A solution of transient rolling contact with velocity

Comments on “A solution of transient rolling contact with velocity dependent friction by the explicit finite element method” Dr.ir. Edwin Vollebregta a

VORtech, P.O. Box 260, 2600 AG Delft, The Netherlands

Abstract Purpose. This paper discusses the reliability of the method presented by Zhao & Li for solving wheel/rail rolling contact with velocity dependent friction. Approach. It is shown that their results are inconsistent with the model used, and that similar inconsistencies can be reproduced using a simple model for a 1D slider. Findings. As a result, it is concluded that the penalty approach with explicit finite elements isn’t suited for dealing with velocity dependent friction in the current configuration. Value. These findings allow to steer research to the development of algorithms that are better suited to the phenomena of concern. Keywords: rolling contact mechanics, wheel/rail contact, velocity dependent friction, stability, explicit finite element method 1. Introduction In 2016, the authors Zhao and Li presented their research on rolling contact with velocitydependent friction (Zhao and Li, 2016). Based on their detailed finite element calculations they conclude that velocity-dependence causes high-frequency oscillations, i.e. rapid fluctuations in the micro-slip velocity and associated slip area. We challenge these findings, and attribute them to numerical artefacts of the computational approach. 2. Background Dry friction between solid bodies is often characterized using Coulomb’s law, postulating a coefficient of friction: F f ≤ µFn (1) ∗

corresponding author; tel. +31-15-285 0128. Email address: [email protected] (Dr.ir. Edwin Vollebregt)

Preprint submitted to Engineering Computations

November 26, 2017

This says that the force of friction F f is bound by a maximum value, that this maximum is proportial to the normal force Fn , acting in the direction perpendicular to the surfaces, and introduces an empirical coefficient of friction µ as the constant of proportionality. Further, the bodies stick (adhere) to each other as long as F f < µFn , and when sliding occurs (F f = µFn ), the friction force opposes this motion. It is generally accepted that this doesn’t represent the true physics in all its detail, but that it works well in an aggregate sense. Within the context of dry, Coulomb friction, there have been many works on the variation of the coefficient of friction µ with time, pressure, sliding velocity, etc. (Dieterich, 1979; Horowitz and Ruina, 1989; Popov et al., 2002). This has led, among others, to the framework of rate- and state-dependent friction (Favreau et al., 1999; Ruina, 1983). This framework has been used mainly for the study of sliding contact situations (Ionescu and Paumier, 1996; Paumier and Renard, 2003), and has recently been introduced in rolling contact situations (Vollebregt, 2014; Zhao and Li, 2016). The distinguishing feature of rolling contact, compared to sliding, is that different parts of the surfaces are in contact at different moments in time, whereas in sliding, all contacting particles more or less undergo the same motion. This introduces the creepage phenomenon in rolling, where one body (railway wheel) may slowly creep over the other (rail) even though there’s no full sliding. This leads to a seeming contradiction with Coulomb’s law of friction (1), allowing for relative motion between the two bodies well before F f reaches µFn . The creep phenomenon is understood by acknowledging that elastic deformations in the contacting bodies may be of the same order of magnitude as the overall motion. The global view of Equation (1) is therefore replaced by a local view where Coulomb’s law is applied to each particle separately: p f ≤ µpn , p f = p f (x) = |pτ (x)|, x ∈ C (2) with C the contact area, pn the normal pressure, and pτ the frictional shear stresses or tractions acting between the two bodies. Now, different regimes may occur in the contact area at the same time. The contact area is thus divided into a slip area S where sliding occurs, and an adhesion (sticking) area H with no sliding. This is captured in the contact conditions as presented in Equations (6)–(7) below. This view has been the cornerstone of rolling contact modelling since the famous work of Carter (Carter, 1926) as discussed for instance by (Johnson, 1985; Kalker, 1990; Knothe, 2008). In railways, measuring the creep force as function of creep velocity, it is often found that the force doesn’t saturate at a maximum value as expressed by Equation (1), but that the force decreases as the creepage is increased further (Polach, 2005; Vollebregt, 2014). This “falling friction” behavior has received a lot of attention of different researchers, among others because it may help understanding the phenomenon of squeal noise (Monk-Steel et al., 2006; Rudd, 1976; Vollebregt and Schuttelaars, 2012). The problem hasn’t been resolved, and therefore it’s important that this receives further attention. 3. The model used by Zhao and Li The paper by Zhao and Li (Zhao and Li, 2016) is the first one presenting the introduction of velocity-dependence in finite elements for rolling contact situations. It builds on the prior construction of the finite element model for wheel-rail contact situations (Zhao and Li, 2011). The model 2

