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Cosserat elastic lattices

  • Mechanics of Extreme Materials
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Abstract

Lattices composed of cubic and triangular prismatic unit cells with polymeric Sarrus linkage rib elements are designed, fabricated via 3D printing and studied experimentally. Size effects in these lattices are observed experimentally; slender specimens appear more rigid in torsion and in bending than expected via classical elasticity. Effects are analyzed via Cosserat elasticity. The magnitude of size effects is sensitive to geometry of the lattices; triangular cells with short ribs revealed the most extreme effects, also the largest characteristic length in relation to cell size. The torsion coupling number is 1, its upper bound, for all lattices. A path to the attainment of arbitrarily large nonclassical effects is delineated.

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References

  1. Timoshenko SP (1983) History of strength of materials. Dover, New York

    Google Scholar 

  2. Schijve J (1966) Note on couple stresses. J Mech Phys Solids 14:113–120

    Article  ADS  Google Scholar 

  3. Cosserat E, Cosserat F (1909) Theorie des Corps Deformables. Hermann et Fils, Paris

    MATH  Google Scholar 

  4. Eringen AC (1968) Theory of micropolar elasticity. In: Liebowitz H (ed) Fracture pp 621–729, vol 1. Academic Press, New York

    Google Scholar 

  5. Gibson LJ, Ashby MF, Schajer GS, Robertson CI (1982) The mechanics of two dimensional cellular solids. Proc R Soc Lond A382:25–42

    Article  ADS  Google Scholar 

  6. Mindlin RD (1963) Effect of couple stresses on stress concentrations. Exp Mech 3:1–7

    Article  Google Scholar 

  7. Gauthier RD, Jahsman WE (1975) A quest for micropolar elastic constants. J Appl Mech 42:369–374

    Article  Google Scholar 

  8. 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

    Article  ADS  Google Scholar 

  9. Lakes RS (1986) Experimental microelasticity of two porous solids. Int J Solids Struct 22:55–63

    Article  Google Scholar 

  10. 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

    Article  ADS  Google Scholar 

  11. Rueger Z, Lakes RS (2016) Experimental Cosserat elasticity in open-cell polymer foam. Philos Mag 96(2):93–111

    Article  ADS  Google Scholar 

  12. Rueger Z, Lakes RS (2016) Cosserat elasticity of negative Poisson’s ratio foam: experiment. Smart Mater. Struct. 25(5):054004

    Article  ADS  Google Scholar 

  13. Rueger Z, Lakes RS (2017) Observation of cosserat elastic effects in a tetragonal negative Poisson’s ratio lattice. Physica Stat Solidi (b) 254(12):1600840

    Article  ADS  Google Scholar 

  14. Minagawa S, Arakawa K, Yamada M (1980) Diamond crystals as Cosserat continua with constrained rotation. Physica Stat Solidi A 57:713–718

    Article  ADS  Google Scholar 

  15. Mora R, Waas AM (2000) Measurement of the Cosserat constant of circular cell polycarbonate honeycomb. Philos Mag A 80:1699–1713

    Article  ADS  Google Scholar 

  16. Askar A, Cakmak AS (1968) A structural model of a micropolar continuum. Int J Eng Sci 6:583–589

    Article  Google Scholar 

  17. Tauchert T (1970) A lattice theory for representation of thermoelastic composite materials. Recent Adv Eng Sci 5:325–345

    Google Scholar 

  18. Adomeit G (1967) Determination of elastic constants of a structured material. In: Continua, EK (ed) Mechanics of generalized, UTAM symposium, Freudenstadt, Stuttgart. Springer, Berlin

  19. Park T, Hwang WS, Hu JW (2009) Plastic continuum models for truss lattice materials with cubic symmetry. J Mech Sci Technol 24(3):657–669

    Article  Google Scholar 

  20. Stolken JS, Evans AG (1998) Microbend test method for measuring the plasticity length scale. J Acta Mater 46:5109–5115

    Article  Google Scholar 

  21. Fearing R (2018) Sarrus linkage, rapid prototyping of millirobots using composite fiber toolkits. https://people.eecs.berkeley.edu/~ronf/DESKTOP/prototyping/linkages.html

  22. Rueger Z, Lakes RS (2018) Strong Cosserat elasticity in a transversely isotropic polymer lattice. Phys Rev Lett 120:065501

    Article  ADS  Google Scholar 

  23. Rueger Z, Lakes RS (2017) Strong Cosserat elastic effects in a unidirectional composite. Z Angew Math Phys 68:54. https://doi.org/10.1007/s00033-017-0796-6

    Article  MATH  Google Scholar 

  24. Lekhnitskii SG (1981) Theory of elasticity of an anisotropic body. Mir, Moscow

    MATH  Google Scholar 

  25. Lakes RS, Drugan WJ (2015) Bending of a Cosserat elastic bar of square cross section—theory and experiment. J Appli Mech 82(9):091002

    Article  Google Scholar 

  26. Drugan WJ, Lakes RS (2018) Torsion of a Cosserat elastic bar of square cross section: theory and experiment. Z Angew Math Phys 69(24):24. https://doi.org/10.1007/s00033-018-0913-1

    Article  MathSciNet  MATH  Google Scholar 

  27. Park HC, Lakes RS (1987) Torsion of a micropolar elastic prism of square cross section. Int J Solids Struct 23:485–503

    Article  Google Scholar 

  28. Zener C (1947) Contributions to the theory of beta-phase alloys. Phys Rev 71:846–851

    Article  ADS  Google Scholar 

  29. Bigoni D, Drugan WJ (2007) Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J Appl Mech 74:741–753

    Article  MathSciNet  Google Scholar 

  30. Rueger Z, Lakes RS (2018) Experimental study of elastic constants of a dense foam with weak Cosserat coupling. J Elast. https://doi.org/10.1007/s10659-018-09714-8

    Article  Google Scholar 

  31. Buechner PM, Lakes RS (2003) Size effects in the elasticity and viscoelasticity of bone. Biomech Model Mechanobiol 1(4):295–301

    Article  Google Scholar 

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Funding

Funding was provided by National Science Foundation (CMMI-1361832).

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Authors and Affiliations

  1. University of Wisconsin Madison, Madison, WI, USA

    Z. Rueger, C. S. Ha & R. S. Lakes

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  1. Z. Rueger
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  2. C. S. Ha
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  3. R. S. Lakes
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Correspondence to R. S. Lakes.

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Rueger, Z., Ha, C.S. & Lakes, R.S. Cosserat elastic lattices. Meccanica 54, 1983–1999 (2019). https://doi.org/10.1007/s11012-019-00968-7

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  • DOI: https://doi.org/10.1007/s11012-019-00968-7

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