack area is given by:PintS T b^ 57Then, for t > tN , the HillMandel Principle is written as follows:

T b^ MjXljZXl

rl e^l dXl; 8 b^; 8 e^l kinematically admissible 58

where the right hand side term represents the RVE total internal vir-tual power per unit of area of a RVE characteristic surface, orthogo-nal to the vector n, and representative of the failure domain in Xl.The area of this surface is given by: jXlj=M .

Considering that the variational equation (58) is satised duringthe previous loading history, see the remark (4.5) below, and thedenition of virtual micro-strain e^l from (56), the HillMandelPrinciple can be expressed in an incremental form, as follows:

dT MjXljZXl

ILy T

drl dXl

" # b^ MjXlj

ZXl

drl rsyu^l dXl 0;

8 b^; 8 u^l 2 VLl 59

4.2.1. First consequence of the HillMandel Principle: the Failure-Oriented homogenized mechanical response

From the variational equation (59), by taking arbitrary varia-tions b^ and u^l 0, it is derived themicro-to-macro Failure-OrientedStress Homogenization Operator HL:

dT HLdrl MjXljZXl

ILy T

drl dXl; for t > tN 60

and by recalling the property of the tensor ILy T

, given by

l. Mech. Engrg. 257 (2013) 221247 233XLldeel dXl 0.This remark justies the item (VI) listed in the introduction

Section.

4.2.2. Second consequence of the HillMandel Principle: the RVEequilibrium problem

From expression (59), by adopting b^ 0 and arbitrary varia-tions u^l, it is derived the variational equilibrium at the micro-level,which is enunciated as follows:

Given the history of macro-generalized strains fetdtR ; btdtg andkinematically admissible increments fdeR; dbg; nd the incremen-tal micro-displacement uctuation eld deul 2 eU Ll, such that:Z

Xl

drl rsyu^l dXl 0; 8 u^l 2 VLl and for t > tN 62

The variational equation (62) determines that the incrementalmicro stress eld, drl, is self-equilibrated. This stress is the reac-tion to the imposed generalized loading system induced by theapplication of the generalized macro-strain increments, deR andI Lydb, in Xl and XLl, respectively. Moreover, notice that the vari-ational equilibrium problem is subjected to special kinematicalrestrictions according to the denitions of eU Ll and VLl, expressedin (53) and (54)), which are consistent with the two strain homog-enization procedures (49) and (50).

For convenience, the relevant expressions dening the CohMMare summarized in Box 3. Note the similarities and differences be-tween the main ingredients of Boxes 2 and 3.

Remark 4.5. The derivation of Eq. (59) in terms ofincrements, from expression (58), requires the time continuity ofmacro-tractions inS. Particularly, this requirementhas tobe fullledatt tN . This is a delicate issuewhichmotivates the specic selection ofthe domain XLl described in the following Section 4.3. This timecontinuity condition can be expressed as follows:

TClaMM TCohMM for t tN 63where TClaMM is the traction vector in S evaluated with the ClaMMformulation:

TClaMM 1jXljZXl

rln dXl 64

and TCohMM is the traction vector in S evaluated with the CohMMformulation:

TCohMM 1jXLlj

ZXLl

rln dXLl 65

Box 3: Cohesive Multi-scale Model (CohMM): given fetdtR ;btg and fdeR;dbg; nd dT (Expressions valid for the points belonging to themacro-cohesive cracks)

1.Kinematics:

del deR I Lydbz}|{

IydeR ;db deel deR /Lly dbsnM rsydeul; 8y 2 Xl/Lly

jXl jjXLl j

Ml

8y 2 XLl0 otherwise

(

I Ly ILy /LlysnM

; 8y 2 Xl

1.a Kinematical Admissibility Concept and Strain Homogenization Procedures:ZXl

deR I Lydbh i

dXl ZXl

del dXl ) deR dbsn

M 1jXlj

ZXl

del dXlZXLl

deR I Lydbh i

dXLl ZXLl

del dXLl ) deR

dbsnl

1jXLlj

ZXLl

del dXLlZ

andZ( )

