PSI - Issue 39

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Davide Leonetti et al. / Procedia Structural Integrity 39 (2022) 9–19 Author name / Structural Integrity Procedia 00 (2019) 000–000

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The mesh of the finite element model used for the determination of the Green’s functions is depicted in Figure 7, following a convergence study. A non-conformal mesh is used between the control volume and the rest of the geometry, whereas a structured mesh is used around the crack tip. Elements of type PLANE183, i.e. 4 or 3 node elements with linear displacement functions, are used and the energy release rate for mode I and mode II of opening are extracted using the virtual crack closure technique. The stress intensity factors are calculated considering a plane state of strain. The Green’s functions are therefore derived by K j,i = 1 √ π a G j,i � a, x a � (11) since the forces are of unitary values. The indices j and i have the same meaning as for Equation 4. The Green’s functions obtained are first compared with unpublished solutions reported in Fett (1997) for a/W =0.6 and ω =22.5 ° . The results are reported in Table 1 where a good agreement can be seen. Therefore, the model is used to derive the Green’s functions for the case modeled by Bogdansky (1999), reported in the last two columns of Table 1.

Table 1. Comparison of Green’s functions with literature and proposed values for the reference case

a/W =0.6 and ω =22.5°

ω=75°, a/W=0.36

Lit.

FE

dev

Lit.

FE

dev

Load location

x/a =0

x/a =0.8

x/a =0

x/a =0.8

16,06

15,85

-1,3% -8,0% -0,5% -2,9%

7,90 0,33 0,20 3,85

7,88 0,30 0,19 4,10

-0,3% -8,0% -6,0% 6,6%

128 9.75 2.89 7.80

34.6 1.98 2.69 8.27

G 1,1 G 2,1 G 1,2 G 2,2

1,09 0,25 3,49

1,00 0,25 3,39

3.2.2. SIF Results The SIFs are reported in Fig.8 where the normalized SIFs are shown as a function of the distance between the center of the contact divided by half the contact length. The weight functions are determined using the direct adjustment method, using Green’s functions for the considered crack configuration, as done in Fett (1997). As shown in Fig. the FE model in Bogdansky (1999) and the proposed analytical model give similar results. Moreover, the proposed approach gives similar trend of some of the analyses reported in Bogdansky (1996). The differences in the SIF trend and peak values are attributed to the difference in the models, namely: the different boundary conditions, and the different load. In the model of Bogdansky a frictional contact is modelled between the disk and the plate, resulting in a different state of stress than the patch load. This difference is further justified when it is considered that different boundary conditions are considered in Bogdansky (1999) as compared to those considered in this paper, as shown in Fig.3.

0.0 0.2 0.4 0.6 0.8 1.0 1.2

1.2

Results

Bogdansky

0.8

0.4

0.0

KI/KImax

KII/KIImax

-0.4

-0.8

-2

-1

0

1

2

3

-2

-1

0

1

2

3

x/c

x/c

(a) mode I SIF (b) mode II SIF Fig. 8. Trend of the SIF as a function of the wheel position, normalized versus the contact semi length. Comparison with Bogdansky (1999).