PSI - Issue 23

Stanislav Žák et al. / Procedia Structural Integrity 23 (2019) 239 – 244 Stanislav Žák et al. / Structural Integrity Procedia 00 (2019) 0 0 – 000

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Fig. 3. J -integral for mode I loading (normalized to Cu material characteristics E Cu and μ Cu with use of plane strain conditions, specimen cross-section area S , loading force F and crack length a ) and crack tip plastic zone radii (normalized to the thickness of remaining Cu film) as functions of crack length normalized to the entire Cu thin film thickness From Fig. 3 one can clearly see that the introduction of significant plastic behavior of the Cu material to the FE model changes the results significantly. The linear solution (dotted black line) shows constant value of the normalized J I (linearly increasing absolute value of J I with increasing crack length a ) for almost the whole range of the crack length. Only for a / t above 0.95, hence for the crack close to the interface, resulting elastic J I starts to decrease, which is in a good agreement with other authors (Cook and Erdogan, 1972; Pan and Amadei, 1999; Romeo and Ballarini, 1997; Zak and Williams, 1963), since in our case the transition on the bi-material interface is from compliant material to the stiffer one. Results from elastic-plastic FE simulation (separated according to the plasticity level described by the ratio σ yy / σ yield ) show increase in J I for cases with a short crack and decrease of J I when the crack is approaching the interface. This effect is more pronounced for higher plasticity levels. Moreover, observed crack tip plastic zone radius r p shows expected increase with increasing a until the plasticity reaches the interface where the growth of the plastic zone stops due to elastic conditions behind the interface (purely elastic model of Si). The resulting dependencies of J I on the crack length for cracks in thin films (when plasticity is taken into account) can be divided into three groups according to the plasticity (or loading) levels and the direct comparison with the linear-elastic solution can show the influence of the crack tip plasticity. The first elastic-plastic solution with a low plasticity ratio of 0.43 shows minimal to no difference in comparison with the elastic solution. Increase of the J I with increasing a is linear. Furthermore, the crack tip plastic zone is small for this case and thus the plasticity has no impact on the crack driving force. Some differences in comparison to the linear solution start to be noticeable for plasticity ratio between 0.5 and 1 (whereas the ratio 1 means fully plasticized Cu thin film). For these cases a small, rather insignificant, increase in normalized J I for short cracks is visible. Moreover, for long cracks (relatively to the Cu film thickness) some shielding effects are starting to play a significant role. This is noticeable by the sudden decrease of normalized J I and 4. Discussion