Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings

ARTICLE IN PRESS

JID: SAS

[m5G;April 5, 2019;18:7]

International Journal of Solids and Structures xxx (xxxx) xxx

Contents lists available at ScienceDirect

International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr

Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings Morteza H. Siboni, Pedro Ponte Castañeda∗ Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, United States

a r t i c l e

i n f o

a b s t r a c t

Article history: Revised 25 February 2019 Available online xxx

This paper is concerned with the development of finite-strain constitutive models for electro-active composites consisting of initially aligned, rigid dielectric fibers of elliptical cross-section that are distributed randomly in a dielectric elastomer matrix. For this purpose, we make use of a variational approach that partially decouples the mechanical and electrostatic contributions to the overall energy and leads to a minimum principle for the average orientation of the fibers. The resulting macroscopic electroelastic constitutive model accounts for electric and mechanical torques on the fibers, as well as for the microstructure and its evolution under finite deformations. In particular, the model characterizes the rotation of the fibers for general externally applied electromechanical loadings, predicting bifurcation instabilities for the special case of aligned loadings. To elucidate the main features of the model, we consider the application to a dielectric elastomer composite actuator and investigate the effects of fiber aspect ratio and initial fiber orientation on its macroscopic response. In addition, the new results are compared with the predictions of an earlier model approximating the fiber rotations by purely mechanical effects.

Keywords: Electrostriction Smart materials Electric torques

© 2019 Elsevier Ltd. All rights reserved.

1. Introduction Electro-active polymers (EAPs) are a class of materials capable of responding to external electric stimuli by spontaneously changing their shape. This special property, known as electrostriction, makes these materials close analogues of biological muscles, and as a consequence they are also known as artificial muscles (BarCohen, 2004). Application areas for EAPs include energy conversion (Ren et al., 2007), haptic devices (Ozsecen et al., 2010), active vibration damping in the automotive industry (Sarban et al., 2009), and tunable optical devices (Aschwanden and Stemmer, 2006). They are also widely used as sensors and actuators (Bar-Cohen, 2004). Understanding the behavior of EAPs at the continuum level requires a thermodynamically consistent constitutive description. The foundations for such a description were laid down in the pioneering work of Toupin (1956), and since then much progress has been made in the development of continuum models for EAPs (e.g., Kovetz, 20 0 0; Dorfmann and Ogden, 2005; McMeeking and Landis, 2005). Dielectric Elastomers (DEs) are an important class of fieldactivated EAPs with huge potential (Cheng and Zhang, 2008). A

∗

Corresponding author. E-mail addresses: [email protected] (P. Ponte Castañeda).

(M.H.

Siboni),

[email protected]

simple dielectric elastomer actuator can be made by sandwiching a thin layer of dielectric elastomer between two compliant electrodes. When stimulated by an external voltage, the dielectric elastomer contracts along the thickness while expanding its area (Brochu and Pei, 2010). Actuators that are capable of producing large electrostrictive strains are highly desirable in applications (Pelrine et al., 1998; Bar-Cohen, 2004). The electrostrictive strains that have been reported (Sundar and Newnham, 1992) for elastomers can be fairly significant (in the order of 10% or even larger for special configurations), but the required operating electric fields are very large (in the order of 107 V/m). Furthermore, the performance of dielectric actuators operating at high voltages is severely restricted by dielectric breakdown, or an electromechanical (pull-in) instability followed by the dielectric breakdown (Stark and Garton, 1955; Zhao and Suo, 2008; Brochu and Pei, 2010). To remedy these limitations on the performance of DEs, different approaches have been proposed in the literature. One such approach to improve the performance of dielectric actuators is to add one or more filler phases, which may have different elastic and electric properties, to a soft elastomeric host in order to make composites, known as Dielectric Elastomer Composites (DECs). Generally speaking, the presence of heterogeneities tends to generate large electric polarizations in such composites, which, in turn, can lead to the development of dipolar forces and electrostatic torques (Shkel and Klingenberg, 1998; Ponte Castañeda and Siboni, 2012). Such

https://doi.org/10.1016/j.ijsolstr.2019.03.027 0020-7683/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS 2

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

interactions, if harnessed by appropriate design of the microstructure, can significantly enhance the electromechanical coupling (Huang et al., 2004). Making use of the classical small-strain formulation of electroelasticity (Landau et al., 1984), micromechanical approaches have been used to investigate the effective electro-active properties of DECs with various types of microstructures. Shkel and Klingenberg (1998) provided results for the electrostriction of isotropic dielectric solids, making use of a mean-field approach to estimate the dipolar interactions at the microscopic level. Using the ‘uniform field’ concept (Benveniste and Dvorak, 1992), micromechanical approaches have also been developed Li and Rao (2004) and Rao and Li (2004) to estimate the effective electrostrictive coefficients of polymer-matrix composites with aligned and randomly oriented ellipsoidal inclusions. However, it was recognized (Li et al., 2004) that the effect of the field fluctuations in the matrix phase, ignored in these early works, could be important. More recently, Tian et al. (2012), still in the context of small strains, developed a rigorous method to compute the effective electro-elastic properties of composites in terms of coupled moments of the electrostatic and elastic fields in the composite, which further highlights the strong influence of field fluctuations. They also provided results for sequentially laminated composites, where such coupled moments could be computed explicitly. In addition, Lefevre and Lopez-Pamies (2014) and Lefevre and Lopez-Pamies (2015) have made use of the homogenization method of Tian et al. (2012) to obtain small-strain estimates for DECs with Differential Coated Sphere and Cylinder microstructures, respectively. Independently, Siboni and Ponte Castañeda (2013) have proposed an alternative homogenization framework for the effective electro-elastic response of DECs with more general distributions of rigid ellipsoidal particles, incorporating the effect of electric torques and particle rotations (Siboni and Ponte Castañeda, 2012), which, unlike dipolar interactions, can have non-trivial effects even at dilute particle concentrations. In addition, they demonstrated that the effective electrostrictive stress for a DEC undergoing infinitesimal strains can be directly related to the first derivative of the effective deformation-dependent permittivity of the composite with respect to the macroscopic strain. Their results suggest that particulate DECs consisting of slightly elongated (in the direction of the applied electric field) spheroidal conducting particles can achieve large electrostriction near the percolation limit. Models for DECs at large deformations have also been advanced recently. For example, building on the classical formulations of finite-strain electroelasticity, deBotton et al. (2007), Bertoldi and Gei (2011) and Rudykh and deBotton (2011) have obtained analytical estimates for the effective response and stability of DECs with layered microstructures. In addition, Ponte Castañeda and Siboni (2012) have recently developed a homogenization method for electro-active composites with periodic distributions of rigid particles, accounting for finite strains and particle rotations and thus providing a consistent generalization of the small-strain work of Siboni and Ponte Castañeda (2013). Results have also been obtained by means of the FEM method by Li and Landis (2012). On the other hand, Lu et al. (2012) have investigated soft dielectrics stiffened by long fibers. In their simplified model, the effect of the fibers on the overall properties of the compound material is neglected. In other words, the fibers act as structural elements in order to constrain the deformation of the elastomeric matrix in the direction of the fibers in such a way that the sample is only allowed to undergo 2D plane strain deformations. Their experimental investigation of the problem demonstrates significant improvements for the maximum achievable electrostrictive strain before dielectric breakdown, consistent with earlier findings of Bolzmacher et al. (2006). By means of the general homogenization framework of Ponte Castañeda and Siboni (2012) and

earlier results for the purely mechanical response of LopezPamies and Ponte Castañeda (20 06a; 20 06b), Siboni and Ponte Castañeda (2014) have recently obtained analytical estimates for a special class of electro-active composites consisting of long, rigid, high-dielectric fibers, which are embedded in a soft ideal dielectric matrix. However, in this work, the contribution of the fibers to the overall properties (i.e., the stiffness, the permittivity, and the electro-mechanical coupling) of the composites is fully accounted for. Furthermore, making use of these results, Siboni and Ponte Castañeda (2014) attempted an optimal design for the microstructure of the fibrous DECs with the objective of achieving large electrostriction, while avoiding possible electro-mechanical instabilities and dielectric breakdown. These authors have shown that—due to the dipolar interactions—increasing the volume fraction or aspect ratio (in the direction of the applied field) of the fibers can lead to significant enhancements in the electromechanical coupling of the fiber-constrained DECs. Thus, DECs with high concentrations of fibers or large aspect ratios for the fibers in general require smaller voltages to achieve a given deformation state. However, high fiber concentrations and/or large fiber aspect ratios can also lead to a dramatic reduction in the overall breakdown field that the DEC can withstand, due to the field-magnification effect of the fibers. In addition, estimates have been computed recently for DECs with sequentially laminated microstructures by Lopez-Pamies (2014), making use of the approach originally developed by deBotton (2005) for elastomers. Several authors have also begun to study instabilities in heterogeneous active materials. In particular, Zhao and Suo (2007) showed that the application of an electric field alone can lead to (snapping) instabilities of the maximum-load type in homogeneous dielectrics undergoing 3D equal bi-axial deformations. They further demonstrated that the application of an equal bi-axial traction loading to the ideal dielectric can delay (or even completely remove) such instabilities. Building on earlier work for the purely mechanical problem (Geymonat et al., 1993), Bertoldi and Gei (2011) investigated loss of positive definiteness, as well as loss of strong ellipticity (see Destrade and Ogden, 2011) for DECs with layered microstructures, while Rudykh and deBotton (2011) studied the loss of strong ellipticity for such composites. More recently, Siboni et al. (2015) have investigated further the possible development of instabilities for the class of fiber-reinforced composites considered by Siboni and Ponte Castañeda (2014). In this work, we implement for the first time the partial decoupling strategy (PDS) of Ponte Castañeda and Siboni (2012) for fiber-constrained DECs with aligned rigid fibers of elliptical crosssection embedded in a ideal dielectric elastomer matrix and subjected to non-aligned loading conditions. In addition, we will compare the predictions of the resulting model with the corresponding estimates obtained by Siboni and Ponte Castañeda (2014) making use of the partial decoupling approximation (PDA) for aligned loading conditions, as well as non-aligned loadings. In particular, we investigate the evolution of the average fiber rotations with the applied electromechanical loadings and its effect on the overall macroscopic response for a DEC actuator with soft electrode boundary conditions. As will be seen, the model predicts the existence of bifurcation instabilities involving the collective reorientation of the fibers for perfectly aligned loading conditions. As a consequence, it is found that the PDA cannot accurately describe the evolution of the microstructure and the macroscopic response of the DECs beyond the onset of these instabilities. The rest of the paper is organized as follows. Section 2 gives a brief overview of the fundamentals of electroelasticity, while Section 3 presents the general variational homogenization framework (Ponte Castañeda and Galipeau, 2011; Ponte Castañeda and Siboni, 2012) to be used in this work. Section 4 provides a detailed description of the class of fiber-constrained DECs of

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

ARTICLE IN PRESS

JID: SAS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

interest and describes the implementation of the PDS for the macroscopic response of the composite under general in-plane loading conditions, in terms of the equilibrium orientation of the elliptical fibers. In addition, the PDA estimates of Siboni and Ponte Castañeda (2014) for the effective electroelastic energy of the DECs are recalled. In Section 5, we provide comparisons between the predictions of this paper and available results from the literature for DECs with cylindrical fibers of circular cross section. Section 6 provides illustrative results for the computation of the electromechanical equilibrium orientation of the fibers and shows that the principal solution for the fiber orientation may bifurcate into lower energy states for sufficiently large electromechanical loadings. Section 7 makes use of the developed constitutive model to analyze the macroscopic response and microstructure evolution for soft electrode DEC actuators under aligned and non-aligned loading conditions. Finally, we conclude the paper in Section 8 by summarizing our findings and providing some possible future research directions. In this paper, scalars are denoted by italic Roman, a, or Greek letters, α ; vectors by boldface Roman letters, b; second-order tensors by boldface italic Roman letters, C, or boldface Greek letters, σ ; and fourth-order tensors by bared letters, P. Where necessary, index notation is adopted—e.g., bi , Cij and Pijkl are respectively the Cartesian components of the vector b, second order-tensor C and fourth-order tensor P. 2. Background on electroelasticity The response of a deformable electro-sensitive material can be described by the theory of electro-elasto-statics (e.g. Toupin, 1956; Eringen and Maugin, 1990; Kovetz, 20 0 0). Consider a homogeneous electroelastic material occupying, in the absence of electric fields and mechanical loadings, a volume 0 in the reference configuration. Under the application of electric fields and mechanical loadings, a material point X in the reference configuration moves to a new x in the deformed configuration of the specimen, denoted by . For simplicity we exclude the possibility of gaps and/or interpenetration regions in the material. This assumptions can be enforced by taking the map x(X), which takes the material forms from the reference configuration to the deformed one, to be continuous and one-to-one. Then, the deformation gradient tensor F = Grad x (with Cartesian components Fi j = ∂ xi /∂ X j ) characterizes the deformation of the material, and it is such that J = det F > 0. Finally, the material satisfies the conservation of mass equation, such that the material density in the deformed configuration becomes (in local form) ρ = ρ0 / det F , where ρ 0 denotes the material density in the reference configuration. The equilibrium equations in Lagrangian forms is given by

