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.

%I European Mathematical Society %G en %U http://hdl.handle.net/1963/6983 %1 6973 %2 Mathematics %4 -1 %$ Submitted by Simone Racca (sracca@sissa.it) on 2013-07-18T08:39:00Z No. of bitstreams: 1 Racca_Toader.pdf: 416939 bytes, checksum: cf459548a10944037e56b7504fe60f51 (MD5) %R 10.4171/IFB/328 %0 Journal Article %J Advances in Calculus of Variations 5 (2012) 433-483 %D 2012 %T A Viscosity-driven crack evolution %A Simone Racca %XWe 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.

%B Advances in Calculus of Variations 5 (2012) 433-483 %I SISSA %G en %U http://hdl.handle.net/1963/5130 %1 4944 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2011-11-29T07:47:42Z\\r\\nNo. of bitstreams: 1\\r\\n63M_Racca.pdf: 473110 bytes, checksum: f4d49c3f7e2b984e9694fbd66a806447 (MD5) %R 10.1515/acv-2011-0012