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We consider a family of linear viscoelastic shells with thickness$2\varepsilon$ ( $\varepsilon$ , small parameter), clamped along a portion oftheir lateral face, all having the same middle surface $S$. We formulate thethree-dimensional mechanical problem in curvilinear coordinates and provideexistence and uniqueness of (weak) solution of the correspondingthree-dimensional variational problem.  We are interested in studying the limit behavior of the three-dimensionalproblems and their solutions (displacements $\mathbf{u}^\varepsilon$ ofcomponents $u_i^\varepsilon$) when $\varepsilon$ tends to zero. To do that, weuse asymptotic analysis methods. First, we formulate the variational problem ina fixed domain independent of $\varepsilon$. Then we assume an asymptoticexpansion of the scaled displacements field$\mathbf{u}(\varepsilon)=(u_i(\varepsilon))$. Identifying the terms of theproposed asymptotic expansion we characterize the zeroth order term as thesolution of a two-dimensional scaled limit problem. Moreover, on one hand, wefind that if the applied body force density is $O(1)$ with respect to$\varepsilon$ and surface tractions density is $O(\varepsilon)$, the limit ofthe field $\mathbf{u}(\varepsilon)$ is the solution of a two-dimensional systemof variational equations called viscoelastic membrane problem. On the otherhand, if the applied body force density is $O(\varepsilon^2)$ and surfacetractions density is $O(\varepsilon^3)$, the limit of the field$\mathbf{u}(\varepsilon)$ is the solution of a system of two-dimensionalvariational equations called viscoelastic flexural problem.  In both cases, we find a model which presents a long-term memory that takesinto account the deformations at previous times. We comment on the existenceand uniqueness of solution for the two-dimensional variational problems foundand announce convergence results in forthcoming papers.

Derivation of models for linear viscoelastic shells by using asymptotic analysis