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Theoretical framework of modeling of ionic EAPs within the Theory of Porous Media

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Abstract

A thermo-electromechanical formulation for the description of ionic electroactive polymers is derived within the framework of the Theory of Porous Media. The model consists of an electrically charged porous solid saturated with an ionic solution. The saturated porous medium is assumed to be incompressible. Different constituents following different kinematic paths are considered such as solid, fluid, anions, cations and free charges. The electromechanical and the electrodynamic field equations are discussed. Based on the second law of thermodynamics, a consistent model is developed. With respect to the closure problem of the model, the needed constitutive relations and evolution equations are presented.

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Abbreviations

\(\mathbf{b}\) :

Gravity (\( \mathrm{m}/\mathrm{s}^2\))

\(\mathbf{b}^\alpha _\mathrm{e}= \rho ^\alpha _\mathrm{e}\, \mathbf{e}^\alpha _\mathrm{e}+ ( \mathrm{grad}\,\mathbf{e}^\alpha _\mathrm{e}) \, \mathbf{p}^\alpha _\mathrm{e}\) :

Electrical body force (\( \mathrm{N}/\mathrm{m}^3\))

\(\mathbf{B}_\alpha = \mathbf{F}_\alpha \, \mathbf{F}_\alpha ^\mathrm{T}\) :

Left Cauchy–Green deformation tensor

\(\mathbf{C}_\alpha = \mathbf{F}_\alpha ^\mathrm{T}\, \mathbf{F}_\alpha \) :

Right Cauchy–Green deformation tensor

\(\mathrm{c}^\alpha _\mathrm{m}\) :

Molar concentration (\(\mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{m}^3\))

\(\mathrm{c}^\alpha _\mathrm{e}\) :

Coefficient of electrostriction (\(\mathrm{C}/\mathrm{V}\, \mathrm{m}\))

\(\mathbf{c}^\alpha _\mathrm{e}\) :

Electro momentum couple density (\(\mathrm{N}/\mathrm{m}^2\))

\(\mathbf{d}^\alpha _\mathrm{e}\) :

Electric displacement (\(\mathrm{C}/\mathrm{m}^2\))

\(\mathbf{D}_\alpha = \displaystyle {\frac{1}{2}} \, ( \, \mathbf{L}_\alpha + \mathbf{L}^\mathrm{T}_\alpha \, )\) :

Symmetric part of the spatial velocity gradient (\(1/\mathrm{s}\))

\(\mathbf{e}^\alpha _{\mathrm{e}0\alpha }= - \, \mathbf{F}^\mathrm{T}_\alpha \, \mathrm{grad}\,_\alpha \phi ^\alpha _\mathrm{e}\) :

Electric field, reference configuration of \(\varphi ^\alpha \) (\(\mathrm{V}/\mathrm{m}\))

\(\mathbf{e}^\alpha _\mathrm{e}= - \, \mathrm{grad}\,\phi ^\alpha _\mathrm{e}\) :

Electric field (\(\mathrm{V}/\mathrm{m}\))

\(\hat{\mathrm{e}}^\alpha \) :

Total production term, energy (\(\mathrm{N}/\mathrm{m}^2 \, \mathrm{s}\))

\(\mathbf{F}^\alpha = \displaystyle {\frac{\partial \varvec{\chi }_\alpha }{\partial \mathbf{X}_\alpha }} \, = \, \mathrm{Grad}\,_\alpha \mathbf{x}\) :

Deformation gradient

\(\mathrm{g}^\alpha _\mathrm{e}= \psi ^\alpha - \displaystyle {\frac{1}{\rho ^\alpha }} \, \mathbf{e}^\alpha _\mathrm{e}\, \cdot \, \mathbf{p}^\alpha _\mathrm{e}\) :

Specific electric Gibbs energy function (\(\mathrm{J}/ \mathrm{k}\mathrm{g}\))

\(\hat{\mathbf{h}}^\alpha \) :

Total production term, moment of momentum (\(\mathrm{N}/\mathrm{m}^2\))

\(\mathbf{I}\) :

Identity tensor

\(\mathbf{j}^\alpha _\mathrm{e}\) :

Electrical current density (\(\mathrm{A}/\mathrm{m}^2\))

\(\mathrm{J}_\alpha = \mathrm{det}\,\mathbf{F}_\alpha \) :