CLl

Ll

ibl

ent

d

uil

at :

234 P.J. Snchez et al. / Comput. Methods Appl. Mech. Engrg. 257 (2013) 221247HLdrl MjXlj XlILy drl dXl jXLlj XLl

drln dXLl

2.b Second consequence of the Hill-Mandel Principle: Micro-Eq

Given fetdtR ;btdtg and fdeR;dbg; find deul 2 eU Ll such thZdT b^ jXlj Xldrl e^l dXl; 8 b^ and 8 e^l kinematically admiss

2.a First consequence of the Hill-Mandel Principle: Failure-Ori

Z T 1 Zill-Mandel Variational Principle of Macro-Homogeneity:

MZMdeul 2 eU Ll; eU Ll deul; deul 2 H1X;Cl

deulsml dCl 01.b Virtual kinematically admissible actions (velocities):

u^l 2 VLl; VLl u^l; u^l 2 H1X;ZCl

u^lsml dCl 0 andZ(

e^l I Lyb^ rsyu^l /Llyb^snrsyu^l; 8 b^ and 8 u^l 2 V

2.HXl

drl rsyu^l dXl 0 8 u^l 2 VLlCLl

deulsmLl dCLl 0

u^lsmLl dCLl 0)

e

ed Stress Homogenization Operator: HL

T

ibrium problem

4.3. Identication of the localization sub-domain XLl in the micro-cell

The so-called Localization sub-domain XLl has to be identied att tN when the macro-crack nucleation condition is detected. Inthis Section, we dene a criterion to perform this task.

First, we recall that the nucleation criterion of Section 2.2,determining tN , is based on the existence of a strain localization-band, normal to the vector n, and the unit vector _ci which is parallelto the initial (pseudo-) velocity crack-opening direction. It is pre-sumed that in the interior points of the macro-band, the strain ratecomponent in the direction _cisn increases. On the contrary, inthe exterior points of the band, this strain rate component de-creases. Then, it is reasonable to postulate that XLl can be identiedby exploiting this feature. The procedure to dene XLl is enunciatedin the following two steps.

Let be given the set: SN ftN;n; _cig determined at the macro-

The L

range

(b) Fig. 5(b) shows the case where the points fullling the crite-rion (66) comprise an inclusion of elastic material, XLel ,which never displays unstable behavior nor softening. Then:XLl XLsl [XLel .

(c) Fig. 5(c) shows the case where the micro-structure has voids.In this case, the voids must be thought of being constitutedof an extremely exible elastic material, with the elasticitycoefcient going to zero. Then, the void interior points canalso be tested through the criterion (66). Those points fulll-ing the criterion are denoted: XLvl . In this case, the localiza-tion sub-domain is dened as: XLl XLsl [XLvl .

Even when we cannot formally prove that the present selectionof XLl guarantees the continuity condition (63) in all conceivablesituations, we show in the numerical tests that traction time con-tinuity at t tN is fullled in the entire range of tested situations,

(b

P.J. Snchez et al. / Comput. Methods Appl. Mech. Engrg. 257 (2013) 221247 235Fig. 5 where the shaded dark gray zone identies the points fulll-ing the requirement (66), and thus, it is identied as the strain-localization band XLl. Also, in this Figure, the points where thematerial is described through a constitutive model with softeningare denoted: XLsl , the additional supra-index s indicates soften-ing. Each situation is characterized as follows:

(a) The simpler and more intuitive case is sketched in Fig. 5(a),where the micro-strain localization band, XLl, is constitutedof materials displaying softening. Evidently, in this case wehave: XLl XLsl .

(a)

Fig. 5.strain-lkinematic based rule (66) dening Xl, describes a wideof different situations, as for example those illustrated incrack nucleation time tN:

(i) Apply to the micro-cell, a uniform strain rate given by _cisn. Compute the corresponding increments of the strainrate uctuation, d _eel, by solving a similar variational equilib-rium problem as it is done in the previous pseudo-timeinstants, see expression (34). Observe that the ClaMM settingis still considered in this step.

