A Fourth Order Formulation of DDM for Crack Analysis in Brittle Solids

A fourth order formulation of the displacement discontinuity method (DDM) is proposed for the crack analysis of brittle solids such as rocks, glasses, concretes and ceramics. A fourth order boundary collocation scheme is used for the discretization of each boundary element (the source element). In this approach, the source boundary element is divided into five sub-elements each recognized by a central node where the displacement discontinuity components are to be numerically evaluated. Three different formulating procedures are presented and their corresponding discretization schemes are discussed. A new discretization scheme is also proposed to use the fourth order formulation for the special crack tip elements which may be used to increase the accuracy of the stress and displacement fields near the crack ends. Therefore, these new crack tips discretizing schemes are also improved by using the proposed fourth order displacement discontinuity formulation and the corresponding shape functions for a bunch of five special crack tip elements. Some example problems in brittle fracture mechanics are solved for estimating the Mode I and Mode II stress intensity factors near the crack ends. These semi-analytical results are compared to those cited in the fracture mechanics literature whereby the high accuracy of the fourth order DDM formulation is demonstrated.