Frictional shakedown and ratcheting of an oscillating cylindrical, elastic contact with Coulomb friction Текст научной статьи по специальности «Физика»

УДК 539.62

Предельное статическое состояние и стационарная циклическая ползучесть в упругом контакте осциллирующего цилиндра

в условиях сухого трения

R. Wetter

Берлинский технический университет, Берлин, 10623, Германия

В работе изучено предельное статическое состояние в упругом контакте цилиндра с плоским основанием. Знакопеременное качение цилиндра с небольшой амплитудой приводит к изменению распределения давления и области контакта. Наряду с касательной нагрузкой, такое колебательное движение инициирует дополнительные процессы скольжения и перемещение тела как целого на макроуровне. При достаточно малой амплитуде колебаний скольжение прекращается после нескольких первых периодов колебаний, тем самым достигается предельное статическое состояние: остаточная сила в контакте успешно противодействует касательной нагрузке. В противоположном случае возникает стационарная циклическая ползучесть: состояния схватывания и проскальзывания поочередно возникают на противоположных сторонах контакта. Это приводит к непрерывному направленному перемещению тела как целого. Найдено распределение касательных напряжений и с использованием гармонических функций Буссинеска-Черрути получены приближения для пределов режима статического предельного состояния по касательной нагрузке и амплитуде колебаний. Это позволяет точно прогнозировать перемещение и понизить уровень касательной нагрузки в предельном статическом состоянии. Результаты находятся в хорошем соответствии с численными и экспериментальными данными.

Ключевые слова: предельный износ, упругий контакт качения, сухое трение, циклическая ползучесть, упругий цилиндр, колебания

Frictional shakedown and ratcheting of an oscillating cylindrical, elastic contact with Coulomb friction

R. Wetter

Technische Universität Berlin, Berlin, 10623, Germany

We examine frictional shakedown of an elastic contact of a cylinder pressed on a flat substrate. Slight oscillatory rolling of the cylinder varies the pressure distribution and the contact region. Together with the tangential load, this rocking motion causes incremental sliding processes and a macroscopic rigid body motion. In case that the oscillation amplitude is sufficiently small, the slip ceases after the first few periods and a safe shakedown occurs: the residual force in the contact withstands the tangential load. Otherwise ratcheting occurs: one side of the contact alternately sticks, while the other slips. This leads to a continuing rigid body motion. By derivation of the tangential stress distribution and use of the Boussinesq and Cerruti potential functions, we find approximations for the shakedown limits for the tangential load and the oscillation amplitude. This allows the accurate prediction of the displacement and the reduced tangential load capacity in the shakedown state. The results show strong agreement with numerical and experimental data.

Keywords: frictional shakedown, elastic rolling contact, Coulomb friction, ratcheting, elastic cylinder, vibrations

1. Introduction

Friction induced connections play an important role in a variety of technical systems, such as screws, bolts and interference fits. The tangential loading capacity of these systems mainly depends on the properties of the tangential contact [1-5]. In case of dry friction the maximal holding force Ftmax is simply determined by Coulomb's law:

Ft, max ~^fn' (1)

where m is the coefficient of friction, and fn is the normal force [6]. If the applied tangential force ft exceeds ft max the contact starts to slip and the contact fails.

In practice, the contact is also affected by external vibrations. For instance, the normal force is cyclic or the pres-

sure distribution varies with constant macroscopic forces [7]. In both cases occurs a stepwise incremental slip of the contact interface, even if the tangential force is insufficient to cause complete sliding, i.e. falls below the maximal force of Eq. (1). This could lead to microslip [8] or fretting [9, 10] of the relevant components.

However, if the residual force in the contact is sufficiently strong, the initial slip stops after a few cycles [11, 12]. Consequently, the entire contact will finally remain in a state of stick, even if the oscillation is continued. Given the similarity to plasticity problems, where shakedown describes a process in which the deformed bodies only show plastic strain in the first few loading cycles and pure elastic

© Wetter R., 2015

response in the steady state, this effect is referred to as fric-tional shakedown. Consequently, the Melan theorem for plastic shakedown [13] was transferred to discrete [14] and continuous systems [15] with Coulomb friction and complete contact, meaning that the contact area does not change during the oscillation.

