Rubber bands and Hooke's Law

  1. 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.
     
  2. jcsd
  3. Gokul43201

    Gokul43201 11,141
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    Chegg
    Are you sure [itex]x = L/L_0~~and~not~~\delta L/L_0~[/itex] ?
     
  4. No, I'm not sure.
     
  5. Pyrrhus

    Pyrrhus 2,276
    Homework Helper

    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?
     
    Last edited: Oct 26, 2004
  6. arildno

    arildno 12,015
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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..
     
  7. PerennialII

    PerennialII 1,102
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    Gold Member

    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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