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L.P. Pook, Frattura ed Integrità Strutturale, 34 (2015) 150-159; DOI: 10.3221/IGF-ESIS.34.16 151 this point, is shown in Fig. 1. A point on the crack tip is the origin of the Cartesian coordinate system and the z axis lies along the crack tip. Displacements of points within the cracked body when the body is loaded are u, v, w in the x, y, z directions. A fundamental fracture mechanics concept is that of crack tip surface displacement [3]. There are three possible modes of crack tip surface displacement, as shown in Fig. 2. These are: mode I where opposing crack surfaces move directly apart in directions parallel to the y axis; mode II where crack surfaces move over each other in the xz plane in directions parallel to the x axis, that is perpendicular to the crack tip; and mode III where crack surfaces move over each other in the xz plane in directions parallel to the z axis, that is parallel to the crack tip. By superimposing the three modes, it is possible to describe the most general case of crack tip surface displacement. Where more precise description is needed Volterra distorsioni can be used [1]. The term mixed mode means that at least one mode, other than mode I, is present. It is matter of observation that, when viewed on a macroscopic scale, and under essentially elastic conditions, cracks in metals tend to grow in mode I, so attention is largely confined to this mode. Crack surfaces are assumed to be smooth, although on a microscopic scale they are generally very irregular. Figure 1 : Notation for crack tip stress field. Figure 2 : Notation for modes of crack tip surface displacement. S TRESS ANALYSIS OF CRACKS he philosophical basis for a fracture mechanics analysis is that for crack growth to take place two conditions need to be satisfied [4]. Firstly, sufficient energy needs to be available to operate a crack growth mechanism (thermodynamic criterion). secondly crack tip stresses must be high enough to operate the mechanism (stress criterion). Stress intensity factors The key concept of stress intensity factor, for mode I and for mode II, arises from a two dimensional linear elastic analysis for a straight crack. Mode III is not possible in two dimensions, so for this mode a quasi two dimensional anti plane analysis is used. In 1957 it was shown by Williams [5] that the stress field in the vicinity of a crack tip is dominated by the leading term of a series expansion of the stress field. This leading term is the stress intensity factor, K , which is a singularity. A particular type of elastic crack tip stress field is associated with each mode of crack tip surface displacement, and subscripts I, II and III are used to denote mode. Other terms are non singular. Individual stress components are proportional to K /  r where r is the distance from the crack tip. Displacements are non singular and proportional to K  r . Once K is known, stress and displacement fields in the vicinity of the crack tip are given by standard equations. For a mode I crack, the second term is a stress parallel to the crack, usually called the T -stress [6]. The T -stress is used in some linear elastic fracture mechanics analyses. The thermodynamic criterion implies that the energy needed to create new crack surfaces must be considered. Stress intensity factors automatically satisfy the stress criterion. In 1957 Irwin [7] showed that they also satisfy the thermodynamic criterion. In 1967 finite element stress analysis software was becoming generally available [8]. Finite elements were first used for the calculation of mode I stress intensity factors in two dimensional geometries in 1969 [9]. A stress intensity factor provides a reasonable description of the crack tip stress field in a K – dominated region at the crack tip, radius r  a /10 where a is crack length, as shown in Fig. 3. An apparent objection to the use of the stress intensity factor approach is the violation, in the immediate vicinity of the crack tip, of the initial linear elastic assumptions, in that strains and displacements are not small. However, as noted by Williams in 1962 [10], if the assumptions are T