Singularityreduced integral equations for displacement discontinuities in threedimensional linear elastic media
 Songshan Li,
 Mark E. Mear
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A systematic procedure is followed to develop singularityreduced 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 weaklysingular, weakform traction integral equation is established. An isolated discontinuity is treated first (including, as special cases, cracks and dislocations) after which singularityreduced 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 singularityreduced 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, singularityreduced 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 weakform traction integral equation is then readily obtained and involves a kernel which is only weaklysingular. 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 weaklysingular, weakform displacement and traction integral equations since these form the basis of an effective numerical procedure for fracture analysis (Li et al., 1998), and such equations are presented for both elastostatics and elastodynamics. A noteworthy aspect of the development is that there is no need to introduce Cauchy principal value integrals much less Hadamard finite part integrals. Finally, the utility of the systematic procedure presented here for use in obtaining singularityreduced integral equations for other unbounded media (viz. the halfspace and bimaterial) is indicated.
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 Title
 Singularityreduced integral equations for displacement discontinuities in threedimensional linear elastic media
 Journal

International Journal of Fracture
Volume 93, Issue 14 , pp 87114
 Cover Date
 19980901
 DOI
 10.1023/A:1007513307368
 Print ISSN
 03769429
 Online ISSN
 15732673
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Discontinuities
 cracks
 integral equations
 boundary elements.
 Industry Sectors
 Authors

 Songshan Li ^{(1)}
 Mark E. Mear ^{(1)}
 Author Affiliations

 1. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX, 78712, USA