Stiffness Matrix for temperature dependency

[corus]

Is there an on-line reference for the stiffness matrix for a thermal conductivity element where the conductivity is temperature dependent? All the references I have found assume constant conductivity.

 

[brad]

Corus,
It is essentially processed in the same manner as temperature-dependent elastic properties. The ABAQUS manuals have a discussion of their implementation (my recollection is that you frequently use ABAQUS).

I am unclear as to the specifics of your question. Can you please clarify where the confusion on your part is?

Regards,
Brad

 

[corus]

Brad,
Thanks for the reply. I shall refer to the Abaqus manuals.
The confusion is because I know that for constant k, the PDE is k.div^2(T)=0 for steady state but for k dependent on T then the equation should be div(k.div(T))=0 which I thought would be treated differently to the text book references I have, as the PDE now involves the differential of k. For stress analysis elastic material properties are simply calculated at the known temperature, but for heat flow it's different, unless it's done iteratively, I think.

 

[brad]

Corus,
Consider this being similar to hyperelastic materials:
Conductivity ~ Stiffness
Flux ~ Load
Temperature ~ Strain

Just as in a Hyperelastic Material, the tangent modulus is dependent on strain, so is the tangent modulus of conductivity dependent on temperature. Iterations occur to solve for a converged solution just as would happen in an elastic solution.

One note--temperature-dependent conductivity is generally a fairly mild form of nonlinearity (as most materials do not exhibit dramatic changes in conductivity, other than for phase changes).

Is this what you were looking for? Thanks for making me think.
Cheers,
Brad

 

[corus]

Brad,
Thanks again for the reply. On further thought an iterative solution would be the right way as applying a defined function for k would be impractical as values are input in tabular form, normally. I'm still not convinced that the correct soultion is obtained though, but as you point out the variation in k with temperature is usually small, particularly within an element, and any terms involving the differential of k will be relatively small. A comparison with an analytical solution is the best way forward to convince myself.
cheers
corus