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L.P. Pook, Frattura ed Integrità Strutturale, 34 (2015) 150-159; DOI: 10.3221/IGF-ESIS.34.16 156 is an important exception to the geometric constraint on permissible mode I crack front families. On a plane the curvature is zero in all directions, so principal directions cannot be defined, and there is no orthogonal net of principal curvature directions. Therefore, for a flat mode I crack there is no geometric constraint on permissible crack front families, and a wide range of crack front families is possible and, indeed, observed [39, 40]. Fatigue crack path prediction In general, fatigue crack paths are difficult to predict. In practice, fatigue crack paths in structures are often determined by large scale structural tests [23, 41]. Theoretical predictions are carried out numerically using finite element analysis or boundary element analysis. The validity of the algorithms used is checked by comparison of predictions with experimental data. Two dimensional mode I fatigue crack path predictions have been carried out by a number of authors, using the same general scheme including Portela [42] in 1993. Calculations are carried out using small increments of straight crack growth. The size of the increment is calculated using an appropriate fatigue crack growth equation. The direction taken by each increment is selected using the criterion that the increment is pure mode I. Agreement between numerical predictions and experimental data obtained using thin sheets is variable [21]. The availability of increasingly powerful computers mean that two dimensional predictions have now largely been superseded by three dimensional predictions. For the special case of a flat mode I initial crack in a symmetrical configuration crack mode I fatigue crack growth is in the same plane as the initial crack, and in a direction perpendicular to the crack front. By definition the crack growth angle, θ , (Fig. 7) is zero. It is implicitly assumed that the crack is directionally stable. Increments of straight fatigue crack growth are calculated for a set of points along the crack front. The tips of these increments, connected by straight lines or a fitted curve, define a new crack front. In 1989 Smith and Cooper [22] carried out some finite element calculations for a flat irregular mode I crack and showed that irregularities rapidly disappeared. In 2005 Dhondt [43] carried out some finite element calculations for a flat mode I fatigue crack path in a model containing geometric discontinuities. The results showed that the crack front intersection angle has a trend towards 100  , which is close to the theoretical critical crack front intersection angle. When the crack crosses a geometric discontinuity there is a large, abrupt change in the crack front intersection angle, but it then converges to the critical crack front intersection angle. Schöllmann et al’s criterion [32] for the initial direction of crack growth in the general three dimensional case of mixed modes I, II and III loading is evaluated on a curved cylinder with centre line along the crack front. The ends of the cylinder are perpendicular to its axis to avoid meshing difficulties. The resulting equations have to be evaluated numerically. The criterion also includes a method of calculating an equivalent mode I stress intensity factor for use in the calculation of fatigue crack growth increments. In the presence of mode III the overall crack path increment is mixed mode. This is compatible with the idea of a twist crack where, on a scale of 1 mm, the crack path is mixed mode (Fig. 8). A fully automatic finite element program incorporating the criterion was used to analyse fatigue crack growth in the frame of a hydraulic press that had failed in service [44]. The results showed good agreement with the observed crack path in the structure. However, there was considerable scatter in the predicted crack front intersection angles. A detailed description of a fully automatic finite element program for the calculation of fatigue crack paths in the general three dimensional case was given by Dhondt [45] in 2014. A particular difficulty is meshing in the vicinity of a corner point when the crack front is not perpendicular to the surface. This is avoided by an approximation in which the surface is locally perpendicular to the crack front. The crack growth direction is determined iteratively using a principal stress criterion. The results obtained are justified by comparison with experimental data. Differences arise where mode III crack surface displacements lead to friction between opposing crack surfaces and to the development of twist cracks. Neglecting friction leads to a conservative result. P LANE STRAIN FRACTURE TOUGHNESS TESTING nterest in assessing the fracture toughness, or resistance to brittle fracture, of metals goes back to 1822 when Tredgold commented on the assessment of cast iron [1]. By 1962 numerous empirical tests had been developed in order to determine whether a steel was brittle or ductile [1]. The best known test is the Charpy impact test using notched test pieces. It is still in use, but it does not provide quantitative fracture toughness data that can be used in design [13]. The situation changed dramatically in the 1960s with the development of standards for plane strain fracture toughness ( K Ic ) testing. This was based on two empirical observations [1, 46]. First, that the sharpest possible machined notch may not adequately represent a crack. Second, that in a test piece of constant thickness, the fracture toughness is a function of test piece thickness. However, if the test piece thickness is above a minimum value, which depends on the material, then the fracture toughness is a minimum, and it is a material constant [46]. This minimum value became known I