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This research is about irreversible domain wall motion leading to a temperature change of the material. The materials are mostly subjected to high frequencies and large electrical loads, which is why insufficient heat dissipation can lead to undesirable temperature developments, so-called self heating. Predictions concerning polycrystalline material behavior are made by the condensed method.

In this paper, the theoretical background of a physically based constitutive model is presented. In addition to the nonlinear ferroelectric behavior, the model considers the nonlinear coupling of thermal and electromechanical fields. Results are presented in terms of a simple analytical solution for a single domain configuration. with the material tensors Cijkl, κ il , γ, e lij , β ij and k l denoting the elasticity tensor, the dielectric tensor, the thermal coefficient, the piezoelectric tensor, the thermal stress and the pyroelectric coefficients, the constitutive equations of a nonlinear thermo-ferroelectric material are obtained aṡwhere σ ij and ij are stresses and strains, D i and E i the electric displacement and electric field and θ andS the temperature and specific entropy. The superscript "rev" refers to reversible quantities and "irr" indicates irreversible changes of state. Balance of energyFor describing a thermoelectromechanical three-field problem, a third field equation is required, which is obtained from the 1 st law of thermodynamics. In the quasistatic case it readṡpostulating that the change of internal energyU is equal to the sum of of the electromechanical work ratėthe specific heat flow ratėwhereq v is an internal heat source andq A j,j a heat flux across the boundary, and the dissipationẊ due to ferroelectric domain switching. The latter requires some considerations concerning its interpretation. The dissipative work rateẊ feeds the irreversible processes of the inelastic zone which, compared to the whole crystal, represents just a very small area being overswept during domain wall motion. The control volume, on the other hand, is reversible and loses this energy, which must therefore be considered by a negative sign [1]:

Weak formulations of boundary value problems are the basis of various numerical discretization schemes. They are classically derived applying the method of weighted residuals or a variational principle. For electrodynamical and caloric problems, variational approaches are not straightforwardly obtained from physical principles like in mechanics. Weak formulations of Maxwell’s equations and of energy or charge balances thus are frequently derived from the method of weighted residuals or tailored variational approaches. Related formulations of multiphysical problems, combining mechanical balance equations and the axioms of electrodynamics with those of heat conduction, however, raise the additional issue of lacking consistency of physical units, since fluxes of charge and heat intrinsically involve time rates and temperature is only included in the heat balance. In this paper, an energy-based approach toward combined electrodynamic–thermomechanical problems is presented within a classical framework, merging Hamilton’s and Jourdain’s variational principles, originally established in analytical mechanics, to obtain an appropriate basis for a multiphysical formulation. Complementing the Lagrange function by additional potentials of heat flux and electric current and appropriately defining generalized virtual powers of external fields including dissipative processes, a consistent formulation is obtained for the four-field problem and compared to a weighted residuals approach.