PSI - Issue 13

Paul Judt et al. / Procedia Structural Integrity 13 (2018) 155–160

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4 Author name / Structural Integrity Procedia 00 (2018) 000–000 a local distinction between the sources of inaccuracy (body, crack faces) is provided. In general F tip 1 underestimates the crack driving force J 1 . In contrast to material forces at nodes α , the forces at nodes β are explicitly related to one crack tip. An improved value of J k is obtained, including the third term in Eq. (6) within a certain region along the crack faces, however excluding the crack tip. As mentioned earlier, material forces at crack faces are composed of real and artificial contributions and only the latter are of interest for the improvement of J k . The real material forces acting at the crack faces are related to the integral of the jump of Eshelby’s tensor across the crack faces and a closed expression for su ffi ciently small r is obtained inserting the dominant terms of the near tip stress and displacement fields into the tensor, i.e. Q k j ( r ) + − n + j = ( Q k j n j ) + ( r ) − ( Q k j n j ) − ( r ) = − 4 K II T 11 E √ 2 π r n + k = q k ( r ) + − , (7) with the mode-II SIF K II and the T -stress T 11 . In the vicinity of the crack tip n j is always perpendicular to the crack’s ligament and therefore Eq. (7) contributes to F 2 only. Integrating Eq. (7) provides a non-linear expression in r and thus a linear extrapolation of F sum 2 ( r ) as introduced in [18] theoretically provides results of limited accuracy. Better results are obtained by not extrapolating the jump of F 2 ( r ) + − but the symmetric part of stresses and strains on the crack faces as these are linearly depending on the distance to the crack tip. Inaccurate stress and strain values in the vicinity of the crack tip resulting from the finite element calculation are replaced by extrapolated values. This approach has been implemented and verified for the accurate calculation of J 2 applying remote integration contours and crack face integrals [6]. Employing extrapolated stress and strain values, the real material forces at the crack faces are calculated and the artificial forces are obtained from β F β, art k = β F β k − β F β, real k , (9) in Eq. (6). The second term in the Eq. (6) is obtained following the global approach according to Denzer et al. [3] as explained earlier. In terms of a simplified local approach, the artificial body forces are not evaluated, thus the second term in Eq. (6) is dropped. Doing so, J 1 is overestimated, while J 2 is obtained comparatively accurate. It appears that the mean value of the under- (no improvements at all) and overestimated variants of the crack driving force provides good results of J 1 . In Fig. 1(a) the investigated numerical specimen with a curved crack is depicted. Di ff erent methods are employed for the crack tip loading analysis and the resulting coordinates of J k are presented in the table in Fig. 1(c). Next to the remote calculation of the J k - and I k -integrals [6, 7], the CTE method [1] is employed for comparison. As expected, the classical local approach of the material force evaluated at the crack tip exhibits the largest deviation to the reference values. Calculating the sum of material forces within the crack tip region and linearly extrapolating F 2 according to [18], see Fig. 1(b), J 1 is accurately calculated and also J 2 is improved well. Skipping the extrapolation of F 2 and replacing material forces along the crack faces by improved forces according to the methodology introduced in this section (novel global approach), J k is even closer to the reference values. Applying the novel local approach 1, thus only improving forces at the crack faces close to the crack tip, J 2 is calculated accurately but J 1 is overestimated. The mean value of J 1 from the underestimated classic calculation and the overestimated novel approach 1 provides good results of J k (novel local approach 2), while the numerical e ff ort is low compared to the global approaches. where F β k are the forces without any improvements, finally inserting β F β, art k σ β 11 ( r ) = 1 2 σ + 11 ( r ) + σ − 11 ( r ) , ε β 11 ( r ) = 1 2 ε + 11 ( r ) + ε − 11 ( r ) (8)