Issue34

P.O. Judt et alii, Frattura ed Integrità Strutturale, 34 (2015) 208-215; DOI: 10.3221/IGF-ESIS.34.22 210   0 0 , lim d lim d kn kj n kj n j kn kj j n kn k n L Q x u n s Q n s x J x                   (6) if the contour   is shrunk to the crack tip. A reduced notation 3 kn kn    is introduced for the Levi-Civita symbol. Applying finite integration contours and considering curved cracks and mixed-mode loading, integration along the crack faces is required to provide path-independence. Special treatment is necessary for the accurate calculation of J 2 and L . Here, numerical inaccuracies are adjusted, e.g. extrapolating the non-singular part of tangential stresses on the crack faces towards the crack tip [10]. (a) (b) Figure 1 : Integration contours, for path-independent J k -, M -, L and I k -integrals, considering physical ( p  ) and auxiliary ( a  ) crack faces and a material interface ( i  ). The M - and L -integrals depend on the origin of the global coordinate system k e  . From Eqs. (5) and (6) and from Fig. 1(a) it becomes clear that if the global coordinate system coincides with the crack tip coordinate system ( ) i k e  , M - and L - integrals vanish as 0 k x   . Otherwise, if the vector 0 k x is pointing from the origin of the global frame to the crack tip and thus 0 k k k x x x    , M - and L -integrals are finite and represent the scalar and vector moments induced by the crack driving force J k . This feature can be applied to the separation of crack tip loadings in two-cracks systems by a global approach, i.e. by calculating the integrals along remote contours including both crack tips [14]. The I k -integral represents the interaction of two loading scenarios at a crack, i.e. the physical (a) and an auxiliary (b) loading [11]. I k is derived by substituting the superimposed stress and displacement fields (a)+(b) ( ) ( ) a b i i i u u u   and (a)+(b) ( ) ( ) a b ij ij ij      into Eq. (4), yielding (a)+(b) (a) (b) k k k k J J J I    . With the interaction energy-momentum tensor (a/b) kj Q the I k -integral reads     (a) ( ) (b) ( ) ( ) ( ) ( ) ( ) (a/b) , , 0 0 1 lim d lim d . 2 b a a b b a k mn mn mn mn kj ij i k ij i k j kj j I u u n s Q n s                             (7) Applying finite integration contours, the integration along both the physical ( p  ) and the auxiliary ( a  ) crack faces is required to provide path-independence [12]. If material interfaces are considered, e.g. i  between material A and B in Fig. 1(b), integration along the interfaces is required in Eqs. (4) - (7) for the sake of path-independence. J k - and I k -integrals accounting for crack face and interface integrals read 0 p i A B d d d , k kj j kj j kj j J Q n s Q n s Q n s                            (8)