Mechanics of Solids Cylinders with Lateral Loads

[Ogmios]

Hi,
I have been searching the web for hours now, and I have had some success, but I have not found everything I am looking for.
The best thing I have found was a similar arrangement in Hibbeler’s Mechanics of Materials (hence the name of the thread). This had a short rod of aluminium in a vice being crushed axially (z). This is a very common problem, and finding the equations for this is simple, many of the texts had these, and the result is a uniform change in diameter in both the x and y directions.
However, I am interested in a rod being crushed diametrically (x), that is, loaded laterally. I have found the equations for the stress in the two dimensions that correspond to the circular cross section, x and y. However, I am also interested in what is happening along the length of the rod, in the z direction.
What I have is for a Force (F) applied in the dimension x, the stress in x ig given by,

σ_x=-6F/πld

where l is the length of the cylinder, or length over which the force is being applied in my actual application (the gauge length for sensing purposes), and d is the diameter of the cylinder. In the y-axis the stress is given by,

σ_y=2F/πld

That is, there is a compressive load from the crushing in the x direction, which results in an expansion in the perpendicular diameter (y), resulting in an elliptical cross section. Here is a link to an image I found

http://what-when-how.com/wp-content/uploads/2011/07/tmp1914_thumb.jpg

What I want is to know what is happening along the length. There must be some strain in this direction according to Poisson's ratio, but I could be wrong. The journal articles I have come across just assume that this is zero for simplicity. I assume they can do this because the length (l) is significantly greater than the diameter (d), but no justification is given for this. The interesting thing is that in their experimental results compared to their theoretical prediction, there is a slight difference, and I think this can be explained by what is happening in the length direction.
Any input would be greatly appreciated.
Kind Regards,
G

 

[OldEngr63]

Is thix crushing between two flat plates, or just how is this load applied to the two siddes of the cylinder? Are you looking for a theory of elasticity solution here?

 

[Ogmios]

Hi,
For my application it is a cylinder being crushed between two flat plates. However, it turns out that this is used a lot in geology and other fields, and is referred to as a diametral tensile test (DTT). They can use curved surface, which have a diameter greater than the diameter of the sample being crushed. This is only to help centre the sample. The same test is used on tablets (pills) for "hardness" measurements, and this is between two flat surfaces.
Here is a better image I found when searching for DTT,

http://www.biomedical-engineering-online.com/content/10/1/44/figure/F2

The important thing is that relative to the circumference this is a point load, and is only a distributed load along the length of the cylinder.
What I would like in the first instance is to know if I am right or wrong. That is, is there some strain induced along the length of the cylinder (the thickness as it is referred to in DTT). Then if there is stain induced, how can I determine this; so I guess I am a looking for a theory of elastic solution...
Kind Regards,
G

 

[OldEngr63]

It has been too long for me to recall the definitions at this point, but you need to look up the terms plane stress and plane strain to see which one of these applies in your case. If the cylinder is infinitely long, one of them will apply, and if it is infinitely short the other will apply, in each case reducing the problem to a 2-D field problem. Get a theory of elasticity book and do some digging.

 

[nvn]

Ogmios: If we assume the rod is free to slip longitudinally, then for the portion of the rod in the vice/vise, eps_z = -(nu/E)(sigma_x + sigma_y), where eps_z = strain (epsilon) in the z direction, nu = Poisson's ratio, and E = tensile modulus of elasticity.