In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e. a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement.

%B Mathematical Models and Methods in Applied Sciences %V 28 %P 1371-1412 %G eng %U https://doi.org/10.1142/S0218202518500379 %R 10.1142/S0218202518500379 %0 Journal Article %J Discrete and Continuous Dynamical Systems - Series A 31 (2011) 1219-1231 %D 2011 %T Crack growth with non-interpenetration : a simplified proof for the pure Neumann problem %A Gianni Dal Maso %A Giuliano Lazzaroni %X We present a recent existence result concerning the quasi-static evolution of cracks in hyperelastic brittle materials, in the frame-work of finite elasticity with non-interpenetration. In particular, here we consider the problem where no Dirichlet conditions are imposed, the boundary is traction-free, and the body is subject only to time-dependent volume forces. This allows us to present the main ideas of the proof in a simpler way, avoiding some of the technicalities needed in the general case, studied in. %B Discrete and Continuous Dynamical Systems - Series A 31 (2011) 1219-1231 %I American Institute of Mathematical Sciences %G en_US %U http://hdl.handle.net/1963/3801 %1 526 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-11-27T10:24:55Z\\r\\nNo. of bitstreams: 1\\r\\nDM-Laz-proc-pre.pdf: 195002 bytes, checksum: 44e408b134f19e4d31e77b3e0762cd73 (MD5) %R 10.3934/dcds.2011.31.1219