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 3D 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.     

Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction

We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V₀ We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V. In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value ? B In V, where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables.Linearized stability analysis predicts constant slip rate motion at V₀ to change from stable to unstable with a decrease in the spring stiffness k below a critical value k_cr. At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k, if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased.Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V₀ is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k:, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations.     

Painlevé paradox during oblique impact with friction

In analyses using non-smooth dynamics, oblique impact of rough bodies in an unsymmetrical configuration can result in self-locking or “jam” at the sliding contact if the coefficient of friction is sufficiently large; this has been termed, Painlevé’s paradox. In the range of configurations and coefficients of friction where Painlevé’s paradox occurs, analyses based on rigid body dynamics give results indicating that either there are multiple solutions or the solution is nonexistent. This conundrum has been resolved by considering that the contact has small normal and tangential compliance which is representative of deformability in a local region around the contact point. An analysis using a hybrid model which includes local compliance of the contact region has calculated the time-dependent changes in relative motion of colliding bodies for a range of incident angles of obliquity, tan^?1[?V₁(0)/V₃(0)] where V₁(0)and V₃(0) are the incident tangential and normal relative velocities at the contact point, respectively. The paradox is shown to result from a negative relative acceleration of the contact points during an initial period of sliding – a negative acceleration that is inconsistent with the assumption of rigid-body contact.     

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.     

Über die instabilität konservativer systeme mit gyroskopischen kräften

Summary Routh's theorem states that a steady motion of a discrete, conservative mechanical system is stable if the dynamic potential W(q)=U(q)–T₀(q) assumes a minimum. This is a generalized version of the theorem on the stability of equilibrium at a minimum of the potential energy, which is due to Dirichlet. It is well known that a steady motion may also be stable if W(q) assumes a maximum instead of a minimum. The stability is then due to the gyroscopic terms in the equations of motion, without which the steady motion would be unstable. Here it is shown that the steady motion is always unstable if not only W(q) but also H ₀(q) assumes a maximum, H ₀(q) being the part of the Hamiltonian that does not depend on the momenta. It is astonishing that this unexpectedly simple criterion was not found before now. In the proof, a variational formulation is used for the problem, and the instability is shown directly from the existence of certain motions which diverge from the trivial solution.

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Vorgelegt von C. Truesdell     

Numerical and experimental analysis of the bi-stable state for frictional continuous system

Unstable friction-induced vibrations are considered an annoying problem in several fields of engineering. Although several theoretical analyses have suggested that friction-excited dynamical systems may experience sub-critical bifurcations, and show multiple coexisting stable solutions, these phenomena need to be proved experimentally and on continuous systems. The present work aims to partially fill this gap. The dynamical response of a continuous system subjected to frictional excitation is investigated. The frictional system is constituted of a 3D printed oscillator, obtained by additive manufacturing that slides against a disc rotating at a prescribed velocity. Both a finite element model and an experimental setup has been developed. It is shown both numerically and experimentally that in a certain range of the imposed sliding velocity the oscillator has two stable states, i.e. steady sliding and stick–slip oscillations. Furthermore, it is possible to jump from one state to the other by introducing an external perturbation. A parametric analysis is also presented, with respect to the main parameters influencing the nonlinear dynamic response, to determine the interval of sliding velocity where the oscillator presents the two stable solutions, i.e. steady sliding and stick–slip limit cycle.

     

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ-method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡λV/R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De_c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De_Lc≡λV/L is shown to be an O(1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.     

“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.     

Instability and friction

A review on the stability analysis of solids in unilateral and frictional contact is given. The presentation is focussed on the stability of an equilibrium position of an elastic solid in frictional contact with a fixed or moving obstacle. The problem of divergence instability and the obtention of a criterion of static stability are discussed first for the case of a fixed obstacle. The possibility of flutter instability is then considered for a steady sliding equilibrium with a moving obstacle. The steady sliding solution is generically unstable by flutter and leads to a dynamic response which can be chaotic or periodic. This dynamic response leads to the generation of stick–slip–separation waves on the contact surface in a similar way as Schallamach waves in statics. Illustrating examples and principal results recently obtained in the literature are reported. Some problems of friction-induced vibration and noise emittence, such as brake squeal for example, can be interpreted in this spirit. To cite this article: Q.S. Nguyen, C. R. Mecanique 331 (2003).     

