# Thermal expansion of a hole in a plate with a temperature gradient

I have a rectangular metal plate with a hole in it (with diameter of 300 mm). The plate has a temperature gradient going from one of its short sides to the other, I can measure the temperature anywhere in the plate.

I want to compute how much the hole is deforming from its round shape.

I know that I can calculate the expansion of the hole at uniform temperature with $$\frac{ΔL}{L_0}=αΔT$$ . So I was wondering if it is correct to measure a bunch temperature points at the edge of the hole and just apply that formula to each of them independently. But since there's a temperature gradient I'm assuming that there will be mechanical stresses between hot and cold zones working against the expansion, is this the case?

Is there a way to compute this by hand?

In the picture $$T1>T2$$. • A sketch might help here. – grfrazee Apr 20 '19 at 22:03
• No, you can’t use that formula. That formula is for uniaxial strain. I don’t think this problem has an analytical solution either, so you’ll need to use something like finite element method on the uncoupled thermoelasticity equations. – Paul Apr 20 '19 at 22:54
• @Paul FEM provides indeed more accurate answer, but i guess if the plate is constrained though that's is not the case here, we can apply Lamé equations for plane stress, of course this would be just an approximation. – Sam Farjamirad Apr 21 '19 at 6:04

So at the stationary state, if we assume the warping of surface limited to small angles, $$\frac{\Delta L_{I,j,k}}{L_{i,j,k}} =\alpha\cdot\Delta T$$
• The expression you post is the relative change in length $\Delta L / L$ for the entire bar after it has experienced an overall temperature change $\Delta T$. The system at hand has a thermal gradient at any point $z$ along the bar. These are not the same cases. – Jeffrey J Weimer May 22 '19 at 2:11