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Stress Intensity Factor and Crack Tip Stresses

Crack tips produce a 1/Sqrt(r) singularity. The stress fields near a crack tip of an isotropic linear elastic material can be expressed as a product of 1/Sqrt(r) and a function of theta with a scaling factor K:

where the superscripts and subscripts I, II, and III denote the three different modes that different loadings may be applied to a crack. The detailed breakdown of stresses and displacements for each mode are summarized in this page. The factor K is called the Stress Intensity Factor.

Stress Intensity Factor in Practice

Engineers are interested in the maximum stress near the crack tip and whether it exceeds the fracture toughness. Thus, the stress intensity factor K is commonly expressed in terms of the applied stresses sigma at r ~ 0 and theta = 0.

For example, for a through crack in an infinite plate under uniform tension sigma, the stress intensity factor is


where a is one half of the width of the through crack. The dimension of K is


In the last few decades, many closed-form solutions of the stress intensity factor K for simple configurations were derived. Some of the common ones are listed in the following three categories: classic, specimen, and structure.

Stress Intensity Factor and Fracture Toughness

Based on the linear theory the stresses at the crack tip are infinity but in reality there is always a plastic zone at the crack tip that limits the stresses to finite values. It is very difficult to model and calculate the actual stresses in the plastic zone and compare them to the maximum allowable stresses of the material to determine whether a crak is going to grow or not.

An engineering approach is to perform a series of experiments and reach at a critical stress intensity factor Kc for each material, called the fracture toughness of the material. One can then determine the crack stability by comparing K and Kc directly.

Kc's for a number of common engineering materials are listed in this page.

Relationship between G and K

Some literature may prefer using strain energy release rate G over stress intensity factor K. These two factors are however directly related by the following formulas:

The dimension of G is