Div S + ρ0 f0 = 0, −T

(1)

where S = J T F is the (first) Piola–Kirchhoff stress tensor, and f0 is the given mechanical body force distribution in the reference configuration. The conservation of angular momentum requires symmetry of the Cauchy stress T = J −1 SF T , implying that SF T = F S T . Note that the deformation gradient F and stress tensor S may be discontinuous across material interfaces, but satisfy the jump conditions [[F ]] = a  N and [[S]]N = 0, where a is a vector which can be determined from the solution of the problem, and N is the normal to the interface in the reference configuration. It is important to emphasize that unlike the mechanical body forces (couples), which are externally prescribed, the electric body forces (couples) are manifestations of the electric fields that develop in the material, and therefore, need to be determined from the solution of the coupled electroelastic problem. Therefore, for the purposes of the present investigation, we include the effects of electric body forces (couples) in the total stress. For this reason,

3

the stresses T and S defined above, also include the electric effects, as it becomes clearer later on when we introduce the constitutive relations. Alternative formulations in which all or part of the electric contributions are described in terms of a body force or body couple in the above equilibrium equations are also available in the literature (Hutter et al., 2006), but such formulations are not considered here. The true (or Eulerian) electric field e and electric displacement field d must satisfy the equations of electrostatics. In terms of the corresponding Lagrangian fields (or “pull-back” of the true electric field e and electric displacement field d), E = F T e, D = J F −1 d, they can be written in the form

Curl E = 0,

and Div D = Q,

(2)

where Q = Jq is the prescribed Lagrangian charge density (per unit volume in 0 ) associated with the true charge density (per unit volume in ) q. The corresponding jump conditions for the electric fields are [[E]] × N = 0 and [[D]] · N =  , where  is the prescribed charge per unit area in the reference configuration. In addition, the appropriate boundary conditions may be obtained by means of the above-mentioned jump conditions for the electric and mechanical fields, taking into account the fact that neither the electric fields nor the stresses are zero outside the electroelastic specimen, even if the specimen is surrounded by empty space (or vacuum). This is due to the fact that vacuum holds electric fields, and therefore also the (self-equilibrated) Maxwell stress. The constitutive behavior of a homogeneous electroelastic material has been described in many different ways (see Kovetz, 20 0 0; Hutter et al., 20 06). However, the form developed by Dorfmann and Ogden (2005) (see also Suo et al., 2008) is most convenient for our purposes here. Thus, we introduce an energydensity function, or a potential W(F, D), such that the first PiolaKirchhoff stress and the Lagrangian electric field may be obtained by

S=

∂W ∂W (F , D ), and E = (F , D ). ∂F ∂D

(3)

The energy function W satisfies objectivity such that W (Q F , D ) = W (F , D ), for all proper orthogonal tensors Q, which implies that W (F , D ) = W (U , D ), with F = RU being the polar decomposition of F. For materials with (internal) incompressibility constraints (i.e. C (F ) = det F − 1 = 0) a hydrostatic pressure term is introduced in the expression for the Piola-Kirchhoff stress (Ogden, 1997) while the expression for the Lagrangian electric field remains unchanged. In this work we are mainly concerned with deriving macroscopic forms for the potentials of heterogeneous EA materials starting from the constitutive behavior of the phases. Thus, next we provide specific forms of the functions W for the matrix and inclusion phases. It will be assumed here that the matrix, labeled with the superscript “1,” is made of a dielectric elastomer, while the inclusions, labeled with the superscript “2,” are made of much stiffer materials. The inclusions are assumed to be very stiff (rigid) compared to the soft elastomeric matrix. In fact, it is helpful for the inclusions to have high dielectric coefficients for stronger electroelastic couplings. However, naturally appearing materials (e.g., ceramics) with high dielectric coefficients also tend to be very stiff mechanically. As a consequence, and for simplicity, the inclusions will be assumed to be perfectly rigid in this work. For simplicity, the matrix phase is an “ideal dielectric elastomer” (McMeeking and Landis, 2005; Zhao and Suo, 2008) with a linear dielectric response described by the isotropic permittivity ε (1) that is taken to be independent of the deformation. Thus, the matrix material will be described here by an energy-density function (in the reference configuration) W (1) of the form (1) W (1) (F , D ) = Wme (F ) + Wel(1) (F , D ),

(4)

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

ARTICLE IN PRESS

JID: SAS 4

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx (1)

where Wme (F ) is the usual (purely) mechanical stored-energy (1) function of the elastomer and Wel (F , D ) is the electrostatic part of the stored-energy function. For the purely mechanical stored energy we adopt the (incompressible) Gent (1996) model as specified by (1) Wme (F ) = −

μ

(1) (1) Jm

2



ln 1 −

I−d





J = det F = 1, I = tr F T F



(5)

where d specifies the dimension of the problem (i.e., d = 2 for 2D plain-strain problems and d = 3 for 3D problems), μ(1) is the (1) shear modulus of the elastomer and Jm , which specifies the limiting value for I − d, is the lock-up parameter. Note that the above purely mechanical energy density function is not convex in F, but is polyconvex and strongly elliptic (Ball, 1976) and reduces to the (1) (incompressible) neo-Hookean model as Jm → ∞. The electrostatic stored energy of the ideal dielectrics can be written as

Wel(1) (F , D ) =

1 (F D ) · (F D ), 2ε (1) J

(6)

with ε (1) constant. This is consistent with the assumption that the dielectric response of the material is linear and independent of the deformation in the current configuration. On the other hand, the behavior of the rigid, polarizable fibers can be described by the energy function (2) W (2) (F , D ) = Wme (F ) + Wel(2) (D ).

(7) (2)

The rigidity constraint is enforced by requiring Wme to be zero when F is a pure rotation R(2) , and infinity otherwise. The electrostatic part of the energy for materials with linear dielectric behavior is taken to be of the standard form

Wel(2) (D ) =

−1 1 D · E (2) D, 2

W ( X, F , D ) =

N 

0(r ) (X ) W (r ) (F , D ),

(9)

r=1

with

(1) Jm

specimen 0 (or the macroscopic scale). In terms of the indicator functions the stored-energy function for the composite can be expressed in the form

(8)

where E (2) is a constant, second-order tensor defining the anisotropic permittivity (or dielectric constant) of the material. Note that, because of objectivity, the tensor E (2) has to be independent of the deformation (or rotations for the special case of rigid particles), and is therefore a constant in the reference configuration. Note that the permittivity in the deformed configuraT

tion is given by ε(2) = R(2) E (2) R(2) and depends on the rotation of the particle R(2) . (This is consistent with the standard law for (2) −1

a linear dielectric e = ε d.) It is also important to mention that the second-order tensors E (2) (and ε(2) ) are positive definite, which is consistent with the assumed convexity of the energy functions W (2) (F , D ) in D. In addition, total stress in the rigid fibers becomes indeterminate.

where W (r ) are are the stored-energy of the phases. In this section, we recall a finite-strain homogenization framework for the above-described electro-active composites with general microstructures in the quasi-static regime (Ponte Castañeda and Siboni, 2012) (see also Ponte Castañeda and Galipeau, 2011, in the magneto-elastic context). Toward this goal, boundary conditions are prescribed that are consistent with “macroscopically uniform” fields in the composite. Here we enforce the conditions

¯ · N, and D · N = D

x = F¯ X,

on

∂ 0 ,

(10)

¯ are a prescribed, constant tensor and vector, rewhere F¯ and D spectively, and N is the outward unit normal to the boundary of the composite specimen ∂ 0 . It then follows, by means of the divergence theorem, that the macroscopic averages (over 0 ) for the deformation gradient and electric displacement fields are given by

F 0 = F¯ , and D0 = D¯ ,

(11)

where ·0 has been used to denote a volume average in the refer¯ can be interpreted as ence configuration. This shows that F¯ and D the macroscopic, or average, deformation gradient and electric displacement field in the composite 0 . Note that it is also possible to specify the electric field on the boundary of the specimen. However, the boundary conditions (10)2 is preferred here since it leads to a minimum-type variational formulation for the homogenization problem, because, as mentioned earlier, the associated potentials W are convex in D and polyconvex in F. Given the boundary conditions (10) and the assumed separation of length scales, it is expected on physical grounds that the composite material will behave like the homogeneous medium with ef˜ . Building on the work fective, or homogenized energy function W of Hill (1972) for purely elastic composites, the homogenized potential for the electro-active composite is defined (Ponte Castañeda and Siboni, 2012) as the volume average of the energy stored in the composite under application of the boundary conditions (10), namely,





¯ = min ˜ F¯ , D W

F ∈K (F¯ )

min W (X, F , D )0 ,

(12)

¯) D∈D0 (D

where W(X, F, D) is defined in terms of the uniform phase potentials W (r ) (F , D ) via expression (9) and where

 



K F¯ = F

 | ∃ x = x(X ) with F = Grad x in 0 , x = F¯ X on ∂ 0 , (13)

3. Homogenization framework We consider a specimen 0 (in the reference configuration) made of the electro-active composite, which consists of N homogeneous phases, occupying sub-domains 0(r ) in 0 . The distribution of the phases is described by the characteristic functions 0(r )

(r = 1, 2), such that 0(r ) is equal to 1 for X ∈ 0(r ) and zero otherwise. Similarly, the specimen in its deformed configuration can be described by the characteristic functions (r ) (r = 1, . . . , N), such that (r ) (x ) = 1 for x ∈ (r ) and zero otherwise, where (r ) is the sub-domain of  (the deformed configuration of the specimen) that is occupied by phase r. Throughout this work, the electroactive composites are assumed to satisfy the separation of length scales hypothesis. In other words, it is assumed that the length scale at which the indicator functions 0(r ) vary (also referred to as the microscopic scale) is very small compared to the size of the

and

 



¯ = D | Div D = 0 in 0 , D · N = D ¯ · N on ∂ 0 D0 D



(14)

are, respectively, sets of admissible deformation gradients and electric displacement fields that are compatible with the boundary conditions (10). It can be readily shown that the Euler–Lagrange equations associated with the variational problem (12) are precisely the equilibrium equation (1) (with f0 = 0) and the electrostatic equation (2). (Note that the energy contributions of the inhomogeneous terms, f0 , Q, and  , are ignored since they have been assumed to vary on the macroscopic length scale, for simplicity, and have no effect on the homogenization problem.) Therefore, the minimizers (assuming that they exist) of the above problem are also solutions of the electroelastic problem (described in the previous section) with boundary conditions (10). To the best of our

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

knowledge, there exist no rigorous mathematical results for the existence of the minimizers for the above variational problem. However, it will be assumed here that the minimizers of the variational problem (12) exist at least for small enough (but not necessarily infinitesimal) deformations and electric displacement fields. In the purely mechanical context, it is known that macroscopic instabilities may arise in these composites via loss of ellipticity (Geymonat et al., 1993; Lopez-Pamies and Ponte Castañeda, 2006b), leading to micro-domain formation beyond these instabilities (Avazmohammadi and Ponte Castañeda, 2016; Furer and Ponte Castañeda, 2018).   ¯ of ˜ F¯ , D Having defined the effective electroelastic energy W the composite, it can be shown by means of appropriate generalization of Hill’s lemma (Ponte Castañeda and Siboni, 2012) that the average stress and average electric field, determined by S¯ = S0 and E¯ = E0 , are given by

S¯ =

∂ W˜ ∂ W˜ , and E¯ = , ∂ F¯ ∂ D¯

(15)

