Frictionally excited thermoelastic instability in disc brakesTransient problem in the full contact regimeAbstractExceeding the critical sliding velocity in disc brakes can cause unwanted forming of hot spots, non-uniform distribution of contact pressure, vibration, and also, in many cases, permanent damage of the disc. Consequently, in the last decade, a great deal of consideration has been given to modeling methods of thermo elastic instability (TEI), which leads to these effects. Models based on the finite element method are also being developed in addition to the analytical approach. The analytical model of TEI development described in the paper by Lee and Barber Frictionally excited thermo elastic instability in automotive disk brakes. ASME Journal of Tribology 1993;115:60714 has been expanded in the presented work. Specific attention was given to the modification of their model, to catch the fact that the arc length of pads is less than the circumference of the disc, and to the development of temperature perturbation amplitude in the early stage of breaking, when pads are in the full contact with the disc. A way is proposed how to take into account both of the initial non-flatness of the disc friction surface and change of the perturbation shape inside the disc in the course of braking.Keywords: Thermo elastic instability; TEI; Disc brake; Hot spots1. IntroductionFormation of hot spots as well as non-uniform distribution of the contact pressure is an unwanted effect emerging in disc brakes in the course of braking or during engagement of a transmission clutch. If the sliding velocity is high enough, this effect can become unstable and can result in disc material damage, frictional vibration, wear, etc. Therefore, a lot of experimental effort is being spent to understand better this effect (cf. Refs.) or to model it in the most feasible fashion. Barber described the thermo elastic instability (TEI)as the cause of the phenomenon. Later Dow and Burton and Burton et al. introduced a mathematical model to establish critical sliding velocity for instability, where two thermo elastic half-planes are considered in contact along their common interface. It is in a work by Lee and Barber that the effect of the thickness was considered and that a model applicable for disc brakes was proposed. Lee and Barbers model is made up with a metallic layer sliding between two half-planes of frictional material. Only recently a parametric analysis of TEI in disc brakes was made or TEI in multi-disc clutches and brakes was modeled. The evolution of hot spots amplitudes has been addressed in Refs. Using analytical approach or the effect of intermittent contact was considered. Finally, the finite element method was also applied to render the onset of TEI (see Ref.).The analysis of nonlinear transient behavior in the mode, when separated contact regions occur, is even accomplished in Ref. As in the case of other engineering problems of instability, it turns out that a more accurate prediction by mathematical modeling is often questionable. This is mainly imparted by neglecting various imperfections and random fluctuations or by the impossibility to describe all possible influences appropriately. Therefore, some effort aroused to interpret results of certain experiments in addition to classical TEI (see, e.g.Ref).This paper is related to the work by Lee and Barber 7.Using an analytical approach, it treats the inception of TEI and the development of hot spots during the full contact regime in the disc brakes. The model proposed in Section 2 enables to cover finite thickness of both friction pads and the ribbed portion of the disc. Section 3 is devoted to the problems of modeling of partial disc surface contact with the pads. Section 4 introduces the term of thermal capacity of perturbation emphasizing its association with the value of growth rate, or the sliding velocity magnitude. An analysis of the disc friction surfaces non-flatness and its influence on initial amplitude of perturbations is put forward in the Section 5. Finally, the Section 6 offers a model of temperature perturbation development initiated by the mentioned initial disc non-flatness in the course of braking. The model being in use here comes from a differential equation that covers the variation of thethermal capacity during the full contact regime of the braking.2. Elaboration of Lee and Barber modelThe brake disc is represented by three layers. The middle one of thickness 2a3 stands for the ribbed portion of the disc with full sidewalls of thickness a2 connected to it. The pads are represented by layers of thickness a1, which are immovable and pressed to each other by a uniform pressure p. The brake disc slips in between these pads at a constant velocity V.We will investigate the conditions under which a spatially sinusoidal perturbation in the temperature and stress fields can grow exponentially with respect to the time in a