Derivation of models for linear viscoelastic shells by using asymptotic analysis

TL;DR

This paper derives simplified two-dimensional models for linear viscoelastic shells by asymptotic analysis as thickness tends to zero, capturing membrane and flexural behaviors with long-term memory effects.

Contribution

It introduces a rigorous asymptotic framework to derive 2D viscoelastic shell models from 3D problems, including membrane and flexural cases, with detailed analysis of limit behaviors.

Findings

Derived 2D viscoelastic membrane and flexural models

Identified conditions for different limit behaviors based on load scaling

Models incorporate long-term memory effects in deformations

Abstract

We consider a family of linear viscoelastic shells with thickness $2\varepsilon$ ( $\varepsilon$ , 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 the three-dimensional problems and their solutions (displacements $\mathbf{u}^\varepsilon$ of components $u_i^\varepsilon$) when $\varepsilon$ tends to zero. To do that, we use asymptotic analysis methods. First, we formulate the variational problem in a fixed domain independent of $\varepsilon$. Then we assume an asymptotic expansion of the scaled displacements field $\mathbf{u}(\varepsilon)=(u_i(\varepsilon))$. Identifying the terms of the proposed asymptotic expansion we characterize the zeroth order term as the solution of a two-dimensional scaled limit problem. Moreover, on one hand, we find that if the applied body force density is $O(1)$ with respect to $\varepsilon$ and surface tractions density is $O(\varepsilon)$, the limit of the field $\mathbf{u}(\varepsilon)$ is the solution of a two-dimensional system of variational equations called viscoelastic membrane problem. On the other hand, if the applied body force density is $O(\varepsilon^2)$ and surface tractions density is $O(\varepsilon^3)$, the limit of the field $\mathbf{u}(\varepsilon)$ is the solution of a system of two-dimensional variational equations called viscoelastic flexural problem. In both cases, we find a model which presents a long-term memory that takes into account the deformations at previous times. We comment on the existence and uniqueness of solution for the two-dimensional variational problems found and announce convergence results in forthcoming papers.