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.

%G eng %U http://preprints.sissa.it:8180/xmlui/handle/1963/35334 %0 Report %D 2018 %T Energy-dissipation balance of a smooth moving crack %A Maicol Caponi %A Ilaria Lucardesi %A Emanuele Tasso %X 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. %G en %U http://preprints.sissa.it/handle/1963/35320 %1 35630 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-08-01T09:36:30Z No. of bitstreams: 1 CLT-preprint.pdf: 455925 bytes, checksum: 931583a37609035d6be19feceb41142d (MD5) %0 Report %D 2018 %T Existence of solutions to a phase field model of dynamic fracture with a crack dependent dissipation %A Maicol Caponi %X 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's criterion. %G en %U http://preprints.sissa.it/handle/1963/35307 %1 35614 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-03-05T16:29:01Z No. of bitstreams: 1 PhFiMo_Caponi.pdf: 418992 bytes, checksum: 6bf6de00d58052411ec3a588e2f7933e (MD5) %0 Report %D 2017 %T Linear hyperbolic systems in domains with growing cracks %A Maicol Caponi %X 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. %G en %U http://urania.sissa.it/xmlui/handle/1963/35271 %1 35577 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-01-30T11:26:46Z No. of bitstreams: 1 Caponi_LinHypSys.pdf: 432435 bytes, checksum: 3b798b4c79150cf0087bc6176258ffc6 (MD5)