Stresses at a Crack Tip and Definition of Stress Intensity Factor

3 Stresses at a Crack Tip and Definition of Stress Intensity Factor 

By using the Westergaard approach and the Airy stress function, the stresses near the tip of a crack may be considered (Fig. 3.2). A set of in-plane Cartesian coordinates X and y, or polar coordinates r and 0, is chosen, with the origin at the crack tip. The boundary conditions are as follows (i) stresses at the crack tip are very large and (ii) the crack surfaces are stress free. 

The physical requirement that strain energy in the elastic body be finite (or bounded) suggests that the order of singularity of stresses at the crack tip can be represented at most by (The basic reasoning is that, with stress and strain proportional to strain energy density would be proportional to The strain 

This condition is satisfied only if g(x) is real along y = 0 (because /x is imaginary for X 0). Taking a Taylor series expansion around the origin, the function g(z) close to the origin then becomes 

Very close to the crack tip, therefore, g(z) g(0) = real constant. The constant is identified with the stress intensity factor Kj, and for convenience 

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