The Influence of Crack-Face Normal and Shear Stress Loading on Hydraulic Fracture-Tip Singular Plastic Fields

We investigated the singular plastic fields at the crack tip of a fracture that is loaded with normal and shear loads due to a viscous flow in a hydraulic fracturing. The level of the expected shear load in comparison with the normal load is examined. The lubrication flow and plastic deformation were decoupled assuming that the relation between applied shear load and normal load follows a linear friction-type relation. This assumption allows to investigate extreme bounds of the solution. Both the applied normal and shear loads are assumed to exhibit singular behavior near the tip which is consistent at the fracture surfaces with the plastic singular stress fields that are investigated. The fractured material is assumed to obey a non-associative Drucker–Prager solid with power law hardening response. The singular values and the corresponding fields were determined over a range of material parameters. For both von Mises material and associative Drucker–Prager material, we found that the level of singularity is given by 1/n where n is the power coefficient of the hardening relation. This level of singularity is stronger than the HRR value, 1/(n + 1), which has been determined for traction free crack surfaces. We found that the shear loading does not influence the level of singularity but it changes the shape of the developed plastic zones with the emergence of a boundary layer near the fracture surface. Deviation from material associativity produces consistent small increases in the level of singularity. The near-tip stress, strain and displacement profiles are illustrated for a few representative cases.