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Mechanics of Materials 42 (2010) 451–468

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Mechanics of Materials

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

Macroscopic response and stability in lamellar nanostructured elastomerswith ‘‘oriented” and ‘‘unoriented” polydomain microstructures

Vikranth Racherla a,1, Oscar Lopez-Pamies b, Pedro Ponte Castañeda c,*

a Laboratoire de Mécanique des Solides, École Polytechnique, 91128 Palaiseau, Franceb Department of Mechanical Engineering, State University of New York, Stony Brook, NY 11794-2300, USAc Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA

a r t i c l e i n f o

Article history:Received 27 July 2009Received in revised form 2 November 2009

Keywords:Lamellar TPEsHomogenizationMicrostructure evolutionMacroscopic instabilities

0167-6636/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.mechmat.2009.11.005

* Corresponding author.E-mail addresses: [email protected]

la), [email protected] (O. Lopez-Pampenn.edu (P. Ponte Castañeda).

1 Present address: Department of Mechanical Engtute of Technology, Kharagpur 721302, India.

a b s t r a c t

We propose a homogenization-based framework to construct constitutive models for themacroscopic response of polycrystalline hyperelastic solids. The theory is presented in arather general context, but attention is primarily focused on its specialization to lamellarthermoplastic elastomers (TPEs). The proposed framework incorporates direct informationon the constitutive properties of the soft and hard blocks, the lamellar nanostructure, aswell as the complete orientation distribution (i.e., the lamination directions) and averageshape of the grains. In addition to providing constitutive models for the macroscopicresponse of lamellar TPEs, the proposed theory also provides information about the evolu-tion of the underlying nano- and micro-structure – including the crystallographic texture –and the associated development of macroscopic instabilities. It is found that for sufficientlylarge stiffness contrast (between the hard and soft blocks) and sufficiently high Poisson’sratio of the soft blocks there is a sudden change in the deformation mode of ‘‘unfavorably”oriented (perpendicular to the tensile direction) layers – from a high-energy triaxial defor-mation mode to a lower-energy rotation and shear-along-the-layers mode, which isresponsible for a reduction in the overall stiffness of the granular aggregate. This reductionin stiffness can lead to the development of shear-band-type instabilities, which are perpen-dicular to the ‘‘unfavorably” oriented layers, and may be precursors to the chevron-typeinstabilities that have been observed in these material systems. The effect of the constitu-tive properties of the blocks and of the initial microstructure on the overall behavior,microstructure evolution and the possible development of these elastic instabilities isinvestigated in some detail.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

TPEs (thermoplastic elastomers) are a technologicallyimportant class of multiphase polymers consisting of mac-romolecular chains with alternating hard and soft blocks.

. All rights reserved.

.ernet.in (V. Racher-ies), [email protected]

ineering, Indian Insti-

Due to chemical incompatibility, the hard and soft blocksself-separate during processing to form distinct morpholo-gies at the nano-scale. These include, depending on thevolume fractions of hard and soft blocks, periodic arrange-ments of lamellae, hexagonally packed rods, and body-cen-tered cubic arrays of spheres of one phase embedded in theother (see, e.g., Fredrickson and Bates, 1996). These nano-morphologies, however, need not be globally aligned. Inparticular, TPEs exhibit a granular structure at the micronlevel, much like metal polycrystals, with different grainsexhibiting different alignment directions. (Some classesof semi-crystalline polymers exhibit this type of dual

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452 V. Racherla et al. / Mechanics of Materials 42 (2010) 451–468

microstructure as well.) Precisely what type of globalalignment can be found in TPEs depends greatly on theprocessing technique employed to make them. While it isstraightforward to generate a random and isotropic distri-bution of domain orientations leading to ‘‘unoriented” TPEs(see, e.g., Gido et al., 1993), it is more challenging (see, e.g.,Honeker and Thomas, 1996) to generate a highly textureddistribution of domains leading to ‘‘oriented” TPEs.

The soft phase in TPEs, at room temperature and quasi-static loading conditions, is generally isotropic and can un-dergo large reversible (elastic) deformations with littlehysteresis. On the other hand, the hard phase typicallyexhibits an initially stiff, elastic response followed by plas-tic yielding with significant hysteresis, or brittle fracture.These generic properties of the phases simply arise fromthe fact that while the glass transition (or melting) temper-ature of the soft phase is lower than room temperature, thecorresponding glass transition (or melting) temperature ofthe hard phase is much higher.

The ability to tailor macroscopic properties – includingelastic stiffness, fracture toughness, and stretchability – bysimply varying the proportions of the hard and soft blocksand the crystallographic texture, combined with superiorprocessability characteristics, has lead to the increasingusage of TPEs in place of standard vulcanized elastomers.In addition, TPEs have also demonstrated considerable po-tential for utilization in a wide variety of emerging tech-nologies, including high-density electronic storagedevices (Cheng et al., 2001), filtration membranes forviruses (Yang et al., 2006) and antireflection coatings (Jooet al., 2006). It is plain that the development of constitutivemodels incorporating detailed information on the underly-ing nanomorphologies, crystallographic texture, and theconstitutive properties of the hard and soft blocks is ofthe essence for the successful implementation of TPEs insuch advanced applications.

Previous theoretical efforts to study TPEs taking into ac-count the underlying microstructure have been mainly de-voted to perfectly oriented TPEs with lamellar structure. Forinstance, Allan et al. (1991) studied the small-strain, lin-ear-elastic response of oriented lamellar styrene-butadi-ene-styrene (SBS) triblock copolymers, arguably the mostcommercially utilized TPE, by modeling these material sys-tems as perfect laminates. In a later contribution, Readet al. (1999) carried out a combined theoretical and fi-nite-element analysis of the behavior of two-dimensional(2D) perfectly oriented TPEs, subjected to tensile loadingperpendicular to the lamellae, focusing on the possiblebuckling of the layers and its connection with the develop-ment of ‘‘chevron” patterns. Unlike Allan et al. (1991), Readet al. (1999) did take into account the finite rotations of thelayers, although only approximately. They found that forperfectly oriented TPEs there is a geometric instability thatleads to a sharp turnover in the macroscopic stress-strainresponse due to the microscopic formation of ‘‘chevronpatterns”. This geometric instability occurs even whenthe phases are taken to be purely elastic; the plastic behav-ior of the hard phase only tends to accentuate the post-bifurcated chevron profile. Motivated by the experimentalfindings of Cohen et al. (2000), Tzianetopoulou and Boyce(2004) conducted plane-strain numerical simulations for

the mechanical response of oriented TPEs subjected to largedeformations normal and parallel to the lamellae, account-ing for lamellar waviness and interface/interphase imper-fections, as well as the effect of clay nano-particles. Inaddition in her Ph.D. thesis, Tzianetopoulou (2007) pro-posed an analytical continuum constitutive model for 2Dhyperelastic laminates composed of compressible Neo-Hookean phases, and used this model in 2D finite elementscalculations to explore the behavior of polycrystalline TPEs.

In the context of the above-mentioned work for per-fectly oriented TPEs, it is relevant to remark that geometri-cally driven instabilities for layered media have beentheoretically examined in the more general framework offinite elasticity by Triantafyllidis and co-workers (Trian-tafyllidis and Maker, 1985; Geymonat et al., 1993; Trian-tafyllidis and Nestorvic, 2005). These works aimed todetermine all possible (elastic) instabilities in such materi-als, including long-wavelength instabilities, which corre-spond to loss of strong ellipticity of the resultingmacroscopic response. In addition, a general constitutivemodel for the response of hyperelastic layered materialsunder finite deformations – including exact relations forthe rotation of the layers – has been developed by Lopez-Pamies and Ponte Castañeda (2009) (see also Lopez-Pamies(2006)). In that work, the possible development of long-wavelength, or ‘‘macroscopic” instabilities was found tobe intimately related to the rotation of the layers under fi-nite-strain loading conditions.

More recently, Lopez-Pamies et al. (2008) have pro-posed a methodology for modeling highly, but imperfectlyoriented lamellar TPEs as ‘‘polycrystalline” aggregates ofgrains consisting of perfect laminates with slightly differ-ent orientations, whose macroscopic properties can becomputed via a two-scale homogenization process. Thisconstitutive framework accounts for finite deformations,and in addition to providing the macroscopic stress-strainresponse of TPEs, it also allows monitoring of the evolutionof the underlying nano-morphologies and crystallographictexture, as well as detecting the onset of macroscopicinstabilities. In their study, these authors worked out re-sults for highly oriented TPEs made up of two families ofgrains with slightly different lamination orientations, andfound that the rotation of the layers within the grains isa key deformation mechanism controlling the macroscopicresponse and stability of lamellar TPEs.

