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Acta Mechanica

, Volume 230, Issue 12, pp 4259–4271 | Cite as

Size-dependent yield function for single crystals with a consideration of defect effects

  • Bo PanEmail author
  • Hiro Tanaka
  • Chao Ling
  • Yoji Shibutani
  • Shufeng Li
Original Paper
  • 107 Downloads

Abstract

In this work, a size-dependent yield function for a single crystal is developed by considering defect effects, including dislocation pile-up, dislocation starvation, and source exhaustion, especially for micro-pillars. It is found that the proposed yield function compares well with the experimental data, and the proposed model is the extension of a single-arm source model to describe the yielding behavior under a more complicated loading case, not only the uniaxial compression test. Our quantitative conclusions suggest that the stacking-fault energy (SFE), the crystallographic orientation, and the slip system are significant factors for the shape of the yield surface: the slip system determines the number of the edges of the yield surface; the crystallographic orientation controls the angles between the adjacent edges but does not change the number of the edges; the low SFE can make sharp corners rounded and contract the shape of the yield surface, or even curve the edge of the yield surface. Moreover, we investigate the explicit relationship among the stacking-fault energy, the dislocation pile-up effect inside the sample, and the shape of the yield surface: materials with a low stacking-fault energy exhibit pronounced dislocation pile-up effects, and their yield surfaces tend to display rounded vertices, corresponding to the v. Mises yield criterion for the single-crystal sample with a {1 1 1} slip system for example; those with a high stacking-fault energy show typical Tresca criterion-type yield surfaces displaying sharp vertices for the single-crystal sample with a {1 1 1} slip system for example. We also show that this yield function can describe the size-dependent yield surface by considering the stochastic length of the dislocation source and the dislocation pile-up length in single-crystalline micro-pillars.

Notes

Acknowledgements

Y. Shibutani gratefully acknowledges the financial support from Grants-in-Aid for Scientific Research (A) (26249002), and H. Tanaka thanks Young Scientists (A) (25709001) for financial support. S. Li thanks the National Natural Science Foundation of China (Grant numbers 51571160 and 51871180) and Natural Science Basic Research Plan in Shaanxi Province of China (Grant number 2015JM5233) for the financial support of this study. The authors thank Mr. Kenta Yukihiro (a former graduate student in Osaka University) who performed the experiments.

