PSI - Issue 13

Danilo D’ Angela et al. / Procedia Structural Integrity 13 (2018) 939–946 Danilo D’Angela and Marianna Ercolino / Structural Integrity Procedia 00 ( 2018) 000 – 000

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Table 1. Mechanical properties of the model.

Elastic response

Fracture criterion

Stage I G thres G C

Stage III c 4 G pl G C G IC G IIC G IIIC

Stage II

ρ

Models

ν

E

c 3

[ K m g 3 ] [-] [ k m N 2 ] [-] [ m/cycles (N/m) c 4 ] [-] [-] [ k m N ] [ k m N ] [ k m N ] 3.24E−12 1.285 80.0 80.0 80.0

[-]

Plane Strain Plane Stress Plane Strain Plane Stress

Room temp. Cryog. Temp.

2.80E−12 2.17E−13 1.86E−13

90.0 80.0 90.0

90.0 80.0 90.0

90.0 80.0 90.0

1E−6

0.99

8050 0.33 2.03E+11

1.355

3. Crack length computation

The numerical crack length ( a ) along the number of cycles ( N ) is computed considering two methods: (a) Φ -based and (b) ASTM-based methods. The former is developed in this study since ABAQUS does not provide for direct history output of the crack (Bergara et al., 2017; Simulia, 2016). It is based on the processing of the distance function Φ (Simulia, 2016). The method is checked and it resulted quite accurate if the mesh is sufficiently homogeneous in the vicinity of the crack. The ASTM-based method is based on an empirical relationship by ASTM (ASTM International, 2015). Eq. 6 and Eq. 7 correlate the displacement of the control point (i.e., A in Fig. 2) to the crack length propagation. The combination factors (i.e., C a to C f ) depend on the location of the control points. E , B and P are the elastic modulus, the thickness and the maximum cyclic load, respectively; ν is the control point measure over the cycles. = ⁄ = + + 2 + 3 + 4 + 5 (6) = {[ ] 0.5 + 1} −1 0.2 ≤ ≤ 0.975 (7) Since ABAQUS performs the analysis up to the complete failure of the component, a conventional fracture limit is established. When the fatigue crack starts growing with a rate significantly larger than the one in Stage II, the onset of Stage III is occurring, and the fracture is conventionally reached. An analytical crack propagation curve is evaluated by numerical integration of Paris law model (Eq. 8) (Santecchia et al., 2016), using Eq. 9 for the specification of ΔK , according to (ASTM International, 2015); a F is found assuming ΔK equal to K C in Eq. 9. The latter is valid for plane stress conditions, but in this context, it has been applied to plane strain conditions too. ( ) = ∫ ( ) 1 0 (8) = (2+ ) √ (1− ) 1.5 (0.886 + 4.64 − 13.32 2 + 14.72 3 − 5.6 4 ) (9) 4. Results The numerical and the analytical crack propagation curves are compared to the experimental results (Fig. 4), carried out by Kim et al. (2015). The numerical curves matched with good agreement the experimental ones, with particular reference to the room temperature tests. ASTM-based method gives the same results for both plane stress and plane strain conditions, except for the pre-failure part of the curves. The results showed that the control point displacement time histories are not significantly affected by the plane stress or plane strain conditions, except for the pre-failure part. On the contrary, the crack length curves estimated using the Φ -based (i.e., actual crack propagation) are more affected by those conditions. However, the differences are much reduced in value, and the pre-failure parts of the respective ASTM-based and Φ -based curves are very similar.