¯ correspond to the respectively. As mentioned earlier, F¯ and D average (or macroscopic) deformation gradient and electric displacement fields in the composite. Therefore, expression (15) provides the macroscopic, or homogenized constitutive relations for the composite. In other words, similar to the local energy functions W (r ) , which characterize the response of the constituent ˜ , as defined by (12), comphases, the effective energy function W pletely describes the macroscopic response of the electro-active composite. Note that although, in general, energy will be stored (via the electric field) in the free space surrounding the specimen, as it is shown above, only the energy stored inside the ˜ ) needs to be considered in the homogenization specimen (i.e., W ˜ is objective, which can problem. In addition, it is noted that W be easily verified by making use of the objectivity of the phase potentials. 4. Applications to fiber-constrained DECs under general in-plane loading conditions We consider a DEC consisting of a random distribution of rigid, dielectric fibers firmly embedded in an ideal dielectric elastomer matrix, as shown in Fig. 1. The matrix is assumed to be isotropic (both mechanically and electrically) and is capable of undergoing finite strains, as described by relations (4) to (6). The fibers are assumed to be aligned, but with arbitrary elliptical cross section and general anisotropic dielectric properties, as characterized by relations (7) and (8). As a consequence of the alignment of the

5

long axes of the fibers with the X3 direction, the DEC can only undergo plane strain deformations in the X1 − X2 plane. It is further assumed that the above-described electro-active composite has a stress-free configuration in the absence of deformation and elec¯ = 0), and that its mechanical betric fields (i.e., when F¯ = I and D havior for small deformations is characterized by the conventional theory of linear elasticity. Under these conditions, we may expect a unique solution to the Euler-Lagrange equations associated with the variational problem (12), at least for sufficiently small deformations and electric fields (i.e., in the neighborhood of F¯ = I and ¯ = 0). However, after a certain amount of loading, this solution D may become unstable and bifurcate into a different (lower energy) solution. In this work, we will refer to the solution before the onset of an instability as the “principal” solution, and to the corresponding solution after the onset of an instability as the “postbifurcation” response. As already discussed in the Introduction, our main objective is to consider non-aligned loading conditions leading to fiber rotations in the plane of the deformation as a consequence of the prescribed finite deformations on the boundary of the DEC specimen, as well as electric torques due to the remotely applied electric fields. For this purpose, we will make use of the “partial decoupling strategy” (PDS) of Ponte Castañeda and Siboni (2012) and compare its predictions for the macroscopic response and microstructure evolution with corresponding results making use of the so-called “partial decoupling approximation” (PDA) that was first used for DECs in the work of Siboni and Ponte Castañeda (2014). As already mentioned, the PDS consists in rewriting the solution of the variational problem (12) for the effective stored-energy function of the electro-active composite in terms of the solutions of “purely mechanical” and “electrostatic” problems, coupled only through the fiber orientations in the deformed configuration—to be determined by means of a finitedimensional optimization process. On the other hand, the PDA consists in the replacement of the electroelastic equilibrium rotation in the PDS by the equilibrium rotation associated with the corresponding purely elastic problem. The use of the PDA in the work of Siboni and Ponte Castañeda (2014) for aligned loadings was justified by the expectation that the electroelastic equilibrium rotation would not be different from the purely elastic equilibrium rotation—at least up to the possible development of instabilities leading to non-aligned conditions in the post-bifurcation regime—due to the fact that the electric torques vanish for aligned loading conditions. On the other hand, for non-aligned conditions the additional effects of electric torques should be taken into account and—based on the recent results of Siboni and Ponte Castañeda (2016) for the effect of torques on the macroscopic response of fiber-reinforced elastomers—would be expected to have significant effects on the equilibrium orientation of the fibers and, hence, on the macroscopic response. For simplicity, we will only provide here the final results and necessary details for the applications to be considered in the next section on results (see Siboni and Ponte Castañeda, 2019, for more details and general results). As already noted, the corresponding PDA estimates for the macroscopic response of the abovedescribed DECs have already been given in the work of Siboni and Ponte Castañeda (2014) and for this reason they will also not be detailed here.

4.1. Initial microstructure

Fig. 1. Schematic of the two-phase fibrous DECs consisting of aligned, rigid, dielectric fibers, embedded firmly in an ideal dielectric matrix.

As depicted in Fig. 2, we consider plane strain deformations of the above-described two-phase fibrous DECs consisting of a matrix phase, denoted by the superscript “1”, and a fiber phase, denoted by the superscript “2”.

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

ARTICLE IN PRESS

JID: SAS 6

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

Fig. 2. Schematics of the transverse plane cross section of the fiber-constrained DEC. (left) Microstructure of the composite in the reference configuration, and (right) microstructure of the composite in the deformed configuration with prescribed rotation ψ¯ (2) for the fibers.

The volume fractions of the matrix and fiber phases in the reference configuration are denoted, respectively, by

c0(1)

= 1 − c0

and

c0(2)

= c0 .

(16)

As discussed in more detail in the work of Siboni and Ponte Castañeda (2014), the fibers are characterized by elliptical regions









I0 = X = (X1 , X2 ), such that Z 0−T X ≤ 1 ,

(17)

where Z0 is a second-order tensor describing the shape and orientation of the fibers in the reference configuration, which are, in turn, determined by the aspect ratio w and the angle β (see Fig. 2). On the other hand, the fibers are distributed with “elliptical symmetry” (Ponte Castañeda and Willis, 1995). This means that the two-point probability function for finding two fibers separated Z = (Z1 , Z2 ) depends on Z only via the combination by  a vector  D−T Z , where D0 is a symmetric, second-order tensor character0 izing a “distributional ellipse” in the reference configuration (refer to dashed ellipse in Fig. 2). More specifically,



 λ¯ ¯  = 0

0

λ¯ −1

.

(20)

As a consequence of their rigidity, the fibers can only undergo rotations in the X1 − X2 plane. Given their identical cross-sections in the reference configuration, we assume that they will rotate with an average rotation denoted by the angle ψ¯ (2) , as depicted in Fig. 2. Thus, in the deformed configuration, the fibers are described by the elliptical regions









I = x = (x1 , x2 ), such that Z −T x ≤ 1 ,

(21)

where Z is the deformed version of Z0 , given by (2)

Z = R¯ Z 0 R¯

(2)

T



(2) cos ψ¯ (2) with R¯ = sin ψ¯ (2)

− sin ψ¯ (2) cos ψ¯ (2)

(22)

It is worthwhile to mention that statistical isotropy is achieved by setting D0 = I , such that the two-point probability function depends on Z only via |Z|, and the distributional ellipse becomes a circle. In this paper, for simplicity, we take the shape of the distributional ellipses in the reference configuration to be identical to the shape of the fibers such that D0 = Z 0 .

In general, ψ¯ (2) will depend on both the macroscopic deformation and the macroscopic electric fields. Similar to the fibers, the evolution of the distribution is expected to depend in general on both the mechanical and electrostatic loadings. However, following earlier work (Ponte Castañeda and Galipeau, 2011; Siboni and Ponte Castañeda, 2014), we assume that the fibers remain distributed with “elliptical” symmetry in the deformed configuration, such that the distributional ellipses evolve solely as a consequence of the macroscopic deformation. Thus, the distribution of the fibers in the deformed configuration is described by the deformed distributional elliptical regions

4.2. Evolution of the microstructure

D = x = (x1 , x2 ), such that D−T x ≤ 1 ,









D0 = X = (X1 , X2 ), such that D0−T X ≤ 1 .

(18)

The kinematics of the fiber-constrained DECs can be described by 2D plane strain deformations in the plane perpendicular to the long axes of the fibers (i.e., X1 − X2 plane). On account of the in¯, compressibility of the DEC, we introduce the loading parameters λ θ¯ , and ψ¯ , defined by the polar decomposition of the macroscopic deformation gradient F¯ = R¯ U¯ , together with the diagonalization of the stretch U¯ , such that



   T ¯ Q¯ , 

F¯ = R¯ Q¯

where



R¯ =



cos ψ¯ sin ψ¯

(19)

− sin ψ¯ , cos ψ¯





cos θ¯ Q¯ = sin θ¯

− sin θ¯ , and cos θ¯









(23)

where

D = F¯ D0 ,

(24)

is the distributional shape tensor in the deformed configuration. In addition, the incompressibility of the matrix phase along with the rigidity of the fibers implies overall incompressibility for the DECs considered in this paper. Therefore, the volume fractions of the phases remain unchanged as the deformation progresses. 4.3. Effective elastic stored-energy function for prescribed in-plane rotation of the fibers For the purpose of obtaining the macroscopic electroelastic stored-energy function of the fiber-reinforced DECs, we make use

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

ARTICLE IN PRESS

JID: SAS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

7

of an explicit expression (Siboni and Ponte Castañeda, 2016) for the effective stored-energy function of fiber-reinforced elastomers subjected to prescribed deformation F¯ and given in-plane fiber rota(2) tions R¯ . Thus, the purely mechanical contribution to the effective

and Siboni (2012) (see also Siboni and Ponte Castañeda, 2019) that the effective electroelastic stored-energy function of the DEC under general non-aligned loading conditions may be expressed in the form

stored-energy function of the above-described DECs when the matrix phase is an (incompressible) Gent elastomer with stored energy of the form given by (5) is given by

¯ , θ¯ , D ¯ , θ¯ , D ¯ = min W ¯ ; ψ¯ (2) , ˜ λ ˜ λ W

˜ me W





(1)    ˆ ¯ , θ¯ ; ϕ = (c0 − 1 ) μ Jm ln 1 − I − 2 , ˜ me λ F¯ ; ψ¯ (2) = W 2 Jm

(25) where

Iˆ =



¯2 c0 1 + λ

2







¯2+λ ¯ 4 w + c0 1 + λ ¯2 + 1 + 2 (c0 − 2 )c0 λ

2

w2

( 1 − c0 ) λ   − sin (ϕ ) sin ϕ + 2β − 2θ¯ 2¯ 2 ( 1 − c0 ) λ w    ¯ 2 1 + w2 2 c0 1 + λ − cos (ϕ ), (1 − c0 )2 λ¯ w





¯ 4 − 1 w2 − 1 c0 λ



2¯ 2 w

(26)

¯ and θ¯ have been defined by expresand the loading parameters λ sions (20), while ϕ denotes the relative rotation of the fibers with respect to the macroscopic rotation, as given by

ϕ := ψ¯ (2) − ψ¯ .

(27)

4.4. Effective electrostatic energy for a given in-plane rotation of the fibers The corresponding estimates for the effective electrostatic energy of DECs with a given in-plane rotation of the fibers have already been obtained in the recent work of Siboni and Ponte Castañeda (2014). The final result for the effective electrostatic energy of the DEC may be written as



 

    1  ¯ , θ¯ , D ¯ ; ψ¯ (2) = W ¯;ϕ = D ¯ · U¯ E˜ −1 U¯ ; ϕ U¯ D ¯ , (28) ˜ el F¯ , D ˜ el λ W 2 where E˜ is the deformation-dependent effective permittivity of the Hashin–Shtrikman–Willis type (Ponte Castañeda and Willis, 1995), given by





E˜ U¯ ; ϕ = ε (1) I +

−1



 −1 pT   c0 1 R p E (2) − ε (1) I R + (1) Pˆ 0 U¯ ε J¯

. (29)

In this last expression,

p R



=

cos ϕ sin ϕ

− sin ϕ cos ϕ









(31)

ψ¯ (2)

where













¯ , θ¯ , D ¯ , θ¯ ; ψ¯ (2) + W ¯ , θ¯ , D ¯ ; ψ¯ (2) := W ¯ ; ψ¯ (2) . ˜ λ ˜ me λ ˜ el λ W

(2) Therefore, in terms of the energy minimizing rotation ψ¯ (2) = ψ¯ eq , it follows that









(2) ¯ , θ¯ , D ¯ , θ¯ , D ¯ =W ¯ ; ψ¯ (2) = ψ¯ eq ˜ λ ˜ λ W .

