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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.


ramdas chennamsetti's picture


'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.


- Ramadas

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

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

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