1 Introduction

Over the last two decades, fueled by their potential to enable a broad spectrum of new technologies ranging from soft robots, to energy harvesters, to speakers, to biomedical and haptic devices [3, 6, 15, 36, 49], increasing efforts have been devoted to gain precise quantitative insight into the electromechanical behavior of dielectric elastomers.

On the theoretical front, most of the reported efforts have restricted attention to the idealization that dielectric elastomers are elastic dielectrics, that is, materials that deform and polarize without incurring dissipation of energy  [9, 11, 27, 28, 32, 34, 43]. While such an idealization may be justified under certain loading conditions, it is not appropriate in general as dielectric elastomers are inherently dissipative solids: they dissipate energy through viscous deformation and through friction in their electric polarization process.

Recognizing their true nature, a handful of dissipative models have been proposed over the last decade following different ad hoc approaches. The majority of such models account only for mechanical—and not for electric—dissipation and, save for a few works [5, 46], most assume ideal dielectric behavior  [2, 18, 47, 51]. The motivation for this bias is that the mechanical relaxation time of dielectric elastomers is much larger that their electric relaxation time. Yet, accounting for electric dissipation, in addition to mechanical dissipation, is critical when dealing with applied alternating currents, which is often the case in applications. Two exemptions that account for both mechanical and electric dissipation are the models presented in  [10] and [38]. The former corresponds to a model for the special case of a non-ideal dielectric elastomer with Gaussian elasticity, constant viscosity, and a single relaxation time due to electric dissipation. The latter corresponds to a model for a different special case, that of an ideal dielectric elastomer with non-Gaussian elasticity, also constant viscosity, and an ad hoc electric dissipation attributed to leakage current that is strictly applicable to a one-dimensional setting. In this context, our purpose in this paper is threefold.

First, we introduce a general constitutive framework to construct models for the dissipative electromechanical behavior of dielectric elastomers that automatically feature the distinguishing even electromechanical coupling of this class of materials as well as seamlessly comply with material frame indifference, material symmetry, and the entropy imbalance requirements. As elaborated in Sect. 2, this framework is nothing more than the celebrated two-potential framework in mechanics  [13, 16, 24, 53, 54] extended to account for the coupling with Maxwell’s equations.

Second, we make use of the general two-potential framework to construct a specific model for the prominent class of isotropic and incompressible dielectric elastomers. As elaborated in Sect. 3, the proposed model corresponds to a generalization of the rubber viscoelastic model of Kumar and Lopez-Pamies  [24]. Accordingly, it accounts for the non-Gaussian elasticity and the deformation-enhanced shear-thinning viscosity typical of dielectric elastomers. It also accounts for their characteristic electrostriction and time- and deformation-dependent polarization. Section 3 also includes a detailed analysis of the specialization of the model in the fundamental limit of small deformations and moderate electric fields.

Third, for demonstration purposes, we deploy the proposed model to describe the electromechanical behavior of the acrylate elastomer VHB 4910 in full. As illustrated in Sect. 4 by means of direct comparisons with experimental data available in the literature, the model is capable both of describing and predicting the electromechanical behavior of this popular and commercially significant dielectric elastomer from 3M.

2 The two-potential framework for dielectric elastomers

2.1 Kinematics

Consider a deformable and polarizable homogeneous solid that occupies in its initial—undeformed, stress-free, and unpolarized—configuration (at time \(t = 0\)) an open bounded domain \({\Omega }_0\subset {{\mathbb {R}}}^3\), with boundary \(\partial {\Omega }_0\) and unit outward normal \({\mathbf{N}}\). We identify material points by their initial position vector \({\mathbf{X}}\in {\Omega }_0\). At a later time \(t\in (0, T]\), due to the applied boundary conditions and source terms described below, the position vector \({\mathbf{X}}\) of a material point moves to a new position specified by

$$\begin{aligned} {\mathbf{x}}={\mathbf{y}}({\mathbf{X}},t), \end{aligned}$$

where \({\mathbf{y}}\) is an invertible mapping from \({\Omega }_0\) to the current configuration \({\Omega }(t)\), also contained in \({{\mathbb {R}}}^3\). We write the associated deformation gradient and its determinant at \({\mathbf{X}}\) and t as

$$\begin{aligned} {\mathbf{F}}=\nabla {\mathbf{y}}({\mathbf{X}},t)=\dfrac{\partial {\mathbf{y}}}{\partial {\mathbf{X}}}({\mathbf{X}},t)\quad \mathrm{and}\quad J=\det {\mathbf{F}}. \end{aligned}$$

2.2 Constitutive behavior

Following the two-potential formalism  [13, 16, 24, 53, 54], absent changes in temperature, the constitutive behavior of the solid can be expediently characterized by two thermodynamic potentials that describe how the solid stores and dissipates energy through deformation and polarization, namely: (i) a free-energy function \(\psi \) and (ii) a dissipation potential \(\phi \).

The functional form of these two potentials must be selected so as to be descriptive of the three basic defining features of dielectric elastomers, to wit:

  • dielectric elastomers exhibit even electromechanical coupling,

  • when all mechanical forces and electric fields are removed after an arbitrary loading path, dielectric elastomers creep to their original undeformed, stress-free, and unpolarized initial configuration, and

  • when subject to mechanical (electric) relaxation and creep loading conditions, dielectric elastomers exhibit a transient response that then evolves into an equilibrium state of deformation (voltage) and stress (polarization).

The last two of the above defining features dictate the following functional form:

$$\begin{aligned} \psi =\psi ^{\mathrm{Eq}}({\mathbf{F}},{\mathbf{E}})+\psi ^{\mathrm{NEq}}({\mathbf{F}},{\mathbf{E}},{\mathbf{F}}{{\mathbf{F}}^{v}}^{-1},{\mathbf{E}}-{\mathbf{E}}^v) \end{aligned}$$
(1)

for the free-energy function and

$$\begin{aligned} \phi =\phi ({\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^v,{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,\dot{{\mathbf{E}}}^v) \end{aligned}$$
(2)

for the dissipation potential; the first feature is summoned further below in Sect. 2.2.1 together with other fundamental constraints that the functions \(\psi \) and \(\phi \) ought to satisfy. In relation (1), \(\psi ^{\mathrm{Eq}}\) characterizes the energy storage in the dielectric elastomer at states of mechanical and electric (i.e., thermodynamic) equilibrium, while \(\psi ^{\mathrm{NEq}}\) characterizes the additional energy storage at non-equilibrium states (i.e., the part of the energy that gets dissipated eventually). Accordingly, the former is a function solely of the deformation gradient \({\mathbf{F}}\) and the Lagrangian electric fieldFootnote 1\({\mathbf{E}}\). On the other hand, the function \(\psi ^{\mathrm{NEq}}\) depends on \({\mathbf{F}}\), \({\mathbf{E}}\), and additionally on the internal variables \({\mathbf{F}}^v\) and \({\mathbf{E}}^v\) via the combinations

$$\begin{aligned} {\mathbf{F}}^e:={\mathbf{F}}{{\mathbf{F}}^{v}}^{-1}\quad \mathrm{and}\quad {\mathbf{E}}^e:={\mathbf{E}}-{\mathbf{E}}^v. \end{aligned}$$

In relation (2), the variables \(\dot{{\mathbf{F}}}^v\) and \(\dot{{\mathbf{E}}}^v\) stand for the time derivatives of the internal variables, that is, \(\dot{{\mathbf{F}}}^v=\partial {\mathbf{F}}^v({\mathbf{X}},t)/\partial t\) and \(\dot{{\mathbf{E}}}^v=\partial {\mathbf{E}}^v({\mathbf{X}},t)/\partial t\).

Remark 1

The specific choice of internal variables \({\mathbf{F}}^v\) and \({\mathbf{E}}^v\) in the formulation (1)–(2) is one among several possible constitutive choices. It is a choice, nonetheless, that has the advantages of being amply general and yet mathematically simple and of having a clear physical meaning. Indeed, \({\mathbf{F}}^v\) and \({\mathbf{E}}^v\) stand for measures of the “dissipative parts” of the deformation gradient \({\mathbf{F}}\) and the electric field \({\mathbf{E}}\). They are consistent with standard and well-tested choices in the separate literatures of mechanics and dielectrics; see, e.g., [24, 33, 39].

Remark 2

In the spirit of the classical linear theories of viscoelasticity  [14] and time-dependent dielectrics [4], it proves instructive to visualize the physical meaning of the two potentials (1) and (2) pictorially in the form of an electrorheological model. Figure 1 provides such a representation, which can be readily identified as the classical Zener model for viscoelastic solids [50] and the Debye model for dielectrics [8] generalized to account for their electromechanical coupling as well as for the constitutive and geometric nonlinearities inherent at finite deformations and finite electric fields.

Fig. 1
figure 1

Electrorheological model of dielectric elastomers. The curly coils and parallel plates represent elastic springs and capacitors, while the dashpot and sharp coil represent a viscous dashpot and a resistor. The elastic springs and capacitors symbolize the ability of dielectric elastomers to store elastic and electric energy, while the viscous dashpot and resistor symbolize their ability to dissipate energy through viscous deformation and molecular friction during their polarization

Full size image

Granted the two potentials (1) and (2), it follows that the “total” first Piola–Kirchhoff stress tensor \({\mathbf{S}}\) (accounting for both, the mechanical and electric contributions [9]) and the Lagrangian electric displacement \({\mathbf{D}}\) at any material point \({\mathbf{X}}\in {\Omega }_0\) and time \(t\in [0,T]\) are given by the relations

$$\begin{aligned} {\mathbf{S}}({\mathbf{X}},t)=\dfrac{\partial \psi ^{\mathrm{Eq}}}{\partial {\mathbf{F}}}+\dfrac{\partial \psi ^{\mathrm{NEq}}}{\partial {\mathbf{F}}} \end{aligned}$$
(3)

and

$$\begin{aligned} {\mathbf{D}}({\mathbf{X}},t)=-\dfrac{\partial \psi ^{\mathrm{Eq}}}{\partial {\mathbf{E}}}-\dfrac{\partial \psi ^{\mathrm{NEq}}}{\partial {\mathbf{E}}}, \end{aligned}$$
(4)

where \({\mathbf{F}}^v({\mathbf{X}},t)\) and \({\mathbf{E}}^v({\mathbf{X}},t)\) are defined implicitly as solutions of the coupled system of ordinary differential equations (ODEs)

$$\begin{aligned} \left\{ \begin{array}{l}\dfrac{\partial \psi ^{\mathrm{NEq}}}{\partial {\mathbf{F}}^v}+\dfrac{\partial \phi }{\partial \dot{{\mathbf{F}}}^v}=0\\ \dfrac{\partial \psi ^{\mathrm{NEq}}}{\partial {\mathbf{E}}^v}+\dfrac{\partial \phi }{\partial \dot{{\mathbf{E}}}^v}=0 \end{array}\right. \end{aligned}$$
(5)

in time subject to the initial conditions \({\mathbf{F}}^v({\mathbf{X}},0)={\mathbf{I}}\) and \({\mathbf{E}}^v({\mathbf{X}},0)=\mathbf 0\).