(30)

where ϕ is given by (27) and Pˆ 0 is a microstructural tensor defined by expressions (58) to (60) in Siboni and Ponte Castañeda (2014) in ¯ and θ¯ . In this context, it terms of the loading parameters λ should be emphasized that—consistent with objectivity—both ex˜ me and W ˜ el depend on the angles ψ¯ and ψ¯ (2) only pressions for W through the combination ϕ = ψ¯ (2) − ψ¯ . For this reason, we can set ψ¯ = 0 and consider only the dependence on the fiber rotation ϕ = ψ¯ (2) . 4.5. Equilibrium rotation of the fibers and effective energy for general non-aligned loading conditions Having obtained explicit expressions for the effective mechanical and electrostatic energies in terms of the in-plane rotation of the fibers ψ¯ (2) , it follows from the work of Ponte Castañeda

(33)

Note that the equilibrium rotation in general depends on the me¯ and θ¯ , as well as the electrostatic chanical loading parameters λ ¯ . Now, as will be seen in the results section, D loading parameter   ¯ , θ¯ , D ¯ ; ψ¯ (2) can become non-convex in ψ¯ (2) , and ˜ λ the energy W (2) therefore the equilibrium value ψ¯ has to satisfy the following eq

conditions for a global minimum

∂ ˜  ¯ ¯ ¯ ¯ (2)  W λ, θ , D; ψ ¯ (2) ¯ (2) = 0, and ∂ ψ¯ (2) ψ =ψeq     (2) ¯ , θ¯ , D ¯ , θ¯ , D ¯ ; ψ¯ eq ¯ ; ψ¯ (2) for all ψ¯ (2) . ˜ λ ˜ λ W ≤W

(34)

Other solutions of the stationary condition (34)1 , if they exist, would correspond to local minima or maxima of the effective   ¯ , θ¯ , D ¯ ; ψ¯ (2) . ˜ λ rotation-dependent energy W 4.6. Constitutive relations for fiber-constrained DECs under non-aligned loading conditions Having obtained the effective energy, as given by (33) we may then obtain the constitutive relations of the DECs under non-aligned conditions by means of relations (15). This requires the computation of the derivatives of the effective energy with respect to the macroscopic deformation gradient F¯ and macro¯ , which will be provided in the scopic electric displacement field D following. For the energy functions of the form given by equation (33), the macroscopic Cauchy stress may be conveniently obtained in terms of the derivatives of the energy function with respect to the load¯ and θ¯ . Thus, we can show that ing parameters λ T T¯ = 2R¯ T¯ B R¯ − p¯ I ,

(32)

(35)

where p¯ is the pressure-like Lagrange multiplier accounting for the overall incompressibility of the DEC, and T¯ B is the Biot-like stress defined by



∂ ˜  ¯ ¯ ¯ ¯ (2)  ∂ λ¯ ¯ ∂ ˜  ¯ ¯ ¯ ¯ (2)  U + W λ, θ , D; ψeq × U¯ W λ, θ , D; ψeq ¯ ¯ ∂C ∂λ ∂ θ¯

∂ θ¯ ¯ U . × U¯ (36) ∂ C¯

T¯ B : =

T 2 Here C¯ = F¯ F¯ = U¯ is the right Cauchy-Green tensor and

1 ∂ λ¯ cos2 θ¯ sin θ¯ cos θ¯ = , 2 ¯ sin θ¯ cos θ¯ sin θ¯ ∂ C¯ 2λ

λ¯ 2 ∂ θ¯ sin 2θ¯ − cos 2θ¯ = . ¯ 4 − cos 2θ¯ − sin 2θ¯ ∂ C¯ 2 − 2λ



(37)

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

ARTICLE IN PRESS

JID: SAS 8

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

Finally, the derivatives of the effective energy with respect to the ¯ and θ¯ can be evaluated as follows loading parameters λ

5.1. DECs with circular fibers

∂ ˜ ¯ ¯ ¯  ∂ ˜  ¯ ¯ ¯ ¯ (2)  ∂ ˜  ¯ ¯ ¯ ¯ (2)  W λ, θ , D = W λ, θ , D; ψeq + W λ, θ , D; ψeq (2) ∂ g¯ ∂ g¯ ∂ ψ¯ eq (2) ∂ ψ¯ eq ∂ ˜  ¯ ¯ ¯ ¯ (2)  × = W λ, θ , D; ψeq for ∂ g¯ ∂ g¯   ¯ , θ¯ . (38) g¯ ∈ λ

Expressions (33) and (34) for the effective energy and equilibrium fiber rotation of DECs consisting of general elliptical fibers can be simplified significantly when the fibers are isotropic and have circular cross section, such that

In obtaining the second line we have used the stationary condition (34)1 . In summary, equations (35) and (36) along with the deriva¯ and θ¯ given by (38) will be tive of the energy with respect to λ used in this work to compute the effective macroscopic stress inside DEC samples. Similarly, we can obtain the macroscopic electric field of the composite as follows

  μ(1) Jm Iˆc − 2 c ˜ me W λ¯ , θ¯ = 0; ϕ = (c0 − 1 ) ln 1 − , 2 Jm

E¯ =

∂ ˜ ¯ ¯ ¯  ∂ ˜  ¯ ¯ ¯ ¯ (2)  W λ, θ , D = W λ, θ , D; ψeq ∂ D¯ ∂ D¯ (2) ∂ ˜  ¯ ¯ ¯ ¯ (2)  ∂ ψ¯ eq ∂ ˜  ¯ ¯ ¯ ¯ (2)  + W λ, θ , D; ψeq × = W λ, θ , D; ψeq , (2) ∂ D¯ ∂ D¯ ∂ ψ¯ eq

and w = 1.



where



¯2 2 c0 1 + λ

2

−



4.7. Corresponding results for the PDA estimates As already mentioned, the corresponding estimates making use of the PDA have already been detailed in the work of Siboni and Ponte Castañeda (2014). However, for completeness, it is useful to recall here that such estimates, which correspond to making use of the purely mechanical energy to determine the equilibrium rotation, may be obtained by simply dropping the electro˜ el in the computation of the minimizing condistatic energy W tions (34) for the equilibrium rotation. With this different  estimate  (2) ¯ , θ¯ , D ¯ = ˜ λ for the equilibrium rotation ψ¯ eq , the total energy is W

 ¯2 4 c0 1 + λ (1 − c0 )2 λ¯

Before presenting the new results for DECs with elliptical fibers, in this section we briefly show some comparisons with previously available results for DECs with circular fibers. It will first be shown that the new estimates for elliptical fibers reduce to the earlier estimates of Siboni and Ponte Castañeda (2014) for circular fibers (w = 1). Note that these results have already been found to be consistent with the earlier results of Siboni and Ponte Castañeda (2013) for infinitesimal deformations. We will first make comparisons with the small-deformation estimates of Lefevre and Lopez-Pamies (2015) for DECS with rigid circular fibers, which were obtained by means of the infinitesimal-deformation homogenization theory of Tian et al. (2012). We will then compare these estimates to finite-strain estimates for rigid circular fibers obtained recently by Lefèvre et al. (2017) by appropriately modifying the sequentially laminated estimates of Lopez-Pamies (2014) to agree with the estimates of Lefevre and Lopez-Pamies (2015) for small strains. Note that the results of Lefèvre et al. (2017) have been developed in the context of magneto-active elastomers, but they can be easily converted to corresponding results for DECs. We begin by specializing the results of the previous section for circular fibers.



(1 − c0 )2 λ¯ 2 cos (ϕ ),

(42)





  1   ¯ , θ¯ , D ¯ = D ¯ · U¯ E˜ −1 ¯, ˜c λ U¯ U¯ D W c el 2

(43)

where E˜ c is the deformation-dependent effective permittivity, given by

 

c0 J¯

E˜ c U¯ = ε (1) I +



ε (2) − ε (1)

−1

I+

1

ε

c

 −1

Pˆ U¯ (1) 0

.

(44)

c

In the above expression, the microstructural tensor Pˆ 0 is given by c

 



Pˆ 0 U¯ =

1 2

−



(2) ¯ , θ¯ , D ¯ ; ψ¯ eq ˜ λ W . The computation of the constitutive equations

5. Comparisons with prior results for DECs with fibers of circular cross section

(41)

and where we have set θ¯ = 0 without loss of generality on account of the transverse isotropy of the microstructure. Correspondingly, the effective electrostatic energy of the DECs consisting of circular fibers is obtained as follows



would then follow a similar procedure as that used in Section 4.6, but using the new stationarity conditions.

¯2+λ ¯4 + 1 + 2 (c0 − 2 )c0 λ



where the stationary condition (34) is used, once again, in order to arrive at the second line.

(40)

Thus, the effective mechanical energy of the DECs consisting of circular fibers can be obtained as follows

Iˆc =

(39)



E (2) = ε (2) I ,

+



¯2 c0 λ λ¯ 2 +1

0

0

¯2−1 c0 λ



λ¯ 2 + 1

1 2

−

c0

λ¯ 2 +1



cos θ¯

 

 

cos θ¯   sin θ¯

sin θ¯   , − cos θ¯

(45)

where we have kept the dependence on θ¯ for generality. Note that the above expression for the effective electrostatic energy is independent of the relative rotation ϕ , and therefore we have that

∂ ˜c W = 0. ∂ ψ¯ (2) el

(46)

Replacing the above identity into equation (34) for the equilibrium rotation of the fibers, it can be shown that ϕ = 0. Finally, on account of the symmetry of the DEC, an expression can be obtained for the effective total energy of the composite in terms of the (transversely) isotropic invariants

I¯1 := F¯ · F¯ ,

¯ ·D ¯, I¯4D := D

as follows



 



  μ(1) Jm Iˆc − 2 ¯ , θ¯ , D ¯ = ( c0 − 1 ) ˜c λ W ln 1 − 2 Jm −



¯ · F¯ D ¯ , and I¯5D := F¯ D

(47)

2  2   ε (2) −ε (1) D ε (2) + ε (1) 2 + I¯1 D   I¯4 +   I¯5 , (48) ε (1) α +β I¯1 2ε (1) α + β I¯1

2c02



where

2  2 α : = 2 (1 + c0 )ε (2) + (1 − c0 )ε (1) + 2c02 ε (2) − ε (1) ,  

 β : = ε (2) + ε (1) (1 + 2c0 )ε (2) + (1 − 2c0 )ε (1) .

(49)

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

ARTICLE IN PRESS

JID: SAS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

9

Fig. 3. (a) Results for the electrostrictive strain ratio ¯ /¯ M , as given by equation (55), for DECs with circular fibers for ε (2) /ε (1) = 10 0 0. The results are compared with the Sequentially Laminated (SL) estimates of Lopez-Pamies (2014), as well as with the Differential Coated Cylinder (DCC) estimates and Finite Element (FE) results of Lefevre and Lopez-Pamies (2015). (b) Corresponding results for the electrostrictive strain ratio for DECs with spherical particles for ε (2) /ε (1) = 10 0 0. The results are compared with the Sequentially Laminated (SL) estimates of Lopez-Pamies (2014), as well as with the Differential Coated Spheres (DCC) estimates and Finite Element (FE) results of Lefevre and Lopez-Pamies (2014) (see also discussion in the text for explanation of Maxwell stress only curves).

This last equation is exactly identical to equation (58) in Lefèvre et al. (2017) in the context of MREs, which is obtained by algebraically manipulating the corresponding results in ˆ in the work of Galipeau and Ponte Castañeda (2013). (Note that λ ¯ in our expressions above when Lefèvre et al. (2017) reduces to λ θ¯ = 0.) They also reduce exactly to the results of Siboni and Ponte Castañeda (2014) for aligned loading of DECs with aligned ellipti¯ is aligned cal fibers when the fibers are taken to be circular and D with the mechanical loading axis. 5.2. The limit of infinitesimal deformations In this subsection, we provide the results for the effective electrostrictive strain of DECs consisting of circular fibers under the “soft electrode” boundary conditions (see Fig. 10). As argued by Siboni and Ponte Castañeda (2014) the electrostrictive strains for DECs under soft electrode boundary conditions can be obtained by solving T¯ = 0, where T¯ is the total stress, as defined by (35). The relevant components for the equilibrium equation are (see Siboni and Ponte Castañeda, 2014, for more details)

T¯11 − T¯22 = 0,

and T¯12 = T¯21 = 0.

