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Multiphysics and Thermodynamic Formulations for Equilibrium and Non-equilibrium Interactions: Non-linear Finite Elements Applied to Multi-coupled Active Materials

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Abstract

Combining several theories this paper presents a general multiphysics framework applied to the study of coupled and active materials, considering mechanical, electric, magnetic and thermal fields. The framework is based on thermodynamic equilibrium and non-equilibrium interactions, both linked by a two-temperature model. The multi-coupled governing equations are obtained from energy, momentum and entropy balances; the total energy is the sum of thermal, mechanical and electromagnetic parts. The momentum balance considers mechanical plus electromagnetic balances; for the latter the Abraham representation using the Maxwell stress tensor is formulated. This tensor is manipulated to automatically fulfill the angular momentum balance. The entropy balance is formulated using the classical Gibbs equation for equilibrium interactions and non-equilibrium thermodynamics. For the non-linear finite element formulations, this equation requires the transformation of thermoelectric coupling and conductivities into tensorial form. The two-way thermoelastic Biot term introduces damping: thermomechanical, pyromagnetic and pyroelectric converse electromagnetic dynamic interactions. Ponderomotrix and electromagnetic forces are also considered. The governing equations are converted into a variational formulation with the resulting four-field, multi-coupled formalism implemented and validated with two custom-made finite elements in the research code FEAP. Standard first-order isoparametric eight-node elements with seven degrees of freedom (dof) per node (three displacements, voltage and magnetic scalar potentials plus two temperatures) are used. Non-linearities and dynamics are solved with Newton-Raphson and Newmark-\(\beta \) algorithms, respectively. Results of thermoelectric, thermoelastic, thermomagnetic, piezoelectric, piezomagnetic, pyroelectric, pyromagnetic and galvanomagnetic interactions are presented, including non-linear dependency on temperature and some second-order interactions.

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Acknowledgments

This research was partially supported by grants CSD2008-00037 Canfranc Underground Physics, Polytechnic University of Valencia under programs PAID 02-11-1828 and 05-10-2674. The first author used the grant Generalitat Valenciana BEST/2014/232 for the completion of this work.

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Authors and Affiliations

  1. Mecánica de Medios Continuos y Teoría de Estructuras, Universitat Politècnica de València, Valencia, Spain

    José L. Pérez-Aparicio

  2. Department of Mechanical Engineering and Construction, Universitat Jaume I, Castellón de la Plana, Spain

    Roberto Palma

  3. Department of Civil and Environmental Engineering, University of California at Berkeley, Berkeley, CA, USA

    Robert L. Taylor

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  1. José L. Pérez-Aparicio
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  2. Roberto Palma
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  3. Robert L. Taylor
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Corresponding author

Correspondence to José L. Pérez-Aparicio.

Appendices

Appendix 1

The EI derivatives in (72) to (74) have to be explicitly calculated and are listed in the first part of this appendix.

First, the derivatives of the Cauchy stress tensor from (72) are

$$\begin{aligned} \frac{\partial {\varvec{T}}^{_C}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} &= {\mathbf {C}}\,{{\varvec{\mathcal {B}}}}^{s}_{b} ;\quad \frac{\partial {\varvec{T}}^{_C}}{\partial {{\mathsf{a}}}_{b}^{_{V}}} = {\varvec{e}}^{_V}{^\top }\,{{\varvec{\mathcal {B}}}}^{}_{b} ; \\ \frac{\partial {\varvec{T}}^{_C}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} &= {\varvec{e}}^\varphi {^\top }\,{{\varvec{\mathcal {B}}}}^{}_{b} ;\quad \frac{\partial {\varvec{T}}^{_C}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}} = -\varvec{\beta }\,{\mathcal{N}}_{b} \end{aligned}$$

The first equation is a \(6\times 3\) matrix; the second to fourth are directly \(6\times 1\) vectors.

The derivatives of the symmetric part of the Maxwell stress tensor from the same stiffnesses are

$$\begin{aligned} \frac{\partial {\varvec{T}}^{_M}_s}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}&= - \frac{1}{2}\, \left \{ \left [ \frac{\partial {\varvec{D}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + \left ( \frac{\partial {\varvec{D}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} \right )^{\top } \right ]\,{{\mathsf{a}}}_{b}^{_{V}} \right.\\&\left.\quad+\, \left [ \frac{\partial {\varvec{B}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + \left ( \frac{\partial {\varvec{B}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} \right )^{\top }\right ]\,{{\mathsf{a}}}_{b}^{_{\varphi }} \right \}\\ \frac{\partial {{\varvec{T}}^{_M}}_s}{\partial {{\mathsf{a}}}_{b}^{_{V}}}&= - \frac{1}{2}\, \left\{ \left [ \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + \left ( \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} \right )^{\top }\right ]\,{{\mathsf{a}}}_{b}^{_{V}}\right. \\&\left.\quad+\, \left [ \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + \left ( \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} \right )^{\top }\right ]\,{{\mathsf{a}}}_{b}^{_{\varphi }}\right. \\&\left.\quad +\, {\varvec{D}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} +({\varvec{D}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b})^\top +2\,\epsilon_0\,{{\mathsf{a}}}_{b}^{_{V}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b}\,{{\varvec{\mathcal {B}}}}^{}_{b}\,{\varvec{I}}\;\;\right\} \\ \frac{\partial {\varvec{T}}^{_M}_s}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}&= - \frac{1}{2}\, \left \{ \left [ \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + \left ( \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} \right )^{\top }\right ]\,{{\mathsf{a}}}_{b}^{_{V}} \right.\\&\left.\quad+\, \left [ \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + \left ( \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} \right )^{\top }\right ]\,{{\mathsf{a}}}_{b}^{_{\varphi }} \right.\\&\left.\quad+\, {\varvec{B}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + ({\varvec{B}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b})^\top +2\,\mu_0\,{{\mathsf{a}}}_{b}^{_{\varphi }}\,{{\varvec{\mathcal {B}}}}^{\top }_{b}\,{{\varvec{\mathcal {B}}}}^{}_{b}\,{\varvec{I}}\right \}\\ \frac{\partial {\varvec{T}}^{_M}_s}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}}&= - \frac{1}{2}\, \left \{ \left [ \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + \left ( \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} \right )^{\top }\right ]\,{{\mathsf{a}}}_{b}^{_{V}} \right.\\&\left.\quad+\, \left [ \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + \left ( \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} \right )^{\top }\right ]\,{{\mathsf{a}}}_{b}^{_{\varphi }} \;\;\right \} \end{aligned}$$

