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.

SN - 1424-3202 UR - https://doi.org/10.1007/s00028-021-00713-2 JO - Journal of Evolution Equations ER - TY - JOUR T1 - A dynamic model for viscoelastic materials with prescribed growing cracks Y1 - 2020 A1 - Maicol Caponi A1 - Francesco Sapio AB -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.

VL - 199 SN - 1618-1891 UR - https://doi.org/10.1007/s10231-019-00921-1 IS - 4 JO - Annali di Matematica Pura ed Applicata (1923 -) ER - TY - JOUR T1 - Energy-dissipation balance of a smooth moving crack Y1 - 2020 A1 - Maicol Caponi A1 - Ilaria Lucardesi A1 - Emanuele Tasso KW - Energy-dissipation balance KW - Fracture dynamics KW - Wave equation in time-dependent domains AB -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 [10] valid for straight fractures.

VL - 483 SN - 0022-247X UR - https://www.sciencedirect.com/science/article/pii/S0022247X19309242 IS - 2 JO - Journal of Mathematical Analysis and Applications ER - TY - JOUR T1 - Existence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation Y1 - 2020 A1 - Maicol Caponi AB -We 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.

VL - 27 SN - 1420-9004 UR - https://doi.org/10.1007/s00030-020-0617-z IS - 2 JO - Nonlinear Differential Equations and Applications NoDEA ER - TY - JOUR T1 - Linear Hyperbolic Systems in Domains with Growing Cracks Y1 - 2017 A1 - Maicol Caponi AB -We 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.

VL - 85 SN - 1424-9294 UR - https://doi.org/10.1007/s00032-017-0268-7 IS - 1 JO - Milan Journal of Mathematics ER -