(ii) In the micro-cell domain, compute the projection of themicro-strain rate uctuation in the direction ( _cisn). Then,dene XLl as follows:

XLl y 2 Xl; d _eely : _cisn > 0 for t tNn o 66It is noted that this methodology is exclusively based on a kine-

matical criterion and does not take into account the specic con-stitutive response of points in XLl.Localization sub-domain XLl for different types of heterogeneous micro-structureocalization sub-domain composed of softening-material and elastic bers, and (cwhich turns to encompass a wide spectrum of possible failure sce-narios. We consider that this issue still remains open in the presentFOMF.

5. Numerical tests

The variational FOMF is implemented into a nite element code.Details of the numerical approximation, as well as several algorith-mic aspects of the model, are exposed in Appendix A.

In this Section we show a series of problems that are focused inthe analysis of the fundamental aspects of the methodology, that isthe nucleation and evolution of strain localized bands in the RVEand the objectivity, with respect to the micro-cell size, of theresulting homogenized constitutive response to be transferred tothe macro-scale. All the numerical tests share the followingcharacteristics:

Micro-structures display a simple topology of heterogeneities. The sub-domains XLl in which strain-localization takes place ispre-induced from the beginning of the analysis by means ofthe material denition at the micro-level; thus the boundaryCLl, following the guidelines given in Fig. 5, is also pre-induced(although criterion (66) is employed in its determination).

In the numerical assessments performed in Sections 5.2, 5.3,5.4, 5.5, the non-linear macro equilibrium equations are notsolved. We dene the RVEmicro-mechanical problem by insert-ing an arbitrary history of generalized macro-strains given bythe successive insertion of fdeR; dbg, at each pseudo-timeinstant. However, in order to demonstrate that the presentapproach works as a multiscale formulation, a simple fully cou-pled (macromicro) stretching numerical test is addressed in

Void

) (c)

s: (a) strain-localization sub-domain composed, entirely, of softening-material, (b)) strain-localization sub-domain composed of softening-material and voids.

Section 5.6. The invariance of the macroscopic solution withrespect to the nite element mesh renement in the micro-scale is also showed in the fully coupled stretching example.

5.1. Preliminaries. General settings

response is computed and compared for the three micro-cellsusing two multi-scale approaches:

The ClaMM, described in Section 3, for the complete loading his-tory. No kinematical enrichment (db 0 8 t) is incorporatedduring the complete analysis. The micro-mechanical problemis entirely driven by deR. In this case, a monotonically increasingvalue of deR is applied during the loading process.

The FOMF, which consists of the ClaMM discussed in Section 3sequentially coupled at time t tN to the CohMM (see Section 4).

236 P.J. Snchez et al. / Comput. Methods Appl. Mech. Engrg. 257 (2013) 221247A generic unit system is adopted: the unit of length is repre-sented by L and the unit of force by F.

Two conventional rate-independent constitutive models areused to simulate the material response at the microscopic level:

A J2 isotropic elasto-plastic model characterized by the follow-ing material parameters: Youngs modulus, El, Poisson ratio,ml, yield stress, rlY and fracture energy per unit of surface area,GlF .

A scalar isotropic elastic-damage model characterized by:El; ml; rlu; GlF , where rlu represents the ultimate uniaxiallimit strength.

Linear and/or exponential strain softening models are assumedin both material responses, see [36,37] for a detailed description ofthese conventional constitutive models. Conventional implicitschemes are used for the integration of such constitutive models.A constitutive regularization method based on the concept ofSmeared Crack Approach (SCA) [31], is adopted. Thus, the initial soft-eningmodulusHl0 which depends, among other parameters, on themicro-structural material characteristic length ch ElGlF=r2lY , isregularizedas shown inTable 1. In the followingexamples, the local-ization bandwidth, l, is chosen in advance according to the size ofthe nite element mesh. However, it is noted that in more generalproblems, this parameter is a result derived from the RVE analysisand the evaluation of the localization domain: XLl.

