On the two-potential constitutive modeling of dielectric elastomers

Abstract

This work lays out the two-potential framework for the constitutive modeling of dielectric elastomers. After its general presentation, where the constraints imposed by even electromechanical coupling, material frame indifference, material symmetry, and entropy imbalance are all spelled out, the framework is utilized to put forth a specific constitutive model for the prominent class of isotropic incompressible dielectric elastomers. The model accounts for the non-Gaussian elasticity and electrostriction typical of such materials, as well as for their deformation-enhanced shear thinning due to viscous dissipation and their time-dependent polarization due to electric dissipation. The key theoretical and practical features of the model are discussed, with special emphasis on its specialization in the limit of small deformations and moderate electric fields. The last part of this paper is devoted to the deployment of the model to fully describe the electromechanical behavior of a commercially significant dielectric elastomer, namely, the acrylate elastomer VHB 4910 from 3M.

Appendix. The reduced dissipation inequality

Denote by ${\mathbf{v}}({\mathbf{x}},t)$ the velocity of the material point that occupies the location ${\mathbf{x}}\in {\Omega }(t)$ at time $t\in [0,T]$ and assume that at any ${\mathbf{x}}\in {\Omega }(t)$ and time $t\in [0,T]$ the following quantities exist and are sufficiently regular both in space and time: the mass density $\rho ({\mathbf{x}},t)$, the mechanical Cauchy stress ${\mathbf{T}}^m({\mathbf{x}},t)$, the mechanical body force ${{\mathbf {f}}}^m({\mathbf{x}},t)$ (per unit volume), the electric force ${{\mathbf {f}}}^e({\mathbf{x}},t)$ (per unit volume), the electric couple ${\mathbf{g }}^e({\mathbf{x}},t)$ (per unit volume), the electric field ${\mathbf{e }}({\mathbf{x}},t)$, the electric displacement ${\mathbf{d }}({\mathbf{x}},t)$, the space charge $q({\mathbf{x}},t)$ (per unit volume), the internal energy $u({\mathbf{x}},t)$ (per unit mass), the heat source $r({\mathbf{x}},t)$ (per unit mass), the heat flux $\widetilde{{{\mathbf {q}}}}({\mathbf{x}},t)$, the entropy $\eta ({\mathbf{x}},t)$ (per unit mass), and the absolute temperature $\theta ({\mathbf{x}},t)$.

1.1 Conservation of mass

Conservation of mass is said to be satisfied provided that

$$\begin{aligned} \frac{\partial \rho }{\partial t}+ \mathrm{div} (\rho {\mathbf{v}})=0,\quad ({\mathbf{x}},t)\in {\Omega }(t)\times [0,T]. \end{aligned}$$

(A.1)

1.2 Balance of linear and angular momenta

Absent inertia, the balance of linear and angular momenta are said to be satisfied provided that

$$\begin{aligned} \mathrm{div}\,{\mathbf{T}}^m+{{\mathbf {f}}}^{m}+{{\mathbf {f}}}^e={\mathbf {0}},\quad ({\mathbf{x}},t)\in {\Omega }(t)\times [0,T] \end{aligned}$$

(A.2)

and

$$\begin{aligned} \widetilde{\varvec{\varepsilon }}\,{{\mathbf{T}}^m}^T={{\mathbf {g}}}^e,\quad ({\mathbf{x}},t)\in {\Omega }(t)\times [0,T], \end{aligned}$$

(A.3)

where $\widetilde{\varvec{\varepsilon }}$ stands for the permutation symbol.

In the context of electro-quasi-statics of interest here, the electric force and couple take the form (see, e.g., Eqs. (7.38) and (7.48) in [35])

$$\begin{aligned} {{\mathbf {f}}}^e=q {\mathbf{e}}+\left( \mathrm{grad}\,{\mathbf{e}}\right) ^T {\mathbf{p }}\quad \mathrm{and}\quad {{\mathbf {g}}}^e=-{\mathbf{p }}\times {{\mathbf {e}}}, \end{aligned}$$

where we recall that ${\mathbf{p }}={{\mathbf {d}}}-\varepsilon _0{{\mathbf {e}}}$ stands for the polarization.

Upon defining the electric stress

$$\begin{aligned} {\mathbf{T}}^e= {{\mathbf {e}}} \otimes {\mathbf{d }} -\dfrac{\varepsilon _0}{2}({{\mathbf {e}}} \cdot {{\mathbf {e}}}){\mathbf{I}}\end{aligned}$$

and invoking Gauss’s (A.6) and Faraday’s (A.7) laws introduced further below, the balance of linear (A.2) and angular (A.3) momenta can be recast as

$$\begin{aligned} \mathrm{div}\,{\mathbf{T}}+{{\mathbf {f}}}^{m}={\mathbf {0}},\quad ({\mathbf{x}},t)\in {\Omega }(t)\times [0,T] \end{aligned}$$

(A.4)

and

$$\begin{aligned} {\mathbf{T}}={\mathbf{T}}^T,\quad ({\mathbf{x}},t)\in {\Omega }(t)\times [0,T] \end{aligned}$$

(A.5)

in terms of the total Cauchy stress ${\mathbf{T}}={\mathbf{T}}^m+{\mathbf{T}}^e$. With help of the connections (6) and the definition ${{\mathbf {f}}}=J{{\mathbf {f}}}^m$, Eqs. (A.4)–(A.5) can be further recast as those summoned in the main body of the text, namely, (11).

