Vibrations of Thin Piezoelectric Shallow Shells: Two-dimensional approximation

Lower dimensional models of shells are preferred in numerical computations to threedimensional models when the thickness of the shells is “very small”. A lot of work has been done on the lower dimensional approximation of boundary value and eigenvalue problem for elastic plates and shells (cf.[2], [3] [4], [5], [6], [8],[9]). Recently some work has been done on the lower dimensional approximation of boundary value problem for piezoelectric shells (cf. [1]) In this paper, we would like to study the limiting behaviour of the eigenvalue problems for thin piezoelectric shallow shells. We begin with a brief description of the problem and describe the results obtained. Let Ω̂ = Φ(Ω),Ω = ω× (−ǫ, ǫ) with ω ⊂ IR, and the mapping Φ : Ω → IR is given by