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Rate-dependent electromechanical behavior of anisotropic fiber-reinforced dielectric elastomer based on a nonlinear continuum approach: modeling and implementation

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Abstract

Dielectric elastomer (DE) characteristics and potentials have led to more efficient actuators in soft robots. Meanwhile, more possibilities could be explored by incorporating other properties, including viscoelasticity and anisotropy. The present study focuses on establishing constitutive laws to capture the electromechanical coupled behavior of anisotropic viscoelastic fiber-reinforced DEs. The proposed model is presented in the framework of the strain energy function, nonlinear electro-elasticity, and nonlinear continuum mechanics approach. The derived model has been calibrated and validated using available experimental results. An Abaqus subroutine has also been developed and used for finite element simulation. Rate-dependent governing equations of a fiber-reinforced layered DE have also been considered as well as the effects of fibers direction and actuating electric field. Obtained results indicate the efficiency of the suggested model and developed subroutine in describing rate-dependency, electro-elasticity, and anisotropy. According to the findings, stiffening the bending actuator with fibers can drastically affect the deflection and force of the sample under different loading conditions.

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Data Availability Statement

All data generated or analyzed during this study are included in this published article.

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Author information

Authors and Affiliations

  1. Passive Safety Research Lab, Mechanical Engineering Faculty, K.N. Toosi University of Technology, Tehran, Iran

    Marzie Majidi & Masoud Asgari

Authors
  1. Marzie Majidi
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  2. Masoud Asgari
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Corresponding author

Correspondence to Masoud Asgari.

Appendix

Appendix

This section is aimed to provide a detailed presentation of the derivation processes of involved parameters and the tangent moduli. It should be noted that if the definition of the Cauchy stress is showed to be correct, the final FEM result obtained using the Cauchy stress and extracted tangent moduli would be correct because the tangent modulus serves as an iterative operator. In this research work, all the stress terms are evaluated using analytical results and appropriate experimental data.