In this paper, we extend the framework of Lopez-Pa-mies et al. (2008) to model lamellar TPEs with completelygeneral crystallographic texture (as opposed to only twofamilies of grains); particular attention is given to the caseof two-phase lamellar TPEs with ellipsoidal grain shapes.Thus, the primary aim of this work is to develop tools forgenerating constitutive models describing the macroscopicresponse of lamellar TPEs. For the first time, such modelssimultaneously take into account direct information onthe properties of the underlying hard and soft blocks, thelamellar nanomorphology, and the complete orientationdistribution of the grains at the micron level. However, itis important to emphasize that the work is directly rele-vant to TPEs with other types of domain structures, as wellas for other types of nanostructured polymeric systems,such as semi-crystalline polymers (e.g., Lee et al., 1993;

V. Racherla et al. / Mechanics of Materials 42 (2010) 451–468 453

Krumova et al., 2006; Nikolov et al., 2006). Three importantassumptions are made in the proposed analysis. First, sep-aration of length scales is assumed at the single-crystal andpolycrystalline levels, allowing the homogenization of theindividual grains and of the polycrystalline specimen tobe performed in two successive steps. Second, neglectingstrain-rate and hysteresis effects (and assuming ‘‘propor-tional” or nearly proportional loading in the hard phase),the hard and soft phases are characterized by hyperelasticstored-energy functions. Finally, building on the works ofRead et al. (1999) and Lopez-Pamies et al. (2008), plasticdeformation is essentially neglected and loss of strongellipticity (LSE) of the macroscopic response of the homog-enized specimen is used to estimate the possible develop-ment of macroscopic elastic instabilities. Plastically driveninstabilities may occur earlier, depending on the loadingconditions, but a careful treatment of these instabilities isbeyond the scope of this work, in which we will be satisfiedwith crude estimates for when plastic deformation maystart to become important.

The paper is organized as follows. In Section 2, we intro-duce some basic notation and define the homogenized(macroscopic) response and stability of a fairly generalclass of hyperelastic polycrystals. In Section 3, we special-ize the formulation of Section 2 to the case of lamellar TPEsand carry out the appropriate computations. More specifi-cally, Section 3.1 deals with the computation of the effec-tive behavior of the individual grains, which are made upof perfect laminates with alternating layers of hard andsoft phases. Sections 3.2 and 3.3 spell out the calculation– based on the tangent second-order (TSO) method (PonteCastañeda, 1996; Ponte Castañeda and Willis, 1999; PonteCastañeda and Tiberio, 2000) – of the macroscopic re-sponse of the polycrystal. In Section 3.4, we identify a setof microstructural variables whose evolution along a fi-nite-deformation loading path help reveal geometricmechanisms that have a major effect on the macroscopicresponse of TPEs. Section 4 presents an application of theconstitutive framework put forward in Section 3 for a mod-el class of TPEs with unoriented (Section 4.1) and oriented(Section 4.2) crystallographic textures. A detailed study ofthe macroscopic response, onset of macroscopic instabili-ties, onset of yielding of the hard phase, and evolution ofmicrostructure, as functions of the properties of the hardand soft phases is provided for these two cases. We con-clude the paper in Section 5 with some theoretical andpractical remarks.

2. Hyperelastic polycrystals: microstructure descriptionand overall behavior

Consider a statistically uniform polycrystalline speci-men occupying a domain X0, with boundary @X0, in thereference configuration.2 It is assumed that there are N dif-ferent crystal orientations characterized by proper orthogo-nal tensors Q ðrÞ0 ðr ¼ 1; . . . ;NÞ relative to a given referencecrystal orientation, and that all the crystals (or grains) are

2 From here on, the subscript 0 shall refer to variables in the undeformedreference configuration.

perfectly bonded. Furthermore, the crystals are anisotropichyperelastic solids, homogeneous at a length scale of the or-der of the grain size, that are characterized by non-convexstored-energy functions

W ðrÞðFÞ ¼WcðFQ ðrÞT0 Þ; r ¼ 1; . . . ;N: ð1Þ

Here, Wc is the stored-energy function of the referencecrystal and FðXÞ ¼ @x=@X (with x, X denoting, respectively,the positions of the same material point in deformed andundeformed configurations) is the deformation gradienttensor. The anisotropic potential for the reference crystalis, of course, taken to be objective so that

WcðFÞ ¼WcðQFÞ; ð2Þ

for all proper orthogonal tensors Q and all deformationgradients F. In addition, for consistency with the classicaltheory of linear elasticity

WcðFÞ ¼12e �Lceþ OðjjF� Ijj3Þ ð3Þ

in the limit of small deformations as F! I. In the aboveexpression, e ¼ ðFþ FT � 2IÞ=2 is the infinitesimal straintensor, and Lc is a positive definite fourth-order modulustensor. Thus, in view of (3), Wc is strongly elliptic for suffi-ciently small deformations, but may lose strong ellipticityat larger deformations.

The initial microstructure, i.e., the initial distribution ofcrystal orientations within the specimen, is characterizedby indicator functions

vðrÞ0 ðXÞ ¼ 1 if X 2 XðrÞ0 and 0 otherwise; ð4Þ

where XðrÞ0 # X0 is the subdomain occupied by crystals withorientation tensor Q ðrÞ0 . In practice, however, the completemicrostructural information embodied in the indicatorfunctions is usually not accessible experimentally. Instead,only certain microstructural features, such as the n-pointprobabilities, are available. In this work we make use ofthe one-point probability function, pðrÞ0 ðXÞ, denoting theprobability of finding phase r at X, and of the two-pointprobability function, pðrsÞ

0 ðX;X0Þ, denoting the probability

of finding phase r at X and phase s at X0 simultaneously.In particular, assuming ergodicity of the specimen and‘‘ellipsoidal symmetry” (Willis, 1977) with no-long rangeorder for the two-point statistics, the initial one- andtwo-point probabilities of the microstructure are given by

pðrÞ0 ðXÞ ¼ cðrÞ0 and

pðrsÞ0 ðX;X

0Þ ¼ pðrsÞ0 ðjZ

T0ðX

0 � XÞjÞ with

pðrsÞ0 ! 0 asjX0 � Xj ! 1;

ð5Þ

where cðrÞ0 are the volume fractions of crystals with orienta-tions Q ðrÞ0 , and Z0 is a second-order shape tensor serving tocharacterize the average shape of the grains.

It should also be pointed out that the polycrystallinetexture or the one-point statistics for the specimen, inthe undeformed configuration, is often specified via theorientation distribution function (ODF) p0 ðakÞ which givesthe probability of finding crystals as a function of orienta-tion (e.g., Euler or Rodrigues) parameters ak. More specifi-cally, if F denotes the domain of orientation parameters ak


for which there is a one-to-one and onto mapping from F

to the set of all distinct crystal orientations C, the initialODF p0 ðakÞ is defined such that the volume fraction c0 ofgrains with orientations in subdomain G#F is given by

c0 ¼ZG

p0 ðakÞ dl; withZF

p0ðakÞ dl ¼ 1; ð6Þ

where dl is the appropriate measure associated with aninfinitesimal element in F. It is also useful to note thatF can be one-, two-, or three-dimensional depending onthe nature of crystal symmetries. Correlation functionscan also be defined to specify two-point statistics (seeAdams and Olson (1998) for details, as well as for relationsbetween the orientation parameters ak and the orientationtensor Q). However, this will not be needed in this work,since the approach adopted here only makes use of the sec-ond-order shape tensor Z0. More specifically, when thetwo-point statistics are assumed to exhibit the form de-fined in Eq. (5), their exact functional form (i.e., the radialdependence) does not influence the macroscopic behaviorof the specimen. It should finally be mentioned that wemake use of both discrete as well as continuous represen-tations of texture. While the discrete representation ofone-point statistics in the undeformed configuration isused for computing the overall behavior of the specimenas a function of initial microstructure, the continuous rep-resentation, provided by the ODF, is used for graphicalillustration of the evolution of the microstructure and forspecifying the initial one-point statistics.

Having defined the constitutive behavior of the crystalsand the initial microstructure, the local constitutive equa-tion, relating the deformation gradient and the first-PiolaKirchhoff stress S(X), can be written as

S ¼ @W@FðX;FÞ; WðX;FÞ ¼

XN

r¼1

vðrÞ0 ðXÞWðrÞðFÞ: ð7Þ

Under the hypothesis that the specimen size is much largerthan the typical grain size, following Hill (1972), the mac-roscopic constitutive equation for the hyperelastic poly-crystal relating the average stress S ¼ hSi to the averagedeformation F ¼ hFi is then given by

S ¼ @fW@FðFÞ; fW ðFÞ ¼ min

FðXÞ2KðFÞh WðX;FÞ i; ð8Þ

where angular brackets h�i denote volume average overX0;fW ðFÞ denotes the effective (average) stored-energyfunction, and K denotes the set of admissible deformationgradients satisfying affine displacement boundary condi-tions x ¼ FX on @X0, i.e.,

KðFÞ ¼ F j9 x ¼ xðXÞ with F ¼ @x@X

;

�J ¼ detðFÞ > 0 in X0; x ¼ FX on @X0

� ð9Þ

Thus, while the stored-energy functions W ðrÞ describethe local (microscopic) behavior and the indicator func-tions vðrÞ0 describe the initial microstructure, the effectivestored-energy function fW computed via the minimum var-iational statement (8) characterizes the macroscopicbehavior of the polycrystal.

