Issue34

P.O. Judt et alii, Frattura ed Integrità Strutturale, 34 (2015) 208-215; DOI: 10.3221/IGF-ESIS.34.22 213 A PPLICATION OF M - AND L - INTEGRALS TO CRACK TIP PLASTIC ZONES he concept of the configurational forces signifies their vanishing in a defect-free body. Within the framework of the FE-method, however, it is observed that, depending on the applied mesh, configurational forces arise though no defects are existent [23]. This phenomenon may be used for the adaption of FE meshes in terms of vanishing configurational forces at internal nodes [23], and as an error indicator of the FE solution [24]. Our investigations confirm that these configurational forces depend on the mesh and the loading. However, configurational forces in an homogeneous body are small compared to those resulting from defects. (a) (b) Figure 3 : (a) Integration contour 0  and c  enclosing the crack tip and the plastic zone, material forces ( ) n k F and the crack driving force J k , (b) shapes of crack tip plastic zones in normalized depiction max p p ( )/ r r  from the von Mises yield criterion. A new application of the configurational forces approach is the separation of the crack driving force and the forces acting at a plastic deformation ( ) n k F , see Fig. 3(a). Evaluating the M - and L -integrals at remote integration contours e.g. along the external boundaries of the model, as presented in Section 2, the contribution of configurational forces in the crack tip plastic zone in general is implicitly included in the result. In contrast to a global approach of the J k -integral, always providing the sum of all configurational forces including J k and ( ) n k F , the global approaches of M - and L -integrals exclusively represent the contribution of the material forces in the plastic zone, if the origin of the global coordinate system is the crack tip. In that case the crack driving force J k has no contribution to M or L . The model of a double cantilever beam (DCB) specimen with different loading conditions is investigated, see Fig. 4(a), including a single mode-I ( F 1 = F 2 = F ), a single mode-II ( F 1 = − F 2 = 0.5 F ) and a plane mixed-mode loading with mixed mode ratio β = K II / K I = −2/3 ( F 1 = F , F 2 = 0). The material model is that of an idealized linear elastic-perfectly plastic behavior with tensile yield strength σ y = 462MPa. A first approximation of the plastic zone p ( ) r  is obtained by substituting the closed-form representation of the asymptotic crack tip field into the von Mises yield criterion for the investigated mode cases, leading to the normalized shapes depicted in Fig. 3(b). With this, the extension of the plastic zone rp along the ligament is calculated according to 2 2 I II p 2 y 3 , 2 K K d    (12) being a simplified measure for the size of the crack tip plastic zone. The global M - and L -integrals are calculated for an increasing load. The calculated values M pl and L pl must be corrected considering the initial configurational forces resulting from mesh and external loading. The correction terms are obtained from the global M - and L -integrals at the linear-elastic regime, and thus pl el M M M   and pl el L L L   . In Fig. 4(b) T