Compact formulae, for bounded and unbounded end conditions, were presented for representing a rectilinear crack in an infinite medium whose plastic zones extended over an arbitrary number of regions having differing frictional stresses. The Navarro-Rios solutions were improved in 3 ways. It was found that the procedure, proposed by them as being a general method (for calculating an arbitrary constant that appeared in the unbounded solution), was erroneous. This was corrected, and it was proved that the extensively used Navarro-Rios expressions were valid for the case where symmetry conditions existed. The need for a constancy of the friction stress over each region was also removed. It was shown that, as long as the variations in the applied and frictional stresses were well-behaved (piece-wise continuous and bounded functions), the solutions remained valid. Formulae were presented which related the relative displacement across the crack plane, and the stresses ahead of the plastic zone, directly to the effective stresses which acted upon the dislocations. This eliminated the need to obtain the dislocation density explicitly.

Compact Formulation for Modelling Cracks in Infinite Solids using Distributed Dislocations. C.Vallellano, A.Navarro, J.Domínguez: Philosophical Magazine A, 2002, 82[1], 81-92