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.
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?
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..
This is the website that I got the information from: http://www.newton.dep.anl.gov/askasci/phy00/phy00525.htm . It's about two-thirds down the page.
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 ?