Except for the coupled macromicro problem of Section 5.6, theincremental strains inserted in the micro-cell domain,fdeR;I Lydbg, are parameterized as follows:deR dvt KsH; K cosK; sinKT ;H cosH; sinHT 67

I Lydwtdb dwt/Llydbsn

M;

db cosb; sinbT ; M l jXljjXLlj68

whereK; H and b are arbitrary angles to be dened in each example,/Lly is the localization function, given by expression (42), andn cosg; singT , with g analytically determined in each case, isthe unit vector normal to themacro-cohesive crackS and parameter-ized through the angle g. The coefcients dv and dw dene the mag-nitude of the regular and localized strain increments, respectively.

Sections 5.25.5 are devoted to analyze the objectivity of theRVE homogenized response with respect to the micro-cell size.Thus, for such numerical examples, we dene: (i) a basic heteroge-neous micro-structural cell, called RVE1 (the term RVE is used in thesense explained in the remark 2.1 of Section 2), and (ii) two addi-tional micro-cells, named RVE2 and RVE3, which are a repetition ofthe basic heterogeneous cell RVE1. The homogenized constitutive

Table 1Initial softening modulus Hl0, according to the adopted constitutive model and theSCA.

Linear softening Exponential softening

J2-plasticity Hl0 El2lch

Hl0 El lchElastic-damage Hl0 12

lch

Hl0 lchIn this case, for t > tN a change in the macro-kinematics, charac-terized by fdb;ng, is induced and inserted into the micro-cells.The evolution of these elds is ruled by expression (68). Fur-thermore, for t > tN , we also control the regular strain incre-ment deR. In some tests we adopt: deR 0 for t > tN . In othersituations, eR is decreased, as it should be expected when thecomplete macroscopic problem is considered. In any case, fort > tN , the localized inserted strain term, fdb;ng, is dominantcompared with deR. Thus, the cohesive homogenized responseis not very sensitive to the particular choice about the insertedregular value deR.1

For all testes, the Minimal Kinematical Restriction is consideredfor the Standard Boundary Condition SBC, dened in Cl throughEq. (51). For the Non-Standard Boundary Condition in CLl, see Eq.(52), the NSBC-Minimal Kinematical Restriction has been adopted,except for the application to situations involving micro-structureswith voids (see Sections 5.4 and 5.6) where the NSBC-LinearBoundary Condition has been applied.2 Plane strain conditions areconsidered in all cases.

5.2. Micro-structures with homogeneous strain localization bands

5.2.1. Case 1-(a): horizontal strain localization bandsThe rst test consists of micro-cells with embedded homoge-

neous strain localization bands. The geometry for the basic heter-ogeneous micro-structure, RVE1, is shown in Fig. 6(a) where wedene: h b 1L and a 0. The additional micro-cell domains,RVE2 and RVE3, which are used in the objectivity analysis, are alsoshown in Fig. 6(b) and (c). The materialM1 is linear-elastic whereasthe material M2 is characterized by an isotropic elastic-damagerelation endowed with linear and exponential strain-based soften-ing models. Table 2 describes the material properties for M1 andM2. The localization bandwidth, l, represents the thickness wherethe micro-strain eld localizes. The value l 0:091 L has beenadopted for the three RVEs, coinciding with the nite element sizein the n-direction. The regularized initial softening modulus, Hl0, iscomputed from expressions given in Table 1.

Fig. 6(f) shows the RVE1-mesh which consists of 3 quadrilateralnite elements, while the discrete model for the RVE2 and RVE3contains 6 and 9 elements, respectively.

The rst numerical simulation is obtained considering:K H 90; dv 1e 5, see Fig. 6(d). Notice that from this

1 In a fully coupled multi-scale environment, the incremental macro-kinematicsfdeR; dbg will be obtained from the solution of the macro-equilibrium problem.

