Issue 8

K. G. Kodancha et alii, Frattura ed Integrità Strutturale, 8 (2009) 45-51; DOI: 10.3221/IGF-ESIS.08.04 46 elements in ABAQUS are referred as C3D20R. This kind of elements is used in the work of Qu and Wang [12] for computation of T-stress. In this analysis, to model the ( 1/ r ) singularity of stress/strain at the crack-front, the nodes of crack front elements are shifted to quarter point as shown in Fig.2. The number of elements in the analysis domain varied with the thickness of the specimens. It is noted that for specimen with thickness 2 mm the number of elements used are 5625 (number of nodes, 25,536) and for 20 mm thick specimen the number of elements used were 7875 (number of nodes, 35, 456). The number of elements in the analysis domain increases with increase in the specimen thickness to maintain the mesh refinement. For specimen with thickness between 2 and 20 mm the numbers of elements were between 5625 – 7875. A typical mesh used in the analysis is shown in Fig.3. The stress distribution and the magnitude of K I and T-stress were obtained by ABAQUS post processor. The method of computation of K I and T-stress from ABAQUS are detailed in the following section. The variation of stress components, T-stress and K I along the crack-front has been studied for varied specimen thickness and crack length to width ratio (a/W=0.4-0.6). The normal stress ( σ ) applied on the specimen as shown in Fig.1 is considered = 50 MPa, which is≈1/3 rd of yield stress to keep the analysis domain approximately under LEFM. The specimen thicknesses (B) considered in this analysis are 2 to 20 mm (B/W=0.1- 1.0) in steps of 2 mm. In these calculations, the material considered is interstitial free steel (IF) possessing yield strength ( σ y ) of 155 MPa and elastic modulus of 197 GPa [14]. Computation of K I and T-stress The computation of stress intensity factor (K I ) is carried out by maximum tangential stress criterion [15] as available in ABAQUS. The T-stress extracted by ABAQUS post processor is as given below. Consider a 3D crack front with a continuously turning tangent as shown in Fig. 4a. Assume a line-load of magnitude f k = f µ k (s) to be applied along the crack front as illustrated in Fig. 4b. In the figure, µ k (s) defines the direction normal to the crack front and in the plane of the crack at point s. The solution for this problem is the case of a plane strain semi-infinite crack with a point force f applied at the crack tip in the direction parallel to the crack. Using superscript ‘L’ to designate the stress and displacement fields, the analytical solution [16] gives: Stress field:     3 11 cos , L f r      2 22 cos sin , L f r      33 cos , L f r     13 23 0 L L      2 12 sin cos , L f r (1) and displacement field :                             2 2 1 1 sin ln 2(1 ) L f r u E d             2 1 (1 2 ) cos sin 2 L f u E  3 0 L u (2) Employing the stress field and displacement field solutions given in Eqns. (1) and (2) as an auxiliary field, Kfouri [17] has extracted the T-stress for 2D crack problems by introducing an interaction J -integral. Nakamura and Parks [18] extended this method of extraction of T-stress to 3D crack problems and provided the domain integral formulations in the form given below:                                         ( ) 1 ( ) L L L i i k k ij ij ij ij c k k j k V s u u q q I s dV A x x x x (3) where V ( s ) is a domain, which encloses the crack front segment between ( s- ε ) and ( s+ ε ) (Refer Fig. 4c ), q k (s) defines the virtual extension of the crack front segment ( s- ε ) ≤ s ≤ ( s+ ε ) and A c is the increase in crack area generated by the virtual