Remark 3

The total Cauchy stress tensor \({\mathbf{T}}\), Eulerian electric field \({\mathbf{e }}\), and Eulerian electric displacement \({\mathbf{d }}\) at the position \({\mathbf{x}}\in {\Omega }(t)\) occupied by the material point \({\mathbf{X}}\) at time \(t\in [0,T]\) are given in terms of their Lagrangian counterparts by the relations

$$\begin{aligned}&{\mathbf{T}}({\mathbf{x}},t)=J^{-1}{\mathbf{S}}{\mathbf{F}}^T, \\&{\mathbf{e }}({\mathbf{x}},t)={\mathbf{F}}^{-T}{\mathbf{E}}, \\&{\mathbf{d }}({\mathbf{x}},t)=J^{-1}{\mathbf{F}}{\mathbf{D}}. \end{aligned}$$
(6)

Moreover, the polarization \({\mathbf{p }}\) at any \({\mathbf{x}}\in {\Omega }(t)\) and \(t\in [0,T]\) is given by

$$\begin{aligned} {\mathbf{p }}({\mathbf{x}},t)={\mathbf{d }}({\mathbf{x}},t)-\varepsilon _0{\mathbf{e }}({\mathbf{x}},t), \end{aligned}$$

where \(\varepsilon _0\) stands for the permittivity of vacuum.

2.2.1 Constraints on the functions \(\psi ^{\mathrm{Eq}}\), \(\psi ^{\mathrm{NEq}}\), \(\phi \)

As is the case with the two-potential modeling of any dissipative solid, any specific choice of functions \(\psi ^{\mathrm{Eq}}\), \(\psi ^{\mathrm{NEq}}\), \(\phi \) in the formulation (1)–(2) must satisfy certain basic physical requirements. We spell out each of these requirements in the sequel, one at a time.

Even electromechanical coupling. As alluded to above, the constitutive relations (3)–(4) for dielectric elastomers must exhibit even electromechanical coupling. This requirement implies that the functions \(\psi ^{\mathrm{Eq}}\), \(\psi ^{\mathrm{NEq}}\), \(\phi \) must satisfy the conditions

$$\begin{aligned}&\psi ^{\mathrm{Eq}}({\mathbf{F}},-{\mathbf{E}})=\psi ^{\mathrm{Eq}}({\mathbf{F}},{\mathbf{E}}), \\&\psi ^{\mathrm{NEq}}({\mathbf{F}},-{\mathbf{E}},{\mathbf{F}}^e,-{\mathbf{E}}^e)=\psi ^{\mathrm{NEq}}({\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^e,{\mathbf{E}}^e), \\&\phi ({\mathbf{F}},-{\mathbf{E}},{\mathbf{F}}^v,-{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,-\dot{{\mathbf{E}}}^v)=\phi ({\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^v,{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,\dot{{\mathbf{E}}}^v), \end{aligned}$$
(7)

for arbitrary \({\mathbf{F}}\), \({\mathbf{E}}\), \({\mathbf{F}}^v\), and \({\mathbf{E}}^v\).

Material frame indifference. Under a change of observer, it is required that the free-energy function \(\psi \) and dissipation potential \(\phi \) remain invariant. A standard calculation shows (see, e.g., Section 2 in [24]) that this requirement implies that the functions \(\psi ^{\mathrm{Eq}}\), \(\psi ^{\mathrm{NEq}}\), \(\phi \) must satisfy the conditions

$$\begin{aligned}&\psi ^{\mathrm{Eq}}({\mathbf{Q}}{\mathbf{F}},{\mathbf{E}})=\psi ^{\mathrm{Eq}}({\mathbf{F}},{\mathbf{E}}), \\&\psi ^{\mathrm{NEq}}({\mathbf{Q}}{\mathbf{F}},{\mathbf{E}},{\mathbf{Q}}{\mathbf{F}}^e,{\mathbf{E}}^e)=\psi ^{\mathrm{NEq}}({\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^e,{\mathbf{E}}^e), \\&\phi ({\mathbf{Q}}{\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^v,{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,\dot{{\mathbf{E}}}^v)=\phi ({\mathbf{Q}}{\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^v,{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,\dot{{\mathbf{E}}}^v) \end{aligned}$$
(8)

for all \({\mathbf{Q}}\in Orth^+\) and arbitrary \({\mathbf{F}}\), \({\mathbf{E}}\), \({\mathbf{F}}^v\), and \({\mathbf{E}}^v\).

Material symmetry. For dielectric elastomers with material symmetry group \(Symm\subseteq Orth^+\), it is required that the free-energy function \(\psi \) and dissipation potential \(\phi \) remain invariant under a change of reference configuration described by any element \({\mathbf{K}}\in Symm\). Again, a standard calculation shows (see, e.g., Section 2 in [24]) that this requirement implies that the functions \(\psi ^{\mathrm{Eq}}\), \(\psi ^{\mathrm{NEq}}\), \(\phi \) must satisfy the conditions

$$\begin{aligned}&\psi ^{\mathrm{Eq}}({\mathbf{F}}{\mathbf{K}},{\mathbf{K}}^T{\mathbf{E}})=\psi ^{\mathrm{Eq}}({\mathbf{F}},{\mathbf{E}}), \\&\psi ^{\mathrm{NEq}}({\mathbf{F}}{\mathbf{K}},{\mathbf{K}}^T{\mathbf{E}},{\mathbf{F}}^e,{\mathbf{K}}^T{\mathbf{E}}^e)=\psi ^{\mathrm{NEq}}({\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^e,{\mathbf{E}}^e), \\&\phi ({\mathbf{F}}{\mathbf{K}},{\mathbf{K}}^T{\mathbf{E}},{\mathbf{F}}^v{\mathbf{K}},{\mathbf{K}}^T{\mathbf{E}}^v,\dot{{\mathbf{F}}}^{v} {\mathbf{K}},{\mathbf{K}}^T\dot{{\mathbf{E}}}^v)\\&= \phi ({\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^v,{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,\dot{{\mathbf{E}}}^v) \end{aligned}$$
(9)

for all \({\mathbf{K}}\in Symm\) and arbitrary \({\mathbf{F}}\), \({\mathbf{E}}\), \({\mathbf{F}}^v\), and \({\mathbf{E}}^v\); in the condition for \(\psi ^{\mathrm{NEq}}\), note that use has been made of the result \({\mathbf{F}}{\mathbf{K}}({\mathbf{F}}^v{\mathbf{K}})^{-1}={\mathbf{F}}{\mathbf{K}}{\mathbf{K}}^T{{\mathbf{F}}^v}^{-1}={\mathbf{F}}{{\mathbf{F}}^v}^{-1}={\mathbf{F}}^e\).

Entropy imbalance. In the context of isothermal processes of interest here, the entropy imbalance or second law of thermodynamics in the form of the reduced dissipation inequality (see the Appendix) imposes the following constraint on the dissipation potential \(\phi \):

$$\begin{aligned}&\left[ \dfrac{\partial \phi }{\partial \dot{{\mathbf{F}}}^v}({\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^v,{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,\dot{{\mathbf{E}}}^v)\right] \cdot \dot{{\mathbf{F}}}^v+ \\&\left[ \dfrac{\partial \phi }{\partial \dot{{\mathbf{E}}}^v}({\mathbf{F}},{\mathbf{E}},{\mathbf{F}}^v,{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,\dot{{\mathbf{E}}}^v)\right] \cdot \dot{{\mathbf{E}}}^v\ge 0 \end{aligned}$$
(10)

for arbitrary \({\mathbf{F}}\), \({\mathbf{E}}\), \({\mathbf{F}}^v\), \({\mathbf{E}}^v\), with equality holding only when \(\dot{{\mathbf{F}}}^v=\mathbf 0\) and \(\dot{{\mathbf{E}}}^v=\mathbf 0\).

2.3 Boundary conditions and source terms

We now specify the external stimuli applied to the solid, which comprise both prescribed mechanical and electric boundary data and source terms in the bulk.

