Matrix in Piezoelectricity

Piezoelectricity is the combined effect of the electrical behavior of the material:

where D is the electric charge density displacement(electric displacement), the epsilon is permittivity and E is electric field strength, and

Hooke's Law:

where S is strain, s is compliance and T is stress.

These may be combined into so called coupled equations, of which the strain-charge form is:

where [d] is the matrix for the direct piezoelectric effect and [dt] is the matrix for the converse piezoelectric effect. The superscript E indicates a zero, or constant, electric filed; the superscript T indicates a zero, or constant. stress field; and the superscript t stands or the transposition of a matrix.

where the first equation represents the relationship for the converse piezoelectric effect and the latter for the direct piezoelectric effect.