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 V₀ 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>V₀ there are some initial conditions from which the contact pressure grows without limit in time, whereas for V<V₀ the system will always tend to one of the stable steady states.     

Separated steady state solutions for two thermoelastic half-planes in contact with out-of-plane sliding

When two materials are placed in contact along an interface, thermoelastic effects can separate the surfaces and create “hot spots” when there is sufficient frictional heating fVp generated at the interface, even if the two surfaces are nominally flat. Additionally, heat can flow because the bodies are generally at different temperatures, and this is an independent cause of separation, generally when heat flows into the less distortive material. These two effects have been considered separately, and here we consider the case with interaction of the two effects, showing possible non-existence, multiplicity and instability of solutions. Approximate Hertzian solutions for the separated contact regime are very limited, particularly for the frictional heating case. Hence, a new efficient full numerical solution is developed, and compared with direct FEM results, the latter permitting also the assessment of stability in the transient regime. Connection to previous results for simple rod models is made. The case of heat flow into the more distortive material is discussed.     

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.     

“Frictionless” and “frictional” ThermoElastic Dynamic Instability (TEDI) of sliding contacts

Recently, we found that a new form of coupled instability, named ThermoElastic Dynamic Instability (TEDI), can occur by interaction between frictional heating and the natural dynamic modes of sliding bodies. This is distinct from the classical dynamic instabilities (DI) which is produced by an interaction between the frictional forces at the sliding interface and the natural modes of vibration of the bodies if the friction coefficient is sufficiently high, and also from ThermoElastic Instability (TEI), which is due to the interaction of frictional heating and thermal expansion, leading for example to low pitched brake noise above some critical speed. This result was relative to an highly idealized system, comprising an elastic layer sliding over a rigid plane including both dynamic and thermoelastic effects, but neglecting shear waves at the interface due to frictional tractions (from which the denomination “frictionless TEDI”). We demonstrate here that including these shear waves destabilizes both the shear and dilatational vibration modes of the system at arbitrarily small friction coefficients and speeds, where DI and TEI are predicted to be stable. A detailed study of the new modes and transient simulations show that for low pressures and high speed, the system tends towards the results of the previous model (“frictionless TEDI”), i.e. the tendency to a state in which the layer bounces over the plane, with alternating periods of sliding contact and separation. In the case of low speeds and high pressures, viceversa, the system is dominated by the modes near the resonance of the shear and dilatational modes, with a resulting complex behaviour, but generally leading to stick-slip regimes, reducing the jumping mode of “frictionless TEDI”, because stick reduces or stops frictional heating production.     

热对流条件下考虑球形夹杂分布的材料热弹接触摩擦热影响分析

机械传动关键活动零部件接触副往往受到力载荷和摩擦热载荷的耦合作用,使得接触界面间的接触力学行为的分析变得极其复杂. 利用基于等效夹杂方法建立的考虑热对流非均质材料热弹接触力学分析模型研究不同摩擦系数、夹杂位置和材料属性等参数对材料表面及内部温升及热应力分布影响规律. 此外,进一步分析了接触副材料中含分布球形夹杂时摩擦热造成的影响. 结果表明:接触副表面温升梯度受热对流系数的影响较大;下表面温升和热应力随摩擦系数增大而增大;分布夹杂则将接触副材料下表面温升及热应力分布变得更为复杂.      

An Eulerian approach for simulating frictional heating in disc-pad systems

Thermal stresses as a result from frictional heating must be considered when designing disc brakes, clutches or other rotating machine components with sliding contact conditions. The rotational symmetry of the disc in these kind of applications makes it possible to model these systems using an Eulerian approach instead of a Lagrangian framework. In this paper such an approach is developed and implemented. The disc is formulated in an Eulerian frame where the convective terms are defined by the angular velocity. By utilizing the Eulerian framework, a node-to-node formulation of the contact interface is obtained, producing most accurate frictional heat power solutions. The energy balance of the interface is postulated by introducing an interfacial temperature. Both frictional power and contact conductances are included in this energy balance. The contact problem is solved by a non-smooth Newton method. By adopting the augmented Lagrangian approach, this is done by rewriting Signorini’s contact conditions to an equivalent semi-smooth equation. The heat transfer in the disc is discretized by a Petrov–Galerkin approach, i.e. the numerical difficulties due to the non-symmetric convective matrix appearing in a pure Galerkin discretization is treated by following the streamline-upwind approach. In such manner a stabilization is obtained by adding artificial conduction along the streamlines. For each time step the thermo-elastic contact problem is first solved for the temperature field from the previous time step. Then, the heat transfer problem is solved for the corresponding frictional power. In such manner a temperature history is obtained sequentially via the trapezoidal rule. In particular the parameter is set such that both the Crank–Nicolson and the Galerkin methods are utilized. The method seems very promising. This is demonstrated by solving a two-dimensional benchmark as well as a real disc brake system in three dimensions.     

