Traction free semi-plane

A semi-plane is exposed to a stationary local smooth thermal perturbation on the boundary:

T|_{y=0} = 10(x^2-0.5)\exp(-x^2)

All the external boundaries of the plane are traction free, and plane strain is assumed.

The temperature distribution at the steady state is:

The computed displacements and the reference solution are:

Reference

Andriy V. Rychahivskyy & Yuriy V. Tokovyy (2008) Correct Analytical Solutions to the Thermoelasticity Problems in a Semi-plane, Journal of Thermal Stresses, 31:11, 1125-1145, DOI: 10.1080/01495730802250854