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Nonlinear flexoelectricity in extended thermodynamics

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Abstract

A nonlinear phenomenological model of flexoelectricity in thermoelastic solids is presented within the frame of continuum mechanics and extended thermodynamics, incorporating the quasi-electrostatic approximation where the time derivative of the electric displacement vector can be neglected in Maxwell–Ampère’s law. An expression for the energy flux is proposed, including many internal variables. An advantage of the proposed model is that it presents a unified approach to study several thermo-electromechanical couplings, including electrostriction, piezoelectricity, dependence of entropy and spontaneous polarization on flexoelectricity, among others. The model may be of interest in revealing the effects of such couplings on the properties of polarizable media, in particular ferroelectrics, when simple boundary conditions prevail. For a fully dynamic description of flexoelectricity, Hamilton’s principle and the variational principle for external forces should be used instead.

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Abbreviations

\({{{x}}}_{{{i}}},{{i}}=1,2,3\) :

Eulerian coordinates or \({K}_{t}\) refer to current configuration

\({\mathcal{X}}_{{{K}}},{{K}}=1,2,3\) :

Lagrangian coordinates or \({K}_{R}\) refer to reference configuration

\({{{\delta}}}_{{{i}}{{j}}}\) & \({{{\delta}}}_{{{K}}{{L}}}\) :

Krönecker symbols in special and material coordinates

\({{{E}}}_{{{i}}}\) & \({\mathbb{E}}_{{{K}}}\) :

Electric field in Eulerian and Lagrangian configuration

\({{{D}}}_{{{i}}}\) & \({\mathcal{D}}_{{{K}}}\) :

Electric displacement in Eulerian and Lagrangian configuration

\({{{P}}}_{{{i}}}\) & \({\mathcal{P}}_{{{K}}}\) :

Polarization in Eulerian and Lagrangian configuration

\(\boxdot_{\left({{K}}{{L}}\right)}\) :

A parenthesis enclosing two tensorial indices \(K\) and \(L\) means symmetry

\({{{u}}}_{{{k}}}\) & \({\mathcal{U}}_{{{K}}}\) :

Displacement field in Eulerian and Lagrangian configuration

\({\mathcal{E}}_{\left({{K}}{{L}}\right)}\) & \({{{\gamma}}}_{{{I}}\left({{K}}{{L}}\right)}\) :

Strain tensor and its derivative in Lagrangian configuration

\({{{q}}}_{{{i}}}\) & \({\mathcal{Q}}_{{{K}}}\) :

Heat flux vector in Eulerian and Lagrangian configuration

\({{\theta}}\) & \({{\Theta}}\) :

Temperature in Eulerian and Lagrangian configuration

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Authors and Affiliations

  1. Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa, 31982, Saudi Arabia

    A. R. El-Dhaba

  2. Department of Mathematics, Faculty of Science, Damanhour University, P.O. Box 22511, Damanhour, El behira, Egypt

    A. R. El-Dhaba

  3. Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

    M. S. Abou-Dina & A. F. Ghaleb

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  1. A. R. El-Dhaba
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  3. A. F. Ghaleb
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El-Dhaba, A.R., Abou-Dina, M.S. & Ghaleb, A.F. Nonlinear flexoelectricity in extended thermodynamics. Arch Appl Mech (2024). https://doi.org/10.1007/s00419-024-02554-0

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