The 3-D weight functions for a quasi-static planar crack

We explicitly evaluate the 3-D weight functions for a planar crack in an isotropic, homogeneous material; these give the full stress intensity factors induced by a static point force applied at an arbitrary position. If we Fourier decompose the 3-D weight functions with respect to the z variable then each Fourier mode satis®es the homogeneous equations of elasticity (except at the crack tip) and the boundary conditions on the crack face. Each Fourier mode diverges like r ÿ1/2 near the crack tip and decays exponentially for non-zero kz. It is proved that these necessary conditions, which hold everywhere in the elastic material excluding the crack tip, are also sucient to determine the 3-D weight functions. In particular, the 3-D weight functions can be calculated without considering an explicit loading problem