2D domain with heat source

In this example, a rectangular domain with the structural and thermal boundary conditions shown in Figure is considered. The problem is analysed under a plane strain condition with:

E = 200\:GPa, \nu = 0.3, \alpha = 11.7\times10^{-6}\:\mathrm{C}, K = 60\:\mathrm{W/mK}

The sides of the rectangle are a= 0.15 m and b = 0.3 m.

The heat source function is:

s = KT_0\left(\frac{2\pi}{b}\right)^2\sin\left(\frac{2\pi y}{b}\right)

with T₀ = 50 deg.

The exact temperature distribution for this problem is:

T = T_0\sin\left(\frac{2\pi y}{b}\right)

Displacement and stress fields are expressed analytically as:

u_x = 0,\:u_y = \left(\frac{1+\nu}{1-\nu}\right)\left(\frac{\alpha T_0b}{2\pi}\right)\left(1-\cos\left(\frac{2\pi y}{b}\right)\right)

\sigma_{xx} = -\frac{E}{1-\nu}\alpha T_0 \sin\left(\frac{2\pi y}{b}\right)

Numerical results are in excellent agreement with the analytical solution.

Reference

Mohammadi et al. (2010) “Boundary element analysis of thermoelastic problems with non-uniform heat sources” DOI: 10.1243/03093247JSA620