International Journal of Solids and Structures 50 (2013) 725–742

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International Journal of Solids and Structures

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A nonuniform TFA homogenization technique based on piecewise interpolationfunctions of the inelastic field

Valentina Sepe, Sonia Marfia ⇑, Elio SaccoDepartment of Civil and Mechanical Engineering, University of Cassino and of the Southern Lazio, 03043 Cassino, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 March 2012Received in revised form 1 November 2012Available online 28 November 2012

Keywords:HomogenizationNonlinear compositesNonuniform TFASMAPlasticity

0020-7683/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijsolstr.2012.11.005

⇑ Corresponding author. Tel.: +39 0 7762993745; faE-mail address: marfia@unicas.it (S. Marfia).

The present paper deals with a homogenization technique based on the Transformation Field Analysis(TFA) for the study of heterogeneous composite media characterized by nonlinear response. Accordingto the TFA, the behavior of the representative volume element (RVE) is studied accounting for the non-linear effects by means of the presence of a uniform inelastic strain distribution in the nonlinear constit-uent of the heterogeneous material. In order to improve the TFA, the assumption of uniformity of theinelastic strain distribution is removed, so that a nonuniform inelastic strain field, better representingthe inelasticity distribution in the composite, is considered. In particular, the inelastic strain is repre-sented as a piecewise linear combination of analytical functions of the spatial variable. The theory, pre-sented in a general framework, can be successfully adopted for deriving the overall constitutive responsefor a wide range of nonlinear composite materials. Furthermore, the procedure is tailored to investigatethe response of composites whose constituents are Shape Memory Alloys (SMA) and materials character-ized by plastic behavior. Finally, numerical applications are developed in order to assess the effectivenessof the proposed nonuniform TFA procedure, comparing the results with the ones carried out performingUniform and Piecewise Uniform TFA homogenizations and nonlinear finite element micromechanicalanalyses.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The increasing use of composite materials for high performanceapplications in several fields of structural engineering has broughtout the need for the development of efficient tools able to providesuitable numerical predictions of their mechanical response.Accordingly, numerous homogenization techniques, which allowto estimate the mechanical properties of heterogeneous mediaand to derive the overall behavior of the equivalent homogenizedmaterial, have been developed. Many homogenization proceduresavailable in literature are based on various effective medium mod-els such as the equivalent eigenstrain method proposed by Eshelby(1957), the Mori and Tanaka approach (Mori and Tanaka, 1973),which used the Eshelby solution, the self-consistent model of Hill(1965), the variational approaches of Hashin and Shtrikman, lead-ing to their well-known bounds (Hashin and Shtrikman, 1963),which have been generalized by Willis (1977, 1983), among manyothers.

All the mentioned homogenization techniques, initially formu-lated to study the linear response of the heterogeneous material,represent the basis for the development of suitable techniques able

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x: +39 0 7762993392.

to consider nonlinear material effects. Of course, the determinationof the mechanical behavior of a composite becomes a more com-plex process, when its constituents are characterized by inelasticphenomena.

A branch of homogenization has focused on the introduction ofrigorous bounds and estimates for the effective properties of non-linear composites, using variational methods. Talbot and Willis(1985) obtained rigorous bounds through the generalization tononlinear media of the Hashin–Shtrikman variational principle.Ponte Casta~neda (1991) proposed an alternative variational struc-ture able to generate bounds and estimates for nonlinear compos-ites from the corresponding information of linear composites withthe same microstructural distribution, and he also provided esti-mates exact to second order in the heterogeneity contrast (PonteCasta~neda, 1996). Lately, Agoras and Ponte Casta~neda (2011) ap-plied the method proposed in Ponte Casta~neda (1991) to multi-scale composites with viscoplastic isotropic constituents and ran-dom sub-structures.

In order to accurately describe the response of a nonlinear het-erogeneous medium, numerical techniques can be adopted to solvethe micromechanical problem. In fact, the finite element method orthe boundary element method, can be successfully adopted; on theother hand, these numerical techniques involve a large number ofinternal variables and, thus, lead to high computational burden.

726 V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742

Simplified homogenization approaches have been proposed withthe aim of reducing the computational complexity of the numericalmicromechanical investigation.

The micromechanical method of cells (Aboudi, 1991), and laterits generalization (Aboudi, 1996), was proposed by Aboudi withthe aim of providing the overall behavior of periodic multiphasematerials of various types.

An interesting approach for solving the nonlinear microme-chanical homogenization problem is the Transformation FieldAnalysis (TFA), originally proposed by Dvorak (1992). It considersthe inelastic strain as a given eigenstrain, assuming the field ofinternal variables to be uniform in each individual constituentof the composite characterized by nonlinear behavior. The effectof eigenstrains is accounted for by solving linear elasticityproblems and it is superimposed to the effect induced by uniformoverall strain. In particular, for two-phase composites the TFAmethod is able to provide exact relations between the elastic strainconcentration tensors and the influence tensors due to the Levin-Rosen-Hashin’s formula (Dvorak and Bahei-El-Din, 1997).

The TFA method was also improved by Dvorak et al. (1994)through the subdivision of each material phase into several sub-sets, developing a PieceWise Uniform TFA (PWUTFA). Consider-ing a piecewise uniform inelastic strain distribution enables toimprove the description of the inelastic strain heterogeneitiesbut, obviously, at the expense of increasing the complexity ofthe technique. However, it has been shown that the TFA andPWUTFA techniques lead to too stiff predictions (Suquet, 1997).Chaboche et al. (2001) also developed a PWUTFA in order to de-rive the nonlinear behavior of damaging composites. In order tobetter represent the complex field of the inelastic strain in therepresentative volume element (RVE) of a nonlinear composite,Michel and Suquet (2003) proposed a nonuniform TFA (NUTFA).The inelastic strain field is considered as nonuniform and de-scribed as the superposition of functions, called inelastic modesand determined numerically by simulating the response of thecomposite along monotone loading paths. The technique wasthen implemented for the study of nonlinear composite materi-als (Michel and Suquet, 2004). Lahellec and Suquet (2007) pro-posed an alternative method for deriving the overall propertiesof nonlinear inelastic composites based on the minimization ofan incremental energy function, within an implicit time-discret-ization scheme; they proved that this approach is equivalent tothe transformation field analysis with a nonuniform eigenstrainfield (Michel and Suquet, 2003). Recently, the nonuniform TFAmodel (Michel and Suquet, 2003) has been usefully applied byFranciosi and Berbenni (2007, 2008) for modeling heterogeneouscrystal and poly-crystal plasticity, characterized by hierarchicalmulti-laminate structures; by Roussette et al. (2009) for thestudy of composites having elastic-viscoplastic and porous elas-tic-viscoplastic constituents; by Fritzen and Böhlke (2011) forthe analysis of the morphological anisotropy of micro-heteroge-neous materials with particle reinforcement; by Jiang et al.(2011) for the analyses of nonlinear composite media made ofporous materials.

Marfia and Sacco (2005), Marfia (2005), and Sacco (2009)developed TFA and PWUTFA homogenization procedures whichconsider the periodicity conditions in order to investigate theoverall nonlinear response of the Shape Memory Alloy (SMA)composites and masonry materials. Marfia and Sacco (2007) alsoproposed a multiscale approach for SMA composite laminatesdeveloping a nonlinear homogenization technique. Addessiet al. (2010) have extended the PWUTFA technique to the Coss-erat continuum to study the mechanical response of masonry.Marfia and Sacco (2012) presented a PWUTFA homogenizationprocedure for the multiscale analysis of periodic masonry,

assuming a bilinear approximation for the inelastic strain ofone subset of the unit cell.

Aim of the present paper is the development of a new homog-enization technique for composite materials characterized by thenonlinear behavior of their constituents in the framework of thetransformation field analysis. Following the proposal discussed inMichel and Suquet (2003) and in Marfia and Sacco (2012), the fieldof the internal variables is considered as nonuniform. The mainnovelties of this work consist in the construction of an approxi-mated nonuniform inelastic strain field in the RVE of the compositemedium and in the derivation of its evolutive problem. The RVE isdivided into subsets; in each subset a nonuniform distribution ofthe inelastic strain, which accounts for all the nonlinear effects,is adopted. In particular, the inelastic strain in each subset is as-sumed as a linear combination of selected analytical functions,called modes, which depend on the spatial variable. The coeffi-cients of the linear combination are tensors and are determinedsolving the evolutive problem. In particular, a new formulation isproposed to compute the evolution of the coefficients of theapproximated form of the inelastic strain on the basis of the con-tinuum evolutive equations.

In the present contribution numerical applications are devel-oped considering different types of periodic composite materials.In particular, plasticity and shape memory alloy models are imple-mented in order to take into account the nonlinear phenomenaoccurring in the material constituents of the examined composites.

In order to verify the efficiency of the developed procedure, thenumerical results obtained by the proposed nonuniform TFAhomogenization technique are compared with the ones carriedout adopting uniform and piecewise uniform TFA procedures andnonlinear finite element micromechanical analyses.

The paper is organized as follows: Section 2 formulates theproblem of the homogenization for nonlinear composites; in Sec-tion 3, the formulation of the proposed nonuniform TFA procedureis presented; in Section 4, the developed nonuniform TFA is tai-lored for plastic and SMA materials, illustrating the numerical pro-cedure; Section 5 deals with the numerical applications.

