On the Justification of Viscoelastic Elliptic Membrane Shell Equations

Abstract

We consider a family of linearly viscoelastic shells with thickness 2ε, clamped along a portion of their lateral face, all having the same middle surface S = θ(ω) ⊂ IR 3 , where ω ⊂ IR 2 is a bounded and connected open set with a Lipschitz-continuous boundary γ. We show that, if the applied body force density is O(1) with respect to ε and surface tractions density is O(ε), the solution of the scaled variational problem in curvilinear coordinates, defined over the fixed domain Ω = ω × (−1, 1), converges in ad hoc functional spaces to a limit u as ε → 0 . Furthermore, the average u(ε) = 1 2 1 −1 u(ε)dx 3 , converges in an ad hoc space to the unique solution of what we have identified as (scaled) two-dimensional equations of a viscoelastic generalized membrane shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

Derivation of models for linear viscoelastic shells by using asymptotic analysis

“…The asymptotic approaches need to be mathematically justified in order to ensure robust results. To this end, guided by the formal analysis developed in this paper, a more deep and robust study including convergence theorems will be presented in forthcoming papers ( [2,3,4]), regarding the different cases that have appeared in this work.…”

“…This will be done by taking into account the type of the middle surface of the family of shells and the subset where the boundary condition of place is considered. Therefore, on one hand, we shall study in [2] the case when S is elliptic and when γ 0 = γ, this is V 0 (ω) = {0} (which implies V F (ω) = {0}). These are known as viscoelastic elliptic membrane shells.…”

“…These are known as viscoelastic elliptic membrane shells. On the other hand, in [4] we shall consider the cases when the membrane is not elliptic or γ 0 = γ but still V F (ω) = {0}. For these cases, additional spaces must be considered in order to obtain well posed problems.…”

“…We shall present the limit problems in a de-scaled form. The details of the convergence and the physical interpretation of the solutions for those problems are subject of forthcoming papers ( [2,3,4]). There we shall see that in fact, the subspace which plays the key role in differentiating viscoelastic membrane shells from viscoelastic flexural shells is V F (ω) instead of V 0 (ω), as happened in the elastic case (see [7]).…”

Full Discretization Scheme for the Dynamics of Elliptic Membrane Shell Model

“…For viscoelastic elliptic membrane shell, there are also very few theoretical results. In [16], for instance, the authors justified the two-dimensional equations of a viscoelastic elliptic membrane shell, which includes a long-term memory that takes into account previous deformations. In [17], Dong et al provided a multi-scale computational method for dynamic thermo-mechanical performance of heterogeneous shell structures.…”