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

摘要

A systematic procedure is followed to developsingularity-reduced integral equations for displacementdiscontinuities in homogeneous linear elastic media. Theprocedure 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-singular, weak-form traction integral equation inestablished. 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 finitedomain. The first step in the development is to regularizeSomiglinan's identity by utilizing a stress function for the stressfundamental solution to effect an integration by parts. Theresulting integral equation is valid irrespective of the choice ofstress function (as guaranteed by a certain 'closure condition'established for the integral operator), but certain particular formsof the stress function are introduced and discussed, including onewhich admits an interpretation as a 'line discontinuity'. Asingularity-reduced integral equation for the displacementgradients is then obtained by utilizing a relation between thestress function and the stress fundamental solution along with theclosure condition. This construction does not rely upon aparticular choice of stress function, and the final integralequation (which is a generalization of Mura's (1963) formula)has a kernel which is a simple function of the stress fundamentalsolution. From this relation, singularity-reduced integralequations for the stress and traction are easily obtained. The keystep in the further development is the construction of analternative stress integral equation for which a differentialoperator has been ' factored out' of the integral. This isaccomplished by, in essence, establishing a stress function for thestress field induced by the discontinuity. A weak-form tractionintegral equation is then readily obtained and involves a kernelwhich is only weakly-singular. The nonuniqueness of this kernelis discussed in detail and it is shown that, at least in a certainsense, the kernel which is given is the simplest possible. Theresults for an isolated discontinuity are then adapted to treatcracks in a finite domain. In doing so, emphasis is given to thedevelopment of weakly-singular, weak-from displacement andtraction integral equations wince these form the basis of aneffective numerical procedure for fracture analysis (Li et al.,1998), and such equations are presented for both elastostatics andelastodynamics. A noteworthy aspect of the development is thatthere is no need to introduce Cauchy principal value integralsmuch less Hadamard finite part integrals. Finally, the utility ofthe systematic procedure presented here for use in obtainingsingularity-reduced integral equations for other unbounded media(viz. The half-space and bi-material) is indicated.