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Continuum Mechanics: Eij and eif given coordinates and displacements

  1. Oct 27, 2011 #1
    1. The problem statement, all variables and given/known data
    http://imageshack.us/photo/my-images/513/selection027.png"
    http://imageshack.us/photo/my-images/513/selection027.png

    2. Relevant equations
    [itex]E_{ij}=\frac{1}{2} \left( \frac{\partial{u_i}}{\partial{X_j}} \frac{\partial{u_j}}{\partial{X_i}} - \delta_{ij}\right) [/itex]
    [itex]\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial{u_i}}{\partial{X_j}}+\frac{\partial{u_j}}{\partial{X_i}}\right)[/itex]

    3. The attempt at a solution
    I'm not exactly sure where to begin. I understand how to find [itex]E_{ij}[/itex] and [itex]\epsilon_{ij}[/itex] given the displacement field, but I am not sure how to construct the field. So far, I came up with:

    [itex]u=0.001x_1+0.003x_2+0.002x_3[/itex]
    [itex]v=0.002x_2+0.001x_3-0.001x_4[/itex]

    But I'm not sure that's right at all, or if I'm supposed to be looking at each individual point, e.g.:

    [itex]u_1=0.001x_1[/itex]
    [itex]v_1=0[/itex]

    [itex]u_2=0.003x_2[/itex]
    [itex]v_2=0.002y_2[/itex]

    [itex]u_3=0.002x_3[/itex]
    [itex]v_3=0.001y_3[/itex]

    [itex]u_4=0[/itex]
    [itex]v_4=-0.001y_4[/itex]

    Am I even remotely close with either idea?

    Thanks!
     
    Last edited by a moderator: Apr 26, 2017
  2. jcsd
  3. Oct 31, 2011 #2
    Not really, sorry..

    Your problem appears to be 2D.
    Furthermore, the two strain tensors that you want to find are Lagrangian and so, by definition, depend on capital [itex]X[/itex], rather than x...

    Unless I misread the problem, you ought to take [itex]X_1=``x''[/itex] and [itex]X_2=``y''[/itex] (that's it.. because it's 2D).

    [itex]u[/itex] and [itex]v[/itex] should depend on [itex]X_1[/itex] and [itex]X_2[/itex].

    You have to figure out the relationship..

    I'd also double check your formula:
    Should read:
    [itex]E_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial X_j}+\frac{\partial u_j}{\partial X_i}+\frac{\partial u_k }{\partial X_i}\frac{\partial u_k}{\partial X_j})[/itex]



    Hope that helps..
     
    Last edited: Oct 31, 2011
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