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European Journal of Mechanics A/Solids 65 (2017) 243e256

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European Journal of Mechanics A/Solids

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

Electromechanical macroscopic instabilities in soft dielectricelastomer composites with periodic microstructures

Artemii Goshkoderia, Stephan Rudykh*

Department of Aerospace Engineering, Technion e Israel Institute of Technology, Haifa 32000, Israel

a r t i c l e i n f o

Article history:Received 26 December 2016Received in revised form14 April 2017Accepted 25 April 2017Available online 4 May 2017

Keywords:Dielectric elastomersStabilityCompositesFinite deformationMacroscopic instabilitiesMicrostructure

* Corresponding author.E-mail address: rudykh@technion.ac.il (S. Rudykh)

http://dx.doi.org/10.1016/j.euromechsol.2017.04.0080997-7538/© 2017 Elsevier Masson SAS. All rights res

a b s t r a c t

We study electromechanical macroscopic instabilities in dielectric elastomer (DE) composites under-going finite strains in the presence of an electric field. We identify the unstable domains for DE com-posites with periodically distributed circular and elliptical inclusions embedded in a soft matrix. Weanalyze the influence of the applied electric field and finite strains, as well as the microstructuregeometrical parameters and material properties, on the stability of the DE composites. We find that theunstable domains can be significantly tuned by an electric field, depending on the electric field directionrelative to pre-stretch and microstructure. More specifically, the electric field aligned with the stretchdirection, promotes instabilities in the composites, and the electric field applied perpendicularly to thestretch direction, stabilizes the composites. Critical stretch decreases with an increase in the volumefraction of circular inclusions. An increase in the contrast between the dielectric properties of the con-stituents, magnifies the role of the electric field, while an increase in the shear modulus contrast resultsin a less stable DE composite. For periodic DE composites with elliptical inclusions, we find that thecritical stretch depends on the inclination angle of the inclusion, and that the critical stretch reaches aunique maximum at an angle defined by the inclusion ellipticity aspect ratio. In the aligned case e whenthe longest side of the inclusion is aligned with the stretch direction e an increase in the ellipticity ratioresults in an increase in critical stretch.

© 2017 Elsevier Masson SAS. All rights reserved.

1. Introduction

Dielectric elastomers (DEs) can achieve large deformationswhen excited by an electric field (Pelrine et al., 1998, 2000b, 2000a).This ability, together with their lightweight, fast response time andflexibility, make DEs attractive for a wide and diverse variety ofapplications, such as artificial muscles (Bar-Cohen, 2001), energy-harvesting and noise canceling devices, soft robotics (Kornbluhet al., 2012; McKay et al., 2010; Carpi et al., 2011; Bortot et al.,2016), and tunable waveguides (Gei et al., 2011; Galich andRudykh, 2016). However, the wide spread usage of DEs has beenlimited due to the extremely high electric fields required to achievelarge strains. Thus, DEs need to operate at the risky edge of elec-tromechanical instabilities (Plante and Dubowsky, 2006; Rudykhet al., 2012; Keplinger et al., 2012; Li et al., 2013). A promisingapproach for reducing the required electric field is to design andfabricate composite materials with an enhanced electromechanical

.

erved.

coupling. Experimental studies show significant enhancements inthe electromechanical coupling in DE composites (Stoyanov et al.,2011; Huang and Zhang, 2004). Moreover, theoretical estimatesand numerical simulations (Tian et al., 2012; Rudykh et al., 2013)predict even more significant improvements in the performance ofDE composites with periodic microstructures. Thus, improvementby orders of magnitude in the electromechanical coupling can beachieved in hierarchically structured composites comprising softerand stiffer phases (Rudykh et al., 2013). Recent advances in themicrostructured material fabrication and 3D printing, allowingrealization of highly structured materials at different length-scales(Kolle et al., 2013; Lee and Fang, 2012; Zheng et al., 2014; Slesarenkoand Rudykh, 2016), provide a great perspective for this approach forenhancing DE performance.

The foundation for the non-linear electroelasticity theory waslaid by the pioneering works by Toupin (1956, 1960) showingthat electromechanical coupling in DEs is characterized by aquadratic dependence on the applied electric field. Recently, theelectroelasticity theory of finite deformations has been refor-mulated by Dorfmann and Ogden (2005, 2010), McMeeking and

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256244

Landis (2005), and Suo et al. (2008); Zhao and Suo (2010), and,more recently, by Liu (2013), and by Li et al. (2016). Itskov andKhiem (2014) and Ortigosa and Gil (2016) considered the as-pects of the convexity of the electro-elastic energy functions.Cohen et al. (2016), Cohen and deBotton (2016) conducted astatistical-mechanics-based analysis of the response of polymerchain networks in DEs. In parallel, significant efforts have beenmade towards the development and implementation of the non-linear electroelasticity framework into numerical schemes (Vuand Steinmann, 2007; Volokh, 2012; Javili et al., 2013; Keipet al., 2014; Galipeau et al., 2014; Jabareen, 2015; Aboudi,2015). The electromechanical instabilities in finitely deformedhomogenous DEs have been analyzed by Zhao and Suo (2007),and Dorfmann and Ogden (2010, 2014), in parallel with theexperimental observations of the failure modes such as pull-ininstabilities (Plante and Dubowsky, 2006), creasing and surfacepatterning (Wang et al., 2011). Based on an exact analytical so-lution available for finitely deformed periodic layered DE com-posites, the studies of the electromechanical instabilities in theperiodic DE laminates have been performed (Bertoldi and Gei,2011; Rudykh and deBotton, 2011; Rudykh and Bertoldi, 2013;Rudykh et al., 2014). These works show the significant depen-dence of DE material stability on the applied electric field andpre-stretch. However, the set of microstructures for which exactanalytical solutions can be derived is limited; as a result, verylittle is known about the instabilities in DE composites withparticulate and periodic microstructures, which showed prom-ising results of significant enhancement in electromechanicalcoupling and actuation (Rudykh et al., 2013). Moreover, theknowledge about the instabilities in these microstructuredelectro-active composites may provide the tools for designingmaterials with switchable functionalities (Bertoldi et al., 2008;Bertoldi and Boyce, 2008; Krishnan and Johnson, 2009; Rudykhand Boyce, 2014; Singamaneni et al., 2008; 2009).

