Wolfgang, Thanks for the question. I'm trying to create a formulation which retains all three components of displacement, but with no dependence on the \phi coordinate. It is not necessarily true that the u_\phi component will be constant, it just has to be free of dependence on \phi.
I have heard in a few places that the case where u_\phi is assumed to be constant (and often zero) is called "torsionless axi-symmetry", and the one that I'm interested in, where u_\phi can depend on r,z, called "generalized axi-symmetry". A test case for the latter, I think, would be a thick-walled cylinder where one wall is fixed, and the other is displaced uniformly in the \phi direction. That should produce a u_\phi which is non-constant but free of dependence on \phi, for an isotropic material. So, I agree with your statement about the form of the gradient if the u_\phi solution is constant, but I think that for my case I need to keep the terms I'm having a problem with. Thanks, Jon On Thu, Aug 5, 2010 at 11:21 PM, Wolfgang Bangerth <[email protected]>wrote: > > Jon, > > > I'm working on an axi-symmetric formulation of elasticity in cylindrical > > coordinates, and I'm a bit stuck when it comes to computing the > gradients. > > The problem is that I need to compute gradients of the shape functions > that > > contain values of shape functions as well. > > > > For example, for a vector-valued shape function v^i, with components > > [v^i_r, v^i_\phi, v^i_z], > > I'm a little confused. You say that you are using an axi-symmetric > formulation, but then you typically only have the r- and z-components of > the > solution, because the solution is constant in phi direction, no? In that > case, the actual 3d gradient in x-y-z-coordinates at a position (r,z) of > the > x-z-plane would be > [\partial_r v^i, 0, \partial_z v^i] > > Or are you instead trying to work in cylindrical coordinates without the > assumption of symmetry? > > W. > > ------------------------------------------------------------------------- > Wolfgang Bangerth email: [email protected] > www: > http://www.math.tamu.edu/~bangerth/<http://www.math.tamu.edu/%7Ebangerth/> > >
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