PSI - Issue 23

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

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its deflection from the elastic solution. When comparing the J I and r p charts in Fig. 3, one can see that this change (deviation) corresponds to crack length when the crack tip plastic zone reaches the interface (i.e. when r p /( t - a ) = 1). When the plastic zone ahead of the crack tip reaches the interface, the stress-strain field starts to be deformed by the discontinuity in both elastic and plastic material properties of used material models. However, simple difference between Young’s moduli of the mentioned material models cannot explain such a difference – the crack tip is much further away from the interface than the difference between E Si and E Cu needs to manifest (for elastic case it was not visible until a was approximately 95% of t ). Therefore, the crack tip plasticity is the governing element. Moreover, this corresponds well with other author’s research for lower levels of loading (Kolednik et al., 2010; Pippan et al., 2000; Pippan and Riemelmoser, 1998; Simha et al., 2003) - the mismatch between yield stresses at the interface causes the J I (or G I ) to increase or decrease for higher to lower or lower to higher yield stress transitions respectively (in presented case the yield stress of Si substrate equals  , i.e. pure elastic behavior - modeled transition is from lower to higher yield stress). The loading levels with plasticity ratio above 1 (full thickness of Cu film is plastically deformed) show completely different J I behavior. The normalized crack driving force is rapidly decreasing with increase of the crack length and there is barely any connection between the J I and r p . The full thickness of the Cu film is plastically deformed and only small area along the crack flanks remains under elastic deformation, therefore the exact crack tip plastic region could not be obtained for the two highest plasticity ratios. However, for the case of σ yy / σ yield = 1.07 the exact crack tip plastic region was still noticeable, but it reached the interface very soon in the simulations. Moreover, it should be mentioned here that the normalization function used in Fig. 3 a is based solely on the elastic solution of J I and thus the highest loading levels are not represented well, but the trend of results should be independent on the used normalization function. The presented case-study of thin elastic-plastic Cu film on the purely elastic Si substrate with a crack approaching the Cu-Si interface showed interesting behavior of the crack driving force (described by J -integral) when the Cu plasticity and high loading levels are considered. Performed FE simulations of the problem revealed that the crack behavior in such a case can be divided into three groups according to the loading/plasticity levels: • In the first group for σ yy / σ yield < 0.5 the crack tip plastic zone is relatively small in comparison with the crack length a and the Cu film thickness t . There is no influence of the Cu plasticity and the elastic-plastic solution merges with purely elastic one. Hence, no plasticity induced shielding is present. • In the second group with nominal stress level as follows: 0.5 < σ yy / σ yield < 1, the plasticity starts to play significant role. A decrease in evaluated J I corresponding to the point when the crack tip plasticity reaches the interface was observed. When this occurs, the crack tip plastic zone starts to stick to the interface and its shape deforms according to the interface, hence a strong plasticity induced shielding was observed for such cases. • In the third group for σ yy / σ yield > 1 the full Cu film thickness is plastically deformed, and the main load-bearing part of the modeled body is the Si substrate. The plasticity induced shielding is strong for such high loading levels. The crack tip plastic zone merges with the overall plasticity of the Cu film (or for plasticity ratios close to 1, the crack tip plasticity is suddenly intensified) and only the increase of the plastic deformation (around the crack tip) above the nominal values in the Cu film could be observed (not the precise r p -value). Therefore, no direct connection between the J I progression and the point when the crack tip plasticity reaches the interface could be observed. The revealed behavior of the crack driving force in the thin elastic-plastic Cu films on the elastic Si substrate can help with understanding of the influence of the elastic-plastic interface on the crack driving force and the crack tip plasticity. Although the loading levels for nominal stress below the Cu yield stress were described and the connection between the plastic zone radius and the J I was revealed, the higher loading above Cu yielding point need more in-depth research and slightly modified approach. 5. Conclusions