poisson's ratio

In this paper we propose different classes of isotropic microstructured media with tunable Poisson's ratio. The elastic periodic systems are continuous porous media and two- and three-dimensional lattices. The microstructural parameters can be tuned in order to have an effective Poisson's ratio equal to zero. The connection between microstructural parameters and effective properties is shown in detail both analytically and numerically.

Continuous system with null Poisson's ratio:

Dear All,

I want to calculate  the limit of the fraction (J-1)/(1-2*nu) when material approach to incompressibility, where nu is the Poisson's ratio and J is the third invariant of the Cauchy-Green deformation tensor. In this case J->1 and nu->0.5, but I can't find any reference, how J depends on nu.

Intuitively, this limit has to be a finite value, as it is known that (J-1)^2/(1-2*nu) -> 0 (under the assumption that there is no hydrostatic pressure). Thanks in advance for any suggestions

Dear Colleagues,

In the literature two definitions of incompressible solid can be found. The first one is: Poisson's ratio =0.5, and the second is: the  third invariant of the Cauchy-Green deformation tensor J=1. I wish to understand the connections between these two definitions. To me, here is something strange.  Poisson's ratio relates strains to stresses,and J as the combination of strain tensor components, doesn't depend on Poisson's ratio. I'll be grateful to you, if you explain me how these two definitions are connected.