Transient behaviour and stability for the thermoelastic contact of two rods of dissimilar materials

Abstract

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 of 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.

Finite Element Analysis of Thermoelastic Contact Stability

“…1 The two-rod problem / solution of the transient heat conduction equation with reasonably fine meshing involves the use of a very small time increment for numerical stability and convergence and (ii) the additional effect of a very pressure-sensitive boundary condition at the contact interface would create considerable computational difficulties of its own. Similar difficulties are encountered with other numerical implementations of transient thermoelastic contact problems and are documented in (Barber and Zhang, 1988;Azarkhin and Barber, 1986).…”

“…The ideas behind the method are best explained in terms of its implementation for the simple two-rod model, which was treated by analytical methods by Barber and Zhang (1988). The results of this latter reference also give a convenient check on the validity of the proposed method.…”

“…In such cases, methods using series solutions might be envisaged, but it seems likely that a direct numerical treatment would be more efficient. In the present paper, we therefore develop one possible numerical implementation for the perturbation problem and illustrate its use in the simple two-rod geometry of Barber and Zhang (1988).…”

Exponential Decay for a Thermoelastic Beam Between two Stops

“…We used a beam of alloy steel, for which thermal and mechanics properties are given in Table 1. We take the values for from [7] and E, , , C and k from [14]. The choice of h have been taken small and the value for T 0 is the ambient temperature for the alloy steel.…”

“…For simplicity, we take f = g = 0 in (14) and (15) and we omit the index . Using (14) and (15), with v = u t and w = , respectively, we arrive at…”