Thermoelastic Contact Research Articles

On the existence and uniqueness of steady state solutions in thermoelastic contact with frictional heating

It is well known that contact and friction in thermoelasticity result in mathematical problems which may lack solutions or have multiple solutions. Previously, issues related to thermal contact and issues related to frictional heating have been discussed separately. In this work, the two effects are coupled. Theorems of existence and uniqueness of solutions in two or three space dimensions are obtained—essentially extending, to frictional heating, results due to Duvaut, which were built on Barber's heat exchange conditions. Two qualitatively different existence results are given. The first one requires that the contact thermal resistance goes to zero at least as fast as the inverse of the contact pressure. The second existence theorem requires no such growth condition, but requires instead that the frictional heating, i.e. the sliding velocity times the friction coefficient, is small enough. Finally, it is shown that a solution is unique if the inverse of the contact thermal resistance is Lipschitz continuous and the Lipschitz constant, as well as the frictional heating, is small enough.

Stability of thermoelastic contact for a rectangular elastic block sliding against a rigid wall

In this note, we determine the stability boundary for the thermoelastic contact of a rectangular elastic block sliding against a rigid wall in the presence of a pressure-dependent thermal contact resistance. This geometry can be seen as intermediate between the idealized ‘Aldo’ rod model and continuum solutions for the elastic half-plane. The solution is obtained by comparing the expression for the perturbed boundary condition including frictional heating with that for purely static loading, already solved by Yeo and Barber (1995). The critical sliding speed is obtained as a function of the temperature difference imposed between the wall and the free end. In most cases, frictional heating tends to destabilize the system. However, for certain forms of the resistance-pressure law, the opposite conclusion is reached and the system can be stable for all sliding speeds.  2004 Elsevier SAS. All rights reserved.

Effects of Thermal Contact Resistance on Transient Thermoelastic Contacts for an Elastic Foundation

The paper presents a numerical solution to the problem of a hot rigid indenter sliding over a thermoelastic Winkler foundation with a thermal contact resistance at constant speed. It is shown analytically that no steady-state solution can exist for sufficiently high temperature or sufficiently small normal load or speed, regardless of the thermal contact resistance. However, the steady-state solution may exist in the same situation if the thermal contact resistance is considered. This means that the effect of the large values of temperature difference and small value of force or velocity which occur at no steady state can be lessened due to the thermal contact resistance. When there is no steady state, the predicted transient behavior involves regions of transient stationary contact interspersed with regions of separation regardless of the thermal contact resistance. Initially, the system typically exhibits a small number of relatively large contact and separation regions, but after the initial transient, the trailing edge of the contact area is only established and the leading edge loses contact, reducing the total extent of contact considerably. As time progresses, larger and larger numbers of small contact areas are established, until eventually the accuracy of the algorithm is limited by the discretization used.

Numerical approximation and error control for a thermoelastic contact problem

A numerical method using finite elements for the spatial discretization and the Crank–Nicolson scheme for the time stepping is applied to a partial differential equation problem involving thermoelastic contact. The Crank–Nicolson scheme is interpreted as a low order continuous Galerkin method. By exploiting the variational framework inherent in this approach, an a posteriori error estimate is derived. This estimate gives a bound on the approximation error that depends on computable quantities such as the mesh parameters, time step and numerical solution. In this paper, the a posteriori estimate is used to develop a time step refinement strategy. Several computational examples are included that demonstrate the performance of the method and validity of the estimate.

Three Dimensional Internal Inclined Crack Growth Behavior due to Repeated Rolling Thermoelastic Contact Considering Friction of Crack Surface

This paper considered three dimensional internal inclined crack growth behaviors due to repeated rolling thermoelastic contact. In this analysis, crack is modeled as an internal inclined planar crack in a three dimensional half-space. Rolling contact is simulated as a long and narrow distributed load with both normal and shear components moving with constant velocity. In comparing the shearing force with the frictional force of crack surface, the phenomenon of stick and slip were considered. At first, stress intensity factors along the crack contour are analyzed using high carbon chromium bearing steels. Next, numerial results of crack growth contours are given based on a modified Paris power low. And fatigue life is estimated by maximum crack growth. The shapes of crack contour and fatigue life are calculated under various conditions, which change sliding ratio, frictional coefficient, angle of inclined crack and friction coefficient of crack surface. As results, when the friction of crack surface is large, the crack progresses greatly in the direction at no friction of crack.