Jacobian, determinant of the deformation gradient

\(\mathrm{k}^\alpha \) :

Compression modulus (\(\mathrm{N}/\mathrm{m}^2\))

\(\mathbf{K}_\alpha = \displaystyle {\frac{1}{2}} \, ( \, \mathbf{B}_\alpha - \mathbf{I}\, )\) :

Karni–Reiner strain tensor

\(\mathbf{L}_\alpha = \displaystyle {\frac{\partial \mathbf{x}^\prime _\alpha }{\partial \mathbf{x}}} \, = \, \mathrm{grad}\,\mathbf{x}^\prime _\alpha \) :

Spatial velocity gradient (\(1/\mathrm{s}\))

\(\mathrm{m}^\alpha _\mathrm{m}\) :

Molar mass (\(\mathrm{k}\mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}\))

\(\hat{\mathbf{m}}^\alpha \) :

Direct production term, moment of momentum (\(\mathrm{N}/\mathrm{m}^2\))

\(\mathrm{n}^\alpha _0\) :

Initial volume fraction

\(\mathrm{n}^\alpha \) :

Volume fraction

\(\mathbf{n}\) :

normal Vector

\(\hat{\mathbf{p}}^\alpha \) :

Direct production term, momentum (\(\mathrm{N}/\mathrm{m}^3\))

\(\hat{\mathbf{p}}^\alpha _\mathrm{E}\) :

Effective direct production term, momentum (\(\mathrm{N}/\mathrm{m}^3\))

\(\mathbf{p}^\alpha _\mathrm{e}\) :

Polarization (\(\mathrm{C}/\mathrm{m}^2\))

\(\mathbf{q}^\alpha \) :

Heat influx vector (\(\mathrm{J}/\mathrm{m}^2 \, \mathrm{s}\))

\(\mathrm{r}^\alpha \) :

External heat supply (\(\mathrm{J}/\mathrm{k}\mathrm{g}\, \mathrm{s}\))

\(\mathrm{R}\) :

Universal gas constant (\(\mathrm{J}/\mathrm{K}\, \mathrm{m}\mathrm{o}\mathrm{l}\))

\(\hat{\mathbf{s}}^\alpha \) :

Total production term, momentum (\(\mathrm{N}/\mathrm{m}^3\))

\(\mathbf{S}^\alpha \) :

Second Piola-Kirchhoff stress tensor (\(\mathrm{N}/\mathrm{m}^2\))

\(\mathbf{t}^\alpha \) :

Traction vector (\(\mathrm{N}/\mathrm{m}^2\))

\(\mathbf{T}^\alpha = \displaystyle {\frac{1}{\mathrm{J}_\alpha }} \, \mathbf{F}_\alpha \, \mathbf{S}^\alpha \, \mathbf{F}_\alpha ^\mathrm{T}\) :

Cauchy stress tensor (\(\mathrm{N}/\mathrm{m}^2\))

\(\mathbf{T}^\alpha _\mathrm{E}\) :

Effective Cauchy stress tensor (\(\mathrm{N}/\mathrm{m}^2\))

\(\mathbf{w}_{\alpha \mathrm{S}}\) :

Seepage velocity (\(\mathrm{m}/\mathrm{s}\))

\(\mathbf{W}_\alpha = \displaystyle {\frac{1}{2}} \, ( \, \mathbf{L}_\alpha - \mathbf{L}^\mathrm{T}_\alpha \, )\) :

Skew-symmetric part of the spatial velocity gradient (\(1/\mathrm{s}\))

\(\mathbf{x}= \mathbf{x}\, ( \, \mathbf{X}_\alpha \, , \, t \, )\) :

Position vector, present configuration (\(\mathrm{m}\))

\(\mathbf{x}^\prime _\alpha \) :

Velocity (\(\mathrm{m}/\mathrm{s}\))

\(\mathbf{x}^{\prime \prime }_\alpha \) :

acceleration (\(\mathrm{m}/\mathrm{s}^2\))

\(\mathbf{X}_\alpha = \mathbf{X}_\alpha \, ( \, \mathrm{X}_\alpha \, , \, t_0 \, )\) :

Position vector, reference configuration (\(\mathrm{m}\))

\(\alpha ^\alpha _{\varTheta }\) :

Heat expansion coefficient (\(1/\mathrm{K}\))