(50)

¯ = 1 + ¯ and In the limit of infinitesimal deformations (i.e., when λ ¯ → 0), we obtain from the results of the previous subsection me me el el − T¯22 + T¯11 − T¯22 , T¯11 − T¯22 = T¯11

me el and T¯12 = T¯12 + T¯12

(51)

where me me T11 − T¯22 =

el el T11 −T¯22

1 + c0 × 4μ(1) ¯ , 1 − c0

me T12 =0

 2 ε (1) ε (1) +ε (2) v20 el = −

 2 × h2 , and T12 = 0. (52) ε (1) +ε (2) −c0 ε (2) −ε (1) 0

Substituting (52) into the equilibrium equation (50), we arrive at

1 − c0 ¯ = × 4μ(1) (1 + c0 )

 2 ε (1) ε (1) + ε (2) v20

 2 × h2 . ε (1) + ε (2) − c0 ε (2) − ε (1) 0

(53)

It is advantageous to normalize the above electrostrictive strain with respect to the electrostrictive strain in the matrix material. Note that the value of the electrostrictive strain for the matrix material can be obtained by setting c0 = 0 in equation (53). Thus, we have

¯ M =

v2 ε (1) × 02 . (1) 4μ h0

(54)

The ratio is then obtained as follows

 (1) 2 ε + ε (2) ¯ 1 − c0 = × . ¯ M 1 + c0 ε (1) + ε (2) − c0 ε (2) − ε (1) 2

(55)

Note that the above ratio can be thought of as a measure of the effect of the dielectric fibers on the overall electrostatic coupling for the DECs. Fig. 3(a) shows the electrostrictive strain ratio ¯ /¯ M of DECs with circular fibers, as given by expression (55), as a function of the concentration c0 , for a fiber-to-matrix dielectric ratio ε (2) /ε (1) = 10 0 0. For comparison purposes, several results taken from the work of Lefevre and Lopez-Pamies (2015) are included in this figure. First, results are shown for the Sequentially Laminated (SL) homogenization method, originated by deBotton (2005) for pure elastomers, and first applied for DECs by Lopez-Pamies (2014) (see also Spinelli et al., 2015, for the corresponding results in the small-deformation limit). Second, results are shown for Differential Coated Cylinder (DCC) microstructures, which were obtained by Lefevre and Lopez-Pamies (2015) making use of the small-deformation homogenization theory of Tian et al. (2012). Finally, results are also shown for the results of finite element (FE) simulations for a distribution of monodisperse circular fibers, obtained by Lefevre and Lopez-Pamies (2015). As can be seen in the figure, both the result (55) of this work and the DCC are in agreement with the FE results for small fiber concentrations. Beyond c0 = 10%, the result (55) and the DCC appear to diverge with the DCC giving better agreement with the FE results. In this context, it should not come as a surprise that the result (55) and the DCC result do not agree beyond the dilute limit—given that underlying linear homogenization estimates,

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

ARTICLE IN PRESS

JID: SAS 10

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

namely, the Hashin–Shtrikman–Willis (HSW) estimates for the result (55) and the DCC estimate, agree only to linear terms in the volume fraction c0 (that is, the two results for the effective shear modulus, as well as for the effective dielectric coefficient, are different to order O(c02 )). The better agreement for the DCC with the FE results is probably related to the fact that the DCC does a better job capturing the stronger fiber interactions for the monodisperse microstructures considered in the FE calculations than the HSW estimates. On the other hand, it can be seen that the SL estimate of Lopez-Pamies (2014) fails to capture the dilute estimate correctly (in other words, the SL estimate predicts the term of order O(c0 ) incorrectly). While this may be surprising given that the SL microstructures are able to capture correctly the dilute terms for the effective conductivity and elasticity of composites with linear properties, it can be shown that the different slope for the electrostrictive strain ratio ¯ /¯ M at c0 = 0 is due to the fact that the SL homogenization method, when applied to DECs, leads to a dielectric susceptibility which depends on the deformation to order O(c0 ). On the other hand, it is known (Siboni and Ponte Castañeda, 2013; Siboni and Ponte Castañeda, 2014) that an isotropic distribution of isotropic circular fibers can only affect the (intrinsic) dielectrostriction to order O(c02 ). This is because for an isotropic distribution of isotropic circular fibers the interactions among the particles are dipolar in nature leading to O(c02 ) effects. These effects are correctly captured (at least qualitatively) by expression (55), as well as by the use of the DCC approximation together with the small-strain homogenization theory of Tian et al. (2012), but not by the SL homogenization estimates. In order to show this point more precisely, we have included in Fig. 3(a) the corresponding estimate for the electrostrictive strain ratio ¯ /¯ M making use only of the Maxwell stress contribution (and ignoring the dielectriction contribution; see Fig. 4 and associated discussion in Siboni and Ponte Castañeda (2013)). Thus, it can be seen that the Maxwell contribution to the electroelastic stress leads to precisely the same electrostrictive strain ratio for dilute volume fractions (i.e., the slope is the same at c0 = 0) as expression (55), which includes dielectriction contributions of order O(c02 ) due to dipolar interactions among the particles. Finally, for completeness, we also provide in Fig. 3(b) the corresponding results for DECs with isotropic distributions of spherical particles, as determined by Siboni and Ponte Castañeda (2013). They are compared with the suitably linearized SL results of Lopez-Pamies (2014) and the Differential Composite Sphere (DCS) assemblages and FE results of Lefevre and Lopez-Pamies (2014). Although the results are quantitatively different, the same qualitative observations apply for this case in terms of comparisons between the different types of estimates. In this context, it is important to emphasize that the main interest of this paper is in situations where the fibers have non-circular cross-sections and the applied electric fields are not aligned with the fiber axes. In these cases, as already discussed, electric torques develop on the fibers, which do lead to stronger effects of order O(c0 ) in the (intrinsic) dielectrostriction of the DECs. For this reason, we anticipate stronger effects for these situations of interest in this paper, but, unfortunately, we are not aware of numerical simulations for DECs with noncircular fibers subjected to non-aligned externally applied electric fields. 5.3. The limit of large deformations Unfortunately, the homogenization method of Tian et al. (2012) is not applicable for finite strains. In fact, the only estimates that are available for DECs with circular fibers at finite strains are those of Siboni and Ponte Castañeda (2014) used in this work, and the SL estimate of Lopez-Pamies (2014). However,

the finite-strain estimates of Lopez-Pamies (2014) can be shown to be subjected to the same limitations as the corresponding estimates for small strains in that they predict inconsistent behavior for dilute concentrations of rigid fibers of circular cross section. While these results are known to be exact for SL microstructures and to give sound estimates for the purely mechanical and purely electrostatic response of the DECs with circular fibers, the SL microstructures appear to incorporate interactions due to effects beyond dipolar, probably due to the special character of the SL microstructures, which may involve electric torques due to the elongated nature of the layers in the SL microstructure. In any event, improved estimates for DECs with circular fibers have been obtained recently by Lefèvre et al. (2017) by modifying the finite-strain estimates of Lopez-Pamies (2014) so that they agree with the small-deformation estimates of Lefevre and LopezPamies (2015) (making use of the theory of the small-strain theory of Tian et al. (2012)). These results, which were presented in the context of analogous MRE systems, were tested against finite-strain FE numerical simulations and found to be quite accurate. In addition, Lefèvre et al. (2017) provided a comparison of their estimate with the estimates of Siboni and Ponte Castañeda (2014) (or, equivalently, the estimates of Galipeau and Ponte Castañeda, 2013, for MREs) in Appendix A of their work. They found “fair qualitative agreement for small volume fractions of particles and small deformations and magnetic fields,” but pointed out that there were “quantitative differences otherwise.” As already pointed out in the previous subsection in the context of small deformations, quantitative differences are expected for non-dilute concentrations due to differences between the underlying HSW and DCC estimates for the linear properties. In an effort to try to quantify differences for large deformations, we assume for simplicity neo-Hookean behavior for the elastomeric matrix and provide below asymptotic re¯ and small concentrations c0 using both essults for large stretch λ timates. Thus, our estimate (55) leads to a results of the form





  1 + 3c0 (1) 2 ε (2) + ε (1) ¯ ,D ¯ ∼ ˜c λ    λ¯ 2 D¯ 2 , W μ λ¯ + (1)  (2) 2 2ε ε + ε (1) + 2c0 ε (2) − ε (1)

(56)

while the corresponding estimate of Lefèvre et al. (2017) leads to





  1 + 2c0 (1) 2 ε (2) + ε (1) ¯ ,D ¯ ∼ ˜c λ    λ¯ 2 D¯ 2 , W μ λ¯ + (1)  (2) 2 2ε ε + ε (1) + 2c0 ε (2) − ε (1)

(57)

where D¯ has been assumed to be aligned with the stretch direction. Thus, it can be seen that the main difference for large stretches is associated with the purely mechanical contribution of the energy which is a little bit stiffer for our estimate (55). On the other hand, in this limit, the coupled electroelastic contribution is precisely the same. In this context, it is useful to recall from the work of deBotton (2005) that the purely mechanical homogenization estimate for the in-plane response of the fiberreinforced elastomers with SL microstructures and neo-Hookean matrix is also neo-Hookean, albeit with the HSW estimate for the effective modulus, while the purely mechanical response of the corresponding elastomers with isotropic distributions of circular fibers, as given by Lopez-Pamies and Ponte Castañeda (2006b), is not neo-Hookean, even if the matrix is neo-Hookean. In the above estimate (57) for the DECs with circular fibers at finite strains, Lefèvre et al. (2017) have replaced the exact HSW estimate for the effective modulus for the SL microstructures with the corresponding linear estimate for the DCC microstructures (while preserving the macroscopic neo-Hookean character of the response)— and hence the differences in the purely mechanical responses at large stretches.

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

ARTICLE IN PRESS

JID: SAS

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

[m5G;April 5, 2019;18:7] 11

Fig. 4. Schematic representation of the non-aligned setup in the reference (a) and deformed (b) configurations.

6. Results for the equilibrium rotation of elliptical fibers in DECs In this section, we consider DECs under general non-aligned conditions, where the initial orientation of the elliptical fibers is characterized by the angle β , as illustrated in Fig. 4. The mechanical loading conditions are as defined by equations (19) and (20) in ¯ and θ¯ , while the Lagrangian and terms of the loading parameters λ Eulerian representations of the electric displacement field are given by

¯ = D¯ Eˆ 2 D +



and





 

¯ −λ ¯ −1 D¯ Eˆ 1 d¯ = sin θ¯ cos θ¯ λ

  λ¯ sin2 θ¯ + λ¯ −1 cos2 θ¯ D¯ Eˆ 2 ,

(58)

respectively. Note that while the Lagrangian electric displacement field is assumed (without loss of generality, given the arbitrary orientation β of the fibers) to be aligned with the vertical direction, the true electric displacement field, obtained via the transforma¯ , is not in general aligned with the X2 direction. tion rule d¯ = J −1 F¯ D The objective of this section is to show how the energy function   ¯ , θ¯ , D ¯ ; ψ¯ (2) depends on the fiber rotation ψ¯ (2) and, in partic˜ λ W ular, how the new (PDS) prescription for the value of the equilibrium fiber rotation ψ¯ (2) compares with the earlier (PDA) prescription based on the purely mechanical part of the energy. We begin the discussion of our results by considering the perfectly aligned case. Thus, Fig. 5 shows the effective energy as a ¯ (2) function of the in-plane rotation of the  fibers ψ for different (1) (1) ¯ values of the electric displacement D/ ε μ = 0, 1, 2, 3, 4, 5. As ¯ = 1) the can be seen in Fig. 5(a) at no deformations (i.e., when λ ˜ is minimized for ψ¯ (2) = ψ¯ (2) = 0, independent of the energy W eq

value of the macroscopic electric displacement field. As we increase the macroscopic stretch this trend continues to hold for ¯ = 1.5 and λ ¯ = 2.0, for the specific example shown in Fig. 5. both λ For larger stretches, however, a more complex behavior is ob¯ = 2.5. For the specific exserved, as can be seen in Fig. 5(d) for λ ˜ ample shown in this  figure, we observe that the effective energy W (1) (1) ¯ is non-convex for D/ ε μ = 0, 1, 2, 3, while it becomes convex for larger values of the electric displacement field. The fact that the energy function for the purely mechanical case (D¯ = 0), or for cases where the electric fields are relatively small, is non-convex ¯ = 2.5), for large values of the macroscopic stretch (in this case λ is consistent with the possible development of a macroscopic instability through loss of strong ellipticity (Lopez-Pamies and Ponte Castañeda, 2006b). On the other hand, the fact that the energy function recovers convexity for sufficiently large values of the electric displacement field D¯ is consistent with the stabilizing effect of this field tending to keep the particles aligned with the field.

