CONCLUSIONES A manera de conclusión:

In this chapter we have provided an overview for the governing equations of elec- troelasticity and constitutive response of homogeneous dielectric elastomers. The formulation used here for describing the governing equations is based on a total stress which includes both the mechanical and electrostatic surface forces. The constitu- tive response of dielectric elastomers were described via a potential energy which is assumed to be a function of the deformation gradient and the Lagrangian electric dis- placement field. We have also provided the incremental equations of electroelasticity along with conditions for predicting material instabilities and failure of electro-active materials.

Chapter 3

A Coupled Homogenization

Framework

Effective medium theories have been used to model the behavior of DECs with vari- ous microstructures at both finite and infinitesimal deformations. For example,Shkel & Klingenberg (1998) have provided estimates for the deformation-dependent per- mittivity, as well as the electrostriction of isotropic dielectric solids. They used a mean-field approach to estimate the dipolar interactions at the microscopic level, and their analytical model for the effective deformation-dependent permittivity is in good agreement with their experiments (Shkel & Klingenberg 1998) for certain polymeric materials. Similarly,Li & Rao(2004) andRao & Li(2004) have developed a microme- chanical approach, which uses the “uniform field” concept of Benveniste & Dvorak

(1992) to provide estimates for the effective electrostrictive coefficients of polymer- matrix composites with aligned and randomly oriented ellipsoidal inclusions. More recently,Tian et al.(2012) have developed a rigorous method to compute the effective electro-elastic properties of composites in terms of coupled moments of the electro- static and elastic fields in the composite. They also provided results for sequentially laminated composites, where such coupled moments could be computed explicitly. In the context of finite deformations, the state of the art for DECs is less advanced. For example, deBotton et al.(2007), Bertoldi & Gei(2011) andRudykh & deBotton

DECs with layered microstructures at finite strains by making use of the classical for- mulations of finite-strain electroelasticity (Toupin 1956). In this chapter we provide a general homogenization theory for the coupled electro-elastic behavior of DECs. In particular, in section 3.1 we provide a variational formulation for the effective energy of DECs, generalizing the energy method of Hill (1972). In section 3.2 we consider two-phase DECs with random microstructures. More specifically, we describe the initial microstructure of the composite and its evolution. Then we introduce the par- tial decoupling strategy for obtaining the effective total electrostatic energy in terms of the effective purely mechanical energy and the effective (deformation-dependent) electrostatic energy. Finally, we provide Hashin-Schtricman type estimaes for the effective (deformation-dependent) permittivity of two-phase DECs with random mi- crostructures , which may in turn be used along with availabe estimates for the purely mechanical energy to obtain the effect total energy of DECs. Section 3.3 provides the corresponding developments in the context of two-phase DECs with periodic mi- crostructures.

3.1

The variational formulation

We consider a specimen Ω0 (in the reference configuration) made of the electro-active composite, which consists ofN homogeneous phases, occupying sub-domains Ω(₀r) in Ω0. The distribution of the phases is described by the characteristic functions Θ(₀r) (r=1, ..., N), such that Θ₀(r) is equal to 1 for X∈Ω(₀r) and zero otherwise. Similarly, the specimen in its deformed configuration can be described by the characteristic functions Θ(r) _(r = _{1, ..., N), such that Θ}(r)(_x) = _{1 for x} ∈_Ω(r) _{and zero otherwise,} where Ω(r) _{is the sub-domain of Ω (the deformed configuration of the specimen) that} is occupied by phase r. Throughout this work, the electro-active composites are assumed to satisfy the separation of length scales hypothesis. In other words, it is assumed that the length scale at which the characteristic functions Θ(₀r) vary (also referred to as the microscopic scale) is very small compared to the the size of the specimen Ω0 (or the macroscopic scale).

