The Elastic Dielectric Response of Elastomers Filled with Liquid Inclusions: From Fundamentals to Governing Equations

Abstract

Over the past decade, soft solids containing electro-and magneto-active liquid—as opposed to solid—inclusions have emerged as a new class of smart materials with promising novel electro- and magneto-mechanical properties. In this context, a recent contribution has put forth a continuum theory that describes the macroscopic elastic behavior of elastomers filled with liquid inclusions under quasistatic finite deformations from the bottom up, directly in terms of their microscopic behavior at the length scale of the inclusions. This chapter presents the generalization of that theory to the coupled realm of the elastic dielectric behavior of such an emerging class of filled elastomers when in addition to undergoing quasistatic finite deformations they are subjected to quasistatic electric fields. The chapter starts with the description of the underlying fundamentals in the continuum—id est, kinematics, conservation of mass, Maxwell’s equations, balance of momenta, and constitutive behavior of both the bulk (the solid elastomer and the liquid inclusions) and the solid/liquid interfaces—and ends with their combination to formulate the resulting governing equations.

Appendices

Appendix A. Gauss’s law in Lagrangian form

On substitution of the definitions $Q=J q$, ${{\textbf {D}}}=J{{\textbf {F}}}^{-1}{\textbf {d}}$, $\widehat{Q}=\widehat{J} \widehat{q}$, and $\widehat{{{\textbf {D}}}}=\widehat{J}\widehat{{{\textbf {F}}}}^{-1}\widehat{{\textbf {d}}}$ in the Eulerian form (10) of Gauss’s law, we have

$$\begin{aligned} \left\{ \begin{array}{ll} \dfrac{\partial }{\partial x_k}\left[ J^{-1}F_{km}D_{m}\right] = J^{-1} Q, &{} {\textbf {x}}\in \Omega \setminus \Gamma \vspace{0.25cm}\\ \dfrac{\partial }{\partial x_l}\left[ \widehat{J}^{-1}\widehat{F}_{km}\widehat{D}_{m}\right] \,\widehat{i}_{kl} - \left[ \left[ J^{-1}F_{km}D_{m} \right] \right] \widehat{n}_{k} = \widehat{J}^{-1} \widehat{Q}, &{} {\textbf {x}}\in \Gamma \vspace{0.25cm}\\ \dfrac{\partial }{\partial x_k}\left[ J^{-1}F_{km}D_{m}\right] = 0, &{} {\textbf {x}}\in \mathbb {R}^3\setminus \Omega \vspace{0.25cm}\\ \left[ \left[ J^{-1}F_{km}D_{m} \right] \right] n_{k} = 0, &{} {\textbf {x}}\in \partial \Omega \end{array}\right. , \end{aligned}$$

(46)

where, again, by ${{\textbf {F}}}$ in $\mathbb {R}^3\setminus \Omega$ we mean any suitably well-behaved extension to $\mathbb {R}^3$ of the deformation gradient ${{\textbf {F}}}$ in the body. Given that $\text {div} \left[ J^{-1} {{\textbf {F}}}^{T}\right] = \widehat{\text {div}}\left[ \widehat{J}^{-1}\widehat{{{\textbf {F}}}}^{T}\right] = \textbf{0}$, Eq. (46) simplify to

$$\begin{aligned} \left\{ \begin{array}{ll} J^{-1}F_{km}\dfrac{\partial D_{m}}{\partial x_k} = J^{-1} Q, &{} {\textbf {x}}\in \Omega \setminus \Gamma \vspace{0.25cm}\\ \widehat{J}^{-1}\widehat{F}_{km}\dfrac{\partial \widehat{D}_{m}}{\partial x_l}\,\widehat{i}_{kl} - \left[ \left[ J^{-1}F_{km}D_{m} \right] \right] \widehat{n}_{k} = \widehat{J}^{-1} \widehat{Q}, &{} {\textbf {x}}\in \Gamma \vspace{0.25cm}\\ J^{-1}F_{km}\dfrac{\partial D_{m}}{\partial x_k} = 0, &{} {\textbf {x}}\in \mathbb {R}^3\setminus \Omega \vspace{0.25cm}\\ \left[ \left[ J^{-1}F_{km}D_{m} \right] \right] n_{k} = 0, &{} {\textbf {x}}\in \partial \Omega \end{array}\right. . \end{aligned}$$