The minimum problem (8) is extremely difficult to solvefor several reasons including the lack of convexity of thelocal stored-energy functions W ðrÞ, as well as the unavail-ability of complete information on the underlying micro-structure vðrÞ0 . Therefore, following the work of Lopez-Pamies and Ponte Castañeda (2006) and Michel et al.(2007) (see Section 2 in these references), the effective po-tential for the polycrystal is instead determined via theassociated stationary variational problem from which themacroscopic constitutive equation is approximated as

S ¼ @cW@FðFÞ; cW ðFÞ ¼ stat

FðXÞ2KðFÞhWðX; FÞi: ð10Þ

Here, the stationary operation formally indicates the selec-tion of the ‘‘principal solution” of the Euler-Lagrange equa-tions associated with the variational principle (8), withoutworrying whether it is the actual minimizing solution. Inessence, the definition (10) ignores the possible develop-ment of bifurcated solutions associated with local instabil-ities. It does contain, however, information about the onsetof ‘‘macroscopic” instabilities with wavelengths compara-ble to the specimen dimensions (Geymonat et al., 1993),as signaled by its loss of strong ellipticity (LSE):

bBðFÞ ¼ minjuj¼jVj¼1

uiVjcLijklðFÞukVl

n o> 0; ð11Þ

where

cLijklðFÞ ¼@2cW@Fij@Fkl

ðFÞ: ð12Þ

Note that loss of strong ellipticity, as detected from failureof condition (11), provides the critical deformation gradi-ents, Fcr , at which the homogenized material becomesmacroscopically unstable, as well as the pairs of unit vec-tors V and u for which these macroscopic instabilities oc-cur. Here, V denotes the normal (in the undeformedconfiguration) to the surface of a weak or strong disconti-nuity of the deformation field, whereas u characterizesthe type of deformation associated with such a discontinu-ity (see, e.g., Knowles and Sternberg, 1975).

3. Application to lamellar TPEs

As already stated in Section 1, lamellar TPEs are consid-ered here as a two-scale material system made up of apolycrystalline aggregate of single crystals, which in turnare made up of laminates of the hard and soft phases. Inthis section, we employ the general formulation of Sec-tion 2 for hyperelastic polycrystals to construct a constitu-tive model for such a class of TPEs. First, in Section 3.1, weformulate and solve the problem for the effective poten-tials W ðrÞ of the individual grains. Having determined thebehavior of the crystals, we then put forward in Section 3.2,by means of the tangent second-order variational method,an estimate for the effective stored-energy function (10) ofthe resulting polycrystal. Section 3.3 spells out the expres-sions for the average deformation gradients in an auxiliarylinear comparison composite (LCC) that are central to thecomputation of the tangent second-order approximationof cW . Finally, in Section 3.4, we write down formulae for


the evolution of microstructural variables that are particu-larly helpful in revealing the main microscopic deforma-tion mechanisms governing the macroscopic response oflamellar TPEs.

3.1. Lamellar single crystals: microstructure and effectivebehavior

In this work, the single crystals that make up the poly-crystalline structure of lamellar TPEs are taken to be per-fect laminates composed of a ‘‘soft” phase with stored-energy function W ð1Þ

c and a ‘‘hard” phase with stored-en-ergy function W ð2Þ

c in initial volume fractions f0 and1� f0, respectively. Assuming piecewise constant fields inthe phases, the effective potential Wc for the reference sin-gle crystal can be shown to be of the form (see, e.g., Trian-tafyllidis and Maker, 1985; Lopez-Pamies and PonteCastañeda, 2009)

WcðFÞ ¼ f0W ð1Þc ðFsÞ þ ð1� f0ÞW ð2Þ

c ðFhÞ; ð13Þ

where, based on the compatibility requirement, the defor-mation gradients in the soft and hard phases are, respec-tively, of the form

Fs ¼ Fþ ð1� f0Þa� N and Fh ¼ F� f0a� N; ð14Þ

with N denoting the initial lamination direction for the ref-erence single crystal. The vector a in (14) is determined bythe equilibrium condition

@W ð1Þc

@FðFsÞ �

@W ð2Þc

@FðFhÞ

" #N ¼ 0: ð15Þ

Note that by assuming piecewise constant fields, we are in-deed following the ‘‘principal solution” to the variationalproblem associated with computation of effective potentialWc. However, this may not correspond to the lowest en-ergy solution, particularly at large deformations, when lo-cal instabilities may take place. Thus, within theframework adopted here, we are implicitly assuming thatlocal instabilities in the grains, which might occur for cer-tain special grain orientations, do not significantly affectthe macroscopic response and stability of the polycrystal.

For later use in the computation of the approximationfor cW , it is useful to note that (see Lopez-Pamies and PonteCastañeda (2009) and deBotton and Ponte Castañeda(1992) for analogous problems in plasticity)

@Wc

@FðFÞ ¼ f0

@W ð1Þc

@FðFsÞ þ ð1� f0Þ

@W ð2Þc

@FðFhÞ; ð16Þ

and

@2Wc

@F@FðFÞ ¼ f0 Lð1Þc þ ð1� f0Þ Lð2Þc Iþ Pð1Þc ðLð2Þc � Lð1Þc Þ

� �; ð17Þ

where

Lð1Þc ¼@2W ð1Þ

c

@F@FðFsÞ; and Lð2Þc ¼

@2W ð2Þc

@F@FðFhÞ ð18Þ

are the incremental modulus tensors of the soft and hardphases, respectively, I is the fourth-order identity tensorwith Cartesian components Iijkl ¼ dikdjl, and Pð1Þc is the rele-vant fourth-order microstructural tensor for layered

microstructures with Cartesian components given byPð1Þc ijkl ¼ ðL

ð1Þc ipkqNpNqÞ�1NjNl.

3.2. Variational approximation for cWHaving defined the constitutive potentials for the lamel-

lar single crystals, we now work out an approximation forthe effective potential cW of the resulting polycrystal, as de-fined by (10)2. This is accomplished here by making use ofthe tangent second-order (TSO) variational procedure ofPonte Castañeda and Tiberio (2000). Thus, the TSO approx-imation for the effective potential cW is given by

cW ðFÞ ¼XN

r¼1

W ðrÞðFðrÞl Þ þ12@W ðrÞ

@FðFðrÞl Þ � ðF� FðrÞl Þ

" #; ð19Þ

where FðrÞl is the average deformation gradient in phase r ofan N-phase linear comparison composite (LCC) with thesame initial microstructure that is subjected to the sameaffine displacement boundary conditions as the hyperelas-tic polycrystal. The behavior of the phases of the LCC ischaracterized by quadratic potentials of the form

W ðrÞl ðFÞ ¼

12ðF� IÞ � LðrÞðF� IÞ þ TðrÞ � ðF� IÞ þ f ðrÞ; ð20Þ

where I is the second-order identity tensor and

LðrÞ ¼ @2W ðrÞ

@F@FFðrÞl

� �;

TðrÞ ¼ LðrÞ I�FðrÞl

� �þ@W ðrÞ

@FFðrÞl

� �; and

f ðrÞ ¼12

I�FðrÞl

� ��LðrÞ I�FðrÞl

� �þ@W ðrÞ

@FFðrÞl

� �� I�FðrÞl

� �þW ðrÞ FðrÞl

� �:

ð21ÞNote that the TSO estimate (19) for the effective stored-en-ergy function of the hyperelastic polycrystal (10)2 reducesultimately to the computation of the phase averages of thedeformation gradient field in the above-defined LCC, whichis a reasonably simple task in view of the linearity of thisproblem. Indeed, the estimation of the phase averages ofthe fields in linear-elastic media is standard and severalhomogenization methods have been devised for this pur-pose (see, e.g., Milton, 2002). The above-defined linearcomparison problem is not a standard problem in linearelasticity – because of the non-symmetry of the stressand strain measures – but it is linear nonetheless and themethods that have been developed for classical linear elas-ticity are equally applicable to these more general linearmaterials (see, e.g., Ponte Castañeda and Tiberio, 2000).

3.3. Phase averages of the deformation gradient in the LCC

Next, we present the linear homogenization scheme forcomputing the phases averages of the deformation gradi-ent in the LCC, needed for the computation of the TSOapproximation (19) for cW . By treating the LCC as a linearthermoelastic composite, and generalizing appropriatelythe work of Laws (1973), the phase averages of the defor-mation gradients can be shown to be of the form.

