Finite deformation crack-line fields in a thin elasto-plastic sheet

Finite deformation in the crack-tip zone of plastic deformation is investigated for Mode-I opening of a crack in a thin sheet of elasto-plastic material. The material obeys the von Mises yield criterion in the true stresses, and the stretching tensor satisfies a flow law of the Prandtl-Reuss type. Incompressibility and a state of generalized plane stress are assumed. It is assumed that linearized elasticity applies outside the zone of plastic deformation. On the crack-line between the crack-tip and the elastic-plastic boundary, two distinct regions have been recognized: the near-tip zone and the intermediate region. In the near-tip zone the fields are controlled by the radius of curvature of the blunted crack-tip. Here the stress field has been approximated by classical plane stress results. It has been assumed that the crack-line stresses may be taken as uniform in the intermediate region. In each region, deformation variables have been determined by the use of the constitutive relations, and the results have been matched to the corresponding quantities in the neighboring region(s). In this manner expressions have been constructed for the deformation gradients on the crack-line, in terms of the distance to the crack-tip in the deformed configuration, the yield stress in shear, and the stress intensity factor of linear elastic fracture mechanics.