Issue34

P.O. Judt et alii, Frattura ed Integrità Strutturale, 34 (2015) 208-215; DOI: 10.3221/IGF-ESIS.34.22 209 providing numerically inaccurate values approaching the crack tip, finally leading to large deviations in J 2 . Improvements for an accurate calculation of J 2 have been presented by Eischen [9] and Judt and Ricoeur [10]. The interaction integral or I -integral is another conservation integral of the J -integral type [11]. Here, the interaction of the physical and an auxiliary crack tip loading is used for the direct calculation of SIF. In general, auxiliary fields are obtained from the asymptotic near tip solution for stresses and displacements. Considering curved cracks, the jump of the auxiliary fields represents the straight auxiliary crack, introducing another strong discontinuity. Calculating the global I -integral, path-independence is maintained considering the integration along both physical and auxiliary crack faces [12]. Due to the fact that the auxiliary fields may be chosen arbitrarily, just satisfying balance, constitutive and kinematic laws, the I -integral can be applied to elegant analyzing multiple cracks systems [13]. The M - and L -integrals are akin to the J -integral and, in terms of the concept of material forces, represent the scalar moment and vector moment induced by the crack driving force. Especially if multiple defects are considered, these integrals provide a distinction between the crack driving force and other material forces [14]. In the paper this concept is adopted for a separation of the crack driving force and the forces acting in the plastic deformation. For an accurate calculation of crack paths, the anisotropic elastic and fracture mechanical properties must also be considered in the model. The elastic anisotropy is related to the crack tip loading analysis [15], whereas the fracture toughness anisotropy requires an extended theory for determining the crack deflection angle [16]. Material interfaces and internal boundaries represent inhomogeneities in the material space, thus producing configurational forces along these boundaries. To obtain the crack driving force, the integration along these boundaries is necessary, providing path-independence. This paper presents a numerical model for the accurate loading analysis and crack path prediction considering anisotropic properties and material interfaces. Calculated crack paths are presented and compared to those obtained from crack growth experiments. T HEORY OF PATH - INDEPENDENT INTEGRALS he Lagrangian density of a system exhibiting a potential U is defined as T U    , with T being the kinetic density. The balance laws in the configurational space are derived from applying the gradient , divergence and curl operator to the Lagrangian  and the Lagrangian moment k x  [17]. Assuming quasi-static conditions, material isotropy and homogeneity, thus neglecting an explicit dependence of  on the location k x , the balance laws read   , , , gard 0 kj ij i k kj j j U u Q            (1)   , , div 1 0, 2 kj k ij i kj j k j m Q x u Q x                   x  (2) , , curl( ) 0, mkn kj n kj n mkn kj j n j Q x u Q x          x    (3) with kj Q being Eshelby's energy-momentum tensor, kj  Kronecker's identity tensor, mkn  the Levi-Civita symbol, ij  the stress tensor, i u the displacements and m = 2 in two-dimensional and m = 3 in three-dimensional space. Eqs. (1) - (3) represent balance equations in a defect-free material. Applying these laws to boundary value problems including a crack, the integrals of a domain V provide finite values and are denoted as J k -, M - and L m -integrals. Considering plane problems, i.e. m = 2, and 3 L L  in the following, the integrals are transformed with Gauss' theorem into line integrals, surrounding the defect or the crack tip, see Fig. 1(a), now reading 0 lim d , k kj j J Q n s       (4)   0 0 lim d lim d , kj k j kj j k k k M Q x n s Q n s x J x              (5) T

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