Stress Intensity Factors

A different approach to estimate whether an existing crack/flaw will grow is by looking at the stress intensity factor (SIF) at the crack tip. In your design classes you have already encountered the concept of stress rise due to stress concentrators.

Under the assumptions of LEFM we can derive the stress field in a cracked body, leading to :

(5)\[ \sigma_{ij} = \left( \frac{k}{\sqrt{r}} \right)f_{ij}(\theta) + \sum_{m=0 }^{\infty}A_Mr^{m/2}g_{ij}^{(m)}(\theta) \]

with \(r\) and \(\theta\) being define at the orifin of the crack tip and counter-clockwise respectively. close to the crack tip, where \(r \to 0\) the second term on the RHS of (8.1) vanishes. As a results, the \(\sigma \propto r^{-1/2}\) relation holds in general for any cracked elastic body.

As you recall we can obtain the displacement directly from the stress field using the elastic constitutive equation to obtain the strains and then integrating them to obtain displacements. This implies that the displacements at the crack tip will be proportional to \(\sqrt{r}\).

As we will see later, when deriving the crack tip fields, it is useful to replace \(k\) in (8.1) with \(K=k\sqrt{2\pi}\).

Moreover, we will add a subscript to \(K\) to differentiate between the different crack opening modes: \(K_I ; K_{II} ; K_{III}\).

\[\begin{split} \lim _{r \to 0} \sigma_{ij}^{(I)} = \frac{K_I}{\sqrt{2 \pi r}}f_{ij}(\theta) \\ \lim _{r \to 0} \sigma_{ij}^{(II)} = \frac{K_{II}}{\sqrt{2 \pi r}}f_{ij}(\theta)\\ \lim _{r \to 0} \sigma_{ij}^{(III)} = \frac{K_{III}}{\sqrt{2 \pi r}}f_{ij}(\theta) \end{split}\]

When encountering a problem of mixed-mode, we can use superposition to find the stresses at the crack tip such that :

\[ \sigma_{ij}^{mixed} = \sigma_{ij}^{I}+\sigma_{ij}^{II}+\sigma_{ij}^{III} \]