# Poissons ratio close-packed spheres

1. Oct 6, 2013

Hello,

See question 7.4 from the link.

"Assuming that atoms are hard elastic spheres, show that Poisson's ratio for a close-packed array of spheres is 1/3"

I am having trouble explaining the proof for this.

I know the that the volume modulus, K, = E(elastic modulus) / ((3(1-2$\upsilon$)) where $\upsilon$ is the elastic modulus.

K = E / (3(1-2$\upsilon$))

When $\upsilon$ = 1/3, K=E.

I'm thinking that since for a hexagonal close packed structure, HCP, the angles between lattice sites is 120°, or 1/3 of the plane of a full crystal structure.

Refer to:
http://www.science.uwaterloo.ca/~cchieh/cact/fig/hcp.gif
http://www.chem.ufl.edu/~itl/2045/lectures/h1.GIF [Broken]

Therefore the elastic properties for a given volume is split in thirds? It seems like a misleading argument, but I can't find a way to explain it with math!

Last edited by a moderator: May 6, 2017