I am looking for an analytical closed form solution for the attached differential equations. The equations are equations of motion and electrostatic potential.
There are solutions for specific cases and there is a solution for an isotropic elastic material in which c's are replaced by Lame constants and Helmholtz decomposition is used. Solutions for the case of axisymmetric loading are available. If you can provide solution for the case of isotropic elastic and radially polarized piezoelectric material then it would be great.
Task 1 (Initial and minor task but necessary)
Develop the differential equations from:
1) The constitutive equations of a radially polarized piezoelectric medium.
2) Physical laws: a) equations of motion and b) Equations of electrostatic discharge (Gauss law)
Write them in terms of mechanical and electrical displacement...you should get the 3 equations
Task 2 (Major work)
Solve the 3 differential equations analytically for non-axisymmetric loading
This is an example of solving the differential equations of a transversely isotropic piezoelectric material.