The first equation cannot directly follow the matrix multiplication convection, as \(\partial {\varvec{T}}^{_M}_s/\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}\) in (72) must be interpreted as a 6 \(\times \) 3 matrix composed of three 6 \(\times \) 1 vectors.

$$\begin{aligned} \frac{\partial {\varvec{T}}^{_M}_s}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} \equiv \left[ \frac{\partial {\varvec{T}}^{_M}_s}{\partial {\mathsf{a}}^{_U}_{b1}},\quad \frac{\partial {\varvec{T}}^{_M}_s}{\partial {\mathsf{a}}^{_U}_{b2}},\quad \frac{\partial {\varvec{T}}^{_M}_s}{\partial {\mathsf{a}}^{_U}_{b3}} \right] \end{aligned}$$

Therefore, in the right hand side the derivatives of \({\varvec{D}}\), \({\varvec{B}}\) are also with respect to each mechanical dof with a resultant dimension of 3 \(\times \) 3, giving one of the components of the previous expression to be converted into the 6 \(\times \) 1 Voigt notation. For instance, for the second mechanical dof, the result is

$$\begin{aligned} \frac{\partial {\varvec{T}}^{_M}_s}{\partial {\mathsf{a}}^{_U}_{b2}}&= {} -\left ( \frac{\partial {\varvec{D}}}{\partial {\mathsf{a}}^{_U}_{b2}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + {{\varvec{\mathcal {B}}}}^{}_{b}\, \frac{\partial {\varvec{D}}^\top }{\partial {\mathsf{a}}^{_U}_{b2}} \right )\, \frac{{{\mathsf{a}}}_{b}^{_{V}}}{2} \\&\quad-\,\left ( \frac{\partial {\varvec{B}}}{\partial {\mathsf{a}}^{_U}_{b2}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} + {{\varvec{\mathcal {B}}}}^{}_{b}\, \frac{\partial {\varvec{B}}^\top }{\partial {\mathsf{a}}^{_U}_{b2}} \right )\, \frac{{{\mathsf{a}}}_{b}^{_{\varphi }}}{2} \end{aligned}$$

The rest of the Maxwell tensor derivatives are directly 3 \(\times \) 3 matrices.

Again for (72), using the third of (71) the derivatives of the ponderomotive forces are

$$\begin{aligned} \frac{\partial {\varvec{f}}^{_{PM}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}&= {} \dot{{\varvec{P}}}_{\times }\, \frac{\partial {\varvec{B}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} + \frac{\partial {\varvec{P}}_{\times }}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}\,\dot{{\varvec{B}}}+\epsilon_0\mu_0\, \frac{\partial {\varvec{M}}_{\times }}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}\,{{\varvec{\mathcal {B}}}}^{}_{b}\,\dot{{\mathsf{a}}}_{b}^{_{V}} \\ \frac{\partial {\varvec{f}}^{_{PM}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}}&= {} \dot{{\varvec{P}}}_{\times }\, \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}} + \frac{\partial {\varvec{P}}_{\times }}{\partial {{\mathsf{a}}}_{b}^{_{V}}}\,\dot{{\varvec{B}}}+\epsilon_0\mu_0\left( \dot{{\varvec{M}}}_{\times } + \frac{\partial {\varvec{M}}_{\times }}{\partial {{\mathsf{a}}}_{b}^{_{V}}} \dot{{\mathsf{a}}}_{b}^{_{V}} \right) {{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial {\varvec{f}}^{_{PM}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}&= {} \dot{{\varvec{P}}}_{\times }\, \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} + \frac{\partial {\varvec{P}}_{\times }}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}\,\dot{{\varvec{B}}}+\epsilon_0\mu_0\, \frac{\partial {\varvec{M}}_{\times }}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} \,{{\varvec{\mathcal {B}}}}^{}_{b}\,\dot{{\mathsf{a}}}_{b}^{_{V}} \\ \frac{\partial {\varvec{f}}^{_{PM}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}}&= {} \dot{{\varvec{P}}}_{\times }\, \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}} + \frac{\partial {\varvec{P}}_{\times }}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}}\,\dot{{\varvec{B}}}+\epsilon_0\mu_0\, \frac{\partial {\varvec{M}}_{\times }}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}}\,{{\varvec{\mathcal {B}}}}^{}_{b}\,\dot{{\mathsf{a}}}_{b}^{_{V}} \end{aligned}$$

where \(\dot{{\varvec{B}}}_{\times }\), \(\dot{{\varvec{M}}}_{\times }\) can directly be calculated from (29). Similar to the first equation of this Appendix, the first derivative of \({\varvec{f}}^{_{PM}}\) is implemented as three 3 \(\times \) 1 column-vectors. The other three derivatives are directly 3 \(\times \) 1 vectors. The subindex \(\times \) implies the allocation of a 3 \(\times \) 1 vector into a 3 \(\times \) 3 antisymmetric matrix, see Sect. 5 and (66).