References

  1. 1.
    Uchic, M.D., Dimiduk, D.M., Florando, J., Nix, W.D.: Sample dimensions influence strength and crystal plasticity. Science 305, 986–989 (2004)CrossRefGoogle Scholar
  2. 2.
    Dimiduk, D.M., Uchic, M.D., Parthasarathy, T.A.: Size-affected single-slip behavior of pure nickel microcrystals. Acta Mater. 53, 4065–4077 (2005)CrossRefGoogle Scholar
  3. 3.
    Greer, J.R., Oliver, W.C., Nix, W.D.: Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Mater. 53, 1821–1830 (2005)CrossRefGoogle Scholar
  4. 4.
    Fleck, N.A., Hutchinson, J.W.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825–1857 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Fleck, N.A., Hutchinson, J.W.: Strain Gradient Plasticity, p. 295. Academic Press, New York (1997)zbMATHGoogle Scholar
  6. 6.
    Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W.: Mechanism-based strain gradient plasticity—I. Theory. J. Mech. Phys. Solids 47, 1239–1263 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Huang, Y., Gao, H., Nix, W.D., Hutchinson, J.W.: Mechanism-based strain gradient plasticity—II. Analyis. J. Mech. Phys. Solids 48, 99–128 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Nix, W.D., Gao, H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425 (1999)zbMATHCrossRefGoogle Scholar
  9. 9.
    Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. 42, 475–487 (1994)CrossRefGoogle Scholar
  10. 10.
    Shan, Z.W., Mishra, R.K., Asif, S.A.S., Warren, O.L., Minor, A.M.: Mechanical annealing and source-limited deformation in submicrometre-diameter Ni crystals. Nat. Mater. 7, 115–119 (2008)CrossRefGoogle Scholar
  11. 11.
    Tang, H., Schwarz, K.W., Espinosa, H.D.: Dislocation escape-related size effects in single-crystal micropillars under uniaxial compression. Acta Mater. 55, 1607–1616 (2007)CrossRefGoogle Scholar
  12. 12.
    El-Awady, J.A., Uchic, M.D., Shade, P.A., Kim, S.L., Rao, S.I., Dimiduk, D.M., Woodward, C.: Pre-straining effects on the power-law scaling of size dependent strengthening in Ni single crystals. Scr. Mater. 68, 207–210 (2013)CrossRefGoogle Scholar
  13. 13.
    Bei, H., Shim, S., Pharr, G.M., George, E.P.: Effects of pre-strain on the compressive stress–strain response of Mo-alloy single-crystal micropillars. Acta Mater. 56, 4762–4770 (2008)CrossRefGoogle Scholar
  14. 14.
    Schneider, A.S., Kiener, D., Yakacki, C.M., Maier, H.J., Gruber, P.A., Tamura, N., Kunz, M., Minor, A.M., Frick, C.P.: Influence of bulk pre-straining on the size effect in nickel compression pillars. Mater. Sci. Eng. A 559, 147–158 (2013)CrossRefGoogle Scholar
  15. 15.
    Xie, K.Y., Shrestha, S., Cao, Y., Felfer, P.J., Wang, Y., Liao, X., Cairney, J.M., Ringer, S.P.: The effect of pre-existing defects on the strength and deformation behavior of a-Fe nanopillars. Acta Mater. 61, 439–452 (2013)CrossRefGoogle Scholar
  16. 16.
    Pan, B., Shibutani, Y., Zhang, X., Shang, F.: Effect of dislocation pile-up on size-dependent yield strength in finite single-crystal micro-samples. J. Appl. Phys. 18, 014305 (2015)CrossRefGoogle Scholar
  17. 17.
    Senger, J., Weygand, D., Gumbsch, P., Kraft, O.: Discrete dislocation simulations of the plasticity of micro-pillars under uniaxial loading. Scr. Mater. 58, 587–590 (2018)CrossRefGoogle Scholar
  18. 18.
    Parthasarathy, T.A., Rao, S.I., Dimiduk, D.M., Uchic, M.D., Trinkle, D.R.: Contribution to size effect of yield strength from the stochastics of dislocation source lengths in finite samples. Scr. Mater. 56, 313–316 (2007)CrossRefGoogle Scholar
  19. 19.
    Pan, B., Shibutani, Y., Tanaka, H.: Dislocation-based constitutive model of crystal plasticity for the size effect of single crystalline micropillar samples. Mech. Eng. J. 4, 00602 (2016)Google Scholar
  20. 20.
    Pan, B., Tanaka, H., Shibutani, Y.: Effect of surface energy upon size-dependent yield strength of single-crystalline hollow micro- and nanopillars. Mater. Sci. Eng. A 659, 22–28 (2016)CrossRefGoogle Scholar
  21. 21.
    Rao, S.I., Dimiduk, D.M., Tang, M., Uchic, M.D., Parthasarathy, T.A., Woodward, C.: Estimating the strength of single-ended dislocation sources in micron-sized single crystals. Philos. Mag. 87, 4777–4794 (2007)CrossRefGoogle Scholar
  22. 22.
    El-Awady, J.A., Wen, M., Ghoniem, N.M.: The role of the weakest-link mechanism in controlling the plasticity of micropillars. J. Mech. Phys. Solids 57, 32–50 (2009)zbMATHCrossRefGoogle Scholar