In this study, we perform an analysis of electromechanicalinstabilities in finitely deformed DE composites with periodicallyarranged active particles embedded in a matrix. In particular, wefocus on the macroscopic stability of periodic two-dimensionalDE composites with circular and elliptical inclusions. We imple-ment the electromechanical instability analysis into a numericalfinite element based tool, and identify the unstable domains forfinitely deformed DE composites in the presence of an electricfield. We analyze the influence of the electric field, pre-stretch,microstructure and material parameters on DE compositestability.

The work is structured as follows: Sec. 2 presents the theoreticalbackground for the finitely deformed dielectric elastomers andelectromechanical instability analysis previously developed byDorfmann and Ogden (2005, 2010) and its specification for a planeproblem reported in Rudykh et al. (2014). The numerical simula-tions, including the electromechanical periodic boundary condi-tions, and the procedure for determination of the electroelasticmoduli are described in Sec. 3. In Sec. 4, we apply the stabilityanalysis to identify the unstable domains for the DE compositeswith periodically distributed circular (4.1) and elliptical (4.2) in-clusions embedded in a matrix. Sec. 5 concludes the paper with asummary and a discussion.

2. Theoretical background

We denote by B 0 and B the regions occupied by a body in thereference and current configurations, respectively. The Cartesianposition vector of a material point in the reference configuration ofa body is X and its position vector in the deformed configuration isx. We introduce a mapping vector function c such that

x ¼ cðXÞ: (1)

The deformation gradient is defined as

F ¼ vcðXÞvX

: (2)

The ratio between the volumes in the current and referenceconfigurations is J ≡ detF>0.

We consider a quasi-static deformation in the absence of amagnetic field, electrical charges or electric currents within thematerial. Consequently, Maxwell equations take the form

DivD0 ¼ 0 and CurlE0 ¼ 0; (3)

where D0 is the electric displacement and E0 is the electric field inthe reference configuration. Note that Div($) and Curl($) are thedifferential operators in the reference configuration, while div($)and curl($) denote the corresponding differential operators in thecurrent configuration. The referential electric field and electricdisplacement are related to their counterpart in the deformedconfiguration (Dorfmann and Ogden, 2005, 2010) via

E0 ¼ FTE and D0 ¼ JF�1D: (4)

We follow the analysis proposed by Dorfmann and Ogden (2005,2010) and consider the elastic dielectrics whose constitutive rela-tion is given in terms of a scalar-valued energy-density functionJðF;E0Þ such that

P ¼ vJ�F;E0

�vF

and D0 ¼ �vJ�F;E0

�vE0

; (5)

where P is the total nominal stress tensor. The correspondingequations for an incompressible material are

P ¼ vJ�F;E0

�vF

� pF�T and D0 ¼ �vJ�F;E0

�vE0

; (6)

where p is an unknown Lagrange multiplier. For an isotropic elec-troelastic material, an energy-density function J can be expressedas a function of the six invariants

J�F;E0

�¼ JðI1; I2; I3; I4e; I5e; I6eÞ; (7)

where

I1 ¼ TrC; I2 ¼ 12

�I21 � TrC2

�; I3 ¼ det C; (8)

I4e ¼ E0$E0; I5e ¼ E0$C�1E0; I6e ¼ E0$C�2E0; (9)

where C ¼ FTF is the right Cauchy-Green strain tensor. In theabsence of body forces the equilibrium equation takes the form

DivP ¼ 0: (10)

The equilibrium equation in the current configuration is

divT ¼ 0; (11)

where the Cauchy stress tensor is related to the first Piola-Kirchhoffstress tensor via T ¼ J�1PFT .

Next we analyze small amplitude perturbations superimposedon the finitely deformation and electric field (Dorfmann and Ogden,2010). The corresponding incremental equations are

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256 245

Div _P ¼ 0; Div _D0 ¼ 0 and Curl _E

0 ¼ 0; (12)

where _P, _D0and _E

0are incremental changes in P, D0 and E0,

respectively. The linearized expressions for the incrementalchanges in the nominal stress tensor and electric displacement are

_Pij ¼ A 0ijkl

_Fkl þ G 0ijk

_E0k and � _D

0i ¼ G 0

jki_Fjk þ E 0

ij_E0j ; (13)

where the tensors of the electroelastic moduli are defined as

A 0iakb ¼

v2JvFiavFkb

; G 0iab ¼

v2J

vFiavE0band E 0

ab ¼v2J

vE0avE0b

: (14)

In general, the electroelastic moduli are functions of the appliedfinite deformation and electric field, namely, A 0 ¼ A 0�F;E0�,G 0 ¼ G 0�F;E0�, and E 0 ¼ E 0�F;E0� , and the specific dependenceis defined by the choice of the energy-density functionJðF;E0Þ. Foran incompressible material, equation (13)1 modifies as

_Pij ¼ A 0ijkl

_Fkl þ G 0ijk

_E0k � _pF�1

ji þ pF�1jk

_FklF�1li ; (15)

where _p is an incremental change in p. The components of theelectroelastic moduli in the current and reference configurationsare related via

A ijkl ¼ J�1FjaFlbA0iakb; G ijk ¼ J�1FjaFkbG

0iab and E ij

¼ J�1FiaFjbE0ab: (16)

It can be shown that these tensors of electroelastic modulipossess the symmetries

A ijkl ¼ A klij; G ijk ¼ G jik and E ij ¼ E ji: (17)

Let _T, _D and _E denote the ‘push-forward’ counterparts of _P, _D0

and _E0, respectively. These incremental changes are given by

G0 ¼ G 2121 � A 2121E 11;

G1 ¼ 2ð � A 2121E 12 þ ðA 1121 � A 2122ÞE 11 þ G 121ðG 221 þ G 122 � G 111ÞÞ;G2 ¼ �A 2121E 22 þ 4ðA 1121 � A 2122ÞE 12 � ðA 1111 � 2A 1122 � 2A 1221 þ A 2222ÞE 11 � 2G 121ðG 112 þ G 121 � G 222Þ

þ ðG 122 þ G 221 � G 111Þ2;G3 ¼ �2ððA 1112 � A 1222ÞE 11 þ ðA 1111 � 2A 1122 � 2A 1221 þ A 2222ÞE 12 þ ðA 2122 � A 1121ÞE 22

þ ðG 121G 122 � ðG 111 � G 122 � G 221ÞðG 112 þ G 121 � G 222ÞÞÞG4 ¼ �ðA 1111 � 2A 1122 � 2A 1221 þ A 2222ÞE 22 � 4ðA 1112 � A 1222ÞE 12 � A 1212E 11 þ ðG 112 þ G 121 � G 222Þ2