The derivatives of \({\varvec{D}}\) in the Maxwell stress expressions and in (73) are from its definitions 3 \(\times \) 1 vectors. Due to the form of the first two equations (71), these derivatives are equal to those of \({\varvec{P}}\) present in \({\varvec{f}}^{_{PM}}\) (third of (71)), except for the one with respect to \({{\mathsf{a}}}_{b}^{_{V}}\). The same must be done for the derivatives of \(\dot{{\varvec{D}}}\) with respect to the time derivatives of the dof.

$$\begin{aligned} \frac{\partial {\varvec{D}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} &= \frac{\partial {\varvec{P}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} = \frac{\partial \dot{{\varvec{P}}}}{\partial \dot{{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}= {\varvec{e}}^{_V}\,{{\varvec{\mathcal {B}}}}^{s}_{b} \\ \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}} &= \left[ \frac{\partial \varvec{\epsilon }}{\partial {\varvec{E}}}\,{{\varvec{\mathcal {B}}}}^{\top }_{b}\,{{\mathsf{a}}}_{b}^{_{V}}-\varvec{\epsilon }\right] \,{{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} &= \frac{\partial {\varvec{P}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} = \frac{\partial \dot{{\varvec{P}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{\varphi }}} = -\varvec{\nu }\,{{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}} &= \frac{\partial {\varvec{P}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}} = \frac{\partial \dot{{\varvec{P}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{T^e}}} = \varvec{\pi }^{_V}\,{\mathcal{N}}_{b} \end{aligned}$$

and those of \({\varvec{D}}^\top \) or \({\varvec{D}}_\times \) (as \({\varvec{B}}^\top , {\varvec{B}}_\times \) in the next equations) are directly the transpose or the cross form of the results. Some of these expressions will be used for the derivatives of the capacity matrices. The remaining derivatives are

$$\begin{aligned} \frac{\partial {\varvec{P}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}} = \frac{\partial {\varvec{D}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}} + \epsilon_0\,{{\varvec{\mathcal {B}}}}^{}_{b} ; \qquad \frac{\partial \dot{{\varvec{P}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{V}}} = \left[ \epsilon_0{\varvec{I}}- \varvec{\epsilon }\right] \,{{\varvec{\mathcal {B}}}}^{}_{b} \end{aligned}$$

The derivatives of \({\varvec{B}}\) and its time derivative for the stresses, ponderomotive forces and (74) are

$$\begin{aligned} \frac{\partial {\varvec{B}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} &= \frac{\partial \dot{{\varvec{B}}}}{\partial \dot{{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} = \mu_0 \frac{\partial {\varvec{M}}}{\partial {{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} = \mu_0 \frac{\partial \dot{{\varvec{M}}}}{\partial \dot{{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} = {\varvec{e}}^\varphi \,{{\varvec{\mathcal {B}}}}^{s}_{b} \\ \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}} &= \frac{\partial \dot{{\varvec{B}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{V}}} = \mu_0 \frac{\partial {\varvec{M}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}} = \mu_0 \frac{\partial \dot{{\varvec{M}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{V}}} = -\varvec{\nu }\,{{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} &= \left( \frac{\partial \varvec{\mu }}{\partial {\varvec{H}}} {{\varvec{\mathcal {B}}}}^{\top }_{b} {{\mathsf{a}}}_{b}^{_{\varphi }} - \varvec{\mu }\right) {{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}} &= \frac{\partial \dot{{\varvec{B}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{T^e}}} = \mu_0 \frac{\partial {\varvec{M}}}{\partial {{\mathsf{a}}}_{b}^{_{T^e}}} = \mu_0 \frac{\partial \dot{{\varvec{M}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{T^e}}} = \varvec{\pi }^\varphi \,{\mathcal{N}}_{b}\end{aligned}$$

Since it is useful in a hysteresis simulation of ferroelectric materials, it has been assumed that the properties \(\varvec{\epsilon }\), \(\varvec{\mu }\) may vary with the corresponding field although not with time. In order to preserve the physical sense of laboratory tests, and since the matrices of \(\varvec{\epsilon }\), \(\varvec{\mu }\) are diagonal (see Appendix 4), the derivatives are performed term by term and may be stored in a 3 \(\times \) 1 vector; for instance the material permittivity derivative results in

$$\begin{aligned} \frac{\partial \varvec{\epsilon }}{\partial {\varvec{E}}} = \left\{\begin{array}{ccc} \frac{\partial \epsilon_1}{\partial E_1}\;,& \frac{\partial \epsilon_2}{\partial E_2}\;,& \frac{\partial \epsilon_3}{\partial E_3} \end{array}\right\}^\top \end{aligned}$$

The remaining derivatives are

$$\begin{aligned} \begin{array}{ll} \frac{\partial {\varvec{M}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} = \frac{1}{\mu_0} \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} + {{\varvec{\mathcal {B}}}}^{}_{b} &{};\qquad \frac{\partial \dot{{\varvec{M}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{\varphi }}} = \left[ {\varvec{I}}- \frac{\varvec{\mu }}{\mu_0} \right] \,{{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial \dot{{\varvec{B}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{V}}} = -\varvec{\mu }\,{{\varvec{\mathcal {B}}}}^{}_{b} &{} \end{array} \end{aligned}$$

The partial derivatives from capacities (76) represent heat dissipation due to electromagnetic dynamics; at the microscopic level, they can be related to friction among dipoles.

