Shakedown and induced microslip of an oscillating frictional contact

УДК 539.62

Shakedown and induced microslip of an oscillating frictional contact

R. Wetter

Berlin University of Technology, Berlin, 10623, Germany

Using the method of reduction of dimensionality, we calculate the microslip motion of a tangentially loaded frictional contact between an elastic sphere and a rigid base. An oscillating rotation of the sphere with a small amplitude leads to a creep motion of the rigid base. Depending on the amplitude and the tangential force, two possible scenarios may occur. For oscillation amplitudes smaller than a critical value, the rigid body shakes down in the sense that the frictional slip ceases after a limited number of rotation cycles. Otherwise, the rigid base starts to slip with a constant mean velocity, which depends on the static displacement and the rotational amplitude.

Frictional contacts play an important role in the generation of solid, releasable and non-releasable connections between technical components. For example, the load capacity of bolted connections [1-3], interference fits [4], and machining fixtures [5] depend on the properties of the tangential contact. In simplified terms, the carrying capacity in the tangential direction is determined by the normal force and the friction coefficient. According to Coulomb&s law, such a connection fails if the applied tangential force exceeds the maximum holding force:

Ft <^Fn. (1)

Static mean forces often predominate in the described applications. Due to vibrations, these forces might be overlapped by oscillating parts, which could lead to fretting fatigue [6, 7] and microslip [8]. This refers to the occurrence of a sliding motion between the two contacting bodies, although the tangential force is far below the maximum holding force of Eq. (1). This effect could lead to an unexpected failure of the connection, since its load capacity will be overestimated, due to the disregard of the structural dynamics response.

The case of a finite tangential slip is referred to as a shakedown process. This means, that a stable equilibrium will be reached after a certain deflection [9, 10].

In the present work we investigate a tangentially loaded contact between a sphere and a plane base in the presence of oscillating rotations in the range of amplitudes covering both the shakedown mode and the stationary creep. We

present both the results of numerical simulations and analytical calculation. For this sake, we use the so-called method of reduction of dimensionality [11-17]. It was shown in [11] that the normal contact between an elastic sphere and an elastic half-plane can be described exactly by a contact with a one-dimensional elastic foundation consisting of a series of independent springs. The reduction method has been exactly proven for contacts of any bodies of revolution with and without adhesion in [15] and was extended to a description of contacts with rough surfaces in [13]. Its applicability was illustrated for frictional contacts with elastomers [16, 17]. Finally it was shown that it is applicable exactly to tangential contacts with friction [14]. The simplicity of the one-dimensional equivalent system allows a complete analytical investigation of the problem to be conducted. An analytical expression for the domain for safe shakedown is identified in Sect. 3, whereas the supercritical case and its associated sliding speed are considered in Sect. 4.

We consider the three-dimensional contact between two smooth surfaces with the relative radius R3D and the effective elastic modulus E . According to the Hertzian contact theory, the relation between the normal load Fn and the indentation depth d reads [12]

Fn = 4 E n E* = * = ™ (2)

© Wetter R., 2012

where G denotes the shear modulus and v, the Poisson&s ratio. By using the method of reduction of dimensionality, it is possible to describe the three-dimensional contact problem using a one-dimensional elastic foundation, as described in Fig. 1.

In this case, the elastic sphere with the radius R1D and the stiffness per length E is pressed against a rigid flat surface. The correlation between the normal force and indentation depth then remains valid if the following condition is satisfied [13]:

R3D = 2 R1D • (3)

The physical background of this relation lies in the proportionality of the stiffness of a three-dimensional contact to the associated contact length instead of the contact surface [13]. This property is valid for both the normal and tangential contact. For being equivalent to the initial three-dimensional system, the normal and tangential stiffness of single springs must be chosen according to the rules

* 2G = E A x =-A x

kt =-A x,

where A x is the distance between adjacent springs [14].

In the framework of the half-space approximation, it is assumed that the main contact region is much smaller than the curvature radius of the sphere. The vertical displacement uz (x) of a spring at the point x within the area of contact is then given by [12]: 2

uz (x) = d--with -a < x < a.

The lateral coordinate x is measured from the center of the contact and a denotes the contact radius [12]:

a = ^J R3D d. (7)

The normal force within a spring at the position x is fz = kn Uz ( x). (8)

As shown in Fig. 1, the rigid base is exposed to a tangential force Ft, which is below the maximum holding force of Eq. (1). The resulting displacement of the rigid base U

causes the springs within the contact region to exhibit a spring-deflection ux in the x-direction. Considering the tangential stiffness kt, the tangential force within a spring at position x yields:

fx = kUx (x). (9)

The resulting displacements will be calculated through a stepwise scheme. In this, Uand ux are increased through small steps Au until the sum of the tangential spring forces corresponds to the tangential force Ft. In each step it is tested whether the tangential force at the springs within the contact region reaches or exceeds the corresponding maximum holding force. If the maximum holding force is not reached, the spring sticks. Otherwise, the tangential displacement is adjusted according to Coulomb&s law. Thus, the balance of forces yields the new displacements of the springs:

stick region: fx < f ^ ux (t + At) = ux (t) + Au,

slip region: fx > f ^ ux (t + At) =jw—uz (t),

rigid body: U (t + At) = U (t) + Au.

