Nonlinear Stability, Thermoelastic Contact, and the Barber Condition

The behavior of a one-dimensional thermoelastic rod is modeled and analyzed. The rod is held fixed and at constant temperature at one end, while at the other end it is free to separate from or make contact with a rigid wall. At this free end we impose a pressure and gap-dependent thermal boundary condition. This condition, known as the Barber condition, couples the thermal and elastic problems. Such systems have previously been shown to undergo a bifurcation from a unique linearly stable steady-state solution to multiple steady-state solutions with alternating stability. Here, the system is studied using the asymptotic matching techniques of boundary layer theory to derive short-time, longtime, and uniform expansions. In this manner, the analysis is extended into the nonlinear regime and dynamic information about the history dependence and temporal evolution of the solution is obtained. @DOI: 10.1115/1.1345699# The analysis of thermal contact problems has revealed a wealth of interesting phenomena. Beginning with J. R. Barber in 1978 ~@1#!, who pointed out that the solution of such problems poses certain difficulties, and continuing to this day, numerous researchers have turned their attention to these problems. Barber observed that the classical assumption of perfect insulation during a separated phase and perfect thermal contact during contact led to models with solutions which were unacceptable on physical grounds. Introducing a pressure and temperature-dependent boundary condition, which would subsequently become known as the Barber condition, he allowed for a smooth transition between the insulated and perfect thermal contact states. Studying a linearized version of a thermal contact problem which included the Barber condition, he showed that the paradoxes inherent in simpler models could be avoided and physically relevant solutions recovered. In 1980, Barber, Dundurs, and Comninou @2# investigated a thermal contact problem using the Barber condition in a onedimensional model of a thermoelastic rod. Imposing a temperature gradient across the rod, they demonstrated that the system underwent a bifurcation from a unique linearly stable steady-state solution to multiple solutions with alternating stability as the magnitude of the thermal gradient was varied. Since that time, various authors have explored the Barber condition and its implications for thermal contact problems ~@3,4#!. While such analyses have been extended to multiple materials ~@3,5,6#!, various geometries ~@7,8#!, and to numerical simulations ~@4#!, most theoretical work to date has relied upon linear stability theory. In a recent article ~@9#! we developed a nonlinear theory which described the history dependence and dynamics of solutions near the bifurcation point for a simplified model of a onedimensional thermoelastic rod. Our model did not, however, include the Barber condition. Since the Barber condition is much more physically realistic than the boundary condition used in ~@9#!, it is desirable to have a nonlinear theory for a model which incorporates the Barber condition. We carry out such an analysis here. While the model studied here differs from the model studied in ~@9#!, only in the use of the Barber condition, the method of analysis differs significantly. In particular, here we use the asymptotic matching techniques of boundary layer theory to derive shorttime, long-time, and uniform asymptotic expansions of the solution. In our prior analysis we used the method of multiple scales, or two-timing, to accomplish similar goals. The switch in techniques is not merely a matter of taste. Rather, any attempt to apply multiple scale techniques to the model considered herein will soon encounter algebraic difficulties. That is, such an attempt becomes analytically intractable. However, as is shown, boundary layer theory may be applied with little difficulty. This not only allows us to carry out the analysis for the one-dimensional rod model with the Barber condition, but gives us hope that similar techniques will yield a nonlinear stability theory for more complicated multidimensional problems. We begin in Section 2 by formulating the governing equations for our model. We make the standard assumption that quasi-static uncoupled thermoplasticity is valid and use the Signorini contact condition to capture periods of separation and contact. We impose the Barber condition on the thermal part of the problem, leaving the contact resistance function unspecified. A solution is constructed for the elastic problem and the system of governing equations is reduced to a nonlocal and nonlinear heat conduction problem. In Section 3, we impose physically realistic constraints on the contact resistance and develop a linear theory. We review the analysis due to Barber @2#, and show that the system studied undergoes a bifurcation from a single linearly stable steady-state solution to multiple steady-state solutions. Finally, in Section 4, we study the behavior of our system near the bifurcation point. That is, we inquire as to what happens when the system is started nearby the now linearly unstable steady-state solution. Using asymptotic matching techniques, we incorporate the effect of stabilizing nonlinear terms into our theory and obtain information about the dynamics and history dependence of the solution. We show that as conjectured, the solution does indeed approach one of the stable solutions obtained in the linear theory.