1.1 Hyperelastic terms

$${\overline{W} }^{hyperelastic}={C}_{10}\left({\overline{I} }_{1}-3\right)+{C}_{01}\left({\overline{I} }_{2}-3\right)$$
(91)
$$\frac{\partial {\overline{W} }^{hyperelastic}}{\partial {{\varvec{C}}}_{ij}}={C}_{10}\left(\frac{\partial {\overline{I} }_{1}}{\partial {{\varvec{C}}}_{ij}}\right)+{C}_{01}\left(\frac{\partial {\overline{I} }_{2}}{\partial {{\varvec{C}}}_{ij}}\right)={C}_{10}\frac{\partial \left({J}^{-\frac{2}{3}}{I}_{1}\right)}{\partial {{\varvec{C}}}_{ij}}+{C}_{01}\frac{\partial \left({J}^{-\frac{4}{3}}{I}_{2}\right)}{\partial {{\varvec{C}}}_{ij}}={C}_{10}\left(\frac{\partial {J}^{-\frac{2}{3}}}{\partial {{\varvec{C}}}_{ij}}{I}_{1}+\frac{\partial {I}_{1}}{\partial {{\varvec{C}}}_{ij}}{J}^{-\frac{2}{3}}\right)+{C}_{01}\left(\frac{\partial {J}^{-\frac{4}{3}}}{\partial {{\varvec{C}}}_{ij}}{I}_{2}+\frac{\partial {I}_{2}}{\partial {{\varvec{C}}}_{ij}}{J}^{-\frac{4}{3}}\right)={C}_{10}\left(\left(-\frac{1}{3}{J}^{-\frac{2}{3}}{{\varvec{C}}}_{ij}^{-1}\right){I}_{1}+{\delta }_{ij}{J}^{-\frac{2}{3}}\right)+{C}_{01}\left(\left(-\frac{2}{3}{J}^{-\frac{4}{3}}{{\varvec{C}}}_{ij}^{-1}\right){I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right){J}^{-\frac{4}{3}}\right)={C}_{10}{J}^{-\frac{2}{3}}\left(-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{\delta }_{ij}\right)+{C}_{01}{J}^{-\frac{4}{3}}\left(-\frac{2}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)$$
(92)
$${{\varvec{S}}}_{ij}^{hyperelastic}=2{C}_{10}{J}^{-\frac{2}{3}}\left(-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{\delta }_{ij}\right)+2{C}_{01}{J}^{-\frac{4}{3}}\left(-\frac{2}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)$$
(93)
$${{\varvec{\sigma}}}_{mn}^{hyperelastic}=\frac{1}{J}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{S}}}_{ij}^{e}=\frac{2}{J}\left({C}_{10}{J}^{-\frac{2}{3}}\left(-\frac{1}{3}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{\delta }_{ij}\right)+{C}_{01}{J}^{-\frac{4}{3}}\left(-\frac{2}{3}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{\delta }_{ij}-{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ij}\right)\right)\right)=\frac{2}{J}\left({C}_{10}{J}^{-\frac{2}{3}}\left(-\frac{1}{3}{\delta }_{mn}{I}_{1}+{{\varvec{B}}}_{mn}\right)+{C}_{01}{J}^{-\frac{4}{3}}\left(-\frac{2}{3}{\delta }_{mn}{I}_{2}+\left({I}_{1}{{\varvec{B}}}_{mn}-{{\varvec{B}}}_{mk}{{\varvec{B}}}_{nk}\right)\right)\right)$$
(94)
$$\frac{\partial {{\varvec{S}}}_{ij}^{hyperelastic}}{\partial {{\varvec{C}}}_{kl}}=2{C}_{10}\frac{\partial \left({J}^{-\frac{2}{3}}\left(-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{\delta }_{ij}\right)\right)}{\partial {{\varvec{C}}}_{kl}}+2{C}_{01}\frac{\partial \left({J}^{-\frac{4}{3}}\left(-\frac{2}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)\right)}{\partial {{\varvec{C}}}_{kl}}=2{C}_{10}\left(\frac{\partial {J}^{-\frac{2}{3}}}{\partial {{\varvec{C}}}_{kl}}\left(-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{\delta }_{ij}\right)+{J}^{-\frac{2}{3}}\frac{\partial \left(-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{\delta }_{ij}\right)}{\partial {{\varvec{C}}}_{kl}}\right)+2{C}_{01}\left(\frac{\partial {J}^{-\frac{4}{3}}}{\partial {{\varvec{C}}}_{kl}}\left(-\frac{2}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)+{J}^{-\frac{4}{3}}\frac{\partial \left(-\frac{2}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)}{\partial {{\varvec{C}}}_{kl}}\right)=2{C}_{10}{J}^{-\frac{2}{3}}\left(-\frac{1}{3}{{\varvec{C}}}_{kl}^{-1}\left(-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{\delta }_{ij}\right)+\frac{\partial \left(-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{\delta }_{ij}\right)}{\partial {{\varvec{C}}}_{kl}}\right)+2{C}_{01}{J}^{-\frac{4}{3}}\left(-\frac{2}{3}{{\varvec{C}}}_{kl}^{-1}\left(-\frac{2}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)+\frac{\partial \left(-\frac{2}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)}{\partial {{\varvec{C}}}_{kl}}\right)=2{C}_{10}{J}^{-\frac{2}{3}}\left(-\frac{1}{3}{{\varvec{C}}}_{kl}^{-1}\left(-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{\delta }_{ij}\right)-\frac{1}{3}\left(\frac{\partial {{\varvec{C}}}_{ij}^{-1}}{\partial {{\varvec{C}}}_{kl}}{I}_{1}+\frac{\partial {I}_{1}}{\partial {{\varvec{C}}}_{kl}}{{\varvec{C}}}_{ij}^{-1}\right)\right)+2{C}_{01}{J}^{-\frac{4}{3}}\left(-\frac{2}{3}{{\varvec{C}}}_{kl}^{-1}\left(-\frac{2}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)-\frac{2}{3}\left(\frac{\partial {{\varvec{C}}}_{ij}^{-1}}{\partial {{\varvec{C}}}_{kl}}{I}_{2}+\frac{\partial {I}_{2}}{\partial {{\varvec{C}}}_{kl}}{{\varvec{C}}}_{ij}^{-1}\right)+\frac{\partial \left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}^{-1}\right)}{{{\varvec{C}}}_{ij}^{-1}}\right)=2{C}_{10}{J}^{-\frac{2}{3}}\left(-\frac{1}{3}{{\varvec{C}}}_{kl}^{-1}\left(-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{1}+{\delta }_{ij}\right)-\frac{1}{3}\left(-\frac{1}{2}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right){I}_{1}+{\delta }_{kl}{{\varvec{C}}}_{ij}^{-1}\right)\right)+2{C}_{01}{J}^{-\frac{4}{3}}\left(-\frac{2}{3}{{\varvec{C}}}_{kl}^{-1}\left(-\frac{2}{3}{{\varvec{C}}}_{ij}^{-1}{I}_{2}+\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)-\frac{2}{3}\left(-\frac{1}{2}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right){I}_{2}+\left({I}_{1}{\delta }_{kl}-{{\varvec{C}}}_{kl}\right){{\varvec{C}}}_{ij}^{-1}\right)+{\delta }_{ij}{\delta }_{kl}-\frac{1}{2}\left({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}\right)\right)=2{C}_{10}{J}^{-\frac{2}{3}}\left(\left(\frac{1}{9}{{\varvec{C}}}_{ij}^{-1}{{\varvec{C}}}_{kl}^{-1}{I}_{1}-\frac{1}{3}{{\varvec{C}}}_{kl}^{-1}{\delta }_{ij}\right)+\left(\frac{1}{6}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right){I}_{1}-\frac{1}{3}{\delta }_{kl}{{\varvec{C}}}_{ij}^{-1}\right)\right)+2{C}_{01}{J}^{-\frac{4}{3}}\left(\left(\frac{4}{9}{{\varvec{C}}}_{ij}^{-1}{{\varvec{C}}}_{kl}^{-1}{I}_{2}-\frac{2}{3}{{\varvec{C}}}_{kl}^{-1}\left({I}_{1}{\delta }_{ij}-{{\varvec{C}}}_{ij}\right)\right)+\left(\frac{1}{3}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right){I}_{2}-\frac{2}{3}\left({I}_{1}{\delta }_{kl}-{{\varvec{C}}}_{kl}\right){{\varvec{C}}}_{ij}^{-1}\right)+{\delta }_{ij}{\delta }_{kl}-\frac{1}{2}\left({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}\right)\right)=2{C}_{10}{J}^{-\frac{2}{3}}\left(\frac{1}{9}{{\varvec{C}}}_{ij}^{-1}{{\varvec{C}}}_{kl}^{-1}{I}_{1}-\frac{1}{3}\left({{\varvec{C}}}_{kl}^{-1}{\delta }_{ij}+{\delta }_{kl}{{\varvec{C}}}_{ij}^{-1}\right)+\frac{1}{6}{I}_{1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right)\right)+2{C}_{01}{J}^{-\frac{4}{3}}\left(\frac{4}{9}{{\varvec{C}}}_{ij}^{-1}{{\varvec{C}}}_{kl}^{-1}{I}_{2}+\frac{2}{3}\left({{\varvec{C}}}_{ij}{{\varvec{C}}}_{kl}^{-1}+{{\varvec{C}}}_{kl}{{\varvec{C}}}_{ij}^{-1}\right)-\frac{2}{3}{I}_{1}\left({\delta }_{ij}{{\varvec{C}}}_{kl}^{-1}+{\delta }_{kl}{{\varvec{C}}}_{ij}^{-1}\right)+\frac{1}{3}{I}_{2}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right)-\frac{1}{2}\left({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}\right)+{\delta }_{ij}{\delta }_{kl}\right)$$
(95)
$$ \begin{gathered} \mathbb{C}_{{mnrs}}^{{hyperelastic}} = \frac{2}{J}{\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} \frac{{\partial {\varvec{S}}_{{ij}}^{{hyperelastic}} }}{{\partial {\varvec{C}}_{{kl}} }} = \frac{2}{J}\left( {2C_{{10}} J^{{ - \frac{2}{3}}} \left( {\frac{1}{9}\delta _{{mn}} \delta _{{rs}} I_{1} - \frac{1}{3}\left( {\delta _{{rs}} {\varvec{B}}_{{mn}} + \delta _{{mn}} {\varvec{B}}_{{rs}} } \right) + \frac{1}{6}I_{1} \left( {\delta _{{mr}} \delta _{{ns}} + \delta _{{ms}} \delta _{{nr}} } \right)} \right)} \right. \hfill \\ \quad \quad \quad \quad \quad + 2C_{{01}} J^{{ - \frac{4}{3}}} \left( {\frac{4}{9}\delta _{{mn}} \delta _{{rs}} I_{2} + \frac{2}{3}\left( {{\varvec{B}}_{{mk}} {\varvec{B}}_{{nk}} \delta _{{rs}} + {\varvec{B}}_{{rk}} {\varvec{B}}_{{sk}} \delta _{{mn}} } \right) - \frac{2}{3}I_{1} \left( {{\varvec{B}}_{{mn}} \delta _{{rs}} + {\varvec{B}}_{{rs}} \delta _{{mn}} } \right) + \frac{1}{3}I_{2} \left( {\delta _{{mr}} \delta _{{ns}} + \delta _{{ms}} \delta _{{nr}} } \right)} \right. \hfill \\ \left. {\left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad - \frac{1}{2}\left( {{\varvec{B}}_{{mr}} {\varvec{B}}_{{ns}} + {\varvec{B}}_{{nr}} {\varvec{B}}_{{ms}} } \right) + {\varvec{B}}_{{mn}} B_{{rs}} } \right)} \right) = \frac{2}{J}\left( {2C_{{10}} \left( {\frac{1}{9}\delta _{{mn}} \delta _{{rs}} \bar{I}_{1} - \frac{1}{3}\left( {\delta _{{rs}} \overline{{\varvec{B}}} _{{mn}} + \delta _{{mn}} \overline{{\varvec{B}}} _{{rs}} } \right) + \frac{1}{6}\bar{I}_{1} \left( {\delta _{{mr}} \delta _{{ns}} + \delta _{{ms}} \delta _{{nr}} } \right)} \right)} \right. \hfill \\ \quad \quad \quad \quad \quad + 2C_{{01}} \left( {\frac{4}{9}\delta _{{mn}} \delta _{{rs}} \bar{I}_{2} + \frac{2}{3}\left( {\overline{{\varvec{B}}} _{{mk}} \overline{{\varvec{B}}} _{{nk}} \delta _{{rs}} + \overline{{\varvec{B}}} _{{rk}} \overline{{\varvec{B}}} _{{sk}} \delta _{{mn}} } \right) - \frac{2}{3}\bar{I}_{1} \left( {\overline{{\varvec{B}}} _{{mn}} \delta _{{rs}} + \overline{{\varvec{B}}} _{{rs}} \delta _{{mn}} } \right) + \frac{1}{3}\bar{I}_{2} \left( {\delta _{{mr}} \delta _{{ns}} + \delta _{{ms}} \delta _{{nr}} } \right)} \right. \hfill \\ \left. {\left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \; - \frac{1}{2}\left( {\overline{{\varvec{B}}} _{{mr}} \overline{{\varvec{B}}} _{{ns}} + \overline{{\varvec{B}}} _{{nr}} \overline{{\varvec{B}}} _{{ms}} } \right) + \overline{{\varvec{B}}} _{{mn}} \overline{{\varvec{B}}} _{{rs}} } \right)} \right) = \frac{1}{J}\left( \begin{gathered} \frac{4}{9}\left( {C_{{10}} \bar{I}_{1} + 4C_{{01}} \bar{I}_{2} } \right)\delta _{{mn}} \delta _{{rs}} - \frac{4}{3}\left( {C_{{10}} + 2C_{{01}} \bar{I}_{1} } \right)\left( {\delta _{{rs}} \overline{{\varvec{B}}} _{{mn}} + \delta _{{mn}} \overline{{\varvec{B}}} _{{rs}} } \right) \hfill \\ \hfill \\ \end{gathered} \right. \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \frac{2}{3}\left( {C_{{10}} \bar{I}_{1} + 2C_{{01}} \bar{I}_{2} } \right)\left( {\delta _{{mr}} \delta _{{ns}} + \delta _{{ms}} \delta _{{nr}} } \right) + \frac{8}{3}C_{{01}} \left( {\overline{{\varvec{B}}} _{{mk}} \overline{{\varvec{B}}} _{{nk}} \delta _{{rs}} + \overline{{\varvec{B}}} _{{rk}} \overline{{\varvec{B}}} _{{sk}} \delta _{{mn}} } \right)\left. { + 4C_{{01}} \left( {\overline{{\varvec{B}}} _{{mn}} \overline{{\varvec{B}}} _{{rs}} - \frac{1}{2}\left( {\overline{{\varvec{B}}} _{{mr}} \overline{{\varvec{B}}} _{{ns}} + \overline{{\varvec{B}}} _{{nr}} \overline{{\varvec{B}}} _{{ms}} } \right)} \right)} \right) \hfill \\ \end{gathered} $$
(96)

