Calculus of Variations and Geometric Measure Theory

V. Crismale - G. Lazzaroni - G. Orlando

Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue

created by lazzaroni on 13 Jul 2016
modified on 23 Jan 2019


Published Paper

Inserted: 13 jul 2016
Last Updated: 23 jan 2019

Journal: Math. Models Methods Appl. Sci.
Volume: 28
Pages: 1371-1412
Year: 2018
Doi: 10.1142/S0218202518500379

ArXiv: 1802.04098 PDF


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.