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Size Effects of Elastic Modulus of FCC Metals Based on the Cauchy-Born Rule and Nanoplate Models

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Abstract

In the present research, a simple quasi-continuum model, the Cauchy-Born rule model, is used to investigate the size effects of elastic modulus for fcc metals. By considering a nanoplate model and calculating the strain energy for the nano-sized plate under tension and bending, the relationship between the elastic modulus and the plate thickness is found. Size effects of the elastic modulus are displayed by the relative differences of the elastic modulus between the nano-sized plate sample and the bulk sample. By comparing the present results with those of others, the effectiveness of the Cauchy-Born rule model in studying the size effects of material properties are shown.

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References

  1. Duan, X.F., Huang, Y., Cui, Y., Wang, J.F. and Lieber, C.M., Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices. Nature, 2001, 409: 66–69.

    Article  Google Scholar 

  2. Friedman, R.S., McAlpine, M.C., Ricketts, D.S., Ham, D. and Lieber, C.M., High-speed integrated nanowire circuits. Nature, 2005, 434: 1085–1085.

    Article  Google Scholar 

  3. Lieber, C.M. and Wang, Z.L., Functional nanowires. MRS Bulletin, 2007, 32: 99–108.

    Article  Google Scholar 

  4. Chen, X.L., Ma, H.S., Liang, L.H. and Wei, Y.G., A surface energy model and application to mechanical behavior analysis of single crystals at sub-micron scale. Computational Materials Science, 2009, 46: 723–727.

    Article  Google Scholar 

  5. Wei, Y.G. and Hutchinson, J.W., Hardness trends in micron scale indentation. Journal of the Mechanics and Physics of solids, 2003, 51: 2037–2056.

    Article  Google Scholar 

  6. Villain, P., Goudeau, Ph., Renault, P.O. and Badawi, K.F., Size effect on intragranular elastic constants in thin tungsten films. Applied Physics Letters, 2002, 81: 4365–4367.

    Article  Google Scholar 

  7. Renault, P.O., Bourhis, E.L., Villain, P., Goudeau, Ph., Badawi, K.F. and Faurie, D., Measurement of the elastic constants of textured anisotropic thin films from x-ray diffraction data. Applied Physics Letters, 2003, 83: 473–475.

    Article  Google Scholar 

  8. Miller, R.E. and Shenoy, V.B., Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 2000, 11: 139–147.

    Article  Google Scholar 

  9. Duan, H.L., Wang, J., Huang, Z.P. and Karihaloo, B.L., Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. Journal of the Mechanics and Physics of Solids, 2005, 53: 1574–1596.

    Article  MathSciNet  Google Scholar 

  10. Sun, C.T. and Zhang, H.T., Size-dependent elastic moduli of platelike nanomaterials. Journal of Applied Physics, 2003, 93: 1212–1218.

    Article  Google Scholar 

  11. Zhou, L.G. and Huang, H.C., Are surface elastically softer or stiffer? Applied Physics Letters, 2004, 84: 1940–1942.

    Article  Google Scholar 

  12. Digreville, R., Qu, J.M. and Cherkaoui, M., Surface free energy and its effect on the elastic behavior of nanosized particles, wires and films. Journal of the Mechanics and Physics of solids, 2005, 53: 1827–1854.

    Article  MathSciNet  Google Scholar 

  13. Dingreville, R., Kulkarni, A.J., Zhou, M. and Qu, J.M., A semi-analytical method for quantifying the size-dependent elasticity of nanostructures. Modelling and Simulation in Materials Science and Engineering, 2008, 16: 1–16.

    Article  Google Scholar 

  14. Wang, G.F., Feng, X.Q., Yu, S.W. and Nan, C.W., Interface effects on effective elastic moduli of nanocrystalline materials. Materials Science and Engineering A, 2003, 363: 1–8.

    Article  Google Scholar 

  15. Zhu, L. and Zheng, X.J., Modification of the elastic properties of nanostructures with surface charges in applied electric fields. European Journal of Mechanics A/solids, 2010, 29: 337–347.

    Article  Google Scholar 

  16. Horstemeyer, M.F., Baskes, M.I. and Plimpton, S.J., Computational nano scale plasticity using embedded atom potentials. Theoretical and Applied Fracture Mechanics, 2001, 37: 49–98.

    Article  Google Scholar 

  17. Huang, X. and Pelegri, A.A., Finite element analysis on nano indentation with friction contact at the film/substrate interface. Composites Science and Technology, 2007, 67: 1311–1319.