þ 2G 122ðG 111 � G 122 � G 221Þ;G5 ¼ 2ððA 1222 � A 1112ÞE 22 � A 1212E 12 þ G 122ðG 112 þ G 121 � G 222ÞÞ;G6 ¼ G 2

122 � A 1212E 22:

(27)

_T ¼ J�1 _PFT ; _D ¼ J�1F _D0

and _E ¼ F�T _E0: (18)

We introduce the notation for the incremental displacement v ¼_x and recall that _F ¼ ðgrad vÞF. By substitution of (15) and (16) into(18), we obtain

_Tij ¼ A ijklvk;l þ G ijk_Ek � _pdij þ pvj;i and � _Di

¼ G jkivj;k þ E ij_Ej: (19)

In the current configuration the incremental equations (12) takethe form

div _T ¼ 0; div _D ¼ 0 and curl _E ¼ 0: (20)

By using equations (19), (20)1 and (20)2 we obtain

A ijklvk;lj þ G ijk_Ek;j � _p;i ¼ 0 and G jkivj;ki þ E ij

_Ej;i ¼ 0: (21)

We seek a solution for equation (21) in the form

vi ¼ ~vif ðba$xÞ; _p ¼ ~qf 0ðba$xÞ; _Ei ¼ ~eif0ðba$xÞ; (22)

where f is a continuous and sufficiently differentiable function, ba ¼a1e1 þ a2e2 þ a3e3 is a unit vector; ~vi, ~ei and ~q are incrementalmacroscopic quantities independent of x.

Next, we restrict our attention to a 2D setting. For this specifi-cation, we follow the stability analysis presented in Rudykh et al.(2014). In particular, for the plane problem we assume theabsence of an electric field in the X3 direction, therefore equation(20)3 reduces to

_E2;1 � _E1;2 ¼ 0: (23)

By using (22)3 together with (23), we find that

~e1 ¼ x�1~e2; (24)

where x ≡ a2=a1. Next, we recall that the incompressibilityconstraint implies that div v ¼ n1;1 þ n2;2 ¼ 0. Then, from (22)1 wehave

~v1 ¼ �x~v2: (25)

Substituting equations (22), (24) and (25) into (21), we obtainthe sextic polynomial equation

G6x6 þ G5x

5 þ G4x4 þ G3x

3 þ G2x2 þ G1xþ G0 ¼ 0; (26)

where the coefficients Gi are given by

The existence of a non-trivial real solution of the polynomialequation (26) is associated with the onset of instability along anelectromechanical loading path. In the following, equation (26)serves as the onset of instability condition for considered DEcomposites.

3. Analysis

In this sectionwe present the analysis for determining the onset

Fig. 1. Schematic representation of two-dimensional rectangular unit cell.

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256246

of electromechanical instabilities in periodic composites undergo-ing finite deformations in the presence of an electric field. We notethat the critical condition (26) can be used to determine the onsetof instabilities in multiphase hyperelastic dielectrics (Rudykh et al.,2014). The analysis is implemented in the finite element codeCOMSOL Multiphysics.

The energy-density function of an n-phase composite is

J�F;X;E

0�¼

Xnr¼1

4ðrÞðXÞJðrÞ�F;E0� (28)

where

4ðrÞðXÞ ¼�1; if X2B ðrÞ

0 ;0; otherwise;

(29)

The volume fraction of the r-phase is

cðrÞ ¼ZB 0

4ðrÞðXÞdV : (30)

The volume fractions are related via

Xnr¼1

cðrÞ ¼ 1: (31)

We impose homogeneous boundary conditions x ¼ F0X on theboundary of the composite vB 0, where F0 is a constant tensor withdet F0 >0 (Hill, 1972; Ogden,1974). It can be shown that the averagedeformation gradient

F ≡1V

ZB 0

FðXÞdV ¼ F0: (32)

The average 1st Piola-Kirchhoff stress tensor, electric displace-ment and electric field are

P ¼ 1V

ZB 0

PðXÞdV ; D0 ¼ 1

V

ZB 0

D0ðXÞdV and E0

¼ 1V

ZB 0

E0ðXÞdV ; (33)

respectively. In the 2-D case the integration is performed over thearea of a periodic unit cell, which occupies the domain

0 � X1 � b and 0 � X2 � a (34)

in the reference configuration.In our finite element numerical simulations, the macroscopic

electro-mechanical loading conditions are applied in terms of pe-riodic boundary conditions of displacement ðu1;u2Þ and electricpotential (U). A representetive 2-D unit cell of a periodic compositeis shown in Fig.1. These periodic boundary conditions are:

The top ðX2 ¼ aÞ and bottom ðX2 ¼ 0Þ sides are related via

8><>:uB2 ¼ uT2 þ �

F22 � 1�a;

uB1 ¼ uT1 þ F12a;

UBðX1;0Þ ¼ UT ðX1; aÞ þ E02a:

(35)

The right ðX1 ¼ bÞ and left ðX1 ¼ 0Þ sides are related via

8><>:uL1 ¼ uR1 þ

�F11 � 1

�b;

uL2 ¼ uR2 þ F21b;

ULð0;X2Þ ¼ URðb;X2Þ þ E01b:

(36)

The application of the periodic boundary conditions allows us todetermine the solution along the electro-mechanical loading path.For simplicity, here we consider the electromechanical loadingdefined as

F ¼ le15e1 þ l�1

e25e2 þ e35e3 and E0 ¼ E

01e1 þ E

02e2:

(37)

We note that the deformation gradient defined in equation (37)does not include macroscopic shear modes of deformation. Forstability consideration of a more general deformation includingnon-zero shear terms, the average deformation gradient with non-zero shear components needs to be transformed into the corre-sponding diagonal average deformation gradient defined in equa-tion (37) by choosing the corresponding principal coordinatesystem. Thus, the components of the electromechanical tensors areto be evaluated in this principal coordinate system; and themacroscopic stability analysis is performed in terms of the modulicalculated in the new principal coordinate system. For simplicity, inthis study we restrict our analysis to the case when the averagedeformation gradient is prescribed by equation (37).

The average response of the periodic composites is calculated byintegration over the corresponding unit cell as in (33)1 and (33)2.The primary solution is valid until a possible instability point alongthe electromechanical loading.