$$\begin{aligned} \frac{\partial {\varvec{f}}^{_{PM}}}{\partial \dot{{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}&= {} \frac{\partial \dot{{\varvec{P}}}_{\times }}{\partial \dot{{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}\,{\varvec{B}}+{\varvec{P}}_{\times }\, \frac{\partial \dot{{\varvec{B}}}}{\partial \dot{{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} +\epsilon_0\mu_0\, \frac{\partial \dot{{\varvec{M}}}_{\times }}{\partial \dot{{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}}\,{{\varvec{\mathcal {B}}}}^{}_{b}\,{{\mathsf{a}}}_{b}^{_{V}} \\ \frac{\partial {\varvec{f}}^{_{PM}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{V}}}&= {} \frac{\partial \dot{{\varvec{P}}}_{\times }}{\partial \dot{{\mathsf{a}}}_{b}^{_{V}}}\,{\varvec{B}}+{\varvec{P}}_{\times }\, \frac{\partial \dot{{\varvec{B}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{V}}} +\epsilon_0\mu_0\left ( \frac{\partial \dot{{\varvec{M}}}_{\times }}{\partial \dot{{\mathsf{a}}}_{b}^{_{V}}}\,{{\mathsf{a}}}_{b}^{_{V}} + {\varvec{M}}_{\times } \right )\,{{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial {\varvec{f}}^{_{PM}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{\varphi }}}&= {} \frac{\partial \dot{{\varvec{P}}}_{\times }}{\partial \dot{{\mathsf{a}}}_{b}^{_{\varphi }}}\,{\varvec{B}}+{\varvec{P}}_{\times }\, \frac{\partial \dot{{\varvec{B}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{\varphi }}} +\epsilon_0\mu_0\, \frac{\partial \dot{{\varvec{M}}}_{\times }}{\partial \dot{{\mathsf{a}}}_{b}^{_{\varphi }}}\,{{\varvec{\mathcal {B}}}}^{}_{b}\,{{\mathsf{a}}}_{b}^{_{V}} \\ \frac{\partial {\varvec{f}}^{_{PM}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{T^e}}}&= {} \frac{\partial \dot{{\varvec{P}}}_{\times }}{\partial \dot{{\mathsf{a}}}_{b}^{_{T^e}}}\,{\varvec{B}}+{\varvec{P}}_{\times }\, \frac{\partial \dot{{\varvec{B}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{T^e}}} +\epsilon_0\mu_0\, \frac{\partial \dot{{\varvec{M}}}_{\times }}{\partial \dot{{\mathsf{a}}}_{b}^{_{T^e}}}\,{{\varvec{\mathcal {B}}}}^{}_{b}\,{{\mathsf{a}}}_{b}^{_{V}} \end{aligned}$$

Finally, the derivatives of the two-way couplings from (77) using (50) are

$$\begin{aligned} \frac{\partial {\mathcal {T}}_0}{\partial \dot{{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}} &= T_0\,\varvec{\beta }{^\top }\,{{\varvec{\mathcal {B}}}}^{s}_{b} ;\quad \frac{\partial {\mathcal {T}}_0}{\partial \dot{{\mathsf{a}}}_{b}^{_{V}}} = -T_0\,\varvec{\pi }^{_V}{^\top }\,{{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial {\mathcal {T}}_0}{\partial \dot{{\mathsf{a}}}_{b}^{_{\varphi }}} &=-T_0\,\varvec{\pi }^\varphi {^\top }\,{{\varvec{\mathcal {B}}}}^{}_{b} \end{aligned}$$

Appendix 2

Related to NEI, the derivatives of the tangent stiffness tensors are obtained using the chain rule. For (81), the Cauchy tensor derivatives are equal to the first and fourth from EI in Appendix 1. The derivatives of the Maxwell vacuum tensor are from (63)

$$\begin{aligned} \frac{\partial {\varvec{T}}^{_M}_v}{\partial {{\mathsf{a}}}_{b}^{_{V}}}&= {} \epsilon_0\,\left [ 2\,{{\varvec{\mathcal {B}}}}^{}_{b}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} - {{\varvec{\mathcal {B}}}}^{\top }_{b}\,{{\varvec{\mathcal {B}}}}^{}_{b}\,{\varvec{I}}\right ]\,{{\mathsf{a}}}_{b}^{_{V}} \\ \frac{\partial {\varvec{T}}^{_M}_v}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}&= {} \mu_0\,\left [ 2\,{{\varvec{\mathcal {B}}}}^{}_{b}\,{{\varvec{\mathcal {B}}}}^{\top }_{b} - {{\varvec{\mathcal {B}}}}^{\top }_{b}\,{{\varvec{\mathcal {B}}}}^{}_{b}\,{\varvec{I}}\right ]\,{{\mathsf{a}}}_{b}^{_{\varphi }} \end{aligned}$$

Again Voigt notation may be applied to these derivatives to obtain a 6 \(\times \) 1 vector. The non-zero derivatives of the electric flux are

$$\begin{aligned} \frac{\partial {\varvec{j}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}}&= {} -\,\varvec{\gamma}\,{{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial {\varvec{j}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}&= {} -\left[ \frac{\partial \varvec{\gamma }}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}\,{{\mathsf{a}}}_{b}^{_{V}} + \left( \varvec{\gamma} \,\frac{\partial \varvec{\alpha}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}+ \frac{\partial \varvec{\gamma}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi}}}\,\varvec{\alpha } \right) \,{{{\mathsf{a}}}_{b}^{_{T^n}}} \right] \,{{\varvec{\mathcal {B}}}}^{}_{b} \\ \frac{\partial {\varvec{j}}}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}}&= {} -\left [ \frac{\partial \varvec{\gamma }}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}}\,{{\mathsf{a}}}_{b}^{_{V}} +\left( \varvec{\gamma}\, \frac{\partial \varvec{\alpha }}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}} + \frac{\partial \varvec{\gamma}}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}}\,\varvec{\alpha} \right){{\mathsf{a}}}_{b}^{_{T^n}} + \varvec{\gamma} \,\varvec{\alpha} \right ]\,{{\varvec{\mathcal {B}}}}^{}_{b} \end{aligned}$$

