which is quadratic in polarization P. Due to the symmetry, the fourth order tensor
Q can be effectively represented by a 6-by-6 coupling matrix. The number of independent components in the matrix depends on the material symmetry. For example, for piezoelectric ceramics, the matrix can be characterized by three independent components:
Q11,
Q12, and
Q44.
where Ps is the saturation polarization.
where E is the applied electric field,
α is a material parameter called the inter-domain coupling, and the mechanics stress is computed assuming mechanically linear material as
where C is the fourth order elasticity tensor. The last term in
Equation 3-163 represents the inverse electrostrictive effect.
where ε0,vac is the electric permittivity of free space, and
is the tangent electric susceptibility matrix. The piezoelectric coupling tensor d is a third order tensor. Due to the symmetry, it can be conventionally represented by a 3-by-6 matrix with only few nonzero components. An important observation from the above formula is that the piezoelectric coefficients should reach their maximum (or minimum) at certain strength of the applied bias field. This is because
P is zero at zero applied field, while
χe tends to zero at large applied field magnitudes because of saturation.
where λs is the saturation electrostriction, and the deviatoric part can be computed for any matrix
A as
Equation 3-164 can be solved using either a time-dependent analysis or a stationary parametric sweep.