concerns the transient (time-dependent) analysis of a scenario involving solid bodies of homogeneous, linearly elastic materials with straight and circular profiles (Figure 1 in (Zhao and Li, 2016): “a disk on a bar”). The finite element model is developed using ANSYS/LS-DYNA, a commercial package that works on the basis of explicit time-stepping. This provides for the basic equations of elasticity, e.g.

 (a)

ρ(a) u¨ (a) = ∇ · σ(a) + f (a) ,

(3)

σ(a) = C(a) :  (a) , i 1h = ∇u(a) + (∇u(a) )T . 2

(4) (5)

Here a = 1, 2 is the body number, e.g. a = 1 for the wheel (z > 0), a = 2 for the rail. u provides the elastic displacements with respect to a reference state, σ is the stress tensor, and  describes the corresponding linearized strains. f (a) represents body forces like gravity, ρ(a) is the mass density, and C(a) a fourth-order stiffness tensor. Equation (3) is the equation of motion (Newton), equation (4) describes the material behavior according to Hooke’s law, and equation (5) is the strain-displacement relation. Equations (3)–(5) constitute an initial/boundary value problem (IBVP). The equations must hold everywhere in the bodies’ interiors x ∈ Ω(a) ⊂ R3 , for a range of times t ∈ [t0 , t1 ], using suitable initial conditions on t0 and boundary conditions on ∂Ω(a) . The boundary conditions consist of prescribed displacements at the bottom of the rail and the wheel center, the absense of stresses at the free surfaces, and the contact conditions that describe the normal and tangential stresses in the region where contact occurs. For brevity we leave the normal part out of consideration (gap function, pressures: see for instance (Wriggers, 2006)), and concentrate on the frictional, tangential part only: in adhesion area H : |s| = 0, p f ≤ µpn , in slip area S : |s| > 0, p f = µpn , arg s = − arg pτ .

(6) (7)

The tractions p used here stand in direct relation to the stresses σ in the contact surface (Wriggers, 2006). The slip velocity s is obtained as the difference of surface velocities, u˙ (1) − u˙ (2) . The extension by Zhao and Li concerns the addition of one equation to the existing model provided by LS-DYNA: µ = µ(t, x) = µ(s(t, x)) = µD + (µS − µD )e−k|s|

(8)

This says that the actual coefficient of friction µ decreases exponentially from the static coefficient µS at |s| = 0 to a dynamic coefficient µD at |s| = ∞ (µD < µS , k > 0). This is implemented by providing µ to LS-DYNA using a user subroutine, for each node of the contacting surfaces, on the basis of the nodal velocities u˙ of a previous time instance. An interesting aspect of the paper is that it employs the full elastodynamic approach. This stands ¨ are in contrast to the common assumption in rolling contact modelling that the inertial terms (ρu) small with respect to the contact stresses (∇ · σ at the surface) and may be ignored (e.g. (Kalker, 2000; Wang and Knothe, 1989)). In those works, Equation (3) is simplified using ρ = 0, f = 0: ∇ · σ(a) = 0. 3

(9)

Figure 1: Figure 8 of (Zhao and Li, 2016): nodal velocities of nodes at the rail top (left) and in the wheel contact surface (right), for the case with velocity dependent friction. The slip velocity is found as the difference, s = vwheel − vrail .

