# Radial variation in Poisson's ratio

Hi,

I am wondering why radial variation in Poisson's ratio exists when a cylinder is compressed? Ie, Poisson's ratio toward the center is lower than what is measured at the circumferential edge.

Thanks

M

## Answers and Replies

Perhaps you could elaborate further, particularly clarifying the axes in relation to the cylinder and the loads.

Say the cylinder axis is in the z direction so circular(?) sections through the cylinder are in planes z = constant.

Is the compression applied uniaxially and to which axis?

Chestermiller
Mentor
Poisson's ratio is a property of the material, and does not vary with spatial location.

The cylinder axis is in the z-direction with applied uniaxial compression in the z also.

Poisson's ratio may be an intrinsic material property, however, during experiments and FE modelling, I find radial variation (ie, along the x axis) with the highest value occuring at the circumferential edge.

This must be the result of the configuration - unconfined compression. I am guessing it may be due to the unconstrained nature of the circumerential edge that this experiences greatest lateral expansion?

Regards

OK we are getting somewhere, but detail is still in short supply.

You appear to be modelling the cylinder compression test (ie short cylinders so no buckling)?

As Chestermiller pointed out, poissons ratio is a material constant, but depending upon your stress model the effect of the loading will produce a variable response due to the confining effect of the material in the cylinder.

Which model are you using, plane stress or plane strain?
If you do not know, just post the equations you are using.

I am convinced what you are observing is not a variation of PR but inherent in you stress-strain relationship.

There is no buckling in the sample at all due to near frictionless specimen-compression platen interface.

As for modelling, I have modelled a cylindrical disk as a neo-hookean elastic material and applied a uniaxial uniform displacement in the z direction upon the superior surface of the disk. I am a complete novice in modelling and mechanics so not sure I can go into anymore detail regarding the equations employed by the FE software.

Regards

Chestermiller
Mentor
There is no buckling in the sample at all due to near frictionless specimen-compression platen interface.

As for modelling, I have modelled a cylindrical disk as a neo-hookean elastic material and applied a uniaxial uniform displacement in the z direction upon the superior surface of the disk. I am a complete novice in modelling and mechanics so not sure I can go into anymore detail regarding the equations employed by the FE software.

Regards

Let me say this back to you so we are sure I understand. The upper and lower platen contacts with the specimen are taken to be frictionless. The cylindrical disk is being compressed by moving the upper platen downward.

You may be a novice, but you owe it to yourself to examine the problem analytically for a
neo-hookean material. The kinematics of this deformation are very simple. Do not surrender to FE software without understanding what it is doing. It may very well be possible that, for a (non-linear) neo-hookean material in this deformation, the quantity that you might define as the Poisson ratio would vary radially.

Start out by examining the kinematics of the deformation. The principal directions will be axial, radial, and circumferential. Write out the equations for the three principal stretch ratios. Write out the components of the finite Green's tensor. Write out the equations for the deformational invariants. Plug these into the neo-hookean equation for the components of the stress tensor. Recognize also that the radial component of the stress tensor is zero at the edge of the cylinder. Write out the stress equilibrium equations, and substitute the stresses into these equations. See if you can solve these equations.