# Continuum Mechanics: Eij and eif given coordinates and displacements

1. Oct 27, 2011

### lanew

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

3. 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!

Last edited by a moderator: Apr 26, 2017
2. Oct 31, 2011

### afreiden

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:
$E_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial X_j}+\frac{\partial u_j}{\partial X_i}+\frac{\partial u_k }{\partial X_i}\frac{\partial u_k}{\partial X_j})$