Calculation of line singularity fields near a crack tip

Abstract

Near-crack-tip elastic fields are of primary importance in all crack propagation theories. Besides external loads, the near-tip field is determined by internal stress sources such as dislocations and other defectsIn particular, recent works (1-5) have pointed out that dislocations shield crack tips against applied loads, because they remarkably alter the local stress inten- sity. Such investigations have been carried out in anti- plane strain conditions, assuming screw dislocations as typical stress sources. This option presents the advantage of obtaining the basic features of the process through a relatively simple mathematical applicationThe drawback is that only single-mode crack loading (mode III) may be considered. Plane elasticity conditions (e.g. edge dislocations), though mathematically more complicated, appear much less limiting. In fact, with mode I and mode II loading, they allow for the treatment of crack growth under combined stresses. This condition is practically unavoidable in order to model real fracture processes, and also to deal with internal stress sources other than dislocations. The present paper proposes a solution for the prob- lem of an internal stress source located near a crack tip in generalized plane stress or plane strain conditions. The resulting formulae differ from those of Rice (6) in that Cauchy integration is avoided, and only the knowledge of the Laurent's expansion coefficients of the bulk-field potentials is required. Because in many cases of interest only a few and directly available bulk coefficients come into play, it is concluded that the obstacle of the mathematical complications, encountered in dealing with plane elasticity, may be noticeably reduced by using the present results. The geometry of the problem is shown in Fig. la. In the complex plane z = x + iy the crack occupies the negative real axis, and the stress source is at the point Zo = xo + iyo. In Fig. lb the z-plane, deprived of the crack line, is mapped on to the half-plane ~ > 0 of the (-plane by the function