# What is incompressible solid?

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.

### Volumetric strain

Hi,

'J' is nothing but the ratio between the volumes in deformed configuation(dv) and reference configuarion (dV). When a material is incompressible, the volume remains the same or change in volume is zero, when a body undergoes deformation. This gives, J = 1. Bulk modulus (K) = -dp/volumetric strain. Since volumetric strain is zero, K tends to infinity. But, K = E/(3(1-2u)). Since, K tends to infinity, u (Poisson ratio) = 0.5.

One more way of explaning this is, when J = 1, tr(d) = 0, where d = rate of deformation tensor. Tr(d) gives rate of volumetric strain. Since the rate of volumetric strain = 0, K tends to infinity. Therefore, u = 0.5.

I hope this helps.

Regards,

Thank you for your answer! Is there way to expilicitely express J in terms of Poisson ratio? As J is the measure of volume change there shold be formula, that relates dV to J. Maybe, you can suggest a book, where this topic is explained in details?

### As I understand, the

As I understand, the relation has to be something like dV=J-1, but maybe to the certain power?

### Volumetric strain and J

Could someone helpwith formula that relates dV to the  third invariant of the Cauchy-Green deformation tensor J ?