Issue 48

A.C. de Oliveria Miranda et alii, Frattura ed Integrità Strutturale, 48 (2019) 611-629; DOI: 10.3221/IGF-ESIS.48.59 612 claim [1]. In fact, even when fatigue cracks initiate from a far-from-elliptical notch tip, they quickly tend to an approximately elliptical front, as shown by the crack that departs from an elongated rectangular notch illustrated in Fig. 1. Figure 1 : Crack that initiated from a rectangular notch, but changed to an elliptical-like front within a short distance from it [2]. Neglecting short crack issues (which, however, may be important in many practical cases, as studied e.g. in [3-4]), fatigue crack growth (FCG) lives can usually be well estimated by integrating the FCG da/dN  K curve of the material, where da/dN is the FCG rate caused by the stress intensity factor (SIF) range  K. Such calculations apply in simple memory-free constant amplitude loading cases. Since FCG curves can be properly measured following standard ASTM procedures, then well-known numerical integration techniques can be used to perform fatigue life predictions of cracked components once the appropriate  K is known for the problem in hand, no matter how complex the SIF expression is. Most fatigue cracks tend to grow in three distinct phases. They usually initiate at a surface as a 2D crack and grow in 2D up to reaching the piece thickness. Then they make a gradual transition from their initial 2D shape to a through 1D crack, and eventually propagate until the final fracture with a crack front shape that can be treated as 1D (albeit they cannot have a straight front, as discussed in [5]). The weak modeling link in this process is the 2D to 1D transition, since there are no standard SIF solutions for this problem, even though there are plenty of handbook expressions to evaluate SIFs for many geometries in which the cracks grow in a single 1D direction, as well as various classic solutions for 2D cracks that propagate maintaining elliptical crack fronts. For practical reasons, these solutions assume an elliptical crack front shape to avoid having to model the free surface singularity [6]. There are classic analytical expressions for quantifying the SIF of elliptical 2D cracks under combined tension and bending, most obtained by proper fitting 3D finite element (FE) solutions. These expressions are complex and difficult to use for quick estimates, but they can be relatively easily integrated by standard numerical methods. For instance, if these cracks occur in a plate of width w (or 2 w , if the subsequent through-crack has two tips) and thickness t , their SIF range  K is a function of the stress range  , of the ratios a/c , a/t and c/w , and of an angle  defined in Fig. 2, because their SIF values vary from point to point along their fronts [7-10]:  K =  (  a)  f  (a/c, a/t, c/w) (1) where f  is a crack shape function and a and c are the ellipsis semi-axes. It usually is a quite reasonable approximation to assume that real 2D crack fronts have a near elliptical shape, but this is not enough to model their FCG behavior and to make FCG life predictions. It is also necessary to assume that the successive fronts of such 2D cracks retain the elliptical shape but may change their a/c ratio at every load cycle, because their SIFs vary along the crack front. Based on the maintenance of an elliptical geometry, the basic modeling idea for 2D FCG predictions is to couple the crack growth in the depth ( a ) and surface width (2 c for surface and c for corner cracks) directions at every load event, as studied e.g. in [10]. However, the transition behavior from a corner or surface flaw into a through-crack usually is not even addressed when modeling them. For example, consider a surface crack as in Fig. 2 (left). If the cracked plate material is tough enough and if 2 w >> 2 c , this surface crack can propagate until reaching the back surface of the plate. Then there is a transition region until the 2D crack can be considered a 1D through crack, whose traces are nearly equal both on the front and on the back face of the plate.