Electrically, consistent with the manner in which electric fields are applied in practice, we take that the solid is immersed in a surrounding space (e.g., air) where there is a heterogeneous electric field \(\overline{{\mathbf{E}}}({\mathbf{X}},t)\) and corresponding electric displacement \(\overline{{\mathbf{D}}}({\mathbf{X}},t)\) that result from the electrodes connected to part of the boundary of the solid together with the nearby presence of space charges and/or other polarizable bodies and the interaction of these with the solid. We can then compactly write the boundary condition as

$$\begin{aligned} {\mathbf{E}}\times {\mathbf{N}}=\overline{{\mathbf{E}}}\times {\mathbf{N}},\quad ({\mathbf{X}},t)\in \partial {\Omega }_0\times [0,T] \end{aligned}$$

or, equivalently,

$$\begin{aligned} {\mathbf{D}}\cdot {\mathbf{N}}=\overline{{\mathbf{D}}}\cdot {\mathbf{N}}-\sigma ({\mathbf{X}},t),\quad ({\mathbf{X}},t)\in \partial {\Omega }_0\times [0,T] \end{aligned}$$

over the entirety of the boundary of the domain occupied by the solid, where \(\sigma ({\mathbf{X}},t)\) stands for the density of surface charges present on \(\partial {\Omega }_0\) (for instance, at electrodes). Throughout \({\Omega }_0\), we also consider that the solid contains a distribution of space charges with density

$$\begin{aligned} Q({\mathbf{X}},t),\quad ({\mathbf{X}},t)\in {\Omega }_0\times [0,T]. \end{aligned}$$

Mechanically, on a portion \(\partial {\Omega }^{{{\mathcal {D}}}}_0\) of the boundary \(\partial {\Omega }_0\), the deformation field \({\mathbf{y}}\) is taken to be given by a known function \(\overline{{\mathbf{y}}}({\mathbf{X}},t)\), while the complementary part of the boundary \(\partial {\Omega }^{{{\mathcal {N}}}}_0=\partial {\Omega }_0\setminus \partial {\Omega }^{{{\mathcal {D}}}}_0\) is subjected to a prescribed mechanical traction \(\overline{{\mathbf{t }}}({\mathbf{X}},t)\). Precisely,

$$\begin{aligned} \left\{ \begin{array}{l} {\mathbf{y}}=\overline{{\mathbf{y}}},\quad ({\mathbf{X}},t)\in \partial {\Omega }^{{{\mathcal {D}}}}_0\times [0,T] \\ {\mathbf{S}}{\mathbf{N}}=\overline{{\mathbf{t }}}+{\mathbf{S}}_{M}{\mathbf{N}},\quad ({\mathbf{X}},t)\in \partial {\Omega }^{{{\mathcal {N}}}}_0\times [0,T]. \end{array}\right. \end{aligned}$$

In this last expression, \({\mathbf{S}}_{M}\) stands for the Maxwell stress outside of the solid. In the case when the solid is surrounded by air,

$$\begin{aligned} {\mathbf{S}}_{M}=\overline{{\mathbf{F}}}^{-T}\overline{{\mathbf{E}}}\otimes \overline{{\mathbf{D}}}-\dfrac{\overline{J}\varepsilon _0}{2}\left( {\mathbf{F}}^{-T}\overline{{\mathbf{E}}}\cdot \overline{{\mathbf{F}}}^{-T}\overline{{\mathbf{E}}}\right) \overline{{\mathbf{F}}}^{-T}, \end{aligned}$$

where \(\overline{{\mathbf{D}}}=\varepsilon _0 \overline{J}\,\overline{{\mathbf{F}}}^{-1}\overline{{\mathbf{F}}}^{-T}\overline{{\mathbf{E}}}\) and where we emphasize that the meaning of the deformation gradient \(\overline{{\mathbf{F}}}\)—as well as the notion of material points \({\mathbf{X}}\)—in the air needs to be interpreted appropriately; see, e.g., Section 6 in  [29]. Finally, throughout \({\Omega }_0\), we also consider that the solid is subjected to a mechanical body force with density

$$\begin{aligned} {\mathbf{f }}({\mathbf{X}},t),\quad ({\mathbf{X}},t)\in {\Omega }_0\times [0,T]. \end{aligned}$$

2.4 Governing equations

Absent inertia and in the context of electro-quasi-statics, the relevant equations of balance of linear and angular momenta read as

$$\begin{aligned} \mathrm{Div}\,{\mathbf{S}}+{\mathbf{f }}={\mathbf{0 }}\quad \mathrm{and }\quad {\mathbf{S}}{\mathbf{F}}^T={\mathbf{F}}{\mathbf{S}}^T,\quad ({\mathbf{X}},t)\in {\Omega }_0\times [0,T] \end{aligned}$$
(11)

while the relevant equations of Maxwell read as

$$\begin{aligned} \mathrm{Div}\,{\mathbf{D}}=Q\quad \mathrm{and }\quad \mathrm{Curl}\,{\mathbf{E}}={\mathbf{0 }},\quad ({\mathbf{X}},t)\in {{\mathbb {R}}}^3\times [0,T]. \end{aligned}$$
(12)

The balance of angular momentum (11)\(_2\) is automatically satisfied by virtue of the material frame indifference granted by condition (8)\(_{1,2}\). Faraday’s law (12)\(_2\) can also be automatically satisfied by introducing a scalar potential \(\varphi \) such that \({\mathbf{E}}=-\nabla \varphi ({\mathbf{X}},t)\). It then follows that the governing equations for the solid reduce to the following coupled system of boundary-value problems:

$$\begin{aligned}&\left\{ \begin{array}{ll} \mathrm{Div}\left[ \dfrac{\partial \psi ^{\mathrm{{Eq}}}}{\partial {\mathbf{F}}}\left( \nabla {\mathbf{y}},-\nabla \varphi \right) +\dfrac{\partial \psi ^{\mathrm{{NEq}}}}{\partial {\mathbf{F}}}\left( \nabla {\mathbf{y}},-\nabla \varphi ,\nabla {\mathbf{y}}{{\mathbf{F}}^v}^{-1},-\nabla \varphi -{\mathbf{E}}^v\right) \right] +{\mathbf{f }}({\mathbf{X}},t)=\mathbf{0}, \quad ({\mathbf{X}},t)\in {\Omega }_0\times [0,T]\\ \det \nabla {\mathbf{y}}({\mathbf{X}},t)>0,\quad ({\mathbf{X}},t)\in {\Omega }_0\times [0,T]\\ {\mathbf{y}}({\mathbf{X}},t)=\overline{{\mathbf{y}}}({\mathbf{X}},t),\quad ({\mathbf{X}},t)\in \partial {\Omega }_0^{{{\mathcal {D}}}}\times [0,T]\\ \left[ \dfrac{\partial \psi ^{\mathrm{{Eq}}}}{\partial {\mathbf{F}}}\left( \nabla {\mathbf{y}},-\nabla \varphi \right) +\dfrac{\partial \psi ^{\mathrm{{NEq}}}}{\partial {\mathbf{F}}}\left( \nabla {\mathbf{y}},-\nabla \varphi ,\nabla {\mathbf{y}}{{\mathbf{F}}^v}^{-1},-\nabla \varphi -{\mathbf{E}}^v\right) -{\mathbf{S}}_M\right] {\mathbf{N}}=\overline{{\mathbf{t }}}({\mathbf{X}},t),\; ({\mathbf{X}},t)\in \partial {\Omega }_0^{{{\mathcal {N}}}}\times [0,T]\end{array}\right. \end{aligned}$$
(13)
$$\begin{aligned}&\mathrm{and} \\&\left\{ \begin{array}{ll} \mathrm{Div}\left[ \dfrac{\partial \psi ^{\mathrm{{Eq}}}}{\partial {\mathbf{E}}}\left( \nabla {\mathbf{y}},-\nabla \varphi \right) +\dfrac{\partial \psi ^{\mathrm{{NEq}}}}{\partial {\mathbf{E}}}\left( \nabla {\mathbf{y}},-\nabla \varphi ,\nabla {\mathbf{y}}{{\mathbf{F}}^v}^{-1},-\nabla \varphi -{\mathbf{E}}^v\right) \right] =-Q({\mathbf{X}},t), \quad ({\mathbf{X}},t)\in {\Omega }_0\times [0,T]\\ \left[ \dfrac{\partial \psi ^{\mathrm{{Eq}}}}{\partial {\mathbf{E}}}\left( \nabla {\mathbf{y}},-\nabla \varphi \right) +\dfrac{\partial \psi ^{\mathrm{{NEq}}}}{\partial {\mathbf{E}}}\left( \nabla {\mathbf{y}},-\nabla \varphi ,\nabla {\mathbf{y}}{{\mathbf{F}}^v}^{-1},-\nabla \varphi -{\mathbf{E}}^v\right) \right] \cdot {\mathbf{N}}=\sigma ({\mathbf{X}},t)-\overline{{\mathbf{D}}}({\mathbf{X}},t)\cdot {\mathbf{N}}, \;({\mathbf{X}},t)\in \partial {\Omega }_0\times [0,T]\end{array}\right. \end{aligned}$$
(14)

together with the coupled system of evolution equations

$$\begin{aligned}&\left\{ \begin{array}{ll} \dfrac{\partial \psi ^{\mathrm{NEq}}}{\partial {\mathbf{F}}^v}\left( \nabla {\mathbf{y}},-\nabla \varphi ,\nabla {\mathbf{y}}{{\mathbf{F}}^v}^{-1},-\nabla \varphi -{\mathbf{E}}^v\right) +\dfrac{\partial \phi }{\partial \dot{{\mathbf{F}}}^v}\left( \nabla {\mathbf{y}},-\nabla \varphi ,{\mathbf{F}}^v,{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,\dot{{\mathbf{E}}}^v\right) =0,\quad ({\mathbf{X}},t)\in {\Omega }_0\times (0,T]\\ {\mathbf{F}}^v({\mathbf{X}},0)={{\mathbf{I}}}, \quad {\mathbf{X}}\in {\Omega }_0\end{array}\right. \end{aligned}$$
(15)
$$\begin{aligned}&\mathrm{and} \\&\left\{ \begin{array}{ll} \dfrac{\partial \psi ^{\mathrm{NEq}}}{\partial {\mathbf{E}}^v}\left( \nabla {\mathbf{y}},-\nabla \varphi ,\nabla {\mathbf{y}}{{\mathbf{F}}^v}^{-1},-\nabla \varphi -{\mathbf{E}}^v\right) +\dfrac{\partial \phi }{\partial \dot{{\mathbf{E}}}^v}\left( \nabla {\mathbf{y}},-\nabla \varphi ,{\mathbf{F}}^v,{\mathbf{E}}^v,\dot{{\mathbf{F}}}^v,\dot{{\mathbf{E}}}^v\right) =0,\quad ({\mathbf{X}},t)\in {\Omega }_0\times (0,T]\\ {\mathbf{E}}^v({\mathbf{X}},0)=\mathbf{0}, \quad {\mathbf{X}}\in {\Omega }_0\end{array}\right. \end{aligned}$$
(16)

for the deformation field \({\mathbf{y}}({\mathbf{X}},t)\), the electric potential \(\varphi ({\mathbf{X}},t)\), and the internal variables \({\mathbf{F}}^v({\mathbf{X}},t)\) and \({\mathbf{E}}^v({\mathbf{X}},t)\).

Remark 4

In general, the boundary data \(\overline{{\mathbf{D}}}({\mathbf{X}},t)\) in (14)\(_2\) is not known a priori. This is because it is implicitly defined by the solution of Maxwell equations (12) in \({{\mathbb {R}}}^3\setminus {\Omega }_0\). Nevertheless, there is a plurality of specific domain geometries \({\Omega }_0\) and specific boundary conditions of practical interest for which the boundary condition (14)\(_2\) can be written in explicit form, often in terms of an electric potential or voltage applied across electrodes.

