  A quasistatic model for a horizontally loaded thin elastic composite at small
strains is studied. The composite consists of two adjacent plates whose
interface behaves in a cohesive fashion with respect to the slip of the two
layers. We allow for different loading-unloading regimes, distinguished by the
presence of an irreversible variable describing the maximal slip reached during
the evolution. Existence of energetic solutions, characterized by equilibrium
conditions together with energy balance, is proved by means of a suitable
version of the Minimizing Movements scheme. A crucial tool to achieve
compactness of the irreversible variable are uniform estimates in H\"older
spaces, obtained through the regularity theory for elliptic systems. The case
in which the two plates may undergo a damage process is also considered.
