Three-dimensional asymptotic analysis of multiple-electroded piezoelectric laminates

The problem of piezoelectric laminates with specified surface tractions and surface and internal electric potentials is studied. By writing the governing equations in the state-space formulation, employing an asymptotic expansion technique, and expressing electric displacement jumps across internal electrodes in terms of basic unknowns, the three-dimensional problem is reduced to a hierarchy of two-dimensional equations with the same homogeneous operators for each order. Different nonhomogeneous terms are only related to the preceding-order solution and can be readily determined by recurrence relations. Moreover, for pure elasticity, the present field equations of the leading order represent the classical thin elastic plate model. The proposed formulation is illustrated by considering a rectangular piezoelectric plate made of an orthotropic material, and with its edges simply supported and grounded. The convergence of the solution is discussed and the repeated averaging technique for partial sums is used to accelerate the convergence of the series solution. Computed results are found to agree well with available analytical results, and new results for electromechanically coupled problems are presented.