Adhesion and debonding in a model of elastic string

We study the problem of adhesion and debonding of an elastic body interacting with a rigid substrate through a layer made of soft breakable adhesive material. The general problem is formulated in the multidimensional vectorial case, while a detailed analysis is carried for the one dimensional case leading to the study of a 1D semilinear wave equation. The initial boundary value problem is affected by the presence of a source term characterizing the interaction of the string withe the adhesive layer. This discontinuous force jumps to zero when a critical value of the displacement is reached. We obtain conditions for the initial boundary value problem ensuring the regularity of the solutions and the attachment–detachment conditions. Finally, we focus on a numerical investigation of the problem by considering regularizations of the source term and different initial conditions.