In this paper we prove the existence of solutions for a class of viscoelastic dynamic systems on time-dependent cracked domains, with possibly degenerate viscosity coefficients. Under stronger regularity assumptions we also show a uniqueness result. Finally, we exhibit an example where the energy-dissipation balance is not satisfied, showing there is an additional dissipation due to the crack growth.

}, url = {http://preprints.sissa.it:8180/xmlui/handle/1963/35334}, author = {Maicol Caponi and Francesco Sapio} } @article {2018, title = {Energy-dissipation balance of a smooth moving crack}, number = {SISSA;31/2018/MATE}, year = {2018}, abstract = {In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in [S. Nicaise, A.M. Sandig - J. Math. Anal. Appl., 2007] valid for straight fractures.}, url = {http://preprints.sissa.it/handle/1963/35320}, author = {Maicol Caponi and Ilaria Lucardesi and Emanuele Tasso} } @article {2018, title = {Existence of solutions to a phase field model of dynamic fracture with a crack dependent dissipation}, number = {SISSA;06/2018/MATE}, year = {2018}, abstract = {We propose a phase-field model of dynamic crack propagation based on the Ambrosio-Tortorelli approximation, which takes in account dissipative effects due to the speed of the crack tips. In particular, adapting the time discretization scheme contained in [Bourdin et al., Int. J. Fracture 168 (2011), 133-143] and [Larsen et al., Math. Models Methods Appl. Sci. 20 (2010), 1021-1048], we show the existence of a dynamic crack evolution satisfying an energy dissipation balance, according to Griffith{\textquoteright}s criterion.}, url = {http://preprints.sissa.it/handle/1963/35307}, author = {Maicol Caponi} } @article {2017, title = {Linear hyperbolic systems in domains with growing cracks}, number = {SISSA;05/2017/MATE}, year = {2017}, abstract = {We consider the hyperbolic system $\ddot u-{\rm div}\,(\mathbb A\nabla u)=f$ in the time varying cracked domain $\Omega\setminus\Gamma_t$, where the set $\Omega\subset\mathbb R^d$ is open, bounded, and with Lipschitz boundary, the cracks $\Gamma_t$, $t\in[0,T]$, are closed subsets of $\overline\Omega$, increasing with respect to inclusion, and $u(t):\Omega\setminus\Gamma_t\to\mathbb R^d$ for every $t\in[0,T]$. We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system $\ddot v-{\rm div}\,(\mathbb B\nabla v)+\mathbf a\nabla v -2\nabla\dot vb=g$ on the fixed domain $\Omega\setminus\Gamma_0$. Under these assumptions, we obtain existence and uniqueness of weak solutions for these two problems. Moreover, we show an energy equality for the functions $v$, which allows us to prove a continuous dependence result for both systems.}, url = {http://urania.sissa.it/xmlui/handle/1963/35271}, author = {Maicol Caponi} }