Following the same procedure, the non-zero derivatives of the thermal flux are

$$\begin{aligned} \frac{\partial {\varvec{q}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}}&= {\mathcal{N}}_{b}\,{{\mathsf{a}}}_{b}^{_{T^n}}\,\varvec{\alpha }\, \frac{\partial {\varvec{j}}}{\partial {{\mathsf{a}}}_{b}^{_{V}}} \\ \frac{\partial {\varvec{q}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}&= -\left[ \frac{\partial \varvec{\kappa }}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}\,{{\varvec{\mathcal {B}}}}^{}_{b} - {\mathcal{N}}_{b}\,\left( \frac{\partial \varvec{\alpha }}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}}\,{\varvec{j}}+ \varvec{\alpha }\, \frac{\partial {\varvec{j}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} \right) \right] \,{{\mathsf{a}}}_{b}^{_{T^n}} \\ \frac{\partial {\varvec{q}}}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}}&= -\varvec{\kappa }\,{{\varvec{\mathcal {B}}}}^{}_{b} +{\mathcal{N}}_{b}\,\varvec{\alpha }\,{\varvec{j}}\\&\quad-\,\left[ \frac{\partial \varvec{\kappa }}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}}\,{{\varvec{\mathcal {B}}}}^{}_{b} -{\mathcal{N}}_{b}\,\left( \frac{\partial \varvec{\alpha }}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}}\,{\varvec{j}}+ \varvec{\alpha }\, \frac{\partial {\varvec{j}}}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}} \right) \right] \,{{\mathsf{a}}}_{b}^{_{T^n}} \end{aligned}$$

For both fluxes and from (60) the derivatives of the Peltier and conductivities’ matrices are, for the symmetric parts

$$\begin{aligned} \frac{\partial \varvec{\alpha }}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} &= \frac{\hbox {d} \varvec{\alpha }_{sk}}{\hbox {d} {{\mathsf{a}}}_{b}^{_{\varphi }}} = -\mu_0\,N\,{{\varvec{\mathcal {B}}}}^{\times }_{b} \\ \frac{\partial \varvec{\alpha }}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}} &= \frac{\hbox {d} \varvec{\alpha }_s}{\hbox {d} {{\mathsf{a}}}_{b}^{_{T^n}}} = {\mathcal{N}}_{b}\, \frac{\hbox {d} \alpha }{\hbox {d} T^n}\,{\varvec{I}}\\ \frac{\partial \varvec{\gamma }}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} &= \frac{\partial \varvec{\gamma }}{\partial \varvec{\rho }}\, \frac{\hbox {d} \varvec{\rho }_{sk}}{\hbox {d} {{\mathsf{a}}}_{b}^{_{\varphi }}} = \mu_0\,R\,\varvec{\gamma }\,{{\varvec{\mathcal {B}}}}^{\times }_{b}\,\varvec{\gamma }\\ \frac{\partial \varvec{\gamma }}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}} &= \frac{\partial \varvec{\gamma }}{\partial \varvec{\rho }}\, \frac{\hbox {d} \varvec{\rho }_s}{\hbox {d} {{\mathsf{a}}}_{b}^{_{T^n}}} = -{\mathcal{N}}_{b}\, \frac{\hbox {d} \rho }{\hbox {d} T^n}\,\varvec{\gamma }\,\varvec{\gamma }\\ \frac{\partial \varvec{\kappa }}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} &= \frac{\hbox {d} \varvec{\kappa }_{sk}}{\hbox {d} {{\mathsf{a}}}_{b}^{_{\varphi }}} = \mu_0\,M\,{{\varvec{\mathcal {B}}}}^{\times }_{b} \\ \frac{\partial \varvec{\kappa }}{\partial {{\mathsf{a}}}_{b}^{_{T^n}}} &= \frac{\hbox {d} \varvec{\kappa }_s}{\hbox {d} {{\mathsf{a}}}_{b}^{_{T^n}}} = {\mathcal{N}}_{b}\, \frac{\hbox {d} \kappa }{\hbox {d} T^n}\,{\varvec{I}} \end{aligned}$$

In the previous expressions the equalities

$$\begin{aligned} \frac{\partial \varvec{\gamma }}{\partial \varvec{\rho }}\,{\mathbf {O}}&= {} \frac{\partial \varvec{\rho }^{-1}}{\partial \varvec{\rho }}\,{\mathbf {O}} = -\varvec{\rho }^{-1}\,{\mathbf {O}}\,\varvec{\rho }^{-1} = -\varvec{\gamma }\,{\mathbf {O}}\,\varvec{\gamma }\\ \frac{\hbox {d} \varvec{\alpha }_s}{\hbox {d} {{\mathsf{a}}}_{b}^{_{T^n}}}&= {} \frac{\hbox {d} \alpha }{\hbox {d} T^n}\, \frac{\hbox {d} T^n}{\hbox {d} {{\mathsf{a}}}_{b}^{_{T^n}}} = \frac{\hbox {d} \alpha }{\hbox {d} T^n}\,{\mathcal{N}}_{b}\end{aligned}$$

have been used, where \({\mathbf{O}}\) is any second-order tensor, symmetric or not. Similar expressions to the second equation can be deduced for \(\varvec{\rho }\), \(\varvec{\kappa }\). The total derivatives \(\hbox {d}\alpha /\hbox {d}T^n\), \(\hbox {d}\rho /\hbox {d}T^n\), \(\hbox {d}\kappa /\hbox {d}T^n\) are directly calculated from their expressions in (92).

