Size effects in lattice-structured cellular materials: material distribution

  • Marcus YoderEmail author
  • Lonny Thompson
  • Joshua Summers
Computation & theory


This work presents a novel mechanism that causes stiffness size effects in periodic cellular materials in bending, but is absent in other boundary conditions. The work begins by demonstrating that in certain circumstances, a single set of micropolar elastic material properties can predict size effects accurately for either bending or shear but not both. This suggests that, for periodic cellular materials, size effects in shear and bending arise from different mechanisms. This work then presents a novel mechanism causing size effects in bending, called a material distribution effect. A sample with only a few unit cells can have its material concentrated close to the neutral axis, or far away, depending on topology and choice of unit cell. A sample with many cells must have its material spread more evenly. This can cause either a stiffening or softening size effect. This work derives formulas to predict the magnitude and direction of these size effects and shows that these formulas are able to predict size effects for a variety of different periodic cellular materials in different types of bending boundary conditions.


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Lakes RS, Nakamura S, Behiri JC, Bonfield W (1990) Fracture mechanics of bone with short cracks. J Biomech 23:967–975. CrossRefGoogle Scholar
  2. 2.
    Goda I, Ganghoffer JF (2015) Identification of couple-stress moduli of vertebral trabecular bone based on the 3D internal architectures. J Mech Behav Biomed Mater 51:99–118. CrossRefGoogle Scholar
  3. 3.
    Goda I, Assidi M, Ganghoffer JF (2014) A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech Model Mechanobiol 13:53–83. CrossRefGoogle Scholar
  4. 4.
    Lakes RS (1991) Experimental micro mechanics methods for conventional and negative Poisson’s ratio cellular solids as Cosserat continua. J Eng Mater Technol 113:148–155. CrossRefGoogle Scholar
  5. 5.
    Andrews EW, Gioux G, Onck P, Gibson LJ (2001) Size effects in ductile cellular solids. Part II: experimental results. Int J Mech Sci 43:701–713. CrossRefGoogle Scholar
  6. 6.
    Yoder M, Thompson L, Summers J (2018) Size effects in lattice structures and a comparison to micropolar elasticity. Int J Solids Struct 143:245–261. CrossRefGoogle Scholar
  7. 7.
    Diebels S, Steeb H (2002) The size effect in foams and its theoretical and numerical investigation. Proc R Soc A Math Phys Eng Sci 458:2869–2883. CrossRefGoogle Scholar
  8. 8.
    Bažant ZP, Christensen M (1972) Analogy between micropolar continuum and grid frameworks under initial stress. Int J Solids Struct 8:327–346. CrossRefGoogle Scholar
  9. 9.
    Wheel MA, Frame JC, Riches PE (2015) Is smaller always stiffer? On size effects in supposedly generalised continua. Int J Solids Struct 67–68:84–92. CrossRefGoogle Scholar
  10. 10.
    Kumar RS, McDowell DL (2004) Generalized continuum modeling of 2-D periodic cellular solids. Int J Solids Struct 41:7399–7422. CrossRefGoogle Scholar
  11. 11.
    Eringen AC (1999) Microcontinuum field theories: I. Foundations and solids. Springer, New YorkCrossRefGoogle Scholar
  12. 12.
    Tekoğlu C (2007) Size effects in cellular solids. University of Groningen, GroningenGoogle Scholar
  13. 13.
    Liebenstein S, Sandfeld S, Zaiser M (2016) Modelling elasticity of open cellular foams: size effects and disorder. Phys Rev B 94:144303. CrossRefGoogle Scholar
  14. 14.
    Liebenstein S, Sandfeld S, Zaiser M (2018) Size and disorder effects in elasticity of cellular structures: from discrete models to continuum representations. Int J Solids Struct 146:97–116. CrossRefGoogle Scholar
  15. 15.
