Fracture Mechanics

Appendix 1 Fracture Mechanics The subject called “fracture mechanics” originated as an essentially macroscopic approach to the solution of engineering problems involving the likelihood of fracture from the unstable propagation of pre-existing cracks. Fracture mechanics derives largely from Griffith’s (1920) theory of brittle failure, generalized in a more or less empirical way to include in the “surface energy” term other types of energy absorp- tion. However, the main impetus for the current development stems from the work of Irwin (Irwin 1948, 1958, and other papers), and the field now comprises a consider- able body of literature covering both theoretical studies and experimental work on many types of materials, including rocks. Succinct introductions to the general principles of fracture mechanics have been given by Eshelby (1971) and Evans and Langdon (1976). More detailed accounts may be found in the books of Tetelman and McEvily (1967), Knott (1973), Lawn and Wilshaw (1975b), Kanninen and Popelar (1985), Broek (1986), and Lawn (1993) and in the seven- volume treatise edited by Liebowitz (1968). Early applications in rock mechanics were made by Bieniawski (1967a,b, 1968b, 1972) and Hardy, Hudson, and Fairhurst (1973), and more recent applications on various scales have been summarized by Rudnicki (1980), Rice (1980), Pollard (1987), Atkinson (1987), Meredith (1990) and Gueguen, Reuschlé, and Darot (1990). There are two aspects to fracture mechanics. The first is the treatment of the local stress distribution in the neighbourhood of the crack tip and the relation of this stress field to the external loading of the body. The second aspect is the response of the material of the body to the state of stress at the crack tip. It is therefore appropriate to discuss fracture mechanics under the two headings, (1) stress intensity analysis and (2) critical parameters for failure, while further comments are added under a third heading, (3) dynamic and kinetic effects. A1.1 Stress Intensity Analysis Although it is clearly acknowledged that non-elastic effects are involved at crack tips in most materials, and that even the elastic behaviour in the most highly stressed regions may be non-linear, the practical analysis of the stress distribution in the neighbourhood of the crack tip is usually done on the basis of the classical linear theory of elasticity. The term linear elastic fracture mechanics is sometimes used to indicate this basis. The linear elastic approach is valid provided that the region of 240 Appendix 1 · Fracture Mechanics non-linear behaviour is negligibly small compared with the length of the crack and other dimensions of the body. In this case the intensity of deformation in the prox- imity of the crack tip is controlled by the singular elastic fields. Some progress has also been made in treating the more complex problems that arise when the plastic region around the crack tip is not negligibly small; in these cases, elastic-plastic, vis- coelastic, or fully plastic analyses are applied (McClintock and Irwin 1965; Williams 1965; Rice 1968; Hellan 1976). Here we shall only consider the linear elastic analysis. The object of stress intensity analysis is to give a measure of the “loading” applied to the crack tip, which is in effect the driving force for crack propagation. The analysis is done in two stages. First, the nature of the stress distribution in the crack tip region alone is considered and, second, the influence on this region of the external loading on the body and of its geometry is taken into account. We now discuss these two aspects in turn. a Stress distribution in the crack tip region. We assume an ideal flat, perfectly sharp crack of zero thickness (the “mathematical crack”, cf. Jaeger and Cook 1979, p. 273). The analysis of the stress distribution in the neighbourhood of the crack tip is approached by first considering three basic plane modes of distortion around the crack tip. These modes, designated I, II, and III, are defined with respect to a reference plane that is normal to a straight-line crack edge. Modes I and II are the plane strain distortions in which the points on the crack surface are displaced in the reference plane, normal and parallel, respectively, to the plane of the crack. Mode III is the anti-plane strain distortion in which the points on the crack surface are displaced normal to the reference plane. The three modes are depicted in Fig. 83, to- gether with commonly used Cartesian coordinate axes. In each case, the displacements are independent of the z coordinate, that is, of the position, along the crack edge, of the reference plane used in defining the distortion modes. The stress and displacement fields around the crack tip can be decomposed into three basic modes of distortion that can be considered as resulting from the application