L.P. Pook, Frattura ed Integrità Strutturale, 34 (2015) 150-159; DOI: 10.3221/IGF-ESIS.34.16 157 as K Ic . The minimum thickness depends on the yield stress, Y , of the material, and is given by 2.5( K I / Y ) 2 . The factor 2.5 is empirical and is a compromise. It was also found that consistent values are obtained for K Ic if it is calculated from the load necessary to cause a significant amount of crack growth, defined as 2 per cent of the initial crack length. Only two dimensional stress intensity factor solutions were available for test pieces of constant thickness, such as the compact tension test piece, shown in Fig. 11 so a limit had to be based on permissible crack front curvature, and an appropriate through thickness average crack length defined. The empirical basis of some aspects of K Ic testing meant that extensive development work was needed to develop workable standards [8, 12]. It also means that standards contain very detailed requirements to ensure reproducibility between laboratories. D ISCUSSION AND CONCLUSIONS he linear elastic analysis of cracked bodies, usually known as linear elastic fracture mechanics (LEFM), is a Twentieth Century development. The first theoretical analysis appeared in 1907, but it was not until the introduction of the stress intensity factor concept in 1957 that widespread application to practical engineering problems became possible. LEFM developed rapidly in the 1960s, with application to brittle fracture and fatigue crack growth, and the development of a standard for the plane strain fracture toughness testing of metals. The first application of finite elements to the calculation of stress intensity factors for two dimensional cases was in 1969. The 1970s were a period of consolidation. LEFM was increasingly used in failure analysis. Analyses were assisted by the publication of stress intensity factor handbooks. Corner point singularities were investigated in the late 1970s. A key finding was that a corner point modes II and III cannot exist in isolation. Hence, the presence of one of these modes always induces the other, and is sometimes called a coupled mode. By 1986 the increasing power of mainframe computers meant that three dimensional finite element analysis of cracked bodies became feasible. Finite element analysis of cracked three dimensional configurations, which started in the late 1980s, confirmed the existence of coupled modes. It was soon found that the existence of corner point effects made interpretation of calculated stress intensity factors difficult, and their validity questionable. In a recent investigation a coupled mode generated by anti-plane loading of a straight through-the-thickness crack in linear elastic discs and plates was studied using accurate three dimensional finite element models. The results make it clear that Bažant and Estenssoro’s analysis of corner point singularities is incomplete. An open question is the need for a new field parameter, probably a singularity, to describe the stresses at surfaces. Finite element analysis had a significant influence on this aspect of the development of LEFM. Numerical two dimensional mode I crack path predictions were carried out in the early 1990s. Agreement between theoretical predictions and experimental data obtained using thin sheets is variable. The availability of increasingly powerful computers means that two dimensional predictions have now largely been superseded by three dimensional predictions. Numerical three dimensional predictions of fatigue crack paths between 2003 and 2014 showed good agreement with observed fatigue crack paths. However, how best to allow for the influence of corner point singularities in three dimensional numerical predictions of fatigue crack paths is an open question. Crack path prediction would not have been possible without the use of finite element analysis or boundary element analysis. In 1998 it was shown that the assumption that crack growth is in mode I leads to geometric constraints on permissible fatigue crack paths. Line tension ensures that crack fronts are smooth curves. A necessary condition for crack growth to be in mode I is that the crack growth surface must be smooth. On a smoothly curved crack growth surface there is an ordered family of smooth curves representing successive positions of the crack front and a family of crack growth trajectories. These are an orthogonal net passing along directions of maximum and minimum curvature. On a plane the curvature is equal in all directions, there is no orthogonal net, and therefore no geometric constraint on permissible crack front families. The biaxiality ratio, a nondimensional function of the T -stress, is sometimes used as a crack path stability criterion, but it is not satisfactory. An alternative approach suggested in 1998, is to use the T -stress ratio, T R , which is a point criterion based on the T -stress. For a particular material, there appears to be a critical value of T R , T Rc , below which a fatigue crack path is directionally stable. However, adequate description of fatigue crack path stability is an open question. Requirements for a valid K Ic fracture toughness test in the latest standard [47] and are essentially unchanged from those in the first standard published in 1970 [48]. In other words, 1960s technology is still being used. This would not matter if the latest standards were satisfactory in practice. However, in 2012 Schijve pointed out that the transferability of fracture toughness test data to practical situations was restricted so structural testing was sometimes needed [49]. Also in 2012 T

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