Issue34

A. Campagnolo et alii, Frattura ed Integrità Strutturale, 34 (2015) 190-199; DOI: 10.3221/IGF-ESIS.34.20 191 mode III c and mode III induces mode II c . These induced modes are sometimes called coupled modes, indicated by the superscript c. The existence of three-dimensional effects at cracks has been known for many years [2-4], but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities [5] and their implications became known [6]. Despite increased understanding, three-dimensional effects are sometimes ignored in situations where they may be important. Within the framework of linear elastic fracture mechanics [7] the stress field in the vicinity of a crack tip is dominated by the leading term of a series expansion of the stress field [8]. This leading term is the stress intensity factor, K. In three dimensional geometries, the derivation of stress intensity factors makes the implicit assumption that a crack front is continuous. This is not the case in the vicinity of a corner point, and the nature of the crack tip singularity changes. The resulting corner point singularities were described in detail in 1979 by Bažant and Estenssoro [5]. Some additional results were given by Benthem in 1980 [9]. For corner point singularities, the polar coordinates 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. In their analysis Bažant and Estenssoro [5] assumed that all three modes of crack tip surface displacement are of the form r λ ρ p F( θ ,  ), where ρ is distance from the crack tip, and p is a given constant. They then calculated λ numerically for a range of situations: λ is a function of Poisson’s ratio, υ . For the antisymmetric mode, λ = 0.598 for υ = 0.3. Benthem [9] made an equivalent assumption but used a different numerical method to calculate  . 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. However, expressions for numerical values of K  , and associated stress and displacement fields, do not appear to be available. At the present state of the art the extent of a corner point singularity dominated region has to be determined numerically. At a corner point stresses are a singularity. They must be either infinite or zero,  is indeterminate, and it is reasonable to speak of stress intensity factors in an asymptotic sense [2,3]. In the limit, as a crack front is approached, displacement fields must be those of a stress intensity factor [9]. Hence, there is an infinitesimal K -dominated region within the core region of a corner point singularity. The stress intensity factors are proportional to s 0.5 - λ where s is the distance from the surface along the z axis [9]. Hence, for the antisymmetric mode K II and K III both tend to infinity as a corner point is approached. Further, as a corner point the ratio K III / K II tends to a limiting value which is a function of v . For v = 0.3 the limiting ratio is 0.5 [9]. Benthem points out that K II and K III lose their meaning at a corner point [9]. Dhondt suggests that modes II c and III c might not be singular [10]. The predicted tendency to infinity is reasonable for K II since relevant stresses are in plane and disclinations are zero [2,3]. From a linear elastic viewpoint the predicted tendency of K III to infinity cannot be correct [4]. At a surface shear stresses perpendicular to the surface are zero, which implies that K III tends to zero as the surface is approached. Mode III is a torsion problem [11]. Under mode III (anti-plane) loading initially plane cross sections, including the surface at a corner point, do not remain plane under load [4] and disclinations appear. It is well known that serious error can arise if warping of non-circular cross sections under torsion is not taken into account in stress analyses [12]. Warping of the surface under mode III means that τ yz at the surface does not have to be zero, and finite values of K III are possible. The implication is that the non linearities cannot be regarded as being in a core region within a corner point singularity dominated region and that Bažant and Estenssoro’s prediction that K III tends to infinity as a corner point is correct. The alternative view, which is supported by a large body of evidence [4], is that apparent values of K III decrease towards the surface in the z direction. This implies that K III tends to zero as a corner point is approached, which is intuitively correct. It also implies that non linearities can be regarded as being within a core region, but does not explain why Bažant and Estenssoro’s analysis does not give the correct limit. Nevertheless, this alternative view may well be adequate when considering practical implications. This paradox needs to be resolved so that the results of finite element analysis can be interpreted correctly. There does not appear to have been a systematic investigation of the extent to which Bažant and Estenssoro’s initial assumption is justified. Their assumption does appear to be satisfactory for the symmetric mode (mode I) in that their analysis leads to useful results [4]. The purpose of the present contribution is to review the study carried out by the same authors in some recent investigations [2,3], in which a coupled fracture mode generated by anti-plane loading of a straight through-the-thickness crack in linear elastic discs and plates has been analysed by means of accurate 3D finite element (FE) models. Due to the uncertainties in the definition of the stress intensity factors on the free surfaces, as stated above, the strain energy density averaged in a control volume (SED) has been employed to quantify the stress intensity through the thickness of the discs [2] and plates [3]. For a review of the SED the reader can refer to [13]. This parameter has been successfully used by Lazzarin and co-authors to assess the fracture strength of a large bulk of materials, characterized by