1.2 Viscoelastic terms

$${\overline{W} }^{viscoelastic}=\frac{1}{4}\left({\overline{I} }_{1}-3\right)\left({\eta }_{1}{\overline{J} }_{2}+{\eta }_{2}{\overline{J} }_{4}^{2}\right)$$
(97)
$$ \begin{gathered} \frac{{\partial \bar{W}^{{viscoelastic}} }}{{\partial \mathop {\varvec{C}}\limits^{.} _{{ij}} }} = \frac{1}{4}\left( {\bar{I}_{1} - 3} \right)\left[ {\eta _{1} \frac{{\partial \bar{J}_{2} }}{{\partial \mathop {\varvec{C}}\limits^{.} _{{ij}} }} + 2\eta _{2} \bar{J}_{4} \frac{{\partial \bar{J}_{4} }}{{\partial \mathop {\varvec{C}}\limits^{.} _{{ij}} }}} \right]\frac{1}{4}\left( {\bar{I}_{1} - 3} \right)\left[ {\eta _{1} \left( {\left( {J^{{ - \frac{2}{3}}} \left( {\frac{1}{2}\left( {\delta _{{ai}} \delta _{{bj}} + \delta _{{bi}} \delta _{{aj}} } \right)\dot{\bar{C}}_{{ab}} } \right) - \frac{1}{3}J^{{ - \frac{2}{3}}} {\varvec{C}}_{{ij}}^{{ - 1}} {\varvec{C}}_{{ab}} \mathop {\overline{{\varvec{C}}} }\limits^{.} _{{ab}} } \right)} \right)} \right. \hfill \\ \left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + 2\eta _{2} \bar{J}_{4} \left( {J^{{ - \frac{4}{3}}} {\varvec{C}}_{{ij}} - \frac{1}{3}J^{{ - \frac{4}{3}}} {\varvec{C}}_{{ij}}^{{ - 1}} \left( {I_{1}^{2} - 2I_{2} } \right)} \right)} \right] = \frac{1}{4}\left( {\bar{I}_{1} - 3} \right)\left[ {\eta _{1} \left( {J^{{ - \frac{2}{3}}} \mathop {\overline{{\varvec{C}}} }\limits^{.} _{{ij}} - \frac{1}{3}{\varvec{C}}_{{ij}}^{{ - 1}} \overline{{\varvec{C}}} _{{ab}} \mathop {\overline{{\varvec{C}}} }\limits^{.} _{{ab}} } \right)} \right.\left. { + 2\eta _{2} \bar{J}_{4} J^{{ - \frac{4}{3}}} \left( {{\varvec{C}}_{{ij}} - \frac{1}{3}{\varvec{C}}_{{ij}}^{{ - 1}} \left( {I_{1}^{2} - 2I_{2} } \right)} \right)} \right] \hfill \\ \end{gathered} $$
(98)
$${{\varvec{S}}}_{ij}^{viscoelastic}=\frac{1}{2}\left({\overline{I} }_{1}-3\right)\left[{\eta }_{1}\left({J}^{-\frac{2}{3}}{\dot{\overline{{\varvec{C}}}} }_{ij}-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}{\overline{{\varvec{C}}} }_{ab}{\dot{\overline{{\varvec{C}}}} }_{ab}\right)+2{\eta }_{2}{\overline{J} }_{4}{J}^{-\frac{4}{3}}\left({{\varvec{C}}}_{ij}-\frac{1}{3}{{\varvec{C}}}_{ij}^{-1}\left({I}_{1}^{2}-2{I}_{2}\right)\right)\right]$$
(99)
$${{\varvec{\sigma}}}_{mn}^{viscoelastic}=\frac{1}{J}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{S}}}_{ij}^{viscoelastic}=\frac{1}{2J}\left({\overline{I} }_{1}-3\right)\left[{\eta }_{1}\left({{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{J}^{-\frac{2}{3}}{\dot{\overline{{\varvec{C}}}} }_{ij}-\frac{1}{3}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ij}^{-1}{\overline{{\varvec{C}}} }_{ab}{\dot{\overline{{\varvec{C}}}} }_{ab}\right)+2{\eta }_{2}{\overline{J} }_{4}{J}^{-\frac{4}{3}}\left({{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ij}-\frac{1}{3}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ij}^{-1}\left(1:\left(C.C\right)\right)\right)\right]=\frac{1}{2J}\left({\overline{I} }_{1}-3\right)\left[{\eta }_{1}\left({{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{J}^{-\frac{2}{3}}\left({J}^{-\frac{2}{3}}{\dot{{\varvec{C}}}}_{ij}-\frac{2}{3}{J}^{-\frac{5}{3}}\dot{J}{{\varvec{C}}}_{ij}\right)-\frac{1}{3}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ij}^{-1}{\overline{{\varvec{C}}} }_{ab}{\dot{\overline{{\varvec{C}}}} }_{ab}\right)+2{\eta }_{2}{\overline{J} }_{4}{J}^{-\frac{4}{3}}\left({{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ij}-\frac{1}{3}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ij}^{-1}\left(1:\left(C.C\right)\right)\right)\right]=\frac{1}{2J}\left({\overline{I} }_{1}-3\right)\left[{\eta }_{1}\left(\left(2{\overline{{\varvec{B}}} }_{mi}{{\varvec{D}}}_{ij}{\overline{{\varvec{B}}} }_{jn}-\frac{2}{3}{J}^{-1}\dot{J}{\overline{{\varvec{B}}} }_{mk}{\overline{{\varvec{B}}} }_{nk}\right)-\frac{1}{3}{\delta }_{mn}{\overline{J} }_{4}\right)+2{\eta }_{2}{\overline{J} }_{4}\left({\overline{{\varvec{B}}} }_{mk}{\overline{{\varvec{B}}} }_{nk}-\frac{1}{3}{\delta }_{mn}\left(\left({\overline{I} }_{1}^{2}-2{\overline{I} }_{2}\right)\right)\right)\right]$$
(100)
