Homogenization of an equation describing linear thin plates excited by piezopatches

In this paper we consider the problem of homogenization of equations describing linear thin plates excited by actuators made of piezoelectric ceramics (see e.g. 1]). It is assumed that the number of actuators goes to innnity whereas their dimension tends to zero. The procedure of homogenization is based on the theory of two-scale convergence studied in 2]. The speciic of the problem considered is the time dependence and the appearance of the forth spatial derivatives in the equation. A result of 3] about two-scale convergence of the second derivatives of subsequences of sequences bounded in L 2 (0; T ; H 2 0 (S)) enables us to handle this case. The paper is illustrated by computer simulations that demonstrate a good approximation of solutions of the original equation by solutions of the homogenized equation if the number of piezoelectric patches is suuciently large.