What is the increase in radius of the disk?

Question

A solid circular disk of radius R is subject to a uniform stress σ on its edges. By how much does the radius of the disk increase?

http://i33.tinypic.com/2j17p6g.jpg

Answer 1

σ/rY(pi)

Y=youngs modulous

Answer 2

Really interesting.

First I started thinking that the inside would be stretched out like a donut.  But that made no sense.  Then I realized that hoop stress would keep this from happening.

Time is tight and there is a lot I have to think about to do this right.  So I may not make it back.  But I like it.

...............

Quick thought, stress/strain is assumed to be linear, so we probably can simplify this.  But I do have to work ....

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Addendum

Give credit where it is due, MA has an excellent answer.  I must learn more from this holy book of his.

But to answer the last question posed by MA, consider a round trampoline.  All points on the clothe fabric have the same strain in all directions of the plane.  Anything else would cause asymmetry.

Answer 3

Stress:

σ_xx =  ÃÂƒ_yy =  ÃÂƒ:

Tensor of stress:

σ_ij =

σ 0 0

0 σ 0

0 0 0

Hooke's law:

σ_ij = λ_ijkl = strain_kl

Isotropic case (I omit zero terms):

strain_xx = 1/E σ_xx - ν/E σ_yy

strain_xx = (1-ν)/E σ

Answer:

ΔR =  (1 - Poisson's ratio)/(Young's modulus) σR

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Simply put, Young's modulus and Poisson's ratio are straight applicable to uniform stress, such that

Tensor of stress:

σ_ij =

σ 0 0

0 0 0

0 0 0

which is stress of a wire streached in x-drection.

In this simple case enlongation of a wire is

strain_xx =  ÃÂƒ/E,

and decrease in diameter is

strain_yy = strain_zz =  -νσ/E.

In the desribed case we have linear combination of streaching in x- and y-direction. And Hooke's law says relation between stress and starin is linear and becuse of that, the solytion is linear combination of strains.

But this description is deceptive and incomplete, because it does not explain, for example, why the described situation is indeed

σ 0 0

0 σ 0

0 0 0

and not for example

σ σ/3 0

σ/3 σ 0

0  0  0