. J J J M aux act (5) Where act J , aux J are the J integrals in the actual and auxiliary fields, respectively, and M is the interaction integral that we are interested in, defined by : aux II II aux I I A i i M x aux u x u ij K K K K E d x q W M i ij aux i ' 1 2 1 1 . (6) With /2 ij aux ij aux ij ij MW is the strain energy of interaction and E E ' in plane stress and ) /(1 2 ' E E in plane strain. Therefore, the stress intensity factor in mode I and II take the form: . 2 ' M E K (7) We take 0 1, aux II aux I K K in mode I and 1 0, aux II aux I K K in mode II. The computing procedure of M is based on the Gauss points within the elements of J domain area A (see Fig 2). 4. Fatigue application We validate the computer software carried out in this study and based on the above developments, in quasi-static (fatigue) loading, we consider a plate (Fig. 3) of size L l mm mm 65 2 2 120 with an edge crack of length a2 , witha mm 10 , and 3 holes (one is of diameter 20 mm and the two others for the load action are both of 13 mm). The material properties are 71.7 10 Pa 9 E , 0.3 . The stress state is plane strain with a mesh of 60x120 elements. The plate is under uniaxial fatigue load with a variation of p KN 20 with 12 increments of da=3 mm. (b) (c) (a) (b) 2w h X Y X1 Y1 X2 Y2 2a D1 D2 p Y3 p Fig.3 the considered validation: (a) Cracked plate containing voids,(b) results using XFEM, (c) experimental results [11] In this case, the crack growth path is followed and compared with that obtained by Giner et al. [11], the results are regrouped in Fig 3. The obtained results as shown in Fig 3b are approximately close to the experimental ones Fig 3c proved so the accuracy of this approach. 5. Dynamic applications For dynamic loads, we are limited to present our results only.
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