Figure 2: Figures 7 (a), 6 and 7 (b) of (Zhao and Li, 2016): distinction of slip and adhesion areas at three successive time instants, t − 0.0057, t and t + 0.0057 ms, in the simulation with velocity dependent friction.

This is the so-called elastostatic approach, where Equation (9) is called the equilibrium equation. Introducing velocity dependence could trigger high-frequency responses in principle, increasing the relevance of the inertial terms. Therefore it would be highly interesting to explore the results of the elastodynamic approach, and learn how much difference this makes in this situation. 4. The results by Zhao and Li The main results of (Zhao and Li, 2016) are reproduced here in Figures 1–3. These figures are used to show that (Zhao and Li, 2016): “In the trailing part of the contact patch where micro-slip occurs, very high-frequency oscillations are excited in the tangential plane by the velocity dependent friction. ... Consequently, the micro-slip distribution varies greatly with time. However, the surface shear stress distribution is quite stable at different instants, even though it significantly changes with the employed friction model.” The current paper is written to discuss the validity of these findings. The reason is that the results are in disagreement with Equations (6)–(8), i.e. the contact conditions are violated. Where slip occurs, the shear stress should equal the friction maximum and should be opposing the slip. If the

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Figure 3: Figure 4 of (Zhao and Li, 2016): pressure (p = pn ) and surface shear stress (τ = pτ ) distributions along the longitudinal axis (y = 0.0) for velocity dependent friction (red) and Coulomb friction (blue).

slip velocity then varies greatly in time, the coefficient of friction µ (denoted f by Zhao & Li) will vary greatly also, such that the surface shear stresses pτ (τ) cannot remain stable. The issue is illustrated in Figure 4, that shows one possible realization for the traction bound µpn on top of the results of Figure 3. The blue lines in this figure are obtained assuming that |s| = 2|vrail x |, where vrail is taken from Figure 1, left. The contact conditions are then violated as follows: x • There appears to be considerable slip |s| (µ  µS ) at many places that are considered to be in the adhesion area; • The surface traction p f falls well below the traction bound µpn at many places that are considered to be in the slip area. The distinction between adhesion and slip is placed somewhat arbitrarily at x = 1 mm in this figure, where the ratio p f /pn reaches a plateau value. No other subdivision of the contact area could be found that goes better with Eqs. (6)–(8). The main problem is that the surface tractions p f of Figure 3 are much lower than the traction bound predicted using the nodal velocities of Figure 1. That would suggest that the nodes could be in the adhesion area the whole time, and consequently, that there should be no slip whatsoever. On the other hand, if there is a lot of slipping, as suggested by the nodal velocities of Figure 1, then the red line should be touching the blue one through most of the contact area. An intriguing feature of Figure 4, right, is the plateau of p f /pn in the region −6 ≤ x ≤ 2. This indicates that the height of the red line in Figure 4, left, is about two-thirds (µD /µS = 0.64) of the height of the black line. This suggests that the model effectively restricts µ to µD instead of computing µ according to the slip dependence of Equation (8).

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Figure 4: Inconsistent results from (Zhao and Li, 2016): large slip |s| (µ  µS ) in the adhesion area, shear stress p f remaining below µpn in the slip area, and fluctuating |s| whereas p f remains stable.

5. Discrepancies introduced by the penalty approach We conjecture that the discrepancies in the results are due to the use of explicit time stepping and to the penalty approach used in LS-DYNA, according to Benson and Hallquist’s surface-to-surface contact algorithm (Benson and Hallquist, 1990). This reads, with notations adapted as in the rest of this paper: p∗τ = pnτ − kt ∆t s∗ , p∗f = |p∗τ |, ( ∗ pτ if p∗f ≤ µpn , n+1 pτ = µpn p∗τ /p∗f if p∗f > µpn

(10) (11)

