A dilemma arising from the conventional boundary conditions for thermoelastic contact was observed by Barber in treating the indentation of an elastic half space by a rigid sphere. If the sphere is colder than the half space, the interface tractions are necessarily tensile near the periphery of the contact region. In order to overcome this difficulty, Barber introduced the idea of an imperfect contact zone. An asymptotic analysis of the transitions between the different zones is carried out in this article. It is found that, if heat flows into the body with the larger distortivity, a direct transition from perfect contact (no resistance to heat flow) to separation (no heat flow) is possible, the zone of imperfect contact (vanishing contact pressure and some resistance to heat flow) is automatically excluded, and the heat flux is square-root singular at the transition. If heat flows in the opposite direction, no direct transition from perfect contact to separation is possible, there must be an intervening zone of imperfect contact, and the heat flux is logarithmically singular at the transition from perfect to imperfect contact. The transition from imperfect contact to separation is always possible, and it is smooth. These conclusions are direct consequences of the inequalities that must be enforced because of the unilateral nature of thermoelastic contact.
