Linear viscoelastic shells: An asymptotic approach

Abstract

We consider a family of linear viscoelastic shells with thickness $2\mathit{\epsilon }$ (where ε is a small parameter), clamped along a portion of their lateral face, all having the same middle surface S. We formulate the three-dimensional mechanical problem in curvilinear coordinates and provide existence and uniqueness of (weak) solution of the corresponding three-dimensional variational problem.

We are interested in studying the limit behavior of both the three-dimensional problems and their solutions when ε tends to zero. To do that, we use asymptotic analysis methods. First, we formulate the variational problem in a fixed domain independent of ε. Then we assume an asymptotic expansion of the scaled displacements field, $\mathit{u}\mathrm{(}\mathit{\epsilon }\mathrm{)}=\mathrm{(}{\mathit{u}}_{\mathit{i}}\mathrm{(}\mathit{\epsilon }\mathrm{)}\mathrm{)}$ , and we characterize the zeroth order term as the solution of a two-dimensional scaled limit problem. Moreover, we find that, depending on the order of the applied forces, the limit of the field $\mathit{u}\mathrm{(}\mathit{\epsilon }\mathrm{)}$ is the solution of one of the two sets of two-dimensional variational equations derived, which can be described as viscoelastic membrane shell and viscoelastic flexural shell problems. In both cases, we find a model which presents a long-term memory that takes into account the deformations at previous times. We finally comment on the existence and uniqueness of solution for the two-dimensional variational problems found and announce convergence results.