Issue 31

R. Citarella et alii, Frattura ed Integrità Strutturale, 31 (2015) 138-147; DOI: 10.3221/IGF-ESIS.31.11 140 where  and E are Poisson’s ratio and Young’s modulus respectively. The displacement u p is evaluated at a point P on the crack front sufficiently close to the crack tip. The displacements p b u , p n u , and p t u are projections of u p on the coordinate directions of the local crack front coordinate system and  = π and  = - π denote upper and lower crack surfaces respectively. K I , K II and K III are the Mode I, II and III SIF’s. In the present work the SIF’s are extracted from the J -integral using the method illustrated in [11], based on the following equation:     2 2 2 1 1 2 I II III J K K K E G    (2) where 2 / (1 ) E E    for plain strain conditions. The quarter-point node technique is used to model the crack-tip singularity. Under mixed mode conditions it is necessary to introduce an equivalent stress intensity factor, K eq , considering Mode I, II and III simultaneously. Several formulae have been proposed for K eq and the most commonly used expression is [12, 13]:   2 2 2 1 eq I II III K K K K      (3) In order to determine new crack front positions, the CPD must be computed. Although expressions exist to calculate the crack growth angle based upon the stress intensity factors, an alternative method is adopted here based on the maximum energy release rate at a crack front point. The G -criterion states that a crack will grow in the direction of maximum energy release rate. The CPD,  =  o , is then determined by: 0 o dG d            ; 2 0 o dG d            The application of a series of virtual crack extensions (Fig. 2) ultimately generates a growth angle at the crack front node that, in the general case, may be out-of-plane. (a) (b) Figure 2 : Energy based calculation of G max (a) and crack grow direction (b) . Remeshing technique A critical issue that must be addressed in 3D FE fracture mechanics analysis is that of mesh generation. In the simplest of geometric cases where symmetry can be used, it may be possible to utilise standard mesh generation tools to produce a crack of the required size. In the general case, however, the use of standard tools leads to several time consuming problems including: • Component geometries are often complex and time consuming to model in their uncracked forms. • Defects often occur at geometrically difficult locations e.g. corners, welds, chamfers. • Initial cracks of the correct size and shape must be inserted into the component at the correct location. • Cracks may develop in a non-planar fashion depending upon the loading. The approach that has been successfully adopted here is the use of ‘crack blocks’ which model the details of the cracked region. Crack-blocks are groups of elements arranged in such a way that they contain a section of crack front. Fig. 3-4 demonstrate the use of the crack-block methodology in generating a cracked mesh from a user-supplied intact