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Scale effect and geometric shapes of grains

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Abstract

The rule-of-mixture approach has become one of the widely spread ways to investigate the mechanical properties of nano-materials and nano-structures, and it is very important for the simulation results to exactly compute phase volume fractions. The nanocrystalline (NC) materials are treated as three-phase composites consisting of grain core phase, grain boundary (GB) phase and triple junction phase, and a two-dimensional three-phase mixture regular polygon model is established to investigate the scale effect of mechanical properties of NC materials due to the geometrical polyhedron characteristics of crystal grain. For different multi-sided geometrical shapes of grains, the corresponding regular polygon model is adopted to obtain more precise phase volume fractions and exactly predict the mechanical properties of NC materials.

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References

  1. Hall E O. The deformation and aging of mild steel: III discussion of results[J]. Proc Phys Soc B, 1951, 64(1):747–753.

    Article  Google Scholar 

  2. Petch N J. The cleavage strength of polycrystals[J]. J Iron Steel Inst, 1953, 174(5):25–28.

    Google Scholar 

  3. Gleiter H. Nanocrystalline materials[J]. Prog Mater Sci, 1989, 33(4):223–315.

    Article  Google Scholar 

  4. Nieman G W, Weertman J R, Siegel R W. Microhardness of nanocrystalline palladium and copper produced by inert-gas condensation[J]. Scripta Metall, 1989, 23(13):2013–2018.

    Article  Google Scholar 

  5. Palumbo G, Erb U, Aust K T. Triple line disclination effects on the mechanical behaviour of materials[J]. Scripta Metall Mater, 1990, 24(12):2347–2350.

    Article  Google Scholar 

  6. Lu K. Nanocrystalline metals crystallized from amorphous solids: nanocrystallization, structure, and properties[J]. Mater Sci Eng R, 1996, 16:161–221.

    Article  Google Scholar 

  7. Mallow T R, Koch C C. Grain growth in nanocrystalline iron prepared by mechanical attrition[J]. Acta Mater, 1997, 45(5):2177–2186.

    Article  Google Scholar 

  8. Sanders P G, Eastman J A, Weertman J R. Elastic and tensile behavior of nanocrystalline copper and palladium[J]. Acta Mater, 1997, 45(17):4019–4025.

    Article  Google Scholar 

  9. Masumura R A, Hazzledine P M, Pande C S. Yield stress of fine grained materials[J]. Acta Mater, 1998, 46(13):4527–4534.

    Article  Google Scholar 

  10. Yamakov V, Wolf D, Phillpot S R, Gleiter H. Grain-boundary diffusion creep in nanocrystalline palladium by molecular-dynamics simulation[J]. Acta Mater, 2002, 50(1):61–73.

    Article  Google Scholar 

  11. Seattergood R O, Koch C C. A modified model for hall-petch behavior in nanocrystalline materials[J]. Scripta Mater, 1992, 27(9):1195–1200.

    Article  Google Scholar 

  12. Hahn H, Padmanabhan K A. A model for the deformation of nanocrystalline materials[J]. Phil Mag B, 1997, 76(44):559–571.

    Google Scholar 

  13. Fedorov A A, Gutkin M Yu, Ovid’ko I A. Transformations of grain boundary dislocation pile-ups in nano- and polycrystalline materials[J]. Acta Mater, 2003, 51(4):887–898.

    Article  Google Scholar 

  14. Fedorov A A, Gutkin M Yu, Ovid’ko I A. Triple junction diffusion and plastic flow in fine-grained materials[J]. Scripta Mater, 2002, 47(1):51–55.

    Article  Google Scholar 

  15. Gutkin M Yu, Kolesnikova A L, Ovid’ko I A, Skiba N V. Disclinations and rotational deformation in fine-grained materials[J]. Phil Mag Lett, 2002, 82(12):651–657.

    Article  Google Scholar 

  16. Ovid’ko I A. Materials science: deformation of nanostructures[J]. Science, 2002, 295(2395):2386–2386.

    Article  Google Scholar 

  17. Gutkin M Yu, Ovid’koy I A. Yield stress of nanocrystalline materials: role of grain boundary dislocations, triple junctions and Coble creep[J]. Philosophical Magazine, 2004, 84(9):847–863.

