ORTHOGONAL POLYNOMIAL SOLUTIONS FOR ELLIPTICAL CRACKS UNDER SHEAR LOADINGS

An infinite solid contains a flat elliptical crack, occupying the surface(in Cartesian coordinates)

The crack faces are subjected to prescribed shear stresses, with Cartesian components q_x and q_y the corresponding crack-face displacements have components ±u_x and ±u_y. We expand u_x, u_y, q_x and q_y as Fourier series in ø and expand each Fourier component as a series of orthodonal polynomials in p. We obtain explicit relations (systems of linear algebraic equations) between the coefficients in these series, and derive simple formulae for the stress-intensity factors. Our systems of equations (i) yield the known analytical solutions for uniform shear and simple torsion, (ii) reduce to those obtained by Krenk for the penny-shaped crack (a=b), and (iii) are computationally attractive for arbitrary polynomial shear loadings.