Issue34

L.P. Pook, Frattura ed Integrità Strutturale, 34 (2015) 150-159; DOI: 10.3221/IGF-ESIS.34.16 152 violated only in a small core region, radius << r , than the general character of the K -dominated region is, to a reasonable approximation, unaffected. Similarly, by this small scale argument, small scale non-linear effects due to crack tip yielding, microstructural irregularities, internal stresses, irregularities in the crack surface, the actual fracture process, etc, may be regarded as within the core region. In the early 1960s there was scepticism about the validity and utility of stress intensity factors [8]. However, collaborative theoretical and experimental work in the late 1960s helped to establish confidence [11, 12]) and by 1974 their use for the solution of practical engineering problems was well established [13], nearly two decades after Williams’ discovery of stress intensity factors. Figure 3 : K -dominated and core regions at a crack tip. Figure 4 : Fracture surface of 19 mm thick aluminium alloy fracture toughness test piece. Corner point singularities In three dimensional geometries, the derivation of stress intensity factors makes the implicit assumption that a crack front is continuous. This is correct for cracks growing from internal defects. It is not correct in the vicinity of a corner point where a crack front intersects a free surface. The nature of the crack tip singularity changes in the vicinity of a corner point, and these corner point singularities, sometimes called vertex singularities, are an important source of three dimensional effects [1]. Many of the cracks observed in service, and in laboratory tests, have corner points, for example Fig. 4. Kinematic considerations for a crack surface intersection angle, γ , (Fig. 5) of 90  , and a crack front intersection angle, β , (Fig. 6) of 90  show immediately that mode II and mode III crack tip surface displacements cannot exist in isolation in the vicinity of a corner point. The presence of one of these modes always induces the other, sometimes called a coupled mode, and indicated by a superscript c. Thus mode II induces mode III c and mode III induces mode II c . This has been demonstrated experimentally by foam plastic models [14]. Hence, in the vicinity of a corner point there are two modes of crack tip surface displacement. One is the symmetric mode, which is mode I. The other is the antisymmetric mode, a combination of modes II and III. For corner point singularities, the polar coordinates in Fig. 1 are replaced by spherical coordinates (r,  ,  ) with origin at the corner point. The angle  is measured from the crack front. There do not appear to be any exact analytic solutions for corner point singularities. An approximate solution was obtained in 1977 by Benthem [15] for the restricted case of a quarter infinite crack in a half space. A more general approximate solution, using essentially the same approach, was obtained in 1979 by Bažant and Estenssoro [16]. In their analysis they assumed that all three modes of crack tip surface displacement are proportional to r λ F( θ ,  ). They then calculated λ numerically for a range of situations. The stress intensity measure, K  , may be used to characterise corner point singularities, where  can be regarded as a parameter defining the corner point singularity. It follows from the initial assumption that stresses are proportional to K  / r  and displacements to K λ r 1- λ , where r is measured from the crack front. Hence, stress and displacement plots are straight lines when plotted using logarithmic scales, and such plots obtained from finite element analyses can be used to determine values of λ . For a crack surface intersection angle,  , of 90  and a crack front intersection angle,  , of 90  there are two modes of stress intensity measure corresponding to the modes of crack tip surface displacement. For the symmetric mode the stress intensity measure is K  S , and for the antisymmetric mode it is K  A . For Poisson’s ratio, ν = 0.3 λ = 0.452 for the symmetric mode and 0.598 for the antisymmetric mode. Recent highly accurate finite element results for discs and plates under anti plane loading [17, 18, 19] do not confirm the latter value. Bažant and Estenssoro’s analysis shows that, for a crack surface intersection angle,  , of 90  ,  is a function of Poisson’s ratio,  , and the crack front intersection angle. At a critical crack front intersection angle,  c ,  = 0.5 and stress intensity factors are recovered. K  S becomes K I , and K  A a combination of K II and K III . For a growing crack the crack front