A second-order theory for piezoelectricity with 6mm and m3 crystal classes

The present paper concentrates on the basic equations of three-dimensional problems for nonlinear piezoelectric materials of hexagonal systems with symmetry class 6mm and of cubic systems with symmetry class m3. Emphasis is placed on developing the nonlinear constitutive relations between extended traction (including elastic stress and polarization) and extended strain (including elastic strain and electric field). The corresponding one-dimensional mathematical models for piezoelectric ceramic with symmetry classes 6mm and m3 are also given. Numerical examples are also carried out for the impact problem to show the important effect of the piezoelectric nonlinearity on the stress wave. Therefore, the derived concise equations can directly be applied to evaluate the nonlinear piezoelectric effects of piezoelectricity by the nonlinear finite element method.