FðrÞl ¼ AðrÞðF� IÞ þ aðrÞ þ I; ð22Þ

where the quantity F� I now plays the role of the infinites-imal strain in the usual calculation. In these expressions,


AðrÞ and aðrÞ are fourth- and second-order ‘‘strain” concen-tration tensors, respectively, depending on the microstruc-ture and elastic moduli of the constituents in the LCC. Theconcentration tensor AðrÞ has major symmetry ðAðrÞijkl ¼ AðrÞklijÞ,but not minor symmetry ðAðrÞijkl – AðrÞjiklÞ. In addition, from thefact that for affine displacement boundary conditionshFli ¼ F, we have that the concentration tensors must satisfyXN

r¼1

cðrÞ0 AðrÞ ¼ I andXN

r¼1

cðrÞ0 aðrÞ ¼ 0: ð23Þ

For linear polycrystals with ‘‘ellipsoidal” microstructures,as defined by relations (5) for the one- and two-point prob-ability functions, self-consistent estimates for the concen-tration tensors, generalizing those of Laws (1973) andWillis (1977, 1981), may be obtained as

AðrÞ ¼ Iþ ePðLðrÞ � eLÞh i�1

aðrÞ ¼ AðrÞ eP XN

s¼1

cðsÞ0 TðsÞAðsÞ � TðrÞ" #

;ð24Þ

where LðrÞ;TðrÞ have been defined in the previous subsec-tion and eL is the effective modulus tensor satisfying theimplicit equation:

eL ¼XN

r¼1

cðrÞ0 LðrÞ Iþ ePðLðrÞ � eLÞh i�1: ð25Þ

In the above expressions, eP is a microstructural tensor,which for the class of ‘‘ellipsoidal” microstructures consid-ered here, may be expressed (Willis, 1977) as

eP ¼ 14pdetZ0

Zjnj¼1jZ�1

0 nj�3 eHðnÞ dS; ð26Þ

where eHijkl ¼ eK�1ik njnl with eK ik ¼ eLijklnjnl and dS is an infini-

tesimal area element on the unit sphere characterized byjnj ¼ 1. Note that both eL and eP have major but not minorsymmetries.

To summarize, the computation of the phase averagesFðrÞl of the deformation gradient in the LCC that are essen-tial for computing the effective potential cW for the poly-crystal via (19), amounts to just solving the system ofcoupled nonlinear algebraic Eqs. (24) and (25), subject to(23), for the concentration tensors AðrÞ and aðrÞ.

3.4. Evolution of microstructure

The above analysis makes use of a Lagrangian descrip-tion of the kinematics. The evolution of the microstructureresulting from the finite changes in geometry are thus al-ready accounted for in the homogenized stored-energyfunction cW . This is in contrast to the corresponding resultsfor viscoplastic polycrystals (see, e.g., Lebensohn et al.,2007), which are usually cast within an Eulerian descrip-tion and do require additional computation of the evolu-tion of the crystallographic and morphological texture.However, although not needed to determine the effectivebehavior, it is still of interest to have direct access to vari-ables characterizing the microstructure evolution, as theyprovide deeper insight into the macroscopic response andpossible development of instabilities.

In this work, we will focus on the study of the averagedeformation behavior of the grains through the evolutionof certain average microstructural variables. Given theinformation that is available from the TSO homogenizationmethod, it proves expedient – as a first approximation – tobase the relevant variables on the average deformationgradient FðrÞl in the grains of the LCC, as determined by rela-tion (22). It should be remarked, however, that a moresophisticated – but also more computationally demanding –approach leading to more accurate estimates for averagesquantities in polycrystals has been recently proposed byIdiart and Ponte Castañeda (2007).

Thus, for the lamellar TPEs considered here, threemicrostructural variables serving to characterize the evo-lution of the deformations in each grain are identified(see Fig. 1 for a graphical illustration of these variables):(i) the average orientation of the layers within the grains,which evolves from NðrÞ to nðrÞ, (ii) the average repeatlength, which evolves from an initial value L0 to a currentvalue LðrÞ, and (iii) the average shear angle along the layers,which starts from a zero initial value and evolves to a cur-rent value bðrÞ. In terms of the phase averages FðrÞl , the equa-tions defining these variables read simply as (Lopez-Pamies and Ponte Castañeda, 2009)

nðrÞ ¼ FðrÞ�Tl NðrÞ

jFðrÞ�Tl NðrÞj

; LðrÞ ¼ jFðrÞ�Tl NðrÞj�1L0; ð27Þ

and

bðrÞ ¼ cos�1 FðrÞl NðrÞ

jFðrÞl NðrÞj� nðrÞ

!: ð28Þ

Here, the evolution of nðrÞ physically describes the averagerotation of the layers within the grains, which can act as adominant geometric softening/stiffening mechanism ofdeformation. Note that, unlike the corresponding resultof Read et al. (1999), the above result for (the evolutionof) nðrÞ is valid for arbitrarily large strains, as long as thesolution remains on the principal path. Moreover, the vari-ables LðrÞ and bðrÞ serve to measure, respectively, how muchthe hard and soft layers deform.

In view of the large number N of grain orientations, itproves useful to introduce a continuous ODF in the de-formed configuration – analogous to the undeformed- con-figuration ODF (6) – serving to characterize the evolutionof the distribution of the average orientations nðrÞ as thedeformation progresses. Thus, denoting by the unit vectorn the continuum version of the discrete lamination direc-tions nðrÞ, the current ODF p is defined via

c ¼ZS

pðnÞdS; withZF

pðnÞdS ¼ 1; ð29Þ

where c is the volume fraction of grains with current orien-tations in the solid angle S#F, with F denoting the unithemisphere (or unit semicircle in two dimensions). Thefunction p provides a convenient way to represent the evo-lution of the crystallographic texture by plotting p as afunction of the current orientation n for increasing valuesof the deformation F. Similarly, the continuum microstruc-tural variables L and b can also be conveniently visualizedas functions of the current orientation n.

0

0(r)

S

Undeformed configurationDeformed configuration

0L

2x (r)

(r)N0

n

gL

sL

1x

L

1x(r)

(r)

(r)

Fig. 1. Schematic of the two-scale microstructure of lamellar TPEs and microstructural variables. The characteristic length of the specimen Ls , the grain sizeLg , and the repeat length L0 are assumed to be well separated in the sense that Ls � Lg � L0.


4. Results and discussion

In this section, the homogenization-based constitutiveframework developed in Sections 2 and 3 is employed toinvestigate the macroscopic response and stability of amodel class of lamellar TPEs with both ‘‘unoriented” and‘‘oriented” crystallographic textures. The focus of thisstudy will be on the elastic constitutive response, and onthe possible development of elastically driven instabilities.As already mentioned, under certain types of loading con-ditions, plastic deformation can take place in the hardphase of lamellar TPEs and control the overall response be-yond a certain elastic regime eventually leading to plasticinstabilities (Cohen et al., 2000). Accordingly, a completeunderstanding of the mechanical behavior of TPEs undergeneral loading conditions requires consideration of theplastic deformation of the hard (glassy or semi-crystalline)blocks. The effect of this plastic deformation is expected,however, to be largely dependent on the specific constitu-tive model (see, e.g. Lopez-Pamies et al., 2008) and is be-yond the scope of this paper. Here, we will be satisfied toprovide only a rough estimate for when the plastic defor-mation initiates and could start playing a significant roleby monitoring the von Mises equivalent stress in the hardphase, as described further below. In summary, the mainaims of this section are to examine in detail the followingissues for lamellar TPEs:

� the key microscopic deformation mechanisms that leadto the onset of macroscopic elastic instabilities,

� the sensitivity of the macroscopic constitutive responseand stability to the initial crystallographic texture,

� the sensitivity of the macroscopic constitutive responseand stability to the heterogeneity contrast between thehard and soft phases, as well as to the compressibilityof the soft phase.

Within the formulation proposed in this work, there arefour specific inputs that are needed to determine the elas-tic response of lamellar TPEs under large strains: (i) the ini-tial shape tensor Z0 that serves to characterize the initial

shape of the grains, (ii) the initial ODF and the associatedinitial grain volume fractions cðrÞ0 that describe the initialdistribution of lamination directions NðrÞ, (iii) the hyper-elastic potentials that characterize the constitutive re-sponse of the hard and soft phases in the lamellar grains,and (iv) the macroscopic deformation gradient F that de-scribes the applied boundary conditions. The specificchoices of each of these inputs are discussed in detail inthe following paragraphs.

Initial shape tensor Z0. As discussed, for instance, inHoneker and Thomas (1996), standard processing methodsfor lamellar TPEs lead to sheets of these materials where theindividual crystals tend to be cylindrical in shape with thelong axis perpendicular to the plane of the sheet and withan average cross-section of elliptical shape (see Fig. 1).Therefore, the average grain shape in the undeformed con-figuration is taken here to be cylindrical, with ellipticalsymmetry in the plane transverse to the cylindrical axis.In this case, the microstructural tensor (26) takes the form

ePijkl ¼w0

2p

Zn2

1þn22¼1

eHijklðn1; n2;0Þn2

1 þw20n

22

dS; ð30Þ

where the components are given with respect to the prin-cipal axes ek of the shape tensor Z0 ¼ w0 e1 � e1 þ e2 � e2.Note that while the unit vector e3 is along the cylindricalaxis, the vectors e1; e2 are along the minor and major axesof the elliptical symmetry in the transverse plane. Whenthe grains are further assumed to be ‘‘equi-axed” in thetransverse plane – as it will be done in the calculationsshown below – the aspect ratio w0 ¼ 1.