2. Homogenization problem for nonlinear composites

The homogenization problem for composite media whose con-stituents can be characterized by nonlinear response, is studied inthe framework of small strain theory. In particular, the interest isdevoted to inelastic phenomena induced by plasticity, damage, vis-co-plasticity and other nonlinear effects.

The mathematical algebraic notations adopted in the followingare briefly introduced. Scalar variables are denoted in italics; vec-tors, second-order tensors and fourth-order tensors in bold; scalarproduct between vectors with symbol ‘� ’; scalar product betweentwo second-order tensors with symbol ‘:’; product between twofourth-order tensors, a fourth-order tensor and a second-order ten-sor, a second-order tensor and a vector (with the relative contrac-tions of two indices, two indices and one index, respectively) withno symbol.

A general representative volume element, able to retain theproperties of the heterogeneous medium characterized by nonlin-ear behavior, is considered. In the following the typical RVE is de-noted with the symbol X.

The constitutive relationships for the constituents of X areformulated in a phenomenological thermodynamic framework(Halphen and Nguyen, 1975). In fact, the existence of a thermody-namic potential is postulated and a free specific energy function isintroduced through a convex potential as:

Wðe;p; T;VkÞ ¼ Weðe;p; TÞ þWpðp;Vk; TÞ; ð1Þ

V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742 727

where:– We represents the elastic strain energy due to the thermo-elas-

tic material deformations, depending on the total strain e, on theinelastic strain p and on the absolute temperature T;

– Wp is the inelastic energy which accounts for all the inelasticphenomena and that is related to the internal variables p andVk and to the absolute temperature T.

Note that internal and state variables e, p, T and Vk depend onthe spatial coordinate x and on the time t. This dependency isexplicitly reported only when it is necessary to clarify theequations.

The Clausius–Duhem inequality, i.e., the second thermody-namic law, takes the following form:

r�@W@e

� �: _e� @W

@Tþ s

� �_T�@W

@p: _p� @W

@Vk: _Vk�q �gradT

TP 0; ð2Þ

with r the stress tensor, s the entropy, and q the heat flux vector.Accordingly, the state laws can be derived from the thermody-

namic potential as:

r ¼ @W@e

; s ¼ � @W@T

; Xp ¼ �@W@p

; Xk ¼ �@W@Vk

: ð3Þ

The relations (3) constitute the thermo-elastic laws related tothe stress tensor r, the entropy s and the thermodynamic forcesXp and Xk associated with the internal variables p and Vk,respectively.

In order to describe the evolution of the internal variables, theexistence of a dissipation potential uð _p; _VkÞ is also postulated.The potential is set as a convex positive function of the flux vari-ables _p and _Vk. The complementary laws are expressed by the nor-mality rule as:

Xp ¼@u@ _p

; Xk ¼@u@ _Vk

: ð4Þ

Introducing /⁄ as the dual potential of / by the Legendre-Fen-chel transformation, the evolution law of the internal variablesresults:

_p ¼ @u�

@Xp; _Vk ¼

@u�

@Xk: ð5Þ

In particular, for the analyzed RVE of the heterogeneous med-ium the thermo-elastic potential We is set as:

Weðe;p; TÞ ¼12

Cðe� pÞ : ðe� pÞ; ð6Þ

where C is the fourth-order elasticity constitutive tensor; thus, thestress-strain relationship is obtained from Eqs. (1) and (3) as:

r ¼ Cðe� pÞ: ð7Þ

Note that the elasticity constitutive tensor C , which is afunction of the spatial variable, in the present work is as-sumed to be a piecewise constant function of the typicalpoint x e X.

In the constituents of the RVE characterized by an elastic behav-ior, both the inelastic energy Wp and the inelastic strain tensor pare set equal to zero.

The homogenization problem is formulated prescribing in X thevalue of the average strain �e, commonly denoted as effective strain.The governing equations in the RVE are:

– compatibility

e ¼ 12ðruþruTÞ ð8Þ

– equilibriumdivr ¼ 0 ð9Þ

– constitutive laws of the RVE constituents defined by Eqs. (1)–(7).The problem is completed by suitable boundary conditions

which are specified in the following. Thus, the homogenizationproblem consists in the determination of the displacement fieldu(x), of the total strain field e(x), of the inelastic strain field p(x) ,of the stress field r(x) and of the average stress �r in X, such that:

�r ¼ 1V

ZXrðxÞdVx; �e ¼ 1

V

ZXeðxÞdVx; ð10Þ

where V is the volume of X and dVx is the infinitesimal volumearound the point individuated by the vector x. Clearly Eqs. (10) holdif the material is not characterized by the presence of holes or pores.

At this stage, assuming the inelastic strain field p(x) as known,the total strain field e(x) can be evaluated by the relation:

eðxÞ ¼ e�eðxÞ þ epðxÞ: ð11Þ

It can be remarked that according to Eq. (11), the local straine(x) is obtained as the sum of the two contributions e�eðxÞ andep(x) , obtained solving the linear elastic localization problemsfor the RVE subjected to the effective strain �e and to a prescribedinelastic strain field p(x) , respectively. Moreover, the strain fieldep(x) is constrained to have null average. Thus, the strain fieldse�eðxÞ and ep(x) can be written in the form:

e�eðxÞ ¼ AðxÞ�e; epðxÞ ¼Z

XQ ðx; yÞpðyÞdVy; ð12Þ

with x and y two typical points of X. In Eq. (12) A(x) is the fourth-order localization tensor, commonly called elastic strain concentra-tion tensor, able to recover the local strain at the point x of the com-posite material when the effective strain �e is prescribed; Q(x, y),representing the modified elastic Green tensor, is the fourth-orderlocalization tensor at the point x associated to the presence of theinelastic strain acting at the point y. It can be highlighted thatEqs. (11) and (12) represent the total strain field, that is the exactsolution of the homogenization problem. Note also that the linearoperator Q(x, y) has to ensure that the average of the strain fieldep(x) is zero, i.e.:

�ep ¼1V

ZXepðxÞdVx ¼

1V

ZX

ZX

Q ðx; yÞpðyÞdVy

� �dVx ¼ 0; ð13Þ

so that the effective strain results:

�e ¼ 1V

ZXe�eðxÞdVx þ

1V

ZXepðxÞdVx ¼

1V

ZX

AðxÞdVx�e

) 1V

ZX

AðxÞdVx ¼ I; ð14Þ

where I is the fourth-order identity tensor.Once the local strain field is evaluated by Eq. (11), the stress

field is computed using the constitutive relation (7), as:

rðxÞ ¼ r�eðxÞ þ rpðxÞ; ð15Þ

where, taking into account formulas (12), it is:

r�eðxÞ ¼ CðxÞ½e�eðxÞ� ¼ CðxÞAðxÞ�e;rpðxÞ ¼ CðxÞ½epðxÞ � pðxÞ� ¼ CðxÞ½

RX Q ðx; yÞpðyÞdVy � pðxÞ�:

ð16Þ

Because of Eq. (10)1 and relation (15), the overall average stress�r is obtained as:

�r ¼ �r�e þ �rp; ð17Þ

728 V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742

with

�r�e ¼ 1V

RX r�eðxÞdV ¼ �C�e;

�rp ¼ 1V

RX rpðxÞdV ¼ 1

V

RX CðxÞ½

RX Q ðx; yÞpðyÞdVy � pðxÞ�dVx:

ð18Þ

In Eq. (18), the overall fourth-order elasticity tensor �C is definedas:

�C ¼ 1V

ZX

CðxÞAðxÞdVx: ð19Þ

Note that, the overall constitutive relation for the RVE can bewritten as:

�r ¼ �Cð�e� �pÞ; ð20Þ

with the overall inelastic strain �p evaluated from Eqs. (17) and (18)as:

�p ¼ ��C�1 �rp: ð21Þ

The evolution of the overall inelastic strain �p is governed by thelaw introduced in Eq. (5)1; taking into account Eqs. (18)2 and (21),it results:

_�p ¼ � 1V

�C�1Z

XCðxÞ

ZX

Q ðx; yÞ _pðyÞdVy � _pðxÞ� �

dVx: ð22Þ

It can be remarked that, although the effective plastic strain �p isnot used in the subsequent computations, its definition allows toformulate the macroscopic constitutive law in the standard repre-sentation form (20). Eq. (21) also puts in evidence that the overallinelastic strain in the RVE is not the trivial average of the inelasticstrains.

3. Nonuniform Transformation Field Analysis

A homogenization procedure, based on the TFA, is presented forthe typical representative volume element X of a composite med-ium characterized by nonlinear response described in the previoussection. In particular, the considered RVE is composed by severalhomogeneous constituents, which can present elastic or inelasticbehavior. The whole RVE is divided in N subsets Xi, each one char-acterized by volume Vi , such that:

X ¼[Ni¼1

Xi; V ¼XN

i¼1

Vi: ð23Þ

Each subset Xi belongs to a single constituent, thus it is charac-terized by homogeneous material. In any subset Xi the inelasticstrain field pi(x, t) , which accounts for all the inelastic phenomenaoccurring in the ith subset, is assumed to be represented as a linearcombination of given analytical functions, called modes, governedby the spatial variable x:

piðx; tÞ ¼XMi

k¼1

pikðtÞli

kðxÞ; ð24Þ

with pikðtÞ the inelastic contribution corresponding to li

kðxÞ, i.e., thekth mode of the ith subset, and Mi the number of active modes ofthe ith subset.

