Electro-Elastic Continuum Models for Electrostrictive Elastomers

Abstract

A continuum finite-deformation model is described for the study of the isothermal electro-elastic deformations of electrostrictive elastomers. The model comprises general balance equations of motion, electrostatics and electro-mechanical energy, along with phenomenological invariant-based constitutive relations. The model is presented in both Eulerian (spatial) and Lagrangian (material) description. Specialization of the considered model is also presented for “Dielectric Elastomers”, which are a specific class of electrostrictive elastomers having dielectric properties independent of deformation.

Appendices

Appendix A. Mathematical Operators

This appendix defines the mathematical operators that have been employed throughout this chapter.

Consider the 3 × 1 vectors, a and b, and the 3 × 3 matrices, A and B,

$$ {\varvec{a}} = \left[ {a_{i} } \right] = \left[ {\begin{array}{*{20}c} {a_{1} } \\ {a_{2} } \\ {a_{3} } \\ \end{array} } \right],\,{\varvec b} = \left[ {b_{i} } \right] = \left[ {\begin{array}{*{20}c} {b_{1} } \\ {b_{2} } \\ {b_{3} } \\ \end{array} } \right],\,{\mathbf {A}} = \left[ {A_{ij} } \right] = \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } & {A_{13} } \\ {A_{21} } & {A_{22} } & {A_{23} } \\ {A_{31} } & {A_{32} } & {A_{33} } \\ \end{array} } \right],\,{\mathbf{B}} = \left[ {B_{ij} } \right] = \left[{\begin{array}{*{20}c} {B_{11} } & {B_{12} } & {B_{13} } \\{B_{21} } & {B_{22} } & {B_{23} } \\ {B_{31} } & {B_{32}} & {B_{33} } \\ \end{array} } \right]$$

(179)

The scalar product of vectors a and b, yielding a scalar quantity, is defined as

$$ \varvec{a} \cdot \varvec{b} = \sum\limits_{l = 1}^{3} {a_{l} b_{l} } . $$

(180)

The tensor product (or the dyad) of vectors a and b, yielding a 3 × 3 matrix, is defined as

$$ \varvec{a} \otimes \varvec{b} = \left[ {\left( {\varvec{a} \otimes \varvec{b}} \right)_{ij} } \right] = \left[ {a_{i} b_{j} } \right] . $$

(181)

The product between a matrix A and a vector b, yielding a 3 × 1 vector, is defined as

$$ {\mathbf{A}}\varvec{b} = \left[ {\left( {{\mathbf{A}}\varvec{b}} \right)_{i} } \right] = \left[ {\sum\limits_{l = 1}^{3} {a_{il} b_{l} } } \right] . $$

(182)

The scalar product between a matrix A and a vector b, yielding a 3 × 1 vector, is defined as

$$ {\mathbf{A}} \cdot \varvec{b} = \left[ {\left( {{\mathbf{A}} \cdot \varvec{b}} \right)_{i} } \right] = \left[ {\sum\limits_{l = 1}^{3} {a_{li} b_{l} } } \right] . $$

(183)

Note that Eqs. (182) and (183) differ in the matrix index used for the summation.

The product between matrices A and B, yielding a 3 × 3 matrix, is defined as

$$ {\mathbf{AB}} = \left[ {\left( {{\mathbf{AB}}} \right)_{ij} } \right] = \left[ {\sum\limits_{l = 1}^{3} {A_{il} B_{lj} } } \right] . $$

(184)

The double contraction between matrices A and B, yielding a scalar, is defined as

$$ {\mathbf{A}}:{\mathbf{B}} = \sum\limits_{m = 1}^{3} {\sum\limits_{l = 1}^{3} {A_{lm} B_{lm} } } .$$

(185)

Given a third-order tensor ${{\mathcal{A}}} = \left[ {\mathcal{A}_{ijk} } \right]$, the double contraction of ${{\mathcal{A}}}$ and a matrix ${\mathbf{B}}$, yielding a 3 × 1 vector, is defined as

$$ \mathbf{{\mathcal A}}:{\mathbf{B}} = \left[ {\left( {{\mathbf { {\mathcal A}}}:{{\mathbf B}}} \right)_{i} } \right] = \sum\limits_{l = 1}^{3} {\sum\limits_{m = 1}^{3} {{\mathcal A}_{ilm} B_{lm} } } .$$

(186)

Given the spatial position coordinate $\varvec{x} = \left[ {\begin{array}{*{20}c} {x_{1} } & {x_{2} } & {x_{3} } \\ \end{array} } \right]^{T}$, the spatial gradient (grad) of a scalar quantity $\phi = \phi \left( {x,t} \right)$, t being the time variable, is defined as

$$ {\text{grad}}\phi = \left[ {\left( {{\text{grad}}\phi } \right)_{i} } \right] = \left[ {{{\partial \phi } \mathord{\left/ {\vphantom {{\partial \phi } {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right]; $$

