PSI - Issue 3

W. Reheman et al. / Procedia Structural Integrity 3 (2017) 477–483 W. Reheman et al. / Structural Integrity Procedia 00 (2017) 000–000

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Figure 3. Displacements u 2 dashed black and stress σ 22 along the symmetry plane x 2 = 0. Elastic result (red) and elastic plastic result for σ Y = 0 . 5 σ s and hardening rate κ = 0 . 01 E (blue).

maintains a crack length of 0 . 2 R in the elastic case and 0 . 06 R in the elastic plastic case. This is less than observed in the experiments, i.e., around 0 . 5 R .

5. Conclusions

A phase field model is used to analyse a precipitate located at the surface of a body. The problem has two length scales, the width of the interface region and the size of the precipitate. Apart from the purely elastic case solutions for yield stresses between 0.1 σ s to 0.5 σ s and strain hardening rates κ from 0 . 01 E to 100 E are examined. The depth of the region with tensile stress decreases with decreasing yield stress. The region with tensile stress becomes very small and it is uncertain if it appears at all for the lowest yield stresses 0 . 1 σ s . It is observed that hardening rates at and below 0.01 give a result almost identical to the perfectly plastic material and at and above 100 may be treated as purely elastic. Lower hardening rates give increasing stress in the symmetry plane x 2 = 0 and the region with tensile stress penetrates deeper into the precipitate. The examination of a crack growing at infinitesimal crack tip load shows that the crack grows a shorter distance in the elastic plastic case. The growth maintains a crack length of 0 . 2 R in the elastic case and 0 . 06 R in the elastic plastic case. This is less than observed in the experiments, i.e., around 0 . 5 R . A shortcoming of the present analysis is that the material is assumed to be isotropic while it is known that the developing precipitate is approximately orthotropic and has ratio of largest to smallest principal expansions of around 1.6. This would require a second phase parameter that keeps track of the direction of the principal expansions, which was not available in the present study.

Acknowledgements

The first two authors gratefully acknowledge the financial support the Swedish Scientific Council under grant no. 2011-5561.

References

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