Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media

• Published in: International journal of fracture 1998-09, Vol.93 (1-4), p.87-114
• Main Authors: Li, Songshan, Mear, Mark E
• Format: Article
• Language: English
• Subjects: Differential equations; Discontinuity; Dislocations; Displacement; Domains; Elastic media; Elastodynamics; Elastostatics; Fracture mechanics; Half spaces; Integral equations; Kernels; Mathematical analysis; Operators (mathematics); Singularity (mathematics); Stress distribution; Stress functions; Traction
• ISSN: 0376-9429; 1573-2673
• DOI: 10.1023/A:1007513307368

A systematic procedure is followed to develop singularity-reduced integral equations for displacement discontinuities in homogeneous linear elastic media. The procedure readily reproduces and generalizes, in a unified manner, various integral equations previously developed by other means, and it leads to a new stress relation from which a general weakly-singular, weak-form traction integral equation is established. An isolated discontinuity is treated first (including, as special cases, cracks and dislocations) after which singularity-reduced integral equations are obtained for cracks in a finite domain. The first step in the development is to regularize Somigliana's identity by utilizing a stress function for the stress fundamental solution to effect an integration by parts. The resulting integral equation is valid irrespective of the choice of stress function (as guaranteed by a certain ‘closure condition’ established for the integral operator), but certain particular forms of the stress function are introduced and discussed, including one which admits an interpretation as a ‘line discontinuity’. A singularity-reduced integral equation for the displacement gradients is then obtained by utilizing a relation between the stress function and the stress fundamental solution along with the closure condition. This construction does not rely upon a particular choice of stress function, and the final integral equation (which is a generalization of Mura's (1963) formula) has a kernel which is a simple function of the stress fundamental solution. From this relation, singularity-reduced integral equations for the stress and traction are easily obtained. The key step in the further development is the construction of an alternative stress integral equation for which a differential operator has been ‘factored out’ of the integral. This is accomplished by, in essence, establishing a stress function for the stress field induced by the discontinuity. A weak-form traction integral equation is then readily obtained and involves a kernel which is only weakly-singular. The nonuniqueness of this kernel is discussed in detail and it is shown that, at least in a certain sense, the kernel which is given is the simplest possible. The results for an isolated discontinuity are then adapted to treat cracks in a finite domain. In doing so, emphasis is given to the development of weakly-singular, weak-form displacement and traction integral equations since these form the basis of an ef