It can be noted that the NUTFA method, proposed by Michel andSuquet (2003), adopted a nonuniform distribution of the inelasticstrains over the whole material phase and not in each subset.

In Eq. (24) the inelastic contributions pikðtÞ (i = 1, . . . , N and

k = 1, . . . , Mi) are the internal tensorial variables of the problem,which are functions of the time, while the modes li

kðxÞ are scalaranalytical functions, regular enough and able to represent a basisfor the inelastic strain field. It can be remarked that the inelastic

modes, which have to be able to provide an approximation of theinelastic strains, are chosen linearly independent and they do nothave to satisfy a priori any other special requirement.

The proposed representation form (24), suggested by the classi-cal approximation of tensorial fields used, for instance, in the finiteelement formulations, is different from the one available in litera-ture (Michel and Suquet, 2003). In fact, in the representation formproposed by Michel and Suquet, the internal variables were scalarfunctions while the inelastic modes were tensors, able to approxi-mate the inelastic deformation in each phase and they were notarbitrarily chosen but determined by nonlinear preanalyses, satis-fying restrictive assumptions (i.e., incompressibility, orthogonality,etc.).

In order to simplify the notation, in the following the depen-dence from the time variable in the inelastic contributions is omit-ted, i.e., pi

kðtÞ ¼ pik.

On the basis of the discretization of the RVE in N subsets Xi, asexpressed in formula (23), and of the representation form (24) forthe inelastic strain, the local strains, given by Eq. (12) evaluated inthe subset Xj, take the form:

ej�eðxÞ ¼ AðxÞ�e; ej

pi ðxÞ ¼XMi

k¼1

HjikðxÞp

ik; ð25Þ

with ej�eðxÞ due to the presence of the effective strain �e and ej

pi ðxÞ dueto the presence of the inelastic strain pi(y) acting in the subset Xi. InEq. (25) Hji

kðxÞ is the fourth-order TFA influence tensor able to givethe strain at the point x e Xj when the mode li

k acts in the subsetXi, resulting as:

HjikðxÞ ¼

ZXi

Q jðx; yÞlikðyÞdVy; ð26Þ

with Qj(x, y) the restriction of the operator Q(x, y) into the subsetXj. Taking into account formula (11), because of the expressions(25), the total strain at the typical point x of the jth subset is ob-tained as:

ejðxÞ ¼ AðxÞ�eþXN

i¼1

XMi

k¼1

HjikðxÞp

ik: ð27Þ

The stress in the subset Xj is given by formulas (15) and (16):

rjðxÞ ¼ rj�eðxÞ þ rj

pðxÞ; ð28Þ

with:

rj�eðxÞ ¼ CjAðxÞ�e;

rjpðxÞ ¼ Cj

XN

i¼1

XMi

k¼1

½HjikðxÞ � djili

kðxÞI�pik;

ð29Þ

being Cj the elastic tensor of the material composing the subset Xj

and dji = 1 if i = j, dji = 0 otherwise. Thus, the overall average stress �ris obtained by formulas (17) and (18); in particular, the expression(18)2 of the stress �rp becomes:

�rp ¼XN

j¼1

CjXN

i¼1

XMi

k¼1

�Sjik pi

k

!; ð30Þ

with

�Sjik ¼

1V

ZXj½Hji

kðxÞ � djilikðxÞI�dVx: ð31Þ

Moreover, Eq. (19), defining the overall elastic tensor, takes thespecific form:

�C ¼ 1V

XN

j¼1

Cj �AjV j; where �Aj ¼ 1

Vj

ZXj

AðxÞdVx: ð32Þ

V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742 729

Finally, because of Eqs. (30)–(32), the explicit form of the over-all average stress �r, defined by Eq. (17), is:

�r ¼ �C�eþXN

j¼1

CjXN

i¼1

XMi

k¼1

�Sjik pi

k: ð33Þ

Formula (21), giving the overall inelastic strain �p, becomes inexplicit form:

�p ¼ ��C�1XN

j¼1

CjXN

i¼1

XMi

k¼1

�Sjik pi

k: ð34Þ

If the evolution of the inelastic strain p occurs, the rate of theoverall inelastic strain �p is governed by Eq. (22), which becomes:

_�p ¼ ��C�1XN

j¼1

CjXN

i¼1

XMi

k¼1

�Sjik

_pik: ð35Þ

It can be pointed out that the developed analysis allows to de-

rive the linear operators A(x) and HjikðxÞ that give the total strain in

a point (see Eq. (27)) when applied on �e and pik , respectively, and

the operators �C and �Sjik that give the overall average stress (see Eq.

(33)) when applied on �e and pik, respectively.

Formula (34) gives the operator that applied on the inelasticstrain pi

k allows to compute the overall inelastic strain �p.It can be remarked that the initial assumption considering the

inelastic strain field as known is introduced only in order toevaluate the operators mentioned above. Once the operatorsare determined, this initial assumption is removed and theinelastic strain field p(x, t) can be regarded as an unknown ofthe problem whose evolution is given by Eq. (5)1. Taking intoaccount formula (24) the evolution of p(x, t) is obtained bycomputing the evolution of the tensorial coefficients pi

kðtÞ. Thisaspect is discussed in details in Section 4 for the specific adoptedconstitutive models.

3.1. Piecewise uniform TFA technique

The subdivision of the inelastic material phases into subsets,adopted in the present contribution, was already introduced forthe PWUTFA homogenization technique which, however, consid-ered only a constant approximation of the inelastic deformationfield in each subset. The PWUTFA can be considered as a simplifiedapproach of the nonuniform TFA homogenization technique pre-sented in the previous section; in fact, it can be obtained assuminga uniform inelastic strain in each subset of a constituent. Thus, itcan be recovered considering a unique and uniform active modein each subset, i.e., setting Mi = 1 and li

1ðxÞ ¼ 1 for i = 1, . . . , N.Accordingly, Eq. (24) becomes:

piðxÞ ¼ pi; ð36Þ

where the tensors pi (i = 1, . . . , N) represent the internal variables inthe RVE, uniform in Xi. Of course, all the formulas (25)–(34) assumea simplified form; in particular, the strain and the stress in a typicalpoint are given by:

ejðxÞ ¼ AðxÞ�eþXN

i¼1

HjiðxÞpi;

rjðxÞ ¼ CjAðxÞ�eþ CjXN

i¼1

½HjiðxÞ � djiI�pi;

ð37Þ

while the overall average stress and the overall inelastic strain takethe form:

�r ¼ �C�eþXN

j¼1

CjXN

i¼1

�Sji pi;

�p ¼ ��C�1XN

j¼1

CjXN

i¼1

�Sji pi:

ð38Þ

3.2. Uniform TFA technique

The Uniform TFA (UTFA) can be obtained as a particular case ofthe presented PWUTFA homogenization technique. The mainassumption made by Dvorak (1992) consists in considering thateach constituent of the RVE characterized by nonlinear responseis discretized by only one subset, where the inelastic strain is as-sumed to be uniform. In other words, the UTFA can be obtainedconsidering a number of subsets equal to the number of the con-stituents. Eqs. (37) and (38) give the local strain, the local stressand the overall response of the composite also in this case, whereN is set equal to the number of the constituents.

4. Constitutive models and numerical procedure

In the following plasticity and Shape Memory Alloy (SMA) mod-els are considered for the RVE constituents.

4.1. Plasticity model with isotropic/kinematic hardening

A well-known plasticity model with combined linear isotropic/kinematic hardening is briefly described (Simo and Hughes, 1998).The thermodynamic internal variables chosen for the modeling arethe plastic strain p and the accumulated plastic strain, denotedby the scalar variable j. Thus, the inelastic energy that accountsfor the hardening effects takes the form:

Wpðp; jÞ ¼12

Hp : pþ 12

Kj2; ð39Þ

where H and K are the kinematic and isotropic hardening parame-ters, respectively. According to the state equations (3), the thermo-dynamic forces associated to the internal variables are:

Xp ¼ r� a Xk ¼ �Kj ð40Þ

with the quantity a ¼ Hp representing the back stress.Adopting the Mises criterion, the limit function depending on

the thermodynamic forces is introduced:

fpðXp;XjÞ ¼ffiffiffi32

rkdevðr� HpÞk � ðry þ KjÞ 6 0; ð41Þ

where ry is the yield stress. Through the evolutive laws theincrements of the plastic strain _p and of the accumulated plasticstrain _j can be assessed:

_p ¼ _fp@fp

@Xp; _j ¼ _fp

@fp

@Xj; ð42Þ

where _fp is the plastic multiplier. The classical Kuhn-Tucker condi-tions are introduced, completing the model:

_fp P 0; f p 6 0; _fpfp ¼ 0: ð43Þ

The plastic multiplier is determined from the consistency con-dition _fp

_f p ¼ 0, as:

_fp ¼@fp

@Xp: C _e

@fp

@Xp: ðCþ HÞ @fp

@Xpþ @fp

@Xj� @fp

@Xj

: ð44Þ

730 V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742

Hence, the plastic strain increment assumes the form:

_p ¼ K _e; K ¼ð @fp

@Xp� @fp

@XpÞC

@fp@Xp

: ðCþ HÞ @fp

@Xpþ @fp

@Xj� @fp@Xj

: ð45Þ

4.2. SMA model

The SMA model, proposed in Souza et al. (1998), Auricchio andPetrini (2004), Evangelista et al. (2009) and Evangelista et al.(2010), is adopted. The model assumes the transformation strain,which is the inelastic strain p, as the internal variable which de-scribes the strain associated to the phase transformation and, inparticular, to the conversion from austenite or multivariant mar-tensite to single-variant martensite. The norm of p is constrainedto satisfy the inequalities 0 6 kpk 6 eL, where eL is a materialparameter indicating the maximum transformation strain reachedat the end of the conversion from austenite or multivariant mar-tensite to single-variant martensite, during a uniaxial test.