(187)

whereas the spatial gradient (grad), divergence (div) and rotation (rot) of a vector $\varvec{a} = \varvec{a}\left( {\varvec{x},t} \right)$, yielding 3 × 3 matrix, a scalar and a 3 × 1 vector respectively, are defined as

$$ {\text{grad}}\varvec{a} = \left[ {\left( {{\text{grad}}\varvec{a}} \right)_{ij} } \right] = \left[ {{{\partial a_{i} } \mathord{\left/ {\vphantom {{\partial a_{i} } {\partial x_{j} }}} \right. \kern-0pt} {\partial x_{j} }}} \right] ,$$

(188)

$$ {\text{div}}\varvec{a} = \sum\limits_{l = 1}^{3} {{{\partial a_{l} } \mathord{\left/ {\vphantom {{\partial a_{l} } {\partial x_{l} }}} \right. \kern-0pt} {\partial x_{l} }}}, $$

(189)

$$ {\text{rot}}\varvec{a} = \left[ {\left( {{\text{rot}}\varvec{a}} \right)_{i} } \right] = \left[ {\sum\limits_{l = 1}^{3} {\sum\limits_{m = 1}^{3} {{{\varepsilon_{ilm} \partial a_{m} } \mathord{\left/ {\vphantom {{\varepsilon_{ilm} \partial a_{m} } {\partial x_{l} }}} \right. \kern-0pt} {\partial x_{l} }}} } } \right], $$

(190)

where $\varepsilon_{ijk}$ is the permutation symbol such that

$$ \varepsilon_{ijk} = \left\{ {\begin{array}{*{20}l} { \, 1 ,\;{\text{ for}}\;{\text{even}}\;{\text{permutation}}\;{\text{of}}\;\left( {i,j,k} \right) ,\; ( {\text{i}} . {\text{e}} .\; 1 2 3 ,\; 2 3 1 ,\; 3 1 2 )} \hfill \\ { - 1 ,\;{\text{for}}\;{\text{odd}}\;{\text{permutation}}\;{\text{of}}\;\left( {i,j,k} \right) ,\; ( {\text{i}} . {\text{e}} .\; 1 3 2 ,\; 2 1 3 ,\; 3 2 1 )} \hfill \\ { 0 ,\;{\text{if}}\;{\text{there}}\;{\text{is}}\;{\text{a}}\;{\text{repeated}}\;{\text{index }}} \hfill \\ \end{array} } \right. .$$

(191)

Besides, the spatial divergence (div) of a matrix ${\mathbf{A}} = {\mathbf{A}}\left( {\varvec{x},t} \right)$ is defined as

$$ {\text{div}}{\mathbf{A}} = \left[ {\left( {{\text{div}}{\mathbf{A}}} \right)_{i} } \right] = \sum\limits_{l = 1}^{3} {{{\partial A_{li} } \mathord{\left/ {\vphantom {{\partial A_{li} } {\partial x_{l} }}} \right. \kern-0pt} {\partial x_{l} }}}. $$

(192)

Given the material position coordinate $\varvec{X} = \left[ {\begin{array}{*{20}c} {X_{1} } & {X_{2} } & {X_{3} } \\ \end{array} } \right]^{T}$, the material gradient (Grad) of a scalar quantity $\phi = \phi \left( {\varvec{X},t} \right)$ is defined as

$$ {\text{Grad}}\phi = \left[ {\left( {{\text{Grad}}\phi } \right)_{i} } \right] = \left[ {{{\partial \phi } \mathord{\left/ {\vphantom {{\partial \phi } {\partial X_{i} }}} \right. \kern-0pt} {\partial X_{i} }}} \right] ; $$

(193)

whereas the material gradient (Grad), divergence (Div) and rotation (Rot) of a vector $\varvec{a} = \varvec{a}\left( {\varvec{X},t} \right)$ are respectively defined as

$$ {\text{Grad}}\varvec{a} = \left[ {\left( {{\text{Grad}}\varvec{a}} \right)_{ij} } \right] = \left[ {{{\partial a_{i} } \mathord{\left/ {\vphantom {{\partial a_{i} } {\partial X_{j} }}} \right. \kern-0pt} {\partial X_{j} }}} \right] , $$

(194)

$$ {\text{Div}}\varvec{a} = \sum\limits_{l = 1}^{3} {{{\partial a_{l} } \mathord{\left/ {\vphantom {{\partial a_{l} } {\partial X_{l} }}} \right. \kern-0pt} {\partial X_{l} }}} , $$

(195)

$$ {\text{Rot}}\varvec{a} = \left[ {\left( {{\text{Rot}}\varvec{a}} \right)_{i} } \right] = \left[ {\sum\limits_{l = 1}^{3} {\sum\limits_{m = 1}^{3} {{{\varepsilon_{ilm} \partial a_{m} } \mathord{\left/ {\vphantom {{\varepsilon_{ilm} \partial a_{m} } {\partial X_{l} }}} \right. \kern-0pt} {\partial X_{l} }}} } } \right] . $$

(196)