In this section, we develop a homogenization framework for the above-described electro-active composites with general microstructures in the quasi-static finite strain regimes. The basic idea is to generalize the heuristic approach ofHill (1972) in finite elasticity. Toward this goal, boundary conditions are prescribed that are consistent with “macroscopically uniform” fields in the composite. Here we enforce the condi- tions

x=F¯X, and D⋅N=D¯ ⋅N, on ∂Ω0, (3.1) where ¯F and ¯D are a prescribed, constant tensor and vector, respectively, and N is the outward unit normal to the boundary of the composite specimen ∂Ω0. It then follows, by means of the divergence theorem, that the macroscopic averages (over Ω0) for the deformation gradient and electric displacement fields are given by

⟨F⟩0 =F¯, and ⟨D⟩0=D¯, (3.2)

where ⟨⋅⟩0 has been used to denote a volume average in the reference configuration. This shows that ¯F and ¯D can be interpreted as the macroscopic, or average, defor- mation gradient and electric displacement field in the composite Ω0. Note that it is also possible to specify the electric field, or the traction on the boundary of the specimen. However, the boundary conditions (3.1) are preferred here since they lead to minimum-type variational formulations for the homogenization problem, because, as mentioned earlier, the associated potentials W are convex in Dand polyconvex in F.

Given the boundary conditions (3.1) and the assumed separation of length scales, it is expected on physical grounds that the composite material will behave like the homogeneous medium with effective, or homogenized energy function ˜W. Following the “energy” method of Hill (1972) for purely elastic composites, we define the ho- mogenized potential for the electro-active composite as the volume average of the energy stored in the composite under application of the boundary conditions (3.1),

namely,

˜

W(F¯,D¯) = min

F∈K(F¯) D∈D0(minD¯)⟨W(X,F,D)⟩0, (3.3)

where W(X,F,D) is defined in terms of the uniform phase potentials W(r)(_F,D) via W(X,F,D) = N ∑ r=1 Θ(₀r)(X) W(r)(F,D), (3.4) and where K (_F¯_{) = {F ∣∃x=x(X) withF =Gradxin Ω0, x=F}¯_{Xon∂Ω0}, (3.5)} and D0(_D¯_{) = {D∣DivD=0in Ω0, D⋅N=D}¯ _{⋅Non∂Ω0} (3.6)} are, respectively, sets of admissible deformation gradients and electric displacement fields that are compatible with the boundary conditions (3.1).

It can be readily shown (see Bustamante et al. 2009) for a more general version of this variational principle including contributions from the surrounding vacuum) that the Euler-Lagrange equations associated with the variational problem (3.3) are precisely the equilibrium equation (2.1)2 (with f0 = 0) and the Maxwell’s equations (2.5). (Note that the energy contributions of the inhomogeneous terms, f0,Q, and Σ, are ignored since they have been assumed to vary on the macroscopic length scale, and have no effect on the homogenization problem.) Therefore, the minimizers (assuming that they exist) of the above problem are also solutions of the electro-elastic problem (described in the previous section) with boundary conditions (3.1). To the best of our knowledge, there exist no rigorous mathematical results for the existence of the minimizers for the above variational problem. However, as was recently argued by

Ponte Casta˜neda & Galipeau (2011) for the analogous magneto-elasticity problem, it will be assumed here that the minimizers of the variational problem (3.3) exist at least for constitutive behaviors of the type discussed in chapter 2 for the ideal dielectric matrix and rigid particles.

of periodic repetitions of a representitive volume element (RVE), the general varia- tional problem (3.3), can be equivalently written in terms of a periodic variational problem. Thus, let us denote the building block (or the elementary unit cell) of the composite by U0 in the reference configuration. For purely elastic composites in the finite-deformation regime, it is known (Braides 1985, M¨uller 1987) that the solution of the homogenization problem (3.3), although periodic, need not have the same pe- riod as the elementary unit cell U0. Thus, while it is expected that the minimum energy solution may be initially periodic on one cell, after a certain level of loading, the solution may develop microscopic (or pattern changing) instabilities and bifur- cate into other solutions that may be periodic on a larger unit cell (or a super cell)

qU0 containing several elementary unit cells (see further below for more details). In such cases, the lowest value for the energy is obtained by “cooperative” interaction among several unit cells, after a certain level of loading. Therefore, the variational problem (3.3) for the effective energy of the electro-active composite with periodic microstructures under arbitrary loading conditions may be written in the form

