Rubber bands and Hooke's Law

Manchot

I have found a website which claims that rubber bands obey a force law
[tex]F=-kT(x-\frac{1}{x^2})[/tex]
[tex]x=\frac{L}{L_0}[/tex]
While this is similar to Hooke's Law in the sense that it *almost* approaches it for large values of x, it is also quite different. Can anyone confirm or deny the formula's reliability? Thanks.

Gokul43201

Are you sure [itex]x = L/L_0~~and~not~~\delta L/L_0~[/itex] ?

Manchot

No, I'm not sure.

Pyrrhus

Well if you're familiar with elasticity you can formulate Hooke's Law in its terms,

Stress = Modulus of Elasticity x Relative Deformation

For a longitudinal deformation, the modulus is called Young's modulus

[tex] \sigma = Y \delta L [/tex]

Since Stress = Force/Area

[tex] \frac{F}{A} = Y \delta L [/tex]

[tex] F = YA \delta L [/tex]

You know

[tex] \delta L = \frac{\Delta L}{L_{o}} [/tex]

[tex] F = YA \frac{\Delta L}{L_{o}} [/tex]

Rearranging

[tex] F = \frac{YA}{L_{o}} \Delta L [/tex]

we have

[tex] F = \frac{YA}{L_{o}} \Delta L [/tex]

Hooke's Law

[tex] F = k \Delta x [/tex]

where k in our equation is (x = L)

[tex] k = \frac{YA}{L_{o}} [/tex]

The people from that page probably tried something similar, can you give us the website?

arildno

The given formula, in order to be meaningful must have [tex]x=\frac{L}{L_{0}}[/tex]

Rewritten slightly, it simply says:
[tex]F=-kT\delta{L}({1+\frac{1}{x}+\frac{1}{x^{2}}})[/tex]

Hence, it predicts a hardening for compression of the rubber.
I don't know if it actually is good, though..

Manchot

This is the website that I got the information from: http://www.newton.dep.anl.gov/askasc...0/phy00525.htm . It's about two-thirds down the page.

PerennialII

Which is what they give under the link. So it looks like a simple uniaxial time-independent hardening mod of sorts ... so is it just a simple made up correction or does it have any theoretical merit ?