Besides, the material divergence (Div) of a matrix ${\mathbf{A}} = {\mathbf{A}}\left( {\varvec{X},t} \right)$ is defined as

$$ {\text{Div}}{\mathbf{A}} = \left[ {\left( {{\text{div}}{\mathbf{A}}} \right)_{i} } \right] = \sum\limits_{l = 1}^{3} {{{\partial A_{li} } \mathord{\left/ {\vphantom {{\partial A_{li} } {\partial X_{l} }}} \right. \kern-0pt} {\partial X_{l} }}} . $$

(197)

With regard to the divergence of the products between vector $\varvec{a}$ and either matrix ${\mathbf{A}}$ or scalar $\phi$

$$ {\text{div}}\left( {{\mathbf{A}}\varvec{a}} \right) = {\text{div}}{\mathbf{A}} \cdot \varvec{a} + {\mathbf{A}}^{T} :{\text{grad}}\varvec{a}, $$

(198)

$$ {\text{div}}\left( {\phi \varvec{a}} \right) = \phi {\text{div}}\varvec{a} + {\text{grad}}\phi \cdot \varvec{a}, $$

(199)

$$ {\text{Div}}\left( {{\mathbf{A}}\varvec{a}} \right) = {\text{Div}}{\mathbf{A}} \cdot \varvec{a} + {\mathbf{A}}^{T} :{\text{Grad}}\varvec{a} , $$

(200)

$$ {\text{Div}}\left( {\phi \varvec{a}} \right) = \phi {\text{Div}}\varvec{a} + {\text{Grad}}\phi \cdot \varvec{a}. $$

(201)

Appendix B. Fundamental Mathematical Theorems

This appendix summarizes the fundamental mathematical theorems that have been employed throughout this chapter.

For any open surface $S\left( t \right)$ with bounding closed curve $L\left( t \right)$, the Stokes’ theorem states

$$ \int\limits_{S\left( t \right)} {{\text{rot}}\varvec{a} \cdot \varvec{n}{\text{ds}}} = \int\limits_{L\left( t \right)} {\varvec{a} \cdot {\text{d}}\varvec{x}} . $$

(202)

where ds is the infinitesimal surface belonging to $S\left( t \right)$ and with unit normal n, whereas $\text{d}\varvec{x}$ is the infinitesimal line element belonging to $L\left( t \right) .$

For any volume $V\left( t \right)$ with bounding closed surface $\partial V\left( t \right)$, the Gauss’ divergence theorem states

$$ \int\limits_{V\left( t \right)} {{\text{div}}\varvec{a}} {\text{dv}} = \int\limits_{\partial V} {\varvec{a} \cdot \varvec{n}{\text{ds}}} , $$

(203)

$$ \int\limits_{V\left( t \right)} {{\text{div}}{\mathbf{A}}} {\text{dv}} = \int\limits_{\partial V} {{\mathbf{A}} \cdot \varvec{n}{\text{ds}}} , $$

(204)

where dv is the infinitesimal volume belonging to $V\left( t \right)$, whereas ds is the infinitesimal surface belonging to $\partial V\left( t \right)$ and with unit normal n. In the presence of a discontinuity surface $\gamma \left( t \right)$, within volume $V\left( t \right)$, across which some vector a and tensor A admit non-continuous values, the Gauss’ divergence theorem states

$$ \int\limits_{V\left( t \right) - \gamma \left( t \right)} {{\text{div}}\varvec{a}{\text{dv}}} + \int\limits_{\partial \gamma \left( t \right)} { \left[\kern-0.15em\left[ \user2{a} \right]\kern-0.15em\right] \cdot \varvec{n}{\text{ds}}} = \int\limits_{\partial V - \gamma \left( t \right)} {\varvec{a} \cdot \varvec{n}{\text{ds}}} , $$

(205)

$$ \int\limits_{V\left( t \right) - \gamma \left( t \right)} {{\text{div}}{\mathbf{A}}} {\text{dv}} + \int\limits_{\gamma \left( t \right)} { \left[\kern-0.15em\left[ \user2{A} \right]\kern-0.15em\right] \cdot \varvec{n}{\text{ds}}} = \int\limits_{\partial V - \gamma \left( t \right)} {{\mathbf{A}} \cdot \varvec{n}{\text{ds}}} , $$

(206)

where $\left[\kern-0.15em\left[ \user2{a} \right]\kern-0.15em\right] \equiv \varvec{a}^{ + } - \varvec{a}^{ - }$ and $\left[\left[ {\mathbf{A}} \right]\right] \equiv {\mathbf{A}}^{ + } - {\mathbf{A}}^{ - }$ indicate the jumps of vector a and matrix A from the positive (+) side to the negative (−) side of the discontinuity.

For any given scalar quantity ϕ (x,t), the Reynolds’ transport theorem states

$$ \frac{\text{d}}{{{\text{d}}t}}\int\limits_{V \left( t \right)} \phi {\text{d}}v = \int\limits_{V \left( t \right)} {\left[ {\dot{\phi } + \phi {\text{div}}\user2{v}} \right]{\text{dv}}} = \int\limits_{V \left( t \right)} {\left[ {\frac{\partial \phi }{\partial t} + {\text{div}}\left( {\phi \user2{v}} \right)} \right]{\text{dv}}} . $$

(207)