Constitutive theory of piezoelectric materials

Within FERRET/MOOSE we have implemented the governing equations of piezoelectrics which allows the user to simulate strongly coupled anisotropic electromechanical phenomena. For a linear piezoelectric material, the stress-divergence equation for mechanical equilibrium reads

where and are the components of the elastic stiffness tensor, elastic strain tensor, and electric field. The direct piezoeletric coefficient is of rank three. Note that the strain tensor is defined in the usual way, .

Additionally, the Poisson equation also includes contributions from the (converse piezo) strain-charge,

with being the components of the elastic stress tensor. With sufficient choice of materials parameters, Equations (1) and (2) can be solved self-consistently under arbitrary mechanical loads or applied electric fields to yield the static configuration of the electrostatic potential and elastic displacements

Equations (1) and (2) can be cast dynamically, to simulate piezeoelectric actuation in real time. This is done by setting the LHS of the first equation to be equal to . The time scale can be set by a prefactor.

Another possible use of this implementation is to rotate all of the tensorial coefficients grain-wise leading to calculations of a polycrystalline piezoelectric. Within our block-restricted polycrystal approach, this allows for the piezoelectric properties to be evaluated in a computational box with a \textit{real} grain structure by rotating the tensors via,

where and are second and third rank tensors and is a rotation operator using the internal RotationTensor operation in MOOSE utils. The rotation operator accepts Euler angles in the standard Bunge sequence ().