1.3 Maxwell’s equations

In the context of electro-quasi-statics of interest here, Maxwell’s equations are said to be satisfied provided that

$$\begin{aligned} \mathrm{div}\,{{\mathbf {d}}}=q,\quad ({\mathbf{x}},t)\in {{\mathbb {R}}}^3\times [0,T] \end{aligned}$$

(A.6)

and

$$\begin{aligned} \mathrm{curl}\,{\mathbf{e}}={\mathbf{0 }},\quad ({\mathbf{x}},t)\in {{\mathbb {R}}}^3\times [0,T]. \end{aligned}$$

(A.7)

With help of the connections (6) and the definition $Q=J q$, Eqs. (A.6)–(A.7) can be recast in the form (12) provided in the main body of the text.

1.4 Balance of energy

Granted the balance Eqs. (A.1), (A.2), (A.3), (A.6), (A.7), balance of energy is said to be satisfied provided that

$$\begin{aligned}&\rho \, \dot{u}+\mathrm{div}\,\widetilde{{{\mathbf {q}}}}-\rho \, r-{\mathbf{T}}^m \cdot {{\mathbf {\Gamma }}} -\dot{{{\mathbf {p}}}}\cdot {\mathbf{e}} \\&\quad -(\mathrm{tr}\,{{\mathbf {\Gamma }}}){{\mathbf {p}}}\cdot {\mathbf{e}}=0,\quad ({\mathbf{x}},t)\in {\Omega }(t)\times [0,T], \end{aligned}$$

(A.8)

where ${{\mathbf {\Gamma }}}=\partial {\mathbf{v}}({\mathbf{x}},t)/\partial {\mathbf{x}}$ stands for the Eulerian velocity gradient.

Throughout this appendix, a superposed “dot” denotes the material time derivative.

1.5 Entropy imbalance

Granted the balance Eq. (A.1), the entropy imbalance is said to be satisfied provided that

$$\begin{aligned} \rho \, \dot{\eta } +\mathrm{div}\left( \frac{\widetilde{{{\mathbf {q}}}}}{\theta }\right) - \frac{\rho \, r}{\theta }\ge 0 ,\quad ({\mathbf{x}},t)\in {\Omega }(t)\times [0,T]. \end{aligned}$$

(A.9)

Upon defining the free energy (per unit mass)

$$\begin{aligned} {{\mathrm {\Psi }}}=u-\theta \eta -\dfrac{1}{\rho }{{\mathbf {p}}}\cdot {\mathbf{e}}\end{aligned}$$

and invoking the balance Eq. (A.8), the entropy imbalance (A.9) can be recast as the reduced dissipation inequality

$$\begin{aligned}&\rho \left( \dot{{{\mathrm {\Psi }}}}+\eta \dot{\theta }\right) -{\mathbf{T}}^m\cdot {{\mathbf {\Gamma }}}+{{\mathbf {p}}}\cdot \dot{{\mathbf{e}}} \\&\quad +\dfrac{1}{\theta }\widetilde{{{\mathbf {q}}}}\cdot (\mathrm{grad}\,\theta )\le 0 ,\quad ({\mathbf{x}},t)\in {\Omega }(t)\times [0,T], \end{aligned}$$

which, in the context of isothermal processes of interest in this work, specializes to

$$\begin{aligned}&\rho \,\dot{{{\mathrm {\Psi }}}}-{\mathbf{T}}^m\cdot {{\mathbf {\Gamma }}}+{{\mathbf {p}}}\cdot \dot{{\mathbf{e}}}\le 0 ,\quad ({\mathbf{x}},t)\in {\Omega }(t)\times [0,T]. \end{aligned}$$

(A.10)

With help of the definition

$$\begin{aligned} \psi =J \rho {{\mathrm {\Psi }}}-\dfrac{\varepsilon _0}{2}J{\mathbf{F}}^{-T}{\mathbf{E}}\cdot {\mathbf{F}}^{-T}{\mathbf{E}}\end{aligned}$$

and the connections (6), the reduced dissipation inequality (A.10) can be further recast as

$$\begin{aligned} \dot{\psi }-{\mathbf{S}}\cdot \dot{{{\mathbf{F}}}}+{\mathbf{D}}\cdot \dot{{\mathbf{E}}} \le 0 ,\quad ({\mathbf{X}},t)\in {\Omega _0}\times [0,T]. \end{aligned}$$

(A.11)

Writing now

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

and making use of the connections (3)–(5), the inequality (A.11) reduces finally to the form (10) provided in the main body of the text.