  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.
