We propose a variational model for the irreversible quasi-static evolution of brittle fractures having fractional Hausdorff dimension in the setting of two-dimensional antiplane and plane elasticity. The evolution along such irregular crack paths can be obtained as $\Gamma$-limit of evolutions along one-dimensional cracks when the fracture toughness tends to zero.

10aVariational models1 aRacca, Simone1 aToader, Rodica uhttp://hdl.handle.net/1963/698301110nas a2200109 4500008004100000245003900041210003600080260001000116520082000126100001800946856003600964 2012 en d00aA Viscosity-driven crack evolution0 aViscositydriven crack evolution bSISSA3 aWe present a model of crack growth in brittle materials which couples dissipative effects on the crack tip and viscous effects. We consider the 2 -dimensional antiplane case with pre-assigned crack path, and firstly prove an existence result for a rate-dependent evolution problem by means of time-discretization. The next goal is to describe the rate-independent evolution as limit of the rate-dependent ones when the dissipative and viscous effects vanish. The rate-independent evolution satisfies a Griffithâ€™s criterion for the crack growth, but, in general, it does not fulfil a global minimality condition; its fracture set may exhibit jump discontinuities with respect to time. Under suitable regularity assumptions, the quasi-static crack growth is described by solving a finite-dimensional problem.

1 aRacca, Simone uhttp://hdl.handle.net/1963/5130