Effects of non-uniform heating on a variable viscosity Rayleigh�CB��nard problem

This study presents a natural convection problem with a temperature-dependent viscosity fluid, driven by buoyancy and influenced by horizontal temperature gradients. A numerical linear stability analysis of the stationary solutions is studied. The horizontal temperature gradients tend to localize motion near the warmer zones and favour pattern formation in the direction perpendicular to the gradient. In fact, the problem is almost 2D in the uniform heating case, but becomes totally 3D in the non-uniform heating case.     

Thermo-elastic dynamic instability (TEDI) in frictional sliding of two elastic half-spaces

In some simplified 1D models, we recently studied the coupling of TEI (thermoelastic instability) and DI (dynamic instability), finding that thermal effects can render unstable the otherwise neutrally stable natural elastodynamic modes of the system, giving rise to a new family of instability which we called TEDI.Here, we study the general case of two sliding elastic half-planes, finding again a relatively weak coupling between thermal and dynamic effects, and the general family of instability TEDI class is found to modify both the otherwise separated TEI and DI classes. The growth factor, the phase velocity and the migrating speeds of the perturbations are wavelength-dependent, and it is difficult to give a complete picture given the high number of materials’ parameters, and the dependence on speed, friction coefficient, and the underlying uniform pressure. However, a set of results are given for “large” and “small” mismatch of shear wave speeds in the materials, and as a function of (i) friction coefficient; (ii) sliding speed V₀; (iii) wavenumber parameter γ. In the case of small mismatch, generalized Rayleigh waves exists already under frictionless conditions, the critical f for instability is zero. DI dominates over TEI typically for large wavenumbers, where the growth factors increase without limit and hence become eventually meaningless, requiring regularizations for example with rate-state dependent friction laws. TEI growth factors vice versa have a maximum at a certain wavenumber and therefore are always well posed. Larger coupling effects are noticed for two materials with large mismatch, but significantly only for sliding speeds comparable with the wave speed. In general, TEI growth factors increase with speed, whereas DI growth factors increase with speed for similar materials and decrease when the mismatch between materials is large.     

Thermoelastic contact instability of a functionally graded layer and a homogeneous half-plane

The conductive heat transfer between two elastic bodies in the static contact can cause the system to be unstable due to the interaction between the thermoelastic distortion and pressure-dependent thermal contact resistance. This paper investigates the thermoelastic contact instability of a functionally graded material (FGM) layer and a homogeneous half-plane using the perturbation method. The FGM layer and half-plane are exposed to a uniform heat flux and are pressed together by a uniform pressure. The material properties of the FGM layer vary exponentially along the thickness direction. The characteristic equation governing the thermoelastic stability behavior is obtained to determine the stability boundary. The effects of the gradient index, layer thickness and material combination on the critical heat flux are discussed in detail through a parametric study. Results indicate that the thermoelastic stability behavior can be modified by adjusting the gradient index of the FGM layer.     

一般各向异性单侧接触界面上波的反射和折射

研究简谐弹性波在一般各向异性介质单侧接触界面上的反射和折射问题．利用Fouier分析方法将非线性Coulomb摩擦接触边界波动问题化为一组代数方程．给出了确定局部分离、滑移和粘着区的思路和方法及各区域的解；讨论了出现界面局部分离和滑移的条件．对特定材料组合情况进行了详细数值计算，给出了界面力、相对滑移速度、张开位移、高频谐波的反射折射系数等特征参量；考察了平面和反平面波动的耦合及整体滑移等．其中关于高频谐波的结果可对已有实验结果给出很好的定性解释．在大多数情况下，即使对摩擦系数无穷大的粘滞接触界面，分离区端部也总是存在一个很小的滑移区。     

