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*European Journal of Applied Mathematics*

**Inserted:** 22 jul 2015

**Last Updated:** 28 jan 2016

**Year:** 2015

**Abstract:**

We consider a viscoelastic body occupying a smooth bounded domain $\Omega\subset{\mathbb R}^3$ under the effect of a volumic traction force $g$. The macroscopic displacement vector from the equilibrium configuration is denoted by $u$. Inertial effects are considered; hence the equation for $u$ contains the second order term $u_{tt}$. On a part $\Gamma_D$ of the boundary of $\Omega$, the body is anchored to a support and no displacement may occur; on a second part $\Gamma_N \subset \partial \Omega$, the body can move freely; on a third portion $\Gamma_C \subset \partial \Omega$, the body is in adhesive contact with a solid support. The boundary forces acting on $\Gamma_C$ due to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a nonlinear ODE settled on $\Gamma_C$ and describing the evolution of the delamination order parameter~$z$. Following the lines of a new approach outlined in \cite{BRSS} and based on duality methods in Sobolev-Bochner spaces, we define a suitable concept of weak solution to the resulting PDE system. Correspondingly, we prove an existence result on finite time intervals of arbitrary length.

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