A bending–stretching model in adhesive contact for elastic rods obtained by using asymptotic methods

This paper is devoted to the derivation and mathematical justification of models for the bending–stretching of an elastic rod in adhesive contact with a deformable foundation. The process is assumed to be quasistatic, and therefore the effects of inertia are neglected. Contact is modeled with normal compliance and the adhesion is modeled by introducing a surface internal variable, the bonding function, the evolution of which is described by an ordinary differential equation. To derive the models we consider the three-dimensional contact problem of an elastic body in adhesive contact with a foundation, introduce a change of variable together with the scaling of the unknowns and parameters of the problem, and we obtain a limit model under the assumption of suitable asymptotic expansions for the scaled unknowns. After that, we obtain error estimates and convergence results which legitimate the limit model. Finally we show that our limit model contains as particular cases models previously considered by other authors. To our knowledge it is for the first time that a rigorous justification and a generalization of those models is provided.