Remark 5

The two-potential framework introduced above is admittedly general, as it applies to dielectric elastomers of arbitrary anisotropy and compressibility featuring a wide range of dissipative mechanical and electric behaviors, and yet relatively simple, as the resulting governing equations (13)–(14) with (15)–(16) for any boundary-value problem of interest are amenable to numerical solution by well-established finite-element and time-integration schemes; see, e.g.,  [25, 28, 30, 40].

Moreover, the handful of models for dielectric elastomers (accounting for dissipative phenomena) that have been proposed in the literature over the past few years following different ad hoc approaches are all special cases of the above two-potential framework, that is, they can be readily generated by appropriate choices of the free-energy functions \(\psi ^{\mathrm{Eq}}\), \(\psi ^{\mathrm{NEq}}\) and the dissipation potential \(\phi \). As noted in the Introduction, however, the majority of such models account only for mechanical dissipation and, save for a few works  [5, 46], most further assume ideal dielectric behavior [2, 18, 47, 51]. Two exemptions that account for both mechanical and electric dissipation are the models presented in [10] and [38].

3 A specific constitutive model for isotropic incompressible dielectric elastomers

In the sequel, we employ the general two-potential constitutive framework laid out in the preceding section to construct a specific model for the prominent class of isotropic and incompressible dielectric elastomers.

The proposed model corresponds to a generalization of the viscoelastic model of Kumar and Lopez-Pamies [24]—which accounts for the non-Gaussian elasticity and the deformation-enhanced shear-thinning viscosity typical of elastomers  [1, 12, 21, 31]—aimed at accounting for the time- and deformation-dependent polarization and electrostriction characteristic of dielectric elastomers  [7, 17, 22, 48].

We begin in Sect. 3.1 by presenting the prescriptions for the functions \(\psi ^{\mathrm{Eq}}\), \(\psi ^{\mathrm{NEq}}\), \(\phi \) and then present the constitutive response that they imply in Sect. 3.2 together with the key theoretical and practical features of the model. We devote Sect. 3.3 to discussing in detail its specialization in the so-called limit of small deformations and moderate electric fields.

3.1 The functions \(\psi ^{\mathrm{Eq}}\), \(\psi ^{\mathrm{NEq}}\), \(\phi \)

The equilibrium free-energy function \(\psi ^{\mathrm{Eq}}\) in the proposed model is given by

$$\begin{aligned} \psi ^{\mathrm{Eq}}=\left\{ \begin{array}{ll} \psi ^{\mathrm{Eq}}_{{{\texttt {LP}}}}(I_1)+\dfrac{m_{{{\texttt {K}}}}-\varepsilon }{2}I_4-\dfrac{m_{{{\texttt {K}}}}}{2}I_5\quad &{} \mathrm{if} \quad J= 1 \\ +\infty &{} \mathrm{otherwise} \end{array} \right. \end{aligned}$$
(17)

with

$$\begin{aligned} \psi ^{\mathrm{Eq}}_{{{\texttt {LP}}}}(I_1)=\sum _{r=1}^2\dfrac{3^{1-\alpha _{r}}}{2\alpha _{r}}\mu _{r} \left[ I_{1}^{\alpha _{r}} -3^{\alpha _{r}}\right] , \end{aligned}$$

where \(I_1\), \(I_4\), \(I_5\) stand for the standard invariants

$$\begin{aligned} I_1=\mathrm{tr}\,{\mathbf{C}},\quad I_4={\mathbf{E}}\cdot {\mathbf{E}},\quad I_5={\mathbf{E}}\cdot {\mathbf{C}}^{-1}{\mathbf{E}}, \end{aligned}$$

written here in terms of the right Cauchy-Green deformation tensor \({\mathbf{C}}={\mathbf{F}}^T{\mathbf{F}}\), and where \(\mu _r\), \(\alpha _r\) (\(r=1,2\)) stand for the material parameters (with unit \(force/length^2\) and unitless) describing the elasticity of the dielectric elastomer at states of mechanical and electric equilibrium, while the material parameters \(\varepsilon \) and \(m_{{{\texttt {K}}}}\) denote, respectively, its initial permittivity and electrostriction coefficient (both with units capacitance/length), also at equilibrium states.

Remark 6

It is a simple matter to check that the equilibrium free-energy function (17) satisfies the requirements of even electromechanical coupling (7)\(_1\), material frame indifference (8)\(_1\), and material symmetry (9)\(_1\) for the case of isotropic dielectric elastomers of interest here.

The non-equilibrium free-energy function \(\psi ^{\mathrm{NEq}}\) is taken to be given by the same functional form as the equilibrium free energy (17) evaluated at appropriate deformation-gradient and electric-field arguments compliant with the constraints (7)\(_2\), (8)\(_2\), (9)\(_2\). Specifically,

$$\begin{aligned} \psi ^{\mathrm{NEq}}=\left\{ \begin{array}{ll} \psi ^{\mathrm{NEq}}_{{{\texttt {LP}}}}(I^e_1)+\dfrac{n_{{{\texttt {K}}}}-\epsilon }{2}I^e_4-\dfrac{n_{{{\texttt {K}}}}}{2}{{\mathcal {I}}}^e_5\quad &{} \mathrm{if} \quad J^e= 1 \\ +\infty &{} \mathrm{otherwise} \end{array} \right. \end{aligned}$$
(18)

with

$$\begin{aligned} \psi ^{\mathrm{NEq}}_{{{\texttt {LP}}}}(I^e_1)=\sum _{r=1}^2\dfrac{3^{1-\beta _r}}{2\beta _r}\nu _{r} \left[ {I^{e}_{1}}^{\beta _r} -3^{\beta _r}\right] , \end{aligned}$$

where \(I^e_1\), \(J^e\), \(I^e_4\), \({{\mathcal {I}}}^e_5\) stand for the pseudo-invariants

$$\begin{aligned} \left\{ \begin{array}{ll} I^e_1&{}=\mathrm{tr}\,{\mathbf{C}}^e={\mathbf{C}}\cdot {{\mathbf{C}}^v}^{-1} \\ J^e&{}=\det {\mathbf{F}}^e= \dfrac{J}{J^v}\\ I^e_4&{}={\mathbf{E}}^e\cdot {\mathbf{E}}^e=({\mathbf{E}}-{\mathbf{E}}^v)\cdot ({\mathbf{E}}-{\mathbf{E}}^v)\\ {{\mathcal {I}}}^e_5&{}={\mathbf{E}}^e\cdot {\mathbf{C}}^{-1}{\mathbf{E}}^e=({\mathbf{E}}-{\mathbf{E}}^v)\cdot {\mathbf{C}}^{-1}({\mathbf{E}}-{\mathbf{E}}^v) \end{array}, \right. \end{aligned}$$
(19)

and where \(\nu _1\), \(\beta _1\), \(\nu _2\), \(\beta _2\), \(\epsilon \), \(n_{{{\texttt {K}}}}\) stand for the six material parameters analogous to \(\mu _1\), \(\alpha _1\), \(\mu _2\), \(\alpha _2\), \(\varepsilon \), \(m_{{{\texttt {K}}}}\) in the equilibrium branch (17); for later use, in these last expressions we have introduced the notation \({\mathbf{C}}^e={{\mathbf{F}}^e}^T{\mathbf{F}}^e\), \({\mathbf{C}}^v={{\mathbf{F}}^{v}}^{T}{\mathbf{F}}^{v}\), \(J^v=\det {\mathbf{F}}^v\).

Remark 7

Similar to (17), it is a simple matter to check that the non-equilibrium free-energy function (18) satisfies the requirements of even electromechanical coupling (7)\(_2\), material frame indifference (8)\(_2\), and material symmetry (9)\(_2\) for \(Symm=Orth^+\).

Remark 8

While in principle natural choices as arguments in \(\psi ^{\mathrm{NEq}}\), the pseudo-invariants \(I^e_5={\mathbf{E}}^e\cdot {{\mathbf{C}}^e}^{-1}{\mathbf{E}}^e\) and \(I^e_6={\mathbf{E}}^e\cdot {{\mathbf{C}}^e}^{-2}{\mathbf{E}}^e\) do not satisfy the material symmetry requirement (9)\(_2\). On the other hand, the pseudo-invariants \({{\mathcal {I}}}^e_5={\mathbf{E}}^e\cdot {{\mathbf{C}}}^{-1}{\mathbf{E}}^e\) and \({{\mathcal {I}}}^e_6={\mathbf{E}}^e\cdot {{\mathbf{C}}}^{-2}{\mathbf{E}}^e\) do satisfy such a constraint, thus the use of \({{\mathcal {I}}}^e_5\) in (18).