Finite deformation and stability behaviour of cylindrical membranes subjected to symmetric line loads2

An ‘exact’ analysis of the complete non-linear load-deflection and stability behaviour of cylindrical membranes without end ‘shear walls’ subjected to longitudinal Symmetric line loads is presented. The analysis includes low as well as high profile structures. In order to determine the lateral stability behaviour, infinitesimal lateral displacements are superimposed on the symmetric finite deflection field.Results indicate that such structures may experience vertical and/or lateral instability depending on their initial geometry. Membranes with initial central angles, θ₀ ? 90° are stable, both vertically and laterally, for all load values. For 42.23° < θ₀ < 90° the structure is laterally stable but becomes vertically unstable at a certain ‘limit load’ W?_V while for θ₀< 42.23°, the structure becomes laterally unstable at a load value w?_l<W?_V.The analysis admits contact between vertical segments of the membrane to either side of the line load as well as between the membrane and the horizontal surface next to the supports.     

Time-dependent plane Poiseuille flow of a Johnson–Segalman fluid

We numerically solve the time-dependent planar Poiseuille flow of a Johnson–Segalman fluid with added Newtonian viscosity. We consider the case where the shear stress/shear rate curve exhibits a maximum and a minimum at steady state. Beyond a critical volumetric flow rate, there exist infinite piecewise smooth solutions, in addition to the standard smooth one for the velocity. The corresponding stress components are characterized by jump discontinuities, the number of which may be more than one. Beyond a second critical volumetric flow rate, no smooth solutions exist. In agreement with linear stability analysis, the numerical calculations show that the steady-state solutions are unstable only if a part of the velocity profile corresponds to the negative-slope regime of the standard steady-state shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to different stable steady states, depending on the initial perturbation. The asymptotic steady-state velocity solution obtained in start-up flow is smooth for volumetric flow rates less than the second critical value and piecewise smooth with only one kink otherwise. No selection mechanism was observed either for the final shear stress at the wall or for the location of the kink. No periodic solutions have been found for values of the dimensionless solvent viscosity as low as 0.01.     

Instability of frictional contact states in infinite layers

The research reported in this paper is focused on the instability of equilibrium and steady sliding states of elastic orthotropic layers in the presence of unilateral obstacles with Coulomb friction with emphasis on a divergence type instability called directional instability of frictional contact states that cannot occur in isotropic layers. Analytic expressions and numerical solutions are provided for the instability mode and for the coefficient of friction at the onset of instability. A parametric study is done to investigate how this coefficient of friction and the instability mode vary with changes of the system parameters. For certain combinations of material data, significantly low coefficients of friction were required for the onset of instability. A finite element model that approximates the continuum and a lumped model that captures some of the features of the continuum are presented.     

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.     