$$\frac{\partial {{\varvec{S}}}_{ij}^{viscoelastic}}{\partial {\dot{{\varvec{C}}}}_{kl}}=\frac{1}{2}\left({\overline{I} }_{1}-3\right)\left[{\eta }_{1}\left({J}^{-\frac{2}{3}}\frac{\partial {\dot{\overline{C}} }_{ij}}{\partial {\dot{{\varvec{C}}}}_{kl}}-\frac{1}{3}{C}_{ij}^{-1}\frac{\partial \left({\overline{C} }_{ab}{\dot{\overline{C}} }_{ab}\right)}{\partial {\dot{{\varvec{C}}}}_{kl}}\right)+2{\eta }_{2}{J}^{-\frac{4}{3}}\frac{\partial {\overline{J} }_{4}}{\partial {\dot{{\varvec{C}}}}_{kl}}\left({C}_{ij}-\frac{1}{3}{C}_{ij}^{-1}\left({I}_{1}^{2}-2{I}_{2}\right)\right)\right]=\frac{1}{2}\left({\overline{I} }_{1}-3\right)\left[{\eta }_{1}\left({J}^{-\frac{2}{3}}\frac{\partial {\dot{\overline{C}} }_{ij}}{\partial {\dot{{\varvec{C}}}}_{kl}}-\frac{1}{3}{C}_{ij}^{-1}{J}^{-\frac{4}{3}}\left({C}_{kl}-\frac{1}{3}{C}_{kl}^{-1}\left({I}_{1}^{2}-2{I}_{2}\right)\right)\right)+2{\eta }_{2}{J}^{-\frac{4}{3}}\left({C}_{kl}-\frac{1}{3}{C}_{kl}^{-1}\left(1:\left(C.C\right)\right)\right)\left({C}_{ij}-\frac{1}{3}{C}_{ij}^{-1}\left(1:\left(C.C\right)\right)\right)\right]=\frac{1}{2}\left({\overline{I} }_{1}-3\right)\left[{\eta }_{1}\left({J}^{-\frac{4}{3}}\frac{1}{2}\left({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}\right)-\frac{1}{3}{J}^{-\frac{4}{3}}\left({C}_{kl}^{-1}{C}_{ij}+{C}_{ij}^{-1}{C}_{kl}\right)+\frac{1}{9}{{C}_{ij}^{-1}C}_{kl}^{-1}\left({\overline{I} }_{1}^{2}-2{\overline{I} }_{2}\right)\right)+2{\eta }_{2}{J}^{-\frac{8}{3}}\left({C}_{ij}{C}_{kl}-\frac{1}{3}\left({C}_{ij}{C}_{kl}^{-1}+{{C}_{ij}^{-1}C}_{kl}\right)\left({I}_{1}^{2}-2{I}_{2}\right)+\frac{1}{9}{{C}_{ij}^{-1}C}_{kl}^{-1}{\left({I}_{1}^{2}-2{I}_{2}\right)}^{2}\right)\right]$$
(101)
$$ \begin{gathered} \mathbb{C}_{{mnrs}}^{{viscoelastic}} = \frac{2}{J}{\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} \frac{{\partial {\varvec{S}}_{{ij}}^{{viscoelastic}} }}{{\partial {\varvec{C}}_{{kl}} }} = \frac{1}{J}\left( {\bar{I}_{1} - 3} \right)\left[ {\eta _{1} \left( {J^{{ - \frac{4}{3}}} \frac{1}{2}\left( {{\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} \delta _{{ik}} \delta _{{jl}} + {\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} \delta _{{il}} \delta _{{jk}} } \right)} \right.} \right. \hfill \\ \quad\quad\quad\quad\quad\quad\quad \left. { - \frac{1}{3}J^{{ - \frac{4}{3}}} \left( {{\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} C_{{kl}}^{{ - 1}} C_{{ij}} + {\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} C_{{ij}}^{{ - 1}} C_{{kl}} } \right) + \frac{1}{9}{\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} C_{{ij}}^{{ - 1}} C_{{kl}}^{{ - 1}} \left( {1:\left( {\bar{C}.\bar{C}} \right)} \right)} \right) \hfill \\ \quad\quad\quad\quad\quad\quad\quad + 2\eta _{2} J^{{ - \frac{8}{3}}} \left( {{\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} C_{{ij}} C_{{kl}} - \frac{1}{3}\left( {{\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} C_{{ij}} C_{{kl}}^{{ - 1}} + {\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} C_{{ij}}^{{ - 1}} C_{{kl}} } \right)\left( {1:\left( {C.C} \right)} \right)} \right. \hfill \\ \quad\quad\quad\quad\quad\quad\quad \left. {\left. { + \frac{1}{9}{\varvec{F}}_{{mi}} {\varvec{F}}_{{nj}} {\varvec{F}}_{{rk}} {\varvec{F}}_{{sl}} C_{{ij}}^{{ - 1}} C_{{kl}}^{{ - 1}} \left( {1:\left( {C.C} \right)} \right)^{2} } \right)} \right] = \frac{1}{J}\left( {\bar{I}_{1} - 3} \right)\left[ {\eta _{1} \left( {J^{{ - \frac{4}{3}}} \frac{1}{2}\left( {{\varvec{B}}_{{mr}} {\varvec{B}}_{{ns}} + {\varvec{B}}_{{ms}} {\varvec{B}}_{{nr}} } \right)} \right)} \right. \hfill \\ \quad\quad\quad\quad\quad\quad\quad \left. { - \frac{1}{3}J^{{ - \frac{4}{3}}} \left( {{\varvec{B}}_{{mk}} {\varvec{B}}_{{nk}} \delta _{{rs}} + {\varvec{B}}_{{rk}} {\varvec{B}}_{{sk}} \delta _{{mn}} } \right) + \frac{1}{9}\delta _{{mn}} \delta _{{rs}} \left( {1:\left( {\bar{C}.\bar{C}} \right)} \right)} \right) \hfill \\ \quad\quad\quad\quad\quad\quad\quad \left. { + 2\eta _{2} J^{{ - \frac{8}{3}}} \left( {{\varvec{B}}_{{mk}} {\varvec{B}}_{{nk}} {\varvec{B}}_{{rl}} {\varvec{B}}_{{sl}} - \frac{1}{3}\left( {{\varvec{B}}_{{mk}} {\varvec{B}}_{{nk}} \delta _{{rs}} + \delta _{{mn}} {\varvec{B}}_{{rk}} {\varvec{B}}_{{sk}} } \right)\left( {1:\left( {C.C} \right)} \right) + \frac{1}{9}\delta _{{mn}} \delta _{{rs}} \left( {1:\left( {C.C} \right)} \right)^{2} } \right)} \right] \hfill \\ \quad\quad\quad\quad = \frac{1}{J}\left( {\bar{I}_{1} - 3} \right)\left( {\frac{{\eta _{1} }}{2}} \right.\left. {\left( {\overline{{\varvec{B}}} _{{mr}} \overline{{\varvec{B}}} _{{ns}} } \right. + \overline{{\varvec{B}}} _{{ms}} \overline{{\varvec{B}}} _{{nr}} } \right) + 2\eta _{2} \overline{{\varvec{B}}} _{{mk}} \overline{{\varvec{B}}} _{{nk}} \overline{{\varvec{B}}} _{{rl}} \overline{{\varvec{B}}} _{{sl}} - \frac{1}{3}\left( {\overline{{\varvec{B}}} _{{mk}} \overline{{\varvec{B}}} _{{nk}} \delta _{{rs}} } \right. \hfill \\ \quad\quad\quad\quad\quad\quad\quad \left. { + \overline{{\varvec{B}}} _{{rk}} \overline{{\varvec{B}}} _{{sk}} \delta _{{mn}} } \right)\left( {\eta _{1} + 2\eta _{2} \left( {\bar{I}_{1}^{2} - 2\bar{I}_{2} } \right)} \right)\left. { + \frac{1}{9}\delta _{{mn}} \delta _{{rs}} \left( {\bar{I}_{1}^{2} - 2\bar{I}_{2} + 2\eta _{2} \left( {\bar{I}_{1}^{2} - 2\bar{I}_{2} } \right)^{2} } \right)} \right) \hfill \\ \end{gathered} $$
(102)

