Poissons ratio close-packed spheres

[barbaadr]

Hello,

See question 7.4 from the link.

http://books.google.com/books?id=0N...CfU3U1qlbfJCRXsRow1xzu7hxEWOjYCow&w=685&w=800

"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

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!

 

[eljose79]

Hello,

Thank you for your question. The proof for Poisson's ratio being 1/3 for a close-packed array of spheres is based on the assumption that the atoms are hard elastic spheres. This assumption allows us to use the concept of elastic deformation, where the atoms can move slightly under stress and return to their original position when the stress is removed.

To understand the proof, we need to first define Poisson's ratio. Poisson's ratio is a measure of the lateral contraction of a material when it is stretched in one direction. It is defined as the negative ratio of the transverse strain to the axial strain. In other words, it measures how much a material gets thinner when it is stretched.

Now, let's consider a close-packed array of spheres. When this structure is subjected to an external force, the spheres will deform slightly in all directions. This deformation can be described by two parameters: the axial strain and the transverse strain. The axial strain is the change in length of the structure in the direction of the applied force, while the transverse strain is the change in diameter of the structure perpendicular to the applied force.

In a close-packed array of spheres, the spheres are arranged in a hexagonal pattern, as shown in the images you provided. This means that when the structure is stretched in one direction, the atoms will move in all three directions. However, due to the hexagonal arrangement, the atoms will move in a way that results in a lateral contraction of 1/3 of the axial strain. This is because the atoms are constrained by the neighboring atoms and cannot move freely in all directions.

Now, let's go back to the definition of Poisson's ratio. We know that Poisson's ratio is the negative ratio of the transverse strain to the axial strain. In this case, the transverse strain is 1/3 of the axial strain. Therefore, Poisson's ratio is -1/3 or 1/3 (since the negative sign is dropped in the final result).

In conclusion, the proof for Poisson's ratio being 1/3 for a close-packed array of spheres is based on the assumption of elastic deformation and the hexagonal arrangement of the atoms. I hope this explanation helps clarify the concept. Please let me know if you have any further questions.