The inelastic potential in the SMA constituent is set as proposedin Evangelista et al. (2009) and it depends on the temperature andon the internal variable p:

Wpðp; TÞ ¼ ½~hðxÞbt þ ð1� ~hðxÞÞbc�hT �Mf ikpk þ hkpk2

þ ðu0 � Ts0Þ þ c ðT � T0Þ � T lnTT0

� �þ IeL ðpÞ; ð46Þ

where x = tr(e) and the function ~hðxÞ is equal to 1 if x > 0, equal to 0if x 6 0; bt and bc are material parameters linked to the dependenceof the transformation stress threshold on the temperature in tensionand in compression, respectively; Mf represents the finishing temper-ature of the austenite-martensite phase transformation evaluated ata stress free state; the symbol h � i indicates the positive part of theargument; h is a material parameter associated to the slope of the lin-ear relation ruling the value of the transformation stress thresholdwith the temperature in the uniaxial case; c is the heat capacity; u0

and s0 are the internal energy and the entropy, respectively, at the ref-erence state characterized by a temperature T0 below which notwinned martensite is observed. The indicator function IeL ðpÞ impliesthe fulfillment of the constraint condition on the transformationstrain norm: IeL ðpÞ ¼ 0 if kpk 6 eL, i.e., when the phase transforma-tion is occurring, and IeL ðpÞ ¼ þ1 if kpk > eL, i.e., when the phasetransformation from austenite or multivariant martensite is com-plete. Accordingly, the norm of this quantity should be bounded be-tween zero for the case of a material without oriented martensite andthe maximum value eL for the case in which the material is fully trans-formed in single-variant oriented martensite.

The thermodynamic force associated to the transformationstrain is obtained from (3)3:

Xp ¼ r� a; ð47Þ

with a, which plays a role similar to the back stress in the plasticitytheory with kinematic hardening, defined as:

a ¼ ½ð~hðxÞbt þ ð1� ~hðxÞÞbcÞhT �Mf i þ hkpk þ c� @kpk@p

; ð48Þ

linearly dependent on the temperature. In Eq. (48) c belongs to thesubdifferential of the indicator function IeL ðpÞ and it is defined asc = 0 if 0 6 kpk < eL and c P 0 if kpk = eL.

The yield function is assumed to depend on the deviatoric partof the thermodynamic force and it is introduced as:

fsðdevðXpÞÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2J2ðdevðXpÞÞ

q� R; R ¼ 2

ffiffiffi23

rrcrt

rc þ rt; ð49Þ

where J2 = 1/2(dev(Xp))2:I is the second invariant of the deviatoricpart of the associate variable Xp and R represents the radius of

the elastic domain in the deviatoric space, with rt and rc the uniax-ial critical stresses in tension and in compression, respectively, eval-uated at T 6 Mf . It can be outlined that R can also be interpreted as aharmonic mean of the critical stress determined in uniaxial tension/compression and it allows to reproduce the different behavior intension and compression of the shape memory alloys. The equa-tions describing the associative normality rule for the internal var-iable p, completed with the Kuhn-Tucker conditions, are:

_p ¼ _fs@fs

@Xp; _fs P 0; f s 6 0; _fsfs ¼ 0; ð50Þ

with _fs the plastic multiplier.

4.3. Numerical procedure

The so-called Voigt notation is adopted, so that second ordertensors are represented as vectors and fourth order tensors asmatrices. In particular, the strains and the stresses are reportedas vectors with 6 components, while symmetric 6�6 matrices de-fine the constitutive matrices. This notation is preferred as it al-lows a straightforward implementation in a numerical code. Inthe following, the quantities in Voigt notations are denoted withthe same symbols adopted for tensor notations; this abuse simpli-fies the description of the numerical procedure, avoiding misun-derstanding and the introduction of further symbols.

For the next computational developments, the reduced vari-ables are introduced as suggested in Michel and Suquet (2004).In particular, the reduced form of the total strain (27) with respect

to the mode ljl , defined in the jth subset, is the vector obtained by

the following expression:

ejl ¼ aj

l�eþ

XN

i¼1

XMi

k¼1

Djilkp

ik; ð51Þ

where the reduced localization matrices are defined as:

ajl ¼

1V

ZXlj

lðxÞAðxÞdV ; Djilk ¼

1V

ZXlj

lðxÞHjikðxÞdV : ð52Þ

Two different approaches could be developed to solve theevolutive problem in order to determine the inelastic strainpj(x) in each subset Xj, i.e., the inelastic strain contributions

pjk with k = 1, . . . , Mj. One approach is based on the reformula-

tion of the evolutive equations in terms of reduced variablesdefined in Eqs. (51) and (52), as proposed in Michel and Suquet(2003), Fritzen and Böhlke (2010) and Fritzen and Böhlke(2011). A further different approach considers the evolutiveequations in the classical continuum framework and usesthe reduced variables to evaluate the best approximation ofthe inelastic field in each subset of the RVE, with respectto the representation form (24). In the following, this newapproach is developed.

In order to recover the evolutive laws for the inelastic straincontributions, the total strain, given by Eq. (27), is represented inthe same form adopted for the inelastic strain (24), as:

ejðxÞ ¼XMj

k¼1

ejkl

jkðxÞ; ð53Þ

being ejk the vector of the total strain contributions corresponding to

the mode ljkðxÞ in the jth subset. Eq. (53) is used in the following

only for the determination of an approximated form of the evolutiveequations of the inelastic strain; of course, the representation form(53) is not adopted in the compatibility (8), the constitutive (7) andthe equilibrium equations (9).

Computing the reduced form of the total strain given by Eq.(53), with respect to mode lj

l , it results:

V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742 731

ejl ¼

1V

XMj

k¼1

ejk

ZXlj

lðxÞljkðxÞdV ¼

XMj

k¼1

Mjlk ej

k; ð54Þ

where

Mjlk ¼

1V

IZ

Xlj

lðxÞljkðxÞdV ; ð55Þ

with I the identity matrix. Introducing the influence matrix Mj andthe vectors ej and ej as:

Mj ¼

Mj11 Mj

12 � � Mj1Mj

Mj21 Mj

22 � � Mj2Mj

��

MjMj1

MjMj2

MjMjMj

266666664

377777775; ej ¼

ej1

ej2

��

ejMj

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

9>>>>>>>=>>>>>>>;; ej ¼

ej1

ej2

��

ejMj

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

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

ð56Þ

Eq. (54) can be rewritten in the inverse form as:

ej ¼ ðMjÞ�1ej; ð57Þ

able to evaluate the vector ej containing the contributions ejk , which

defines the total strain in the subset Xj, according to Eq. (53). Notethat the matrix Mj results not singular if the selected modes,adopted to represent the inelastic fields, are linearly independent.Furthermore, if the mode functions satisfy the conditionR

X ljlðxÞl

jkðxÞdV ¼ 0 when l – k, that represents an orthogonality

condition between the functions ljlðxÞ and lj

kðxÞ, the formula (55)gives Mj

lk ¼ 0 when l – k, so that the matrix Mj becomes diagonal;moreover, when the condition

RX lj

lðxÞljlðxÞdV ¼ 1, that represents

a normality condition, is also satisfied, it results ej ¼ ej.Before approaching the nonlinear evolutive problem, the re-

duced localization matrices ajl and Dji

lk, defined in Eqs. (52), andthe influence matrices Mj given by Eq. (56), have to be computed.In fact, the reduced localization and influence matrices are nottime-dependent. In particular, the matrices aj

l and Djilk are evaluated

solving linear elastic problems, each one characterized by the RVEsubjected to only one nonzero component of:

– �e, in order to obtain the components of the localization matricesaj

l reduced with respect to the mode ljl (with l = 1, . . . , Mj) in

each subset Xj (with j = 1, . . . , N);– pi

k (with i = 1, . . . , N and k = 1, . . . , Mi), in order to derive thecomponents of the localization matrices Dji

lk reduced withrespect to the mode lj

l (with l = 1, . . . , Mj) in each subset Xj

(with j = 1, . . . , N).

Once the elastic pre-analyses are performed, the evolutive non-linear problem is approached adopting a backward Euler integra-tion scheme, with a return map technique to solve the nonlineartime step. The procedure is strain driven, i.e., the average strain �eis assigned at each time step. In particular, once the solution atthe previous time step tn is determined, the solution at the currentstep tn+1 is evaluated using an implicit integration procedure. Here-inafter, all the quantities related to the previous time step are de-noted with the subscript ‘|n’, while the ones referred to the currenttime step are indicated with no subscript.