\(\varepsilon ^\alpha = \psi ^\alpha + {\varTheta }^\alpha \, \eta ^\alpha \) :

Specific internal energy (\(\mathrm{J}/\mathrm{k}\mathrm{g}\))

\(\varepsilon ^\alpha _\mathrm{e}\) :

Specific electric energy (\(\mathrm{N}/\mathrm{m}^2 \, \mathrm{s}\))

\(\hat{\varepsilon }^\alpha \) :

Direct production term, energy (\(\mathrm{N}/\mathrm{m}^2 \, \mathrm{s}\))

\(\eta ^\alpha \) :

Specific entropy (\(\mathrm{J}/\mathrm{K}\, \mathrm{k}\mathrm{g}\))

\({\varTheta }^\alpha _0\) :

Initial temperature (\(\mathrm{K}\))

\({\varTheta }^\alpha \) :

Absolute temperature (\(\mathrm{K}\))

\(\hat{\kappa }^\alpha _\mathrm{e}\) :

Total production, electric flux (\(\mathrm{C}/\mathrm{m}^3\))

\(\lambda ^\alpha \) :

Lam\(\acute{\mathrm{e}}\) constants (\(\mathrm{N}/\mathrm{m}^2\))

\(\mu ^\alpha \) :

Lam\(\acute{\mathrm{e}}\) constants (\(\mathrm{N}/\mathrm{m}^2\))

\(\mu ^\alpha _{\mathrm{c}\mathrm{p}}\) :

Chemical potential (\(\mathrm{J}/\mathrm{k}\mathrm{g}\))

\(\mu ^\alpha _{\mathrm{m}0}\) :

Chemical potential, molar (\(\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}\))

\({\varvec{\pi }}^\alpha _\mathrm{e}= 1 / \rho ^\alpha \, \mathbf{p}^\alpha _\mathrm{e}\) :

Polarization per unit mass (\(\mathrm{C}\,\mathrm{m}/\mathrm{k}\mathrm{g}\))

\(\rho ^{\alpha \mathrm{R}}\) :

Real density (\(\mathrm{k}\mathrm{g}/\mathrm{m}^3\))

\(\rho ^\alpha = \mathrm{n}^\alpha \, \rho ^{\alpha \mathrm{R}}\) :

partial density (\(\mathrm{k}\mathrm{g}/\mathrm{m}^3\))

\(\hat{\rho }^\alpha \) :

Total production term, mass (\(\mathrm{k}\mathrm{g}/\mathrm{m}^3 \, \mathrm{s}\))

\(\hat{\rho }^\alpha _\mathrm{e}\) :

Production of electric charge (\(\mathrm{A}/\mathrm{m}^3\))

\(\rho ^\alpha _\mathrm{e}\) :

Electric charge density (\(\mathrm{C}/\mathrm{m}^3\))

\(\phi ^\alpha _\mathrm{e}\) :

Electrical potential (\(\mathrm{V}\))

\({\varvec{\chi }}_\alpha \) :

Function of motion (\(\mathrm{m}\))

\(\psi ^\alpha \) :

Specific Helmholtz free energy (\(\mathrm{J}/\mathrm{k}\mathrm{g}\))

\(\mathcal{P}\) :

Lagrangean multiplier (pressure) (\(\mathrm{N}/\mathrm{m}^2\))

\(\mathcal{E}\) :

Lagrangean multiplier (related to electrical potential) (\(\mathrm{V}\))

\(\mathcal{SC}\) :

Set of constitutive relations

\(\mathcal{SP}\) :

Set of process variables

\(\mathcal{SU}\) :

Set of unknown fields

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Acknowledgments

This work has been supported by the German Research Society (DFG) within the Priority Program SPP 1713 “Modeling of Ionic Electroactive Polymers–Consistent Formulation of the thermo-electro-chemo-mechanical coupling effects and Finite-Element Discretization.”

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  1. Institute of Mechanics, University of Duisburg-Essen, Universitätsstr. 15, 45141, Essen, Germany

    J. Bluhm, S. Serdas & J. Schröder

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Bluhm, J., Serdas, S. & Schröder, J. Theoretical framework of modeling of ionic EAPs within the Theory of Porous Media. Arch Appl Mech 86, 3–19 (2016). https://doi.org/10.1007/s00419-015-1110-8

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