Affiliations: [a] Departamento de Matemática Aplicada, Univ. de Santiago de Compostela, Spain. E-mail: [email protected] | [b] Departamento de Matemáticas, Univ. da Coruña, Spain. E-mail: [email protected]
Abstract: We consider a family of linear viscoelastic shells with thickness 2ε (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, u(ε)=(ui(ε)), 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 u(ε) 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.
Keywords: Asymptotic analysis, viscoelasticity, shells, membrane, flexural, time dependent