2 In the context of kinematically-based multi-scale formulations, the application ofthe Minimal Kinematical Constraint is equivalent to imposing an Uniform TractionBoundary Condition on the micro-cell boundary. Whenever the stiffness of a micro-constituent intersecting the RVE-boundary goes to zero, the uniform traction alsogoes to zero and so, the minimally constrained multi-scale model provides an invalidhomogenized response. The situation is worse if a pore reaches the RVE-boundary. Insuch cases, alternative multi-scale models are preferred. This argument, and takinginto account the possibility of nding pores being cut by the strain-localization bandXLl , justies the use of the Linear multi-scale model that is proposed for the NSBC.

Independently of the previous comment and after various numerical experiments ourconclusion is that, in general, the Linear Boundary Condition is an appropriate selectionfor the NSBC.

bM1

h

2h

3h

b

RVE1M1

M2

RVE2 RVE3

(e)(d)

te el) v-

P.J. Snchez et al. / Comput. Methods Appl. Mech. Engrg. 257 (2013) 221247 237(a) (b)Fig. 6. Micro-structures with homogeneous strain localization bands: (a) RVE1 (3 niRVE3 (9 nite elements in the discrete model), (d) v-function used for the ClaMM, (eRVE1.setting, we model micro-structures with horizontal softening-bands subjected to a purely axial regular strain in the verticaldirection, i.e. only eRyy 0, see Eq. (67).

Fig. 7(a) and (b) display the homogenized stressstrain consti-tutive relation in terms of the axial components (ryy vs. eRyy) thatare obtained with the ClaMM and considering both, linear andexponential, softening models. During the elastic range, indicatedas AB in these gures, the macro mechanical responses are iden-tical for the three considered RVE-sizes. Thus, the constitutiveClaMM is objective with respect to this parameter. However, a

Table 2Material properties for the micro-structures displayed in Fig. 6.

El F=L2 ml rlu F=L2 GlF F=L ch L

Material: M1 2.1e4 0.2 Material: M2 2.5e3 0.2 42 0.60 0.85

Ryy

yy[

]F/

L2

(a) (

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

5

10

15

20

25

30

35

40

45RVE1RVE2RVE3

RVE3RVE2

SBC

RVE1

A

B

C1

C2

C3

Fig. 7. Homogenized stressstrain relation (ryy vs. eRyy) obtained using theregular macro-strain increase after crossing the point B originates,for different RVE sizes, a very marked non-objective mechanical re-sponse, as displayed by curves denoted BC1, BC2 and BC3.

Point B marks the (pseudo-) time instant in which the Classicalhomogenized response loses objectivity. It agrees with the nucle-ation time, tN , obtained using the procedure based on the spectralproperties of the Homogenized Tangent Constitutive Tensor, C,

b

(c) (f)

RVE mesh1

ements in the discrete model), (b) RVE2 (6 nite elements in the discrete model), (c)function and w-function used for the FOMF, (f) Finite element discretization for theintroduced in Section 2.2. The vector n, also given by such a crite-rion, is consistent with the direction of the softening bands denedin the micro-structure.

Fig. 8 shows the results obtained with the Cohesive Multi-scaleModel (CohMM) after crossing the point B (t > tN). The homoge-nized cohesive responses are depicted in terms of the cartesiancomponents Ty vs. by, see Fig. 8(a) and (b). For t > tN , the (incre-mental) inserted strain localized mode, I Lydb, is characterizedby: b 90; dw 1:5e 4 and g 90. In addition, the regularcomponent, eR, remains constant and equal to eRtN. Fig. 6(e) de-picts a schematic representation of the functions vt and wt that

Ryy

b)

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

5

10

15

20

25

30

35

40

45RVE1RVE2RVE3

yy[

]F/

L2

RVE3RVE2

SBC

RVE1

A

B

C2

C1

C3

ClaMM: (a) linear softening model, (b) exponential softening model.

