PLANE-STRESS MIXED-MODE NEAR-TIP FIELDS IN ELASTIC PERFECTLY PLASTIC SOLIDS

Within the context of the small-strain approach, the plane-stress mixed-mode near-tip fields of a stationary crack in an elastic perfectly plastic solid under small-scale yielding conditions are examined by finite element methods. The finite element results show that asymptotically at the crack tip two elastic sectors exist under near mode I mixed-mode loading conditions, and one elastic sector exists under near mode II mixed-mode loading conditions. The fully yielded near-tip field, plastically deformed at all angles, is obtained only under pure mode II loading conditions. The corresponding asymptotic crack-tip solutions (consisting of constant stress sectors, curved fan sectors, and elastic sectors) are also constructed. The asymptotic crack-tip stress solutions agree well with the finite element results for the complete range of mixed-mode loadings. Some similarities and differences between the near-tip fields under plane-stress and plane-strain conditions are also discussed. A SUBSTANTIAL understanding of the near-tip structures for power-law hardening materials obeying a deformation plasticity theory has been achieved in recent years. Representative works along this line are those of Hutchinson( 1,2), Rice(3), and Rice and Rosengren(4) for a crack under pure mode I and pure mode II loading conditions, and of Shih(S, 61 under mixed-mode loading conditions. For power-law hardening materials, the asymptotic crack-tip stress and strain fields possess the well-known HRR singularity. The corresponding crack-tip field solutions for perfectly plastic materials were also proposed by the above authors, with the assumption that the material surrounding the crack-tip is fully yielded at all angles. These solutions agree with the perfectly plastic limits of the corresponding asymptotic solutions for power-law hardening materials, and contain radial stress discontinuities under plane-strain near mode I mixed-mode conditions and under plane-stress mode I and mixed-mode conditions (for anisotropic perfectly plastic materials, see Pan(7,8)). Within the framework of rigid perfectly plastic theory (where the elastic strain is neglected), a line of discontinuity in the stress field may be viewed as a mathematical idealization of an infinitively thin elastic region separating two plastic regions. However, for perfectly plastic materials when the material elasticity is considered, it seems reasonable to ask: how does the elastic strain affect the near-tip field structures? To address the above issue under plane-strain conditions, Gao(9) proposed the crack-tip stress fields that contain two elastic sectors under mixed-mode loading conditions. However, the finite element computations carried out by Saka et aL(lO) showed that only one elastic sector exists around the crack tip under mixed-mode loading and small-scale yielding conditions, with Poisson's ratio being nearly l/2. A systematic investigation of the plane-strain mixed-mode crack-tip fields has been conducted by Dong and Pan(l 1, 121. Their computational results and asymptotic analysis show that, as the limiting steady-stress state near the crack tip, the near mode I crack-tip fields do contain an elastic sector, but they differ from the crack-tip fields proposed by Saka et aL(lO) by a constant stress sector separating the elastic sector and a neighboring fan sector. The corresponding conditions for the existence of the elastic sector were examined by asymptotic analyses and verified by finite element computations. Very little information is available in the literature pertaining to the above issues for elastic perfectly plastic materials, particularly under plane-stress mixed-mode loading conditions, despite its practical importance to structural problems. To our best knowledge, a detailed finite