Westergaard Stress Function Approach

The function Z(z) is analytic and therefore satisfies the relationship V Z(z) = 0. The derivatives of the function Z(z) are defined as follows  

Because the derivatives of analytic functions are also analytic and are harmonic, the chosen function th(z) satisfies the biharmonic equation = 0. Note also that Eqn. (3.21) is a special case of Eqn. (3.15), in which only the first and third functions are retained, namely  

Based on the definition of stresses in terms of the Airy stress function, given in Eqn. (3.10), one obtains from the Westergaard function  

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To illustrate the use of the Westergaard stress function approach, the case of a central crack of length 2a in an infinitely large thin plate (i.e., for generalized plane... 

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. 

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