Numerical analysis of end effects in laminated piezoelectric circular cylinders

Abstract

A method to quantify Saint-Venant’s principle for laminated piezoelectric circular cylinders is described. An algebraic eigensystem is constructed based on homogeneous equations of equilibrium with displacement and electric potential (voltage) fields stated in exponential form. The end effects for displacements and voltages, as well as stresses and electric displacements of self-equilibrated states are represented by the eigendata extracted from the eigensystem. The real parts of the eigenvalues convey information on the inverse decay lengths and their corresponding eigenvectors are displacement and voltage distributions of the self-equilibrated states. Stress and electric displacement eigenvectors are formed by appropriate differentiation of the eigenvectors for the displacement and voltage fields and through the coupled electromechanical constitutive laws. The right and left-eigenvectors, which are obtained from the eigensystem and its adjoint, are related through bi-orthogonality relationships. The end effects due to arbitrary displacements, voltages, tractions or electric displacements prescribed at the cylinder’s ends are obtained by means of an expansion theorem based on these bi-orthogonality relationships, or by a least-squares method. A number of verification examples are provided to demonstrate that the present results compare well with known analytical solutions and numerical results obtained via three-dimensional finite element analyses.