Fracture Mechanics Online Class

Numerical Methods

This chapter introduces some existing numerical methods dedicated to the simulation of crack propagation:

Numerical Methods > Reminder of LEFM

Definition of elastic fracture

Up to now we have mainly considered elastic fracture mechanics. Strictly speaking, elastic fracture means that the only changes at the material level during the failure are atomic separations. As this way of thinking is too restrictive for real life applications, a pragmatic definition should be: The process zone, which is the region where the inelastic deformations happen, is a small region compared to the specimen sizes (including crack size), and is located at the crack tip. The inelastic deformations may include, among others, plastic flow, micro-fractures, or void growth.

Summary of the previous lectures

The Stress Intensity Factor

For linear elastic materials, the linear elastic stress analysis and the asymptotic solution therefore describe the fracture process with accuracy. In particular for the three modes of fracture (I for opening, II for in-plane sliding and III for out-of-plane shearing) the asymptotic solution is expressed in terms of the SIFs following

\begin{equation}\begin{cases} \mathbf{\sigma}^\text{mode i} = \frac{K_i}{\sqrt{2\pi r}} \mathbf{f}^\text{mode i}(\theta) \\ \mathbf{u}^\text{mode i} = K_i\sqrt{\frac{r}{2\pi}} \mathbf{g}^\text{mode i}(\theta) \end{cases},\label{eq:fandg}\end{equation}

The crack closure integral

The crack closure integral represents the energy required to close the crack on an infinitesimal length $da$.

The J-integral concept

Crack Growth

As previously detailed, the growth can be analyzed as follows: