The stress field and energy of a three-dimensional dislocation loop at a crack tip

The self stress field and self energy are estimated for a planar 3D dislocation loop emanating from a half-plane crack tip. While the problem is of greatest interest for analysis of shear loops nucleating from the crack tip in the concentrated stress field there due to applied loadings, it is addressed here in the interest of tractability for 3D prismatic loops lying in the same plane as the crack. Exact elastic calculations for that case are based on recent developments of 3D crack weight function theory and specific results are given for induced stress fields, intensity factors and energy of semicircular and rectangular prismatic dislocation loops. Also, self stresses and energy expressions are derived for the 2D case of a line dislocation lying parallel to the crack for arbitrary Burgers vector type and general orientation of the dislocated plane relative to the crack plane, and those results are used together with the 3D prismatic loop results to estimate approximately the self energy for 3D shear dislocation loops emanating from the tip on planes inclined to the crack plane. Energy results are given in terms of a correction factor m to the usual estimate of energy for an emergent crack tip loop as half the energy of a full loop (identified as the emergent loop and its image relative to the crack tip) in an uncracked solid. That is, if the energy of a full circular loop of radius r in an uncracked solid is 2πrA 0 1n ( 8r e2r0 ), with r 0 = core cut-off and A 0 = energy factor, then the energy of a semicircular loop of radius r emerging from the crack tip is shown to take the form πrA 0 In ( 8mr e2r0 ) and the constant m is calculated here as 2.2 for a prismatic loop ahead of a crack and estimated approximately to range from about 1.2 to 1.9 for representative shear loops inclined to the crack plane. The self energy exceeds the half-full-loop value, corresponding to m = 1, and it is observed that this effect increases by √m the predicted loads to nucleate a dislocation loop of the assumed shape from a crack tip.