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Derivation problems for dilation

  1. Sep 7, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that, if there are no body forces, the dilation e ([itex] e=e_{xx} + e_{yy} + e_{zz} = div \;\vec{u}[/itex]) must satisfy the condition [itex]\nabla^2 e = 0[/itex]


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

    [itex] e_{xx} = \frac{\partial U_x}{\partial x}[/itex]
    [itex] e_{yy} = \frac{\partial U_y}{\partial y}[/itex]
    [itex] e_{zz} = \frac{\partial U_z}{\partial z}[/itex]

    [itex] e = e_{xx} + e_{yy} + e_{zz}[/itex]


    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) [itex](\lambda + \mu)\frac{\partial^2 e}{\partial x^2} + \mu \nabla^2\frac{\partial U_x}{\partial x} = 0[/itex]
    (2) [itex](\lambda + \mu)\frac{\partial^2 e}{\partial y^2} + \mu \nabla^2\frac{\partial U_y}{\partial y} = 0[/itex]
    (3) [itex](\lambda + \mu)\frac{\partial^2 e}{\partial z^2} + \mu \nabla^2\frac{\partial U_z}{\partial z} = 0[/itex]

    [itex] (\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[/itex]

    [itex] (\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[/itex]

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

    [itex] (\lambda + \mu)\nabla^2e+\mu \nabla^2e=0[/itex]

    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

    [itex] (\lambda + \mu)\nabla^2e+\mu \nabla^2 div \;\vec{U}=0[/itex]

    [itex]div \;\vec{U} = 0[/itex]

    Which leads me to [itex]\nabla^2 e = 0[/itex]
     
    Last edited: Sep 7, 2013
  2. jcsd
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