In the author's recent work [16] the oscillating elastic rolling contact was introduced. This is a model for an incomplete or advancing contact, in which the contact area changes during oscillation. The exact analytical shakedown limits were formulated and it was shown that shakedown is accompanied with a significant decrease of the maximum tangential load capacity. In the present work, we extend the geometry of the model and examine a cylindrical contact configuration. Again we use numerical and experimental analysis to give the shakedown limits and to determine the influence on the load capacity.

2. Models and methods

Firstly, we consider a tangentially loaded contact of a rigid cylindrical body of length L and an elastic half space, as sketched in Fig. 1. We assume dry friction with constant The cylinder is pressed on the substrate by the normal force Fn and is additionally loaded with a tangential force Ft. The system is assumed to be uncoupled, meaning that variations in normal forces will not induce any tangential displacement and vice versa.

As a consequence, the integral equations describing the normal and tangential loading are uncoupled [7]. This requires Dundurs' constant b = 0 as it is the case for friction-less contacts, similar materials, incompressible materials or if one body is rigid and the other one is incompressible [17]. The radius of the cylinder R

R =

1 1

— + —

R

(2)

1 1

as well as the effective elastic moduli of the substrate and G*

\-i

E =

(3)

Fig. 1. Tangentially loaded contact of a rigid cylinder with radius R and length L and elastic halfspace. Cross section shows elliptical pressure p(x) and indentation depth d

are chosen as effective quantities of a contact consisting of two elastic cylinders with particular radii Ri, shear moduli G and Poisson ratios vt. According to the Hertz theory [6, 17], we assume an elliptic pressure distribution in the area of the contact, which is constant in the j-direction:

P( x) = Po

/ \ 2 ^

1 - 1 x 1

a

V v y

1/2

-a < x < a.

(4)

Here p0 denotes the maximum pressure in the center and 2a is the width of the contact area:

Po =-

2 Fn

4 RFn

(5)

naL v nE L With p(x) as in Eq. (4) the indentation depth of the cylinder d reads [18, 19]:

2F„

nE L

^(1 + ln4) + ln -2 a

(6)

The tangential stress was derived independently by Cattaneo [20] and Mindlin [21]. A good description of their procedure is given in [6] or [22]. In case that the tangential force is insufficient to cause complete sliding, i.e. Ft < |Fn, slipping will only occur at the boundary region of the contact area, whereas the center region —c < x < c remains in a state of stick, where the half stick width reads:

c = a

1 --F_

mFn

12

(7)

To maintain this condition, known as incipient sliding, the traction t(x) in the contact must be a superposition of two elliptical traction distributions of the Hertzian type as in Eq. (4). In the complete contact region -a < x < a applies the traction:

T1( x) = mPo

/ / \ 2 \

1 - x

a

v J

V

12

(8)

while in the sticking region —c < x < c the second traction is added:

T2( x) = -TC

c / \ 2 \

1 - x

c

v v ) )

1/2

(9)

Po p(x)

/ 2c ^ \

\ \\

L 2a *

Fig. 2. Tangential contact with pressure p(x) and traction distribution t(x). Incipient sliding: contact width 2a and stick region with width 2c in the center

Fig. 3. Oscillating contact of rigid cylinder and elastic halfspace. Center of the cylinder rolls with amplitude W. Macroscopic forces are constant: rocking of bodies

The correctional term t0 reads

c a

(10)

and is given by the condition that the displacement in the stick region must be constant. Finally, the traction is distributed as sketched in Fig. 2:

t(x) = t1(x) for c < |x| < a, (11)

t(x) = t1(x) + t2(x) for 0 < |x| < c. (12)

However, no exact solution is known for the tangential displacement of the substrate Ustat, one exception being elliptical contacts, where the displacements are expressed in terms of elliptical integrals [23]. Hence, using numerical integration of the potential functions of Boussinesq and Cerruti [22] and regression analysis it is possible to give an approximation for the tangential displacement:

\0.92

U stat = Uo

1 -

i-A

mF,

\\

y J

(13)

Here U0 denotes the maximum displacement which is also an approximation based on the indentation depth of Eq. (6):

2 F

U o

nE L

1

(1 + ln4) + ln

1.8

E_L_

*

G a

w

(14)

J J

This static tangential contact is superposed by a slight oscillatory rolling of the cylinder. Here the amplitude W denotes the lateral movement of the center of the cylinder as depicted in Fig. 3.