The increment of the inelastic strain contributions Dpjk, with

k = 1, . . . , Mj, into a subset Xj, with j = 1, . . . , N, is evaluated devel-oping the following predictor-corrector procedure.

In the subset Xj, the trial solution is determined setting all theinternal variables equal to the ones corresponding to the previoustime step tn, e.g. pj

k ¼ pjk jn (with j = 1, . . . , N), so that the reduced

trial strain contributions ejðtrialÞl (with l = 1, . . . , Mj) are obtained

by formula (51) and then assembled in the vector ejðtrialÞ. The vector

ej(trial) is given by expression (57), from which the vectors ejðtrialÞk are

extracted.Then, at any point Pj of Xj, individuated by the position vector

xj, the inelastic strain increment Dpj(xj) is determined. In fact, fromEq. (53), the trial strain at Pj results:

ejðtrialÞðxjÞ ¼XMj

k¼1

ejðtrialÞk lj

kðxjÞ: ð58Þ

The trial stress is computed by Eq. (7):

rjðtrialÞðxjÞ ¼XMj

k¼1

Cj½ejðtrialÞk � pjðtrialÞ

k �ljkðx

jÞ: ð59Þ

The trial yield function f ðtrialÞ� ðxjÞ for plasticity (⁄ = p) or SMA

(⁄ = s) constitutive equation can be evaluated from formula (41),(49), respectively. If f ðtrialÞ

� ðxjÞ < 0 the step is elastic and the trialquantities are the solution at Pj, so that Dpj(xj) = 0. Otherwise,the correction phase is performed allowing the computation ofDpj(xj) for the plastic and SMA model. For a sake of brevity, thespecific algorithms characterizing the corrector phase are not re-ported herein, as they are described in details in Simo and Hughes(1998) and in Auricchio and Petrini (2004), Evangelista et al.(2009). Finally, the new inelastic strain vector is determined at Pj

as:

pjðxjÞ ¼XMj

k¼1

pjðtrialÞk lj

kðxjÞ þ DpjðxjÞ: ð60Þ

Once the distribution of Dpj(xj) is determined in Xj, an approx-imation of the increment of the inelastic strains Dpj

k, withk = 1, . . . , Mj, is evaluated minimizing the following error function:

EðDpj1;Dpj

2; . . . ;DpjMj Þ ¼

ZXj

DpjðxjÞ �XMj

k¼1

Dpjkl

jkðx

j�����

�����dVx: ð61Þ

Minimizing the error function E with respect to all the incre-ments of the inelastic strains, the quantities Dpj

1;Dpj2; . . . ;Dpj

Mj

are obtained and, thus, the updated values of the inelastic strainspj

1;pj2; . . . ;pj

Mj are computed.With the values of the inelastic strain contributions determined

at the iteration r, i.e., pjðrÞ1 ;pjðrÞ

2 ; . . . ;pjðrÞMj , a new trial state is com-

puted and a new correction, if needed, is performed, leading tonew values for the inelastic strain contributionspjðrþ1Þ

1 ;pjðrþ1Þ2 ; . . . ;pjðrþ1Þ

Mj . The iterative procedure goes on until theresidual res is lower than a prefixed tolerance, with

res ¼XN

j¼1

XMj

k¼1

kpjðrþ1Þk � pjðrÞ

k k : ð62Þ

In the present work, the reduced localization matrices ajl and

Djilk, defined in Eqs. (52), and the influence matrices Mj, given by

Eq. (56), are evaluated performing linear elastic finite elementanalyses. Thus, the integral in Eq. (61) is evaluated according tothe Gauss quadrature as:

EðDpj1;Dpj

2; . . . ;DpjMj Þ ¼

Xnj

e¼1

Xng

g¼1

wl DpjðxjgÞ �

XMj

k¼1

Dpjkl

jkðx

jgÞ

����������; ð63Þ

where nj is the number of finite elements included in the sub-domain Xj, ng is the number of Gauss quadrature points in eachelement and wl is the Gauss weight. Of course, other techniquescan be adopted to evaluate the matrices aj

l , Djilk and Mj, and the error

function. The flow-chart of the numerical procedure for the evalua-tion of the increments of the inelastic strains is given in Table 1.

Table 1Flow-chart of the numerical procedure for the evaluation of the increments of theinelastic strains.

� Perform the linear elastic pre-analyses to evaluate ajl , Dji

lk and Mj withi, j = 1, . . . , N, l = 1, . . . , Mj and k = 1, . . . , Mi;

� Time step [tn, t]s Assign �es While res > tol do iteration r

j For each Xj with j = 1, . . . , N

� set pjk ¼ pj

k jn with k = 1, . . . , Mj

� evaluate ejðtrialÞl (with l = 1, . . . , Mj), Eq. (51)

� assemble ejðtrialÞl in the vector ejðtrialÞ

� determine ej(trial), Eq. (57)

� extract ejðtrialÞk k = 1, . . . , Mj, from ej(trial)

� set E = 0� for e = 1 to nj

s for g = 1 to ng

j evaluate ejðtrialÞðxjgÞ, Eq. (58)

j compute rjðtrialÞðxjgÞ, Eq. (59)

j determine f ðtrialÞ� ðxj

gÞj if f ðtrialÞ

� ðxjgÞ < 0! Dpjðxj

gÞ ¼ 0

j if f ðtrialÞ� ðxj

gÞP 0 ! DpjðxjgÞ–0

j update E, Eq. (63)s next g� next e

�minimize E with respect to Dpj1;Dpj

2; . . . ;DpjMj

� update pj1;p

j2; . . . ;pj

Mj

j next js evaluate res, Eq. (62)� next time step

732 V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742

5. Numerical applications

5.1. Periodic composites

The nonlinear homogenization procedure, presented in the pre-vious section, is adopted to study the overall mechanical responseof periodic composite media. The periodic microstructure of theconsidered materials allows to analyze a repetitive unit cell (UC)subjected to suitable boundary conditions in order to determinethe homogenized overall behavior of the whole composite.

In the following, the UC is denoted as X, since it plays for peri-odic media the equivalent role of the RVE for random media. Thediscussion is limited to the case of 2D problems.

For periodic media, introducing a Cartesian reference systemðO;x1; x2Þ in the UC, the displacement field u ¼ fu1 u2 gT in thetypical point x ¼ f x1 x2 gT of the unit cell is expressed by the fol-lowing representation form:

u1 ¼ �e11x1 þ 12

�c12x2 þ ~u1;

u2 ¼ 12

�c12x1 þ �e22x2 þ ~u2;ð64Þ

where �e11, �e22 and �c12 are the components of the effective strain vec-tor �e acting on the unit cell, ~u1 and ~u2 are the components of thevector ~uðxÞ that represents the periodic part of the displacement.From formula (64), the strain vector is given by:

eðxÞ ¼ �eþ ~eðxÞ; ð65Þ

where ~eðxÞ is the periodic part of the strain, with null average in X,associated to the displacement ~uðxÞ.

The classical periodicity and continuity conditions are pre-scribed to the displacement field (Suquet, 1987; Luciano and Sacco,1998).

5.2. Numerical results

Periodic composites made of elastic, plastic and SMA constitu-ents are studied in the framework of 2D plane strain analysis. Lin-ear elastic pre-analyses are performed to evaluate the reducedlocalization matrices aj

l and Djilk and the influence matrices Mj,

using 2D four-node finite elements. Furthermore, nonlinear micro-mechanical analyses (FEA) are carried out implementing 2D planestrain four-node quadrilateral finite elements, characterized byelastic, plastic and SMA constitutive models, in the code FEAP (Tay-lor, 2008).

In order to verify the effectiveness of the proposed homogeniza-tion technique two different composite materials are studied, bothcomposed by only two constituents. Each composite material ischaracterized by a specific UC, which is studied using the threetypes of homogenization techniques (UTFA, PWUTFA, NUTFA) de-scribed in the previous section. In particular, when one of thetwo constituents, occupying a set Xe X, is characterized by linearelastic behavior, the inelastic strain can be different from zero onlyin the complementary part of the UC, denoted as ~X ¼ X=Xe.

The results obtained using the UTFA, PWUTFA and NUTFA arecompared with the ones carried out from the micromechanicalanalysis. In particular, the considered UCs are analyzed accordingto the following approaches:

1. UTFA: the ith constituent of the whole UC, characterized bynonlinear behavior, is considered as a subset, in which theinelastic strain is assumed uniform.

2. PWUTFA: each constituent of the whole UC, characterized bynonlinear behavior, is subdivided in several subsets; in the typ-ical ith subset, the inelastic strain is assumed uniform.

3. NUTFA: each constituent of the whole UC, characterized bynonlinear behavior, is subdivided in several subsets, in the typ-ical ith subset, the inelastic strain is assumed nonuniform.

The modes lik forming a basis for the representation of the

inelastic strain in each subset ith are chosen. In particular, linearlyindependent polynomial functions are adopted to approximate theinelastic strain in the following applications.

The prescribed modes in the UC reference system ðO;x1; x2Þ arechosen as:

1. UTFA: Mi = 1 and li1 ¼ 1.

2. PWUTFA: Mi = 1 and li1 ¼ 1.

3. NUTFA: Mi = 3 and li1 ¼ 1, li

2 ¼ x1, li3 ¼ x2.