of uniform loadings at infinity, which correspond, in the absence of the perturbations due to the crack, respectively to a uniform tensile stress σ normal to the crack (I), a uniform shear stress τx parallel to the crack in the x direction (II), and a uniform shear stress τz parallel to the crack in the z direction (III), as shown also in Fig. 83. A variety of mathematical methods can then be applied to calculate Fig. 83. The three basic plane modes of distortion at a crack tip. Mode I: displacement normal to crack plane; mode II: displacement parallel to crack plane and normal to crack edge; mode III: displacement parallel to crack plane and to crack edge A1.1 · Stress Intensity Analysis 241 the local stress and displacement fields around the crack tip (for resumés of the methods, see Paris and Sih 1965; Rice 1968; Sih and Liebowitz 1968; Jaeger and Cook 1979). The calculations show that the stress components all vary inversely as the square root of the distance from the crack tip in all radial directions from the crack tip. This, of course, implies singularities at the crack tip but the strain energy at any finite volume surrounding the crack tip is bounded. For each mode of distortion, each of the stress components and displacements can be expressed as a product of a spatial distribution function that is independent of the actual value of the applied stress and a scaling factor that depends only on the applied stress and the crack length. The same scaling factor applies for each of the stress components and displacements components in a given mode and is known as the stress intensity factor for that mode. The stress intensity factors are designated respectively KI, KII, and KIII. The formalism can be simply demonstrated with the particular case of the values of the σyy stress component in mode I at points near the crack tip within the material in the plane of the crack; the elastic solution is where 2c is the length of the crack (the body being assumed to be large compared with the crack length), x is the distance from the crack tip, and σ is the macroscopic stress applied normal to the crack. This expression can be written in the form where The factor is the distribution function along the x axis and is the scaling factor or stress intensity factor. The full expressions for all the components of stress and displacement at all points in the reference plane in each of the three basic modes of distortion can be found in Paris and Sih (1965), Sih and Liebowitz (1968), Eshelby (1971), and many other papers on fracture mechanics, as listed above. b Stress intensity factors. So far, we have only considered the general modes of distortion I, II, and III that arise under the particular systems of uniform loading at infinity defined above. Because of the linear properties of elastic behaviour, all more general cases of distortion around crack tips can be treated as combinations of these basic modes. Even in three-dimensional problems, the same consideration applies at each point along a crack edge. Therefore, in the general case, the elastic stress and displacement fields around the crack tip can be obtained from a superposition of 242 Appendix 1 · Fracture Mechanics the fields for the three basic modes, scaled according to an appropriate set of three stress intensity factors, KI, KII, and KIII. The stress intensity factors thus can be said to give a measure of the intensity of loading of the crack tip. The values of KI, KII, and KIII in a particular case depend both on the external loading or macroscopic stress field and on the geometry of the specimen (external dimensions and crack dimensions). They have been calculated for many practical cases; a few cases are described by Sih and Liebowitz (1968) and by Eshelby (1971) and comprehensive listings are given by Paris and Sih (1965), Rooke and Cartwright (1976), and Tada, Paris and Irwin (1985). A1.2 Critical Parameters for Failure The type of analysis just given thus enables us, for a particular loading on a specimen of particular shape, to describe the mechanical situation around a crack tip in terms of stress intensity factors and the basic distortion modes. The question next arises as to what aspect of this local situation around the crack tip determines when the crack will begin to spread under the action of increasing the external loading. The simplest view is that the crack will begin to spread when a certain critical intensity of loading is reached at its tip, as measured by the stress intensity factors. That is, the failure cri- terion is expressed in terms of critical stress intensity factors. These factors are material parameters, depending on the type of material, its particular physical condition of grain size, strain-hardening, etc., and the conditions of temperature and pressure.

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