The system is assumed to be quasi-static, meaning that we assume a constant m and neglect inertia effects. This is valid as long as the excitation is slower than the propagation speed of elastic waves within the bodies. The overall macroscopic normal and tangential forces will both be kept constant in magnitude. According to this, the problem setting is equivalent to a frictional contact, which is exposed to an oscillatory back and forth movement of the normal force, i.e. a rocking of the contacting bodies. The pure rolling does not lead to any additional friction force or momentum but changes the local pressure and the stick and slip regions.

Fig. 4. Experimental setting: steel cylinder, silicone rubber substrate, weight, drive (PI-M 403-DG) and laser-vibrometer (Polytec OFV-5000)

2.1. Experimental setting

The experimental setting is depicted in Fig. 4. It consists of a cylinder made of ST-52 steel and a silicon-rubber substrate. Thus, the system is almost uncoupled as b ~ 0. Other important parameters are listed in Table 1. For minimization of external influences, the substrate is put on a low friction cross roller table. Its resistance force of Fr = = 0.1 N lowers the actual tangential force, which itself is controlled by a single weight mg that is connected to the substrate through a string. Hence, the tangential force results to Ft = mg - Fr.

The weight of the sphere acts as the normal force Fn and the rolling motion of the sphere is generated by a lever arm construction. Its main bearing is located exactly on the same level as the contact point between sphere and substrate. As this point corresponds to the instantaneous center of motion, the oscillations of the lever arm result in a pure rolling of the sphere and additional influences on the macroscopic load regime are minimized. The back and forth motion of the lever arm is generated by a high-precision linear drive. Finally, the rigid body displacement of the substrate U is measured using a high resolution laser vibrometer.

Table 1

Properties of the experimental setting

Radius R, mm

Length L, mm

Friction coefficient m

Young's modulus Ej E2, MPa

Poisson ratio vJ v 2

Normal force Fn, N

Contact width 2a, mm

Indentation depth d, mm

40

80

0.79

206-103/3.6

0.3/0.5

32.2 N

4.2

0.26

Fig. 5. Numerical model using CONTACT software

2.2. 3D simulation using CONTACT

The well-known CONTACT software package is used to conduct a two dimensional quasi static simulation of the problem [24, 25]. The program uses constant element discretization and nested iteration processes to solve the transient problem of rolling. The geometry is entered using a so called non-Hertzian approach, where the distance between the undeformed surfaces of the two bodies is specified through the quadratic function:

h( x) = —. 2R

(15)

We model the cylinder with length L as a truncated 3D problem. This means that we use just one row of elements and all the contact quantities such as pressure or traction are constant along the y-axis. The problem is thus two dimensional as shown in Fig. 5.

We use a world fixed coordinate system, in which the rolling is simulated as a stepwise incremental shift of the profile. Thus, after n rolling steps the actual profile reads: \2

h( x) =

( x - n AW )2 2 R

(16)

with DW being the step length. We use the following parameters for the simulation:

R = 40 mm, L = 500 mm, Fn / L = 0.39 N/mm, E* = 6.8 N/mm2, G * = 4.5 N/mm2, m = 0.58. (17) The number of elements varies with the amplitude, due to the world fixed approach. With a side length of Dx = = 0.02 mm and an incremental step length of DW = 0.02 mm, we get 220-307 discretization elements. We computed 10 periods of rolling for each combination of tangential loading and amplitude resulting in 881-2641 computation steps for each case.

3. Results and discussion

The parameters of influence Ft and W are normalized with the maximum holding force and the half contact width: ft = Ft/^Fn, " = W/a. (18)

We restrict ourselves to tangential forces below the maximum holding force, and oscillation amplitudes smaller than the half contact width:

ft < 1, w < 1. (19)

In other words, without oscillatory rolling, no complete sliding will occur and the center of the sphere will not be moved beyond the initial area of contact at any time. Thus,

; 0.09

<D

| 0.07

Oh

0.05

4Msd 3

Jr 1

wstat 1

0

4 6 Periods n

10

Fig. 6. Displacement u for different oscillation amplitudes w = = 0.26 (1), 0.52 (2), 0.77 (3) and f = 0.08. Shakedown: displacement stops and system reaches a new equilibrium

taken by itself, neither of the two factors leads to a failure of the contact.