It can be remarked that the chosen modes for the NUTFA are lin-early independent and do not satisfy the orthogonality and nor-mality conditions for the subsets adopted in the applicationsillustrated in the following.

It is worth noting that different coordinate system could beintroduced in each subsets. The chosen modes can satisfy the con-dition of orthogonality with a suitable choice of these local refer-ence systems. In particular, if a local reference system ðO;xi

1; xi2Þ

with the origin in the center of the area is introduced in the ith sub-set (with i = 1, . . . , N) the orthogonality conditions between thechosen modes (li

1 ¼ 1, li2 ¼ xi

1, li3 ¼ xi

2) are:

ZXi

1xi1dVx ¼ 0;

ZXi

1xi2dVx ¼ 0;

ZXi

xi1 xi

2dVx ¼ 0: ð66Þ

Conditions (66)1 and (66)2 are trivially satisfied as they repre-sent the static moment of area of the ith subset, while the principalinertia axes should be chosen as reference axes in order to fulfillcondition (66)3.

V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742 733

The homogenization results are compared with the FEA micro-mechanical ones, in order to show the effectiveness of the pro-posed NUTFA technique and to demonstrate the improvedperformances with respect to the UTFA and PWUTFA procedures.

In the following applications, the CPU times (evaluated on apersonal computer with the following characteristics: Intel Core2Duo CPU P8400 @ 2.26Ghz 2.27GHz, 3.00GB RAM) are computedfor the performed homogenization and nonlinear micromechanicalanalyses. It can be pointed out that the computational times for thepre-analyses are very short and they are included in the CPU timesreported for the homogenization analyses.

Table 3

5.3. Plastic matrix and elastic inclusion

The first application aims to determine the overall mechanicalresponse of a periodic heterogeneous material made of a plasticmatrix, represented by ~X composed of N plastic subsets, and byan elastic inclusion, denoted by the elastic subset Xe.

The UC geometry is shown in Fig. 1; the geometrical parametersare a1 ¼ 15 mm and a2 ¼ 9 mm, a unit thickness is considered andthe inclusion volume fraction is set equal to 15%. In particular thedimensions of the inclusion are 5 mm along x1-direction and 4 mmalong x2-direction.

The constituent material properties are defined in Table 2, indi-cating with the symbols E and m the Young modulus and the Pois-son ratio, respectively, and being ry the yield stress and K thelinear isotropic hardening parameter of the plastic matrix. Obvi-ously, the same material properties are adopted for each inelasticsubset composing ~X.

The UC is discretized with a regular mesh composed by 128 ele-ments for the matrix and 16 elements for the inclusion. This dis-cretization is adopted both for the elastic pre-analyses and forthe nonlinear micromechanical analysis.

Characteristics for each homogenization analysis.

Type ofanalysis

No. ofsubsets

No. of activemodes

No. of scalar internalvariables

UTFA 1 1 3PWUTFA (i) 3 1 9PWUTFA (ii) 12 1 36

5.3.1. Loading along the x1-axisThe response of the UC subjected to an average strain history �e11

is investigated. In particular, a monotonic increasing value of theaverage strain is prescribed until the value �e11 ¼ 0:005 is reached.

Fig. 1. Scheme of the periodic UC made of plastic matrix with elastic inclusion.

Table 2Material properties for the constituents of the first UC.

Elastic material E ¼ 25000 MPa m = 0.15Plastic material E ¼ 2500MPa m = 0.15

ry ¼ 3 MPa K ¼ 1 MPa

The homogenization analyses are developed considering for thematrix, which represents the inelastic part ~X, the followingchoices:

UTFA: one subset, N = 1 with Mi = 1;

PWUTFA: (i) three subsets, N = 3 with Mi = 1,

(ii) twelve subsets, N = 12 with Mi = 1;

NUTFA:(i) one subset, N = 1 with Mi = 3,(ii) four subsets, N = 4 with Mi = 3.

The number of subsets, of active inelastic modes and of scalarinternal variables, i.e., the number of components of the coeffi-cients pi

k, of each homogenization analysis, are summarized inTable 3.

Accordingly, the four different schemes reported in Fig. 2 areemployed for the described analyses. In particular, the scheme(a) is adopted for UTFA and NUTFA(i); the scheme (b) for PWUT-FA(i); the scheme (c) for NUTFA(ii); the scheme (d) for PWUTFA(ii).

It is worth noting that the analyses NUTFA(i) and PWUTFA(i) arecharacterized by the same computational burden, because thesame number of history variables has to be stored for both cases.The same consideration can be made for the analyses NUTFA(ii)and PWUTFA(ii) since they are both characterized by the samenumber of internal variables.

The mechanical responses of the UC, in terms of the averagestress �r11 versus the average strain �e11, obtained carrying out thefive homogenization analyses, defined above, are compared withthe one derived by performing a nonlinear finite element microme-chanical analysis.

NUTFA (i) 1 3 9NUTFA (ii) 4 3 36

Fig. 2. UC made of: (a) one inelastic subset, (b) three inelastic subsets, (c) fourinelastic subsets, (d) twelve inelastic subsets.

Fig. 3. UC mechanical response adopting the homogenization techniques and the micromechanical analysis.

734 V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742

From Fig. 3 some observations can be remarked:

– the results achieved by the UTFA are significantly differentfrom the micromechanical ones and, moreover, the UTFA solutionis the worst approximation of the FEA results;– as expected, the increasing of the number of the plastic subsets,

representing the matrix ~X, leads to an improvement of the homog-enized results, both for the PWUTFA and the NUTFA;– the comparisons between the analyses characterized by the

same computational burden, i.e., NUTFA(i) with PWUTFA(i) andNUTFA(ii) with PWUTFA(ii), clearly show that the proposed NUTFAgives better results than the PWUTFA. The nonuniform homogeni-zation analyses provide responses that are in very good agreementwith the FEA micromechanical results;– the results accessed by the NUTFA(i) are even better than the

ones corresponding to the PWUTFA(ii), although the PWUTFA(ii)is characterized by a larger computational effort than the NUTFA(i).This interesting result further highlights the capabilities of the pro-posed NUTFA with respect to the PWUTFA.

In order to better appreciate the differences between the devel-oped homogenization analyses with respect to the micromechani-cal one, the final values of the average stress �r11, achieved for�e11 ¼ 0:005 and the computational times required by the differentanalyses, are compared in Table 4. The estimation of the error ismade with respect to the value of the average stress�r11 ¼ 2:83MPa assessed by the FEA analysis for the same valueof the prescribed tensile strain �e11 ¼ 0:005.

Table 4Comparison of the average stress and of the CPU times for the different analyses.

Analysis �r11 (MPa) Error (%) CPU times (s)

FEA 2.83 – 200UTFA 4.62 63.25 5PWUTFA(i) 3.69 30.39 11PWUTFA(ii) 3.61 27.56 25NUTFA(i) 2.93 3.53 10NUTFA(ii) 2.89 2.12 24

For what concerns the CPU times, it can be highlighted that theUTFA analysis completes in only 5 s giving the larger error value,while both the NUTFA analyses, which provide the best results,are characterized by CPU times of 10 s and 24 s (1 s less than thecomputation times of the corresponding PWUTFA analyses).

The CPU times show the advantage to use the homogenizationtechniques instead of solving the nonlinear micromechanicalproblem.

5.3.2. Biaxial tensile loadingThe case of a biaxial tensile loading is also performed and an

increasing value of the average strains �e11 and �e22 is prescribed inthe UC until the value �e11 ¼ �e22 ¼ 0:005 is reached, keeping the ra-tio �e11=�e22 ¼ 1 during the whole analysis. In this case, the UTFA, theNUTFA(ii) and the PWUTFA(ii) analyses, introduced in the previoussubsection, are performed, considering the schemes represented inFig. 2(a), Fig. 2(c) and Fig. 2(d), respectively. The results obtainedby the homogenization analyses are compared with the FEA ones.In Fig. 4(a) and in Fig. 4(b) the mechanical response of the UC isrepresented in terms of average normal stress �r11 and �r22 versusaverage normal strain �e11 and �e22, respectively.

The comparison between the different analyses clearly puts inevidence the NUTFA ability of reproducing results in very goodagreement with the FEA solution along both the directions. Onthe contrary the results obtained by the PWUTFA provide a goodapproximation of the micromechanical response along x2-direc-tion, while they are significantly different from the FEA solutionalong x1-direction. The different PWUTFA estimates along the x1

and x2 directions are due to the choice for the shapes and volumesof the plastic subsets, which are smaller and more numerous alongx2 while they are coarser along x1. It can be stressed that the spe-cific subdivision, adopted in the inelastic region, doesn’t affect sig-nificantly the NUTFA response, which is less influenced by theshapes of the subsets.

Also for the biaxial loading history, the UTFA analysis providesthe worst approximation of the micromechanical solution alongboth the directions.

These remarks can be further highlighted through the compar-ison between the final values of the average stresses �r11 and �r22

obtained performing the UTFA, PWUTFA(ii) and NUTFA(ii) analyses

Fig. 4. UC mechanical response along x1-direction and x2-direction for a biaxial test.

