Poissons ratio close-packed spheres

[barbaadr]

Hello,

See question 7.4 from the link.

http://books.google.com/books?id=0N0...ow&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(1-2υ)) where υ is Poisson's ratio

K = E / (3(1-2υ))

When υ = 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/~cch...ct/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!

 

[KataKoniK]

Hi there,

The reason why Poisson's ratio is 1/3 for a close-packed array of spheres can be explained using the concept of shear strain. When a force is applied to a material, it causes a deformation or strain in the material. Shear strain specifically refers to the deformation that occurs when a force is applied parallel to a surface, causing the material to slide or shear in that direction.

In a close-packed array of spheres, the spheres are tightly packed together, leaving very little space between them. When a force is applied parallel to the surface of the array, the spheres will try to slide past each other, causing a shear strain. However, because the spheres are hard and elastic, they will resist this deformation and try to return to their original positions. This resistance to shear strain is known as shear modulus, G.

Now, when a material is subjected to a shear strain, it also experiences a deformation in the perpendicular direction. This is known as normal strain, and it is related to the shear strain by Poisson's ratio, which is defined as the ratio of the normal strain to the shear strain. In other words, Poisson's ratio describes how much a material will deform in the perpendicular direction when it is subjected to a shear strain.

In the case of a close-packed array of spheres, because the spheres are tightly packed together, they will resist the shear strain and try to maintain their original positions. This means that the normal strain will be very small compared to the shear strain. In fact, as the spheres are perfectly elastic, the normal strain will be zero. This leads to a Poisson's ratio of 0, which is the case for most perfectly elastic materials.

However, if we take into account the fact that the spheres are not just randomly packed, but rather arranged in a close-packed array, we can see that there is a preferred direction for the spheres to move in when subjected to a shear force. This is the direction of the closest packing, which is at an angle of 120 degrees in a hexagonal close-packed structure. This means that as the spheres are sheared, they will move in this direction, causing a normal strain in the perpendicular direction. But because the spheres are perfectly elastic, this normal strain will be equal to the shear strain, resulting in a Poisson's ratio of 1/3.

In summary, the close-packed arrangement of spheres allows for a preferred direction of movement during shear strain, which results in