Necking Near a Crack Tip in a Plate: a Plane Problem

Abstract

The nonlinear plane problem of the tension of a plate with a crack and a neck at the crack tip is solved. The stress-strain curve is assumed piecewise-linear with a constant bulk modulus. By applying the Fourier transform and discretization, the problem is reduced to a system of nonlinear algebraic equations. The distribution of the neck depth near the crack tip and the effect of nonlinearity on the crack-tip opening displacement are studied Introduction. The primary task of fracture mechanics is to analyze the stress-strain state near a crack and to establish criteria and laws governing its growth [2,9,23]. It is quite obvious that the adequate solution of this problem is possible based on a comprehensive study of the physics of the phenomenon and accompanying processes, a rigorous formulation of a chosen mathematical model describing the phenomenon, and the exact solution of the corresponding equations. Simplifications in the formulation of the mathematical model and analytic or numerical problem-solving methods may substantially distort the description of the real situation.Problems and criteria of fracture mechanics are known [12, 13] to be classical and nonclassical. The former include problems of tension or shear near cracks provided that deformation behavior does not change sharply before fracture. Failure criteria in the former case are formulated for quantities describing the stress-strain state at the crack tip.Nonclassical problems of fracture mechanics involve search for new failure mechanisms that can then be used to solve certain classes of problems. Fundamental areas of fracture mechanics are problems of the fracture of materials compressed along cracks [5,8,10,13,14] and dynamic problems for homogeneous and composite materials with cracks [6,7,12].Unlike most problems in mechanics where the strength of a body is assessed by comparing the maximum stresses or strains to their ultimate levels for a certain material, the calculated stresses and strains at the crack tip in linear fracture mechanics are infinite, which eliminates the possibility of using this procedure and creates formulational and computational difficulties [2]. Therefore, classical fracture mechanics was developed by formulating alternative concepts and approaches based on additional simplifying assumptions on the stress distribution and failure mechanism at the crack tip. These approaches are known as energy-, stress-, and strain-based failure criteria [2].Griffith's energy criterion for brittle fracture [11] and its modification for quasibrittle fracture by Orowan and Irwin [15,22] are considered fundamental failure criteria in classical fracture mechanics. These criteria are based on the assumption of balance between the elastic strain energy released as the crack grows and either the increment of the surface energy in the case of brittle fracture or the energy dissipated in the plastic zone at the crack tip in the case of qusibrittle fracture. This assumption cannot be considered correct because the increment of surface energy equ...