A mathematical analysis of a one dimensional model for dynamic debonding

In this thesis we develop a mathematical analysis for a dynamic model of peeling test in dimension one. In the first part we give existence and uniqueness results for dynamic evolutions. In the second part we study the quasistatic limit of such evolutions, i.e., the limit as inertia tends to zero. In the model the wave equation $u_{tt}-u_{xx}=0$ is coupled with a Griffith's criterion for the propagation of the debonding front. Our first results provide existence and uniqueness for the solution to this coupled problem under different assumptions on the data. This analysis is extended when we study the initiation of the debonding process. We also give an existence and uniqueness result for solutions to the damped wave equation $u_{tt}-u_{xx}+u_t=0$ in a time-dependent domain whose evolution depends on the given debonding front. We then analyse the quasistatic limit without damping. We find that the limit evolution satisfies a stability condition; however, the activation rule in Griffith's (quasistatic) criterion does not hold in general, thus the limit evolution is not rate-independent. This behaviour is due to the oscillations of the kinetic energy and of the presence of an acceleration term in the limit. The same phenomenon is observed even in the case of a singularly perturbed second order equation $\eps^2 \ddot{u}_\eps + V_{x}(t,u_\eps(t))=0$, where $V(t,x)$ is a potential. We assume that $u_0(t)$ is one of its equilibrium points such that $V_x(t,u_0(t))=0$ and $V_{xx}(t,u_0(t))>0$. We find that, under suitable initial data, the solutions $u_\eps$ converge uniformly to $u_0$, by imposing mild hypotheses on $V$. However, a counterexample shows that such assumptions cannot be weakened. Thus, inertial effects can not, in general, be captured by a pure quasistatic analysis.