We prove an existence result for the fractional Kelvin{\textendash}Voigt{\textquoteright}s model involving Caputo{\textquoteright}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{\textendash}Voigt{\textquoteright}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.

}, isbn = {1424-3202}, url = {https://doi.org/10.1007/s00028-021-00713-2}, author = {Maicol Caponi and Francesco Sapio} } @article {2020, title = {A dynamic model for viscoelastic materials with prescribed growing cracks}, volume = {199}, year = {2020}, month = {2020/08/01}, pages = {1263 - 1292}, abstract = {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.

}, isbn = {1618-1891}, url = {https://doi.org/10.1007/s10231-019-00921-1}, author = {Maicol Caponi and Francesco Sapio} } @article {2020, title = {Energy-dissipation balance of a smooth moving crack}, volume = {483}, year = {2020}, month = {2020/03/15/}, pages = {123656}, 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 [10] valid for straight fractures.

}, keywords = {Energy-dissipation balance, Fracture dynamics, Wave equation in time-dependent domains}, isbn = {0022-247X}, url = {https://www.sciencedirect.com/science/article/pii/S0022247X19309242}, author = {Maicol Caponi and Ilaria Lucardesi and Emanuele Tasso} } @article {2020, title = {Existence of solutions to a phase{\textendash}field model of dynamic fracture with a crack{\textendash}dependent dissipation}, volume = {27}, year = {2020}, month = {2020/02/11}, pages = {14}, abstract = {We propose a phase{\textendash}field model of dynamic fracture based on the Ambrosio{\textendash}Tortorelli{\textquoteright}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{\textendash}1048, 2010), we show the existence of a dynamic crack evolution satisfying an energy{\textendash}dissipation balance, according to Griffith{\textquoteright}s criterion. Finally, we analyze the dynamic phase{\textendash}field model of Bourdin et al. (Int J Fract 168:133{\textendash}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{\textendash}140) with no dissipative terms.

}, isbn = {1420-9004}, url = {https://doi.org/10.1007/s00030-020-0617-z}, author = {Maicol Caponi} } @article {2017, title = {Linear Hyperbolic Systems in Domains with Growing Cracks}, volume = {85}, year = {2017}, month = {2017/06/01}, pages = {149 - 185}, abstract = {We consider the hyperbolic system {\"u}$${ - {\rm div} (\mathbb{A} \nabla u) = f}$$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.

}, isbn = {1424-9294}, url = {https://doi.org/10.1007/s00032-017-0268-7}, author = {Maicol Caponi} }