Fracture Mechanics Online Class

The $J$-integral

This chapter covers the methodologies developed to evaluate the $J$-integral, which was shown to be of importance in non-linear fracture models such as in the HRR theory. Indeed, in elasto-plasticity, the $J$-integral characterizes the crack tip loading, and thus can be used to evaluate failure onset and, under some assumptions, the tearing properties, i.e. the resistance curve during crack growth.

$J$-integral > HRR theory (continuation)

The understanding of the HRR theory is essential to understand the practical use of the $J$-integral theory. The related theory is described in the beforehand chapter.

$J$-integral > Application for elasto-plastic materials

Irreversibility vs. reversibility

 

Picture IX.1: Two cracked samples with two different non-linear material behaviors: non-linear reversible (left) and non-linear irreversible (right).
Picture IX.2: Two different non-linear material behaviors: non-linear reversible (red) and non-linear irreversible (blue).

Let us consider two cracked samples, see Picture IX.1, made of two different materials exhibiting the apparent same behavior during proportional and monotonic loading:

Therefore, in the absence of crack propagation, the two samples have the same stress distributions with $G_2=G_1$ and $J_2=J_1$. As a result $J_2=G_2$ for elasto-plasticity at the conditions, on the one hand, that the loading remains proportional and monotonic (i.e. there is no crack propagation) and, on the other hand, that the crack will subsequently growth straight ahead.

The interest in evaluating the $J$-integral for a given cracked body configuration is thus high in elasto-plasticity since it allows

Evaluation of the $J$-integral

In the following we will thus be interested, on the one hand on how to evaluate the $J$-integral:

and, on the other hand, on how to experimentally evaluate the critical $J_C$: