PSI - Issue 13

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

159

Author name / Structural Integrity Procedia 00 (2018) 000–000

5

100MPa

15

12

50 mm

9

Method J 2 F k classic local 8.85 -4.60 F k global [18] 9.95 -9.68 F k novel global 9.91, -9.46 F k novel local 1 11.35 -9.46 F k novel local 2 10.11 -9.46 J k -Integral [6] 9.97 -9.27 I k -Integral [7] 9.94 -9.06 CTE [1] 9.91 -9.08 (c) J k resulting from di ff erent methods J 1

6

F 1 F 2 F Ext 1 F Ext 2

3

25 mm

0

60 ◦

F sum k / (N/mm)

− 3 − 6 − 9

− 12 − 15

100 mm

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

r / mm (b) Extrapolation of F sum k

(a) Specimen

according to [18]

Fig. 1. Numerical example of a curved crack and application of di ff erent methods for crack loading analysis

3. Crack initiation applying material forces

Di ff erent from the material force at the crack tip, the forces calculated following [13] at other defects such as material interfaces or free surfaces, e.g. crack faces, holes or notches, depend on the local mesh size. To obtain a mesh-independent loading quantity, material forces at these defects are divided by the local element edge length and thickness, providing the material traction vector q k according to Eq. (2). Quadratic isoparametric 8-node elements are employed in the calculations, requiring the application of specific weighting factors, depending on the element node position, to obtain a physical distribution of the material tractions [8]. Material forces at free surfaces in general are perpendicular with respect to the surface. It is assumed, that new cracks emerge into the direction of q k at the location where q k reaches a critical value. At the numerical model of a flat bar tensile test specimen with inclined central U-shaped notch (notch inclination angle γ = 45 ◦ ) and externally applied displacement loading as experimentally investigated in [14], material tractions are calculated at free surfaces. The largest material tractions are observed at the notch and the distribution of the absolute value of q k along the notch surface is plotted in Fig. 2(a). The maximum value of | q k | is observed at the angle ϕ = − 0 . 25 π and coincides with the maximum of the average ERR G ∗ = − ∆Π / ∆ a calculated from explicit total energy states of the specimen with and without initiated crack, see [8]. Furthermore, the same crack initiation angle was observed in experiments [14]. In Fig. 2(b) the average ERR is plotted vs. the crack initiation angle ϑ for five initiation positions ϕ . It is obvious, that the maximum of G ∗ for each ϕ is located at ϑ = 0. Thus G ∗ is maximized if a crack initiates perpendicular to the notch surface. The global maximum (in the investigated cases) is provided for ϕ = − 45 ◦ and ϑ = 0 which is also predicted by the maximum material tractions, see Fig. 2(a).

Acknowledgment

The authors would like to acknowledge the financial support of the ”Landes-O ff ensive zur Entwicklung Wissenschaftlich-o¨konomischer Exzellenz (LOEWE)” research funding program ”Safer Materials”.

References

[1] Barsoum, R., 1976. On the use of isoparametric finite elements in linear fracture mechanics. International Journal of Numerical Methods in Engineering 10, 25–37. [2] Bergez, D., 1974. Determination of stress intensity factors by use of path-independent integrals. Mechanics Research Communications 1, 179–180.