Here p∗τ is a trial value for the frictional shear stress that’s computed using the available stress pnτ of the previous time step n, the relative displacement ∆t s∗ between the two surfaces (for which the precise definition is unclear to us at this stage), times the penalty parameter kt . pn+1 is the frictional τ shear stress at the new time step n + 1 that must be computed. Note that these equations converge to the desired equations (6)–(7) when kt → ∞. On the one hand, p∗τ will tend to infinity where |s∗ | > 0, in the direction opposite to s∗ , such that pn+1 gets the τ ∗ right magnitude and direction from the second part of (11). On the other hand, if |s | = 0 then pn+1 will satisfy the traction bound by the first part of equation (11). However, some discrepancies τ between (10)–(11) and (6)–(7) are built into the algorithm and accepted for practical values of kt . We conjecture that the discrepancies of the penalty approach become unacceptably large when the slip velocity exhibits rapid fluctuations. This could be tested by Zhao and Li using simulations with different values of kt , reporting how much this affects their results. If the results truly reflect the solution of the model formed by Equations (3)–(8) then they will be more or less the same irrespective of the value of kt . If, on the other hand, the results do change with the value of kt , affecting patterns of Figures 1–3, it would imply that the penalty approach isn’t working well for the problem under consideration. 6

Figure 5: Schematic overview of the 1-D test-model for a block on a plane. The plane moves to the left at constant speed vre f . The block is dragged along by friction and pulled to the right by the spring force.

vre f k f ric Fn

−1 6 20

m µS t

0.01 0.5 [0, 5]

k spring µD δt

10 0.32 0.0001

cdamp sadhes x0 , v0

0.04 0.001 0, vre f

Table 1: Parameter values for the 1-D test-model.

6. Results for a 1-D slider The important role of numerical mathematics is illustrated using a 1-D test-model that could almost be analyzed analytically. This concerns a simple slider on a plane with falling friction as illustrated in Figure 5. The plane is a belt moving to the left with velocity vre f . The slider is dragged to the left by friction and held back by a spring. The unknowns are its position x(t) and velocity v(t). The equations used are: = = = = = =

x˙(t), v˙ (t), F spring + F f ric , −cdamp v(t) − k spring x(t), v(t) − vre f , µD + (µS − µD )e−k f ric |s| , ( s if |s| ≤ sadhes −µFn sadhes F f ric (t) = s −µFn |s| if |s| > sadhes

v(t) a(t) ma(t) F spring (t) s(t) µ(t)

(12) (13) (14) (15) (16) (17) (18)

Equation (18) presents a regularized form of Coulomb friction. It uses threshold sadhes to distinguish between adhesion and slip, and replaces the vertical slope of the Coulomb model at s = 0 (Eqs. (6)– (7)) by a steep slope between −sadhes and +sadhes . The parameter values are listed in Table 1.

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6.1. Implicit solver The true behavior of the test-model is explored using an implicit solver on the basis of the backward Euler approach. Equations (12)–(18) are discretized as follows: xn+1 − xn = vn+1 δt vn+1 − vn m = −cdamp vn+1 − k spring xn+1 + F f ric (sn+1 ), δt sn+1 = vn+1 − vre f ,

(19) (20) (21)

|sn+1 |

µn+1 = µ(sn+1 ) = µD + (µS − µD )e−k f ric , n+1 s n+1 F n+1 ) = −µn+1 Fn . f ric = F f ric (s max(sadhes , |sn+1 |)

(22) (23)

For given values at time n, the main unknown is vn+1 at time n + 1:   m + cdamp δt + k spring δt2 vn+1 = mvn − δt k spring xn + δt F f ric (vn+1 − vre f )

(24)

In sliding, F f ric is only weakly dependent on vn+1 , such that simple Picard iteration appears to work well: (0)

v

=v , v n

(i+1)

= f

(i)

mvn − δt k spring xn + δt F f ric (v(i) − vre f ) . = f (v ) = m + cdamp δt + k spring δt2 (i)

(25)

This is repeated until |v(i+1) − v(i) | < . However, this doesn’t work in the adhesion regime, where F f ric is practically linear in sn+1 . This strong dependency is attenuated by adding a linear term based on the static coefficient of friction µS : ! δt µS Fn n+1 2 v = m + cdamp δt + k spring δt + sadhes vn+1 mvn − δt k spring xn + δt F f ric (vn+1 − vre f ) + δt µS Fn . (26) sadhes In this regime, Picard iteration is implemented as (i)

(0)

v

=v , v n

(i+1)

= g(v ) = (i)

v mvn − δt k spring xn + δt F f ric (v(i) − vre f ) + δt µS Fn sadhes

m + cdamp δt + k spring δt2 +

δt µS Fn sadhes

.