Fig. 6 shows the corresponding results for DECs with β = 0◦ , when a pure shear deformation is applied at an angle θ¯ = 90◦ (i.e., when the stretch is aligned with the large axis of the inclusion). For this specific loading conditions both deformation and electric field have an stabilizing effect on the fibers, and therefore the en(2) ˜ is always minimized for ψ¯ eq ergy function W = ψ¯ (2) = 0 for all values of the deformation and electric displacement field. However, the energy function can still become non-convex for sufficiently large values of D¯ , which is a consequence of the fact that the electrostatic component of the energy function has minima also at ψ¯ (2) = ±π . Fig. 7 shows the corresponding results for DECs with β = 0, when a pure shear deformation, characterized by the macroscopic ¯ , is applied at an angle θ¯ = 30◦ . For the purely mechanstretch λ ical loadings (when D¯ = 0), we observe that the equilibrium rotations are negative (i.e., fibers undergo clock-wise rotation) and they increase in magnitude as the macroscopic stretch becomes larger. This is consistent with the fact that the fibers tend to align themselves with the tensile axis of the deformation (Lopez-Pamies and Ponte Castañeda, 2006b). As the Lagrangian electric displacement field, D¯ , increases the equilibrium rotation decreases (becomes ¯ shown in this figmore negative) monotonically for all values of λ ure. This can be explained by noting that the equilibrium electrostatic orientation of the fibers is determined by the direction of the true electric displacement field. As explained earlier (see equation (58)2 for more details) under non-aligned loading conditions the true electric displacement field may have a component in the X1 direction, despite the fact that the Lagrangian electric displacement field is taken to be in the X2 direction. For the specific example of Fig. 7 the direction of the true electric displacement is such that it causes additional negative rotations of the fibers. In this context, it should be noted that for this case involving non(2) aligned electro-mechanical loading, the equilibrium rotation ψ¯ eq according to the PDS depends on the electric displacement field and is different from the purely mechanical equilibrium rotation based on the PDA. However, for this case where the electric field is largely aligned with the particles, the difference is relatively small. Fig. 8 shows the corresponding results for DECs with β = 90◦ , under a pure shear deformation, characterized by the macroscopic ¯ , applied at θ¯ = 90◦ . For this specific loading conditions stretch λ both the deformation and the electric field have a destabilizing effect on the fibers. More specifically, as can be seen in Fig. 8(a), the effective energy is convex and the equilibrium rotation of the fibers is identically zero when there is no applied deformation and  for small electric fields (in this case for D¯ / ε (1) μ(1) = 0, 1, 2). As the electric field increases further, the effective energy becomes non-convex with two identical minima at ψ¯ (2) = ±ψ¯ eq . Moreover,

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS 12

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx





¯ , θ¯ , D ¯ ; ψ¯ (2) on the in-plane rotation of the fibers for β = 0◦ , θ¯ = 0◦ , and different values of the electric ˜ λ Fig. 5. This figure shows the dependence of the effective energy W ¯ = 1.5, (c) λ ¯ = 2.0, and (d) λ ¯ = 2.5. In this figure w = 2, Jm = 100, ¯ = 1.0, (b) λ displacement field. The minima of the energy for each case are marked by the symbol “•”. (a) λ

c = 0.2, and E (2) /ε (1) = 10 0 0.

as can be seen in parts (b) through (d) of Fig. 8, the destabilizing effect of the applied electric field tends to happen at smaller values for the field, as the mechanically imposed deformation increases. Fig. 9 shows the corresponding results for DECs with β = 60◦ , under a pure shear deformation, characterized by the macroscopic ¯ , applied at θ¯ = 30◦ . Note that for this specific examstretch λ ple the mechanical loading (pure shear at 30◦ ) tends to rotate the fibers in the positive direction, while the electrostatic loading (electric displacement field in the X2 direction) tends to rotate the fibers in the negative direction. Thus, as can be seen in Fig. 9(a), in ¯ = 1) the equilibthe absence of mechanical loadings (i.e., when λ rium rotation of the fibers is always negative and its magnitude increases by increasing the electrostatic loading. This shows the tendency of the fibers to align themselves with the external electric fields. Fig. 9(b) shows the corresponding results for the case where both mechanical and electrostatic loadings are present at the same ¯ = 1.5 and D¯ = 0, the time. As can be seen in this figure when λ equilibrium rotation of the fibers is in the positive direction. As the electric displacement field increases, the equilibrium rotation of the fibers increases, up to a critical value, while staying on the positive side. However, for values of the electric displacement field

larger than this critical value, the effective energy becomes nonconvex with the global minimum on the negative side. Thus increasing the electric displacement beyond this critical value causes the orientation of the fibers to suddenly jump form the positive side to the negative side. For larger values of the deformation (i.e., ¯ = 2, 2.5) the fibers always tend to attain positive rotations for λ (for the shown values of D¯ ) even after the energy function becomes non-convex—see Fig. 9(c) and (d). For this case involving non-aligned electrical and mechanical loading, the equilibrium ro(2) tation ψ¯ eq according to the PDS is found to depend rather strongly on the electric displacement field and can be quite different from the purely mechanical equilibrium rotation based on the PDA for large values of D¯ . In summary, we have found that the interplay between the mechanical and electrostatic effects determines the equilibrium rotation of the fibers in a DEC under generally non-aligned conditions. Depending on the specific parameters chosen, the electrostatic and elastic effects may tend to reorient the fibers in the same or opposite directions. In particular, it was found that there are cases where—as the electric fields increase from zero to a large value while the macroscopic stretch is held fixed—the equilibrium

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx



¯ , θ¯ , D ¯ ; ψ¯ (2) ˜ λ Fig. 6. This figure shows the dependence of the effective energy W

13



on the in-plane rotation of the fibers for β = 0◦ , θ¯ = 90◦ , and different values of the ¯ = 1.5, (c) λ ¯ = 2.0, and (d) λ ¯ = 2.5. In this figure w = 2, ¯ = 1.0, (b) λ electric displacement field. The minima of the energy for each case are marked by the symbol “•”. (a) λ Jm = 100, c = 0.2, and E (2) /ε (1) = 10 0 0.

rotation of the fibers may undergo a sudden jump from a set of orientations controlled by the mechanical effects to another set of orientations controlled by the electrostatic effects.

form



F¯ F¯ = 11 0

F¯12 , 1/F¯11

(59)

where 7. Results for DECs with elliptical fibers under soft electrode boundary conditions In this section, we consider a dielectric actuator made of a DEC with initially non-aligned (with the coordinate axes) microstructure, as depicted in Fig. 10. The ideal dielectric elastomer matrix will be taken to be of the Gent type with Jm = 100, while the rigid dielectric fibers will be chosen such that E (2) /ε (1) = 10 0 0, and with volume fraction c = 0.2 and aspect ratio w. In the absence of external tractions and after the application of the voltage v0 , the composite is expected to undergo an in-plane deformation of the



F¯11 =







¯ 4 − 1 cos2 θ¯ 1+ λ

and F¯12 =

λ¯

¯ −1 ¯3−λ sin θ¯ cos θ¯ λ









¯ 4 − 1 cos2 θ¯ 1+ λ

.

(60) On the other hand, due to the presence of the conducting electrodes, the Eulerian and Lagrangian electric fields are aligned with the X2 direction, such that

e¯ =

F¯11 v0 Eˆ 2 h0

and E¯ =

v0 ˆ

h0

E2 ,

(61)

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS 14

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx



¯ , θ¯ , D ¯ ; ψ¯ (2) ˜ λ Fig. 7. This figure shows the dependence of the effective energy W



on the in-plane rotation of the fibers for β = 0◦ , θ¯ = 30◦ , and different values of the ¯ = 1.5, (c) λ ¯ = 2.0, and (d) λ ¯ = 2.5. In this figure w = 2, ¯ = 1.0, (b) λ electric displacement field. The minima of the energy for each case are marked by the symbol “•”. (a) λ Jm = 100, c = 0.2, and E (2) /ε (1) = 10 0 0.

where h0 denotes the height of the sample in the reference configuration. Note that both the Eulerian and Lagrangian electric displacement fields will have components in the X1 direction when β = 0, π /2, since for such cases the principal axes of the effective permittivity will no longer be aligned with the coordinate axes. In this section we will make use of the above-described PDS to determine the equilibrium orientation of the fibers and macroscopic response of the DEC actuator under non-aligned loadings. As discussed in the previous section, at certain points in a given loading path the electroelastic energy of the composite may become non-convex as a function of the fiber orientation. For such non-convex energies, the stationary condition (34)1 can

have multiple solutions. In this section, following the PDS, we will make use of the global minimum of (33) for general nonaligned loading conditions. However, for aligned loading cases, we will also consider—for reference purposes—the “principal” solution obtained by continuing the corresponding solution for small strains/small fields beyond the point where the electroelastic energy becomes non-convex and the “principal” solution no longer gives the global minimum of the energy. In addition, for comparison purposes, we will also include the PDA solutions (Siboni and Ponte Castañeda, 2014) based on the purely mechanical equilibrium rotations for general non-aligned loading conditions. In this respect, it should be noted that the “principal” solution associated with the PDA for aligned loadings coincides with the

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS

ARTICLE IN PRESS M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx



¯ , θ¯ , D ¯ ; ψ¯ (2) ˜ λ Fig. 8. This figure shows the dependence of the effective energy W

[m5G;April 5, 2019;18:7] 15



on the in-plane rotation of the fibers for β = 90◦ , θ¯ = 90◦ , and different values of the ¯ = 1.5, (c) λ ¯ = 2.0, and (d) λ ¯ = 2.5. In this figure w = 2, ¯ = 1.0, (b) λ electric displacement field. The minima of the energy for each case are marked by the symbol “•”. (a) λ Jm = 100, c = 0.2, and E (2) /ε (1) = 10 0 0.

above-mentioned “principal” solution for the PDS—and, for this reason, only one “principal” solution will appear in the results to be shown below. We begin our discussion by considering the perfectly aligned case (i.e., β = 0◦ ). Thus, Fig. 11 shows plots of the effective response of a DEC consisting of fibers of aspect ratio w = 2 under the perfectly aligned conditions. In particular, Fig. 11(a) and (b) show the Lagrangian electric and electric displacement fields as ¯ , while Figs. 11(c) and (d) show the correfunctions of the stretch λ sponding results for the loading angle θ¯ and the shear component of the macroscopic deformation F¯12 . In addition, Fig. 11(e) and (f) show the evolution of the microstructure as determined by the inplane rotation of the fibers ψ¯ (2) , as well as the relative rotation ψ¯ (2) − ψ¯ . As expected, both the PDS (shown in solid blue lines) and the PDA (red solid lines) agree identically with the “principal” solution (black dashed lines) for small values of the applied electric field. This is because, for this aligned loading case, the equilibrium value for the particle rotation—both according to the PDS and PDA—is initially identically zero (in agreement with the corresponding value of the particle rotation for the principal solution). However, the PDA solution is seen to bifurcate from the

¯ larger than a cerprincipal solution for a value of the stretch λ ¯ tain value (λbr 2.2 for this value of w). As can be seen from the fiber orientation plots shown in Fig. 11(e) and (f), this is due to the spontaneous rotation of the fibers, due to the loss of convexity of mechanical energy as a function of the fiber orientation. In fact, the purely mechanical energy can be shown to become non(2) convex with two local minima at ψ¯ eq = ±ψ ∗ , corresponding to the two branches seen in the figures for the PDA. Note that the two branches have the same responses for E¯ and D¯ 2 as functions ¯ , as can be seen in Fig. 11(a) and (b), even though of the stretch λ these two branches correspond to the two different microstruc(2) tural states defined by ψ¯ eq = ±ψ ∗ , as can be seen in Fig. 11(e) and (f). On the other hand, for this particular value of w, it can be seen that prediction for the response of the DEC generated by the PDS scheme (solid blue curves) does not bifurcate from the principal solution, at least for the range of stretches shown in the figure. This result can be explained by the stabilizing effect of the electric fields, which tend to keep the fibers aligned with the externally applied electric fields. As a consequence, it is observed that the (approximate) PDA response is “stiffer” than the corresponding PDS (and principal solution) response, since the electric

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS 16

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx



¯ , θ¯ , D ¯ ; ψ¯ (2) ˜ λ Fig. 9. This figure shows the dependence of the effective energy W



on the in-plane rotation of the fibers for β = 60◦ , θ¯ = 30◦ , and different values of the ¯ = 1.5, (c) λ ¯ = 2.0, and (d) λ ¯ = 2.5. In this figure w = 2, ¯ = 1.0, (b) λ electric displacement field. The minima of the energy for each case are marked by the symbol “•”. (a) λ Jm = 100, c = 0.2, and E (2) /ε (1) = 10 0 0.