The derivatives of \({\varvec{B}}\) with respect to \({{{\varvec {\mathsf{{a}}}}}}_{b}^{_U}\), \({{\mathsf{a}}}_{b}^{_{T^n}}\) are zero since neither piezomagnetic nor pyromagnetic interactions are considered; in addition, due to the use of magnetic scalar potential \(\partial {\varvec{B}}/ \partial {{\mathsf{a}}}_{b}^{_{V}} \approx {{\varvec{0}}}\). The only non-zero term is

$$\begin{aligned} \frac{\partial {\varvec{B}}}{\partial {{\mathsf{a}}}_{b}^{_{\varphi }}} = \frac{\partial \dot{{\varvec{B}}}}{\partial \dot{{\mathsf{a}}}_{b}^{_{\varphi }}} = -\mu_0\,{{\varvec{\mathcal {B}}}}^{}_{b} \end{aligned}$$

Sub, supraindex

Description

\((\cdot )_{1,2,3}\)

Cartesian directions/FEM const.

\((\cdot )^\infty \)

Surrounding

\((\cdot )_{_T}\)

Thermal

\((\cdot )^e\)

Equilibrium

\((\cdot )^{k,l},(\cdot )_{k,l}\)

Free index, iteration

\((\cdot )^n\)

Non-equilibrium

\((\cdot )^\top \)

Transpose

\((\cdot )^*\)

Critical size

figure gh

First time derivative

\((\cdot )_{tt}\)

Total

\((\cdot )_{_{F}}\)

Driving force related

\((\cdot )^f\)

Free charge related

\(||\cdot ||\)

Norm

\((\cdot )_s,\,(\cdot )^s\)

Symmetric part

\((\cdot )_{sk},\,(\cdot )^{sk}\)

Skew-symmetric part

\((\cdot )_p\)

Natural boundary condition

\((\cdot )_u\)

Essential boundary condition

figure gi

Prescribed variable

figure gk

Second time derivative

\((\cdot )_{_U},(\cdot )^{_U}\)

Mechanical

\((\cdot )^b\)

Bound charge related

\((\cdot )_{_V},(\cdot )^{_V}\)

Electrical

\((\cdot )_\varphi ,(\cdot )^\varphi \)

Magnetic

\((\cdot )^{-1}\)

Inverse

\((\cdot )_{_P}\)

Poynting theorem related

\((\cdot )^{_{Mi}}\)

Minkowski formalism related

\((\cdot )^{_{Ab}}\)

Abraham formalism related

\((\cdot )^{_M}\)

Maxwell tensor related

\((\cdot )^{_{EM}}\)

Lorentz electro-magnetic

\((\cdot )^{_{PM}}\)

Ponderomotive

\((\cdot )^{_C}\)

Cauchy tensor related

\((\cdot )^{_R}\)

Residual

\((\cdot )_v\)

Vacuum

\((\cdot )_\times \), \((\cdot )^{\times} \)

Cross product transformed

\((\cdot )^h\)

FEM approximation

\((\cdot )^{\mathbf{x}}\)

Nodal coordinates

\((\cdot )_{a,b}\)

Local node number

\(I,J\)

Dof index

\((\cdot )_{_N}\)

Newmark integration related

\(|\cdot |\)

Determinant

Appendix 3

Description of symbols used for EI and NEI are given in the tables. The base units for the SI system have been used, with the addition of Newton [N], Joule [J], and Volt [V], of special relevance in mechanical and electrical engineering.

If a symbol represents several magnitudes (as for instance \({\mathcal {S}}\)), central dots \([\cdots ]\) are used for the units if there is not enough space to list all of them. Only diagonal submatrices are included, the off-diagonal ones from Appendices 1, 2 represent interactions between these fields.

Appendix 4

Material properties of a representative material (units in Appendix 3) to be applied to the EI cases are listed in this Appendix. They have been obtained from several references, mostly for BaTiO\(_3\)–CoFeO\(_4\) but also for other similar materials; to the best of our knowledge, no complete characterization of any material has been published probably due to the experimental difficulty and also to the absence of a general formulation. Mass density, heat capacity, thermal expansion and thermal conductivity are obtained from [56]. The pyroelectric properties are from [54], the pyromagnetic from [17] and the rest from [119].

Abbreviations

Description

FEAP

FE Analysis Program

EI

Equilibrium interaction

NEI

Non-equilibrium interaction

FEM

Finite element method

SS

Second Sound

dof

Degree of freedom

ET

Equilibrium thermodynamics

NET

Non-equilibrium thermodynamics

ENET

Extended NET

MEMS

Micro-electro-mechanical sensor

NEMS

Nano-electro-mechanical sensor

PDE

Partial differential equation

FE

Finite element

\(nel\)

# nodes per element

As mentioned in Sect. 4.2.2 for \(\alpha_{_T}\), the material properties reflect the transverse isotropy of Fig. 18. The exception in the listed data is in \(\varvec{\pi }^{_V}\), \(\varvec{\pi }^\varphi \), for which properties are assumed isotropic due to the lack of data.

