A numerical scheme based on the Fourier series method is developed for the solution of Laplace’s equation on a polygonal domain. This scheme is applied to the solution of the pure mode III fracture mechanics problem. For the single edge cracked strip problem under out-of-plane loading, the value of the J-Integral evaluated from the Fourier solution was found to be in good agreement with classical solution. The real power of the scheme lies in its applicability to general polygonal shapes. This capability is illustrated by three sample problems that involve strips with a slanted crack, multiple cracks and branching cracks under mode III loading.

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