    Tekoğlu C, Onck P (2008) Size effects in two-dimensional Voronoi foams: a comparison between generalized continua and discrete models. J Mech Phys Solids 56:3541–3564. CrossRefGoogle Scholar
  16. 16.
    Lakes RS (1993) Strongly Cosserat elastic lattice and foam materials for enhanced toughness. Cell Polym 12:17–30Google Scholar
  17. 17.
    Rueger Z, Lakes RS (2016) Experimental Cosserat elasticity in open-cell polymer foam. Philos Mag 96:93–111. CrossRefGoogle Scholar
  18. 18.
    Anderson WB, Lakes RS (1994) Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam. J Mater Sci 29:6413–6419. CrossRefGoogle Scholar
  19. 19.
    Brezny R, Green DJ (1990) Characterization of edge effects in cellular materials. J Mater Sci 25:4571–4578. CrossRefGoogle Scholar
  20. 20.
    Andrews EW, Gioux G, Onck P, Gibson LJ (2001) Size effects in ductile cellular solids. Part I: modeling. Int J Mech Sci 43:681–699CrossRefGoogle Scholar
  21. 21.
    Yoder M, Thompson L, Summers J (2019) Size effects in lattice-structured cellular materials: edge softening effects. J Mater Sci 54:3942–3959. CrossRefGoogle Scholar
  22. 22.
    Waseem A, Beveridge AJ, Wheel MA, Nash DH (2013) The influence of void size on the micropolar constitutive properties of model heterogeneous materials. Eur J Mech A/Solids 40:148–157. CrossRefGoogle Scholar
  23. 23.
    Beveridge AJ, Wheel MA, Nash DH (2012) The micropolar elastic behaviour of model macroscopically heterogeneous materials. Int J Solids Struct 50:246–255. CrossRefGoogle Scholar
  24. 24.
    McGregor M, Wheel MA (2014) On the coupling number and characteristic length of micropolar media of differing topology. Proc R Soc A Math Phys Eng Sci 470:20140150. CrossRefGoogle Scholar
  25. 25.
    Dunn MA, Wheel MA (2016) Computational analysis of the size effects displayed in beams with lattice microstructures. In: Altenbach H, Forest S, Krivtsov A (eds) Generalized continua as models for materials. Springer, Berlin, pp 129–144CrossRefGoogle Scholar
  26. 26.
    Liu S, Su W (2009) Effective couple-stress continuum model of cellular solids and size effects analysis. Int J Solids Struct 46:2787–2799. CrossRefGoogle Scholar
  27. 27.
    Fish J, Belytschko T (2007) A first course in finite element analysis. Wiley, ChichesterCrossRefGoogle Scholar
  28. 28.
    Zhang H, Wang H, Liu G (2005) Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies. Acta Mech Sin 21:388–394. CrossRefGoogle Scholar
  29. 29.
    Gibson LJ, Ashby MF (1999) Cellular materials: structure and properties, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  30. 30.
    Stronge WJJ, Wang XL (1999) Micropolar theory for two-dimensional stresses in elastic honeycomb. Proc R Soc A Math Phys Eng Sci 455:2091–2116. CrossRefGoogle Scholar
  31. 31.
    Dos Reis F, Ganghoffer JF (2011) Construction of micropolar continua from the homogenization of repetitive planar lattices. Springer, Berlin HeidelbergGoogle Scholar
  32. 32.
    Gauthier RD (1974) Analytical and experimental investigations in linear isotropic micropolar elasticity. University of Colorado, BoulderGoogle Scholar
  33. 33.
    Lakes RS (1986) Experimental microelasticity of two porous solids. Int J Solids Struct 22:55–63. CrossRefGoogle Scholar
  34. 34.
    Diebels S, Scharding D (2011) From lattice models to extended continua. In: Markert B (ed) Advances in extended and multifield theories for continua. Lecture notes in applied and computational mechanics. Springer, Berlin. Google Scholar
  35. 35.
    Nakamura S, Lakes RS (1988) Finite element analysis of stress concentration around a blunt crack in a Cosserat elastic solid. Comput Methods Appl Mech Eng 66:257–266. CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringClemson UniversityClemsonUSA

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