Skip to main content
Log in

Size effects of effective Young’s modulus for periodic cellular materials

  • Published:
Science in China Series G: Physics, Mechanics and Astronomy Aims and scope Submit manuscript

Abstract

With the wide demands of cellular materials applications in aerospace and civil engineering, research effort sacrificed for this type of materials attains nowadays a higher level than ever before. This paper is focused on the prediction methods of effective Young’s modulus for periodical cellular materials. Based on comprehensive studies of the existing homogenization method (HM), the G-A meso-mechanics method (G-A MMM) and the stretching energy method (SEM) that are unable to reflect the size effect, we propose the bending energy method (BEM) for the first time, and a comparative study of these four methods is further made to show the generality and the capability of capturing the size effect of the BEM method. Meanwhile, the underlying characteristics of each method and their relations are clarified. To do this, the detailed finite element computing and existing experimental results of hexagonal honeycombs from the literature are adopted as the standard of comparison for the above four methods. Stretch and bending models of periodical cellular materials are taken into account, respectively for the comparison of stretch and flexural displacements resulting from the above methods. We conclude that the BEM has the strong ability of both predicting the effective Young’s modulus and revealing the size effect. Such a method is also able to predict well the variations of structural displacements in terms of the cell size under stretching and bending loads including the non-monotonous variations for the hexagonal cell. On the contrary, other three methods can only predict the limited results whenever the cell size tends to be infinitely small.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Gibson L J, Ashby M F. Cellular Solids: Structure and Properties. 2nd ed. Cambridge: Cambridge University Press, 1997

    Google Scholar 

  2. Hassani B, Hinton E. A review of homogenization and topology optimization II-analytical and numerical solution of homogenization equations. Comput Struct, 69, 1998: 719–738

    Article  Google Scholar 

  3. Bendsøe M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng, 1988, 71(2): 197–224

    Article  Google Scholar 

  4. Guedes J M, Kikuchi N. Pre and post processing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng, 1990, 83: 143–198

    Article  MATH  MathSciNet  Google Scholar 

  5. Sigmund O. Materials with prescribed constitutive parameters: An inverse homogenization problem. Int J Solids Struct, 1994, 31(17): 2313–2329

    Article  MATH  MathSciNet  Google Scholar 

  6. Fujii D, Chen B C, Kikuchi N. Composite material design of two-dimensional structures using the homogenization method. Int J Numer Methods Eng, 2001, 50: 2031–2051

    Article  MATH  MathSciNet  Google Scholar 

  7. Hohe J, Becker W. Effective stress-strain relations for two-dimensional cellular sandwich cores: Homogenization, material models, and properties. Appl Mech Rev, 2002, 55(1): 61–87

    Article  Google Scholar 

  8. Zhang W H, Dai G M, Wang F W, et al. Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures. Acta Mech Sin, 2007, 23(1): 77–89

    Article  ADS  MathSciNet  Google Scholar 

  9. Andrews E W, Gioux G, Onck P, et al. Size effects in ductile cellular solids. Part II: Experimental results. Int J Mech Sci, 2001, 43: 701–713

    Article  MATH  Google Scholar 

  10. Onck P R, Andrews E W, Gibson L J. Size effects in ductile cellular solids. Part I: Modeling. Int J Mech Sci, 2001, 43: 681–699

    Article  MATH  Google Scholar 

  11. Tantikom K, Aizawa T, Mukai T. Symmetric and asymmetric deformation transition in the regularly cell-structured materials Part I: Experimental study. Int J Solids Struct, 2005, 42: 2199–2210

    Article  Google Scholar 

  12. Lestari W, Qiao P Z, Song G B, et al. Evaluation of bending and shear moduli of sandwich structures by dynamic response based technique. 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, 2003

  13. Yan J, Cheng G D, Liu S T, et al. Prediction of equivalent elastic properties of truss materials with periodic microstructure and the scale effects (in Chinese). ACTA Mech Solida Sin, 2005, 26(4): 421–428

    Google Scholar 

  14. Dai G M, Zhang W H. Size effects of basic cell in static analysis of sandwich beams. Int J Solids Struct, 2008, 45: 2512–2533

    Article  Google Scholar 

  15. Zhang W H, Sun S P. Scale-related topology optimization of cellular materials and structures. Int J Numer Methods Eng, 2006, 68: 993–1011

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to WeiHong Zhang.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 50775184), the National Basic Research Program of China (Grant No. 2006CB601-205), the Aeronautical Science Foundation (Grant No. 2008ZA53007), the Doctorate Foundation of Northwestern Polytechnical University (Grant No. CX200610), and the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University (Grant No. 30715003)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dai, G., Zhang, W. Size effects of effective Young’s modulus for periodic cellular materials. Sci. China Ser. G-Phys. Mech. Astron. 52, 1262–1270 (2009). https://doi.org/10.1007/s11433-009-0151-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11433-009-0151-9

Keywords

Navigation