Finally, the dissipation potential \(\phi \) is prescribed as

$$\begin{aligned} \phi&=\dfrac{1}{2}\dot{{\mathbf{F}}}^v{{\mathbf{F}}^v}^{-1}\cdot \left[ \varvec{{\mathcal {A}}}\left( {\mathbf{F}},{\mathbf{F}}^e\right) \dot{{\mathbf{F}}}^v{{\mathbf{F}}^v}^{-1}\right] \\&\quad +\dfrac{1}{2}{\mathbf{F}}^{-T}\dot{{\mathbf{E}}}^v\cdot \left[ \varvec{{\mathcal {B}}}\left( {\mathbf{F}},{\mathbf{E}},{\mathbf{E}}^e\right) {\mathbf{F}}^{-T}\dot{{\mathbf{E}}}^v\right] \end{aligned}$$
(20)

with

$$\begin{aligned} {{\mathcal {A}}}_{ijkl}({\mathbf{F}}, {\mathbf{F}}^{e})=\,&2\eta _{{{\texttt {K}}}}(I_1^e,I_2^e,I_1^v){{\mathcal {K}}}_{ijkl}+3\eta _{{{\texttt {J}}}}{{\mathcal {J}}}_{ijkl} \end{aligned}$$
(21)

and

$$\begin{aligned} {{\mathcal {B}}}_{ij}\left( {\mathbf{F}},{\mathbf{E}},{\mathbf{E}}^e\right) = \zeta \delta _{ij}, \end{aligned}$$
(22)

where \(\delta _{ij}\) denotes the Kronecker delta, and \(\varvec{{\mathcal {K}}}\) and \(\varvec{{\mathcal {J}}}\) stand for the standard shear and hydrostatic orthogonal projection tensors

$$\begin{aligned} {{\mathcal {K}}}_{ijkl}=\frac{1}{2}\left[ \delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}-\frac{2}{3}\delta _{ij}\delta _{kl}\right] , \;{{\mathcal {J}}}_{ijkl}=\frac{1}{3}\delta _{ij}\delta _{kl}. \end{aligned}$$
(23)

In the constitutive prescriptions (21)–(22), \(\eta _{{{\texttt {K}}}}\), \(\eta _{{{\texttt {J}}}}\), and \(\zeta \) are material functions/parameters that describe, respectively, the viscosity and the polarization friction of the dielectric elastomer. The two viscosity coefficients are given by

$$\begin{aligned} \eta _{{{\texttt {K}}}}(I_1^e,I_2^e,I_1^v)=\eta _{\infty }+\frac{\eta _{0}-\eta _{\infty }+K_{1}\left[ {I_{1}^{v}}^{\gamma _{1}}-3^{\gamma _{1}}\right] }{1+\left( K_{2}\, J_2^{\mathrm{NEq}}\right) ^{\gamma _{2}}} \end{aligned}$$

with

$$\begin{aligned} J_2^{\mathrm{NEq}}=\left( \dfrac{I_1^{e\,2}}{3}-I_2^{e}\right) \left( \sum _{r=1}^2 3^{1-\beta _{r}}\nu _{r} I_{1}^{e\,\beta _{r}-1} \right) ^{2} \end{aligned}$$

and

$$\begin{aligned} \eta _{{{\texttt {J}}}}=+\infty , \end{aligned}$$

where we recall that \(I_1^e\) is given by (19)\(_1\) in terms of \({\mathbf{C}}\) and \({\mathbf{C}}^v\),

$$\begin{aligned} \left\{ \begin{array}{ll} I^e_2&{}=\dfrac{1}{2}\left[ (\mathrm{tr}\,{\mathbf{C}}^e)^2-\mathrm{tr}\,{{\mathbf{C}}^e}^2\right] \\ &{}=\dfrac{1}{2}\left[ ({\mathbf{C}}\cdot {{\mathbf{C}}^v}^{-1})^2-{{\mathbf{C}}^v}^{-1}{\mathbf{C}}\cdot {\mathbf{C}}{{\mathbf{C}}^v}^{-1}\right] \\ I_1^v&{}=\mathrm{tr}\,{\mathbf{C}}^v \end{array} \right. , \end{aligned}$$

and \(\eta _0>\eta _\infty \ge 0\), \(\gamma _1\ge 0\), \(\gamma _2\ge 0\), \(K_1\ge 0\), \(K_2\ge 0\) are material parameters; the units of \(\eta _{{{\texttt {J}}}}\), \(\eta _0\), \(\eta _\infty \), \(K_1\) are \(force\times time/length^2\), \(\gamma _1\), \(\gamma _2\) are unitless, while \(K_2\) has units of \(length^4/force^2\). The polarization-friction parameter \(\zeta \) is taken to be a non-negative material constant with unit \(capacitance\times time/length\).

Remark 9

It is not difficult to verify that the dissipation potential (20) satisfies the requirements of even electromechanical coupling (7)\(_3\), material frame indifference (8)\(_3\), and material symmetry (9)\(_3\) for \(Symm=Orth^+\). Noting the inequalities \(\eta _{{{\texttt {K}}}}\ge 0\), \(\eta _{{{\texttt {J}}}}> 0\), \(\zeta \ge 0\), it is also straightforward to check that the dissipation potential (20) complies as well with the reduced dissipation inequality (10).

Remark 10

The polarization-friction coefficient \(\zeta \) in the dissipation potential (20) can be vastly generalized to be a function of the deformation gradient \({\mathbf{F}}\), the electric field \({\mathbf{E}}\), and/or the internal variable \({\mathbf{E}}^v\), instead of just a constant. For instance, any (suitably well-behaved) non-negative function of the generic form

$$\begin{aligned} \zeta =\zeta (I_1,I_4,I_5,I_4^e,{{\mathcal {I}}}_5^e) \end{aligned}$$

may be employed. Because of the current lack of guiding experimental results, we choose \(\zeta \) to be a constant in this work.

3.2 The constitutive response

Having introduced the free-energy functions (17) and (18) and the dissipation potential (20), it is now a simple matter to spell out the constitutive relations (3)–(4) with evolution equations (5) that they imply. Indeed, the first Piola-Kirchhoff stress \({\mathbf{S}}\) and Lagrangian electric displacement \({\mathbf{D}}\) are given simply by

$$\begin{aligned} {\mathbf{S}}&= \left[ \sum \limits _{r=1}^2 3^{1-\alpha _{r}}\mu _{r} I_{1}^{\alpha _{r}-1} \right] {\mathbf{F}}-p\, {\mathbf{F}}^{-T} \\&\quad + \left[ \sum \limits _{r=1}^2 3^{1-\beta _{r}} \nu _{r} ({\mathbf{C}}\cdot {{\mathbf{C}}^{v}}^{-1} )^{\beta _{r}-1} \right] {\mathbf{F}}{{\mathbf{C}}^{v}}^{-1} \\&\quad + m_{{{\texttt {K}}}}{\mathbf{F}}^{-T}{\mathbf{E}}\otimes {\mathbf{F}}^{-1}{\mathbf{F}}^{-T}{\mathbf{E}} \\&\quad + n_{{{\texttt {K}}}}{\mathbf{F}}^{-T}({\mathbf{E}}-{\mathbf{E}}^v)\otimes {\mathbf{F}}^{-1}{\mathbf{F}}^{-T}({\mathbf{E}}-{\mathbf{E}}^v) \end{aligned}$$
(24)

and

$$\begin{aligned} {\mathbf{D}}&=\left( \varepsilon -m_{{{\texttt {K}}}}\right) {\mathbf{E}}+m_{{{\texttt {K}}}}{\mathbf{F}}^{-1}{\mathbf{F}}^{-T}{\mathbf{E}} \\&\quad + \left( \epsilon -n_{{{\texttt {K}}}}\right) ({\mathbf{E}}-{\mathbf{E}}^v)+n_{{{\texttt {K}}}}{\mathbf{F}}^{-1}{\mathbf{F}}^{-T}({\mathbf{E}}-{\mathbf{E}}^v), \end{aligned}$$
(25)

where p stands for the arbitrary hydrostatic pressure associated with the incompressibility constraint \(J=1\), and where \({\mathbf{C}}^{v}\) and \({\mathbf{E}}^v\) are defined implicitly as solutions of the evolution equations

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{{\mathbf{C}}} ^{v}=\frac{\sum \nolimits _{r=1}^2 3^{1-\beta _{r}} \nu _{r} ({\mathbf{C}}\cdot {{\mathbf{C}}^{v}}^{-1} )^{\beta _{r}-1}}{\eta _{{{\texttt {K}}}}(I_1^e,I_2^e,I_1^v)} \left[ {\mathbf{C}}- \frac{1}{3} ({\mathbf{C}}\cdot {{\mathbf{C}}^{v}}^{-1}) {\mathbf{C}}^v\right] \\ {\mathbf{C}}^v({\mathbf{X}},0)={\mathbf{I}}\end{array} \right. \end{aligned}$$
(26)

and

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{{\mathbf{E}}} ^{v}=-\left( \dfrac{n_{{{\texttt {K}}}}}{\zeta }{\mathbf{I}}+\dfrac{\epsilon -n_{{{\texttt {K}}}}}{\zeta } {\mathbf{C}}\right) ({\mathbf{E}}-{\mathbf{E}}^v) \\ {\mathbf{E}}^v({\mathbf{X}},0)=\mathbf{0} \end{array} \right. ; \end{aligned}$$
(27)

note that the dependence on the internal variable \({\mathbf{F}}^v\) enters only through the combination \({\mathbf{C}}^{v}={{\mathbf{F}}^v}^T{\mathbf{F}}^v\).

The remainder of this section is dedicated to describing key features of the proposed constitutive model (24)–(27).

The Eulerian description. It follows from the constitutive relations (24)–(25) and the connections (6) that the Cauchy stress \({\mathbf{T}}\) and Eulerian electric displacement \({\mathbf{d }}\) are given by

$$\begin{aligned} {\mathbf{T}}&= \left[ \sum \limits _{r=1}^2 3^{1-\alpha _{r}}\mu _{r} I_{1}^{\alpha _{r}-1} \right] {\mathbf{F}}{\mathbf{F}}^T -p\, {\mathbf{I}}\\&\quad + \left[ \sum \limits _{r=1}^2 3^{1-\beta _{r}} \nu _{r} ({\mathbf{C}}\cdot {{\mathbf{C}}^{v}}^{-1} )^{\beta _{r}-1} \right] {\mathbf{F}}{{\mathbf{C}}^{v}}^{-1}{\mathbf{F}}^T\\&\quad + m_{{{\texttt {K}}}}{\mathbf{e }}\otimes {\mathbf{e }} + n_{{{\texttt {K}}}}({\mathbf{e }}-{\mathbf{e }}^v)\otimes ({\mathbf{e }}-{\mathbf{e }}^v) \end{aligned}$$

and

$$\begin{aligned} {\mathbf{d}}&=\left[ \left( \varepsilon -m_{{{\texttt {K}}}}\right) {\mathbf{F}}{\mathbf{F}}^T+m_{{{\texttt {K}}}}{\mathbf{I}}\right] {\mathbf{e }} \\&\quad + \left[ \left( \epsilon -n_{{{\texttt {K}}}}\right) {\mathbf{F}}{\mathbf{F}}^T+n_{{{\texttt {K}}}}{\mathbf{I}}\right] ({\mathbf{e }}-{\mathbf{e }}^v) \end{aligned}$$
(28)

in terms of the Eulerian electric field \({\mathbf{e }}={\mathbf{F}}^{-T}{\mathbf{E}}\), where use has been made of the notation \({\mathbf{e }}^v={\mathbf{F}}^{-T}{\mathbf{E}}^v\).