$$\begin{aligned} {\mathbf {C}}&= {} \left[\begin{array}{cccccc} 116&77&78&0&0&0 \\&116&78&0&0&0 \\&162&0&0&0 \\ {\scriptstyle -sym-}&&89&0&0 \\&&86&0 \\&&&86 \end{array}\right] \; \times 10^9\\ {\varvec{e}}^{_V}&= {} \left[\begin{array}{cccccc} 0&0&0&0&0&11.6 \\&0&0&0&11.6&0 \\ -4.4&-4.4&18.6&0&0&0 \end{array}\right]\\ {\varvec{e}}^\varphi&= {} \left[\begin{array}{cccccc} 0&0&0&0&0&5.5 \\ 0&0&0&0&5.5&0 \\ 5.8&5.8&7&0&0&0 \end{array}\right] \; \times 10^2\\ \varvec{\epsilon }&= {} \left[\begin{array}{ccc} 11.2&0&0 \\ 0&11.2&0 \\ 0&0&12.6 \end{array}\right] \; \times 10^{-9}\\ \varvec{\mu }&= {} \left[\begin{array}{ccc} 5&0&0 \\ 0&5&0 \\ 0&0&10 \end{array}\right] \; \times 10^{-6}\\ \varvec{\nu }&= {} \left[\begin{array}{lll} 5.37&0&0 \\ 0&5.37&0 \\ 0&0&2737.5 \end{array}\right] \; \times 10^{-12}\\ \varvec{\pi }^{_V}&= {} \left\{\begin{array} {lll} 58.3,&58.3,& 58.3 \end{array}\right\}^\top \times 10^{-5}\\ \varvec{\pi }^\varphi&= {} \left\{\begin{array} {lll}5,& 5,&5 \end{array}\right\}^\top \times 10^{-2}\\ \varvec{\beta }&= {} \left\{\begin{array} {llllll}1.67,& 1.67, & 1.96,& 0,& 0,& 0 \end{array}\right\}^\top \times 10^6 \\ \end{aligned}$$
$$\begin{aligned} \kappa &= 2.61 ; \quad T_0 = 293 \\ c &= 434 ; \quad d = 5.5 \times 10^{-21} \\ \epsilon_0 &= 8.8542 \times 10^{-12} ; \quad \mu_0= 4\pi \times 10^{-7} \\ \rho_m &= 5.7 \times 10^3 \end{aligned}$$

Smbl

SI

Description

Dim.

\({\varvec{B}}\)

[V s/m\(^2\)]

Magnetic induction

3 \(\times \) 1

\({\varvec{x}}\)

[m]

Eulerian coordinate

3 \(\times \) 1

\(\nabla \)

[1/m]

Gradient operator

3 \(\times \) 1

\(V\)

[V]

Elect. scalar pot./voltage

 

\({\varvec{j}}\)

[A/m\(^2\)]

Electric flux

3 \(\times \) 1

\(T\)

[K]

Temperature

 

\(\varOmega \)

[m\(^3\)]

Thermodynamic domain

 

\(\varGamma \)

[m\(^2\)]

Boundary of domain

 

\({\mathcal {S}}\)

[\(\cdots \)]

State variable

 

\({\mathcal {I}}\)

[\(\cdots \)]

Intensive variable

 

\({\mathcal {E}}\)

[\(\cdots \)]

Extensive variable

 

d

 

Exact differential

 

\({\mathsf {E}}\)

[J]

Internal energy

 

\({\mathsf {Q}}\)

[J]

System heat

 

\(\delta \)

 

Variation

 

\({\mathsf {W}}\)

[J]

System mechanical work

 

\({\mathsf {S}}\)

[J/K]

ET entropy

 

\({\varvec{T}}\)

[N/m\(^2\)]

Mechanical stress

6 \(\times \) 1

\({\varvec{S}}\)

[m/m]

Mechanical strain

6 \(\times \) 1

\({\varvec{E}}\)

[V/m]

Electric field

3 \(\times \) 1

\({\varvec{P}}\)

[A s/m\(^2\)]

Polarization

3 \(\times \) 1

\(\mu_0\)

[V s/A\(\cdot \)m]

Vacuum permeability

 

\({\varvec{H}}\)

[A/m]

Magnetic field

3 \(\times \) 1

\({\varvec{M}}\)

[A/m]

Magnetization

3 \(\times \) 1

\({\varvec{X}}\)

[m]

Lagrangian coordinate

3 \(\times \) 1

\({\varvec{U}}\)

[m]

Lagrangian displacement

3 \(\times \) 1

\({\varvec{u}}\)

[m]

Eulerian displacement

3 \(\times \) 1

\(\partial \)

 

Partial differential

 

\({_{\mathcal {P}}}\)

[\(\cdots \)]

Eulerian continuum prprt.

 

\(t\)

[s]

Time

 

\({\mathbf{v}}\)

[m/s]

Eulerian velocity

3 \(\times \) 1

\(\rho \)

[\(\cdots \)]

Density (w/ subindex)

 

\(\varDelta \)

 

Increment

 

\(m\)

[kg]

Mass

 

\(q\)

[A]

Electric charge

 

\(Kn\)

 

Knudsen number

 

\(De\)

 

Deborah number

 

\({\mathcal {P}}\)

[\(\cdots \)]

Thermodynamic variable

 

\({\mathsf {e}}\)

[J/m\(^3\)]

NET energy density

 

\({\mathsf {s}}\)

[J/K m\(^3\)]

NET entropy density

 

\({\mathbf {n}}\)

 

Outward boundary normal

3 \(\times \) 1

\(\sigma \)

[J/s K]

Entropy production

 

\({\varvec{F}}\)

[\(\cdots \)]

Driving forces

3 \(\times \) 1

\({\varvec{q}}\)

[J/s m\(^2\)]

Thermal flux

3 \(\times \) 1

\(L^{kl}\)

[\(\cdots \)]

1st-order material property

 

\({\varvec{W}}\)

[m/m]

Small rotation strain

6 \(\times \) 1

\({\varvec{f}}\)

[N/m\(^3\)]

Volume force

3 \(\times \) 1

\({\varvec{t}}\)

[N/m\(^2\)]

Boundary pressure

3 \(\times \) 1

\(\epsilon_0\)