(27)

An infinite loop occurs when the iterands v(i) skip over the entire adhesion area from negative to positive slip and vice versa. This is detected and prevented by setting v(i+1) = vre f in such cases, which works well. Slow convergence is only found in certain regions of low sliding. This is accomodated for by allowing many iterations to be performed in order to get an accurate solution to the equations.

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6.2. Explicit solver An explicit time stepping strategy is implemented as follows: n

µn+1 = µ(sn ) = µD + (µS − µD )e−k f ric |s | , sn n n+1 F n+1 = F (s ) = −µ F , f ric n f ric max(sadhes , |sn |) vn+1 − vn = −cdamp vn − k spring xn + F n+1 m f ric , δt xn+1 − xn vn + vn+1 = δt 2 n+1 n+1 s = v − vre f .

(28) (29) (30) (31) (32)

These equations are evaluated from top to bottom, after which the next time step is started. 6.3. Results for the implicit method Results for the implicit method are shown in Figure 6, left, and Figure 7, for the parameter values as listed in Table 1. These results are understood as follows: • The block starts in adhesion with s = 0, µ = µS = 0.5. • Friction keeps the block in adhesion while x(t) > −0.998 = −µ(sadhes )Fn /k spring , with s increasing slowly to s = sadhes = 0.001. • Friction enters the falling regime when s increases above sadhes . Due to the small mass and strong negative slope of µ(s), the block accelerates strongly. • The net force on the block reverses where F f ric = −F spring , after which overshoot occurs because of inertia. Without damping, the force reverses at about x = −0.64 (large s, µ ≈ µD ), and adhesion is regained at twice the jump, at about x = −0.28. After this, the process starts repeating itself. • The motion is restricted by damping, reversing the net force earlier and diminishing the overshoot of the equilibrium position. The stability of the implicit solver is illustrated in Table 2 using the convergence of characteristic values as the time step is made smaller and smaller. Using the values of δt = 1 · 10−6 as the true solution, the errors of larger time steps are found as C · δt (1st order convergence), with C ≈ 4 for xmin , −17 for xmax and −350 for vmax . Table 3 shows the effect of the mass and damping parameters on the characteristics of the block motion. On the one hand, these may be used to get better insight into the system behavior. On the other hand, they underpin the robustness and reliability of the results.

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Figure 6: Left: results for slider using an implicit time stepping algorithm. Right: erroneous results for the explicit solver.

Figure 7: Realized coefficient of friction µ for the slider model using explicit and implicit solvers. For the explicit solver, the results deviate from Equation (17).

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Table 2: Convergence of results for the implicit solver at timestep refinement.

m 0.01

c 0.04

δt 1 · 10−3 1 · 10−4 1 · 10−5 1 · 10−6

xmin −0.99456 −0.99782 −0.99816 −0.99820

xmax , t > 0.7 vmax −0.36352 9.951 −0.34823 10.260 −0.34664 10.292 −0.34648 10.295

Table 3: Effects of mass and damping parameter values on the characteristics of the block motion computed with the implicit solver.

m 0.01 0.001 0.0001 0.01

0.01 0.001 0.0001

c 0.04

δt 1 · 10−4

0.04 1 · 10−4 0.004 0.0004 0.04 1 · 10−4 0.004 0.0004

xmin −0.99782 −0.99391 −0.99380 −0.99782 −1.00141 −1.00177 −0.99782 −0.99754 −0.99780

xmax , t > 0.7 vmax −0.34823 10.260 −0.45659 26.574 −0.61447 53.302 −0.34823 10.260 −0.28746 11.288 −0.28074 11.400 −0.34823 10.260 −0.30947 34.394 −0.30623 109.294