((a) (]

/L2 35

40

45RVE1RVE2RVE3

B

d C

238 P.J. Snchez et al. / Comput. Methods App[]

F/L2

(a)

35

40

45RVE1RVE2RVE3

B

Hom

ogen

ized

Trac

tion

Ty

[F

y [ ]L

GF

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

5

10

15

20

25

30

NSBC

SBC

RVE3RVE2

RVE1

D

Fig. 8. Homogenized cohesive response (Ty vs. by) obtained using the proposecontrol the generalized strains during the loading history, from thebeginning of the analysis. Note that the selected set of parametersdenes a pure Mode I of fracture for the macro-cohesive crack.

Fig. 8 evidences that the CohMM provides an objective cohesiveresponse during the post-critical regime. This property holds forboth strain-softening damage models, linear and exponential, seethe curves denoted as BD in Fig. 8(a) and (b), respectively.

The area under the curve Tyby, i.e. the gray shaded zones inFig. 8(a) and (b), denes the fracture energy per unit of surface areaat macro-level, here denoted as GF . Note that the new multi-scalecohesive formulation returns an objective measure of GF , irrespec-tive of the RVE-size. In this particular case, due to the simplicity ofthe test3, the macro-fracture energy obtained by the integration ofthe curve Tyby, agrees with the fracture energy dened for thematerial at the micro-level (GF GlF), for both softening-based dam-age models.

The whole constitutive response, given by the proposed varia-tional FOMF, provides two homogenized responses for differenttime intervals: (i) the ClaMM solution, for t < tN , range AB inFig. 7, and (ii) the CohMM solution, for t > tN , range BD in Fig. 8.

3 The onset of the non-linear behavior at the micro-scale coincides with theactivation of the crack at macro-level and the softening-band is homogeneous in themicro-scale.

Ryy

yy

0 1 2 3 4 5 6 7 8 9-3

0

5

10

15

20

25

30

RVE3RVE2

SBC

RVE1

A

C1

C2

C3

Fig. 9. Homogenized stressstrain response obtained using the ClaMM: (a) Axial stb)

[]

F/L2 3

3.5

4RVE1RVE2RVE3

B

b)

Hom

ogen

ized

Trac

tion

Ty

[]

F/L2

y [ ]L

GF

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

5

10

15

20

25

30

35

40

45RVE1RVE2RVE3

NSBC

SBC

RVE3RVE2

RVE1

D

B

ohMM for t > tN: (a) linear softening model, (b) exponential softening model.

l. Mech. Engrg. 257 (2013) 2212475.2.2. Case 1-(b): inclined strain localization bandsA second set of numerical simulations, considering identical

material properties forM1 andM2, is analyzed. For the sake of sim-plicity, only an exponential softening rule is adopted. Inclinedbands of material with softening, characterized by an anglea 20, dene the topology of the heterogeneous micro-structure.

The increments of regular strains, deR, and the localized compo-nent, Iydb, inserted into the micro-scale, are dened by:H b 60; K 110 and g, with g determined with the cracknucleation criterion. As a result, once the cohesive crack is acti-vated at the macro-level, a Mixed Mode of fracture is simulated.Minimal kinematical restriction is applied on the boundary Cl ofXl. Observe that this type of kinematical constraint is consistentwith the nucleation and subsequent development of inclined soft-ening bands, whereas others types of boundary conditions, suchas Linear or Periodic, do not admit such failure pattern.

The coefcients that control the generalized loading history,dvt and dwt, are similar to those dened in the previous case1-(a), see Fig. 6(d) and (e).

Numerical results obtained with the ClaMM are shown in Fig. 9.Fig. 9(a) depicts the homogenized stressstrain constitutive re-sponse in terms of the axial components, ryy vs. eRyy, whileFig. 9(b) displays the curve rxy vs. eRxy. In the pre-critical regime,denoted as AB in the plots, the macro-mechanical response isobjective with respect to the RVE-size. However, after crossingthe point B, this p