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A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints

We consider a viscoelastic body occupying a smooth bounded domain of $R^3$ under the effects of volumic traction forces. Inertial effects are considered: hence, the equation describing the evolution of displacements is of the second order in time. On a part of the boundary of the domain, the body is anchored to a support and no displacement may occur; on a second part, the body can move freely; on a third portion of the boundary, the body is in adhesive contact with a solid support. The boundary forces acting there as a byproduct of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a boundary variable that represents the local fraction of active bonds and is assumed to satisfy a doubly nonlinear ODE. Following the lines of a new approach based on duality methods in Sobolev-Bochner spaces, we define a suitable concept of weak solution to the resulting system of equations. Correspondingly, we prove an existence result on finite time intervals of arbitrary length.