1.3 Volumetric terms

$${W}^{vol}=\frac{K}{2}{\left(J-1\right)}^{2}$$
(103)
$$\frac{\partial {W}^{vol}}{\partial {{\varvec{C}}}_{ij}}=K\left(J-1\right)J{{\varvec{C}}}_{ij}^{-1}$$
(104)
$${{\varvec{S}}}_{ij}^{vol}=K\left(J-1\right)J{{\varvec{C}}}_{ij}^{-1}$$
(105)
$${{\varvec{\sigma}}}_{mn}^{vol}=\frac{1}{J}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{S}}}_{ij}^{vol}={{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}K\left(J-1\right){{\varvec{C}}}_{ij}^{-1}=K\left(J-1\right){\delta }_{mn}$$
(106)
$$\frac{\partial {{\varvec{S}}}_{ij}^{vol}}{\partial {{\varvec{C}}}_{kl}}=\frac{\partial \left(K\left(J-1\right)J{{\varvec{C}}}_{ij}^{-1}\right)}{\partial {{\varvec{C}}}_{kl}}=K\left(\left(2J-1\right){{\varvec{C}}}_{ij}^{-1}\frac{\partial J}{\partial {{\varvec{C}}}_{kl}}+\left(J-1\right)J\frac{\partial {{\varvec{C}}}_{ij}^{-1}}{\partial {{\varvec{C}}}_{kl}}\right)=\frac{KJ}{2}\left(\left(2J-1\right){{\varvec{C}}}_{ij}^{-1}{{\varvec{C}}}_{kl}^{-1}-\left(J-1\right)\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right)\right)$$
(107)
$${\mathbb{C}}_{mnrs}^{vol}=\frac{2}{J}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{F}}}_{rk}{{\varvec{F}}}_{sl}\left(\frac{KJ}{2}\left(\left(2J-1\right){{\varvec{C}}}_{ij}^{-1}{{\varvec{C}}}_{kl}^{-1}-\left(J-1\right)\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right)\right)\right)=K\left(\left(2J-1\right){\delta }_{mn}{\delta }_{rs}-\left(J-1\right)\left({\delta }_{mr}{\delta }_{ns}+{\delta }_{ms}{\delta }_{nr}\right)\right)$$
(108)

1.4 Anisotropic terms

$${W}^{anisotropic}=\frac{{k}_{1}^{\left(1\right)}}{2{k}_{2}^{\left(1\right)}}\left\{exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]-1\right\}+\frac{{k}_{1}^{\left(2\right)}}{2{k}_{2}^{\left(2\right)}}\left\{exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]-1\right\}$$
(109)
$$\frac{\partial {W}^{anisotropic}}{\partial {{\varvec{C}}}_{ij}}=\frac{{k}_{1}^{\left(1\right)}}{2{k}_{2}^{\left(1\right)}}\frac{2\left({I}_{4}-1\right){k}_{2}^{\left(1\right)}\partial \left({I}_{4}-1\right)}{\partial {{\varvec{C}}}_{ij}}\left\{exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right\}+\frac{{k}_{1}^{\left(2\right)}}{2{k}_{2}^{\left(2\right)}}\frac{2\left({I}_{6}-1\right){k}_{2}^{\left(1\right)}\partial \left({I}_{6}-1\right)}{\partial {{\varvec{C}}}_{ij}}\left\{exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right\}={k}_{1}^{\left(1\right)}{{\varvec{A}}}_{i}^{(1)}{{\varvec{A}}}_{j}^{(1)}\left({I}_{4}-1\right)\left\{exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right\}+{k}_{1}^{\left(2\right)}{{\varvec{A}}}_{i}^{(2)}{{\varvec{A}}}_{j}^{(2)}\left({I}_{6}-1\right)\left\{exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right\}$$
(110)
$${{\varvec{S}}}_{ij}^{anisotropic}=2{k}_{1}^{\left(1\right)}{{\varvec{A}}}_{i}^{(1)}{{\varvec{A}}}_{j}^{(1)}\left({I}_{4}-1\right)\left\{exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right\}+2{k}_{1}^{\left(2\right)}{{\varvec{A}}}_{i}^{(2)}{{\varvec{A}}}_{j}^{(2)}\left({I}_{6}-1\right)\left\{exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right\}$$
(111)
$${{\varvec{\sigma}}}_{mn}^{anisotropic}=\frac{1}{J}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{S}}}_{ij}^{anisotropic}=\frac{2}{J}\left({k}_{1}^{\left(1\right)}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{A}}}_{i}^{(1)}{{\varvec{A}}}_{j}^{(1)}\left({I}_{4}-1\right)\left\{exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right\}+{k}_{1}^{\left(2\right)}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{A}}}_{i}^{(2)}{{\varvec{A}}}_{j}^{(2)}\left({I}_{6}-1\right)\left\{exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right\}\right)=\frac{2}{J}\left({k}_{1}^{\left(1\right)}{{\varvec{a}}}_{m}^{(1)}{{\varvec{a}}}_{n}^{(1)}\left({I}_{4}-1\right)\left\{exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right\}+{k}_{1}^{\left(2\right)}{{\varvec{a}}}_{m}^{(2)}{{\varvec{a}}}_{n}^{(2)}\left({I}_{6}-1\right)\left\{exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right\}\right)$$
(112)
$$\frac{\partial {{\varvec{S}}}_{ij}^{anisotropic}}{\partial {{\varvec{C}}}_{kl}}=2{k}_{1}^{\left(1\right)}{{\varvec{A}}}_{i}^{(1)}{{\varvec{A}}}_{j}^{(1)}\left(\left({I}_{4}-1\right)\frac{\partial \left\{exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right\}}{\partial {{\varvec{C}}}_{kl}}+\frac{\partial {I}_{4}}{\partial {{\varvec{C}}}_{kl}}\left\{exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right\}\right)+2{k}_{1}^{\left(2\right)}{{\varvec{A}}}_{i}^{(2)}{{\varvec{A}}}_{j}^{(2)}\left(\left({I}_{6}-1\right)\frac{\partial \left\{exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right\}}{\partial {{\varvec{C}}}_{kl}}+\frac{\partial {I}_{6}}{\partial {{\varvec{C}}}_{kl}}\left\{exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right\}\right)=2{k}_{1}^{\left(1\right)}{{\varvec{A}}}_{i}^{(1)}{{\varvec{A}}}_{j}^{(1)}\left((2{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}+1)\frac{\partial {I}_{4}}{\partial {{\varvec{C}}}_{kl}}exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right)+2{k}_{1}^{\left(2\right)}{{\varvec{A}}}_{i}^{(2)}{{\varvec{A}}}_{j}^{(2)}\left((2{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}+1)\frac{\partial {I}_{6}}{\partial {{\varvec{C}}}_{kl}}exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right)=2{k}_{1}^{\left(1\right)}{{\varvec{A}}}_{i}^{(1)}{{\varvec{A}}}_{j}^{(1)}{{\varvec{A}}}_{k}^{(1)}{{\varvec{A}}}_{l}^{(1)}\left((2{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}+1)exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right)+2{k}_{1}^{\left(2\right)}{{\varvec{A}}}_{i}^{(2)}{{\varvec{A}}}_{j}^{(2)}{{\varvec{A}}}_{k}^{(2)}{{\varvec{A}}}_{l}^{(2)}\left((2{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}+1)exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right)$$
(113)
$${\mathbb{C}}_{mnrs}^{anisotropic}=\frac{2}{J}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{F}}}_{rk}{{\varvec{F}}}_{sl}\left(2{k}_{1}^{\left(1\right)}{{\varvec{A}}}_{i}^{(1)}{{\varvec{A}}}_{j}^{(1)}{{\varvec{A}}}_{k}^{(1)}{{\varvec{A}}}_{l}^{(1)}\left((2{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}+1)exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right)+2{k}_{1}^{\left(2\right)}{{\varvec{A}}}_{i}^{(2)}{{\varvec{A}}}_{j}^{(2)}{{\varvec{A}}}_{k}^{(2)}{{\varvec{A}}}_{l}^{(2)}\left((2{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}+1)exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right)\right)=\frac{4}{J}\left({k}_{1}^{\left(1\right)}{{\varvec{a}}}_{m}^{(1)}{{\varvec{a}}}_{n}^{(1)}{{\varvec{a}}}_{r}^{(1)}{{\varvec{a}}}_{s}^{(1)}\left((2{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}+1)exp \left[{k}_{2}^{\left(1\right)}{\left({I}_{4}-1\right)}^{2}\right]\right)+{k}_{1}^{\left(2\right)}{{\varvec{a}}}_{r}^{(2)}{{\varvec{a}}}_{s}^{(2)}{{\varvec{a}}}_{m}^{(2)}{{\varvec{a}}}_{n}^{(2)}\left((2{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}+1)exp \left[{k}_{2}^{\left(2\right)}{\left({I}_{6}-1\right)}^{2}\right]\right)\right)$$
(114)

