Issue 45

G. Gomes et alii, Frattura ed Integrità Strutturale, 45 (2018) 67-85; DOI: 10.3221/IGF-ESIS.45.06 69 In the above equation, HPV denotes Hadamard principal value. The tensors S ijk (x’,x) and D ijk (x’,x) contain derivatives of T ij (x’,x) and U ij (x’,x) , respectively. Similarly to eq. (1), when r tends to zero, S ijk presents a hipersingularity of order 1/r 2 while D ijk has a strong singularity of order 1/r . Thus, we can write the traction components t j on smooth parts of the boundary in the form                   1 ' ' ', ' ', 2 j i ijk k i ijk k t x n x HPV S x x u x d x n x CPV D x x t x d x         (3) where n i represents the ith component of the unit outward normal to the boundary at point x’. Eqns. (1) and (3) form the basis of the DBEM and are applied here by using the standard BEM approach. After discretising the boundary into elements, assuming a quadratic variation of the displacements and tractions within each element, and applying the discrete version of Eqns. (1) and (3) at a number of collocation points equal to the number of boundary nodes, a linear system of algebraic equations is obtained which allows the calculation of the initially unknown displacements and tractions at the boundary nodes, using the system Hu=Gt (4) where H and G contain the integrals of T ij and U ij , respectively, given by Eqn. (1), or the integrals of S ijk and D ijk given by Eqn. (3), respectively. The vectors t and u contain the traction and displacement components at the boundary nodes, respectively. The system described by Eqn. (4) can be assembled as, Ax=By=f (5) where x is a vector containing the unknown values of t i and u i , while y is a vector containing the boundary conditions ప ഥ and ప ഥ . The matrices A and B are obtained by rearranging the matrices H and G in the conventional BEM manner. Determination of Stress Intensity Factors The elastic stress field at the crack tip can be determined by a correlation between the size, geometry and a Stress Intensity Factor (SIF), which defines their magnitude and also is used in the propagation angle prediction and growth increments. Therefore, a material can resist for crack growth without brittle fracture while the SIF falls below a critical value KIc, which is the material fracture toughness. The stress intensity factors are generally obtained using a displacement or stress extrapolation technique, which requires sufficient mesh refinement at the crack tip in conventional FEM approaches [12]. An alternative is to use methods relying on an energy approximation, therefore avoiding singularities close to the crack tip. The evaluation of SIFs using a path- independent integral is particularly efficient with the BEM as displacements, displacement derivatives and stresses at internal points are directly obtained from the boundary integral equations. In this paper, the J-integral technique for obtaining SIFs is employed. According to [2], this is a very efficient post- processing technique for solving crack problems in general and is based on integrals along a path-independent boundary [13]. Here, the decomposition technique for uncoupling the SIFs with the J-integral was implemented considering a circular contour path around each crack tip, as illustrated in Fig. 1, whose reference system has its origin at the tip of a traction-free crack. According to [13], the relationship between the J-Integral and the SIFs is given by, 2 2 ' I II K K J E   (6) where K I and K II are the SIFs at the crack tip for modes I and II, respectively, E’ is the Young’s modulus, with E’=E for plane stress and E’=E /(1- v 2 ) for plane strain, where v is the Poisson ratio. Considering the SIFs uncoupling, the J-integral can be given by the sum of the two integrals, I II J J J   (7)