Abstract
Micromorphic (microstructure) elastic constants are considered within the context of experimental results for foams and rib lattices and of subsets such as Cosserat elasticity, void elasticity and reduced micromorphic elasticity. Experimentally, longitudinal wave dispersion and cutoff frequencies reveal several of the micromorphic b coefficients. In static experiments, no size effects are evident in compression for the materials studied. The corresponding micromorphic a coefficients are not distinguishable from zero. By contrast, Cosserat effects are pronounced in these materials.
Similar content being viewed by others
References
Sokolnikoff, I.S.: Theory of Elasticity. Krieger, Malabar (1983)
Koiter, W.: Couple stresses in the theory of elasticity: I and II. Proc. Koninkl. Ned. Akad. Wetensch. Ser. B 67, 17–44 (1964)
Cosserat, E., Cosserat, F.: Theorie des Corps Deformables. Hermann et Fils, Paris (1909)
Mindlin, R.D.: Stress functions for a Cosserat continuum. Int. J. Solids Struct. 1, 265–271 (1965)
Mindlin, R.D.: Stress functions for Cosserat elasticity. Int. J. Solids Struct. 6, 389–398 (1970)
Eringen, A.C.: Theory of micropolar elasticity. In: Liebowitz, H. (Ed.) Fracture, Vol. 1, pp. 621–729. Academic Press, New York (1968)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78 (1964)
Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13, 125–147 (1983)
Eringen, A.C.: Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci. 28(12), 1291–1301 (1990)
Neff, P., Ghiba, I., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Continu. Mech. Thermodyn. 26(5), 639–681 (2014)
Askar, A., Cakmak, A.S.: A structural model of a micropolar continuum. Int. J. Eng. Sci. 6, 583–589 (1968)
Tauchert, T.: A lattice theory for representation of thermoelastic composite materials. Recent Adv. Eng. Sci. 5, 325–345 (1970)
Bigoni, D., Drugan, W.J.: Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech. 74, 741–753 (2007)
Prall, D., Lakes, R.S.: Properties of a chiral honeycomb with a Poisson’s ratio of \(-1\). Int. J. Mech. Sci. 39(3), 305–314 (1997)
Spadoni, A., Ruzzene, M.: Elasto-static micropolar behaviour of a chiral auxetic lattice. J. Mech. Phys. Solids 60, 156–171 (2012)
Minagawa, S., Arakawa, K., Yamada, M.: Diamond crystals as Cosserat continua with constrained rotation. Phys. Status Solidi A 57, 713–718 (1980)
Chen, Y., Lee, J.D.: Determining material constants in micromorphic theory through phonon dispersion relations. Int. J. Eng. Sci. 41, 871–886 (2003)
Dillard, T., Forest, S., Ienny, P.: Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams. Eur. J. Mech. A Solids 25, 526–549 (2006)
Cowin, S.C.: An incorrect inequality in micropolar elasticity theory. Zeitschrift fur Angewandte Mathematik und Physik 21, 494–497 (1970)
Krishna Reddy, G.V., Venkatasubramanian, N.K.: On the flexural rigidity of a micropolar elastic circular cylinder. J. Appl. Mech. 45, 429–431 (1978)
Gauthier, R.D., Jahsman, W.E.: A quest for micropolar elastic constants. J. Appl. Mech. 42, 369–374 (1975)
Rizzi, G., Hutter, G., Khan, H., Ghiba, I.D., Madeo, A., Neff, P.: Analytical solution of the cylindrical torsion problem for the relaxed micromorphic continuum and other generalized continua (including full derivations), (arXiv:2104.11322). Math. Mech. Solids (2021). https://doi.org/10.1177/10812865211023530