To identify the onset of the coupled electromechanical in-stabilities, we utilize the critical condition (26). The analysis of theelectromechanical stability of DE composites requires determina-tion of the instantaneous tensors of the electroelastic moduli A ijkl,G ijk and E ij. To this endwe use a numerical perturbation technique(see for example for the mechanical case (Riks, 1984; Ecer, 1973;Hangai and Kawamata, 1972)). In particular, we perform a set ofincremental changes in deformation and electric field from afinitely deformed state in the presence of an electric field. Theseincremental changes lead to a macroscopic response of the DEcomposite, resulting in variations in the average nominal stress andelectric displacement. We perform the following five small ampli-tude numerical tests, namely

F½1� ¼ le15e1 þ dge15e2 þ l

�1e25e2 and F

½2�

¼ le15e1 þ l�1

e25e2 þ dge25e1; (38)

F½3� ¼

�lþ dl

�e15e1 þ

�lþ dl

��1e25e2; (39)

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256 247

E0½1� ¼

�E01 þ dE

�e1 þ E

02e2 and E

0½2� ¼ E01e1 þ

�E02 þ dE

�e2;

(40)

where dg; dl and dE are sufficiently small quantities. By per-forming these tests and calculating the corresponding changes inthe first Piola-Kirchhoff stress tensor and in the referential electricdisplacement, the electroelastic moduli in the reference configu-ration are determined through the use of relations (13). The cor-responding equations for the electroelastic moduli are

A 0ijkl ¼

dP½m;0�ij

dF½m�kl

¼

�P�F½m�

;E0�� P

�F;E

0��

ij�F½m� � F

�kl

¼

�P�F½m�

;E0�� P

�F;E

0��

ij

dg; ksl;m ¼ 1;2; (41)

G 0ijk ¼

dP½0;n�ij

dE0½n�k

¼

�P�F;E

0½n��� P�F;E

0��

ij�E0½n� � E

0�k

¼

�P�F;E

0½n��� P�F;E

0��

ij

dE;n ¼ 1;2; (42)

E 0ij ¼ � dD

0½0;n�i

dE0½n�j

¼ �

�D0�F;E

0½n��� D0�F;E

0��

i�E0½n� � E

0�j

¼ �

�D0�F;E

0½n��� D0�F;E

0��

i

dE;n ¼ 1;2: (43)

Note that in equation (41) only shear tests F½1�

and F½2�

areapplied, consequently m ¼ 1;2. To determine the components ofelectroelastic moduli A 0

ijkk (no summation), isj, we use the sym-metry A 0

ijkk ¼ A 0kkij and apply equation (41) again. However, when

i ¼ j and k ¼ l, the components A 0iikk (no summation) cannot be

determined from the tension test (39). This is due to the fact thatthe following equation system stemming from equation (13)(dP

½3;0�11 ¼ A 0

1111dF½3�11 þ A 0

1122dF½3�22;

dP½3;0�22 ¼ A 0

2211dF½3�11 þ A 0

2222dF½3�22;

(44)

does not have a solution for A 01111, A

02222, and A 0

1122 ¼ A 02211.

Clearly, it is impossible to fully characterize the elastic properties ofa material on the basis of plane tests alone (Ogden, 2008). However,the corresponding coefficients in the critical condition for the onsetof instabilities (26) can be fully determined through certain com-binations of the terms A 0

iikk (no summation). These combinationscan be obtained from the information provided by the tension test(39). The incompressibility constraint implies

dF : F�T ¼ 0: (45)

For the tension test (39), the incompressibility constraint reads

dF½3�11 ¼ �dF

½3�22F11F22

¼ � dF½3�22�

F22�2 : (46)

Thus, the combinations ðA 02222 � A 0

2211Þ and ðA 01111 � A 0

2211Þcan be determined by applying the tension test (39) together with(16)1, (44) and (46). In particular, we obtain

A 02222 � A 0

2211 ¼ F22F22A 2222 � F22F11A 2211

¼ F222�A 2222 � A 2211F11

�F22

�¼ F

222dP

½3;0�22

.dF

½3�22; (47)

and

A 01111 � A 0

2211 ¼ F11F11A 1111 � F22F11A 2211

¼ F211�A 1111 � A 2211F22

�F11

�¼ F

211dP

½3;0�11

.dF

½3�11; (48)

where dP½3;0�ii ¼ ðPðF½3�;E0Þ � PðF;E0ÞÞii, no summation.

By utilizing equations (16), (17), (27), (41e43), (47), (48) and(27), all the components of the electro-elastic tensors required forcalculating of the coefficients Gi can be determined. To perform thestability analysis, we implement these equations in our MATLABcode to evaluate the components of electroelastic moduli from thenumerical tests, and, then, to check the condition for the onset ofinstability (26) at each point of the electromechanical loading path.Thus, the critical stretch and electric field are determined. Results ofthe stability analysis of specific periodic DE composites are dis-cussed in the next section.

4. Results

The stability analysis presented in Sec. 3 is rather generic,however, in this section the examples are presented for DE com-posites which phases are described by an ideal dielectric elastomermodel. The corresponding energy-density function is

JðrÞ�F;E0� ¼ mðrÞ

2ðTrC� 3Þ � ε0ε

ðrÞJ2

E0C�1E0 � mðrÞ ln J

þLðrÞ

2ðJ � 1Þ2; (49)

where m is the shear modulus andL is the Lam�e’s first parameter, ε0is the permittivity of vacuum, and ε is the relative permittivity. Inthe numerical simulations, we maintain the nearly incompressiblebehavior of the phases by assigning a high ratio between the firstLam�e parameter and shear modulus, in particular, LðrÞ=mðrÞ ¼ 100.Note that the macroscopic deformation is enforced to be incom-pressible, while locally the materials are allowed to develop somecompressible (nearly incompressible) deformations.

To determine the response of periodic DE composites prior toinstability, we utilize (35) and (36) to impose the periodicboundary conditions on a unit cell. The equilibrium equation andthe Maxwells equations are solved separately in the FE codeCOMSOL Multiphysics, and the electromechanical coupling isestablished via the dependence of the resulting total stress onthe electric field, while the Maxwells equations are solved in thecurrent configuration, which is a result of deformations. Theresulting output is calculated in terms of the average nominalstress tensor (33)1 and electric displacement (33)2. By perform-ing the small amplitude numerical tests (38e40) and making useof relations (41e43), (47) and (48), we determine the elec-troelastic moduli in the reference configuration. Then, thecounterparts of the electroelastic moduli in the current config-uration are evaluated by using equation (16). To determine thecritical stretch lc, we check the condition for the onset ofmacroscopic instability (26) at each point of the electrome-chanical loading path (37).

