A crack embedded in a half-plane solid traction-free on the infinite straight boundary is analyzed. The response of the material is linear elastic. A system of singular integral equations for the unknown dislocation densities defined on the crack faces is derived. These equations are then specialized to the problem of a crack located arbitrarily in an orthotropic material which are found to depend on two material parameters only. For a crack oriented either perpendicular or parallel to the infinite straight boundary, the kernel functions appearing in the singular integral equations are obtained in real form which are valid for arbitrary alignment of the orthotropic material. Furthermore, these kernel functions are found to be valid even for degenerate materials and can directly lead to those kernel functions for isotropic materials. Numerical results have been carried out for horizontal or vertical crack problems to elucidate the effect of material parameters on the stress intensity factors. The effect of the alignment of the material on the stress intensity factors is also presented for degenerate materials.