# Simplest meaningful PDE/FEM calculation for mechanical stress due to heat

W have a complicated structure on which we do some FEM calculations regarding electrical potentials and heat distribution. The equations have the form

$\nabla\kappa\nabla u = f + g\rvert_{N}$

where $u$ is the temperature or electrical potential, $\kappa$ is a material parameter and $f$ and $g$ are heat or current densities that enter the investigated object over the whole volume given as a density-per-volume or over the surface $N$, given as density-per-area, respectively.

Now, we'd like to calculate the mechanical stress induced by heating of the material. In literature, this is usually described with vector valued functions and using nontrivial tensors. We wonder whether it is possible to formulate a meaningful calculation of the mechanical stress in the same form as the equation above, in order to re-use our code that we wrote to solve it. We are aware that this is not possible for all materials and larger stresses/displacements.

Is it possible to formulate a meaningful calculation of the mechanical stress in the same form as the equation above? For which type of materials and other restrictions would this form be valid?

• I believe the answer provided here(scicomp.stackexchange.com/questions/29583/…) is essentially what you need. – Bill Greene Jun 9 '18 at 12:56
• @BillGreene: No, because it will exactly lead to a vector-valued equation. What I'm asking about is the maximal simplification of thermal stresses that leads to something meaningful and can be written in the above form with just altering the material parameters and loads. – Michael Jun 9 '18 at 12:59
• I think that the answer is no. If you have a body without boundary conditions you could have stain without stress due to thermal expansion. If you confine the body you could have an hydrostatic (scalar) stress. But in general, I would say that you can't just describe it as a single scalar. – nicoguaro Jun 9 '18 at 16:06
• This is really a modeling question. I don't think this is the right forum for it. – Wolfgang Bangerth Jun 14 '18 at 22:48