Fig. 2. Schematic representation of DE composites with square and rectangular periodic microstructures with circular (a) and elliptical (b) inclusions; the periodic unit cell (c).

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256248

4.1. DE composites with periodically distributed circular inclusions

We examine the response of DE composites with periodicallydistributed circular particles embedded in a soft matrix. Fig. 2(a)shows a representative 2-D periodic composite with a square pe-riodic unit cell of circular inclusions embedded in a matrix. Thecontrasts in the shear moduli and relative permittivities of thephases are km ¼ mðiÞ=mðmÞ and kε ¼ ε

ðiÞ=εðmÞ, where superscripts ðiÞand ðmÞ correspond to the inclusion and matrix, respectively. The

Fig. 3. Normalized deviatoric Cauchy stress vs stretch ratio for DE composites with the cont50 ((c),(d)). The volume fraction of the inclusion phase is cðiÞ ¼ 0:283 (corresponds to the d

average dimensionless electric fields are bE0 ¼ E0 ffiffiffiffiffiffiffiffiffiffiffiffi

ε0ε=mp

(where

ε ¼ cðmÞεðmÞ þ cðiÞεðiÞ and m ¼ cðmÞmðmÞ þ cðiÞmðiÞ) and

~E0 ¼ E

0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε0ε

ðmÞ=mðmÞp. The volume fraction of the inclusion phase is

defined by cðiÞ ¼ pd2=4a2.We start from considering the macroscopic response of the pe-

riodic DE composites. Fig. 3 shows the dependence of the meandeviatoric part of the normalized Cauchy stress, s0 ¼ devs=mðmÞ as afunction of the stretch ratio for different values and directions of the

rasts in the shear moduli and relative permittivities km ¼ kε ¼ 10 ((a),(b)) and km ¼ kε ¼iameter d ¼ 0:6).

Fig. 4. Critical stretch ratio vs electric field for DE composites with the contrasts in the shear moduli and relative permittivities km ¼ kε ¼ 10. (a) - the electric field is aligned withthe stretch direction, (b) - the electric field applied perpendicularly to the stretch direction.

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256 249

applied electric field. The results are shown for DE composites withthe contrasts in the shear moduli and relative permittivitieskm ¼ kε ¼ 10 (Fig. 3(a) and (b)) and km ¼ kε ¼ 50 (Fig. 3(c) and (d)).

The composite volume fraction is cðiÞ ¼ 0:283 (corresponds to thediameter d ¼ 0:6a). The stress-stretch behavior for the purely me-

chanical case (in the absence of an electric field, bE0 ¼ 0) is denotedbydashed curves. The continuous curvecorresponds to the responseof the DE composites subjected to an electric field of magnitudebE01 ¼ 0:5; the applied electric field is aligned with the stretch di-

rection ðe1Þ. The dash-dotted curve represents the response of theDE composites in the presence of an electric field applied perpen-

dicularly to the stretch direction, E ¼ bE02e2 with bE0

2 ¼ 0:5. The valuesof the critical stretches corresponding to the onset of instability are

denoted by circles. We observe that s10ðlÞ is a monotonically

increasing function of l,whiles20ðlÞ is a decreasing function. Clearly,

the associate electrical stresses in the unit cell depend on the spatialelectric field and the effective permittivity. Meanwhile, e1 and e2components of the average electricfield in the current configuration

depend on the stretch as E1 ¼ l�1

E01 and E2 ¼ lE

02. The effective

permittivity ~ε11 is a decreasing function of the stretch ratio, while~ε22 is a monotonic increasing function of the stretch ratio.1 This,together, with the fact that E1 decreases and E2 increases with an

increase in l, results in the observed behavior of the stress-stretchcurves in Fig. 3. In particular, the corresponding continuous curves

approach the dashed ones with an increase in l, while the dash-dotted curves shift away from the dashed ones with an increase inthe applied stretch.

Next we examine the stability of the DE periodic compositesalong the applied electromechanical loading (39). The dependenceof the critical stretch on the electric field is shown in Fig. 4 for thecomposite with the contrasts in the shear moduli and relativepermittivities km ¼ 10 and kε ¼ 10, respectively. The dash-dotted,continuous and dashed curves are for the volume fractionscðiÞ ¼ 0:283 (corresponds to the diameter d ¼ 0:6a), 0.196

1 This is based on our numerical observations and analytical predictions foreffective permittivity of the 2D unit cell, where the circular inclusion is assumed tobe significantly stiffer than the matrix phase (for example, km ¼ 50; 100); thus, onecan obtain an explicit expression for effective permittivity as functions of stretchbased on the results by Sihvola and Lindell, 1992, K€arkk€ainen et al., 2000.

(corresponds to the diameter d ¼ 0:5a) and 0.126 (corresponds tothe diameter d ¼ 0:4a), respectively. The arrows indicate the tran-sition from the stable to unstable domains. Indeed, in the absenceof an electric field the critical stretches correspond to the case of thepurely mechanical loading. Fig. 4(a) shows the results for the casewhen the electric field is aligned with the stretch direction, whileFig. 4(b) corresponds to the case when the electric field is appliedperpendicularly to the stretch direction. We observe that the crit-ical stretch is a monotonically decreasing function of an electricfield for the aligned case (Fig. 4(a)). When the electric field isapplied perpendicularly to the stretch direction (Fig. 4(b)), thecritical stretch increases with an increase in the electric field, thus,manifesting in a stabilizing effect of the electric field in this case. DEcomposites become more stable with a decrease in the volumefraction of the inclusion for both directions of the applied electricfield. The dashed curve, corresponding to the DE with the lowestconsidered volume fraction cðiÞ ¼ 0:196 (corresponds to the diam-eter d ¼ 0:4a), is above the solid and dash-dotted curves for allvalues of the applied electric field (see Fig. 4).

