CRACK INDUCED STRESS FIELD IN AN ELASTIC-PLASTIC PLATE
R. N. Dubey
Department of Mechanical Engineering, University of Waterloo, Ontario, Canada N2L 3GL
E-mail: r4dubey@yahoo. com
This paper presents a solution for stress and deformation fields induced by a central crack in an elasticplastic plate subject to tensile load. The solution is controlled by a crack opening parameter related to material modulus and far-field stress.
Cracks in a structure cause stress at the tip to increase to a level that could lead to structural failure. Hence, knowledge of the stress distribution around crack tips is important to engineers and designers. Design requirement and need for a failure analysis was the reason for a rapid development in crack analyses especially since the World War II. Early work used small deformation theory, linear elastic behavior and relied on un-deformed geometry to satisfy traction boundary conditions. The result led to singular stress and strain fields at the crack tip. The contributions on this topic are part of fracture mechanics now known as linear elastic fracture mechanics (LEFM). Because of the inherent contradiction between small deformation and singular stress and strain fields at the tip, there is an implicit understanding that the result obtained under such assumptions does not apply at or close to the tip. LEFM solution does not apply at large distances from the crack tip either. The applicability of linear elastic fracture mechanics was thus limited to a finite domain surrounding but excluding the crack tip. However, crack tip analyses did produce the concept of stress intensity factor. This factor controls the stress field around the tip and its value was found to depend on crack length, far field stress and also on structural geometry. Naturally, determination of stress intensity factor became the focus of research and its critical value became a basis for structural design.
When stresses in metals exceed yield limit, they undergo plastic deformation. Since LEFM predicts high value for stresses in an area around crack tip, part of that area is subject to yielding and plastic deformation. Because the yield criterion limits stresses to remain within a finite value, the stress field within plastic zone cannot be singular. A new approach is therefore required to accommodate plastic behavior near crack tip.
It is possible to estimate plastic zone size on the basis of elastic analyses. Irwin (1957) proposed that the actual plastic zone is greater than this estimate. To obtain the actual size, he evaluated the load between the tip and yield point from the elastic analysis and redistributed it over the plastic zone. The proposal of Dugdale and Barenblatt (Barenblatt, 1962) to remove stress singularity at the tip is based on canceling two singularities, one from the elastic analysis and other associated with the wedge force due to yield stress. Both of these proposals considered perfectly plastic solid that allows no strain hardening. They also assume blunting of the crack tip. There is thus an implicit recognition that crack blunting in plastic deformation and non-singular stress field in plastic zone go hand in hand. In other words, singular stress field is incompatible with a blunt crack tip. The conclusion is obvious: stress at the crack tip is reduced to a finite value because of blunting of crack caused due to deformation. If that is the case, an analysis formulated in terms of deformed geometry that allows for blunting is expected to result in finite crack tip stress.
M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 493-504.
© 2006 Springer. Printed in the Netherlands.
Two analytical crack tip analyses for plastic solids involving work hardening nonlinear material are due to Hutchinson (1968) and Rice and Rosenberg (1968). They obtained singular stress field near crack tip using finite deformation theory of plasticity. They assume a stress field, known as HRR solution, consistent with singular strain energy density. Just like the LEFM solution, HRR solution is not valid at the tip because singular stress it predicts at the crack tip is incompatible with the limitation on stresses imposed by plasticity. HRR solution is not valid at large distance from the tip either. A finite element analysis by Mcmeeking et al. identifies the area over which HRR solution applies.
Singh et al. (1994) obtained stress, strain and displacement field around a crack in an infinite, isotropic and linear elastic plate. They used a non-classical small deformation theory. The classical theory assumes the displacement to be small such that replacing the deformed position of a particle by its initial un-deformed position is likely to induce negligible error in the solution. Singh et al. used the reverse argument that in an analysis of a problem involving small displacement, it is equally justified to use the deformed position of a particle, rather than its un-deformed position. Further, the consistency of analysis requires the use of deformed geometry if the boundary value problem is formulated in terms of true stress and true traction. On this basis, Singh et al. (1994) obtained a solution for the entire plate. They obtained the geometry of the deformed crack surface as part of their solution. In this presentation, we use their methodology to obtain stress and deformation field in an elastic-plastic plate. Solution uses deformed geometry and linearized stress-strain behavior.