Abstract
All the five independent elastic properties/constants of micro- and nano-structured hierarchical and self-similar random irregular honeycombs with different degrees of cell regularity are obtained by analysis and computer simulation in this paper. Cell wall bending, stretching, and transverse shearing are the main deformation mechanisms of hierarchical honeycombs. The strain gradient effects at the micro-meter scale, and the surface elasticity and initial stress effects at the nano-meter scale are incorporated into all the deformation mechanisms in the analysis and finite element simulations. The results show that the elastic properties of hierarchical random irregular honeycombs strongly depend on the thickness of the first-order cell walls if it is at the micro-meter scale, and that if the thickness of the first-order cell walls is at the nano-meter scale, the elastic properties of hierarchical random irregular honeycombs are not only size-dependent, but are also tunable and controllable over large ranges. In addition, the geometrical properties of nano-structured hierarchical random irregular honeycombs are also tunable and controllable.
Similar content being viewed by others
References
Warren WE, Kraynik M (1987) Foam mechanics: the linear elastic response of two-dimensional spatially periodic cellular materials. Mech Mater 6:27–37
Silva MJ, Hayes WC, Gibson LJ (1995) The effects of non-periodic microstructure on the elastic properties of two-dimensional cellular solids. Int J Mech Sci 37:1161–1177
Masters IG, Evans KE (1996) Models for the elastic deformation of honeycombs. Compos Struct 35:403–422
Gibson LJ, Ashby MF (1997) Cellular solids. Pergamon Press, Oxford
Chen C, Lu TJ, Fleck NA (1999) Effect of imperfections on the yielding of two-dimensional foams. J Mech Phys Solids 49:2245–2271
Zhu HX, Mills NJ (2000) The in-plane non-linear compression of regular honeycombs. Int J Solids Struct 37:1931–1949
Zhu HX, Hobdell JR, Windle AH (2001) The effects of cell irregularity on the elastic properties of 2D Voronoi honeycombs. J Mech Phys Solids 49:857–870
Zhu HX, Thorpe SM, Windle AH (2006) The effect of cell irregularity on the high strain compression of 2D Voronoi honeycombs. Int J Solids Struct 43:1061–1078
Zhu HX, Chen CY (2011) Combined effects of relative density and material distribution on the mechanical properties of metallic honeycombs. Mech Mater 43:276–286
Zhu HX (2010) Size-dependent elastic properties of micro- and nano-honeycombs. J Mech Phys Solids 58:696–709
Cammarata RC, Sieradzki K (1994) Surface and interface stresses. Annu Rev Mater Sci 24:215–234
Miller RE, Shenoy VB (2000) Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11:139–147
Wang J, Duan HL, Huang ZP, Karihaloo BL (2006) A scaling law for properties of nano-structured materials. Proc R Soc A462:1355–1363
Zhu HX, Karihaloo BL (2008) Size-dependent bending of thin metallic films. Int J Plast 24:956–972
Zhu HX (2008) The effects of surface and initial stresses on the bending stiffness of nanowires. Nanotechnology 19:405703
Zhu HX, Wang J, Karihaloo BL (2009) Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms. J Mech Mater Struct 4:589–604
Toupin RA (1962) Elastic materials with couple stresses. Arch Ration Mech Anal 11:385–414
Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11:415–448
Mindlin RD (1963) Influence of couple-stresses on stress concentration. Exp Mech 3:1–7
Fleck NA, Hutchinson JW (1993) A phenomenological theory for strain gradient effects in plasticity. J Mech Phys Solids 41:1825–1857
Nix WD, Gao H (1998) Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 46:411–425
Aifantis EC (1999) Strain gradient interpretation of size effects. Int J Fract 95:299–314
Gao H, Huang Y, Nix WD, Hutchinson JW (1999) Mechanism-based strain gradient plasticity—I. Theory. J Mech Phys Solids 47:1239–1263
Anthoine A (2000) Effect of couple-stresses on the elastic bending of beams. Int J Solids Struct 37:1003–1018
Lubarda VA, Markenscoff X (2000) Conservation integrals in couple stress elasticity. J Mech Phys Solids 48:553–564
Haque MA, Saif MTA (2003) Strain gradient effect in nanoscale thin films. Acta Mater 51:3053–3061
Lam DCC, Yang C, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508
McFarland AW, Colton JS (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15:1060–1067
Biener J, Wittstock A, Zepeda-Ruiz LA, Biener MM, Zielasek V, Kramer D, Viswanath RN, Wessmuller J, Baumer M, Hamza AV (2009) Surface-chemistry-driven actuation in nanoporous gold. Nat Mater 8:47–51
Raiteri R, Butt HJ (1995) Measuring electrochemically induced surface stress with an atomic force microscope. J Phys Chem 99:15728–15732
Weissmuller J, Viswanath RN, Kramer D, Zimmer P, Wurschum R, Gleiter H (2003) Charge-induced reversible strain in a metal. Science 300:312–315
Kramer D, Viswanath RN, Weissmuller J (2004) Surface-stress induced macroscopic bending of nanoporous gold cantilevers. Nano Lett 4:793–796
Haiss W, Nichols RJ, Sass JK, Charle KP (1998) Linear correlation between surface stress and surface charge in anion adsorption on Au(111). J Electroanal Chem 452:199–202
Lakes R (1993) Materials with structural hierarchy. Nature 361:511–515
Taylor CM, Smith CW, Miller W, Evans KE (2011) The effects of hierarchy on the in-plane elastic properties of honeycombs. Int J Solids Struct 24:1330–1339
Zhu HX, Yan L, Zhang R, Qiu XM (2012) Size-dependent and tunable elastic properties of hierarchical honeycombs with square and equilateral triangular cells. Acta Mater 60:4927–4939
Zhu HX, Wang ZB (2013) Size-dependent and tunable elastic and geometric properties of hierarchical nano-porous materials. Sci Adv Mater 5:677–686
Zhu HX, Wang ZB, Fan TX, Zhang D (2012) Tunable bending stiffness, buckling force and natural frequency of nanowires and nanoplates. World J Nano Sci Eng 2:161–169
Diao J, Gall K, Duan ML, Zimmerman JA (2006) Atomistic simulations of the yielding of gold nanowires. Acta Mater 54:643–653
ANSYS finite element software
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, H.X., Zhang, H.C., You, J.F. et al. The elastic and geometrical properties of micro- and nano-structured hierarchical random irregular honeycombs. J Mater Sci 49, 5690–5702 (2014). https://doi.org/10.1007/s10853-014-8288-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10853-014-8288-y