Application of the distributed dislocation technique for calculating cyclic crack tip plasticity effects

Abstract

A B S T R A C T This paper describes a method for modelling cyclic crack tip plasticity effects based on the distributed dislocation technique (DDT). A strip-yield model is utilised to allow for the determination of the crack opening displacement, size of the plastic zones and in the case of a fatigue crack, the wake of plasticity. The DDT can be easily implemented for a wide range of cracked geometries with reliable control over the accuracy and convergence. Thickness effects can also be incorporated through a recently obtained solution for an edge dislocation in an infinite plate of finite thickness. Results for finite length cracks that have had limited growth, such that there is no plastic wake, are presented for a range of applied loads and R-ratios. Further results are provided for a steady-state fatigue crack in a plate of finite thickness. The present results are compared with analytical solutions and they show an excellent agreement. a = half crack length B y (x) = dislocation density function E = Young's modulus g(x) = crack opening displacement and plastic stretch G yy (x), G zz (x) = singular kernel functions for an edge dislocation in an infinite plate of finite thickness h = half plate thickness i, j, k, m, p and q = index ranges for integration and collocation points K = remotely applied stress intensity factor K (x, ξ ) = non-singular geometry kernel function K t = stress intensity factor at crack tip or end of plastic zone K max , K min = maximum and minimum values of applied stress intensity factor n = number of discrete integration points n rp = index number for edge of direct plastic zone n rpc = index number for edge of reverse plastic zone R = applied stress ratio r p = direct plastic zone size r p = direct plastic zone size normalised by plane stress value r p,c = reverse plastic zone size r p,c = reverse plastic zone size normalised by plane stress value s i = ith discrete integration point t k = kth discrete collocation point α = plastic constraint factor β = half size of the crack surface contact region δ 0 = initial crack tip opening displacement

A fatigue crack growth model for interacting cracks in a plate of arbitrary thickness

“…According to the dependences in this figure, a higher maximum stress and a lower load ratio increase the length of the reverse plasticity region ahead of the crack tip. This general trend is in line with the previous theoretical finding by Codrington and Kotousov . The length of the reverse plasticity region at the minimum applied stress may have a significant impact on crack closure and opening behaviours .…”

A crack closure model of fatigue crack growth in plates of finite thickness under small-scale yielding conditions

“…It is assumed in the present analysis that plane stress conditions will prevail remote from the crack tip (Codrington and Kotousov, 2007;Yang and Freund, 1985) and therefore (24) is in the plane stress form. The resulting system of equations can be solved via the usual methods.…”

Out-of-plane stress and displacement for through-the-thickness cracks in plates of finite thickness

“…A full description of the technique has already been given by Codrington and Kotousov [2007]; however, a brief review will be presented here for completeness. It is assumed that a throughthe-thickness crack lies along the x axis (−∞ < x < 1) in an infinite plane and is subjected to a remotely applied mode I stress intensity factor K .…”