Asymptotic finite deformation analysis of growing crack fields in elastic-perfectly plastic materials (Q1261035)

The small-displacement-gradient formulation of the mode-I growing crack problem results in an intrinsic inconsistency, since velocity solutions predict large deformation and rotation in the near-tip region. Asymptotic solutions for the stress and velocity fields are derived that are valid within a finite deformation context. The following kinematic description of a macroscopically homogeneous material is used: plastic flow is assumed to occur by crystalline slip only, and the remaining response of the material is due to rotation and elastic deformation of the crystalline lattice; the elastic response of the material is assumed to be hyperelastic; the material is assumed to be non-hardening with yield stress much less than the elastic moduli, and to be isotropic with elastic moduli remaining constant irrespective of the plastic deformation; the material is assumed also to be perfectly plastic. A rigorous finite deformation generalization of the small-displacement- gradient Prandtl-Reuss equation is derived and rewritten in terms of non- dimensional variables. The general three-dimensional equations are simplified by assuming plane strain conditions and are made explicit. The solutions obtained are represented by a singular perturbation series. The finite strain solutions presented are closely related to the previously developed small-displacement-gradient solution by \textit{W. J. Drugan} and \textit{X.-Y. Chen} [J. Mech. Phys. Solids 37, No. 1, 1-26 (1989; Zbl 0666.73068)]. The two theories are valid in distinct regions with a different non-dimensional radii.