Continuum Mechanics: Eij and eif given coordinates and displacements

[lanew]

Homework Statement

http://imageshack.us/photo/my-images/513/selection027.png"
http://imageshack.us/photo/my-images/513/selection027.png

Homework Equations

$E_{ij}=\frac{1}{2} \left( \frac{\partial{u_i}}{\partial{X_j}} \frac{\partial{u_j}}{\partial{X_i}} - \delta_{ij}\right)$
$\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial{u_i}}{\partial{X_j}}+\frac{\partial{u_j}}{\partial{X_i}}\right)$

The Attempt at a Solution

I'm not exactly sure where to begin. I understand how to find $E_{ij}$ and $\epsilon_{ij}$ given the displacement field, but I am not sure how to construct the field. So far, I came up with:

$u=0.001x_1+0.003x_2+0.002x_3$
$v=0.002x_2+0.001x_3-0.001x_4$

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

$u_1=0.001x_1$
$v_1=0$

$u_2=0.003x_2$
$v_2=0.002y_2$

$u_3=0.002x_3$
$v_3=0.001y_3$

$u_4=0$
$v_4=-0.001y_4$

Am I even remotely close with either idea?

Thanks!

 

[afreiden]

Am I even remotely close with either idea?

Thanks!

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 $X$, rather than x...

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

$u$ and $v$ should depend on $X_1$ and $X_2$.

You have to figure out the relationship..

I'd also double check your formula:
Should read:
$E_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial X_j}+\frac{\partial u_j}{\partial X_i}+\frac{\partial u_k }{\partial X_i}\frac{\partial u_k}{\partial X_j})$
Hope that helps..