This question is designed to illustrate the difference between the accurate solution for crack tip stresses, and the approximate solution (using only the first term in the expression, which contains the singularity) on which the stress intensity factor, K, is based. It should take about 15 minutes to complete.

For a through-thickness crack in an infinite plate, the tensile stress distribution ahead of the crack tip is accurately described by the equation:

where x is distance along the crack from its centre-line, as shown in the diagram.

In terms of the stress intensity factor, the stress very near to the crack tip can be approximated as:

where r is distance ahead of the crack tip.

What is the difference between the stress predictions of these two equations at a distance r = 0.02a ahead of the crack tip?