Asymptotic crack tip fields for pressure-sensitive materials and dynamic crack growth under plane stress conditions

Abstract

For the mathematical description of the yield behaviour of elastic-plastic materials the von Mises yield criterion proves to be suitable for the interpretation of the stress and strain curves of metals. However, many materials exhibit a pressure-sensitive yielding. These material properties are not adopted in the von Mises yield criterion. Hence a generalization of the yield criterion is necessary to examine pressure-sensitive materials. In this contribution, a generalization of the von Mises yield criterion is used to investigate the crack tip fields of pressure-sensitive materials like porous metals and certain polymers [1,2]. This extension, which includes a share of the hydrostatic stresses, leads to the so-called Drucker-Prager yield criterion. The factor ~ in this yield criterion is the measure of the pressure-sensitivity and describes the strength of the influence of the hydrostatic stresses on the yield process. A first asymptotic analysis of stress and displacement fields for dynamic crack problems in elastic-plastic solids was performed by Amazigo and Hutchinson [3] by assuming the validity of the J2-flow theory. Achenbach, Kanninen and Popelar [4], Ponte Castaneda [5], Ponte Castarleda and Mataga [6], t3stlund and Gudmundson [7] and Yuan, Yuan and Schwalbe [8,9] generalized those investigations for a dynamic crack propagation including the plastic reloading region on the crack surfaces for a mixed mode loading situation. Studies concerning pressure-sensitive materials using the HRR-field theory were performed by Li and Pan [10], Li [11] and Yuan and Lin [12]. Bigoni and Radi [13,14] advanced the before mentioned investigation [7-9] for pressure-sensitive materials. However, that analysis was restricted to quasistatic crack extension as well as mode I loading. In this paper, the asymptotic stress and velocity crack tip fields for fast running cracks in an elastic-plastic, pressure-sensitive material are determined. The behavior of the singularity exponent s of the stress-singularity was examined for different pressure-sensitivity parameters kt, crack-tip velocities v and hardening coefficients a. The hardening coefficient t~ describes the relation between the shear and the tangent shear modulus. The assumption of homogeneous, isotropic material behaviour showing linear hardening was used. Further, the incremental theory was applied and stationary crack growth under plane stress conditions has been adopted.