Issue 43

L.C.H. Ricardo, Frattura ed Integrità Strutturale, 43 (2018) 57-78; DOI: 10.3221/IGF-ESIS.43.04 58 of the influence of loading sequences on fatigue crack growth [2]. Investigations covering the effects of particular interest, after high overload, loading in the growth rate region, called crack growth retardation, seem to have little interest nowadays. Stouffer & Williams [3] and other researchers show a number of attempts to model this phenomenon through manipulation of the constants and stress intensity factors in the Paris-Erdogan equation however little appears to have been done in the effort to develop a completely rational analysis of the problem. Probably, the only one reason that the existing models of retarded crack growth are not satisfactory is that these models are deterministic whereas the fatigue crack growth phenomenon shows strong random features. In addition, most of the reported theoretical descriptions of the retardation are based on data fitting techniques, which tend to hide the behaviour of the phenomenon. If the retarding effect of a peak overload on the crack growth is neglected, the prediction of the material lifetime is usually very conservative [4]. Accurate predictions of the fatigue life will hardly become possible before the physics of the peak overload mechanisms is better clarified. According to the existing findings, the retardation is a physically very complicated phenomenon which is affected by a wide range of variables associated with loading, metallurgical properties, environment, etc., and it is difficult to separate the contribution of each of these variables [5]. C RACK PROPAGATION CONCEPTS rwin [6,7] defines in his work a release energy rate G, which is a measure of the available energy, dП- potential of energy and A- crack area, to provoke crack propagation as shown in Eq. (2.1). The term rate as employed is not related to a derivate in relation to the time but is referred to a change in the potential energy rate in the crack area. Later, this quantity has been called K , and is used to characterize the stress state ("stress intensity") near a crack tip caused by a remote load or residual stress in isotropic and elastic bodies. The stress field in the crack tip is given by Eq. (2.2),   d G dA (2.1)           1/2 1/2 2 3 (2 ) ( ) ( ) ( ) ...... ij ij ij ij K r f A g A h r (2.2) where K is the stress intensity factor; r and  are the distance from the crack tip and the angle between the crack tip and the plane of the crack, respectively; A i is a constant of the material; f ij (  ) , g ij (  ) and h ij (  ) are functions of  . After years, the stress-intensity factors for a large number of crack configurations have been generated; and these have been collated into several handbooks (see, for example, Refs [8,9]). The use of K is meaningful only when small-scale yielding conditions exist. Plasticity and nonlinear effects will be covered in the next section.Because fatigue-crack initiation is, in general, a surface phenomenon, the stress-intensity factors for a surface- or corner-crack in a plate or at a hole, such as those developed by Raju and Newman [10,11], are solutions that are needed to analyze small-crack growth. Some of these solutions are used later to predict fatigue-crack growth and fatigue lives for notched specimens made of a variety of materials [12]. Paris & Erdogan [13] conducted a revision on the crack propagation approach from Head [14] and others and discussed the similarity of these theories and the differences of results between them, isolated and in group tests. Paris suggested that, for a cyclical load variation, the stress field in the crack tip for a cycle can be characterized by a variation of the stress intensity factor,    max min K K K (2.3) where K max and K min are the maximum and the minimum stress intensity factors, respectively. In the crack propagation curve, the linear part represents the Paris - Erdogan law, when plotting the values of  K vs da/dN in logarithmic scale. Fatigue crack initiation and growth under cyclic loading conditions is controlled by the plastic zones that result from the applied stresses and exist in the vicinity (ahead) of a propagating crack and in its wake or flanks of the adjoining surfaces. For example, the fatigue characteristics of a cracked specimen or component under a single overload or variable amplitude loading situations are significantly influenced by these plastic zones. In modelling the fatigue crack growth rate this is accounted by the incorporation of accumulative damage cycle after cycle and should include plasticity effects. During the crack propagation the plastic zone should grown and the plastic wake will have compressive plastic zones that can help to keep the crack close. Hairman & Provan [15] discuss the problems pertaining to fatigue loading of engineering structures under single overload and variable amplitude loading involving the estimation of plasticity affected zones ahead of the crack tip. I