1.5 Dielectric terms

$${W}^{electric}=-\frac{1}{2}\varepsilon J{{\varvec{C}}}_{pq}^{-1}{\mathbb{E}}_{p}{\mathbb{E}}_{q}$$
(115)
$$\frac{\partial {W}^{electric}}{\partial {{\varvec{C}}}_{ij}}=-\frac{1}{2}\varepsilon {\mathbb{E}}_{p}{\mathbb{E}}_{q}\frac{\partial \left(J{{\varvec{C}}}_{pq}^{-1}\right)}{\partial {{\varvec{C}}}_{ij}}=-\frac{1}{2}\varepsilon {\mathbb{E}}_{p}{\mathbb{E}}_{q}\left(J\frac{\partial {{\varvec{C}}}_{pq}^{-1}}{\partial {{\varvec{C}}}_{ij}}+{{\varvec{C}}}_{pq}^{-1}\frac{\partial J}{\partial {{\varvec{C}}}_{ij}}\right)=-\frac{1}{2}\varepsilon {\mathbb{E}}_{p}{\mathbb{E}}_{q}\left(-\frac{1}{2}J\left({{\varvec{C}}}_{ip}^{-1}{{\varvec{C}}}_{jq}^{-1}+{{\varvec{C}}}_{iq}^{-1}{{\varvec{C}}}_{jp}^{-1}\right)+\frac{1}{2}J{{\varvec{C}}}_{pq}^{-1}{{\varvec{C}}}_{ij}^{-1}\right)=-\frac{1}{4}\varepsilon J{\mathbb{E}}_{p}{\mathbb{E}}_{q}\left({{\varvec{C}}}_{pq}^{-1}{{\varvec{C}}}_{ij}^{-1}-\left({{\varvec{C}}}_{ip}^{-1}{{\varvec{C}}}_{jq}^{-1}+{{\varvec{C}}}_{iq}^{-1}{{\varvec{C}}}_{jp}^{-1}\right)\right)$$
(116)
$${{\varvec{S}}}_{ij}^{electric}=-\frac{1}{2}\varepsilon J{\mathbb{E}}_{p}{\mathbb{E}}_{q}\left({{\varvec{C}}}_{pq}^{-1}{{\varvec{C}}}_{ij}^{-1}-\left({{\varvec{C}}}_{ip}^{-1}{{\varvec{C}}}_{jq}^{-1}+{{\varvec{C}}}_{iq}^{-1}{{\varvec{C}}}_{jp}^{-1}\right)\right)$$
(117)
$${{\varvec{\sigma}}}_{mn}^{electric}=\frac{1}{J}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{S}}}_{ij}^{electric}=-\frac{1}{2}\varepsilon {\mathbb{E}}_{p}{\mathbb{E}}_{q}\left({{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{pq}^{-1}{{\varvec{C}}}_{ij}^{-1}-\left({{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ip}^{-1}{{\varvec{C}}}_{jq}^{-1}+{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{iq}^{-1}{{\varvec{C}}}_{jp}^{-1}\right)\right)=-\frac{1}{2}\varepsilon \left({\delta }_{mn}{{\varvec{C}}}_{pq}^{-1}{\mathbb{E}}_{p}{\mathbb{E}}_{q}-\left({{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{ip}^{-1}{{\varvec{C}}}_{jq}^{-1}{\mathbb{E}}_{p}{\mathbb{E}}_{q}+{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{C}}}_{iq}^{-1}{{\varvec{C}}}_{jp}^{-1}{\mathbb{E}}_{p}{\mathbb{E}}_{q}\right)\right)=-\frac{1}{2}\varepsilon \left({\delta }_{mn}{\mathbbm{e}}_{l}{\mathbbm{e}}_{l}-2{\mathbbm{e}}_{m}{\mathbbm{e}}_{n}\right)=\varepsilon \left({\mathbbm{e}}_{m}{\mathbbm{e}}_{n}-\frac{1}{2}{\delta }_{mn}{\mathbbm{e}}_{l}{\mathbbm{e}}_{l}\right)$$
(118)
$$\frac{\partial {{\varvec{S}}}_{ij}^{electric}}{\partial {{\varvec{C}}}_{kl}}=-\frac{1}{2}\varepsilon J{\mathbb{E}}_{p}{\mathbb{E}}_{q}\left(\frac{\partial \left({{\varvec{C}}}_{pq}^{-1}{{\varvec{C}}}_{ij}^{-1}\right)}{\partial {{\varvec{C}}}_{kl}}-\left(\frac{\partial \left({{\varvec{C}}}_{ip}^{-1}{{\varvec{C}}}_{jq}^{-1}\right)}{\partial {{\varvec{C}}}_{kl}}+\frac{\partial \left({{\varvec{C}}}_{iq}^{-1}{{\varvec{C}}}_{jp}^{-1}\right)}{\partial {{\varvec{C}}}_{kl}}\right)\right)=-\frac{1}{2}\varepsilon J{\mathbb{E}}_{p}{\mathbb{E}}_{q}\left(\frac{\partial {{\varvec{C}}}_{pq}^{-1}}{\partial {{\varvec{C}}}_{kl}}{{\varvec{C}}}_{ij}^{-1}+\frac{\partial {{\varvec{C}}}_{ij}^{-1}}{\partial {{\varvec{C}}}_{kl}}{{\varvec{C}}}_{pq}^{-1}-\left(\frac{\partial {{\varvec{C}}}_{ip}^{-1}}{\partial {{\varvec{C}}}_{kl}}{{\varvec{C}}}_{jq}^{-1}+\frac{\partial {{\varvec{C}}}_{jq}^{-1}}{\partial {{\varvec{C}}}_{kl}}{{\varvec{C}}}_{ip}^{-1}+\frac{\partial {{\varvec{C}}}_{iq}^{-1}}{\partial {{\varvec{C}}}_{kl}}{{\varvec{C}}}_{jp}^{-1}+\frac{\partial {{\varvec{C}}}_{jp}^{-1}}{\partial {{\varvec{C}}}_{kl}}{{\varvec{C}}}_{iq}^{-1}\right)\right)=\frac{1}{4}\varepsilon