Initial ODF. Experimental observations also suggest thatthe layers within the grains of typical lamellar TPEs tend tobe aligned with the cylindrical axis of the grains (and per-pendicular to the sheet). Because of this and for simplicity,the initial stacking directions of the lamellae are restrictedhere to lie in x1 � x2 plane such that

NðrÞ ¼ cosðhðrÞ0 Þe1 þ sinðhðrÞ0 Þe2 ðr ¼ 1; . . . ;NÞ; ð31Þ

where hðrÞ0 are the initial grain orientation angles (see Fig. 1).In the calculations below, detailed investigations will be


conducted for two kinds of initial distribution of stackingdirections within the x1 � x2 plane: ‘‘unoriented” corre-sponding to an isotropic distribution in the plane (see Sec-tion 4.1), and ‘‘oriented” corresponding to a Gaussiandistribution of orientations in the plane (see Section 4.2).In all the calculations, ðN ¼Þ40 uniformly distributed initialgrain orientation angles hðrÞ0 are used. Numerical experi-mentation shows that larger numbers of initial laminationdirections ðN > 40Þ, or the way they are chosen (uniformlyor randomly) has a marginal effect on the macroscopicstress-strain response and microstructure evolution, andonly a small effect on the onset of macroscopic instabilities.

Constitutive behavior for the phases. For actual lamellarTPEs, fairly elaborate potentials (see, e.g., Lopez-Pamieset al., 2008) are necessary to accurately account for plasticdeformations in the hard (glassy or semi-crystalline) phaseand the strongly non-linear elastic deformations in the soft(rubbery) phase. However, since the primary aim of thispaper is to investigate the generic effect of the elastic prop-erties of the phases in the lamellar grains, and of the micro-structure, on the overall behavior, microstructureevolution, and the onset of macroscopic elastic instabili-ties, simple compressible Neo-Hookean potentials allow-ing for thorough parametric analyses are chosen todescribe the hard and soft phases. Thus, the potentials forthe phases are taken to be of the form

W ðpÞc ðFÞ ¼

lðpÞ

2ðF � F� 3Þ � lðpÞ lnðdet FÞ

þ kðpÞ

2� lðpÞ

3

!ðdet F� 1Þ2 ðp ¼ 1;2Þ; ð32Þ

where lðpÞ; kðpÞ ðp ¼ 1;2Þ are the initial shear and bulkmoduli of the soft and hard phases, respectively. For laterreference, it is useful to define the heterogeneity contrastparameter t ¼ lð2Þ=lð1Þ P 1 and the ‘‘Poisson’s ratio”mðpÞ ¼ ð3kðpÞ � 2lðpÞÞ=ð6kðpÞ þ 2lðpÞÞ. Throughout this sec-tion, the Poisson’s ratio of the hard phase will be taken tobe mð2Þ ¼ 0:33, while that of the soft phase mð1Þ will be al-lowed to vary in the range mð1Þ 2 ½0;0:5�, with the limitingvalue mð1Þ ¼ 0:5 corresponding to the case of incompress-ible behavior.

Even though the constitutive potentials (32) are purelyelastic, they still allow to compute an estimate for whenthe onset of plastic deformation in the hard blocks of thelamellae may become important. Indeed, since the averagedeformation gradient in grains with orientation r areapproximately given by FðrÞl , the corresponding averagedeformation gradient in the hard phase FðrÞh can be readilycomputed from Eqs. (14) and (15) with F ¼ FðrÞl . Then, theaverage Piola-Kirchhoff and Cauchy stresses in the hardphase of orientation r may be computed, to a first approx-imation, via

SðrÞh ¼@W ð2Þ

c

@FðFðrÞh Þ; r

ðrÞh ¼

1

detFðrÞh

SðrÞh FðrÞTh ; ð33Þ

from which the von Mises equivalent stress in the hardphase of orientation r can in turn be computed via

rhðrÞe ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32

r0ðrÞh � r

0ðrÞh

rwith r

0ðrÞh ¼ r

ðrÞh �

13rðrÞh kkI: ð34Þ

Thus, by comparing the von Mises equivalent stress rhðrÞe

with the value of the yield stress ry of the hard (glassy orsemi-crystalline) phase, we may obtain a ‘‘crude” estimatefor when the plastic effects start to play a role, and possiblymodify the conclusions of our analyses based on purelyelastic deformations. In this work, based on the normalizedyield stress for polystyrene in SBS – the most widely usedthermoplastic elastomer – plastic yielding in grains with agiven orientation r is estimated to take place when

rhðrÞe P ry � 7lð1Þ ð35Þ

We conclude this remark by re-emphasizing that the con-stitutive framework proposed here applies equally well tomore sophisticated potentials (not just Neo-Hookean)including those that are commonly used in the context offinite-strain J2 deformation theory of plasticity (Hutchin-son and Neale, 1981), which would permit a more detailedstudy of the plastic deformation of the hard phase. Indeed,preliminary calculations show that the results for the‘‘principal” solution and the associated macroscopic insta-bilities are not qualitatively affected by the constitutivemodel chosen for the hard phase.

Applied loading conditions. Motivated by the availableexperimental data (see, e.g., Honeker and Thomas, 1996;Garcia, 2006), we focus on macroscopic deformation gradi-ents that are consistent with a uni-axial state of stress inthe x1-axis. Therefore,

F ¼ Fij ei � ej ði; j ¼ 1;2;3Þ ð36Þ

is chosen such that the macroscopic first Piola-Kirchhoffstress

S ¼ @cW@FðFÞ ¼ S e1 � e1: ð37Þ

Within this context, for convenience, F11 ¼ k is chosen asthe loading parameter and the rest of the eight compo-nents of the deformation gradient are determined suchthat the corresponding eight components of the stress ten-sor are identically zero and the stress state is therefore uni-axial. From a computational point of view, it is importantto recognize that for the material systems with the micro-structure dictated by (30) and (31), Neo-Hookean hard andsoft phases (32), and uni-axial loading conditions (37) ofinterest here, the problem to be solved becomes two-dimensional (generalized plane strain).

4.1. Unoriented samples

In this subsection we examine in detail the behavior ofthe model class of lamellar TPEs described above withunoriented crystallographic texture. Specifically, we con-sider samples with cylindrical grains with circular crosssection and an isotropic distribution of stacking directionsin the transverse ðx1 � x2Þ plane, for which the initial ODFis simply given by

p0ðh0Þ ¼1p

with 0 6 h0 6 p: ð38Þ

The initial grain angles and the associated grain volumefractions may then be taken as


hðrÞ0 ¼r � 1=2

Np; cðrÞ0 ¼

1Nðr ¼ 1; . . . ;NÞ: ð39Þ

This subsection is organized as follows. The theoreticalpredictions for the macroscopic stress–stretch responseand onset of macroscopic (long-wavelength) instabilities,as a function of elastic properties of the phases, are pre-sented in Figs. 2 and 3. The deformation mechanismsresponsible for the onset of macroscopic instabilities andthe effect of elastic properties of the phases on the defor-

a b

Fig. 2. Uniaxial stress–stretch curves for an unoriented specimen subjected to unof the soft phase mð1Þ . (Loss of strong ellipticity (LSE) is indicated by the full circ

Fig. 3. Critical stretch and critical stress at LSE as a function of contrast parameteof Poisson’s ratio mð1Þ . (c and d) Effect of initial volume fraction of the soft phase

mation mechanisms, are then investigated in Figs. 4–6. Itmust be remarked that macroscopic instabilities, whoseonset is marked by the loss of strong ellipticity (LSE) ofthe homogenized specimen, typically manifest themselvesin the form of localized bands of deformation. Though nar-row as compared to the specimen dimensions, the bandsare much wider than the average grain size. In addition,microscopic instabilities, e.g. of buckling type with wave-lengths on the order of grain size or less, may occur earlierthan the long wavelength instabilities but they cannot be

iaxial tension. (a) Effect of contrast parameter t. (b) Effect of Poisson’s ratioles.)

r t in unoriented specimens subjected to uniaxial tension. (a and b) Effectf0.

a b

dc

Fig. 4. Evolution of microstructural variables and average effective stress in the hard phase of the lamellae, for three different grains, in an unorientedspecimen subjected to uniaxial tension for the soft phase initial volume fraction f0 ¼ 0:5, contrast in shear moduli of the phases t ¼ lð2Þ=lð1Þ ¼ 10, initialPoisson’s ratio of the soft phase mð1Þ ¼ 0:45, and initial Poisson’s ratio of the hard phase mð2Þ ¼ 0:33. (a) Average current lamination angle h in three grainscomposed of lamellae that are nearly perpendicular ðh0 ¼ 2:25Þ, diagonal ðh0 ¼ 42:75Þ and parallel ðh0 ¼ 87:75Þ to the stressing direction x1 in theundeformed configuration. (b) Average repeat length ratio L=L0. (c) Average shear angle b. (d) Average effective stress in the hard phase of the lamellae rh

e

normalized with respect to the initial shear modulus of the soft phase lð1Þ . The dashed line represents yielding for the hard phase based on the yield stressfor polystyrene in SBS.

a b

Fig. 5. Evolution of current lamination angle in an unoriented sample subjected to uniaxial tension for three grains that are initially (nearly) perpendicularðh0 ¼ 2:25Þ, diagonal ðh0 ¼ 42:75Þ and parallel ðh0 ¼ 87:75Þ to the stressing direction x1. (a) Effect of contrast parameter t for f0 ¼ 0:5; mð1Þ ¼ 0:45, andmð2Þ ¼ 0:33. (b) Effect of Poisson’s ratio of the soft phase mð1Þ for t ¼ 10; f0 ¼ 0:5, and mð2Þ ¼ 0:33.


captured using the framework adopted here. Nevertheless,LSE provides an upper limit for the onset of these instabil-ities (Geymonat et al., 1993).

