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Young's modulus in microscopic terms

  1. Jan 17, 2010 #1
    1. The problem statement, all variables and given/known data
    By considering the force-separation curve for two adjacent atoms in a solid, f(x), show that the Young’s modulus can be expressed on the microscopic scale as:
    [tex]Y = - \frac{1}{x_0} \frac{df}{dx}\right| |_{x=x_0}[/tex]
    (the | is meant to go allt he way form the top to bottom of df/dx)
    where [tex]x_0[/tex] is the equilibrium seperation of the atoms

    2. Relevant equations

    [tex]f(x) = - \left(\frac{A}{x}\right)^7 + \left(\frac{B}{x}\right)^{13}[/tex]
    (I'm assuming A and B should be 1 angstrom so 1E-10m)

    3. The attempt at a solution
    [tex]x_0[/tex] is found by doing f(x).dx = 0 to find where f(x) crosses the x axis.
    [tex] E \equiv \frac{\mbox {tensile stress}}{\mbox {tensile strain}} = \frac{\sigma}{\varepsilon}= \frac{F/A_0}{\Delta L/L_0} = \frac{F L_0} {A_0 \Delta L} [/tex]
    Y is the gradient of stress/straing graph

    hmmmm
    :(
     
  2. jcsd
  3. Jan 17, 2010 #2
    I can't exactly read your equations but a long time ago I remember doing an analysis based on a typical diamond lattice structure whereby you could show that Posson's ration came out correctly ( approximatley 1/3) if you assumed that the bond lengths stayed the same and only the bond angles were deformed.
     
  4. Jan 18, 2010 #3

    Mapes

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    What's [itex]E =F L_0/A_0 \Delta L [/itex] in differential form? What's a reasonable estimate for [itex]A_0[/itex] on the atomic scale?
     
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