A nonlinear spectral rate-dependent constitutive equation for electro-viscoelastic solids

Abstract

In this communication a spectral constitutive equation for nonlinear viscoelastic-electroactive bodies with short-term memory response is developed, using the total stress formulation and the electric field as the electric independent variable. Spectral invariants, each one with a clear physical meaning and hence attractive for use in experiment, are used in the constitutive equation. A specific form for constitutive equation containing single-variable functions is presented, which are easy to analyze compared to multivariable functions. The effects of viscosity and an electric field are studied via the results of boundary value problems for cases considering homogeneous distributions for the strains and the electric field, and some these results are compared with experimental data.

Appendix: P-property

The description of the P-property uses the eigenvalues ($\lambda _i$) and eigenvectors (${\textit{\textbf{u}}}_i$) of the symmetric tensor ${\textit{\textbf{U}}}$. A general anisotropic scalar function $\Phi$, such as that given in (37) and (45), where its arguments are expressed in terms spectral invariants with respect to the basis $\{{\textit{\textbf{u}}}_1,{\textit{\textbf{u}}}_2, {\textit{\textbf{u}}}_3 \}$ can be written in the form

$$\begin{aligned} \Phi = {\tilde{W}}(\lambda _1,\lambda _2,\lambda _3,{\textit{\textbf{u}}}_1,{\textit{\textbf{u}}}_2,{\textit{\textbf{u}}}_3), \end{aligned}$$

(130)

with the symmetrical property

$$\begin{aligned} {\tilde{W}}(\lambda _1,\lambda _2,\lambda _3,{\textit{\textbf{u}}}_1,{\textit{\textbf{u}}}_2,{\textit{\textbf{u}}}_3) = {\tilde{W}}(\lambda _2,\lambda _1,\lambda _3,{\textit{\textbf{u}}}_2,{\textit{\textbf{u}}}_1,{\textit{\textbf{u}}}_3) = {\tilde{W}}(\lambda _3,\lambda _2,\lambda _1,{\textit{\textbf{u}}}_3,{\textit{\textbf{u}}}_2,{\textit{\textbf{u}}}_1) . \end{aligned}$$

(131)

In view of the non-unique values of ${\textit{\textbf{u}}}_i$ and ${\textit{\textbf{u}}}_j$ when $\lambda _i=\lambda _j$, a function ${\tilde{W}}$ should be independent of ${\textit{\textbf{u}}}_i$ and ${\textit{\textbf{u}}}_j$ when $\lambda _i=\lambda _j$, and ${\tilde{W}}$ should be independent of ${\textit{\textbf{u}}}_1$, ${\textit{\textbf{u}}}_2$ and ${\textit{\textbf{u}}}_3$ when $\lambda _1=\lambda _2=\lambda _3$. Hence, when two or three of the principal stretches have equal values the scalar function $\Phi$ must have any of the following forms

$$\begin{aligned} \Phi = \left\{ \begin{array}{ll} {W}_{(a)}(\lambda ,\lambda _k,{\textit{\textbf{u}}}_k) , &{} \hbox {when} \, \, \lambda _i=\lambda _j=\lambda , i\ne j \ne k \ne i \\ {W}_{(b)}(\lambda ) , &{} \hbox {when} \, \, \lambda _1=\lambda _2=\lambda _3=\lambda \end{array}\right. \end{aligned}$$

(132)

As an example of (132), consider ${\displaystyle \Phi ={{{\textit{\textbf{a}}}}}\bullet {{{\textit{\textbf{C}}}}}{{{\textit{\textbf{a}}}}}=\sum _{i=1}^3 \lambda _i^2 ({{{\textit{\textbf{a}}}}}\bullet {{{\textit{\textbf{u}}}}}_i)^2 }$, where ${\textit{\textbf{a}}}$ is a fixed unit vector and ${\displaystyle \sum _{i=1} ({{{\textit{\textbf{a}}}}}\bullet {{{\textit{\textbf{u}}}}}_i)^2 = 1}$. If $\lambda _1=\lambda _2=\lambda$, we have ${\displaystyle \Phi = {W}_{(a)}(\lambda ,\lambda _3,{{{\textit{\textbf{u}}}}}_3) =\lambda ^2 + (\lambda _3^2 - \lambda ^2)({{{\textit{\textbf{a}}}}}\bullet {{{\textit{\textbf{u}}}}}_3)^2}$ and in the case of $\lambda _1=\lambda _2=\lambda _3=\lambda$, $\Phi ={W}_{(b)}(\lambda ) = \lambda ^2$. Note that, for example, ${\displaystyle {{{\textit{\textbf{C}}}}}= \sum _{i=1}^3 \lambda _i^2 {{{\textit{\textbf{u}}}}}_i\otimes {{{\textit{\textbf{u}}}}}_i}$ (or ${\textit{\textbf{U}}}$ ) and all the classical invariants described in Spencer [41] , satisfy the P-property. In Refs.  [35] and [40], the P-property described here is extended to non-symmetric tensors such as the two-point deformation tensor ${\textit{\textbf{F}}}$.