从规划法求解看有摩擦接触解的不唯一性

以单点接触问题为例研究了有摩擦接触问题的几个特点，给出了这类问题出现不唯一解的条件。进而从二次规划求解角度分析了解的不稳定性与不唯一性的数学原理，文中给出了规划法求解有摩擦接触问题应注意的问题     

Mechanics of non-slipping adhesive contact on a power-law graded elastic half-space

In previous work about axisymmetric adhesive contact on power-law graded elastic materials, the contact interface was often assumed to be frictionless, which is, however, not always the case in practical applications. In order to elucidate the effect of friction and the coupling between normal and tangential deformations, in the present paper, the problem of a rigid punch with a parabolic shape in non-slipping adhesive contact with a power-law graded half-space is studied analytically via singular integral equation method. A series of closed-form analytical solutions, which include the frictionless and homogeneous solutions as special cases, are obtained. Our results show that, compared with the frictionless case, the interfacial friction tends to reduce the contact area and the indentation depth during adhesion. The magnitude of the coupling effect depends on both the Poisson ratio and the gradient exponent of the half-space. This effect vanishes for homogeneous incompressible as well as for linearly graded materials but becomes significant for auxetic materials with negative Poisson’s ratio. Furthermore, influence of mode mixity on the adhesive behavior of power-law graded materials, which was seldom touched in literature, is discussed in details.     

岩石试件端面摩擦效应数值模拟研究

试件端面摩擦效应直接影响试件内的塑性等效应变、侧向位移的分布和单元应力应变曲线。本文运用ANSYS中的接触单元模拟了平面应变状态下端面摩擦效应对塑性等效应变、侧向位移和单元应力应变曲线的影响,得到了不同摩擦系数时塑性等效应变及侧向位移的渐进变化形式。当接触面摩擦较小时,塑性等效应变图案为上下两个X形网络,侧向位移上下分布均匀;当接触面摩擦增大时,塑性等效应变网络向中部靠拢并且明显增大,侧向位移上下分布不均匀,中部较上下端面位移大;当试件端面侧向位移被限制,即摩擦力很大时,塑性等效应变网络变为一个X形局部化带,侧向位移分布更加不均匀,中部明显隆起。     

Stability analysis of numerical interface conditions in fluid–structure thermal analysis

This paper analyses the numerical stability of coupling procedures in modelling the thermal diffusion in a solid and a fluid with continuity of temperature and heat flux at the interface. A simple one-dimensional model is employed with uniform material properties and grid density in each domain. A number of different explicit and implicit algorithms are considered for both the interior equations and the boundary conditions. The analysis shows that in general these are stable provided that Dirichlet boundary conditions are imposed on the fluid and Neumann boundary conditions are imposed on the solid; in each case the imposed values are obtained from the other domains. © 1997 John Wiley & Sons, Ltd.     

Interfacial slip and creep in rolling contact incorporating a cylinder with an elastic layer

A systematic approach for investigating the interfacial behaviour of tyred systems is proposed. A two-dimensional contact model of an elastic strip, shrink-fitted onto a wheel, and subjected to different rolling contact conditions, has been adopted to illustrate the method. The model combines existing techniques to explore individual elastic contact problems and it enables us to characterise the behaviour at the strip/substrate interface caused by loads induced by a quasi-static application of stationary and moving loads on the surface of the layer. The solution is compared to the stationary load case and regimes of local slip, full stick, separation and frictional creep are identified and collated for a variety of loading conditions, materials and geometries. Further, this article presents an investigation of frictional shakedown for layered systems subjected to periodic contact loading. The term shakedown is here referred to as the possibility of developing interfacial residual stresses at the layer/substrate interface such that frictional slip, originally activated by the applied external contact load, ceases after a few loading cycles. The possible applicability of the Melan’s theorem for elastic frictional system is investigated and preliminary results presented.     

Fully-developed heat transfer in annuli for viscoelastic fluids with viscous dissipation

Analytical solutions are obtained for heat transfer in concentric annular flows of viscoelastic fluids modeled by the simplified Phan-Thien–Tanner constitutive equation. Solutions for thermal and dynamic fully developed flow are presented for both imposed constant wall heat fluxes and imposed constant wall temperatures, always taking into account viscous dissipation.Equations are presented for the normalized temperature profile, the bulk temperature, the inner and outer wall temperatures and, through their definitions for the inner and outer Nusselt numbers as a function of all relevant non-dimensional parameters. Some special results are discussed in detail. Given the complexity of the derived equations, for ease of use compact exact expressions are presented for the Nusselt numbers and programmes to calculate all quantities are made accessible on the internet. Generally speaking, fluid elasticity is found to increase the heat transfer for imposed heating at the wall, especially in combination with internal heat generation by viscous dissipation, whereas for imposed wall temperatures it reduces heat transfer when viscous dissipation is weak.     

