Some Implications of Dundurs' Theorem for Thermoelastic Contact and Crack Problems

It is well known that a simply-connected isotropic elastic body in a state of plane strain and with traction-free boundaries remains free of stress if it is subject to steady-state heat conduction. A recent theorem due to Dundurs shows that in this state the curvature of any initially straight line element is proportional to the heat flux across the line element. A closely related three dimensional result is proved for the sum of the principal curvatures of planes parallel to the faces of an infinite thick plate. These results have certain implications for thermoelastic crack and contact problems. For example: (i) thermal distortion has no effect on the contact pressure distribution at an insulated interface or at an interface between two similar materials, (ii) the thermal stress in a cracked solid depends on the temperature field only through the value of a certain constant related to the average temperature difference across the crack, (iii) steady-state heat flow induces no stresses in an axisymmetric thick plate containing an external crack.