    Article  Google Scholar 

  18. Xiao, S.P. and Belytschko, T., A bridging domain method for coupling continua with molecular dynamics. Computer Methods in Applied Mechanics and Engineering, 2004, 193: 1645–1669.

    Article  MathSciNet  Google Scholar 

  19. Born, M. and Huang, K., Dynamical Theory of Crystal Lattices, first ed. Oxford University Press, 1962.

  20. Ericksen, J.L., The Cauchy Born Hypotheses for Crystals, Phase Transformations and Material Instabilities in Solids. In: Gurtin, M.E. (Ed.). Academic Press, 1984, 61–77.

    Google Scholar 

  21. Park, H.S. and Liu, W.K., An introduction and tutorial on multiple-scale analysis in solids. Computer Methods in Applied Mechanics and Engineering, 2004, 193: 1733–1772.

    Article  MathSciNet  Google Scholar 

  22. Xiao, S. and Yang, W., Temperature-related Cauchy–Born rule for multi-scale modeling of crystalline solids. Computational Material Science, 2006, 37: 374–379.

    Article  Google Scholar 

  23. Steinmann, P., Elizondo, A. and Sunyk, R., Studies of validity of the Cauchy-Born rule by direct comparison of continuum and atomistic modeling. Modelling and Simulation in Materials Science and Engineering, 2007, 15: 271–281.

    Article  Google Scholar 

  24. Park, H.S., Klein, P.A. and Wagner, G.J., A surface Cauchy-Born model for nanoscale materials. International Journal for Numerical Methods in Engineering, 2006, 68: 1072–1095.

    Article  MathSciNet  Google Scholar 

  25. Park, H.S. and Klein, P.A., Surface Cauchy-Born analysis of surface stress effects on metallic nanowires. Physical Review B, 2007, 75: 085408.

    Article  Google Scholar 

  26. Park, H.S. and Klein, P.A., A surface Cauchy-Born model for silicon nanostructures. Computer methods in applied mechanics and engineering, 2008, 2008: 3249–3260.

    Article  MathSciNet  Google Scholar 

  27. Huang, Y.C., Shu, X.L., Kong, Y., Wang, L.L. and Hu, W.Y., Lattice dynamics of fcc transition metals by modified analytic embedded-atom method. The Chinese Journal of Nonferrous Metals, 2002, 12: 36–40. (in Chinese)

    Google Scholar 

  28. Chen, L., Calculation and applicability analysis for elastic constants of fcc crystal. Chinese Journal of Mechanical Engineering, 2005, 41: 46–50. (in Chinese)

    Article  Google Scholar 

  29. Pandya, C.V., Vyas, P.R., Pandya, T.C., Rani, N. and Gohel, V.B., An improved lattice mechanical model for FCC transition metals. Physica B, 2001, 307: 138–149.

    Article  Google Scholar 

  30. Foiles, S.M., Baskes, M.I. and Daw, M.S., Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. Physical Review B, 1986, 33: 7983–7991.

    Article  Google Scholar 

  31. Simmons, G. and Wang, H., Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook. MIT Press, Cambridge, 1971.

    Google Scholar 

  32. Kittel, C., Introduction to Solid State Physics. 7th edition. New York: Wiley, 1997.

    MATH  Google Scholar 

  33. Wu, B., Liang, L.H., Ma, H.S. and Wei, Y.G., A trans-scale model for size effects and intergranular fracture in nanocrystalline and ultra-fine polycrystalline metals. Computational Materials Science, 2012, 57: 2–7.

    Article  Google Scholar 

  34. Shenoy, V.B., Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Physical Review B, 2005, 71: 094104.

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Authors and Affiliations

  1. State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, 100190, Beijing, China

    Jianyun Liu, Jingru Song & Yueguang Wei

Authors
  1. Jianyun Liu
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  2. Jingru Song
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  3. Yueguang Wei
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Corresponding authors

Correspondence to Jianyun Liu or Yueguang Wei.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11021262, 10932011 and 91216108) and the National Basic Research Program of China (2012CB937500). The first author (J. Liu) would like to thank the helpful advice of Dr. Wei Xu from Institute of Mechanics, Chinese Academy of Sciences.

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Liu, J., Song, J. & Wei, Y. Size Effects of Elastic Modulus of FCC Metals Based on the Cauchy-Born Rule and Nanoplate Models. Acta Mech. Solida Sin. 27, 111–121 (2014). https://doi.org/10.1016/S0894-9166(14)60021-5

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  • DOI: https://doi.org/10.1016/S0894-9166(14)60021-5

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