˜ W(F¯,D¯) =min q ˜ WqU0(F¯,D¯), for q= (q1, q2, q3) ∈N3, (3.7) where ˜ WqU0(_F¯_,D¯) = _min u′∈_qU# 0 min A′∈_qU# 0 {_∣ 1 qU0∣∫qU0W(X; ¯F + ∇u ′_,D¯ _{+ ∇ ×A}′_{)dX} (3.8)}

is the effective energy associated with the super-cell qU0. In this last expression, u′

andA′areqU0-periodic fluctuation functions, such that the deformation gradient and electric displacement trial fields are given by F =F¯+Gradu′ and D =D¯ +CurlA′. Note that the minimizations over the sets of qU0-periodic displacement and vector

potential fluctuation fields (i.e., u′ and A′∈qU₀#) in expression (3.8) is equivalent to minimizations over the admissible sets (3.5) and (3.6) in expression (3.3).

Having defined the effective electro-elastic energy ˜W(F¯,D¯) of the composite, it can be shown by means of appropriate generalization of Hill’s lemma (see, for

example, Casta˜neda & Suquet 1997) that the average stress and average electric field, determined by ¯S = ⟨S⟩0 and ¯E= ⟨E⟩0, are given by

¯ S= ∂W˜ ∂F¯, and ¯ E=∂W˜ ∂D¯ , (3.9)

respectively. As mentioned earlier, ¯F and ¯D correspond to the average (or macro- scopic) deformation gradient and electric displacement fields in the composite. There- fore, expression (3.9) 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 phases, the effective energy function ˜W, as defined by (3.3), completely 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 specimen (i.e., ˜W) needs to be considered in the homog- enization problem. In addition, it is noted that ˜W is objective, which can be easily verified by making use of the objectivity of the phase potentials.

The Eulerian counterparts of the above effective constitutive equations for electro- active composites can be obtained in terms of the volume averages, denoted by ⟨⋅⟩, of the true mechanical and electrical fields over the deformed configuration Ω of the composite. Thus, we define ¯T = ⟨T⟩, ¯d = ⟨d⟩ and ¯e = ⟨e⟩, which can be shown to satisfy the following relations:

¯

T =J¯−1S¯F¯T, e¯=F¯−TE¯, and d¯=J¯−1F¯D¯. (3.10) Furthermore, we define the effective Eulerian energy-density function ˜w(F¯,d¯) =

˜

W(F¯,J¯F¯−1_d¯) /_J¯_{. It then follows from equation (3.10)}

1 that the average Cauchy stress is given by

¯ T = ∂w˜

∂FF

T +(_w˜−_¯e⋅_d¯)_I+_¯e⊗_{d¯. (3.11)} Note that ¯T is symmetric, which can be shown from objectivity using arguments completely analogous to those used in connection with equation (2.10) to show the

symmetry of the local stress tensor T. Also, it is easy to show, by means of (3.10)2, that

¯

e= ∂w˜

∂d¯. (3.12)

We remark that, we are not aware of mathematically rigorous results for the ex- istence of solutions of the variational problem (3.3) (or (3.7) in the periodic case). However, at least for the material models discussed in Section 2.2 for the elastomeric matrix and rigid inclusion phases, which ensure quasi-convexity inF (for fixedD) and convexity inD (for fixedF), together with appropriate growth conditions, for the lo- cal potentialW, it would be expected that results generalizing those ofM¨uller(1987) and Geymonat et al. (1993) for purely mechanical systems should hold. Naturally, non-unique solutions would be expected to arise as a consequence of the possible de- velopment of “microscopic” and “macroscopic” instabilities (e.g. Michel et al. 2010a) as the deformation progresses.

In the next section, building on earlier works (Ponte Casta˜neda & Galipeau 2011,

Ponte Casta˜neda & Siboni 2012), we propose a “partial decoupling strategy” in order to decouple the mechanical and electrostatic effects for electro-active composites with periodic and random microstructures. This will allow us to express the solution of the variational problem (3.3) (or (3.7) in the periodic case) 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 a finite set of mi- crostructural variables, corresponding to the particle positions and orientations in the deformed configuration, which may be obtained by means of a finite-dimensional optimization process.