Note that in this work we prescribe the deformation togetherwith the applied electric field, and identify the stable and unstabledomains in terms of these two electromechanical loading param-eters. Alternatively, one can specify electric field and applied trac-tions t ¼ sn (where n is the outward normal), and find a loadingpath for these conditions in terms of calculated deformations oractuations. The reported in Fig. 4 curves separate the stable andunstable domains for a variety of applied combinations of electricfield and tractions. To illustrate this we plot in Fig. 5 the depen-dence of the stretch on the applied electric field for the DE com-posites subjected to different constant tractions. The dimensionlesstraction is normalized by the shear modulus of the matrix, namely~t ¼ t=mðmÞ. The dashed, dotted and dash-dotted curves correspondto the tractions ~t2 ¼ �3:8, ~t2 ¼ �3:7 and ~t2 ¼ �3:5, respectively.We observe that stretch increases with an increase in the electricfield. A decrease in the absolute value of traction shifts the curve tothe left side. The continuous curve in Fig. 5 corresponds to the onsetof instability for the composite (which is also shown by the dashedcurve in Fig. 4(b)). The shaded region above the continuous curverepresents the unstable domain. The intersection points of the

continuous and dashed ðbE02 ¼ 0:15; lc ¼ 1:77Þ, continuous and

dotted ðbE02 ¼ 0:25; lc ¼ 1:78Þ, and continuous and dash-dotted

Fig. 5. Stretch vs electric field for the DE composite subjected to different constanttractions. The contrast in the shear moduli and relative permittivities is km ¼ kε ¼ 10.The volume fraction of the inclusion is cðiÞ ¼ 0:126 (corresponds to the diameterd ¼ 0:4a). The continuous curve corresponds separates the stable and unstable (greyarea) domains.

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256250

ðbE02 ¼ 0:38; lc ¼ 1:81Þ curves correspond to the onset of the elec-

tromechanical instabilities along the corresponding loading pathswith different tractions. The continuous curve represents the set ofonset of instability points for all possible applied tractions.

Next, we present the dependence of the critical stretch on thecontrast in the shear moduli km. The contrast in the relative per-mittivities of the phases is kept fixed at the value kε ¼ 10. The dash-dotted, continuous and dashed curves correspond to the electric

fields ~E01 ¼ 1; ~E

01 ¼ 0:5 and ~E

0 ¼ 0 in Fig. 6(a), and ~E02 ¼ 0:2; ~E

02 ¼

0:1 and ~E0 ¼ 0 in Fig. 6(b), respectively. We observe that the crit-

ical stretch decreases with an increase in the contrast in the shearmoduli. We find that, for any contrast in the shear moduli, theelectric field aligned with the stretch direction (e1-axis) promotesinstabilities (see Fig. 6(a)), whereas the electric field appliedperpendicularly to the stretch direction stabilizes the DE compos-ites (see Fig. 6(b)). We also observe a rapid increase in the critical

Fig. 6. Critical stretch vs contrast in the shear moduli for composites with the contrast in re(b) - the electric field applied perpendicularly to the stretch direction. The volume fraction

stretch as the contrast in the shear moduli decreases towards themechanically homogeneous composites (km ¼ 1). The correspond-ing curves flatten with a further increase in the shear modulikma60.

The dependence of the critical stretch on the contrast in bothshear moduli and electric permittivities is shown in Fig. 7. Thedash-dotted, continuous and dashed curves correspond to the

electric fields ~E02 ¼ 0:4; ~E

01 ¼ 1 and ~E

0 ¼ 0, respectively. The vol-ume fraction of the inclusion is cðiÞ ¼ 0:283 (corresponds to thediameter d ¼ 0:6a). We observe that an increase in the phasecontrast leads to a decrease in the critical stretch. Consistently withthe previous observations, we find that the critical stretch dashed-dotted curve for the case when the electric field applied perpen-dicularly to the stretch direction, appears above the dashed curvecorresponding to the purely mechanical loading (E ¼ 0), and alsoabove the curve corresponding to the case when the electric field isaligned with the stretch direction in Fig. 7.

Next, we examine the influence of the relative permittivity ratiokε on the critical stretch for composites with the fixed contrast inthe shear moduli km ¼ 10. The dependence of the critical stretch onthe contrast in relative permittivities is shown in Fig. 8(a). Thedash-dotted, continuous and dashed curves correspond to theresponse of DE composites subjected to the electric fields~E02 ¼ 0:4; ~E

01 ¼ 1 and ~E

0 ¼ 0, respectively. The volume fraction ofthe inclusion phase is cðiÞ ¼ 0:283 (corresponds to the diameterd ¼ 0:6a). The electric field stabilizes composite when appliedperpendicularly to the stretch direction, and the electric field pro-motes instabilities when applied in the stretch direction (e1).

However, we observe that the curves lcðkεÞ flatten for both di-rections of the applied electric field with an increase in relativepermittivity ratio. To provide more details on this behavior we plotthe dependence of the average electric field in the matrix and in-clusion phases on the relative permittivity ratio in Fig. 8(b). The

results are presented for the applied electric field is ~E02 ¼ 0:4 and

stretch l ¼ 1:1. The dotted horizontal line corresponds to the

average electric field ~E2 ¼ l ~E02 ¼ 0:44 in the deformed state. The

dash-dotted and continuous curves correspond to the currentelectric fields in the matrix and inclusion phases, respectively. Weobserve that for a high relative permittivity ε

ðiÞ, the electric field

almost does not penetrate into the inclusion phase. When εðiÞ

lative permittivities kε ¼ 10. (a) - the electric field is aligned with the stretch direction,of the inclusion phase is cðiÞ ¼ 0:126 (corresponds to the diameter d ¼ 0:4a).

Fig. 8. (a) - critical stretch vs relative permittivity ratio, (b) - the dependence of the averagethe shear moduli km ¼ 10. The volume fraction of the inclusion phase is cðiÞ ¼ 0:283 (corres

Fig. 9. Critical stretch vs volume fraction of the inclusion phase for DE composites wkm ¼ kε ¼ 50 (b).

Fig. 7. Critical stretch vs phase material property contrast. The volume fraction of theinclusion is cðiÞ ¼ 0:283 (corresponds to the diameter d ¼ 0:6a).

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256 251

approaches infinity, the electric field and the associated electro-static stresses in the inclusion phase vanish while the corre-sponding electrostatic part of the total stresses in the matrix phasereach a saturation value. Consequently, the influence of the contrastin the phase permittivities on the critical stretch ratio reduces, sothe corresponding curves (see the dash-dotted and continuouscurves in Fig. 8(a)) asymptotically approach a constant value. Wenote that for permittivity ratio ε2=ε1 ¼ 1we obtain the same criticalstretch regardless the direction and magnitude of the appliedelectric field; in such case, the stability is fully determined by themechanical properties of the phases.

