# [aspect-devel] compressibility in 2d

Max Rudolph maxwellr at gmail.com
Mon Feb 13 11:46:18 PST 2017

In 2D calculations based on 3D conservation laws, you have to decide
whether to make a plane stress or plane strain assumption. Other
possibilities exist too, like formulations that allow out-of-plane
displacements but force d/dz=0 where z is the out of plane coordinate. If
you make the plane strain approximation, deviatoric stresses act in the
out-of-plane direction (as Jonathan pointed out) whose role is to keep flow
confined to the plane. See for instance Turcotte and Schubert section 3.6
(elasticity, but similar equations).

This is also related to the discussion that Wolfgang and I had a couple of
months back about the correct units of 'heat flux' or 'heat flux density',
whichever term you prefer, in 2D vs 3D.

On Mon, Feb 13, 2017 at 11:28 AM, Menno Fraters <menno.fraters at outlook.com>
wrote:

> Hey all,
>
> Thanks for the quick reply and answers. We have been discussing this
> (with Wim and Cedric), and at first we still found it strange that the 2D
> case of the Stokes equation yields a different deviator in the matrix (1/2
> vs 1/3). But then we realised that in the truly 2D case all the units
> should also be different anyway (e.g. density kg/m2, pressure N/m, etc) and
> one should actually derive the whole thing from the beginning in 2D. In
> Aspect and other codes we always work in 2D with true physical units, e.g.
> density in kg/m^3, pressure N/m2 etc, implicitly assuming a finite width of
> the 2D slice, i.e. we implicitly  work with 3D physics and therefore the
> “⅓” is okay.
>
> I guess something like this should be added somewhere in the manual, but
> for now I think my question is answered.
>
> Thanks for the help!
>
> Cheers,
>
> Menno, Wim and Cedric
>
>
> ------------------------------
> *From:* Aspect-devel <aspect-devel-bounces at geodynamics.org> on behalf of
> Wolfgang Bangerth <bangerth at tamu.edu>
> *Sent:* Friday, February 10, 2017 11:04:11 PM
> *To:* aspect-devel at geodynamics.org
> *Subject:* Re: [aspect-devel] compressibility in 2d
>
> On 02/10/2017 02:18 PM, cedric thieulot wrote:
> > So we’ve been talking about it with Menno today and it looks indeed
> > that the literature is unanimous on this. However, if \dot{\bm
> > \epsilon} is a 2x2 tensor/matrix the formula with 1/3 does not make
> > it traceless. I/we suppose that there is a somewhat subtle argument
> > here, but so far it has eluded us.
>
> The argument is that you really have a 3-dimensional velocity vector,
> but that the assumption that flow is 2-dimensional means that (i) the
> z-component of the velocity is zero, and (ii) all derivatives of the
> first two components with respect to z are also zero. So the gradient
> (and symmetric gradient) of the real velocity has the form
>
>    [ s_xx s_xy 0 ]
>    [ s_yx s_yy 0 ]
>    [ 0    0    0 ]
>
> If you use this and subtract 1/3 of the trace times the identity, then
> the resulting 3x3 tensor is trace-free.
>
> > Furthermore, if one considers an infinite domain in, say, the z
> > direction , dv_z/dz will be for sure nul, so that \dot{epsilon}_{zz}
> > is nul but if we now look at the deviatoric strainrate,
> > \dot{epsilon}_{zz}^{dev} will receive the value -1/3 (e_xx+e_yy).
> > What does this mean ?
>
> Yes, it is nonzero. But if you then again take the divergence of it
> (where all z-derivatives are zero), then it doesn't matter that there
> was something in the zz-spot.
>
> Best
>   W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth          email:                 bangerth at colostate.edu
>                             www: http://www.math.colostate.edu/~bangerth/
> Wolfgang Bangerth's Homepage - math.colostate.edu
> <http://www.math.colostate.edu/~bangerth/>
> www.math.colostate.edu
> Homepage of Wolfgang Bangerth, links to publications, etc
>
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