Transient Behaviour and Stability for the Thermoelastic Contact of Two Rods of Dissimilar Materials

-The paper investigates the transient behaviour and stability of a system consisting of two thermally conducting elastic rods in contact on their end faces, the other ends of the rods being built-in to two rigid walls which are maintained at different temperatures. It is assumed that there is a thermal resistance at the interface between the rods which is a known function of contact pressure or gap. A perturbation method is used to analyse the stability of the system and it is shown that there is a range o f conditions under which the steady-state solution is unique but unstable. A finite difference method is then used to model the transient behaviour; it shows that under such conditions the system always tends to a steady oscillatory state in which the contact pressure varies periodically with time, possibly with periods of separation. 1. I N T R O D U C T I O N It is well known that mathematical difficulties can arise in the solution of steady-state thermoelastic contact problems if conventional idealized boundary conditions are applied [1-31. Difficulties over existence of solution can be circumvented by postulating a more physically realistic boundary condition involving a pressure or gap dependent thermal resistance at the interface [4, 5], but multiple solutions are still possible with this formulation. The stability of the steady-state solutions has been investigated for various onedimensional systems which exhibit multiple solutions, using perturbation methods [6-8"1. In all cases, it was found that when the steady-state solution was unique, it was also stable, whereas when multiple solutions were obtained, they were alternately stable and unstable. Furthermore, in each case a variational statement of the stability criterion could be formulated which was mathematically equivalent to that prescribed by the perturbation solution, but it proved impossible to justify the variational formulation from first principles in view of the non-conservative nature of the system. More recently, an investigation into the stability of the two dimensional contact of two half-planes [9] has shown that the behaviour is considerably more complex than previous results suggested. The problem was solved by examining the conditions under which a sinusoidal perturbation in the contact pressure can grow with time, following techniques developed by Dow and Burton [10"1 and Richmond and Huang [11"1. As in the onedimensional cases cited above, a characteristic equation is obtained whose zeros correspond to the exponential growth rates of physically admissible perturbations. However, in contrast to the previous solutions, the characteristic equation involves the thermal diffusivities of the two materials as well as the 'steady-state' properties such as conductivity and thermal expansion coefficient. It is therefore possible to envisage material combinations for which the steady-state solution---determined by steady-state properties--is unique, but unstable. The transient behaviour in such cases remains to be determined, but the most likely possibility would seem to be some kind of non-linear oscillation about the unstable steady state. This conclusion has important consequences for the interpretation of previous solutions of steady-state thermoelastic contact problems, such as the classical thermoelastic Hertz problem [12"1, which have always been implicitly assumed to be stable. If one of the half-planes in I-9] is replaced by a non-conducting rigid body, the characteristic equation reduces to a simpler form and the stability criterion shares many of the features of those from earlier one-dimensional analyses. We therefore conclude that the