Issue34

P.O. Judt et alii, Frattura ed Integrità Strutturale, 34 (2015) 208-215; DOI: 10.3221/IGF-ESIS.34.22 211 0 p a i A (a/b) (a/b) (a/b) (a/b) B d d d d , k kj j kj j kj j kj j I Q n s Q n s Q n s Q n s                                         (9) and kj Q         denotes the jump of the energy-momentum tensor across crack faces and interfaces. A local coordinate system * k e  is introduced, with * 1 e  pointing into the tangential direction of the interface. The stress vector i t and the tangential derivatives of displacements ,1 i u are continuous at the interface. The integrand related to the J k -integral reads       AB A A (A) (B) ( ) ( ) (A B ) (B) ( ) (A) (B) 2 11 11 1,1 12 1,2 1,2 22 2,2 2,2 B AB 1 , 2 j j Q n u u u u u                        (10) with (AB) denoting values that are continuous across the material interfaces. The additional integrand related to the I k - integral reads         A a/b (b,A) (b,B) (a,AB) (a,AB) (b,A) (b,B) (a,AB) (b,A) (b,B) 2 11 11 1,1 12 1,2 1,2 22 2,2 2,2 B . j j Q n u u u u u                        (11) Bearing in mind that the auxiliary fields do not have to exhibit an interface, an appropriate choice is (b, A) = (b, B), providing   A a/b 2 B 0 j j Q n          . Therefore, according to Yu et al. [18], the integration along the interface should be superfluous. Our investigations reveal, however, that the integral along the interface according to Eq. (9) must not be neglected. Path-independence is investigated at a model with a straight slant crack, hole and interface as presented by Judt and Ricoeur [19], where the values J 2 were obtained without special treatment of the crack face integrals [10] thus being inaccurate. The J k - and I k -integrals along small circular contours enclosing the crack tip (denoted as local) are compared to the values obtained by the approach as presented in this section according to Eqs. (8) and (9) (denoted as global). Furthermore, SIF are calculated for a comparison of J k - and I k -integrals. Three different choices of the auxiliary fields are considered in the region B (see Fig. 1(b)): a) the auxiliary fields are chosen according to the material constants in B, respectively; b) the auxiliary fields are chosen according to the material constants in A and the integration along the interface is neglected in Eq. (9); c) the auxiliary fields are chosen according to the material constants in A and the I k -integral is calculated according to Eq. (9). Method J 1 or I 1 / (N/mm) J 2 or I 2 / (N/mm) K I / MPa  mm K II / MPa  mm J k local 0.05721 -0.03304 104.46 33.21 J k global 0.05731 -0.03301 104.58 33.15 I k local 0.0009954 -0.0003162 104.52 33.21 I k global 0.0009948 -0.0003168 104.46 33.27 I k global 0.0008242 -0.0004517 86.54 47.43 I k global 0.0009725 -0.0003119 102.12 32.75 Table 1 : Numerical values of local and global J k - and I k -integrals and SIF, three different choices of the auxiliary fields a – c . From the results in Tab. 1 it is obvious that path-independence is only retained considering interface integrals. Nevertheless, values for the non-physical choice of the auxiliary fields differ slightly from those of the physically motivated choice with dissimilar elastic constants in A and B. A NISOTROPY IN ELASTIC PROPERTIES AND FRACTURE TOUGHNESS he extrusion, drawing or rolling process applied to wrought metal products, in general yields a specified geometric orientation of the microstructure (texture) and thus of the material’s mechanical properties. The J k -integral according to Eq. (8) may be applied straight forwardly to anisotropic elastic material [15]. As the near-tip stress T