Limit models in the analysis of three different layered elastic strips

The mechanical behavior of three types of laminated strips is investigated. They are made of three layers filled with homogeneous, isotropic and elastic materials; the upper and lower layer are called adherents, the middle layer is called adhesive. The first model studies a strip consisting of three layers made of materials with similar stiffness; the second one concerns with a strip in which the adhesive is soft; in particular, we suppose that the elastic stiffness of the middle layer is two orders of magnitude smaller than that of the upper and lower layers; the third case is a strip in which the core is thinner and stiffer than the two adherents: the elastic modula of the adherents are one order of magnitude bigger that those of the adhesive. After identifying a parameter of smallness ε (which measures the thickness and the stiffness of each layer), the limit of the solution when ε tends to zero has been considered. Afterwards, it has been shown that each solution of the simplified models verifies the so-called limit problems, written using a “weak” and a “strong” formulation. The existence and uniqueness of the solutions of each limit problem have been established. The strong convergence of the exact solutions towards the solution of the limit problem of the first model has been established, too.