    Article  Google Scholar 

  18. Kocks U F. Relation between polycrystal deformation and single-crystal deformation[J]. Metal Trans, 1970, 1(55):1121–1143.

    Google Scholar 

  19. Carsley J E, Ning J, Milligan W W, Hackney S A, Aifantis E C. A simple, mixtures-based model for the grain size dependence of strength in nanophase metals[J]. Nanostruct Mater, 1995, 5(4):441–448.

    Article  Google Scholar 

  20. Konstantinidis D A, Aifantis E C. On the “anomalous” hardness of nanocrystalline materials[J]. Nanostruct Mater, 1998, 10(7):1111–1118.

    Article  Google Scholar 

  21. Benson David J, Fu Hsueh-hung, Meyers Marc André. On the effect of grain size on yield stress: extension into nanocrystalline domain[J]. Mat Sci Eng A, 2001, 319–321:854–861.

    Article  Google Scholar 

  22. Song H W, Guo S R, Hu Z Q. A coherent polycrystal model for the inverse Hall-Petch relation in nanocrystalline materials[J]. Nanostruct Mater, 1999, 11(2):203–210.

    Article  Google Scholar 

  23. Xiang Qing, Guo Xingming. The scale effect on the yield strength of nanocrystalline materials[J]. Internat J Solids Struct, 2006, 43(9):7793–7799.

    Google Scholar 

  24. Wang N, Wang Z, Aust K T, Erb U. Effect of grain size on mechanical properties of nanocrystalline materials[J]. Acta Metal Mater, 1995, 43(2):519–528.

    Article  Google Scholar 

  25. Kim H S. A composite model for mechanical properties of nanocrystalline materials[J]. Scripta Mater, 1998, 39(8):1057–1061.

    Article  Google Scholar 

  26. Kim H S, Bush M B. The effects of grain size and porosity on the elastic modulus of nanocrystalline materials[J]. Nanostruct Mater, 1999, 11(3):361–367.

    Article  Google Scholar 

  27. Kim H S, Estrin Y, Bush M B. Plastic deformation behaviour of fine-grained materials[J]. Acta Mater, 2000, 48(2):493–504.

    Article  Google Scholar 

  28. Kim H S, Estrin Y. Phase mixture modeling of the strain rate dependent mechanical behavior of nanostructured materials[J]. Acta Mater, 2005, 53(3):765–772.

    Article  Google Scholar 

  29. Gutkin M Yu, Ovid’ko I A, Pande C S. Theoretical models of plastic deformation process in nanocrystalline materials[J]. Rev Adv Mater Sci, 2001, 2(1):80–102.

    Google Scholar 

  30. Tjong S C, Chen Haydn. Nanocrystalline materials and coating[J]. Mat Sci Eng R, 2004, 45(1):1–88.

    Article  Google Scholar 

  31. Zhou Y, Erb U, Aust K T, Palumbo G. The effects of triple junctions and grain boundaries on hardness and Young’s modulus in nanostructured Ni-P[J]. Scripta Mater, 2003, 48(6):825–830.

    Article  Google Scholar 

  32. Zhao M, Li J C, Jiang Q. Hall-Petch relationship in nanometer size range[J]. J Alloy Compd, 2003, 361(2):160–164.

    Article  Google Scholar 

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, P. R. China

    Guo Hui  (郭辉) & Guo Xing-ming Doctor  (郭兴明) (Professor)

Authors
  1. Guo Hui  (郭辉)
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  2. Guo Xing-ming Doctor  (郭兴明)
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Corresponding author

Correspondence to Guo Xing-ming Doctor  (郭兴明).

Additional information

Contributed by GUO Xing-ming

Project supported by the National Natural Science Foundation of China (No.10472061), Key Project of Shanghai Municipal Commission of Science and Technology (No.04JC14034), the Doctoral Foundation of Ministry of Education of China (No.20060280015) and Shanghai Leading Academic Discipline Project (No.Y0103)

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Guo, H., Guo, Xm. Scale effect and geometric shapes of grains. Appl Math Mech 28, 141–149 (2007). https://doi.org/10.1007/s10483-007-0201-1

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  • DOI: https://doi.org/10.1007/s10483-007-0201-1

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Chinese Library Classification

2000 Mathematics Subject Classification

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