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

TL;DR

This paper establishes the existence of quasistatic evolutions in a cohesive fracture model with fatigue effects, where energy dissipation depends on the total variation of crack opening, capturing fatigue phenomena in small-strain elasticity.

Contribution

It introduces a novel model incorporating fatigue via energy dissipation depending on crack jump variation and proves the existence of quasistatic evolutions using Young measures.

Findings

Existence of quasistatic evolutions in the model.

Fatigue phenomena can cause complete fracture through small oscillations.

Weak formulation with Young measures effectively handles jump variation control.

Abstract

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.