6.4. Results for the explicit method The explicit method of Eqs. (28)–(32) doesn’t reproduce the results correctly, as shown in Figure 6, right, and Figure 7. 1. The collapse occurs too soon, at x = −0.78 instead of −1, and adhesion is recovered too early, at x = −0.53 instead of −0.35; 2. Significant slip is observed in the adhesion regime, with µ ∈ [0.38, 0.5] instead of µ = µS = 0.5 (bottom part of Figure 6, right). 3. The friction dependence of Equation (17) isn’t satisfied precisely by the coefficients of friction µn and slip velocities sn (Figure 7). The mechanisms behind this are illustrated in Figure 8: • There’s quite some slipping at all times, instead of honouring |s| < sadhes while the block is in adhesion; • The block appears to be kicked to the right at isolated time instances, by sudden forces F f ric ≈ 10; By these kicks, the slip velocity increases to about 0.2, by which µ falls to about 0.39; • In the next time step, a negative force is obtained of about F f ric = −7.8 as prescribed by the sliding regime, stronger than the spring force while x > −0.78. This decelerates the block with respect to the plate, increasing the coefficient of friction µ;

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Table 4: Regaining stability of the results for the explicit solver at timestep refinement.

m 0.01

c 0.04

δt 1 · 10−4 5 · 10−5 3 · 10−5 2 · 10−5 1 · 10−5

xmin −0.77922 −0.85854 −0.90435 −0.99825 −0.99823

xmax , t > 0.7 vmax −0.52563 4.006 −0.46069 6.285 −0.42321 7.601 −0.34628 10.299 −0.34637 10.297

• When the velocity v reaches vre f = −1, the slip reverses sign and the process starts repeating itself. The unwanted mechanisms may be suppressed by reducing the time step used as shown in Table 4. The maximum permitted time step appears to be linear in m/k spring as expected from the Euler forward method, for a harmonic oscillator with damping. Further testing has shown that it also depends on the friction curve (k f ric and µS −µD ), the vertical load Fn , and the Coulomb regularization parameter sadhes , that aren’t considered in the common stability analysis. This analysis reveals a potential problem in the penalty approach in combination with explicit time stepping. The problem is introduced by the strong variation of F f ric with small changes in |s|, which turns the adhesion regime into a delicate equilibrium. The penalty approach of Equations (10)–(11) aims for this state using kind of a rough control algorithm: if the slip is positive, try to make the tractions more negative, and in case of negative slip, increase the tractions to the positive side. Depending on the choice of time step and penalty parameter kt , this may overshoot at irregular times and be the cause of artificial fluctuations. 7. Conclusions This paper questions the reliability of the method presented by Zhao and Li for solving wheel/rail rolling contact with velocity dependent friction. The authors reported that “rapid fluctuations” were found in the micro-slip velocity and associated slip area. At the same time, the associated shear stress distribution remained smooth and nearly constant in time. We have shown that these results are in disagreement with the equations used, i.e. the contact conditions are violated. Where slip occurs, the coefficient of friction should be described by the equation for the velocity dependence, but this seems to be largely ignored in the model. The discrepancies in the results by Zhao and Li are conjectured to be due to the use of a penalty approach in combination with explicit time stepping. This is supported by the results for a 1-D testmodel of a slider on a plane with falling friction. “Rapid fluctuations” are found in the results when explicit time stepping is used, that disappear when an auxilliary time step restriction is honoured or when the model is solved using an implicit solution approach. As a consequence, one should be particularly cautious when attempting to resolve falling friction using the explicit penalty method. The method should be tested rigorously with different settings of the penalty parameter kt , in order to detect numerical artefacts, before it can be concluded that “rapid fluctuations are caused by the velocity dependence”. 12

Figure 8: Detailed results for the explicit solver, zooming in on t ∈ [0.78, 0.80].

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