Viscous dissipation effects on thermal entrance region heat transfer in pipes with uniform wall heat flux

The analytical solution to Graetz problem with uniform wall heat flux is extended by including the viscous dissipation effect in the analysis. The analytical solution obtained reduces to that of Siegel, Sparrow and Hallman neglecting viscous dissipation as a limiting case. The sample developing temperature profiles, wall and bulk temperature distributions and the local Nusselt number variations are presented to illustrate the viscous dissipation effects. It is found that the role of viscous dissipation on thermal entrance region heat transfer is completely different for heating and cooling at wall. In the case of cooling at wall, a critical value of Brinkman number, Br _c=−11/24, exists beyond which (−11/24<Br<0) the fluid bulk temperature will always be less than the uniform entrance temperature indicating the predominance of cooling effect over the viscous heating effect. On the other hand, with Br < Br _c the bulk temperature T _b will approach the wall temperature T _w at some downstream position and from there onward the bulk temperature T _b becomes less than the wall temperature T _w with T _w > B _b > T ₀ indicating overall heating effect for the fluid. The numerical results for the case of cooling at wall Br < 0 are believed to be of some interest in the design of the proposed artctic oil pipeline.     

Flow and heat transfer in a second grade fluid over a stretching sheet

An analysis is carried out to study the flow and heat transfer characteristics in a second grade fluid over a stretching sheet with prescribed surface temperature including the effects of frictional heating, internal heat generation or absorption, and work due to deformation. In order to solve the fourth-order non-linear differential equation, associated with the flow problem, a fourth boundary condition is augmented and a proper sign for the normal stress modulus is used. It is observed that for a physical flow problem the solution is unique. The solutions for the temperature and the heat transfer characteristics are obtained numerically and presented by a table and graphs. Furthermore, it is shown that the heat flow is always from the stretching sheet to the fluid.     

Heat transfer by Hagen-Poiseuille flow in the thermal development region with axial conduction

The heat transfer in the region of circular pipes close to the beginning of the heating section is investigated for low-Péclet-number flows with fully developed laminar velocity profile. Axial heat conduction is included and its effect on the temperature distribution is studied not only for the region downstream of the start of heating but also for that upstream. The energy equation is solved numerically by a finite difference method. Results are presented graphically for various Péclet numbers between 1 and 50. The boundary conditions are uniform wall temperature and uniform wall heat flux with step change at a certain cross-section. For the latter case, also some results for the region near the end of the heating section are reported. The solutions are applicable for the corresponding mass transfer situations where axial diffusion is important if the temperature is replaced by the concentration andPe byReSc.     

Optimal shapes of contact interfaces due to sliding wear in the steady relative motion

The transient wear process at contact frictional interface of two elastic bodies in relative steady motion induces evolution of shape of the interface. A steady wear state may be reached with uniform wear rate and fixed contact surface shape. In this paper, the optimal contact shape is studied by formulating several classes of shape optimization problems, namely minimization of generalized wear volume rate, friction dissipation power and wear dissipation rate occurring in two bodies. The wear rule was assumed as a nonlinear dependence of wear rate on friction traction and relative sliding velocity, similar to the Archard rule. The wear parameters of two bodies may be different. It was demonstrated that different optimal contact shapes are generated depending on objective functional and wear parameters. When the uniform wear rate is generated at contact sliding surfaces, the steady state is reached. It was shown that in the steady state the wear parameters of two bodies cannot be independent of each other. The solution of nonlinear programming problem was provided by the iterative numerical procedure. It was assumed that the relative sliding velocity between contacting bodies results from translation and rotation of two bodies. In general, both regular and singular regimes of wear rate and pressure distribution may occur. The illustrative examples of drum brake, translating punch and rotating annular punch (disc brake) provide the distribution of contact pressure and wear rate for regular and singular cases associated with the optimality conditions. It is shown that minimization of the generalized wear dissipation rate provides solutions assuring existence of steady wear states.