Lower dimensional approximation of eigenvalue problem for thin piezoelectric shells with non-uniform thickness.
Mathematics
We first consider thin piezoelectric shallow shells with variable thickness and show that the eigenvalues of the three dimensional problem are order square of thickness the corresponding scaled eigensolutions converge to the eigensolutions of a two dimensional model. We then consider flexural shells (for eq: if the middle surface is a plate or if it is flat in a small region or cylindrical shells etc) with variable thickness and show that if the space of inextensional displacement is infinite dimensional, then the eigenvalues are of order square of thickness and the corresponding eigensolutions converge to the eigensolutions of two dimensional flexural shell model.
Ref: Mathai, Job, Sabu, N.; Lower dimensional approximation of eigenvalue problem for piezoelectric shells with nonuniform thickness. Indian J. Math. 63 (2021), no. 1, 1–33.