Rizzi, G., Ghiba, I.D., Madeo, A., Neff, P.: Analytical solution of the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua (including full derivations). Arch. Appl. Mech. (2021). https://doi.org/10.1007/s00419-021-02064-3
Rizzi, G., Khan, H., Ghiba, I.D., Madeo, A., Neff, P.: Cosserat micropolar elasticity: classical Eringen vs. dislocation form, arxiv:2206.02473v1
Hoff, N.J.: The applicability of Saint Venant’s principle to airplane structures. J. Aeronaut. Sci. 12, 455–460 (1945)
Fung, Y.C.: Foundations of Solid Mechanics. Prentice Hall, NJ (1968)
Stephen, N.G., Wang, P.J.: On Saint Venant’s principle in pin jointed frameworks. Int. J. Solids Struct. 33, 79–97 (1996)
Chen, C.P., Lakes, R.S.: Dynamic wave dispersion and loss properties of conventional and negative Poisson’s ratio polymeric cellular materials. Cell. Polym. 8(5), 343–359 (1989)
Yang, J.F.C., Lakes, R.S.: Experimental study of micropolar and couple stress elasticity in bone in bending. J. Biomech. 15, 91–98 (1982)
Lakes, R.S.: Size effects and micromechanics of a porous solid. J. Mater. Sci. 18, 2572–2581 (1983)
Mora, R., Waas, A.M.: Measurement of the Cosserat constant of circular cell polycarbonate honeycomb. Philos. Mag. 80, 1699–1713 (2000)
Rueger, Z., Lakes, R.S.: Experimental Cosserat elasticity in open cell polymer foam. Philos. Mag. 96(2), 93–111 (2016)
Rueger, Z., Lakes, R.S.: Cosserat elasticity of negative Poisson’s ratio foam: experiment. Smart Mater. Struct. 25. 054004 (2016)
Rueger, Z., Lakes, R.S.: Experimental study of elastic constants of a dense foam with weak Cosserat coupling. J. Elast. 137, 101–115 (2019)
Rueger, Z., Lakes, R.S.: Strong Cosserat elasticity in a transversely isotropic polymer lattice. Phys. Rev. Lett. 120, 065501 (2018)
Merkel, A., Tournat, V.: Experimental evidence of rotational elastic waves in granular phononic crystals. Phys. Rev. Lett. 107(22), 225502 (2011)
Lakes, R.S., Gorman, D., Bonfield, W.: Holographic screening method for microelastic solids. J. Mater. Sci. 20, 2882–2888 (1985)
Lakes, R.S.: Reduced warp in torsion of reticulated foam due to Cosserat elasticity: experiment. Zeitschrift fuer Angewandte Mathematik und Physik (ZAMP) 67(3), 1–6 (2016)
Lakes, R.S., Drugan, W.J.: Bending of a Cosserat elastic bar of square cross section—theory and experiment. J. Appl. Mech. 82(9), 091002 (2015)
Lakes, R.S.: Cosserat shape effects in the bending of foams. Mech. Adv. Mater. Struct. https://doi.org/10.1080/15376494.2022.2086328
Reasa, D.R., Lakes, R.S.: Nonclassical chiral elasticity of the gyroid lattice. Phys. Rev. Lett. 125, 205502 (2020)
Brezny, R., Green, D.J.: Characterization of edge effects in cellular materials. J. Mater. Sci. 25(11), 4571–4578 (1990)
Burteau, A., NGuyen, F., Bartout, J.D., Forest, S., Bienvenu, Y., Saberi, S., Naumann, D.: Impact of material processing and deformation on cell morphology and mechanical behavior of polyurethane and nickel foams. Int. J. Solids Struct. 49, 2714–2732 (2012)
Chen, C.P., Lakes, R.S.: Dynamic wave dispersion and loss properties of conventional and negative Poisson’s ratio polymeric cellular materials. Cell. Polym. 8(5), 343–359 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lakes, R.S. Experimental evaluation of micromorphic elastic constants in foams and lattices. Z. Angew. Math. Phys. 74, 31 (2023). https://doi.org/10.1007/s00033-022-01923-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-022-01923-5