[A s/V m]

Vacuum permittivity

 

\(\varvec{\chi }\)

[\(\cdots \)]

Material susceptibility

3 \(\times \) 3

\({\varvec{D}}\)

[A s/m\(^2\)]

Electric displc./induction

3 \(\times \) 1

\(\varvec{\epsilon }\)

[A s/V m]

Material permittivity

3 \(\times \) 3

\(\varvec{\mu }\)

[V s/A m]

Material permeability

3 \(\times \) 3

\({\varvec{I}}\)

 

Identity matrix

3 \(\times \) 3

\({\varvec{A}}\)

[V s/m]

Magnetic vector

3 \(\times \) 1

\(\varphi \)

[A]

Magnetic scalar potential

 

\(\varvec{V}\)

[A s/m]

Electric vector

3 \(\times \) 1

\(c_{_P}\)

[\(\cdots \)]

Poynting constant

 

\(\dot{r}\)

[V A/m\(^3\)]

Scalar Poynting residual

 

\({\varvec{G}}\)

[N s/m\(^3\)]

Momentum density

3 \(\times \) 1

\(T_0\)

[K]

Reference temperature

 

\(c\)

[J/kg K]

Heat capacity

 

\({\mathcal {T}}_0\)

[N/s m\(^2\)]

Two-way thermoelast. terms

 

\(\varvec{\beta }\)

[N/K m\(^2\)]

Thermal expansion tensor

6 \(\times \) 1

\(\varvec{\pi }\)

[\(\cdots \)]

Pyro-\(V\), -\(\varphi \) couplings

3 \(\times \) 1

\(d\)

[m\(^2\)]

Two-temp. constraint

 

\(\nabla ^2\)

[1/m\(^2\)]

Laplace operator

 

\(\varvec{\mathcal {Q}}\)

[J/s m\(^2\)]

Two-temperature heat flux

3 \(\times \) 1

\({\mathbf {C}}\)

[N/m\(^2\)]

Elasticity tensor

6 \(\times \) 6

\(\alpha_{_T}\)

[1/K]

Thermal expansion coeff.

 

\(\Pi \)

[J]

Electromagnetic enthalpy

 

\({\varvec{e}}\)

[\(\cdots \)]

Piezo-\(V\), -\(\varphi \) couplings

3 \(\times \) 6

\(\varvec{\nu }\)

[s/m]

Magnetoelectric coupling

3 \(\times \) 3

\(\kappa \)

[J s/m K]

Thermal conductivity

 

\(\lambda \)

[N/m\(^2\)]

First Lamé parameter

 

\(\mu \)

[N/m\(^2\)]

Shear modulus

 

\(\delta_{ij}\)

 

Kronecker delta

 

\({\varvec{L}}\)

[\(\cdots \)]

NEI material properties

 

\(\varvec{\alpha }\)

[V/K]

Seebeck tensor

3 \(\times \) 3

\(\alpha \)

[V/K]

Seebeck coefficient

 

\(\varvec{\rho }\)

[m V/A]

Electric resistivity tensor

3 \(\times \) 3

\(\rho \)

[m V/A]

Electric resistivity

 

\(\varvec{\kappa }\)

[J s/m K]

Thermal conductivity tensor

3 \(\times \) 3

\(\varvec{\gamma }\)

[J s/m K]

Electric conductivity tensor

3 \(\times \) 3

\(N\)

[m\(^2\)/K s]

Nernst coefficient

 

\(R\)

[m\(^3\)/A s]

Hall coefficient

 

\(M\)

[m\(^2\)/V s]

Righi-Leduc coefficient

 

\({\mathcal{N}}\)

 

Local shape function

 

\(\varvec{\xi }\)

[m]

Local FE coordinates

3 \(\times \) 1

\({{\varvec {\mathsf{{a}}}}}\)

[m]

Nodal elastic dof

3 \(\times \) 1

\({{\mathsf{a}}}_{}^{_{}}\)

[\(\cdots \)]

Other nodal dof

 

\({{\varvec{\mathcal {B}}}}^{s}_{}\)

[1/m]

Local FE mech. gradient

6 \(\times \) 3

\({{\varvec{\mathcal {B}}}}^{}_{}\)

[1/m]

Local FE potential gradient

3 \(\times \) 1

\({\varvec{\mathcal {R}}}, {\mathcal{R}}\)

[\(\cdots \)]

Residual

 

\({\varvec{\mathcal {K}}}, {\mathcal{K}}\)

[\(\cdots \)]

Tangent stiffness

 

\({\varvec{\mathcal {C}}}, {\mathcal{C}}\)

[\(\cdots \)]

Tangent capacity

 

\({\varvec{\mathcal {M}}}\)

[\(\cdots \)]

Tangent mass

3 \(\times \) 3

\(\beta , \gamma \)

[\(\cdots \)]

Newmark-\(\beta \) parameters

 

\(l_{i}\)

[m]

Dimension in \(x_i\) direction

 

\(\varUpsilon \)

[K,V,A]

Any of 3 scalar potentials

 

\(\varvec{\Xi }\)

[\(\cdots \)]

Associated field for \(\varUpsilon \)

3 \(\times \) 1

\(\nu \)

 

Poisson ratio

 

\({\mathbf{O}}\)

[\(\cdots \)]

Any second-order tensor

3 \(\times \) 3

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Pérez-Aparicio, J.L., Palma, R. & Taylor, R.L. Multiphysics and Thermodynamic Formulations for Equilibrium and Non-equilibrium Interactions: Non-linear Finite Elements Applied to Multi-coupled Active Materials. Arch Computat Methods Eng 23, 535–583 (2016). https://doi.org/10.1007/s11831-015-9149-9

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