Rubber bands and Hooke's Law


by Manchot
Tags: bands, hooke, rubber
Manchot
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#1
Oct25-04, 09:51 PM
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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.
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Gokul43201
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#2
Oct26-04, 11:12 AM
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Are you sure [itex]x = L/L_0~~and~not~~\delta L/L_0~[/itex] ?
Manchot
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#3
Oct26-04, 11:26 AM
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No, I'm not sure.

Pyrrhus
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#4
Oct26-04, 11:34 AM
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Rubber bands and Hooke's Law


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
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#5
Oct26-04, 12:01 PM
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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
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#6
Oct26-04, 01:54 PM
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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
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#7
Oct26-04, 02:50 PM
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The given formula, in order to be meaningful must have x=L/L0 ...
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 ?


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