Table 5Comparison of the average stress along x1-direction and x2-direction and of the CPUtimes for the different analyses.

Analysis �r11 (MPa) Error (%) �r22 [MPa] Error (%) CPU times (s)

FEA 2.53 – 2.51 – 225UTFA 4.42 74.70 4.46 76.28 5PWUTFA(ii) 3.30 30.43 2.65 4.74 27NUTFA(ii) 2.64 4.35 2.62 4.38 26

V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742 735

for the prescribed average strain values �e11 ¼ �e22 ¼ 0:005, as re-ported in Table 5.

The errors along x1-direction and x2-direction are estimatedwith respect to the final values of the average stresses derived bythe FEA analysis, which are: �r11 ¼ 2:53 MPa and �r22 ¼ 2:51 MPa.

Table 5 also shows the CPU times related to the four differentanalyses, putting in evidence that the NUTFA(ii) analysis requiresa computation time smaller than 1 s compared to the PWUTFA(ii)case and provides the minimum value of the error.

5.4. Plastic matrix and SMA inclusion

The periodic composite material analyzed in this example con-sists of an Aluminum matrix (6061 Al) and embedded sphericalNitinol inclusions (TiNi) with the volume fraction equal to 20%.Both the matrix and the inclusions are characterized by nonlinearresponses and are divided into inelastic subsets Xi. In particularplastic subsets for the matrix ðXi

AlÞ and SMA subsets for the inclu-sions ðXi

TiNiÞ are considered. Fig. 5 illustrates the UC geometry,characterized by a unit thickness.

The properties of the Aluminum and Nitinol materials are theones specified by Lee and Taya (2004) and Birman (2010), andare reported in Table 6. In particular, the Young’s modulus forthe Nitinol inclusions is taken as the arithmetic mean of the valuesof the elastic moduli of martensite and austenite given by Lee andTaya (2004), since the SMA model adopted in this work takes intoaccount a unique modulus for both the phases.

The shear behavior of the UC of the SMA composite is investi-gated and two different applications are carried out considering

Fig. 5. Scheme of the periodic UC made of Aluminum matrix with Nitinolinclusions.

Table 6Material properties for the constituents of the second UC.

6061 Al EAl ¼ 70000 MPa mAl ¼ 0:33ry ¼ 245 MPa KAl ¼ 85 MPa

TiNi ETiNi ¼ 46650 MPa mTiNi ¼ 0:43hTiNi ¼ 1000 MPa bTiNi ¼ 2:1 MPa K�1

eL TiNi ¼ 0:04 Mf TiNi ¼ 280 Krc TiNi ¼ 72 MPa rt TiNi ¼ 56 MPa

736 V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742

for the SMA inclusions the pseudoelastic effect and the shapememory effect.

Because of the UC symmetry and having considered a shearloading, a half of the unit cell can be analyzed prescribing appropri-ate periodicity conditions to the displacement field. A mesh com-posed of 304 quadrilateral elements is adopted for the elasticpre-analyses and for the nonlinear micromechanical analysis.

In both the applications, the proposed nonuniform TFA analysisis developed considering one subset for the SMA inclusionsðXi

TiNi i ¼ 1Þ and four subsets for the plastic matrixðXi

Al i ¼ 1; . . . ;4Þ, as illustrated in Fig. 6(a).In particular, the NUTFA is performed assigning three active

modes (li1 ¼ 1, li

2 ¼ x1, and li3 ¼ x2) to the Aluminum subsets

ðXiAl i ¼ 1; . . . ;4Þ, and only one active mode (li

1 ¼ 1) to the Nitinolsubset ðXi

TiNi i ¼ 1Þ. Thus, a linear distribution of the inelastic strainfield is considered in the plastic matrix subsets along x1 and x2

directions (MiAl ¼ 3), while a uniform inelastic strain field is consid-

ered for the SMA inclusions (MiTiNi ¼ 1). The choice to approximate

the inelastic strain in the NiTi inclusions as a uniform field, simpli-fies the computations without influencing significantly the analy-sis, as the inclusions represent only the 20% of the UC.

The mechanical response of the composite, when the pseudo-elastic effect in the SMA inclusions is activated, is investigated. In

Fig. 6. UC with Nitinol and Aluminum subsets adop

fact, an increasing average shear strain �c12 is prescribed in theUC until the value 0.08 is reached at the constant temperatureT ¼ 313 K, greater than Af (temperature at which the austeniticphase is stable). Then, the prescribed strain is removed allowingthe recovery of the transformation strain in the SMA inclusions,exploiting the TiNi pseudoelasticity.

The results obtained with the NUTFA homogenization tech-nique are compared with the ones determined performing the non-linear micromechanical analysis (FEA). Fig. 7 shows the UCbehavior in terms of the average shear stress �s12 versus the averageshear strain �c12 for the two different analyses.

It can be pointed out that the proposed NUTFA is able to repro-duce SMA composite response and leads to results in very goodagreement with the micromechanical analysis.

In order to better appreciate the capability of the proposedhomogenization technique to provide a good approximation ofthe UC behavior, the averaged mechanical responses of the Alumi-num and Nitinol phases obtained by the NUTFA, are compared tothe averaged responses of each phase assessed by the FEA. Thus,in Fig. 8 the average shear stresses �s�12, with � ¼ Al; SMA, evaluatedfor the plastic matrix and for the SMA inclusions, versus the aver-age shear strains �c12 are plotted for both the NUTFA and FEA anal-yses. Fig. 8 shows that the NUTFA procedure is able to reproducethe mechanical behavior of the single material phases of the com-posite, giving results that are essentially in agreement with theones of the micromechanical analysis. It can be noted that thegreater difference is in the response of Nitinol material. In orderto investigate this mismatch the solution is improved consideringa finer approximation of the inelastic field in the area of the plasticmatrix surrounding the Nitinol inclusions. In fact, the number ofthe plastic subsets in this area is increased as represented inFig. 6(b). Thus, an improved NUTFA computation is performed con-sidering twelve Aluminum subsets with three active modes(li

1 ¼ 1, li2 ¼ x1, and li

3 ¼ x2) and one Nitinol subset with onlyone active mode (li

1 ¼ 1). The results of the improved analysis, de-noted by NUTFA+, are reported in Fig. 7 in terms of the UC overallresponse and in Fig. 8 in terms of the average response of the Alu-minum matrix and of the Nitinol inclusions.

It can be noted that for what concerns the overall response, theincrease of the number of subsets does not lead to a significantimprovement of the solution, as the results obtained with a re-duced number of subsets already give a very good approximationof the micromechanical solution as remarked above. On the con-trary, the increase of the number of subsets significantly improvesthe results only in terms of the average SMA response in the finalpart of the loading phase and in the unloading phase, as illustratedin Fig. 8. During the loading phase, when the average shear strain isabout 0.06, the austenite-martensite phase transformation is com-pleted in the inclusions and a restiffening occurs in the Nitinol. Thiseffect is smoothed in the micromechanical analysis because of theplastic behavior in the material surrounding the inclusions thatlimits the stress value in the SMA material, as a consequence of

ted for the NUTFA (a) and for the NUTFA+ (b).

Fig. 7. UC mechanical response adopting the NUTFA homogenization technique and the micromechanical analysis.

Fig. 8. Mechanical responses of the Aluminum phase and of the Nitinol phase adopting the nonuniform homogenization procedure and the nonlinear micromechanicalanalysis.

V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742 737

the equilibrium condition at the interface. On the other hand, inthe NUTFA analyses if the stress is not well computed in the mate-rial surrounding the inclusion, i.e., when a reduced discretizationin subsets of this area is adopted, the stress in the SMA is not suf-ficiently limited and it tends to increase leading to that overstiff-ness effect. In order to underline that the overall SMA response issignificantly influenced by the approximation adopted for theinelastic strain in the plastic matrix, the inelastic shear strain p12

distributions, evaluated adopting the FEA, the NUTFA and the NUT-FA+ are investigated. In particular, in Fig. 9 p12 distributions areplotted and compared at the end of the loading phase(�c12 ¼ 0:08), where the mismatch between the FEA and the homog-enization results in terms of the overall SMA response is more sig-nificant. From Fig. 9, it can be pointed out that:

– the NUTFA, the NUTFA+ and the FEA results are in very goodaccordance for what concerns the uniform inelastic strain dis-tribution in the Nitinol inclusions;

– the NUTFA+ improves the results obtained by the NUTFA forwhat concerns the distribution of the inelastic strain in the plas-tic matrix with respect to the FEA ones.

The mismatch between the NUTFA+ and the FEA results could befurther reduced by adopting a greater number of Aluminum subsetsor by choosing different modes to represent the inelastic field.

The CPU time for the NUTFA analysis is of 312 s, when the totalsubsets are five, and of 616 s, when the subsets are thirteen, whilethe computational time required by the FEA analysis is of 2563 s.The increase of the number of subsets leads to a higher computa-

Fig. 9. Inelastic shear strain evaluated by FEA (a) NUTFA (b) and NUTFA+ (c) at the end of the loading phase (�c12 ¼ 0:08).

738 V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742

tional time that in any case remains lower than the CPU time of thenonlinear micromechanical analysis.

