We perform an analytical first study of the influence of a uniform porosity distribution, for the entire range of porosity level, on the stress field near a plane strain tensile crack tip in ductile material. Such uniform porosity distributions (approximately) arise in incompletely sintered or previously deformed (e.g., during processing) ductile metals and alloys. The elastic-plastic Gurson-Tvergaard constitutive formulation is employed. This model has a sound micromechanical basis, and has been shown to agree well with detailed numerical finite element solutions of, and with experiments on, voided materials. To facilitate closed-form analytical results to the extent possible, we treat nonhardening material with constant, uniform porosity. We show that the assumption of singular plastic strain in the limit as the crack tip is approached renders the governing equations statically determinate with two permissible types of near-tip angular sector: one with constant Cartesian components of stress (“constant stress”); and one with radial stress characteristics (“generalized centered fan”). The former admits an exact asymptotic closed-form stress field representation, and although we prove the latter does not, we derive a highly accurate closed-form approximate representation. We show that complete near-tip solutions can be constructed from these two sector types for the entire range of porosity. These solutions are comprised of three asymptotic sector configurations: (i) “generalized Prandtlfield”for low porosities (0 ≤ f ≤ .02979), similar to the plane strain Prandtl field of fully dense materials, with a fully continuous stress field but sector extents that vary with porosity; (ii) “plane-stress-like field” for intermediate porosities (.02979 < f < .12029), resembling the plane stress solution for fully dense materials, with a ray of radial normal stress discontinuity but sector extents that vary with porosity; (iii) two constant stress sectors for the remaining high porosity range, with a ray of radial normal stress discontinuity and fixed sector extents. Among several interesting features, the solutions show that increasing porosity causes significant modification of the angular variation of stress components, particularly for a range of angles ahead of the crack tip, while also causing a drastic reduction in maximum hydrostatic stress level.
The zero-normal-stress condition in plane-stress and shell elastoplasticity