By employing now the chain rule and the identities ${\textbf {n}}= |J {{\textbf {F}}}^{-T}{\textbf {N}}|^{-1} \, J {{\textbf {F}}}^{-T}{\textbf {N}}$, $\widehat{{{\textbf {F}}}}^{-1}={{\textbf {F}}}^{-1}\,\widehat{{{\textbf {i}}}}$, and $\widehat{{\textbf {n}}} = \widehat{J}^{-1} \, J {{\textbf {F}}}^{-T}\widehat{{\textbf {N}}}$ together with the fact that $J^{\texttt {i}} {{{\textbf {F}}}^{\texttt {i}}}^{-T}\widehat{{\textbf {N}}}=J^{\texttt {m}} {{{\textbf {F}}}^{\texttt {m}}}^{-T}\widehat{{\textbf {N}}}$, we obtain

$$\begin{aligned} \left\{ \begin{array}{ll} \dfrac{\partial D_{m}}{\partial X_n}F^{-1}_{nk}F_{km}= Q, &{} {{\textbf {X}}}\in \Omega _0\setminus \Gamma _0 \vspace{0.25cm}\\ \dfrac{\partial \widehat{D}_{m}}{\partial X_n}\,\widehat{F}^{-1}_{nk}\widehat{F}_{km} - \left[ \left[ D_{k} \right] \right] \widehat{N}_{k} =\widehat{Q}, &{} {{\textbf {X}}}\in \Gamma _0 \vspace{0.25cm}\\ \dfrac{\partial D_{m}}{\partial X_n}F^{-1}_{nk}F_{km} = 0, &{} {{\textbf {X}}}\in \mathbb {R}^3\setminus \Omega _0\vspace{0.25cm}\\ \left[ \left[ D_{k} \right] \right] \widehat{N}_{k} = 0, &{} {{\textbf {X}}}\in \partial \Omega _0\end{array}\right. \end{aligned}$$

Finally, recognizing that $\widehat{{{\textbf {F}}}}^{-1}\widehat{{{\textbf {F}}}}=\widehat{{\textbf {I}}}$, Gauss’s law in Lagrangian form (11) readily follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \dfrac{\partial D_{m}}{\partial X_m}= Q, &{} {{\textbf {X}}}\in \Omega _0\setminus \Gamma _0 \vspace{0.25cm}\\ \dfrac{\partial \widehat{D}_{m}}{\partial X_n}\,\widehat{I}_{mn} - \left[ \left[ D_{k} \right] \right] \widehat{N}_{k} =\widehat{Q}, &{} {{\textbf {X}}}\in \Gamma _0 \vspace{0.25cm}\\ \dfrac{\partial D_{m}}{\partial X_m} = 0, &{} {{\textbf {X}}}\in \mathbb {R}^3\setminus \Omega _0\vspace{0.25cm}\\ \left[ \left[ D_{k} \right] \right] \widehat{N}_{k} = 0, &{} {{\textbf {X}}}\in \partial \Omega _0\end{array}\right. \end{aligned}$$

Appendix B. Faraday’s law in Lagrangian form

Direct use of the definition ${\textbf {E}}={{\textbf {F}}}^{T}{{\textbf {e}}}$ and the transformation rule $\text {d}{\textbf {x}}={{\textbf {F}}}\text {d}{{\textbf {X}}}$ for material line elements allows to recast the integral form (12) of Faraday’s law as

$$\begin{aligned} \int _{\partial \Sigma } {{\textbf {e}}}\cdot \text {d}{\textbf {x}}=\int _{\partial \Sigma _0} {\textbf {E}}\cdot \text {d}{{\textbf {X}}}=0. \end{aligned}$$

(47)

By making use of Stokes’s theorem

$$\begin{aligned} \int _{\Sigma _0}\left( \text {Curl}\,{\textbf {E}}\right) \cdot \widetilde{{\textbf {N}}}\,\text {d}{{\textbf {X}}}=\int _{\partial \Sigma _0} {\textbf {E}}\cdot \text {d}{{\textbf {X}}}+\int _{\partial \mathcal {S}_0} \left[ \left[ {\textbf {E}} \right] \right] \cdot \text {d}{{\textbf {X}}}, \end{aligned}$$

this time around in the initial configuration, the Lagrangian localized form (14) of Faraday’s law readily follows from (47).