AbstractIn this article, the boundary element method is implemented for solving linear elastic fracture mechanics problems, using new complex Fourier shape functions which include constant and linear complex Fourier shape functions. They, unlike the classic Lagrange shape functions, can satisfy the exponential and trigonometric function fields in addition to the polynomial ones. Six numerical examples are provided to investigate the new shape functions and also, the stress intensity factors (SIF) are calculated for each so that the accuracy of the present study is illustrated. The results are compared with those obtained by classic Lagrange ones and the existing methods.