Crack Tip Interpolation, Revisited

It is well known that the near tip displacement field on a crack surface can be represented in a power series in the variable √r, where r is the distance to the tip. It is shown herein that the coefficients of the linear terms on the two sides of the crack are equal. Equivalently, the linear term in the crack opening displacement vanishes. The proof is a completely general argument, valid for an arbitrary (e.g., multiple, nonplanar) crack configuration and applied boundary conditions. Moreover, the argument holds for other equations, such as Laplace. A limit procedure for calculating the surface stress in the form of a hypersingular boundary integral equation is employed to enforce the boundary conditions along the crack faces. Evaluation of the finite surface stress and examination of potentially singular terms lead to the result. Inclusion of this constraint in numerical calculations should result in a more accurate approximation of the displacement and stress fields in the tip region, and thus a more accurate evaluation of stress intensity factors.