To illustrate the influence of phase volume fractions on theonset of the electromechanical instability, we plot the criticalstretch as a function of inclusion volume fraction in Fig. 9. The dash-dotted, continuous and dashed curves correspond to the electric

fields ~E02 ¼ 0:3, ~E

01 ¼ 0:6 and ~E

0 ¼ 0, respectively. The contrasts inthe shear moduli and relative permittivities are km ¼ kε ¼ 10 inFig. 9(a) and km ¼ kε ¼ 50 in Fig. 9(b). In a manner similar to thepurely mechanical case, we observe that the critical stretch de-creases with an increase in the inclusion volume fraction. Thisdecrease is greater at lower volume fractions cðiÞ(0:2 than at

electric field in the matrix and inclusion phases for DE composites with the contrast inponds to the diameter d ¼ 0.6a).

ith the contrasts in shear moduli and relative permittivities km ¼ kε ¼ 10 (a) and

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256252

moderate ones cðiÞa0:4. Similar to the previous observations, wefind that the electric field stabilizes the composite if appliedperpendicularly to the stretch direction, while the electric fieldapplied in the stretch direction promotes instabilities.

4.2. DE composites with periodically arranged elliptical inclusions

Nextwe investigate the behavior of periodic DE composites withelliptical inclusions embedded in a soft matrix. The schematicillustration of the periodic composites and corresponding unit cellare shown in Fig. 2(b) and (c). The ratio between the length of thesides of the rectangular unit cell is a=b ¼ 1:5. The volume fraction ofthe inclusion is defined by cðiÞ ¼ pr1r2=ðabÞ. In the non-alignedcase, when the inclination angle of the elliptical inclusion in theundeformed configuration is as0;p=2, the particle rotates with theapplied tension. As a result, more complicated distributions of thestress and electric field components is induced by a macroscopi-cally applied electromechanical loading, as compared to the alignedcase ða ¼ 0;p=2Þ. An example of the inhomogeneous distributionsof the deviatoric parts of the Cauchy stress components and electricfield are shown in Fig. 10. The DE composite with the volumefraction of the inclusion cðiÞ ¼ 0:136 is subjected to the stretch

l ¼ 1:55. The aspect ratio and inclusion inclination angle arew ¼ r1=r2 ¼ 2 ðr2 ¼ 0:18aÞ and the inclusion inclination angle isa ¼ p=4. The contrast ratios are km ¼ kε ¼ 50

Fig. 10. Distribution of the deviatoric Cauchy stress components (aed) and electric field (e((b),(d),(f),(h)). The contrasts in the shear moduli and relative permittivities are km ¼ kε ¼aspect ratio of the particles are cðiÞ ¼ 0:136 and w ¼ 2 (corresponds to the semi-major axis

ðmðmÞ ¼ 1 MPa and εðmÞ ¼ 5Þ. The field distributions are presented

for the applied electric fields bE01 ¼ 1 in Fig.10(a), (c), (e) and (d), and

for bE02 ¼ 0:3 in Fig. 10(b), (d), (f) and (h). We note that the applied

electromechanical loading (bE01 ¼ 1, l ¼ 1:55) corresponds to the

onset of the electromechanical instabilities. We observe that thehighest stresses appear in the regions near the inclusion for bothdirections of the applied electric field (see Fig. 10(a)-(d)). Note thatrelatively high concentrations of the electric fields are found at theregions near the inclusion. This is despite the fact that the corre-sponding average values of these fields equal to zero, E1 ¼ 0 inFig. 10(f), and E2 ¼ 0 in Fig. 10(g). However, the electric field almostdoes not penetrate the inclusion. In Fig. 10(e) and (h) we observethat the electric field in the inclusion is homogeneous and its ab-solute magnitude within the inclusion is smaller than the absolutevalue of the electric field in the matrix.

Finally, we investigate the stability of the DE composites withthe periodically arranged elliptical inclusions. To illustrate the in-fluence of the applied electromechanical loading on the stability ofthe DE composites, we plot the dependence of the critical stretchon the electric field for different inclination angles and aspect ratiosof the elliptical inclusion in Fig. 11. The contrasts in shear moduliand relative permittivities are km ¼ 50 and kε ¼ 10, respectively.The volume fraction of the inclusion is kept constant at the valuecðiÞ ¼ 0:136 (corresponds to r1r2 ¼ 0:0648a2). The aspect ratio is

eh) in the deformed unit cell under electric fields bE01 ¼ 1 ((a),(c),(e),(g)), and bE0

2 ¼ 0:350; mðmÞ ¼ 1 MPa and ε

ðmÞ ¼ 5. The volume fraction of inclusion phase and ellipticityr1 ¼ 0:36a).

Fig. 11. Critical stretch ratio vs electric field for DE composites with the contrasts in shear moduli and relative permittivities km ¼ 50 and kε ¼ 10, respectively. (a) and (c) - theelectric field is aligned with the stretch direction, (b) and (d) - the electric field is applied perpendicularly to the stretch direction. The aspect ratio is w ¼ 2 in (a) and (b). Theinclination angle is a ¼ p=2 in (c) and (d). The volume fraction of the inclusion is cðiÞ ¼ 0:136 (corresponds to r1r2 ¼ 0:0648a2).

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256 253

w ¼ 2 in Fig. 11(a) and (b), and the inclination angle is a ¼ p=2 inFig. 11(c) and (d). Similar to the observed behavior of the DEcomposites with circular inclusions (see Sec. 4.1), here we find thatthe electric field stabilizes the composite if applied perpendicularlyto the stretch direction, and the electric field aligned with thestretch direction promotes instabilities. At the beginning, the crit-ical stretch increases with an increase in the inclination angle. Thedash-dotted curve (corresponding to a ¼ p=3) is above thecontinuous (corresponding to a ¼ p=6) and dashed (correspondingto a ¼ 0) curves for all values of the applied electric field (seeFig. 11(a) and (b)). However, a further increase in the inclinationangle leads to a rapid decrease in the critical stretch. We find thatthe critical stretches for the DE composite with a ¼ p=2 (dash-dotted curves in Fig. 11(c) and (d)) are significantly smaller than forthe DE composites with a ¼ 0 (dashed curves in Fig. 11(a) and (b)).The critical stretch decreases with an increase in the aspect ratio.The dash-dotted (w ¼ 2) curve is below the solid (w ¼ 1:5) anddashed curves (w ¼ 1) for all values of the applied electric field (seeFig. 11(c) and (d)).

