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Mathematics of Multiscale Materials pp 129–153Cite as

Elastic Freedom in Cellular Solids and Composite Materials

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 99))

Abstract

The question of how much freedom is to be incorporated in a continuum theory must ultimately be decided by experiment. There are several theories which describe behavior of materials. An early uniconstant theory was proposed based on atomic interaction theory; it was abandoned since it predicted a Poisson’s ratio of 1/4 for all materials. The elasticity theory currently accepted as classical allows Poisson’s ratios in isotropic materials in the range -1 to 1/2. Common materials exhibit a Poisson’s ratio from 1/4 to nearly 1/2. We have prepared materials with a Poisson’s ratio as small as -0.8. Deformation mechanisms in these materials include relative rotation of micro-elements, and non-affine micro-deformation. The relation between properties and structure is exploited to prepare viscoelastic composites with high stiffness combined with high damping. Generalized continuum theories exist with more freedom than classical theory. For example, in Cosserat elasticity there are characteristic lengths as additional engineering elastic constants. Recent experimental work discloses a variety of cellular and fibrous materials to exhibit such freedom, and the characteristic lengths have been measured. In hierarchical solids structural elements themselves have structure. Several examples of natural structural hierarchy are considered, with consequences related to optimality of material properties.

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Author information

Authors and Affiliations

  1. Department of Biomedical Engineering, Department of Mechanical Engineering, Center for Laser Science and Engineering, University of Iowa, Iowa City, IA, 52242, USA

    Roderic Lakes

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  1. Roderic Lakes
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Editors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Kenneth M. Golden  & Graeme W. Milton  & 

  2. Statistical Laboratory, University of Cambridge, Cambridge, CB2 1SB, England

    Geoffrey R. Grimmett

  3. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455, USA

    Richard D. James

  4. Schlumberger-Doll Research Center, Old Quarry Road, Ridgefield, CT, 06877, USA

    Pabitra N. Sen

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Lakes, R. (1998). Elastic Freedom in Cellular Solids and Composite Materials. In: Golden, K.M., Grimmett, G.R., James, R.D., Milton, G.W., Sen, P.N. (eds) Mathematics of Multiscale Materials. The IMA Volumes in Mathematics and its Applications, vol 99. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1728-2_9

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  • DOI: https://doi.org/10.1007/978-1-4612-1728-2_9

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