PSI - Issue 17

Siegfried Frankl et al. / Procedia Structural Integrity 17 (2019) 51–57 Siegfried Frankl / Structural Integrity Procedia 00 (2019) 000 – 000

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and after each  u , the energy release rate G is computed according to equation 2 from the FE model of the current displacement load u for the current crack length a as well as a +  a . The crack propagates by  a if the G value is higher than G c . The model checks for further crack propagation, until G < G c or the maximum crack length a max is reached. Then, the model runs for u +  u with the updated crack length. Fig. 2a shows this procedure by plotting the strain energy plus the frictional dissipation ( U + W f ) over the displacement u . Applying an initial displacement  u , the difference between the U + W f curves for the crack lengths a 0 and a 0 +  a is used to compute G according to equation 2. Since G is below G c , the crack is not extended and u is increased by  u . For a value of u = 2  u , G is computed in a similar way and is above G c . The crack thus grows by  a , but G for a +  a, is too small for the crack to grow another  a . So u is increased again, and it takes three  u steps until the crack grows again by  a . These  u and  a steps are shown in Fig. 2b. Fig. 2c shows the resulting force-displacement curve. This concept can also be employed with more complex crack growth criteria or even competing crack growth.

Fig. 2: Scheme of the iterative concept with (a) the strain energy U and the frictional dissipation W f , (b) the crack length a and (c) the tensile force F plotted over the applied displacement u .

3. Results and discussion

3.1. Study of necessary mesh size in FE model

Fig. 3 shows a contour plot of the von-Mises stress,  mis , in the model with a crack length of a = 4 mm, an applied displacement of u = 0.4 mm and a mesh size of m = 0.1 mm. The end of the fibre bundle is pulled to the left, and the rubber part is pressed against the rigid punch. Severe deformations can be seen at the crack tip, with maximum  mis values of 8.5 MPa. These maximum stresses of course highly depend on the mesh-size. Close to the rigid punch, the crack faces are pressed together. For computing stiffness and energies, it must be ensured that the mesh size has no influence on the energy output of the simulation. To investigate this, the mesh size m is varied between 0.025 mm and 0.6 mm in 8 steps with a logarithmic distribution. In general, the necessary mesh size has to be derived for all crack lengths. In this section, three representative crack lengths of a = 0.25 mm, 1 mm and 4 mm are used. For the simulations, the parameters from Table 2 are used and a displacement at the fibre bundle end of u = 0.4 mm is applied. Fig. 4 shows how the external work W e , the strain energy U , the frictional dissipation W f , and the tensile force F change with varied mesh size m . Since those values strongly depend on the crack length, they are shown in a normalized way, i.e. divided by their value for an m value of 0.025 mm. Table 3 shows their absolute values for all three crack lengths. It can be seen that the W f value lays two orders of magnitudes below W e and U . As mentioned in section 2.2, the path-dependence of the crack opening is not regarded in the model, i.e. the W f values might not be very accurate. This is indicated in the W f results, since W f with a value of 2.09 mJ for a crack length of 4 mm is smaller than W f = 6.22 mJ for a crack length of 1 mm. These unrealistic changes, however, are considerably smaller