# Derivation problems for dilation

1. Sep 7, 2013

### roldy

1. The problem statement, all variables and given/known data
Show that, if there are no body forces, the dilation e ($e=e_{xx} + e_{yy} + e_{zz} = div \;\vec{u}$) must satisfy the condition $\nabla^2 e = 0$

2. Relevant equations
(1) $(\lambda + \mu)\frac{\partial e}{\partial x} + \mu \nabla^2U_x = 0$
(2) $(\lambda + \mu)\frac{\partial e}{\partial y} + \mu \nabla^2U_y = 0$
(3) $(\lambda + \mu)\frac{\partial e}{\partial z} + \mu \nabla^2U_z = 0$

$e_{xx} = \frac{\partial U_x}{\partial x}$
$e_{yy} = \frac{\partial U_y}{\partial y}$
$e_{zz} = \frac{\partial U_z}{\partial z}$

$e = e_{xx} + e_{yy} + e_{zz}$

3. The attempt at a solution
The homework gives hints to differentiate (1) with respect to x, (2) with respect to y, and (3) with respect to z. Then I add these up.

(1) $(\lambda + \mu)\frac{\partial^2 e}{\partial x^2} + \mu \nabla^2\frac{\partial U_x}{\partial x} = 0$
(2) $(\lambda + \mu)\frac{\partial^2 e}{\partial y^2} + \mu \nabla^2\frac{\partial U_y}{\partial y} = 0$
(3) $(\lambda + \mu)\frac{\partial^2 e}{\partial z^2} + \mu \nabla^2\frac{\partial U_z}{\partial z} = 0$

$(\lambda + \mu)\left(\frac{\partial^2e}{\partial x}+\frac{\partial^2e}{\partial y}+\frac{\partial^2e}{\partial z}\right)+\mu(\nabla^2\frac{\partial U_x}{\partial x}+\nabla^2\frac{\partial U_y}{\partial y}+\nabla^2\frac{\partial U_z}{\partial z})=0$

$(\lambda + \mu)\left(\frac{\partial^2e}{\partial x}+\frac{\partial^2e}{\partial y}+\frac{\partial^2e}{\partial z}\right)+\mu(\nabla^2e_{xx}+\nabla^2e_{yy}+\nabla^2e_{zz})=0$

$(\lambda + \mu)\left(\frac{\partial^2e}{\partial x}+\frac{\partial^2e}{\partial y}+\frac{\partial^2e}{\partial z}\right)+\mu \nabla^2e=0$

$(\lambda + \mu)\nabla^2e+\mu \nabla^2e=0$

From this I do not know how to arrive at the result. I can't see anything that I did wrong.

Edit:

Found my problem. The result should be

$(\lambda + \mu)\nabla^2e+\mu \nabla^2 div \;\vec{U}=0$

$div \;\vec{U} = 0$

Which leads me to $\nabla^2 e = 0$

Last edited: Sep 7, 2013