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..