The question of how much freedom is to be incorporated in an elasticity theory must ultimately be decided by experiment. However, during the development of the theory of elasticity, it was by no means obvious how much freedom was necessary to describe materials. For example, the early uniconstant theory of Navier is based upon the assumption that forces act along the lines joining pairs of atoms and are proportional to changes in distance between them. This theory entails a Poisson's ratio of 1/4, for all materials. Experimental measurements (about a century ago) of Poisson's ratio of about 1/3 in common materials led to the replacement of uniconstant elasticity by the more general classical elasticity, following the continuum view of Green, which allows Poisson's ratios between -1 and 1/2. This range is predicted by energy analysis of stability of a block with free surfaces. Cellular solids have been developed which exhibit a controlled negative Poisson's ratio as small as -0.8. Such materials may be called anti-rubber, auxetic, or dilational. Deformation mechanisms in these materials include relative rotation of micro-elements, and non-affine micro-deformation. Negative elastic moduli are possible but are associated with instability. The Poisson's ratio of unstable or metastable phases can exceed the classic bounds [-1, 1/2], and composite moduli can exceed classic composite theory bounds. Theoretical and experimental examples will be presented.
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