Application of curvilinear coordinates to dislocation density fields around cracks in linear elastic–plastic solids

Dislocation density fields can be found from either gradients of the stress field components (including the rotation pseudo-stress component) or from gradients of the plastic strain components. The latter method is most easily done if the plastic strain components are expressed in an orthogonal curvilinear coordinate system. In this paper are presented appropriate curvilinear coordinate systems for the three types of cracks in plane strain and antiplane strain. The crack is considered to be in a linear elastic–plastic solid with no yield stress. Using these curvilinear coordinates it is possible to complete the earlier dislocation based solution for modes I, II and III cracks in such solids and place on firmer ground the surprising dislocation crack tip anti-shielding result for the mode I crack. Also given in this paper are the equations, in plane strain and antiplane strain, for strain compatibility and stress functions expressed for a general orthogonal curvilinear coordinate system.