Discretization of the plane problem for a cracked body with nonlinear stress–strain diagram under tension

Abstract

The plane problem for a cracked body with a piecewise-linear stress-strain diagram under tension is reduced by the Fourier transformation to a system of nonlinear algebraic equations. The system is numerically solved for plane strain and stress states of a perfect elastoplastic material to study plastic zones, stress and strain distributions, and displacements of crack faces Keywords: crack, discretization of a nonlinear problem, plastic zone, stress and strain distributions, crack opening displacement Introduction. One of the main divisions of fracture mechanics is the mechanics of cracks that studies the stress-strain state around a crack to establish criteria and patterns of its development. A rigorous approach to determining the limiting equilibrium state of a solid with a crack is known [3] to employ classical failure criteria for stresses or strains at the crack tip and allow for all the segments of the real stress-strain diagram. However, the difficulties related to the problem formulation and computation gave rise to alternative approaches based on additional assumptions on stress distribution and crack-tip fracture mechanism. Among them, the most popular are energy-, stress-, and strain-based failure criteria.The use of approximate approaches raises the question whether the simplifying assumptions and associated failure criteria adequately describe the real physical and mechanical processes at the crack tip and fit the rigorous mathematical formulation of the problem. This question can only be answered after a thorough analysis of the simplified failure criteria, experimental data, and the exact solution of the rigorously formulated problem.Griffith's energy criterion for brittle fracture [9], including its modification for quasibrittle fracture by Orowan and Irwin [11,19], is considered a basic failure criterion in crack mechanics. This criterion is based on the idea 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 quasibrittle fracture. However, this idea is incorrect [14]. Indeed, solving the nonlinear problem leads to infinite stresses at the crack tip, whereas the increment of surface energy and the work of plastic deformation are expressed in terms of finite stresses consistent with the real stress-strain diagram. This means that the stresses discontinue at the point separating the elastic and fracture ranges, despite what continuum mechanics tells us. Griffith's criterion will fail if we consider that the stresses in the range where the elastic strain energy is released are finite. Moreover, the increment of surface energy equal to the work done to separate two adjacent atomic layers is at least an order of magnitude less than the released elastic strain energy stored in at least ten atomic layers (concept of continuity), i.e., these quantities are incommensurable.The J-integral or the Cherepanov-Rice criterion [8] is a ...

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

“…It should be noted that the plastic zones and the curves of the normal stress s 22 0 ( , ) x i versus the coordinate x hardly depend on whether the neck is present or not and have the form as in [20]. Figure 5 shows, according to (2.10), curves of the residual crack opening displacement at the crack tip v( ) As can be seen, the curves have a peak at the crack tip, and it moves away from the crack tip with increasing load.…”

“…The length and angle of the cut were found after formulating additional unjustified assumptions. Actually [20], fracture process zones can be determined by solving the nonlinear problem [20,21], an inhomogeneous combined stress state holding within them and their aspect ratio being of the order of 0.7-0.8 for plane strain state and 0.8-1.0 for plane stress state. It also makes no physical sense to define the singular components of the asymptotic approximation of the linear elastic problem at infinity instead of homogeneous loads, which has been natural conventional practice.…”