Stefanosv
(Stefanos Voulgaris)
October 20, 2022, 12:01pm
#1
I’ve noticed the usage of 2 equations to calculate the energy of each node when solving using the elasticity code

```
Energy1=dx(uu)+dy(vv)
Energy2=2*mu*(dx(uu)^2+dy(vv)^2+((dx(vv)+dy(uu))^2)/2.)+lamda*(dx(uu)+dy(vv))^2;
```

I need to calculate the energy for using it as a parameter to be minimized on an optimization problem.
Which one do you suggest using?
I can’t seem to find any bibliography about them though.

Dear Stefans,

The elastic energy density is \frac{1}{2}(\sigma : \epsilon) = \frac{1}{2}(\lambda (Trace(\epsilon))^2+2\mu(\epsilon : \epsilon)) =\frac{1}{2}\left[\lambda (\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y})^2+2\mu((\frac{\partial u}{\partial x})^2+(\frac{\partial v}{\partial y})^2+\frac{1}{2}(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})^2\right].

It seems to me that both formulae in your post are wrong…

Best regards, Alex