Elasticity theory allows negative compressibility in a constrained object, but thermodynamic presentations of stability suggest negative compressibility cannot occur. In the context of thermodynamics, it is claimed that the compressibility (inverse bulk modulus) must be positive.

Since that claim contradicts the result of elasticity theory, it is of interest to study the assumptions made. The continuum has a non-denumerable infinite number of degrees of freedom, while a solid made of atoms has a finite, albeit large, number of degrees of freedom. If, however the atoms are in vibratory motion due to non-zero temperature, then the solid of atoms has a form of freedom not present in the continuum.

The proof of Wallace of positive compressibility depends on the notion of a positive definite matrix. This is understood in the context of elasticity as a restrictive condition corresponding to a specified load boundary condition at the surfaces. The Wallace result corresponds to stability for an unconstrained object and does not contradict the broader elasticity result for a constrained solid.

Negative compressibility or negative stiffness, with stiffness considered as the bulk modulus, is anticipated in the context of the Landau theory of phase transformations. If the surface of the material is free of constraint, a negative modulus entails instability. In the context of phase transformations, such instability of unconstrained objects is manifested as a change in crystal structure or a change in volume, or both.

Lakes, R., Wojciechowski, K. W.,

Negative incremental compressibility is experimentally observed in foams under volumetric constraint. Because this sort of foam is compliant, it is easy to apply such a constraint.

Moore, B., Jaglinski, T., Stone, D. S., and Lakes, R. S.,

Negative compressibility may be inferred from the behavior of composites containing inclusions subject to partially constrained phase transformations as presented in our negative stiffness page. Extreme properties via stored energy are achieved. We have not called such materials metamaterials and we have not called them architected materials.

There are other unstated assumptions within thermodynamics; they are dealt with elsewhere.