Crack Induced Stress and Deformation Field

This paper presents a solution for stress and deformation field induced by a central crack in an elastic infinite plate subjected to uniaxial and biaxial far field loading. Effects of plasticity on the stress and deformation field near the crack tip is analyzed for the case of elastic, rigid-plastic material subjected to uniaxial loading. Analytical model is derived to characterize the stress and deformation field, deformed crack geometry in the plastic zone, size of plastic zone and crack tip radius of curvature. INTRODUCTION Cracks in a structure may increase stress value to a level, that could lead to structural failure. A knowledge of the crack induced stress distribution around crack tip is therefore necessary for structural assessment. The classical elastic analysis of stress field around a crack leads to singular stress and strain field at the crack tip. Because of singular stress and strain field, it is generally understood that the classical solution does not apply at or close to the crack tip. However, crack tip analysis [1] did produce the concept of stress intensity factor. This factor describes the stress singularity around the crack tip and forms the basis of failure assessment. Experimental observation indicates blunting of the crack tip in both elastic and elastic-plastic materials [2, 3]. The final shape of the deformed crack geometry defined by exact theoretical solution in an elastic plate is an ellipse [4, 5] as shown in Fig. 1. But singular solution does not predict any displacement of the crack tip during deformation along the crack axis. Thus classical stress analysis implicitly assumes that the position of the crack tip, before and during loading, is unchanged. Thus an initial crack (Fig. 1a) characterized by its half crack length ai deforms in to an ellipse (Fig. 1b) with semi major and semi minor axis of lengths af and bf , respectively. Singh et al. [6] argued that an elastic analysis involving small displacement is equally valid with reference to the deformed configuration of the crack. On this basis, Singh at al. [6] obtained a solution for an infinite plate with an central crack, under uni-axial tension and showed that the stress at the crack tip is high but finite. Due to high stress concentration most materials experience plastic deformation around crack tip, once stress value exceeds yield limit. It is clear that the material yield criterion would limit the stresses to finite values, thus avoiding the singularity. Two analytical crack tip solution for plastic material are due to Hutchinson [7] and Rice and Rosengren [8], using path independent J integral derived by Rice [9], known as HRR solution. But, Rice’s path independent line integral does not apply when the stress strain relation has a discontinuity and hence HRR solution is not applicable for ideal elastic, rigid plastic material [7]. To accommodate the crack tip plasticity effect in a elastic, rigid plastic solid, approximate solutions have been developed that attempt to reconcile elastic solution and plastic behavior in a heuristic manner [10]. The main objective of this paper is to present a consistent analysis of the crack tip stress and deformation field and plasticity effects in an infinite plate of elastic, rigid-plastic material under plane stress conditions.