# Longitudinal and transversal strain

The Poisson ratio of the material equals 0,33. If I load a cube or a cylinder that is fixed at one end with compression force in ANSYS I can confirm that the ratio is 0,33 by dividing the lateral by longitudinal strain.

But when I load a geometry that is hollow with changing cross section the lateral/longitudinal ratios across the geometry change in unreasonable intervals that make no sense to me, because I was convinced that the ratio is always constant and does not depend on cross-section or shape. My question is, what does the ratio depend on and how? If somebody could direct me to some useful literature, it would be greatly appreciated.

• For a hollow geometry with changing cross section, the stress is not uniaxial anymore, this is probably why you are not measuring the same ratio along the rod. – user190081 Nov 5 '18 at 16:38

For simplicity lets assume you have a cylinder of length 20 cm which is made up of two short 10 cm cylinders, both hollow and have a 1 cm diameter hole drilled out of them, one has the cross section surface are of 2 cm^2 and the other thinner one has a cross section surface 1 cm.

Now we apply P = 2kN to the top of the cylinder.

the stress on the thick part will be

$$\sigma = 2\, \text{kN}/ 2\, \text{cm}^2 = \ 1\, \text{kN/cm}^2$$ but for the thinner part $$\sigma = 2\, \text{kN}/ 1\, \text{cm}^2 = 2\,\text{kN/cm}^2.$$

So the thin section vertically strains twice and so horizontally its strain is two times more than the thick part.

Same idea applies to random changes in cross section of a bar, and even more complication arises when these changes have secondary stress concentration and distort the surface of strain plane along the length of the test sample.

• If the horizontal and vertical components of strain say increase by factor 2, does that mean that the ratio lateral/longitudinal stays the same? Is it possible that, in case of the complicated random cross-sections in the structure and axial loading, the lateral strain is greater than the longitudianal strain? – breza Nov 6 '18 at 12:00

The Poisson ratio is defined for a sufficiently uniform material (i.e., one for which the continuum assumption reasonably holds). If your material is not uniform, then what you're obtaining is some type of effective Poisson ratio, which could be just about anything and doesn't hold broad meaning. This is covered in most textbooks on elasticity or mechanics of materials.

• Material I use is homogeneous. – breza Nov 8 '18 at 13:43