The special case when \({\mathbf{E}}={\mathbf{E}}^v=\mathbf{0}\). In the absence of electric fields when \({\mathbf{E}}={\mathbf{E}}^v=\mathbf{0}\), the model (24)–(27) reduces to the viscoelastic model of Kumar and Lopez-Pamies [24], precisely, the first Piola-Kirchhoff stress (24) reduces to

$$\begin{aligned} {\mathbf{S}}&= \left[ \sum \limits _{r=1}^2 3^{1-\alpha _{r}}\mu _{r} I_{1}^{\alpha _{r}-1} \right] {\mathbf{F}}-p\, {\mathbf{F}}^{-T} \\&\quad +\left[ \sum \limits _{r=1}^2 3^{1-\beta _{r}} \nu _{r} ({\mathbf{C}}\cdot {{\mathbf{C}}^{v}}^{-1} )^{\beta _{r}-1} \right] {\mathbf{F}}{{\mathbf{C}}^{v}}^{-1}, \end{aligned}$$

where \({\mathbf{C}}^{v}\) is still defined by the evolution equation (26). Because such a model accounts for the non-Gaussian elasticity and the deformation-enhanced shear-thinning viscosity typical of elastomers, it is expected to be descriptive and predictive of the viscoelastic behavior of most standard elastomers; see Section 3.1 in [24] for comparisons with experiments.

The special case when \({\mathbf{F}}={\mathbf{F}}^v={{\mathbf{Q}}}\in Orth^+\). In the absence of stretch when \({\mathbf{F}}={\mathbf{F}}^v={{\mathbf{Q}}}\in Orth^+\), the model (24)–(27) reduces to the classical Debye model [8], precisely, the electric displacement (25) reduces to

$$\begin{aligned} {\mathbf{D}}=&\varepsilon \,{\mathbf{E}}+\epsilon \left( {\mathbf{E}}- {\mathbf{E}}^v\right) , \end{aligned}$$

where \({\mathbf{E}}^v\) is solution of the evolution equation

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{{\mathbf{E}}} ^{v}=-\dfrac{\epsilon }{\zeta }({\mathbf{E}}-{\mathbf{E}}^v) \\ {\mathbf{E}}^v({\mathbf{X}},0)=\mathbf{0} \end{array} \right. . \end{aligned}$$

This constitutive relation describes reasonably well the time-dependent polarization response of most standard dielectric elastomers all the way up to their electric breakdown; see, e.g., the experimental results presented in  [7, 17, 22, 52].

Numerical solution of the evolution equations (26) and (27). The evolution equations (26) and (27) for the internal variables \({\mathbf{C}}^v\) and \({\mathbf{E}}^v\) are decoupled from one another and hence can be solved independently. Moreover, while the evolution equation (27) for the electric internal variable \({\mathbf{E}}^v\) does depend on both the deformation and the electric field in the dielectric elastomer via \({\mathbf{C}}\) and \({\mathbf{E}}\), the evolution equation (26) for the mechanical internal variable \({\mathbf{C}}^v\) does not depend on \({\mathbf{E}}\).

From a mathematical point of view, both evolution equations are systems of first-order ODEs. Whereas the system (27) for \({\mathbf{E}}^v\) is linear, the system (26) for \({\mathbf{C}}^v\) is nonlinear. An efficient and robustFootnote 2 numerical scheme to generate solutions for both of these systems is the explicit fifth-order Runge-Kutta scheme with extended region of stability due to Lawson [25]. For a generic system of nonlinear first-order ODEs

$$\begin{aligned} {\dot{{\mathbf{A}}}}(t)={\mathbf{G }}(t,{\mathbf{A}}(t)), \end{aligned}$$

denoting by \({\mathbf{A}}_k\) the numerical approximation of the solution \({\mathbf{A}}(t_k)\) at the discretized time \(t_k\), this scheme provides the solution \({\mathbf{A}}_{n+1}\) in terms of the solution at the previous time step \({\mathbf{A}}_{n}\), for a given discretized time interval \([t_n,t_{n+1}]\), by the rule

$$\begin{aligned} {\mathbf{A}}_{n+1}= {\mathbf{A}}_{n}+ \frac{{{\mathrm {\Delta }}} t}{90}(7{\mathbf{k }}_{1}+ 32{\mathbf{k }}_{3}+ 12{\mathbf{k }}_{4}+ 32{\mathbf{k }}_{5}+ 7{\mathbf{k }}_{6}) \end{aligned}$$

with

$$\begin{aligned} {\mathbf{k }}_{1}=\,&{\mathbf{G }}\left( t_{n}, {\mathbf{A}}_n\right) \\ {\mathbf{k }}_{2}=\,&{\mathbf{G }}\left( t_{n}+ \frac{{{\mathrm {\Delta }}} t}{2}, {\mathbf{A}}_n+ {\mathbf{k }}_{1}\, \frac{{{\mathrm {\Delta }}} t}{2}\right) \\ {\mathbf{k }}_{3}=\,&{\mathbf{G }}\left( t_{n}+ \frac{{{\mathrm {\Delta }}} t}{4}, {\mathbf{A}}_n+ (3{\mathbf{k }}_{1}+ {\mathbf{k }}_{2})\, \frac{{{\mathrm {\Delta }}} t}{16}\right) \\ {\mathbf{k }}_{4}=\,&{\mathbf{G }}\left( t_{n}+ \frac{{{\mathrm {\Delta }}} t}{2}, {\mathbf{A}}_n+ {\mathbf{k }}_{3}\, \frac{{{\mathrm {\Delta }}} t}{2}\right) \\ {\mathbf{k }}_{5}=\,&{\mathbf{G }}\left( t_{n}+ \frac{3{{\mathrm {\Delta }}} t}{4}, {\mathbf{A}}_n+ 3 (-{\mathbf{k }}_{2}+ 2{\mathbf{k }}_{3}+ 3{\mathbf{k }}_{4})\, \frac{{{\mathrm {\Delta }}} t}{16}\right) \\ {\mathbf{k }}_{6}=\,&{\mathbf{G }}\left( t_{n}+ {{\mathrm {\Delta }}} t, {\mathbf{A}}_n+ ({\mathbf{k }}_{1}+ 4{\mathbf{k }}_{2} + 6{\mathbf{k }}_{3}- 12{\mathbf{k }}_{4}+ 8{\mathbf{k }}_{5})\, \frac{{{\mathrm {\Delta }}} t}{7}\right) , \end{aligned}$$

where \({{\mathrm {\Delta }}} t=t_{n+1}-t_{n}\).

Material parameters and their determination from experiments. The model (24)–(27) contains nineteen material parameters:

  • four (\(\mu _1\), \(\alpha _1\), \(\mu _2\), \(\alpha _2\)) describing the non-Gaussian elasticity at states of mechanical and electric equilibrium,

  • four (\(\nu _1\), \(\beta _1\), \(\nu _2\), \(\beta _2\)) describing the additional non-Gaussian elasticity at non-equilibrium states,

  • six (\(\eta _0\), \(\eta _\infty \), \(\gamma _1\), \(\gamma _2\), \(K_1\), \(K_2\)) describing the viscous dissipation that stems from the motion of the underlying polymer chains,

  • two (\(\varepsilon \), \(m_{{{\texttt {K}}}}\)) describing the polarization and electrostriction at states of mechanical and electric equilibrium,

  • two (\(\epsilon \), \(n_{{{\texttt {K}}}}\)) describing the additional polarization and electrostriction at non-equilibrium states, and

  • one (\(\zeta \)) describing the frictional dissipation that stems from the process of polarization.

The fourteen purely mechanical parameters (\(\mu _1\), \(\alpha _1\), \(\mu _2\), \(\alpha _2\), \(\nu _1\), \(\beta _1\), \(\nu _2\), \(\beta _2\), \(\eta _0\), \(\eta _\infty \), \(\gamma _1\), \(\gamma _2\), \(K_1\), \(K_2\)) can be determined by simply fitting (e.g., by means of least squares) the model simultaneously to a set of uniaxial relaxation data and a set of uniaxial tension/compression data at constant stretch rate. Alternatively, they can be determined by fitting simultaneously two sets of uniaxial tension/compression data at two sufficiently different constant deformation rates; see Section 3.1 in  [24].

The three purely electric parameters (\(\varepsilon \), \(\epsilon \), \(\zeta \)) can be determined from conventional broadband dielectric spectroscopy measurements; see, e.g., Chapter 2 in [23].

Finally, the two remaining electromechanical parameters (\(m_{{{\texttt {K}}}}\), \(n_{{{\texttt {K}}}}\)) can be determined from electrostriction experiments and from broadband dielectric spectroscopy measurements in pre-stretched specimens; see, e.g.,  [22, 28, 37, 48].

Dielectric spectroscopy on pre-stretched specimens. As just mentioned, a standard experimental technique to probe the electromechanical behavior of dielectric elastomers is the dielectric spectroscopy of pre-stretched specimens.

Customarily, as a first step, a thin layer of the dielectric elastomer of interest is stretched biaxially, say \(\lambda _1\) in the \({\mathbf{e }}_1\) direction and \(\lambda _2\) in the perpendicular direction \({\mathbf{e }}_2\), and held in place for long enough (usually, about a day) to reach mechanical equilibrium so that, according to the model (24)–(27), the deformation gradient \({\mathbf{F}}\) and internal variable \({\mathbf{C}}^v\) are of the spatially homogeneous and constant-in-time forms

$$\begin{aligned} F_{ij}=\left[ \begin{array}{ccc} \lambda _1 &{} 0 &{} 0 \\ 0 &{} \lambda _2 &{} 0 \\ 0 &{} 0 &{} \frac{1}{\lambda _1 \lambda _2} \end{array} \right] \quad \mathrm{and} \quad C^v_{ij}=C_{ij}=\left[ \begin{array}{ccc} \lambda ^2_1 &{} 0 &{} 0 \\ 0 &{} \lambda ^2_2 &{} 0 \\ 0 &{} 0 &{} \frac{1}{\lambda ^2_1 \lambda ^2_2} \end{array} \right] \end{aligned}$$
(29)

with respect to the Cartesian laboratory axes \(\{{{\mathbf{e }}}_1\), \({\mathbf{e }}_2\), \({\mathbf{e }}_3\}\), where \({\mathbf{e }}_3\) denotes the thin layer normal.