3.1. Influence of the oscillatory rolling

Using the numerical model, we compute the rigid body displacement between remote points within the cylinder and the substrate as the sum of the elastic displacement and the accumulated shift per time step, i.e. the displacement U.

As one expects, the system response corresponds to oscillating contact with spherical rolling body [16]. In combination with the constant macroscopic load, the oscillatory rolling leads to an increased rigid body displacement in relation to the static value, as depicted in Figs. 6 and 7. Here, u is the normalized displacement:

u = U/U0 , (20)

where U0 is the static displacement for ft = 1 as stated in Eq. (14). Figure 6 shows that the displacement stops after a few periods and the contact holds, even if the rolling continues. The system reaches a new equilibrium and the according displacement refers to the constant time independent shakedown displacement u sd [14]. For this effect to occur, ft and w must fall below the shakedown limits. Otherwise the contact fails.

In this case, the accumulated shift per time step leads to a continuing displacement, as depicted in Fig. 7. This effect is referred to as ratcheting or induced microslip [7, 16].

Fig. 7. Displacement u for different oscillation amplitudes w = = 0.26 (1), 0.52 (2), 0.77 (3) and f = 0.58. Ratcheting: displacement continues

Fig. 8. Pressure distributionp(x) and contact region in the reversal points, caused by the oscillatory rolling

3.2. Contact configuration after shakedown

In the beginning of the process the system is in equilibrium and the entire contact sticks. Now, if the cylinder starts rolling, a pressure drop occurs at the actual trailing edge, whereas the pressure is increased at the leading edge. If the cylinder reaches one of the reversal points, i.e. ± W, the actual pressure distribution in the actual contact yields:

P (x) = Po

1-

x m w

a

(21)

with - a ± W < x < a ± W, as depicted in Fig. 8. At the trailing edge the decreasing pressure causes slipping and thus a decreasing tangential stress. In contrast, at the leading edge, the tangential stress initially remains constant. The resulting imbalance in the tangential direction increases the rigid body displacement between cylinder and substrate u within every back and forth movement. However, assuming sufficiently small ft and w, shakedown occurs and a saturation level is reached after a certain number of periods.

In the new equilibrium, there remain three characteristic contact widths, as depicted in Fig. 9. Firstly, in the center region, the displacement must be constant because of the sticking condition (stick region). Secondly, adjacent thereto occurs a region, where the tangential stress equals the traction bound (slip region). Thirdly, at the outside of the contact, the cylinder is periodically released and the tangential traction is zero (zero traction). In summary we get:

Fig. 9. Traction distribution and contact regions (stick region, slip region, zero traction) after shakedown

stick region: 0 < | x | < csd ^ u — const, slip region: csd < | x | < b ^ t = |p, zero traction: b<|x |< a ^t — 0. Here csd denotes the half stick width after shakedown and for b applies b = a - W.

In order to deduce the traction, we start with the slip region. Here t(x) equals the friction coefficient times the pressure, which appears at the reversal points of the oscillation, as stated in Eq. (21):

ti( x) = |Po

1 -

x + W

x2V/2

(22)

with csd < | x | < b. In contrast to the static case, the traction in the center will be from a different type, as this region is neither released nor slipping at any time. However, we proceed in the Cattaneo [20], Mindlin [21] manner and propose that the traction in the stick region 0 < | x | < csd is again a superposition of two Hertzian distributions:

Î , .W/2 ( / x2V/2

T 2 ( x):

L21

1

( x ^

KbJ

22

1

csd

(23)

Each of these tractions causes a parabolic tangential displacement [23]:

Uxsd(x) = Cl x2 + C2 +

bE csdE

22 x2 x,

(24)

which must be constant in accordance to the no slip condition. This yields:

^ (25)

T 22 — T

21"

csd

Further, the traction must be continuous at the edge of the sticking region x = csd. This gives:

/ Y/2 ( , -12

T 21 = IPO

1 -

csd

1 -

csd

(26)

The complete traction t(x) is distributed as shown in Fig. 9:

stick region: t(x) = t2 (x) for 0 < | x | < csd, (27) slip region: t(x) = t (x) for csd < | x | < b. (28)

After shakedown, the integral of the traction over the contact width must match the tangential loading per length:

F b csd

= Jt(x)dx = 2 Jxj(x) dx + 2 Jt2(x)dx. (29)

L s Csd 0

With equations (22), (23), (25) and (26) this gives:

ft — — arccos n

/ \ ( / \ 2 ^

csd 2 + 1- csd

a n a

V J \ v J /

arcsm

csd

csd

1-

12

csd

- w

. (30)

Through numerical solving, Eq. (30) allows the determination of the unknown half stick width csd. Figure 10

Fig. 10. Normalized half stick width c, function of ft for different w

sd

after shakedown as a

Fig. 12. Tangential traction distribution t(x) after shakedown for ft = 0.08 and different w

shows the half stick width as a function of ft for different amplitudes w. For a mutual verification csd is also computed using the CONTACT model. Lines depict the results using Eq. (30), whereas marks depict the solutions gained via simulation. Both solutions show strong agreement. As one expects, the half stick width decreases if ft and w are increased.

With these results, it is now possible to compute the missing traction constants t21 and t22 and thus t(x). Again, we use the simulation for a mutual verification. The traction distribution for different ft is shown in Fig. 11.

Here, the dashed lines show the numerical solution, whereas the analytical expression is denoted by the solid lines and just given for one half of the symmetrical contact. Both solutions show strong agreement. As one can see, the stick region decreases and the maximal traction increases with ft. Higher amplitudes w have the same effect. The stick region decreases, whereas the periodically released area increases, as depicted in Fig. 12. Additionally, the magnitude of the remaining traction is amplified, as the tangential loading is the same for all three cases shown. This refers to the shakedown effect, where the residual force must be sufficiently strong to prevent any further sliding [14].

3.3. Shakedown

With the traction being known, it is possible to compute the tangential displacement in case of shakedown Usd using the potential functions of Boussinesq and Cerruti [22]:

Fig. 11. Tangential traction distribution t(x) after shakedown for w = 0.26 and different ft = 0.22 (1), 0.44 (2), 0.66 (3)

Usd =■

nG

nG

l/2 b

J J

- L2 csd L 2 csd /

J J

- l/2 0

1-

-v v£

7"

-v vc

_+ 3

T1(C, n)d^dn + T2(C, n)dCdn.

where

and

G =

1 1 —+ —

G1 G2

-1

1

v = —

G

v, v

1

(31)

(32)

(33)

However, since the exact solution for the half stick width csd remains unknown, it is not possible to give a closed-form solution for Usd. Instead we use Eq. (30) to compute

csd

for different ft and w. Subsequently, we insert c

sd

into Eq. (31) and perform a numerical integration to get the normalized Usd. Finally we use regression analysis to give an approximation for the normalized displacement usd =

= U sd/ U 0:

092 (34)

usd

= 1 - (1 - ft)0 92 + 0.3ftw.

Fig. 13. Shakedown displacement of the substrate usd as a function of the tangential force ft for different oscillation amplitudes w. The oscillatory rolling increases the displacement by comparison with its static value u^

Figure 13 shows the shakedown displacement u sd for different /t and w. It is increased in comparison to the corresponding static value ustat. Solid lines give the results using Eq. (34) and asterisks give the 3D simulation results. Finally error-bars and marks indicate the experimental results. The bandwidth of the displacements is narrower than for the spherical contact examined in [16].