Three Dimensional Internal Inclined Crack Growth Behavior due to Repeated Rolling Thermoelastic Contact

Three Dimensional Internal Inclined Crack Growth Behavior due to Repeated Rolling Thermoelastic Contact

TRANSIENT ANALYSIS OF THERMOELASTIC CONTACT BEHAVIORS IN COMPOSITE MULTIDISK BRAKES

The transient analyses of the thermoelastic contact problem are performed for carbon–carbon composite multidisk brakes subject to mechanical and frictionally excited thermal loads. The finite element method, based on the coupled theory in which displacement and temperature fields are mutually affected, is applied for the numerical simulations. In the present study, to improve the accuracy of computations and stabilize the algorithm, the implicit transient scheme is used for the thermoelastic analysis. The law of action and reaction, Signorini's law of contact, Coulomb's law of friction, and Archard's law of wear are applied to be valid locally at each point for friction surfaces. The computation results are obtained for the antiskid brake conditions and presented for the transient evolution of contact pressure, temperature on each friction surface between the bodies, and thermoelastic deformations. It is also found that the high thermal stresses due to considerable temperature difference during the landing process occur in multidisk brakes.

Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem

We study the controllability properties of a nonlinear parabolic system that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case the system is proved to be exactly null controllable provided the parameters of the system are smooth. The proof is based on changing the control variable and using Aubin’s Compactness Lemma to obtain an invariant set for the linearized controllability map. Then, by proving that the found solution is sufficiently smooth, we get the null controllability for the original system.

Instability of thermoelastic contact for two half-planes sliding out-of-plane with contact resistance and frictional heating

Instability of thermoelastic contact for two half-planes sliding out-of-plane with contact resistance and frictional heating

CONTACT PHENOMENA IN BRAKING AND ACCELERATION OF BUSH-SHAFT SYSTEM

In this work a new one-dimensional model of thermoelastic inertial contact of a braking pad and a rotating inertial shaft, in the braking or accelerating process, is presented and analyzed. Besides dry friction, frictional heat is generated between the two given bodies in contact. The dynamics of the considered system is governed by two non-linear differential equations and one non-linear integral equation. The stability of static solutions, as well as the behavior of contact characteristics in braking and acceleration are analysed.

Instability of thermoelastic contact for two half-planes sliding out-of-plane with contact resistance and frictional heating

Thermoelastic contact is known to show instabilities when the heat transmitted across the interface depends on the pressure, either because of a pressure-dependent thermal contact resistance R( p) or because of frictional heating due to the product of friction coefficient, speed, and pressure, fVp. Recently, the combined effect of pressure-dependent thermal contact resistance and frictional heating has been studied in the context of simple rod models or for a more realistic elastic conducting half-plane sliding against a rigid perfect conductor “wall”. Because R( p) introduces a non-linearity even in full contact, the “critical speed” for the uniform pressure solution to be unstable depends not just on material properties, and geometry, but also on the heat flux and on pressure. Here, the case of two different elastic and conducting half-planes is studied, and frictional heating is shown to produce significant effects on the stability boundaries with respect to the Zhang and Barber (J. Appl. Mech. 57 (1990) 365) corresponding case with no sliding. In particular, frictional heating makes instability possible for a larger range of prescribed temperature drop at the interface including, at sufficiently high speeds, the region of opposite sign of that giving instability in the corresponding static case. The effect of frictional heating is particularly relevant for one material combinations of the Zhang and Barber (J. Appl. Mech. 57 (1990) 365) classification (denominated class b here), as above a certain critical speed, the system is unstable regardless of temperature drop at the interface. Finally, if the system has a prescribed heat flow into one of the materials, the results are similar, except that frictional heating may also become a stabilizing effect, if the resistance function and the material properties satisfy a certain condition.

Indentation of an Anisotropic Half-Space by a Heated Flat Punch

The two-dimensional thermoelastic contact problem of an anisotropic half-space indented by a heated rigid flat punch is studied using the extended version of Stroh’s formalism. Two cases, where the contact interface is nonslip and frictionless, have been considered. In the first case, the contact is perfect throughout the punch face. In the second case, separation is assumed to occur at the edges of the punch.

摩擦熱を伴う転がり接触を受ける三次元内部傾斜き裂の進展挙動(OS3-7 圧電材料、き裂,OS3 先端機能性構造材料の力学的挙動に関する数理解析とマルチスケールアナリシスへの進展)

摩擦熱を伴う転がり接触を受ける三次元内部傾斜き裂の進展挙動(OS3-7 圧電材料、き裂,OS3 先端機能性構造材料の力学的挙動に関する数理解析とマルチスケールアナリシスへの進展)

Thermoelastic Contact of a Rotating Inertial Cylinder with a Bush under the Conditions of Friction Self-Oscillations

We construct the solution of the problem of thermoelastic contact of an inertial cylinder with a rigid bush under the conditions of friction heating and wear. It is assumed that the friction coefficient depends on the relative velocity of contacting bodies. We analyze the stability of stationary solutions, determine contact parameters in the process of nonuniform rotation of the cylinder, and demonstrate the possibility of friction self-oscillations.