Steady and transient sliding under rate-and-state friction

The physics of dry friction is often modelled by assuming that static and kinetic frictional forces can be represented by a pair of coefficients usually referred to as μ_s and μ_k, respectively. In this paper we re-examine this discontinuous dichotomy and relate it quantitatively to the more general, and smooth, framework of rate-and-state friction. This is important because it enables us to link the ideas behind the widely used static and dynamic coefficients to the more complex concepts that lie behind the rate-and-state framework. Further, we introduce a generic framework for rate-and-state friction that unifies different approaches found in the literature.We consider specific dynamical models for the motion of a rigid block sliding on an inclined surface. In the Coulomb model with constant dynamic friction coefficient, sliding at constant velocity is not possible. In the rate-and-state formalism steady sliding states exist, and analysing their existence and stability enables us to show that the static friction coefficient μ_s should be interpreted as the local maximum at very small slip rates of the steady state rate-and-state friction law.Next, we revisit the often-cited experiments of Rabinowicz (J. Appl. Phys., 22:1373–1379, 1951). Rabinowicz further developed the idea of static and kinetic friction by proposing that the friction coefficient maintains its higher and static value μ_s over a persistence length before dropping to the value μ_k. We show that there is a natural identification of the persistence length with the distance that the block slips as measured along the stable manifold of the saddle point equilibrium in the phase space of the rate-and-state dynamics. This enables us explicitly to define μ_s in terms of the rate-and-state variables and hence link Rabinowicz's ideas to rate-and-state friction laws.This stable manifold naturally separates two basins of attraction in the phase space: initial conditions in the first one lead to the block eventually stopping, while in the second basin of attraction the sliding motion continues indefinitely. We show that a second definition of μ_s is possible, compatible with the first one, as the weighted average of the rate-and-state friction coefficient over the time the block is in motion.     

Flow Bifurcations in a Thin Gap Between Two Rotating Spheres

This paper presents the use of a parameter continuation method and a test function to solve the steady, axisymmetric incompressible Navier–Stokes equations for spherical Couette flow in a thin gap between two concentric, differentially rotating spheres. The study focuses principally on the prediction of multiple steady flow patterns and the construction of bifurcation diagrams. Linear stability analysis is conducted to determine whether or not the computed steady flow solutions are stable. In the case of a rotating inner sphere and a stationary outer sphere, a new unstable solution branch with two asymmetric vortex pairs is identified near the point of a symmetry-breaking pitchfork bifurcation which occurs at a Reynolds number equal to 789. This solution transforms smoothly into an unstable asymmetric 1-vortex solution as the Reynolds number increases. Another new pair of unstable 2-vortex flow modes whose solution branches are unconnected to previously known branches is calculated by the present two-parameter continuation method. In the case of two rotating spheres, the range of existence in the (Re ₁ , Re ₂ ) plane of the one and two vortex states, the vortex sizes as a function of both Reynolds numbers are identified. Bifurcation theory is used to discuss the origin of the calculated flow modes. Parameter continuation indicates that the stable states are accompanied by certain unstable states. Received 26 November 2001 and accepted 10 May 2002 Published online 30 October 2002 Communicated by M.Y. Hussaini     

On the stability of the simple shear flow of a Johnson–Segalman fluid

We solve the time-dependent simple shear flow of a Johnson–Segalman fluid with added Newtonian viscosity. We focus on the case where the steady-state shear stress/shear rate curve is not monotonic. We show that, in addition to the standard smooth linear solution for the velocity, there exists, in a certain range of the velocity of the moving plate, an uncountable infinity of steady-state solutions in which the velocity is piecewise linear, the shear stress is constant and the other stress components are characterized by jump discontinuities. The stability of the steady-state solutions is investigated numerically. In agreement with linear stability analysis, it is shown that steady-state solutions are unstable only if the slope of a linear velocity segment is in the negative-slope regime of the shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to a stable steady state. The number of the discontinuity points and the final value of the shear stress depend on the initial perturbation. No regimes of self-sustained oscillations have been found.     

Dynamic anti-plane sliding of dissimilar anisotropic linear elastic solids

The stability of steady, dynamic, anti-plane slipping at a planar interface between two dissimilar anisotropic linear elastic solids is studied. The solids are assumed to possess a plane of symmetry normal to the slip direction, so that in-plane displacements and normal stress changes on the slip plane do not occur. Friction at the interface is assumed to follow a rate and state-dependent law with velocity weakening behavior in the steady state. The stability to spatial perturbations of the form exp(ikx₁), where k is the wavenumber and x₁ is the coordinate along the interface is studied. The critical wavenumber magnitude, |k|_cr, above which there is stability and the corresponding phase velocity, c, of the neutrally stable mode are obtained from the stability analysis. Numerical plots showing the dependence of |k|_cr and c on the unperturbed sliding velocity, V_o, are provided for various bi-material combinations of practical interest.     

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

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.