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Novel implementation of homogenization method to predict effective properties of periodic materials

Acta Mechanica Sinica volume 29, pages 550–556 (2013)Cite this article

Abstract

Representative volume element (RVE) method and asymptotic homogenization (AH) method are two widely used methods in predicting effective properties of periodic materials. This paper develops a novel implementation of the AH method, which has rigorous mathematical foundation of the AH method, and also simplicity as the RVE method. This implementation can be easily realized using commercial software as a black box, and can use all kinds of elements available in commercial software to model unit cells with rather complicated microstructures, so the model may remain a fairly small scale. Several examples were carried out to demonstrate the simplicity and effectiveness of the new implementation.

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References

  1. Hill, R.: A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13, 213–222 (1965)

    Article  Google Scholar 

  2. Christensen, R.M., Lo, K.H.: Solutions for effective shear properties in three phase sphere and cylinder models. Journal of the Mechanics and Physics of Solids 27, 315–330 (1979)

    MATH  Article  Google Scholar 

  3. Kanit, T., Forest, S., Galliet, I., et al.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. International Journal of Solids and Structures 40, 3647–3679 (2003)

    MATH  Article  Google Scholar 

  4. Hill, R.: The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society. Section A 65, 349–354 (1952)

    Article  Google Scholar 

  5. Yan, J., Cheng, G.D., Liu, S.T., et al.: Comparison of prediction on effective elastic property and shape optimization of truss material with periodic microstructure. International Journal of Mechanical Sciences 48, 400–413 (2006)

    MATH  Article  Google Scholar 

  6. Voigt, W.: On the relation between the elasticity constants of isotropic bodies. Ann. Phys. Chem. 274, 573–587 (1889)

    Article  Google Scholar 

  7. Reuss, A.: Determination of the yield point of polycrystals based on the yield condition of sigle crystals. Z. Angew. Math. Mech. 9, 49–58 (1929)

    MATH  Article  Google Scholar 

  8. Bakhvalov, N., Panasenko, G.: Homogenisation: Averaging Process in Periodic Media. Kluwer Academic Publ, Dordrecht (1989)

    MATH  Book  Google Scholar 

  9. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North Holland Publ, Amsterdam (1978)

    MATH  Google Scholar 

  10. Sanchez-Palencia, E., Zaoui, A.: Homogenization Techniques for Composite Media. Springer Verlag, Berlin (1987)

    MATH  Book  Google Scholar 

  11. Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering 71, 197–224 (1988)

    MathSciNet  Article  Google Scholar 

  12. Fujii, D., Chen, B.C., Kikuchi, N.: Composite material design of two-dimensional structures using the homogenization design method. International Journal for Numerical Methods in Engineering 50, 2031–2051 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  13. Nishiwaki, S., Frecker, M.I., Min, S., et al.: Topology optimization of compliant mechanisms using the homogenization method. International Journal for Numerical Methods in Engineering 42, 535–559 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  14. Qiao, P., Fan, W., Davalos, J.F., et al.: Optimization of transverse shear moduli for composite honeycomb cores. Composite Structures 85, 265–274 (2008)

    Article  Google Scholar 

  15. Sigmund, O., Torquato, S.: Design of smart composite materials using topology optimization. Smart Materials and Structures 8, 365–379 (1999)

    Article  Google Scholar 

  16. Sigmund, O.: Materials with prescribed constitutive parameters: an inverse homogenization problem. International Journal of Solids and Structures 31, 2313–2329 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  17. Hassani, B., Hinton, E.: A review of homogenization and topology optimization I-homogenization theory for media with periodic structure. Computers and Structures 69, 707–717 (1998)

    MATH  Article  Google Scholar 

  18. Hassani, B., Hinton, E.: A review of homogenization and topology opimization II-analytical and numerical solution of homogenization equations. Computers and Structures 69, 719–738 (1998)

    Article  Google Scholar 

  19. Hassani, B., Hinton, E.: A review of homogenization and topology optimization III-topology optimization using optimality criteria. Computers and Structures 69, 739–756 (1998)

    Article  Google Scholar 

  20. Muñoz-Rojas, P.A., Carniel, T.A., Silva, E.C.N., et al.: Optimization of a unit periodic cell in lattice block materials aimed at thermo-mechanical applications. In: Heat Transfer in Multi-Phase Materials. Springer Berlin, Heidelberg, 301–345 (2011)

    Google Scholar 

  21. Deshpande, V.S., Fleck, N.A., Ashby, M.F.: Effective properties of the octet-truss lattice material. Journal of the Mechanics and Physics of Solids 49, 1747–1769 (2001)

    MATH  Article  Google Scholar 

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Authors and Affiliations

  1. Dept. of Engineering Mechanics, Dalian University of Technology, 116023, Dalian, China

    Geng-Dong Cheng, Yuan-Wu Cai & Liang Xu

Authors
  1. Geng-Dong Cheng
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  2. Yuan-Wu Cai
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  3. Liang Xu
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Corresponding author

Correspondence to Geng-Dong Cheng.

Additional information

The project was supported by the National Natural Science Foundation of China (91216201).

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Cheng, GD., Cai, YW. & Xu, L. Novel implementation of homogenization method to predict effective properties of periodic materials. Acta Mech Sin 29, 550–556 (2013). https://doi.org/10.1007/s10409-013-0043-0

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  • DOI: https://doi.org/10.1007/s10409-013-0043-0

Keywords

  • Effective property
  • Periodic material
  • Homogenizatio
  • RVE method