Finite element analysis of transient thermoelastic behaviors in disk brakes

A transient analysis for thermoelastic contact problem of disk brakes with frictional heat generation is performed using the finite element method. To analyze the thermoelastic phenomenon occurring in disk brakes, the coupled heat conduction and elastic equations are solved with contact problems. The numerical simulation for the thermoelastic behavior of disk brake is obtained in the repeated brake condition. The computational results are presented for the distributions of pressure and temperature on each friction surface between the contacting bodies. Also, thermoelastic instability (TEI) phenomenon (the unstable growth of contact pressure and temperature) is investigated in the present study, and the influence of the material properties on the thermoelastic behaviors (the maximum temperature and contact ratio on the friction surfaces) is investigated to facilitate the conceptual design of the disk brake system. Based on these numerical results, the thermoelastic behaviors of the carbon–carbon composites with excellent mechanical and thermal properties are also discussed.

The thermoelastic Aldo contact model with frictional heating

In the study of the essential features of thermoelastic contact, Comninou and Dundurs (J. Therm. Stresses 3 (1980) 427) devised a simplified model, the so-called “Aldo model”, where the full 3 D body is replaced by a large number of thin rods normal to the interface and insulated between each other, and the system was further reduced to 2 rods by Barber's Conjecture (ASME J. Appl. Mech. 48 (1981) 555). They studied in particular the case of heat flux at the interface driven by temperature differences of the bodies, and opposed by a contact resistance, finding possible multiple and history dependent solutions, depending on the imposed temperature differences. The Aldo model is here extended to include the presence of frictional heating. It is found that the number of solutions of the problem is still always odd, and Barber's graphical construction and the stability analysis of the previous case with no frictional heating can be extended. For any given imposed temperature difference, a critical speed is found for which the uniform pressure solution becomes non-unique and/or unstable. For one direction of the temperature difference, the uniform pressure solution is non-unique before it becomes unstable. When multiple solutions occur, outermost solutions (those involving only one rod in contact) are always stable. A full numerical analysis has been performed to explore the transient behaviour of the system, in the case of two rods of different size. In the general case of N rods, Barber's conjecture is shown to hold since there can only be two stable states for all the rods, and the reduction to two rods is always possible, a posteriori.

Interaction of thermal contact resistance and frictional heating in thermoelastic instability

Thermoelastic contact problems can posess non-unique and/or unstable steady-state solutions if there is frictional heating or if there is a pressure-dependent thermal contact resistance at the interface. These two effects have been extensively studied in isolation, but their possible interaction has never been investigated. In this paper, we consider an idealized problem in which a thermoelastic rod slides against a rigid plane with both frictional heating and a contact resistance. For sufficiently low sliding speeds, the results are qualitatively similar to those with no sliding. In particular, there is always an odd number of steady-state solutions; if the steady-state is unique it is stable and if it is non-unique, stable and unstable solutions alternate, with the outlying solutions being stable. However, we identify a sliding speed V 0 above which the number of steady states is always even (including zero, implying possible non-existence of a steady-state) and again stable and unstable states alternate. A parallel numerical study shows that for V> V 0 there are some initial conditions from which the contact pressure grows without limit in time, whereas for V< V 0 the system will always tend to one of the stable steady states.

TRANSIENT THERMOELASTIC ANALYSIS OF DISK BRAKES IN FRICTIONAL CONTACT

The transient analysis of the thermoelastic contact problem of automotive disk brakes with frictional heat generation is performed using the finite element method. To analyze the thermoelastic phenomenon occurring in disk brakes, the coupled heat conduction and elastic equations are solved with contact problems. In the present work, the fully implicit transient scheme for the thermoelastic analysis is used to improve the accuracy of computations at every time step. The numerical simulation for the thermoelastic behavior of disk brakes is obtained in drag brake condition. The computational results are presented for the distributions of pressure and temperature on each friction surface between the contacting bodies. Also, the thermoelastic instability (TEI) phenomenon (the unstable growth of contact pressure and temperature) is investigated in the present study. The effects of the rotating speed of the disk on thermoelastic behaviors, such as the temperature distribution and contact ratio of the friction surfaces, are investigated.

Generation of hot spots in a wet multidisk clutch during short-term engagement

Hot spots observed in clutches and brakes result from unstable thermoelastic behavior of the frictional system. The phenomenon has been a subject of numerous experimental and theoretical studies, most of which considered steady-state operating conditions. However, most of the clutches and brakes used in automotive and heavy-duty equipment operate in transient mode, at rapidly varying sliding speed. In the paper, generation of hot spots during short-term engagement with varying speed is investigated. Both theoretical analysis based on finite element model of thermoelastic contact problem and experimental stand tests are presented. The study shows that severe hot spots can be produced during short-term clutch operation with high initial sliding speed and that small geometric imperfections of the disks are able to effectively trigger that process. Tests performed with two different friction materials and with different geometric parameters of the disks indicate how elastic properties and geometry influence hot spotting. It is demonstrated, in particular, that the problem can be effectively mitigated even if an instantaneous sliding at high speed (exceeding the critical speed for thermoelastic instability) cannot be avoided.

Global existence and decay for the semilinear thermoelastic contact problem

In this paper a class of semilinear thermoelastic contact problems is considered and the existence and exponential decay of the weak solutions are obtained.