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.
In this paper, we present a geometric discretization scheme for incompressible linearized elasticity. We use ideas from discrete exterior calculus (DEC) to write the action for a discretized elastic body modeled by a simplicial complex. After characterizing the configuration manifold of volume-preserving discrete deformations, we use Hamilton's principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field.