The dash-dot line shows the maximum displacement ulim that is reached before complete sliding occurs and the contact fails. If so, the stick width csd goes to zero. In combination with Eq. (30) this gives the exact relation between maximum tangential load ft lim and maximum amplitude

wlim:

2

ft,lim = 2 (arCCOs wlim - wlim (1 - WL ^ ). (35)

n

Linear regression analysis again gives approximations for the inverse relation:

Wlim = 1 - ftOlim ' (36)

as well as for the maximum displacement:

ulim = /aim. (37)

The equations (35) and (36) enable one to specify the highest possible amplitude to maintain a safe shakedown for a given tangential force and vice versa. Again, we use the CONTACT model for verification. We start with rather small amplitudes and increase stepwise, until ratcheting occurs. Figure 14 depicts the maximum amplitude as a function of the tangential force. Equation (36) is shown as a solid line. Triangles give the simulation results, where upward pointing ones depict the last shakedown amplitude, and downward pointing ones depict the first ratcheting amplitude. It shows that for the same amplitude, the cylindrical contact could theoretical bear lower tangential forces than the spherical one, which is illustrated by the dash-dotted line [16].

Also experimentally, the maxima were identified by a stepwise increase of w. The according results are shown by asterisks and error-bars. The deviations between Eq. (36)

Maximum tangential force ^ lim

Fig. 15. Maximum displacement

ulim as a function of the tangential force /t lim. Approximation (solid), simulation (marks) and experiment (error-bars)

and the experiments are relatively high. The good agreement with the numerical values indicates a faulty experiment. One explanation for this might be a deviation of the actual pressure distribution from the assumed cylindrical one. A local pressure peak would increase /t,lim in comparison with the theoretical value.

Figure 15 shows the according maximum displacements. Formula (37) (solid line), simulation (triangles) and experiment (asterisks and error-bars) are in good agreement.

3.4. Ratcheting

Once /t and w exceed the shakedown limits stated in Eqs. (35) and (36) the contact fails. In dependence on the actual rolling direction, one side of the contact sticks, while the other slips. This accumulated displacement results in a rigid body motion referred to as ratcheting or walking [7, 16]. Figure 16 gives the experimental values for the incremental displacement per period Du. It shows that Du increases with /t and w. The solid lines depict the approximation function:

au = 0.64/t (w - Wlim), (38)

that has been derived from a linear regression analysis of

g 0.4-_ Equation (36)^0^. %

S ---Sphere ^¿X x _ A Sim-shakedown <|> ^ V Sim-ratcheting ^^ q q ^ Experiment t_|_

' 0.0 0.4 0.8

Maximum tangential force ft lim

Fig. 14. Maximum amplitude wlim as a function of the tangential force ft lim. Approximation (solid), simulation (marks) and experiment (error-bars)

Tangential force ft

Fig. 16. Incremental displacement Du as a function of the tangential force /t for different amplitudes. Approximation (marks) and experiment (error-bars)

numerical data with ustat as regressor. It shows that Du is lower for the cylindrical contact than for the spherical one [16]. The ratcheting effect can be used for the generation and control of small displacements in case that an increase of the tangential loading is not possible or high accuracy is needed as in MEMS devices.

4. Conclusion

As an extension of the model introduced in the author's recent work [16], we examined a quasi-static frictional system that consists of a rigid cylinder of length L and an elastic substrate. Again Coulomb friction with constant m and a steady macroscopic load regime was assumed. The system was from an uncoupled type, meaning that a varying normal force will not induce a displacement in the tangential direction and vice versa.

Slight oscillatory rolling of the cylinder varies the pressure distribution and the contact region. In turn this leads to partial slip and a macroscopic rigid body displacement. As for the rolling sphere, this displacement stops, if oscillation amplitude and tangential force fall below the shakedown limits. Otherwise, the contact fails and the displacement continues as a consequence of the ratcheting effect.

We derived the exact pressure and traction distribution and approximations for the shakedown limits. Again, shakedown is accompanied with a significant decrease of the tangential load capacity. In theory, the load capacity is lower for cylindrical contacts in comparison to spherical ones. In addition, we gave approximations for the rigid body displacement in case of shakedown and the incremental displacement in case of ratcheting. It shows that the bandwidth of shakedown displacement is narrower and the incremental displacement is lower for the cylindrical contacts.

Future research is needed, to examine other rolling bodies as cones or ellipsoids. Additionally, the interaction of oscillating normal forces and the rolling should be considered.

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Поступила в редакцию 26.03.2015 г.

Сведения об авторе

Wetter Robbin, Dipl.-Ing., Technische Universität Berlin, r.wetter@tu-berlin.de