Analytical bounds on the physical properties of multiphase media provide limits on properties attainable with variation of phase geometry. One may synthesize, both conceptually and experimentally, material microstructures which permit bounds to be approached or attained. It has become common to refer to such materials as metamaterials.
For isotropic materials the range of Poisson's ratio is from -1 to 1/2. Rubbery materials have Poisson's ratios approaching the upper limit. By contrast, foam materials with a negative Poisson's ratio  can have Poisson's ratio as small as -0.8, and have been referred to as anti-rubber . The cause of the negative Poisson's ratio is non-affine deformation. Theoretically, hierarchical laminates can exhibit extremal Poisson's ratio approaching -1 . These laminates have structure on several levels of scale; they are hierarchical. By appropriate choice of constituent properties one can achieve Poisson's ratios approaching the lower limit of -1. One may also achieve a Poisson's ratio of -1 by a two-dimensionally chiral honeycomb structure .
In composite materials with inclusions of negative stiffness one can attain extreme viscoelastic behavior and exceed conventional bounds. One can also achieve extremely large values of Young's modulus exceeding the modulus of diamond. These composites make use of inclusions capable of phase transformations. The transformation is partially constrained by the surrounding matrix. High damping can also be achieved in lumped systems in which negative stiffness heterogeneity is on a larger, even a macroscopic, scale.
Material microstructures are presented which can exhibit coefficients of thermal expansion larger than that of either constituent  download pdf . We conceptualize cellular solids as square or hexagonal lattices with two-layer rib elements and determine the thermal expansion coefficient. Thermal expansion increases without bound as the rib elements are made more slender. These cellular solids contain considerable void space. Moreover, we present dense extremal structures which substantially exceed the bounds for thermal expansion of a two-phase composite, by allowing slip at interfaces between phases  download pdf . New classes of extremal materials with extreme properties are envisaged, based on slip interfaces and void space tending to zero. Thermal expansion generated in this way compares favorably in expansion-density maps  with homogeneous materials including those with negative expansion.
Bounds on the viscoelastic properties of two phase composite materials are developed  and presented in stiffness-loss maps. One can achieve high stiffness combined with high damping by proper choice of material microstructure and constituent properties , as demonstrated by experiment. Development of practical damping layers based on this concept is in progress.
Strength to density ratio
One can achieve a high ratio of strength to weight in cellular materials and structures with structural hierarchy . Both honeycombs and foams can be provided with hierarchical structure and both give rise to much improved strength for given density.
One can achieve giant piezoelectric properties; even approach singular values of properties, by several methods. Methods may be applied on different length scales. See Negative stiffness inclusions
Strongly Cosserat elastic solids are a subject of current interest in our laboratory. Cosserat solids offer freedom not present in classically elastic solids. Stress concentration factors are reduced, and there is a pathway to greater toughness.
References. Articles to download are available elsewhere on this site.
 R. S. Lakes, "Foam structures with a negative Poisson's ratio", Science , 235 1038-1040, 1987
 J. Glieck, The New York Times, 14 April 1987.
 G. Milton, "Composite materials with Poisson's ratios close to -1", J. Mech. Phys. Solids, 40, 1105-1137, 1992
 Prall, D. and Lakes, R. S., "Properties of a chiral honeycomb with a Poisson's ratio -1", Int. J. of Mechanical Sciences, 39, 305-314, 1996.
 Lakes, R. S., "Cellular solid structures with unbounded thermal expansion", Journal of Materials Science Letters, 15, 475-477, 1996.
 Lakes, R. S., "Dense solid microstructures with unbounded thermal expansion", J. Mechanical Behav. Mts., 7, 85-92, 1996.
 Lakes, R. S., "Solids with tunable positive or negative thermal expansion of unbounded magnitude", Applied Phys. Lett. 90, 221905, 2007.
 Gibiansky, L.V. and Lakes, R. S., "Bounds on the complex bulk modulus of a two-phase viscoelastic composite with arbitrary volume fractions of the components", Mechanics of Materials, 16, 317-331, 1993.
 Brodt, M. and Lakes, R. S., "Composite materials which exhibit high stiffness and high viscoelastic damping", J. Composite Materials, 29, 1823-1833, 1995.
 Lakes, R. S., "Materials with structural hierarchy", Nature, 361, 511-515 (1993). Cover article.