Derivation problems for dilation

[roldy]

Homework Statement

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 $\nabla^2 e = 0$

Homework 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) $(\lambda + \mu)\frac{\partial e}{\partial z} + \mu \nabla^2U_z = 0$

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

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

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 $\nabla^2 e = 0$

 

[mighty2000]

.

This is the correct result, but it is important to note that the divergence of the displacement vector (div U) must also be equal to zero in order for this condition to hold. This makes sense, as a non-zero divergence would indicate the presence of body forces, which contradicts the initial assumption.