We prove an existence result for the fractional Kelvin–Voigt’s model involving Caputo’s derivative on time-dependent cracked domains. We first show the existence of a solution to a regularized version of this problem. Then, we use a compactness argument to derive that the fractional Kelvin–Voigt’s model admits a solution which satisfies an energy-dissipation inequality. Finally, we prove that when the crack is not moving, the solution is unique.

1 aCaponi, Maicol1 aSapio, Francesco uhttps://doi.org/10.1007/s00028-021-00713-200903nas a2200157 4500008004100000020001400041245007800055210006900133260001500202300001600217490000800233520041700241100001900658700002100677856004700698 2020 eng d a1618-189100aA dynamic model for viscoelastic materials with prescribed growing cracks0 adynamic model for viscoelastic materials with prescribed growing c2020/08/01 a1263 - 12920 v1993 aIn 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.

1 aCaponi, Maicol1 aSapio, Francesco uhttps://doi.org/10.1007/s10231-019-00921-100964nas a2200205 4500008004100000020001400041245005600055210005500111260001600166300001100182490000800193520032700201653003100528653002200559653004400581100001900625700002200644700002000666856007200686 2020 eng d a0022-247X00aEnergy-dissipation balance of a smooth moving crack0 aEnergydissipation balance of a smooth moving crack c2020/03/15/ a1236560 v4833 aIn 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.

10aEnergy-dissipation balance10aFracture dynamics10aWave equation in time-dependent domains1 aCaponi, Maicol1 aLucardesi, Ilaria1 aTasso, Emanuele uhttps://www.sciencedirect.com/science/article/pii/S0022247X1930924201243nas a2200145 4500008004100000020001400041245010900055210007100164260001500235300000700250490000700257520076800264100001901032856004601051 2020 eng d a1420-900400aExistence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation0 aExistence of solutions to a phase–field model of dynamic fractur c2020/02/11 a140 v273 aWe 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.

1 aCaponi, Maicol uhttps://doi.org/10.1007/s00030-020-0617-z01454nas a2200145 4500008004100000020001400041245006100055210006100116260001500177300001400192490000700206520103000213100001901243856004601262 2017 eng d a1424-929400aLinear Hyperbolic Systems in Domains with Growing Cracks0 aLinear Hyperbolic Systems in Domains with Growing Cracks c2017/06/01 a149 - 1850 v853 aWe 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.

1 aCaponi, Maicol uhttps://doi.org/10.1007/s00032-017-0268-7