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## Re: [Getfem-users] Stiffness matrix for linear elasticity

 From: Roman Putanowicz Subject: Re: [Getfem-users] Stiffness matrix for linear elasticity Date: Wed, 29 Dec 2010 02:16:36 +0100 User-agent: Mutt/1.5.13 (2006-08-11)

```Dear Yves,
> This formula implements in fact a weak term. This represents
>
> y
> where A:B is the Frobenius (scalar) product of matrices.
>
Thank you for your comment, it was much helpful. Somehow I couldn't
see the forest for the trees.

Now I can link the implementation and the formal derivations.
Below I include the explanation of this link I have built for myself.
It is rather informal so please correct me if I to brutally violate
some math rules :-)

we have:

Cauchy stress:
sigma =   lambda div(u) I + 2 mu (1/2 (grad(u) + grad^T(u)))

The force balance equation (neglecting inertia term and body forces)
div(sigma) = 0

Weak form of weighted residual equation
int{div(sigma):v} ->  int{sigma:grad(v)} + boundary term

Considering the term under the integral sign

Then we get:

Well now I see the correspondence with

lambda.t(:,i,i,:,j,j) + mu.t(:,i,j,:,i,j) + mu.t(:,j,i,:,i,j)

Somehow I couldn't get this using the index notation (maybe was not patient
enough :).

I decided to write this (now obvious :) elaborate derivation because
the above way the to calculate the stiffness matrix for linearised elasticity
is a bit different from what one can find in the elementary books on FEM.

Is it true that this particular way of calculation is driven
b) avoidance of problems with incompressible materials ?

Thanks again and best regards,

Roman
--
Roman Putanowicz, PhD  < address@hidden  >
Institute for Computational Civil Engng (L-5)
Dept. of Civil Engng, Cracow Univ. of Technology
www.l5.pk.edu.pl, tel. +48 12 628 2569, fax 2034

```