# Solution of plate with a hole

1. Jul 7, 2016

### kajalschopra

1. The problem statement, all variables and given/known data
Physical understanding about the variation of hoop stresses in a plate with a hole

2. Relevant equations

I have been reading about the solution of stresses in solution of plate with a hole from here: http://www.fracturemechanics.org/hole.html

. I’m unable to physically understand the following:

1. How is that the hoop stresses become infinity at theta = + or -90? I’m unable to sense this physically? I understand that there is a hole there but at theta = 0, the hoop stresses are compressive (this also corresponds to the hole position since r = a). I’m unable to sense the two scenarios physically. That is: how the hoop stresses become infinity at r = a and theta = +90 or -90 and the hoop stresses become compressive at theta = 0?
3. The attempt at a solution

The question is to do with physical understanding.

2. Jul 7, 2016

### Staff: Mentor

The hoop stress is not infinite at $\theta = \pm 90$. It is $3\sigma_{\infty}$. At $\theta = \pm 90$, the hoop stress is acting in the same direction as the far field stress $\sigma_{\infty}$. But, because there is no material in the hole to help support this tensile load, the hoop stress has to pull harder.

With regard to $\theta = 0$, the hoop stress here is acting in the direction perpendicular to the applied load. Because the stretching is uniaxial in the far field, the far field strain is negative in this transverse direction (by the Poisson effect). The radial stress is zero here, so the tendency for the material to contract in width near the hole is less than in the far field where the radial stress is tensile. So a compressive stress in necessary here to try to get the material adjacent to the hole to try to contract by an amount on the same order as the far field.

3. Jul 7, 2016

### kajalschopra

Thanks a lot Sir, that was extremely useful. One forgets the Poissons effect when thinking in polar coordinates.

Sir, I have another basic question:

1) Why are closed form solutions starting with infinite plates? What is so simple in infinite plates?

With warm regards
Kajal

4. Jul 7, 2016

### Staff: Mentor

The boundary conditions are much simpler to apply.

5. Jul 7, 2016

### kajalschopra

Sorry Sir, I do ot quite understand. Can you explain reference to the plate with circular hole problem (Kirsch solution)? I do understand that since the plate is infinite the hole dimensions do not come into play in the expressions for stresses sigma_rr and sigma_theta around the hole. But, do not quite understand how the boundary conditions are easier to apply.

6. Jul 7, 2016

### Staff: Mentor

Maybe I didn't express my answer quite right. In this problem, having the boundary condition at infinity simplified the form of the solution to stress equilibrium equations. Just imagine what the solution would be like if, rather than the boundary conditions being applied at infinity, the boundary were located about parallel lines only about 1 hole diameter away on either side of the hole .

7. Jul 7, 2016

### The Bill

If the plate weren't infinite, then the length and width of the plate and the hole's horizontal and vertical position in the plate would all be part of the solution. Also, the free edges at the left and right would have to be taken into account as boundary conditions. When you pull on a strip of material at both ends, it pinches in the middle, and modeling that with boundary conditions on the sides is difficult.

8. Jul 8, 2016

### kajalschopra

Thank you very much, sir's.

http://www.fracturemechanics.org/hole.html

As I see, there is not much logic in deriving the solution of finite width plates using the solution of infinite width plate. As I see the result of stress concentration factor Kt is completely empirical. The nominal stress cited in the above example is but a result of experimental observations.

Am I right?

9. Jul 8, 2016

### Staff: Mentor

I don't think so. I think the empirical equation was derived by fitting a polynomial to the mathematical solution of the elastic stress equilibrium equations for the case of a finite width. The mathematical solution was probably obtained numerically using finite element.

10. Jul 8, 2016

### kajalschopra

ok..last question, sir./

Why is that the solution for stresses is expressed and derived in polar (in case of circle) or elliptical (in case of ellipse) coordinates? Why is it difficult to derive in Cartesian coordinates?

11. Jul 8, 2016

### Staff: Mentor

The equations turn out to be easier to solve that way.