Hello Dr. Bangerth,
Thank you stating this!
I was able to go through the paper from Jean-Paul more thoroughly a few
times. I do have a few questions regarding the paper.
1) On equation 23, where do the subscripts ijkl come from? I understand
that subscripts ij are from equation 5. But I do not see where the
subscripts k and l factor into the problem.
2) Why is the shape function for the problem in the form as eqn. 23? Is
this the basic definition of the shape function in deal.ii? Or is this
specific to the problem?
3) How dos equation 27 reduce down from this:
to this:
Is this because of the Kronecker delta?
4) Also, I am not familiar with the abbreviation eps. Which math function
does this relate to?
5) Since -div (C eps(u)) = b is equivalent to -div (K grad u) = , I am
almost thinking I should do the Mixed Laplace approach for the problem?
But, after reading the paper by Jean-Paul a few times, it would seem that
the -div (C eps(u)) = b approach does not require Mixed Laplace.
On Tuesday, November 6, 2018 at 11:15:32 PM UTC-5, Wolfgang Bangerth wrote:
>
>
> > I have been reading through you paper a couple of times now but I od not
> see
> > how it relates to anisotropic materials. I could be missing something or
> > misunderstanding some parts. From what I can tell from the paper, the
> strong
> > form of linear elasticity is:
> >
> > ∇· σ + b = 0 on Ω.
> >
> > where σ is the stress tensor which is modeled by σ = σ m + σ f. The
> isotropic
> > linear constitutive law is given by: σm = C m : ε. Which is basically
> Hooke's
> > law. Now, hooke's law does have an analogue for the polarization of a
> > dielectric material when the material is exposed to an electric field. I
> am
> > not very sure if this is the direction that I need to go in.
> >
> >
> > I have been looking up anisotropic materials for linear elasticity. I
> might be
> > missing something but I am having a hard time seeing how this relates to
> the
> > electrostatic case? Maybe I need to spend a little bit more time on this
> > topic. I do see that there is a similarity between div *[* *\epsilon*
> *e*] =
> > \rho^{f} and ∇· σ + b = 0 where σ = *\epsilon* *e *and b = - \rho^{f}.
>
> You have
> sigma = C eps(u)
> and
> -div sigma = b
> which together yields
> -div (C eps(u)) = b
> and is the equivalent of
> -div (K grad u) = f
>
> The elastic material is anisotropic if the stress-strain tensor C
> represents
> an anisotropic material. In the elasticity case, it isn't just a dxd
> matrix,
> but a dxdxdxd (rank-4) tensor. So it looks more complicated, but it isn't
> really.
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> <javascript:>
> www: http://www.math.colostate.edu/~bangerth/
>
>
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