Fig. 2 shows the macroscopic stress–stretch response ofunoriented specimens, subjected to uniaxial tension, as a

function of elastic properties of the phases. It is plain thatthe macroscopic stiffness increases with increasing con-trast but is fairly insensitive to the Poisson’s ratio of thesoft phase, particularly close to the incompressibility limit.Furthermore, the stiffness is a non-trivial function of the

a b

dc

Fig. 6. Evolution of (a) orientation distribution function p, (b) average repeat length ratio L=L0, (c) average shear angle b, and (d) Average effective stress inthe hard phase of the lamellae rh

e normalized with respect to the initial shear modulus of the soft phase lð1Þ in grains in an unoriented specimen subjectedto uniaxial tension. The dashed line represents yield surface for the glassy phase based on the normalized yield stress for polystyrene in SBS.


elastic properties of the phases and the evolution of themicrostructure – and not just a simple rule of mixtures.Remarkably, the model predicts LSE at a finite stretch �kcr ,and no results are shown beyond this point. Furthermore,for all unoriented specimens subjected to uniaxial tensionalong the x1 axis, at LSE, unit vectors V ¼ e2 and u ¼ e1 arefound to minimize the product in Eq. (11), i.e. the localizedband at LSE is found to be aligned with the tensile x1-direc-tion or perpendicular to the compressive x2-direction(compression is induced by the Poisson effect). Thus,unoriented specimens typically lose strong ellipticity at�kcr , when the incremental modulus L1212 (shear modulusalong the loading direction) vanishes. In this context itmust be emphasized (see, e.g., Vogler et al., 2001) thatexperimental observations of instabilities are usually postmortem and therefore correspond to the final configurationof the specimen, which necessarily involve the post-bifur-cation solution and may look quite different from theabove-described initial instability mode, which is basedon the ‘‘principal” solution. Nevertheless, the critical defor-mation predicted by this approach should still provide areasonable estimate for the possible onset of the experi-mentally observed instabilities.

Fig. 3 shows the critical stretch �kcr and the correspond-ing critical stress Scr at which LSE ensues, as a function ofthe contrast t, and Poisson’s ratio and initial volume frac-tion of the soft phase, mð1Þ and f0, respectively. Note that

no instabilities are predicted by the analysis at small con-trasts (t less than about 5), and that, in general, the largerthe contrast the smaller the stretch and stress at which lossof strong ellipticity ensues. In general, the critical stretchseems to be more sensitive to the contrast than the criticalstress, while the critical stress is more sensitive to the vol-ume fraction of the hard and soft phases than the criticalstretch. On the other hand, both the critical stress andstretch are equally sensitive to the Poisson’s ratio of thesoft phase.

As mentioned earlier, the proposed theory may also beused to investigate deformation mechanisms, leading toLSE, by analyzing the evolution of the microstructural vari-ables introduced in Section 3.4. In addition, the Von Misesequivalent stress in the hard phase of the lamellar grainsmay be used to estimate as to when the plastic effectsmay start to become important. In this regard, we depictin Fig. 4 the evolution of the microstructural variablesh; L; b and the von Mises equivalent stress in the hard phaserh

e in an unoriented specimen subjected to uniaxial stressalong the x1 axis. We focus our attention on three particu-lar initial grain orientations: perpendicularly orientedðh0 ¼ 2:25Þ, diagonally oriented ðh0 ¼ 42:75Þ, and parall-elly oriented ðh0 ¼ 87:75Þ. The results shown are for con-trast t ¼ lð2Þ=lð1Þ ¼ 10, Poisson’s ratio of the soft phasemð1Þ ¼ 0:45, and initial volume fraction of the soft phasef0 ¼ 0:5. As expected, the deformation of the grains is

Fig. 7. Comparison of homogenization estimates (TSO) with FEM simu-lations of Tzianetopoulou (2007) and Taylor- and Sachs-type estimatesfor plane strain extension: macroscopic Cauchy stress versus macroscopicstretch. FEM results labeled 1, 4 and 5 are for 3 different realizationssubjected to plane strain tension along two perpendicular directions(directions 1 and 2).


highly dependent on the initial grain orientation. Parallellyoriented grains ðh0 ¼ 87:75Þ deform mainly throughstretching of lamella, resulting in a monotonic increase inthe equivalent stress in the hard phase and a steady de-crease in repeat length ratio, while diagonally orientedgrains ðh0 ¼ 42:75Þ deform mainly via a combination of ri-gid rotation and shear along the lamella. In contrast, theperpendicularly oriented grains ðh0 ¼ 2:25oÞ deform differ-ently at small and large stretches. At small overallstretches, these grains deform predominantly by stretchingalong the stacking direction, resulting in a steady increasein the repeat length ratio and an increase in the equivalentstress in the hard phase. However, this is an energeticallyunfavorable mode of deformation as an increasingly largeramount of energy is required to deform the hard phase,and a new deformation mode ensues, consisting of a com-bination of rigid rotation and shear along lamella, whichresults in a much slower increase in repeat length ratioand equivalent stress in the hard phase of the lamellae.This transition from a ‘‘high-energy” to a ‘‘lower-energy”deformation mode is responsible for the drop in the shearstiffness (i.e., in L1212) of the material along the tensiledirection, which results in LSE. Here, it is important to re-mark that the results shown beyond the LSE correspond tothe ‘‘principal” solution, and are shown only to highlightthe trends in the principal solution, but are not meant tobe interpreted as the actual solution beyond LSE.

To better understand the effect of the elastic propertiesof the phases on the onset of the macroscopic instabilities,Fig. 5 shows results for the evolution of grain orientations –for the same set of grains as in Fig. 4 (i.e. for h0 ¼ 2:25,42.75, and 87.75�) – for different values of the contrastand Poisson’s ratio of the soft phase. It can be seen fromthe figure that evolution of grain orientation h, particularlyfor perpendicularly oriented grains, is strongly dependenton the elastic properties of the phases. More specifically,the larger the contrast t or the larger the Poisson’s ratioof the soft phase mð1Þ, the earlier and sharper the transitionin deformation mode. This correlates well with the corre-sponding plots in Fig. 3 that also show earlier instabilities(LSE) for larger contrast or larger Poisson’s ratio. Interest-ingly, there is very little effect of t and mð1Þ on the evolutionof initially parallelly and diagonally oriented grains. Thisfact, which is plausible from a physical view point, seemsto corroborate the hypothesis that the development ofthe elastic macroscopic instabilities is directly linked tothe change in deformation mode of the perpendicularlyoriented grains.

A more complete picture of the evolution of the miscro-structural variables is provided in Fig. 6, which shows plotsfor the ODF p, microstructural variables L and b, and vonMises equivalent stress in the hard phase rh

e , as a functionof the current grain orientation h. Fig. 6a shows the ODFs,indicating the probability p of finding grains with a certaingrain angle h in the deformed configuration, for three dif-ferent stretches for the same set of material parametersand loading conditions as in Fig. 4. The ODFs evolve quiterapidly, particularly for deformations near LSEð�kcr ¼ 1:55Þ. At and beyond LSE, the majority of the lamel-lae exhibit approximately a 20 misalignment with thestressing direction resulting in a ‘‘double-bump” pattern

in the ODF. This remarkable result is consistent with therecent experimental observations of Garcia (2006) (seeFig. V.16), which also show the transition from a single-mode to a bimodal distribution of orientations. As seenfrom Fig. 4b–d, this transition is clearly due to the fact thatwhile lamellae aligned with the stretching direction de-form by stretching the hard phase (there is no alternativefor these orientations), ‘‘unfavorably” oriented lamellae de-form mainly via shearing of the soft phase, which is ener-getically preferable. It is relevant to remark that theseobservations are also in very good qualitative agreementwith uniaxial tension experiments on isotropic lamellarTPEs (Séguéla and Prud’homme, 1981) where at relativelysmall stretches, the initially randomly oriented lamella ro-tate such that majority of them align at around ±20 degreesfrom the stressing direction, resulting in the characteristicfour-point small angle X-ray scattering (SAXS) patterns.

In addition, it is noted that the dashed lines in Figs. 4dand 6d correspond to the above-mentioned estimate (35)for plastic yield in the hard phase. They show that, aswould be expected, parallelly oriented grains are the firstto undergo plastic yielding followed by the perpendicularlyoriented grains. Plastic yielding occurs in these grains justbefore the LSE. However, since the volume fraction ofgrains that undergo plastic yielding before the onset ofmacroscopic instabilities is relatively small, the estimatesfor microstructure evolution and macroscopic responsepresented above are expected to be reasonably accurateuntil the LSE.