J{\mathbb{E}}_{p}{\mathbb{E}}_{q}\left({{\varvec{C}}}_{ij}^{-1}\left({{\varvec{C}}}_{pk}^{-1}{{\varvec{C}}}_{ql}^{-1}+{{\varvec{C}}}_{pl}^{-1}{{\varvec{C}}}_{qk}^{-1}\right)+{{\varvec{C}}}_{pq}^{-1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right)-\left({{\varvec{C}}}_{jq}^{-1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{pl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{pk}^{-1}\right)+{{\varvec{C}}}_{ip}^{-1}\left({{\varvec{C}}}_{jk}^{-1}{{\varvec{C}}}_{ql}^{-1}+{{\varvec{C}}}_{jl}^{-1}{{\varvec{C}}}_{qk}^{-1}\right)+{{\varvec{C}}}_{jp}^{-1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{ql}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{qk}^{-1}\right)+{{\varvec{C}}}_{iq}^{-1}\left({{\varvec{C}}}_{jk}^{-1}{{\varvec{C}}}_{pl}^{-1}+{{\varvec{C}}}_{jl}^{-1}{{\varvec{C}}}_{pk}^{-1}\right)\right)\right)$$
(119)
$${\mathbb{C}}_{mnrs}^{electric}=\frac{2}{J}{{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{F}}}_{rk}{{\varvec{F}}}_{sl}\left(\frac{1}{4}\varepsilon J{\mathbb{E}}_{p}{\mathbb{E}}_{q}\left({{\varvec{C}}}_{ij}^{-1}\left({{\varvec{C}}}_{pk}^{-1}{{\varvec{C}}}_{ql}^{-1}+{{\varvec{C}}}_{pl}^{-1}{{\varvec{C}}}_{qk}^{-1}\right)+{{\varvec{C}}}_{pq}^{-1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right)-\left({{\varvec{C}}}_{jq}^{-1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{pl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{pk}^{-1}\right)+{{\varvec{C}}}_{ip}^{-1}\left({{\varvec{C}}}_{jk}^{-1}{{\varvec{C}}}_{ql}^{-1}+{{\varvec{C}}}_{jl}^{-1}{{\varvec{C}}}_{qk}^{-1}\right)+{{\varvec{C}}}_{jp}^{-1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{ql}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{qk}^{-1}\right)+{{\varvec{C}}}_{iq}^{-1}\left({{\varvec{C}}}_{jk}^{-1}{{\varvec{C}}}_{pl}^{-1}+{{\varvec{C}}}_{jl}^{-1}{{\varvec{C}}}_{pk}^{-1}\right)\right)\right)\right)=\frac{1}{2}\varepsilon {{\varvec{F}}}_{mi}{{\varvec{F}}}_{nj}{{\varvec{F}}}_{rk}{{\varvec{F}}}_{sl}\left({\mathbb{E}}_{p}{\mathbb{E}}_{q}\left({{\varvec{C}}}_{ij}^{-1}\left({{\varvec{C}}}_{pk}^{-1}{{\varvec{C}}}_{ql}^{-1}+{{\varvec{C}}}_{pl}^{-1}{{\varvec{C}}}_{qk}^{-1}\right)+{{\varvec{C}}}_{pq}^{-1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{jl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{jk}^{-1}\right)-\left({{\varvec{C}}}_{jq}^{-1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{pl}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{pk}^{-1}\right)+{{\varvec{C}}}_{ip}^{-1}\left({{\varvec{C}}}_{jk}^{-1}{{\varvec{C}}}_{ql}^{-1}+{{\varvec{C}}}_{jl}^{-1}{{\varvec{C}}}_{qk}^{-1}\right)+{{\varvec{C}}}_{jp}^{-1}\left({{\varvec{C}}}_{ik}^{-1}{{\varvec{C}}}_{ql}^{-1}+{{\varvec{C}}}_{il}^{-1}{{\varvec{C}}}_{qk}^{-1}\right)+{{\varvec{C}}}_{iq}^{-1}\left({{\varvec{C}}}_{jk}^{-1}{{\varvec{C}}}_{pl}^{-1}+{{\varvec{C}}}_{jl}^{-1}{{\varvec{C}}}_{pk}^{-1}\right)\right)\right)\right)=\varepsilon \left({\mathbbm{e}}_{r}{\mathbbm{e}}_{s}{\delta }_{mn}+{\mathbbm{e}}_{m}{\mathbbm{e}}_{n}{\delta }_{rs}-\left({\mathbbm{e}}_{r}{\mathbbm{e}}_{n}{\delta }_{ms}+{\mathbbm{e}}_{n}{\mathbbm{e}}_{s}{\delta }_{mr}+{\mathbbm{e}}_{m}{\mathbbm{e}}_{r}{\delta }_{ns}+{\mathbbm{e}}_{m}{\mathbbm{e}}_{s}{\delta }_{nr}\right)\right)$$
(120)

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Majidi, M., Asgari, M. Rate-dependent electromechanical behavior of anisotropic fiber-reinforced dielectric elastomer based on a nonlinear continuum approach: modeling and implementation. Eur. Phys. J. Plus 138, 73 (2023). https://doi.org/10.1140/epjp/s13360-023-03688-w

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