The static electromechanical coupling factor in linear piezoelectric material

We show that the electromechanical coupling factor is the solution of two eigenvalue problems which depends on the piezoelectric, elastic and dielectric properties of the materials. We show that there are at most three eigenvalues and three associated strain eigentensors which render extremum the ratio of mechanical energy converted into electric energy: their value is the same if the three eigenvalues whose associated electric field eigenvectors render extremum the ratio of electrical energy converted into mechanical.