# Solid discs and coeff. thermal expansion mismatch

Consider a solid disc of radius $$R$$ and height $$h$$ which we glue onto an infinite plane surface. If we now change the temperatur and the assume that

1. the (linear) thermal expansion coefficient of the substrate vanishes, $$\alpha_{substrate} = 0 \mu m /(K m)$$,
2. the (linear) thermal expansion coefficient of the disc is given by $$\alpha>0$$, and
3. that the glue hold the disc and the substrate perfectly in place,

does anybody know a formula to describe the (free) surface of the disc? I'm not looking for finite element simulations, but a descriptive formula, which gives me some intuition about the shape. I expect, that the shape depends on all kind of properties, like e.g. Young's module $$E$$, and the height $$h$$. However, this problem seems to be fundamental for all kind of engineering work.

Throughout the majority of the disc, the strains in the radial and hoop directions will be essentially zero, and the strain in the thickness direction will be uniform. Only at the very edge, within about 2h radially from the edge will this picture be disturbed. Within the broad central region, if you solve Hooke’s law 3D equations for this situation , you will find that the thickness strain is $$\epsilon =\frac{(1+\nu)}{(1-\nu)}\alpha \Delta T$$where $$\nu$$ is the Poisson ratio of the disc material.