The dependence of the critical stretch on the inclusion inclina-tion angle is shown in Fig. 12. The results are presented for DEcomposites with the volume fraction of inclusion cðiÞ ¼ 0:136(corresponds to r1r2 ¼ 0:0648a2), and with the contrasts in theshear moduli and relative permittivities are km ¼ 50 and kε ¼ 10,

respectively. The dash-dotted, continuous and dashed curves

correspond to the electric fields ~E02 ¼ 0:3; ~E

01 ¼ 1 and ~E

0 ¼ 0,respectively. The aspect ratio is w ¼ r1=r2 ¼ 1:5 in Fig. 12(a) andw ¼ 2 in Fig. 12(b). For these non-aligned cases, we also observethat the electric field stabilizes the composite if applied perpen-dicularly to the stretch direction, and the electric field aligned withthe stretch direction promotes instabilities. We observe a slightincrease in the critical stretch before a point amax where the curve

lcðaÞ reaches a maximum. This increase is followed by a relativelyfast decrease in the critical stretch with a further increase in theinclination angle. The critical stretch reaches aminimum at a ¼ p=2corresponding the most unstable configuration. The value of theinclination angle, associated with the maximum stretch, changessignificantly with the change in the inclusion aspect ratio(amaxz0:19p for w ¼ 1:5 and amaxz0:3p for w ¼ 2). However, thevalue of the corresponding angle barely changes with a change inthe electric field. An increase in the aspect ratio leads to a morepronounced dependence of the critical stretch on the inclination

angle. For example, ðlcÞmax � ðlcÞminz0:16 for w ¼ 1:5, and

ðlcÞmax � ðlcÞminz0:3 for w ¼ 2.The dependence of the critical stretch on the aspect ratio of the

elliptical particles is shown in Fig. 13. The examples are presentedfor DE composites with the contrasts in the shear moduli and

Fig. 13. Critical stretch vs aspect ratio of the elliptical inclusion for DE composites with the contrasts in the shear moduli and relative permittivities km ¼ 50 and kε ¼ 10,respectively. The inclination angles of the elliptical inclusions are a ¼ 0 (a) and a ¼ p=6 (b). The volume fraction of inclusion phase is cðiÞ ¼ 0:136 (corresponds to r1r2 ¼ 0:0648a2).

Fig. 12. Critical stretch vs inclination angle of the elliptical inclusion. The contrasts in the shear moduli and relative permittivities are km ¼ 50 and kε ¼ 10, respectively. The aspectratios are w ¼ 1:5 (a) and w ¼ 2 (b). The volume fraction of the inclusion is cðiÞ ¼ 0:136 (corresponds to r1r2 ¼ 0:0648a2).

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relative permittivities km ¼ 50 and kε ¼ 10, respectively. The vol-

ume fraction is kept constant at the value cðiÞ ¼ 0:136 (correspondsto r1r2 ¼ 0:0648a2). The dash-dotted, continuous and dashed

curves are for the electric fields ~E02 ¼ 0:3; ~E

01 ¼ 1 and ~E

0 ¼ 0. Theinclination angles are a ¼ 0 in Fig. 13(a) and a ¼ p=6 in Fig. 13(b).Consistently with the previous observations, we find that theelectric field stabilizes the composite if applied perpendicularly tothe stretch direction, and the electric field aligned with the stretchdirection promotes instabilities. We observe that for the alignedcase the critical stretch monotonically increases with an increase inaspect ratio w (see Fig. 13(a)). However, the behavior changesdramatically, when a non-aligned case ða ¼ p=6Þ is examined (seeFig. 13(b)). The critical stretch decreases fast and reaches a mini-mum for the aspect ratio wminz0:9, and, then, the critical stretchincreases with a further increase in aspect ratio. The electric fieldalmost does not influence the position of the most unstableconfiguration corresponding to wmin.

5. Conclusions

In this work, we examined the stability of periodic DE

composites subjected to finite strains in the presence of an electricfield. We implemented the stability analysis into a numerical finiteelement based tool, and through the numerical evaluation of thetensors of electroelastic moduli, the unstable domains were iden-tified for periodic DE composites with circular and with ellipticalinclusions embedded in a matrix. We studied the influence of theelectromechanical loading (large deformations and electric fieldexternally applied), and the geometrical and material parameterson the stability of the DE composites with particulate microstruc-tures. We found that an electric field destabilized the compositeswhen it was applied parallel to the stretch direction, and an electricfield stabilized the composites when it was applied perpendicularlyto the stretch direction. For periodic DE composites with circularinclusions, critical stretch was found to decreasewith an increase inthe volume fraction of the inclusion phase. An increase in the shearmoduli contrast was found to promote instabilities in the DEcomposites. We observed that the role of the electric field becamemore significant with an increase in the dielectric constant contrastof the phases. For the DE composites with elliptical inclusions, wefound that the inclusion inclination angle influenced the stability ofthe DE composites. In particular, the critical stretch curve wasfound to possess a unique maximum at a certain inclination angle.

A. Goshkoderia, S. Rudykh / European Journal of Mechanics A/Solids 65 (2017) 243e256 255

The corresponding inclination angle is defined by the aspect ratio ofthe elliptical inclusion. When the elliptical inclusion was alignedwith the stretch direction (the semi-major axis coincided with thestretch direction), we observed that an increase in the ellipticityaspect ratio resulted in a more stable behavior of the DE compos-ites. In addition, we observed some deformation induced tunabilityof the components of the effective relative dielectric permittivitytensor in the periodic DE composites.

The analysis of the electromechanical instabilities takes intoaccount the microstructure characteristics, which affect the effec-tive electroelastic moduli, and, hence, the stability of the DE com-posites. However, microscopic instabilities may develop at themicrostructure length-scales. To analyze the microscopic in-stabilities, a more involved analysis is required such as Bloch-Floquet type of analysis including the Bloch-Floquet conditionsimposed on both mechanical and electric field variations; asopposite to the standard microscopic instability analysis for purelymechanical problems. The coupled microscopic instability analysisrequires development of a new numerical tool, which can bedeveloped in the future to predict the onset of instabilities at finelength scales.

These findings can be useful in further design of DE compositeswith enhanced properties and switchable functionalities. In addi-tion, the presented analysis of the electromechanical instabilitiescan be applied to more complicated and richer multiphase periodiccomposites.

Acknowledgments

This research was supported by the Israel Science Foundation(grant 1550/15 and 19713/15). SR gratefully acknowledges thesupport of Taub Foundation through the Horev Fellowship e

Leaders in Science and Technology.

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