Fig. 10. Schematic of a dielectric actuator made out of a DEC with non-aligned (with the coordinate axes) microstructure sandwiched between two compliant electrodes. (a) The actuator in the reference configuration and (b) the actuator in the deformed configuration (i.e., after the application of the voltage). Note that the external tractions are assumed to be zero.

field required to generate the same overall stretch is larger for the PDA. For completeness, Fig. 12 shows the corresponding results for a DEC consisting of fibers with aspect ratio w = 3. As can be seen from these figures, the branching of the PDA response curves from

the principal solution, for this case (w = 3), occurs at a smaller ¯ =λ ¯ Br 1.7) than for the previous case (w = 2). This can stretch (λ be explained by noting that fiber-constrained composites consisting of fibers with larger aspect ratios are mechanically less stable under the loading conditions considered here. For the same reason,

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS

ARTICLE IN PRESS M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

[m5G;April 5, 2019;18:7] 17

Fig. 11. The response of a DEC with β = 0◦ under the soft electrode boundary condition and zero external tractions. (a) Electric field, (b) electric displacement field, (c) ¯ . In this figure Jm = 100, E (2) /ε (1) = 10 0 0, and w = 2, c = 0.2. loading angle, (d) F¯12 , (e) equilibrium rotation, and (f) relative rotation as functions of λ

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS 18

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

Fig. 12. The response of a DEC with β = 0◦ under the soft electrode boundary condition and zero external tractions. (a) Electric field, (b) electric displacement field, (c) ¯ . In this figure Jm = 100, E (2) /ε (1) = 10 0 0, and w = 3, c = 0.2. loading angle, (d) F¯12 , (e) equilibrium rotation, and (f) relative rotation as functions of λ

it is found that the PDS solution also bifurcates in this case into a lower energy solution at sufficiently large values of the applied fields. Note that even though the differences between the PDS and principal solutions in Fig. 12(a) and (b) are very small, the branching of the PDS solution can be easily observed in Fig. 12(c)–(f). It

is also observed that the branching of the PDS results happens at larger stretches (compared to the PDA results) due to the stabilizing effects of the electric fields. Furthermore, the branching of the PDS solutions is seen to happen more smoothly. Therefore, it is concluded that—consistent with the previous results—the PDA

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS

ARTICLE IN PRESS M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

leads to a stiffer response relative to the more accurate PDS results. Fig. 13 shows the results for DECs subjected to non-aligned loadings with β = 22.5◦ and fiber aspect ratio w = 2. As can be seen in Fig. 13(a) to (d), the macroscopic response of the DECs, as predicted by the PDA scheme, is initially in good agreement with the corresponding response obtained by means of the more accurate PDS scheme. This suggests that for small values of the fields (i.e., when the mechanical effects are dominant) the PDA scheme can provide relatively accurate estimates for the effective response of the DECs. However, as the electric fields increase, and therefore the electrostatic effects become stronger, the PDA scheme loses its accuracy. In addition, it should be noted that the predictions of the PDS and PDA schemes for the microstructure evolution are quite different, even for very small electric fields. In particular, as can be seen in Fig. 13(e) to (f), the PDA scheme leads to positive in-plane rotations for the fibers, while the PDS leads to negative rotations for the fibers. In this context, it should be recalled that the fiber rotations in the PDA scheme are determined by the purely mechanical energy, and that, in the purely mechanical problem, the fibers tend to align their longer in-plane axes with the direction of the larger principal stretch, i.e., the X1 direction for the example shown here. For the specific example of Figs. 13, this leads to positive rotations for the fibers in the purely mechanical problem, and therefore the rotations of the fibers as determined by the PDA scheme are positive. On the other hand, the rotations according to the PDS scheme are obtained from the coupled electromechanical energy in which both mechanical and electrical effects are present. In particular, in the presence of both mechanical and electrical loadings, the equilibrium configurations for the mechanical and electrical energy contributions tend to be different—the mechanical energy tends to align the longer in-plane axes of the fibers with the direction of the larger principal stretch (i.e., the X1 direction for the example shown here), while the electrostatic contribution tends to align the longer axes of the fibers with the direction of the applied electric field (i.e., the X2 direction for the example shown here). Thus, the equilibrium orientation of the fibers is the result of the complex interplay between these two tendencies and depends on the relative strengths of the mechanical and electrical loadings. For the specific example of Figs. 13, the electrostatic effects appear to become dominant, even for very small fields— and the PDS scheme predicts that the fibers undergo negative rotations, in contrast to the positive rotations observed for the PDA. In conclusion, it is found that while the predictions of the PDA for the macroscopic response are relatively accurate (compared to the PDS) for small values of the externally applied fields, they become increasingly less accurate for larger values of the fields. Moreover, the predictions of the PDA and PDS for the microstructural evolution are qualitatively different even for small values of the external fields. Figs. 14 and 15 show plots of the corresponding results for DECs with β = 45◦ and β = 67.5◦ , respectively. As can be seen from the figures, the PDA predictions for the macroscopic response are quite similar to the corresponding predictions using the more accurate PDS scheme for small enough values of the externally applied fields, but they become increasingly different with increasing values of the externally applied fields. In addition, it is observed that the PDA and PDS responses are qualitatively similar to the corresponding results for DECs with β = 22.5◦ . However, as β increases to 45◦ and then 67.5◦ , the magnitude of the fiber rotations as obtained by the PDA scheme get progressively smaller than the corresponding values for DECs with β = 22.5◦ . This can be explained by noting that in the purely mechanical problem fibers tend to align their longer axes with the X1 direction, and therefore smaller initial misalignment with the X1 direction (i.e., larger β ) results in a smaller rotation. On the other hand, as β increases

[m5G;April 5, 2019;18:7] 19

to 45◦ and then 67.5◦ , the magnitude of the fiber rotations as obtained by the PDS scheme become progressively larger than the corresponding values for DECs with β = 22.5◦ . This is due to the fact that in the PDS scheme the rotations are largely controlled by the direction of the applied electric fields (i.e., X2 direction), and therefore larger misalignments with this direction (i.e., larger β ) results in a larger electrostatic torque and consequently larger rotations. Next, Fig. 16 shows the corresponding results for DECs with β = 90◦ and w = 2. As can be seen in this figure, the PDA predictions for the macroscopic response and microstructure evolution of the DEC coincide exactly with the principal solution. This is because, according to the PDA scheme, the rotation of the fibers is obtained from the purely mechanical energy contribution, and for DECs with β = 90◦ (i.e., when the longer axes of the fibers is aligned with the X1 direction) the rotation of the fibers is identically zero. On the other hand, the PDS results are found to coincide with the principal solution only up to a certain critical ¯ =λ ¯ 2.4). As the deformation progresses beyond this stretch (λ br value, the PDS solution branches out into two lower energy solutions with identical electric fields but different microstructural evolutions (depending on whether the fibers “flip” to one side or the other). These new lower energy solutions correspond to situations where the equilibrium (in-plane) rotations of the fibers are approximately equal to ± π2 , as can be seen in Fig. 16(e). This result is physically consistent with the tendency of the fibers to reorient their longer in-plane axes with the external electric fields. Finally, Fig. 17 shows the corresponding results for a DEC with β = 90◦ and w = 3. The main observation from this figure is that the branching of the PDS solutions happens earlier ¯ 1.9, as opposed to λ ¯ 2.4 for w = 2). This is ex(i.e., λ br br plained by the fact that fibers with larger aspect ratios experience larger electrostatic torques, and are therefore (electrostatically) less stable. Thus far we have studied the response of DECs with both aligned and non-aligned microstructures as obtained by the PDA and PDS schemes. For the perfectly aligned case when β = 0◦ , we have shown that both PDA and PDS solutions agree with the principal solution for small enough fields, but then branch out from the principal solution with the PDA leading to a stiffer response. For the perfectly aligned case when β = 90◦ , it was found that only the PDS solutions branches out from the principal solution with the PDA remaining identical to the principal solution. On the other hand, for non-aligned loadings, it was found that the PDA estimates for the macroscopic response are fairly close (but no identical) to the more accurate PDS estimates for small values of the externally applied field, but differ quite significantly for larger values of the fields. This phenomenon could be traced back to the different evolutions for the fiber orientations under the two different schemes. In conclusion, the approximate PDA scheme should only be used for small values of the electric fields relative to the mechanical stiffness of the matrix material, as measured by the dimensionless parameter

κ :=

1

μ

(1)

×

D¯ 22

ε (1)

.

(62)

When κ  1, the mechanical effects are stronger (relative to the electrostatic effects), and therefore the difference between the response curves, as obtained by the PDA and PDS schemes, are negligible. On the other hand, when κ ࣡ 1 the electrostatic effects are comparable with or stronger than the mechanical effects, and therefore the difference between the PDA and PDS schemes may not be ignored.

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS 20

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

Fig. 13. The response of a DEC with β = 22.5◦ under the soft electrode boundary condition and zero external tractions. (a) Electric field, (b) electric displacement field, (c) ¯ . In this figure Jm = 100, E (2) /ε (1) = 10 0 0, and w = 2, c = 0.2. loading angle, (d) F¯12 , (e) equilibrium rotation, and (f) relative rotation as functions of λ

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS

ARTICLE IN PRESS M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

[m5G;April 5, 2019;18:7] 21

Fig. 14. The response of a DEC with β = 45◦ under the soft electrode boundary condition and zero external tractions. (a) Electric field, (b) electric displacement field, (c) ¯ . In this figure Jm = 100, E (2) /ε (1) = 10 0 0, and w = 2, c = 0.2. loading angle, (d) F¯12 , (e) equilibrium rotation, and (f) relative rotation as functions of λ

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS 22

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

Fig. 15. The response of a DEC with β = 67.5◦ under the soft electrode boundary condition and zero external tractions. (a) Electric field, (b) electric displacement field, (c) ¯ . In this figure Jm = 100, E (2) /ε (1) = 10 0 0, and w = 2, c = 0.2. loading angle, (d) F¯12 , (e) equilibrium rotation, and (f) relative rotation as functions of λ

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS

ARTICLE IN PRESS M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

[m5G;April 5, 2019;18:7] 23

Fig. 16. The response of a DEC with β = 90◦ under the soft electrode boundary condition and zero external tractions. (a) Electric field, (b) electric displacement field, (c) ¯ . In this figure Jm = 100, E (2) /ε (1) = 10 0 0, and w = 2, c = 0.2. loading angle, (d) F¯12 , (e) equilibrium rotation, and (f) relative rotation as functions of λ

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS 24

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

Fig. 17. The response of a DEC with β = 90◦ under the soft electrode boundary condition and zero external tractions. (a) Electric field, (b) electric displacement field, (c) ¯ . In this figure Jm = 100, E (2) /ε (1) = 10 0 0, and w = 3, c = 0.2. loading angle, (d) F¯12 , (e) equilibrium rotation, and (f) relative rotation as functions of λ

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS

ARTICLE IN PRESS M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

8. Concluding remarks In this work we have investigated the effective response and evolution of the microstructure for fiber-constrained DECs under general non-aligned loading conditions. We have considered both the PDA scheme, in which the rotations are obtained from the solution of the purely mechanical problem, and the more accurate PDS scheme, in which the rotations are obtained via the more general equilibrium conditions (34). We have also considered the possible development of instabilities for aligned loadings by way of bifurcations into lower energy states. The results for the evolution of the microstructure (i.e., the rotations of fibers in the fiber-constrained DECs) show the complex interplay between the mechanical and electrostatic effects, especially, for the non-aligned loading conditions. More specifically, we have seen that the equilibrium rotations predicted by the PDA and PDS schemes may be in opposite directions. Furthermore, we have observed that, for non-aligned loading conditions, there may be a sudden jump in the orientation of the fibers as the electric fields increase, while the deformation is held fixed, due to the change in relative strength of electrostatic effects compared to the elastic effects. The results for the response of DECs under the electrode boundary condition and for non-aligned loadings show that the difference between the PDA and PDS solutions are small for small deformations. This is in spite of the fact that, as noted above, the predictions for the fiber rotation may be quite different. For larger deformations, the difference between the PDA and PDS solutions becomes more significant. To distinguish between this two cases, we have identified the dimensionless parameter κ which characterizes the relative strength of the electrostatic and elastic effects. Thus, when κ < < 1, the easier-to-implement PDA scheme can be safely used to obtain the effective response of the DECs. On the other hand, for cases where κ ࣡ 1, the more accurate PDS scheme must be used since for such cases the electrostatic effects are comparable to (or stronger than) the elastic effects and therefore they cannot be ignored. In addition, it has been found that, while the PDA and PDS principal solutions are identical for aligned loadings, there can be significant differences in the post-bifurcation responses. The results show that both PDA and PDS schemes bifurcate from the principal solution for the cases when β = 0◦ (i.e., when the fibers are favorably aligned for electrical loading, but unfavorably aligned for mechanical loadings) and for large enough deformations; however, the PDS scheme bifurcates from the principal solution at a larger deformation (compared to the PDA scheme) due to the stabilizing effect of the electric fields. For cases when β = 90◦ (i.e., when the fibers are favorably aligned for mechanical loading, but unfavorably aligned for electrical loadings), the results show that the PDS solutions bifurcate from the principal solution, due to the tendency of the fibers to align their longer in-plane axes with the direction of the applied field. (In this case, the PDA solution does not bifurcate.) Finally, it is important to remark in this context that the instabilities observed in this work appear at a more microscopic level than the corresponding instabilities in the work of Siboni and Ponte Castañeda (2014) and Siboni et al. (2015), which were determined by loss of ellipticity of the overall response generated by the principal solution (see also Avazmohammadi and Ponte Castañeda, 2016; Furer and Ponte Castañeda, 2018). Further work will be necessary to assess the differences and links between these two different approaches. Acknowledgments This work was begun with the support of the Applied Computational Analysis Program of the Office of Naval Research under Grant

