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.

%V 199 %P 1263 - 1292 %8 2020/08/01 %@ 1618-1891 %G eng %U https://doi.org/10.1007/s10231-019-00921-1 %N 4 %! Annali di Matematica Pura ed Applicata (1923 -) %0 Journal Article %D 2020 %T Energy-dissipation balance of a smooth moving crack %A Maicol Caponi %A Ilaria Lucardesi %A Emanuele Tasso %K Energy-dissipation balance %K Fracture dynamics %K Wave equation in time-dependent domains %XIn 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 [10] valid for straight fractures.

%V 483 %P 123656 %8 2020/03/15/ %@ 0022-247X %G eng %U https://www.sciencedirect.com/science/article/pii/S0022247X19309242 %N 2 %! Journal of Mathematical Analysis and Applications %0 Journal Article %D 2020 %T Existence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation %A Maicol Caponi %XWe propose a phase–field model of dynamic fracture based on the Ambrosio–Tortorelli’s approximation, which takes into account dissipative effects due to the speed of the crack tips. By adapting the time discretization scheme contained in Larsen et al. (Math Models Methods Appl Sci 20:1021–1048, 2010), we show the existence of a dynamic crack evolution satisfying an energy–dissipation balance, according to Griffith’s criterion. Finally, we analyze the dynamic phase–field model of Bourdin et al. (Int J Fract 168:133–143, 2011) and Larsen (in: Hackl (ed) IUTAM symposium on variational concepts with applications to the mechanics of materials, IUTAM Bookseries, vol 21. Springer, Dordrecht, 2010, pp 131–140) with no dissipative terms.

%V 27 %P 14 %8 2020/02/11 %@ 1420-9004 %G eng %U https://doi.org/10.1007/s00030-020-0617-z %N 2 %! Nonlinear Differential Equations and Applications NoDEA %0 Journal Article %D 2017 %T Linear Hyperbolic Systems in Domains with Growing Cracks %A Maicol Caponi %XWe consider the hyperbolic system ü$${ - {\rm div} (\mathbb{A} \nabla u) = f}$$in the time varying cracked domain $${\Omega \backslash \Gamma_t}$$, 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 $${\bar{\Omega}}$$, increasing with respect to inclusion, and $${u(t) : \Omega \backslash \Gamma_t \rightarrow \mathbb{R}^d}$$for every $${t \in [0, T]}$$. We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system v̈$${ - {\rm div} (\mathbb{B}\nabla v) + a\nabla v - 2 \nabla \dot{v}b = g}$$on the fixed domain $${\Omega \backslash \Gamma_0}$$. 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. The same study has already been carried out in [3, 7] in the scalar case.

%V 85 %P 149 - 185 %8 2017/06/01 %@ 1424-9294 %G eng %U https://doi.org/10.1007/s00032-017-0268-7 %N 1 %! Milan Journal of Mathematics