  23. 23.
    Tang, H., Schwarz, K.W., Espinosa, H.D.: Dislocation-source shutdown and the plastic behavior of single-crystal micropillars. Phys. Rev. Lett. 100, 185503 (2008)CrossRefGoogle Scholar
  24. 24.
    Montheillet, F., Gilormini, P., Jonas, J.J.: Relation between axial stresses and texture development during torsion testing: a simplified theory. Acta Metall. 33, 705–718 (1985)CrossRefGoogle Scholar
  25. 25.
    Houtte, P.V.: Calculation of the yield locus of textured polycrystals using the Taylor and the related Taylor theories. Texture Microstruct. 7, 29–72 (1987)CrossRefGoogle Scholar
  26. 26.
    Lequeu, Ph, Gilormini, P., Montheillet, F., Bacroix, B., Jonas, J.J.: Yield surfaces for textured polycrystals—I. Crystallographic approach. Acta Metall. 35, 439–451 (1987)CrossRefGoogle Scholar
  27. 27.
    Lequeu, Ph, Gilormini, P., Montheillet, F., Bacroix, B., Jonas, J.J.: Yield surfaces for textured polycrystals—II. Analytical approach. Acta Metall. 35, 1159–1174 (1987)CrossRefGoogle Scholar
  28. 28.
    Arminjon, M.: A regular form of the Schmid law. Application to the ambiguity problem. Texture Microstruct. 14–18, 1121–1128 (1991)CrossRefGoogle Scholar
  29. 29.
    Darrieulat, M., Piot, D.: A method of generating analytical yield surfaces of crystalline materials. Int. J. Plast. 12, 575–610 (1996)zbMATHCrossRefGoogle Scholar
  30. 30.
    Gambin, W.: Plasticity of crystals with interacting slip systems. Eng. Trans. 39, 303–324 (1991)MathSciNetGoogle Scholar
  31. 31.
    Gambin, W.: Refined analysis of elastic–plastic crystals. Int. J. Solids Struct. 29, 2013–2021 (1992)zbMATHCrossRefGoogle Scholar
  32. 32.
    Gambin, W., Barlat, F.: Modeling of deformation texture development based on rate independent crystal plasticity. Int. J. Plast. 13, 75–85 (1997)zbMATHCrossRefGoogle Scholar
  33. 33.
    Zamiri, A., Pourboghrat, F., Barlat, F.: An effective computational algorithm for rate-independent crystal plasticity based on a single crystal yield surface with an application to tube hydroforming. Int. J. Plast. 23, 1126–1147 (2007)zbMATHCrossRefGoogle Scholar
  34. 34.
    Zamiri, A., Pourboghrat, F.: A novel yield function for single crystals based on combined constraints optimization. Int. J. Plast. 26, 731–746 (2010)zbMATHCrossRefGoogle Scholar
  35. 35.
    Guan, Y., Pourboghrat, F., Barlat, F.: Finite element modeling of tube hydroforming of polycrystalline aluminum alloy extrusions. Int. J. Plast. 22, 2366–2393 (2006)zbMATHCrossRefGoogle Scholar
  36. 36.
    Panin, V.E., Armstrong, R.W.: Hall–Petch analysis for temperature and strain rate dependent deformation of polycrystalline lead. Phys. Mesomech. 19, 35–40 (2016)CrossRefGoogle Scholar
  37. 37.
    Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Interdisciplinary Applied Mathematics, pp. 10–11. Springer, New York (1998)Google Scholar
  38. 38.
    Kreisselmeier, G., Steinhauser, R.: Systematic control design by optimizing a vector performance index. In: Proceedings of IFAC Symposium on Computer Aided Design of Control Systems, Zürich, Switzerland (1979)Google Scholar
  39. 39.
    Hartford, J., Sydow, B., Wahnström, G., Lundqvist, B.I.: Peierls barriers and stresses for edge dislocations in Pd and Al calculated from first principles. Phys. Rev. B 58, 2487–2496 (1998)CrossRefGoogle Scholar
  40. 40.
    Jennings, A.T., Burek, M.J., Greer, J.R.: Microstructure versus size: mechanical properties of electroplated single crystalline Cu nanopillars. Phys. Rev. Lett. 104, 135503 (2010)CrossRefGoogle Scholar
  41. 41.
    Grujicic, M., Dong, P.: Computer simulation of martensitic transformation in Fe–Ni face-centered cubic alloys. Mater. Sci. Eng. A 201, 194–204 (1995)CrossRefGoogle Scholar
  42. 42.
    Kowalczyk, K., Gambin, W.: Model of plastic anisotropy evolution with texture-dependent yield surface. Int. J. Plast. 20, 19–54 (2004)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Xi’an Thermal Power Research Institute Co. Ltd.Xi’anChina
  2. 2.Department of Mechanical EngineeringOsaka UniversitySuitaJapan
  3. 3.Southern University of Science and TechnologyShenzhenChina
  4. 4.School of Materials Science and EngineeringXi’an University of TechnologyXi’anChina
  5. 5.Joining and Wedding Research InstituteOsaka UniversityIbarakiJapan

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