[m5G;April 5, 2019;18:7] 25

N0 0 0141110708 and completed with the support of the Applied Math Program of the National Science Foundation under Grant No. DMS-1613926. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijsolstr.2019.03.027. References Aschwanden, M., Stemmer, A., 2006. Polymeric, electrically tunable diffraction grating based on artificial muscles. Opt. Lett. 31, 2610–2612. doi:10.1364/OL.31. 002610. Avazmohammadi, R., Ponte Castañeda, P., 2016. Macroscopic constitutive relations for elastomers reinforced with short aligned fibers: instabilities and postbifurcation response. J. Mech. Phys. Solids 97, 37–67. doi:10.1016/j.jmps.2015.07. 007. Ball, J.M., 1976. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403. doi:10.10 07/BF0 0279992. Electroactive polymer (EAP) actuators as artificial muscles: Reality, potential, and challenges, 2004. In: Bar-Cohen, Y. (Ed.), 2nd. SPIE—The International Society for Optical Engineering, Bellingham, WA doi:10.1117/3.547465.ch1. Benveniste, Y., Dvorak, G.J., 1992. Uniform fields and universal relations in piezoelectric composites. J. Mech. Phys. Solids 40 (6), 1295–1312. Bertoldi, K., Gei, M., 2011. Instabilities in multilayered soft dielectrics. J. Mech. Phys. Solids 59 (1), 18–42. doi:10.1016/j.jmps.2010.10.001. Bolzmacher, C., Biggs, J., Srinivasan, M., 2006. Flexible dielectric elastomer actuators for wearable human-machine interfaces. Proc. SPIE 6168, 616804–616804– 12. doi:10.1117/12.677981. Brochu, P., Pei, Q., 2010. Advances in dielectric elastomers for actuators and artificial muscles. Macromol. Rapid Commun. 31, 10–36. doi:10.10 02/marc.20 090 0425. Cheng, Z., Zhang, Q., 2008. Field-activated electroactive polymers. MRS Bull. 33, 183–187. doi:10.1557/mrs2008.43. deBotton, G., 2005. Transversely isotropic sequentially laminated composites in finite elasticity. J. Mech. Phys. Solids 53 (6), 1334–1361. doi:10.1016/j.jmps.2005. 01.006. deBotton, G., Tevet-Deree, L., Socolsky, E.A., 2007. Electroactive heterogeneous polymers: analysis and applications to laminated composites. Mech. Adv. Mater. Struct. 14 (1), 13–22. doi:10.1080/15376490600864372. Destrade, M., Ogden, R.W., 2011. On magneto-acoustic waves in finitely deformed elastic solids. Math. Mech. Solids 16 (6), 594–604. doi:10.1177/ 1081286510387695. Dorfmann, A., Ogden, R.W., 2005. Nonlinear electroelasticity. Acta Mech. 174, 167– 183. doi:10.10 07/s0 0707-0 04-0202-2. Electrodynamics of continua. volume 1 - Foundations and solid media. volume 2 Fluids and complex media, 1990. In: Eringen, A.C., Maugin, G.A. (Eds.), 1. Furer, J., Ponte Castañeda, P., 2018. Macroscopic instabilities and domain formation in neoHookean laminates. J. Mech. Phys. Solids 118, 98–114. doi:10.1016/j.jmps. 2018.05.006. Galipeau, E., Ponte Castañeda, P., 2013. A finite-strain constitutive model for magnetorheological elastomers: magnetic torques and fiber rotations. J. Mech. Phys. Solids 61 (4), 1065–1090. doi:10.1016/j.jmps.2012.11.007. Gent, A.N., 1996. A new constitutive relation for rubber. Rubber Chem. Technol. 69 (1), 59–61. doi:10.5254/1.3538357. Geymonat, G., Müller, S., Triantafyllidis, N., 1993. Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Ration. Mech. Anal. 122 (3), 231–290. doi:10.10 07/BF0 0380256. Hill, R., 1972. On constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. Soc. Lond. A. Math. Phys. Sci. 326 (1565), 131–147. doi:10.1098/ rspa.1972.0 0 01. Huang, C., Zhang, Q.M., deBotton, G., Bhattacharya, K., 2004. All-organic dielectricpercolative three-component composite materials with high electromechanical response. Appl. Phys. Lett. 84, 4391–4393. doi:10.1063/1.1757632. Hutter, K., Ursescu, A., van de Ven, A.A.F., 2006. Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids. Springer. Kovetz, A., 20 0 0. Electromagnetic Theory with 225 Solved Problems. Oxford University Press, USA. Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P., 1984. Elecrodynamics of Continuous Media. Pergamon Press. Lefèvre, V., Danas, K., Lopez-Pamies, O., 2017. A general result for the magnetoelastic response of isotropic suspensions of iron and ferrofluid particles in rubber, with applications to spherical and cylindrical specimens. J. Mech. Phys. Solids 107, 343–364. Lefevre, V., Lopez-Pamies, O., 2014. The overall elastic dielectric properties of fa suspension of spherical particles in rubber: an exact explicit solution in the small-deformation limit. J. Appl. Phys. 116, 134106. Lefevre, V., Lopez-Pamies, O., 2015. The overall elastic dielectric properties of fiber-strengthened/weakened elastomers. J. Appl. Mech. 82 (11), 111009. Li, J.Y., Huang, C., Zhang, Q., 2004. Enhanced electromechanical properties in allpolymer percolative composites. Appl. Phys. Lett. 84, 3124–3126. doi:10.1063/1. 1702127.

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027

JID: SAS 26

ARTICLE IN PRESS

[m5G;April 5, 2019;18:7]

M.H. Siboni and P. Ponte Castañeda / International Journal of Solids and Structures xxx (xxxx) xxx

Li, J.Y., Rao, N., 2004. Micromechanics of ferroelectric polymer-based electrostrictive composites. J. Mech. Phys. Solids 52, 591–615. doi:10.1016/S0022-5096(03) 00117-0. Li, W., Landis, C.M., 2012. Deformation and instabilities in dielectric elastomer composites. Smart Mater. Struct. 21, 094006. Lopez-Pamies, O., 2014. Elastic dielectric composites: theory and application to particle-filled ideal dielectrics. J. Mech. Phys. Solids 64, 61–82. Lopez-Pamies, O., Ponte Castañeda, P., 2006a. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I–theory. J. Mech. Phys. Solids 54 (4), 807–830. doi:10.1016/j.jmps.2005. 10.006. Lopez-Pamies, O., Ponte Castañeda, P., 2006b. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: II–Application to cylindrical fibers. J. Mech. Phys. Solids 54 (4), 831–863. doi:10.1016/j.jmps.2005.10.010. Lu, T., Huang, J., Jordi, C., Kovacs, G., Huang, R., Clarke, D.R., Suo, Z., 2012. Dielectric elastomer actuators under equal-biaxial forces, uniaxial forces, and uniaxial constraint of stiff fibers. Soft Matter 8 (22), 6167–6173. doi:10.1039/ c2sm25692d. McMeeking, R.M., Landis, C.M., 2005. Electrostatic forces and stored energy for deformable dielectric materials. J. Appl. Mech. 72, 581–590. doi:10.1115/1.1940661. Ogden, R.W., 1997. Non-linear Elastic Deformations. Courier Dover Publications. Ozsecen, M.Y., Sivak, M., Mavroidis, C., 2010. Haptic interfaces using dielectric electroactive polymers. Proc. SPIE 7647, 764737. doi:10.1117/12.847244. Pelrine, R.E., Kornbluh, R.D., Joseph, J.P., 1998. Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation. Sens. Actuators, A: Phys. 64 (1), 77–85. doi:10.1016/S0924-4247(97)01657-9. Ponte Castañeda, P., Galipeau, E., 2011. Homogenization-based constitutive models for magnetorheological elastomers at finite strain. J. Mech. Phys. Solids 59 (2), 194–215. doi:10.1016/j.jmps.2010.11.004. Ponte Castañeda, P., Siboni, M.H., 2012. A finite-strain constitutive theory for electro-active polymer composites via homogenization. Int. J. Non-Linear Mech. 47 (2), 293–306. doi:10.1016/j.ijnonlinmec.2011.06.012. Ponte Castañeda, P., Willis, J.R., 1995. The effect of spatial distribution on the effective behavior of composite materials and cracked media. J. Mech. Phys. Solids 43 (12), 1919–1951. doi:10.1016/0 022-5096(95)0 0 058-Q. Rao, N., Li, J.Y., 2004. The electrostriction of p(VDF-Trfe) copolymers embedded with textured dielectric particles. Int. J. Solids Struct. 41, 2995–3011. doi:10.1016/j. ijsolstr.2003.12.021. Ren, K., Liu, Y., Hofmann, H., Zhang, Q.M., Blottman, J., 2007. An active energy harvesting scheme with an electroactive polymer. Appl. Phys. Lett. 91, 132910. doi:10.1063/1.2793172. Rudykh, S., deBotton, G., 2011. Stability of anisotropic electroactive polymers with application to layered media. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 1–12. doi:10.10 07/s0 0 033- 011- 0136- 1.

Sarban, R., Oubaek, J., Jones, R.W., 2009. Closed-loop control of a core free rolled EAP actuator. Proc. SPIE 7287, 72870G–1+. doi:10.1117/12.815433. Shkel, Y.M., Klingenberg, D.J., 1998. Electrostriction of polarizable materials: comparison of models with experimental data. J. Appl. Phys. 83 (12), 7834–7843. doi:10.1063/1.367958. Siboni, M.H., Avazmohammadi, R., Ponte Castañeda, P., 2015. Electromechanical instabilities in fiber-constrained, dielectric-elastomer composites subjected to all-around dead-loading. Math. Mech. Solids 20, 729–759. doi:10.1177/ 1081286514551501. Siboni, M.H., Ponte Castañeda, P., 2012. A magnetically anisotropic, ellipsoidal inclusion subjected to a non-aligned magnetic field in an elastic medium. C. R. Mécanique 340 (4–5), 205–218. doi:10.1016/j.crme.2012.02.003. Siboni, M.H., Ponte Castañeda, P., 2013. Dielectric elastomer composites: smalldeformation theory and applications. Philos. Mag. 93 (21), 2769–2801. doi:10. 1080/14786435.2013.788258. Siboni, M.H., Ponte Castañeda, P., 2014. Fiber-constrained, dielectric-elastomer composites: finite-strain response and stability analysis. J. Mech. Phys. Solids 68, 211–238. doi:10.1016/j.jmps.2014.03.008. Siboni, M.H., Ponte Castañeda, P., 2016. Macroscopic response of particle-reinforced elastomers subjected to prescribed torques or rotations on the particles. J. Mech. Phys. Solids 91, 240–264. doi:10.1016/j.jmps.2016.02.028. Siboni, M.H., Ponte Castañeda, P., 2019. Finite-strain Homogenization Models for Anisotropic Dielectric Elastomer Composites. In: Merodio, J., Ogden, R.W. (Eds.), Solid Mechanics and Its Applications: Constitutive Modelling of Solid Continua. Springer, p. to appear. Spinelli, S.A., Lefèvre, V., Lopez-Pamies, O., 2015. Dielectric elastomer composites: a general closed-form solution in the small-deformation limit. J. Mech. Phys. Solids 83, 263–284. Stark, K., Garton, C., 1955. Electric strength of irradiated polythene. Nature 176, 1225–1226. Sundar, V., Newnham, R.E., 1992. Electrostriction and polarization. Ferroelectrics 135 (1), 431–446. doi:10.1080/00150199208230043. Suo, Z., Zhao, X., Greene, W., 2008. A nonlinear field theory of deformable dielectrics. J. Mech. Phys. Solids 56 (2), 467–486. doi:10.1016/j.jmps.2007.05.021. Tian, L., Tevet-Deree, L., deBotton, G., Bhattacharya, K., 2012. Dielectric elastomer composites. J. Mech. Phys. Solids 60 (1), 181–198. doi:10.1016/j.jmps.2011.08. 005. Toupin, R.A., 1956. The elastic dielectric. J. Ration. Mech. Analysis 5, 849–915. Zhao, X., Suo, Z., 2007. Method to analyze electromechanical stability of dielectric elastomers. Appl. Phys. Lett. 91, 061921. doi:10.1063/1.2768641. Zhao, X., Suo, Z., 2008. Electrostriction in elastic dielectrics undergoing large deformation. J. Appl. Phys. 104 (12), 123530. doi:10.1063/1.3031483.

Please cite this article as: M.H. Siboni and P. Ponte Castañeda, Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019. 03.027