How to solve elasticity problem for plane stress condition

[Deepika Kushwah]

Hello Everyone,

I'm trying to solve elasticity problem refereeing step 8 and have got solution for plane strain problem. Now I wish to solve the problem with plane stress condition for which matrix D (constitutive matrix) is to be modified to get stiffness (K = B^T*D*B). 

Not clear where it has been defined in the code and how it can be modified. 

Please suggest.

Thanks & Regards,

Deepika

[Wolfgang Bangerth]

On 11/20/22 07:20, Deepika Kushwah wrote:
>
> I'm trying to solve elasticity problem refereeing step 8 and have got solution
> for plane strain problem. Now I wish to solve the problem with plane stress
> condition for which matrix D (constitutive matrix) is to be modified to get
> stiffness (K = B^T*D*B).
> Not clear where it has been defined in the code and how it can be modified.

Deepika:
assemble_system() is always the function where the system matrix is built. We
do not generally assemble systems via B^T*D*B, however, but rather based on
the weak formulation of the equation -- that is, the bilinear form that
corresponds to the equation you want to solve.

A good first step would be to understand what equation step-8 solves, and how
that corresponds to what that program's assemble_system() does. Then you'll
have to identify what the bilinear form is for the equation you care about.
The final step is to translate that in the same way as what the program
currently does.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Deepika Kushwah]

Thank you, Prof Bangerth

I understood that dealii software uses the weak form to solve any kind of problem. 

But my question is that in step 8 the weak form for elasticity is for a general case or it is specifically for the plane strain case?

Thanks & Regards

Deepika

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[Wolfgang Bangerth]

On 11/23/22 02:07, Deepika Kushwah wrote:
>
> I understood that dealii software uses the weak form to solve any kind of
> problem.
> But my question is that in step 8 the weak form for elasticity is for a
> general case or it is specifically for the plane strain case?
>

Deepika:
the introduction of step-8 contains this note, which I believe is the best
description of what equation the program is solving:

As written, the equations above are generally considered to be the right
description for the displacement of three-dimensional objects if the
displacement is small and we can assume that Hooke's law is valid. In that
case, the indices i,j,k,l above all run over the set {1,2,3} (or, in the C++
source, over {0,1,2}). However, as is, the program runs in 2d, and while the
equations above also make mathematical sense in that case, they would only
describe a truly two-dimensional solid. In particular, they are not the
appropriate description of an x−y cross-section of a body infinite in the z
direction; this is in contrast to many other two-dimensional equations that
can be obtained by assuming that the body has infinite extent in z-direction
and that the solution function does not depend on the z coordinate. On the
other hand, there are equations for two-dimensional models of elasticity; see
for example the Wikipedia article on plane strain, antiplane shear and plan
stress.