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.

%8 2021/06/04 %@ 1424-3202 %G eng %U https://doi.org/10.1007/s00028-021-00713-2 %! Journal of Evolution Equations %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