Keeping the stretched configuration (29) fixed, as a second step, electrodes are placed on the opposite surfaces of the specimen and these are then connected to a power source that sets a time-harmonic voltage across. Neglecting fringe effects, the Lagrangian electric field \({\mathbf{E}}\) and internal variable \({\mathbf{E}}^v\) in the specimen specialize to be of the forms

$$\begin{aligned} E_i=\left[ \begin{array}{c} 0 \\ 0 \\ E(t) \end{array} \right] \quad \mathrm{with} \quad E(t)=E_0 e^{i\omega t} \end{aligned}$$
(30)

and

$$\begin{aligned} E^v_i=\left[ \begin{array}{c} 0 \\ 0 \\ E^v(t) \end{array} \right] \quad \mathrm{with} \quad E^v(t)=\dfrac{E_0}{1+i \omega \tau }\left( e^{i\omega t}-e^{-\frac{t}{\tau }}\right) , \end{aligned}$$
(31)

where \(\omega \) is the angular frequency of the applied harmonic voltage, \(i=\sqrt{-1}\), \(E_0\) is the amplitude of the induced Lagrangian electric field, and

$$\begin{aligned} \tau =\dfrac{\lambda _1^2\lambda _2^2\zeta }{(1-\lambda _1^2\lambda _2^2)n_{{{\texttt {K}}}}-\epsilon } \end{aligned}$$
(32)

has been introduced for notational convenience. Physically, this last quantity characterizes the electric relaxation time of the dielectric elastomer at the stretched configuration (29).

The third and last step consists in running the experiment for a sufficiently long period of time (usually, a few miliseconds suffice) so that the initial transient part of the response—characterized by the term \(e^{-\frac{t}{\tau }}\) in (31)—vanishes and a harmonic relationship can be established between the Eulerian electric displacement \({\mathbf{d }}={\mathbf{F}}{\mathbf{D}}= d(t) {\mathbf{e }}_3\) and the Eulerian electric field \({\mathbf{e }}={\mathbf{F}}^{-T}{\mathbf{E}}=\lambda _1\lambda _2 E_0 e^{i\omega t}{\mathbf{e }}_3=e_0 e^{i\omega t}{\mathbf{e }}_3\). It immediately follows from (28), (29)\(_1\), (30), and (31) that that harmonic relationship is given by

$$\begin{aligned} d(t)=\varepsilon ^{\star }(\omega ;\lambda _1,\lambda _2) e_0 e^{i\omega t} \end{aligned}$$

with

$$\begin{aligned} \varepsilon ^{\star }(\omega ;\lambda _1,\lambda _2)&= m_{{{\texttt {K}}}}+n_{{{\texttt {K}}}}+\dfrac{\varepsilon +\epsilon -m_{{{\texttt {K}}}}-n_{{{\texttt {K}}}}}{\lambda _1^2\lambda _2^2} \\&\quad -\dfrac{1}{1+i\omega \tau }\left( n_{{{\texttt {K}}}}+\dfrac{\epsilon -n_{{{\texttt {K}}}}}{\lambda _1^2\lambda _2^2}\right) . \end{aligned}$$
(33)

Clearly, the quantity (33) can be thought of as the apparent permittivity of the dielectric elastomer at the stretched configuration (29) and this is how (33) is precisely interpreted in the experimental literature. Using the convention \(\varepsilon ^{\star }(\omega ;\lambda _1,\lambda _2)={\varepsilon ^{\star }}^{\prime }(\omega ;\lambda _1,\lambda _2)-i{\varepsilon ^{\star }}^{\prime \prime }(\omega ;\lambda _1,\lambda _2)\), its real \({\varepsilon ^{\star }}^{\prime }(\omega ;\lambda _1,\lambda _2)\) and imaginary \({\varepsilon ^{\star }}^{\prime \prime }(\omega ;\lambda _1,\lambda _2)\) parts read as

$$\begin{aligned} {\varepsilon ^{\star }}^{\prime }(\omega ;\lambda _1,\lambda _2)&= m_{{{\texttt {K}}}}+n_{{{\texttt {K}}}}+\dfrac{\varepsilon +\epsilon -m_{{{\texttt {K}}}}-n_{{{\texttt {K}}}}}{\lambda _1^2\lambda _2^2} \\&\quad -\dfrac{1}{1+\omega ^2 \tau ^2}\left( n_{{{\texttt {K}}}}+\dfrac{\epsilon -n_{{{\texttt {K}}}}}{\lambda _1^2\lambda _2^2}\right) \end{aligned}$$
(34)

and

$$\begin{aligned} {\varepsilon ^{\star }}^{\prime \prime }(\omega ;\lambda _1,\lambda _2)=-\dfrac{\omega \tau }{1+\omega ^2 \tau ^2}\left( n_{{{\texttt {K}}}}+\dfrac{\epsilon -n_{{{\texttt {K}}}}}{\lambda _1^2\lambda _2^2}\right) , \end{aligned}$$
(35)

where we recall that \(\tau \) is given by (32). Relations (34)–(35) make it plain that the dielectric spectroscopy of pre-stretched specimens is indeed an expedient approach to determine the electromechanical material parameters \(m_{{{\texttt {K}}}}\) and \(n_{{{\texttt {K}}}}\) of any dielectric elastomer of interest. Section 4 below presents an example of such a calibration process for the acrylate elastomer VHB 4910.

3.3 The limit of small deformations and moderate electric fields

The distinguishing even electromechanical coupling of dielectric elastomers comes forth most plainly in the so-called limit of small deformations and moderate electric fields. Such a limit has been studied extensively in the restricted setting of elastic dielectrics, when both mechanical and electric dissipation are absent; see, e.g.,  [26, 41, 42, 44, 45]. In this subsection, we work out the limit of small deformations and moderate electric fields for the proposed model (24)–(27) in the general setting when mechanical and electric dissipation are both present.

Consider \(\xi \) to be a non-negative scalar parameter of choice and take the deformation gradient and the Lagrangian electric field to be of the orders \(||{\mathbf{F}}-{\mathbf{I}}||=O(\xi )\) and \(||{\mathbf{E}}||=O(\xi ^\frac{1}{2})\). In the limit as \(\xi \searrow 0\), making use of the notation \({\mathbf{H}}={\mathbf{F}}-{\mathbf{I}}\) and \({\mathbf{H}}^v={\mathbf{F}}^v-{\mathbf{I}}\), it follows that the constitutive response (24)–(27) reduces asymptotically to leading order to

$$\begin{aligned} {\mathbf{S}}&= {\mathbf{L}}^{\mathrm{Eq}} {\mathbf{H}}+{\mathbf{L}}^{\mathrm{NEq}} ({\mathbf{H}}-{\mathbf{H}}^v)-p {\mathbf{I}} \\&\quad + {\mathbf{M}}^{\mathrm{Eq}} {\mathbf{E}}\otimes {\mathbf{E}}+{\mathbf{M}}^{\mathrm{NEq}} ({\mathbf{E}}-{\mathbf{E}}^v)\otimes ({\mathbf{E}}-{\mathbf{E}}^v), \end{aligned}$$
(36)

subject to the constraint \(\mathrm{tr}\,{\mathbf{H}}=0\), and

$$\begin{aligned} {\mathbf{D}}=\varvec{\varepsilon }^{\mathrm{Eq}}{\mathbf{E}}+\varvec{\varepsilon }^{\mathrm{NEq}} ({\mathbf{E}}-{\mathbf{E}}^v) \end{aligned}$$
(37)

with

$$\begin{aligned}&{\mathbf{L}}^{\mathrm{Eq}}=2(\mu _1+\mu _2)\varvec{{\mathcal {K}}},\quad {\mathbf{L}}^{\mathrm{NEq}}=2(\nu _1+\nu _2)\varvec{{\mathcal {K}}}, \\&{\mathbf{M}}^{\mathrm{Eq}}=m_{{{\texttt {K}}}}\left( \varvec{{\mathcal {K}}}+\varvec{{\mathcal {J}}}\right) ,\quad {\mathbf{M}}^{\mathrm{NEq}}=n_{{{\texttt {K}}}}\left( \varvec{{\mathcal {K}}}+\varvec{{\mathcal {J}}}\right) , \\&\varvec{\varepsilon }^{\mathrm{Eq}}=\varepsilon \, {\mathbf{I}},\quad \varvec{\varepsilon }^{\mathrm{NEq}}=\epsilon \, {\mathbf{I}}, \end{aligned}$$
(38)

where p stands for the arbitrary hydrostatic pressure associated with the incompressibility constraint \(\mathrm{tr}\,{\mathbf{H}}=0\), \(\varvec{{\mathcal {K}}}\) and \(\varvec{{\mathcal {J}}}\) stand, again, for the orthogonal projection tensors (23), and \({\mathbf{H}}^{v}\) and \({\mathbf{E}}^v\) are defined by the evolution equations

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{{\mathbf{H}}} ^{v}=\dfrac{\nu _1 + \nu _2}{\eta _0} \left( {\mathbf{H}}-{\mathbf{H}}^v \right) \\ {\mathbf{H}}^v({\mathbf{X}},0)={\mathbf {0}} \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{{\mathbf{E}}} ^{v}=-\dfrac{\epsilon }{\zeta }({\mathbf{E}}-{\mathbf{E}}^v) \\ {\mathbf{E}}^v({\mathbf{X}},0)=\mathbf{0} \end{array} \right. , \end{aligned}$$

which admit the closed-form solutions

$$\begin{aligned} H^v_{ij}({\mathbf{X}},t)=\int _0^t\dfrac{e^{-\frac{t-s}{\tau _{{\texttt {m}}}}}}{\tau _{{\texttt {m}}}}H_{ij}({\mathbf{X}},s)\mathrm{d}s \end{aligned}$$
(39)

and

$$\begin{aligned} E^v_{i}({\mathbf{X}},t)=\int _0^t\dfrac{e^{-\frac{t-s}{\tau _{{\texttt {e}}}}}}{\tau _{{\texttt {e}}}}E_{i}({\mathbf{X}},s)\mathrm{d}s. \end{aligned}$$
(40)

In these last expressions, we have introduced the notation \(\tau _{{\texttt {m}}}=\eta _0/(\nu _1+\nu _2)\) and \(\tau _{{\texttt {e}}}=-\zeta /\epsilon \). Physically, these quantities characterize the mechanical and electric relaxation times of the dielectric elastomer in the limit of small deformations and moderate electric fields. Typically, \(\tau _{{\texttt {m}}}\) is in the order of \(10^2\) seconds, while \(\tau _{{\texttt {e}}}\) is in the much faster range of \([10^{-6},10^{-3}]\) seconds.