Finally, in Fig. 7, comparisons are presented with theFEM results of Tzianetopoulou (2007) for plane strainextension of several realizations of polygranular systemswith random orientations and hexagonal grains. (In thisplot and for this plot only, we have changed the uniaxialloading condition described by expression (37) to matchthe plane strain extension conditions used in the numeri-cal simulations). More specifically, our model predictionsfor the average Cauchy stress versus the average stretchare compared in this figure with the FEM results for 3


different realizations (RVEs 1, 4, and 5) subjected to planestrain tension along two perpendicular directions (direc-tions 1 and 2). As can be seen from the figure, the resultsfor each realization are not quite isotropic, and they differfrom one realization to the other. This is, of course, to beexpected given the statistical nature of the problem. De-tailed comparisons with our model predictions, which areessentially isotropic by construction, would require a lar-ger set of realizations and ensemble averages to be per-formed. However, it can be seen from the figure that,although a little bit stiffer than the rough average of the gi-ven FEM realizations, the model predictions are consistentwith these results. Certainly, they exhibit very good agree-ment for small strain, when the effect of differences in thevarious realizations are expected to be relatively small, andconsistent with the FEM results, they are much softer thanthe Taylor-type estimate, and also quite a bit stiffer thanthe Sachs-type estimate. Also, it should be pointed out thatas already discussed in the context of our model predic-tions, the FEM simulations of Tzianetopoulou (2007) alsoexhibit the development of extra peaks in the ODFs withincreasing deformation. However, detailed comparisonswere not possible due to apparent differences in the defini-tions used for the ODFs in this work. Of course, the FEMsimulations also include valuable information about the lo-cal distributions of the fields in the polycrystals, and thereader is referred to Tzianetopoulou (2007) for more de-tails on this.

4.2. Oriented samples

Next, we examine the behavior of lamellar TPEs withoriented crystallographic texture. In this case, the initialdistribution of lamination directions is characterized by aGaussian-like ODF of the form (see Fig. 11)

p0 ¼1z0

e�N0 �N02 r2 ; N0 ¼ N� l; ð40Þ

where l ¼ cosðhm0 Þ e1 þ sinðhm

0 Þ e2 denotes the (bias) meanstacking direction, hm

0 denotes the angle between thestressing direction and the mean stacking direction, rcharacterizes the width of initial distribution of stackingdirections, N ¼ cosðh0Þ e1 þ sinðh0Þ e2 with ðhm

0 � p=2Þ 6 h0 6

ðhm0 þ p=2Þ, and the constant

z0 ¼Z hm

0 þp=2

hm0 �p=2

e�N0 �N02 r2 dh0; with

Z hm0 þp=2

hm0 �p=2

p0dh0 ¼ 1: ð41Þ

The initial grain angles and the associated grain volumefractions are then taken to be

hðrÞ0 ¼ hm0 � hs

0 þ ðr � 1=2Þ dh0 and

cðrÞ0 ¼Z hm

0 �hs0þrdh0

hm0 �hs

0þðr�1Þdh0

p0ðh0Þdh0;ð42Þ

where hs0 ¼ 4r and dh0 ¼ 8r=N for r2 << 1 and hs

0 ¼ �p=2and dh0 ¼ p=N otherwise. Note that while hm

0 ¼ 0 corre-sponds to the limiting case when the stressing directionis perpendicular to the lamella, hm

0 ¼ 90 corresponds tothe case when the lamella are aligned with the stressingdirection. Also note that r! 0 corresponds to the extreme

case of a perfect laminate. In the other extreme case whenr!1, (40) reduces to the unoriented distribution (38)studied in the preceding subsection.

Fig. 8 shows results for the Young’s ModulusE ¼ dS=d�kð�k ¼ 1Þ, as a function of the distribution parame-ter r and loading angle hm

0 . From the figure, it is evident thatE is extremely sensitive to the initial distribution of stack-ing directions when evaluated along the lamella, i.e., forhm

0 ¼ 90. By contrast, there is relatively little effect of theinitial distribution of lamination directions on the Young’smodulus along other loading directions ðhm

0 6 75Þ, evenfor reasonably broad initial distributions ð0:05 6 r 6 0:1Þ.These observations indicate that assuming an orientedspecimen to be a perfect laminate, while attempting to inferthe elastic properties of the underlying soft and hard blocksfrom measurements of the macroscopic Young’s moduli atdifferent loading angles (as attempted, for instance, by Al-lan et al., 1991), can lead to erroneous results – namely, aconsiderable underestimation of properties of the hardphase, particularly its shear modulus – especially whenthe initial distribution of orientations is not narrow.

Fig. 9 shows plots of the macroscopic stress versus themacroscopic stretch in oriented specimens, subjected touniaxial tension along the mean stacking direction, withinitial distribution of stacking directions characterized byr ¼ 0:1 (see Fig. 11b). For comparison purposes, resultsare also included in dashed lines for a perfect crystalðr ¼ 0Þ with layer normal aligned with the tensile direc-tion. Note that softening of the overall tensile response isobserved only for imperfect samples with r ¼ 0:1. Thesoftening in imperfect samples is attributed to the evolu-tion of microstructure in this specimen, as will seen inmore detail further below. Once again, loss of ellipticityis observed at a critical stretch �kcr , beyond which no resultsare shown. The model predicts that �kcr depends strongly onthe contrast t, and a little less strongly on the Poisson’s ra-tio of the soft phase mð1Þ. The results show that as was thecase for the unoriented samples, the material loses elliptic-ity because the shear modulus L1212 (the shear modulusperpendicular the compressive direction) vanishes at �kcr .

Fig. 10 shows the critical stretch and correspondingcritical stress at which macroscopic instabilities take placein oriented systems, under tensile loading perpendicular tothe average layer direction, as a function of the elasticproperties of the phases and initial distribution of stackingdirections. Note from Fig. 10a and b that increasing thecontrast t and increasing the Poisson’s ratio of the softphase mð1Þ leads to smaller critical stretches and stresses.Fig. 10c and d show that the initial width r of the distribu-tion of stacking directions has a moderate effect on boththe critical stretch and stress for strongly aligned systemswith r < 0:1. However, the effect is more significant as rincreases beyond this level and results in a sizable delayin the possible onset of instabilities, which is consistentwith the unoriented systems considered earlier (whichcorrespond to the limit as r!1). Note also that highercontrasts leads to smaller (larger) critical stresses forr < 1ðr > 1Þ, while higher contrasts leads to smaller criti-cal stretches that are independent of the value of r.

Fig. 11 shows results for the evolution of grain orienta-tion h and von Mises equivalent stress rh

e in the hard phase

a b

Fig. 8. Young’s modulus of oriented specimen as a function of the variance r of the initial orientation distribution for several angles hm0 between mean

lamination direction and the stressing direction.

a b

Fig. 9. Uniaxial stress–stretch curves for oriented specimens subjected to uniaxial tension along the mean stacking direction. (a) Effect of contrast in shearmoduli of the phases t. (b) Effect of Poisson’s ratio of the soft phase mð1Þ .


of the lamellae – in a perpendicularly oriented grainðh0 � 2Þ – for oriented specimens subjected to uniaxialtension along the mean stacking direction, for three differ-ent initial distributions of stacking directions withr ¼ 0:04, 0.1, and 0.2. Clearly, the deformation behaviorof the perpendicularly oriented grains in oriented speci-mens is similar to that of the unoriented specimens. Thatis, while these grains deform mainly by stretching perpen-dicular to the lamellae at small stretches, they deform pre-dominantly via a combination of rigid rotation and shearalong the lamellae close to and beyond LSE. Therefore wecan conclude that the transition from one deformationmode to a different one is once again the source for thepredicted elastic instabilities. Furthermore, the narrowerthe initial distribution of stacking directions the sharperand earlier the transition in deformation mode for the per-pendicularly oriented grains. Similar observations reveal-ing such deformation mechanisms were also made byLopez-Pamies et al. (2008) in the simpler setting of a nearlyoriented specimen composed of only two slightly misori-ented stacks of lamellae. Note from Fig. 11b that, at leastuntil LSE, the von Mises equivalent stress in the hard oflamellae in perpendicularly oriented grains is much belowthe yield stress for the hard phase. Thus, the elastic insta-bilities are predicted to precede plastic instabilities in per-

pendicularly loaded oriented specimen, in agreement withexperimental observations for this type of loading (Garcia,2006).