The shape memory effect in the TiNi inclusions is also investi-gated for the considered SMA composite. For this purpose two dif-

Table 7First loading history for the analysis of the shape memory effect.

t (s) 0 1 2 3 4 5 6 7 8

�c12 0.0000 0.0300 0.0257 0.0257 0.0247 0.0247 0.0208 0.0208 0.0000T (K) 280 280 280 320 320 420 420 820 820

Fig. 10. UC mechanical response adopting the nonuniform homogenization procedure and the nonlinear micromechanical analysis.

Fig. 11. Comparison between the transformation strain average norm evaluated by the FEA and by NUTFA.

V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742 739

ferent loading histories are assigned in terms of the average shearstrain �c12 and the temperature T.

The scheme of the unit cell adopted in both the applications isthe one represented in Fig. 6(a). The first loading history is givenas reported in Table 7.

As the procedure is strain driven, the thermomechanical loadingpath is prescribed in order to keep the average shear stress �s12 al-most equal to zero during the unloading phase for 2 s < t < 7 s.

The comparison between the mechanical response of the UCachieved adopting the NUTFA homogenization procedure and theone obtained with the micromechanical analysis demonstratesthe ability of the proposed technique to capture the shape memoryeffect in a SMA composite, as illustrated in Fig. 10.

In fact, the specific loading history, with a heating process thatreaches temperatures above Af ¼ 308 K, allows the phase transfor-mation from detwinned martensite to austenite in the SMA inclu-

Table 8Second loading history for the analysis of the shape memory effect.

t (s) 0 1 2 3

�c12 0.0000 0.0300 0.0257 0.0000T (K) 280 280 280 400

740 V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742

sions. It can be noted that the results assessed by the FEA analysisshow a nearly constant distribution of the inelastic strain for theTiNi inclusions. The choice of a unique and constant active modein the SMA subset in this case seems to be effective for approximat-ing the transformation strain field, as confirmed by the good agree-ment of the UC responses obtained by the two analyses.

The complete recovery of the transformation strain in the Niti-nol material, is shown in Fig. 11, where the Euclidian norm of thetransformation strain average kpTiNik, evaluated through the two

Fig. 12. UC mechanical response adopting the nonuniform homogen

Fig. 13. Comparison between the transformation st

analyses, is plotted in function of time. Thus, the quantity kpTiNikrepresents:

– in the micromechanical analysis, the norm of the transforma-tion strain average evaluated in the Nitinol constituent as:

kpTiNik ¼ 1

Vi

ZXi

pðxÞdVx

��������

– in the NUTFA analysis, the norm of the uniform inelastic strain(as only one mode li

1 ¼ 1 is activated) obtained in the Nitinolconstituent as:

kpTiNik ¼ kpi1k:

It’s worth noting that FEA and NUTFA provide the same trend ofkpTiNik:

ization procedure and the nonlinear micromechanical analysis.

rain norm evaluated by the FEA and by NUTFA.

V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742 741

– for 0 < t < 1, during the loading phase, the austenite-martens-ite phase transformation occurs and the value kpTiNik increasesuntil a maximum value kpTiNik < eL = 0.04 is reached;

– for 1 < t < 2 the transformation strain norm kpTiNik remainsconstant for both the FEA and the NUTFA;

– for 2 < t < 6 a decreasing of the transformation strain normoccurs as a consequence of the phase transformation fromdetwinned martensite to austenite;

– for 6 < t < 7 a steep reduction of kpTiNik occurs, so that it goesrapidly to zero for both the analyses (FEA and NUTFA); thus,the martensite-austenite reverse transformation is completedin the SMA inclusions.

The NUTFA analysis is characterized by a CPU time of 293 swhile the nonlinear micromechanical analysis takes 7025 s to becompleted.

The second thermomechanical loading path is applied as shownin Table 8. Hence, at a constant temperature, T = Mf, the cell isloaded until the value �c12 ¼ 0:0300 is reached and unloaded tillthe value �c12 ¼ 0:0257 at which the average stress �s12 is almostequal to zero; then, the SMA composite is heated above Af and un-loaded until a null value of the average shear strain is reached.

The obtained results are presented in Fig. 12, in terms of aver-age shear stress �s12 versus average shear strain �c12, and they showagain a good agreement between the NUTFA and the FEA analyses.The CPU times for this loading history are 212 s for the homogeni-zation analysis and 2613 s for the micromechanical one.

The thermomechanical loading history allows the total recoveryof the transformation strain in the TiNi inclusions, by transformingback into the parent austenitic phase. From Fig. 13 it can bepointed out that the transformation strain norm kpTiNik has thesame trend in both the FEA and NUTFA analyses:– for 0 < t < 1 an increasing value of kpTiNik can be observed in the

TiNi inclusions, due to the phase transformation from austeniteto detwinned martensite;

– for 1 < t < 2 the transformation strain norm remains constant ata value lower than eL for both the FEA and NUTFA analyses;

– for 2 < t < 3 the value of kpTiNik decreases and rapidly goes tozero denoting that the transformation from detwinned mar-tensite to austenite occurs and is completed when t ¼ 3 s,because of the heating and unloading processes.

6. Conclusions

A nonuniform TFA homogenization procedure has been pro-posed in order to derive the mechanical response of compositematerials characterized by the nonlinear behavior of their constit-uents. The main idea presented in this work consists in the defini-tion of the nonuniform inelastic strain field as a linear combinationof given analytical functions, whose coefficients have to be evalu-ated solving the evolutive equations of the material constituentsof the typical RVE. The analytical functions are selected in orderto give a proper approximation, i.e., a basis of representation, ofthe inelastic strain field, without any other special requirement.In the framework of the proposed formulation it has been also pos-sible to derive the uniform and piecewise uniform TFA techniquesas particular and simplified cases. Plasticity and shape memoryalloy models have been considered for the RVE constituents char-acterized by nonlinear response.

Numerical applications have been performed for the case ofperiodic composites. In the first example a unit cell made of aplastic matrix with an elastic inclusion has been considered.The numerical results have remarked the capability of theproposed NUTFA technique to provide results in very goodagreement with the micromechanical ones. Furthermore, thecomparison with the UTFA and PWUTFA procedures has

highlighted the enhancement introduced with the new formula-tion with respect to the previous ones, as it approximates themicromechanical solution better than the UTFA and PWUTFAhomogenization analyses. In the second application, the mechan-ical response of a UC consisting of an Aluminum matrix (6061Al) and spherical Nitinol inclusions (TiNi) has been investigated.The comparison with the micromechanical results has shown theability of the presented procedure to well capture the overallbehavior of the special composite, correctly reproducing thekey features of the shape memory alloys: the pseudoelasticand the shape memory effects.

The proposed nonuniform TFA homogenization techniqueappears accurate and effective in the resolution of the microme-chanical problem and in the evaluation of the overall nonlinearresponse of different heterogeneous media. Thus, the presentedprocedure can be regarded as an efficient, i.e., simple and accurate,numerical tool for the design and the tailoring of innovative com-posite materials for which the nonlinear effects can play significantrole. Moreover, it can be remarked that the developed technique isready to be implemented at the Gauss point level in a finiteelement code in order to perform multiscale analyses for structuresmade of nonlinear composite materials.

An interesting issue could be the comparison between the re-sults obtained by the proposed NUTFA and the ones accessed bythe ‘original’ NUTFA, first developed by Michel and Suquet (2003)and then modified and adopted in numerous works (Roussetteet al., 2009, Michel and Suquet, 2009; Fritzen and Böhlke, 2010,2011, among the others). A significant difference is the approxima-tion used for the inelastic strain field. The proposed techniqueadopts a nonuniform distribution of the inelastic strain in eachsubset while the ‘original’ NUTFA considers a nonuniform inelasticstrain over the whole material phase. Furthermore, in the presentwork the choice of the modes (i.e., of the approximation functions)does not require any nonlinear micromechanical preanalysis of theRVE as in the original method and hence it does not require thedetermination of the necessary number, direction, versus and en-tity of the pre-analyses. Another significant difference betweenthe two NUTFA approaches consists in the way to recover the evo-lution of the coefficients of the approximation form adopted for theinelastic strain.

The expected results from the ‘original’ NUTFA could also bebetter than the ones obtained from the proposed method, mainlywhen the RVE is subjected to monotone loading histories, closeto the ones used in the pre-analyses. But the proposed procedurecould guarantee satisfactory results in any case even for cyclicloading and for materials characterized by significantly differentresponse in tension and in compression.

Numerical comparisons between the two NUTFAs are not easyto perform as they would depend on the specific examined case,i.e., with respect to the proposed technique the original one couldprovide better results in a case and worse in another one.

For this reason no comparison between the two NUTFA tech-niques has been presented, but it could be interesting to performsome comparisons in the future developing Round Robin Tests incollaboration with researchers that use the ‘original’ NUTFA inorder to verify the main advantages and disadvantages of thetwo techniques.

Acknowledgements

This research has been developed thanks to the financial sup-port of the Italian Ministry of Education, University and Research(MIUR) through PRIN projects ‘‘Shape-memory-alloy advancedmodeling for civil, industrial and biomedical engineering applica-tions’’ (code 2008MRKXLX) and ‘‘Multi-scale modelling of materi-als and structures’’ (code 2009XWLFKW).

742 V. Sepe et al. / International Journal of Solids and Structures 50 (2013) 725–742

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