Reduced set of material parameters. Out of the total of nineteen in the full model (24)–(27), only a combination of eight material parameters shows up in the asymptotic constitutive response (36)–(40):

  • one (\(\mu =\mu _1+\mu _2\)) describing the initial shear modulus at states of mechanical and electric equilibrium,

  • one (\(\nu =\nu _1+\nu _2\)) describing the additional initial shear modulus at non-equilibrium states,

  • one (\(\tau _{{\texttt {m}}}=\eta _0/(\nu _1+\nu _2)\)) describing the initial mechanical relaxation time,

  • two (\(\varepsilon \), \(m_{{{\texttt {K}}}}\)) describing the initial permittivity and electrostriction at states of mechanical and electric equilibrium,

  • two (\(\epsilon \), \(n_{{{\texttt {K}}}}\)) describing the additional initial permittivity and electrostriction at non-equilibrium states, and

  • one (\(\tau _{{\texttt {e}}}=-\zeta /\epsilon \)) describing the initial electric relaxation time.

Electrostriction. As alluded to above, a fairly accessible and thus popular experiment to probe the coupled electromechanical behavior of dielectric elastomers consists in measuring their deformation—commonly referred to as electrostriction—that results from an imposed spatially homogeneous electric field in the absence of stresses. This is routinely achieved by sandwiching a thin layer of the dielectric elastomer between two compliant electrodes connected to a battery. In such a setup, the stress is indeed roughly zero everywhere (inside the dielectric elastomer as well as in the surrounding space), while the electric field is roughly uniform within the dielectric elastomer and zero outside of it.

Consider hence a thin layer of dielectric elastomer subject to electromechanical states where the stress \({\mathbf{S}}\) and electric field \({\mathbf{E}}\) are of the spatially homogeneous—albeit time dependent—forms

$$\begin{aligned} S_{ij}=\left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right] \qquad \mathrm{and} \qquad E_i=\left[ \begin{array}{c} 0 \\ 0 \\ E(t) \end{array} \right] \end{aligned}$$

with respect to some Cartesian laboratory axes \(\{{{\mathbf{e }}}_1\), \({\mathbf{e }}_2\), \({\mathbf{e }}_3\}\), where, as in the preceding section, \({\mathbf{e }}_3\) denotes the thin layer normal.

It follows from the constitutive relations (36) through (40) that

$$\begin{aligned} H_{ij}=\left[ \begin{array}{ccc} -\frac{1}{2}H(t) &{} 0 &{} 0 \\ 0 &{} -\frac{1}{2}H(t) &{} 0 \\ 0 &{} 0 &{} H(t) \end{array} \right] \quad \mathrm{and} \quad D_i=\left[ \begin{array}{c} 0 \\ 0 \\ D(t) \end{array} \right] , \end{aligned}$$

where the electrostriction strain H(t) in the direction of the applied electric field is defined implicitly by the Volterra integral equation

$$\begin{aligned}&S_{33}=(\mu _1+\mu _2+\nu _1+\nu _2)H(t)+\dfrac{m_{{{\texttt {K}}}}}{3}E^2(t)- \\&(\nu _1+\nu _2)\int _0^t\dfrac{e^{-\frac{t-s}{\tau _{{\texttt {m}}}}}}{\tau _{{\texttt {m}}}}H(s)\mathrm{d}s+ \\&\dfrac{n_{{{\texttt {K}}}}}{3}\left( E(t)-\int _0^t\dfrac{e^{-\frac{t-s}{\tau _{{\texttt {e}}}}}}{\tau _{{\texttt {e}}}}E(s)\mathrm{d}s\right) ^2=0, \end{aligned}$$
(41)

while the sole non-trivial component D(t) of the Lagrangian electric displacement is given explicitly by the relation

$$\begin{aligned} D(t)=(\varepsilon +\epsilon )E(t)-\epsilon \int _0^t\dfrac{e^{-\frac{t-s}{\tau _{{\texttt {e}}}}}}{\tau _{{\texttt {e}}}}E(s)\mathrm{d}s. \end{aligned}$$

The result (41) generalizes the classical result for the electrostriction of elastic dielectrics with even electromechanical coupling to account for both mechanical and electric dissipation. Indeed, for the case of applied electric fields E(t) with a fixed constant value for times larger than a certain threshold time, say \(E(t)=E_{{{\texttt {f}}}}\) for \(t\ge t_{{{\texttt {f}}}}\), the electrostriction strain H(t) defined by (41) reduces asymptotically to the classical result (see, e.g., Section 2.25 in [42] and Eq. (60) in [41])

$$\begin{aligned} H(t)=H_{{{\texttt {f}}}}=-\dfrac{m_{{{\texttt {K}}}}}{3(\mu _1+\mu _2)}E^2_{{{\texttt {f}}}} \end{aligned}$$

for \(t\gg t_{{{\texttt {f}}}}\), once the mechanical and electric dissipation has taken place and the dielectric elastomer has reached its equilibrium state.

4 Application to VHB 4910

In this last section, for demonstration purposes, we determine the material parameters in the proposed model (24)–(27) for the acrylate elastomer VHB 4910 from 3M by fitting experimental data available in the literature. To illustrate the model’s predictive capabilities, we also present comparisons between its predictions and additional experimental data.

Due in part to its growing commercial significance, a plurality of experimental works aimed at characterizing different aspects of the electromechanical behavior of such a dielectric elastomer have been reported in the literature over the past fifteen years. Presumably because of the technical difficulties in carrying out the experiments, however, the reported results are not entirely consistent with one another. Here, for definiteness, we choose subsets of the experimental data reported in [19] and  [37] in order to determine the nineteen material parameters in (24)–(27).

Precisely, we make use of the results reported in [19] for the equilibrium states in a multi-step relaxation test together with those from two uniaxial tension loading/unloading tests at the constant stretch rates of 0.01 and 0.05 \(s^{-1}\) in order to determine the fourteen purely mechanical parameters (\(\mu _1\), \(\alpha _1\), \(\mu _2\), \(\alpha _2\), \(\nu _1\), \(\beta _1\), \(\nu _2\), \(\beta _2\), \(\eta _0\), \(\eta _\infty \), \(\gamma _1\), \(\gamma _2\), \(K_1\), \(K_2\)) in the model. The resulting values from such a calibration process are listed in Table 1. Figures 5(a)–(b) in  [24] compare the three sets of experimental data and the fitted response described by the model.

Table 1 Material parameters for VHB 4910 determined from experiments in [19] and [37]
Full size table

Furthermore, we make use of the dielectric spectroscopy results reported in [37] for an undeformed specimen and a biaxially pre-stretched specimen, with an equal biaxial stretch of 2, in order to determine the remaining three purely electric parameters (\(\varepsilon \), \(\epsilon \), \(\zeta \)) and the two electromechanical parameters (\(m_{{{\texttt {K}}}}\), \(n_{{{\texttt {K}}}}\)). The results of such a calibration process are also reported in Table 1. Figure 2 compares those two sets of experimental data and the fitted response described by the model. There it is of note that the experimental data for intermediate frequencies is not accurately captured by the model suggesting that the polarization-friction coefficient \(\zeta \) should not be assumed to simply be constant but rather a function of the electric field \({\mathbf{E}}\). We shall explore this generalization in future work.

Fig. 2
figure 2

The model response (34) fitted to the experimental dielectric spectroscopy data of Qiang et al. [37] for the real part of the apparent permittivity of undeformed (\(\lambda _1=\lambda _2=1\)) and biaxially stretched (\(\lambda _1=\lambda _2=2\)) VHB 4910 specimens. The results are shown as functions of the frequency \(f=\omega /2\pi \), from \(f=0.1\) to \(10^5\) Hz

Full size image

The results presented by Figs. 5(c) and (d) in [24] have already shown that the model (24)–(27) with the material parameters listed in Table 1 can predict the purely mechanical response of VHB 4910 reasonably well. The results presented by Figs. 3 and 4 here indicate that this is also the case more generally for its coupled electromechanical response.

Fig. 3
figure 3

Real part of the apparent permittivity of biaxially pre-stretched (\(\lambda _2=\lambda _1\)) VHB 4910, at the frequency \(f=\omega /2\pi =10\) Hz, as a function of the pre-stretch \(\lambda _1\). The solid circles correspond to the experimental data of Qiang et al. [37], while the solid line corresponds to the response (34) predicted by the model with the material parameters listed in Table 1

Full size image
Fig. 4
figure 4

Load/unload response of uniaxially pre-stretched (\(\lambda _1=3\)) VHB 4910 subject simultaneously to a uniaxial stretch \(\lambda _2\) perpendicular to the pre-stretch direction, applied at the constant stretch rate of \(|\dot{\lambda }_2|=0.01\) \({\hbox {s}}^{-1}\), and a constant Lagrangian electric field \(E_3=8.6\) MV/m. The solid circles correspond to the experimental data of Hossain et al. [20], while the solid line corresponds to the response predicted by the model with the material parameters listed in Table 1

Full size image

Specifically, Fig. 3 confronts the response (34) predicted by the model with the experimental data of Qiang et al. [37] for the real part of the apparent permittivity of biaxially pre-stretched (\(\lambda _2=\lambda _1=1, 1.5, 2, 2.5, 3, 3.5\), and 4) VHB 4910 specimens at the frequency \(f=\omega /2\pi =10\) Hz; see Fig. 6 in [37].

On the other hand, Fig. 4 compares the model prediction with the experimental data of Hossain et al. [20] for the stress-stretch (\(S_{22}\)–\(\lambda _2\)) response of a uniaxially pre-stretched (\(\lambda _1=3\)) specimen that is loaded/unloaded in the direction perpendicular to the pre-stretch at the constant stretch rate of 0.01 \({\hbox {s}}^{-1}\) in the presence of a 5 kV voltage applied across its thickness resulting in a constant Lagrangian electric field of \(E_3=8.6\) MV/m; see Fig. 5 in [20].