For completeness, Fig. 12 shows evolution plots of theODF as the deformation �k progresses all the way up toLSE at �kcr . Results are shown for two different values ofthe initial width r (0.04 and 0,1) of the orientation distri-bution. It can be seen from these figures that the behavioris qualitatively similar for the two values of r, consisting inthe broadening and flattening of the peak, eventually lead-ing to the development of a bimodal distribution about thesymmetry direction ðh ¼ 0Þ. The development of the dou-ble-bump in the distribution is again seen to be closely re-lated to the above-mentioned sudden rotation of theperpendicularly oriented grains (with small values of h0)near the LSE point. Comparing with the corresponding re-sults shown in Fig. 6a for the unoriented samples, it isinteresting to remark that although the starting point isquite different – a flat distribution for the unoriented sys-tems versus a highly peaked distribution for the orientedsystems – the elastic instability is driven by the same phys-ical mechanism of shear-and-rotation of the ‘‘unfavorably”oriented layers (i.e., those perpendicular to the tensiledirection), leading to a transition from a unimodal distri-bution to a bimodal one.

a b

dc

Fig. 10. Effect of contrast in shear moduli of the phases, distribution of initial lamination directions, and Poisson’s ratio of the soft phase on critical stretchand critical stress at loss of strong ellipticity in perpendicularly loaded oriented specimens.

a b

Fig. 11. Deformation behavior of a perpendicularly oriented grain in oriented specimens subjected to uniaxial tension along the mean stacking direction. (a)Evolution of grain orientation. (b) Evolution of von Mises equivalent stress in the hard phase of the lamellae.


Fig. 13 shows a comparison of the evolution of the ODFand macroscopic stress-strain response for uniaxial com-pression (C) with the prior results for tension (T) in an ori-ented sample. Fig. 13a again shows that the ‘‘unfavorably”oriented lamellae tend to rotate towards the stressingdirection for uniaxial tension ð�k > 1Þ, leading to theformation of a double-bump pattern. On the contrary, foruniaxial compression ð�k < 1Þ, there are essentially no ‘‘unfa-vorably” oriented lamellae since most grains are alignedwith the tensile direction that is induced by the Poisson ef-

fect in the direction perpendicular to the tensile direction.In fact, the layers still tend to rotate to align themselveswith the tensile direction, but only slightly, and in this caseit only leads to an increasingly sharper peak (in the ODF) ath ¼ 0 as the deformation progresses. By comparing themacroscopic responses for tension and compression inFig. 13b, it becomes even clearer that the stress-strainsoftening and LSE observed for uniaxial tension are theresult of the comparatively large rotation and shear of the‘‘unfavorably” oriented grains and not of the nonlinear

a b

Fig. 13. Results for a perpendicularly loaded oriented specimen in tension (T) and compression (C). (a) Evolution of the ODF. (b) Uniaxial stress–stretchcurves.

a b

Fig. 12. Evolution of the ODF in oriented specimens subjected to uniaxial tension along the mean stacking direction for two different initial distributions oforientations.


constitutive behaviors of the phases. On the other hand, foruniaxial compression, most layers are subjected locally totension along their long dimension, which leads to plasticyielding and eventual failure of the layers. The criticalstrain for plastic yielding of layers, which happens almostsimultaneously in nearly all the grains, is indicated by a fullsquare in Fig. 13b.

5. Conclusions

A homogenization-based framework has been proposedto model hyperelastic polycrystals. The analytical frame-work allows for the efficient computation of variationalestimates for the evolution of the microstructure and themacroscopic constitutive response of lamellar TPEs makingdirect use of the properties of the constituent blocks, initialdistribution of lamination directions, average grain shape,and initial volume fractions of the blocks. Motivated byexperimental observations for oriented sheet specimensprepared using the roll casting process (Honeker and Tho-mas, 1996) – and for simplicity – the average grain shapein the undeformed configuration was assumed here to be

cylindrical with circular cross-section in the transversex1 � x2 plane (the plane of the sheet). Furthermore, the ini-tial lamination directions were restricted to lie in thex1 � x2 plane so that for loadings within the plane, theproblem becomes two-dimensional (generalized planestrain). Two kinds of initial distributions of stacking direc-tions were considered: unoriented corresponding to an iso-tropic distribution of lamination directions in the planeand oriented corresponding to a Gaussian distribution oflamination directions. In line with experiments on TPEs re-ported in the literature, the analysis is restricted to uniax-ial loading in the x1 � x2 plane, the focus being the study ofthe generic effect of the elastic properties of the phases andof the initial microstructure on the macroscopic responseof lamellar TPEs up to the possible onset of macroscopicelastic instabilities.

In unoriented samples, grains with lamellae that arenearly perpendicular, diagonal, and parallel to the stressingdirection (in the undeformed configuration) exhibit con-trasting behaviors. While the initially parallel and diago-nal grains respectively deform by stretching along thelamellae and a combination of rigid rotation and shear alongthe lamellae, the initially perpendicular grains deform


predominantly by stretching along the stacking directionat small strains and via a combination of rigid rotation andshear along the lamellae soon after. The transition fromthe hard deformation mode (controlled by stretching ofthe hard phase) to the soft mode (controlled by shear ofthe soft phase and rotation of the layers) in the ‘‘unfavor-ably” oriented grains (i.e., those that are initially perpen-dicular to the tensile direction) leads to the onset of amacroscopic elastic instability. Furthermore, the lamellaein the grains tend to develop preferential orientations ofabout 20 degrees, symmetrically about the loading axis,leading to a transition from a ‘‘unimodal” to a ‘‘bimodal”distribution of orientations. (Similar observations havebeen made recently in the Ph.D. thesis of Tzianetopoulou(2007), who carried out detailed FEM simulations for themacroscopic response of polygranular specimens usingvarious realizations of unoriented specimens subjected toplane strain conditions.) This reorganization of the orienta-tion distribution of the grains leads to a shear-band insta-bility at the macroscopic level that can be relatedmicroscopically to the formation of zig-zag chevron pat-terns in isotropic styrene-butadiene-styrene (SBS) blockcopolymers (Séguéla and Prud’homme, 1981; Garcia,2006). This deformation mechanism and the associatedmacroscopic elastic instabilities were found to be stronglydependent on the heterogeneity contrast of the hard andsoft blocks, as well as on the Poisson’s ratio of the softblocks, but weakly dependent on the volume fraction ofthe blocks. In addition, the critical stress at the onset ofthe macroscopic instabilities is found to be weakly depen-dent of the properties of the hard phase, particularly forlarge elastic contrasts. The situation is similar to that of fi-ber-reinforced elastomers, subjected to compression alongthe fibers, where for extremely stiff fibers the critical stressdepends only on the properties of the elastomeric matrixand initial volume fraction of fibers (Agoras et al., 2009; Lo-pez-Pamies and Idiart, 2009). Furthermore, nearly all thegrains behave elastically until the onset of macroscopic(elastic) instabilities indicating that plastic yielding of thehard (glassy or semi-crystalline) phase is expected to havea significant effect on the overall response only after theonset of the macroscopic instabilities.

In oriented specimens, the Young’s modulus was foundto be quite sensitive to the initial distribution of orienta-tions, especially for loadings that are aligned with themean layer direction: the broader the initial distribution,the greater the deviation of the Young’s modulus (alongthe mean layer direction) relative to the correspondingperfect laminate. This could provide a possible explanationfor the large scatter in experimentally measured (Allanet al., 1991; Cohen et al., 2000) Young’s modulus alongthe lamellae in oriented SBS specimens. In oriented speci-mens subjected to uniaxial tension along the mean lamina-tion direction N, herein referred to as ‘‘perpendicularlyloaded” specimens, the microstructure evolution in thegrains and their sensitivity to the elastic properties and ini-tial volume fractions of the phases is similar to that inunoriented specimens. However, the transition in thedeformation mode for the ‘‘unfavorably” oriented grains,leading to the development of macroscopic instabilities,is strongly dependent on the variance of the orientation

distribution function, occurring much earlier and beingmore sensitive to the elastic properties of the phases for(perpendicular loading of) the oriented specimens thanfor the unoriented specimens. On the other hand, for paral-lel loading of the oriented specimens the proportion ofunfavorably oriented grains is minimal and the evolutionof the microstructure is quite different tending to increase(instead of decrease) the overall stiffness of the material, sothat no elastic instabilities were detected. Instead the lay-ers are expected to yield plastically and eventually breakup at sufficiently large strains leading to a different typeof instability which cannot be appropriately modeled withthe present theory.

Although no direct comparisons with experiments havebeen attempted in this work, the encouraging predictionsof the theoretical model for somewhat idealized constitu-ent properties and microstructures suggest that the theorydoes contain many of the basic ingredients necessary to beable to handle more realistic (3D) microstructures and con-stitutive models for the constituents blocks of TPEs withlamellar microstructures. In addition, the general theorycan be applied – with suitable extensions – for TPE withother domain structures, such as the cylindrical and spher-ical geometries (Honeker and Thomas, 1996), as well as toother polymeric systems with similar two-scale structure,such as certain types of semi-crystalline polymers, wheresimilar chevron-type instabilities have also been observed(Krumova et al., 2006).

Acknowledgements

The work of V.R. and O.L.P. was supported by theAgence Nationale de la Recherche (France) and that ofP.P.C. by the National Science Foundation (USA) throughGrant CMMI–0654063. The authors would like to thankan anonymous reviewer for pointing us to the Ph.D. thesisof T. Tzianetopoulou. Some comparisons to this work wereincluded in Fig. 7 at the suggestion of this reviewer.

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