Cohesive zone model

Definition

Let a body contain a surface $\Gamma$ along which a crack may open. Let $\delta = [u] \cdot \nu$ be the normal opening, that is the jump of the displacement $u$ across $\Gamma$ in the direction of the unit normal $\nu$. A traction–separation law is a function $T \colon [0, \infty) \to [0, \infty)$ and a number $\delta_c > 0$ such that:

1. $T(\delta) = 0$ for $\delta \ge \delta_c$;
2. $T$ is integrable, and $G_c = \int_0^{\delta_c} T(\delta) \, d\delta$ is positive.

A cohesive zone model is the boundary value problem of the body with the following condition on $\Gamma$. Wherever $\delta > 0$, the normal traction $\sigma \nu \cdot \nu$ on both faces of $\Gamma$ equals $T(\delta)$. The cohesive zone is the part of $\Gamma$ where $0 < \delta < \delta_c$, and the crack is fully open where $\delta \ge \delta_c$. The number $G_c$ is the fracture energy and $\max T$ is the cohesive strength.