Since the position of the upper elastic sphere is kept constant, the tangential motion of the rigid body does not influence the vertical displacements of the spring attachments. Thus, there exists no coupling in the sense that tangential slip in the contact area has no effect on the normal component of the contact reactions. This refers to the no normal-tangential coupling case of Klarbring et al. [9], which is one important condition for a shakedown behavior. Furthermore, the inertial forces of the contacting bodies are not considered. Hence, the problem is quasi-static.

The system described above is additionally exposed to an oscillating rotation of the upper elastic sphere. This rotation is realized by a triangular wave with the period T and the amplitude cp as shown in Fig. 2.

The rotary motion corresponds to a rolling wheel and is again computed through a stepwise approach. Initially, the head of the most deflected spring P is selected as the instantaneous center of rotation:

xP = x(max(uz)). (10)

Hence, the new displacement uz (t + At) of a spring is given

//)///?? M ; ?///h/s???;;/////y/ u

Fig. 1. Three-dimensional contact (a) and the equivalent one-dimensional model (b)

♦ Rigid

////// /////////////;;

|a = 0 u

Fig. 2. Oscillating rotation of the elastic sphere (a) and rotational angle versus time (b)

as a function of its distance from the point P and the rotational step Ap:

uz (t + At) =uz (t) + (x-xP) Acp. (11)

In this way, the elastic sphere rolls on the rigid base, i.e., P is changing, and in each step, the new vertical and tangential displacements are calculated as described above.

For the purpose of limitation of the number of parameters, it is convenient to introduce a non-dimensional representation. According to this, the variables are replaced by dimensionless counterparts and related characteristic variables:

Ft = Ft0/t, U = U0u, cp = p00. (12)

The maximum holding force without rotation [12] is chosen to be the characteristic tangential force:

Fto =MF =^r3D2d3/2 ^

•Ft = uRD d 32 /f

The according maximum displacement before complete slippage occurs [12] serves as the characteristic lateral displacement:

TT K J 2-v ,

U0 d = -X d ^

kt 2 (1 -v) 2-v

^ U = u—-T du. (14)

It is also useful to choose the ratio of the indentation depth and the contact radius to be the characteristic rotational amplitude:

d-d. p0 =— ^ cp = -0. (15)

Thus, the case 0 > 2 describes the situation where the most deflected spring of the sphere is rotated beyond the original area of contact. Additionally, the time is normalized by the period:

t = tT. (16)

According to Klarbring et al., a system has reached a state of shakedown at some time t0 if no further frictional slip occurs for times t> t0 and all springs remain in a state of stick. The necessary condition for this to occur is the existence of a time-independent displacement called shakedown displacement such that the springs maximum holding

force is not exceeded at any time. Klarbring et al. state that the existence of such a displacement provides a sufficient condition for the system to shake down if there exists no normal-tangential coupling [9, 10]. Their theorem places no restriction on the number of load cycles or the time required to achieve shakedown. It also allows the possibility that the final displacement of the system might differ from the theoretical shakedown displacement.

As there exist no normal-tangential coupling in the system described above, the only determining factor for whether or not a shake-down will occur is the existence of such a theoretical shake-down displacement. For our system, this is the case as long as the amplitude of the rotational angle does not exceed a certain value. If so, the reciprocating motion of the sphere initially leads to an additional displacement of the rigid body. However, after a limited number of rotation periods, a stable equilibrium is reached, from which a fixed displacement will not be exceeded. This shakedown displacement is a function of the tangential force as well as the rotational amplitude, as shown in Figs. 3 and 4.

It can be seen that an amplification of the tangential force or the rotational amplitude leads to a higher displacement of the rigid body. Additionally, it takes more periods to reach the shakedown displacement.

As the contact region remains constant after a certain number of rotational periods, we can refer to the shakedown concept stated by Klarbring et al., since their work is restricted to contact problems in which the contact area is known a priori and no separation occurs as a result of the external forces [10]. Additionally, we can determine analytical expressions for the relation between the different limiting values. In case of a safe shakedown, there remain three different constant radii, regardless of the current rotational motion, as depicted in Fig. 5.

The radii b and c can be derived from simple kinematic relations. The slip radius b delimits the region where the springs are not deflected horizontally. This is caused by the periodic detachment of the springs due to the rotation and leads the radius b to be a function of the rotational amplitude Cp:

b = a -1R3DCp. (17)

The stick radius c delimits the region where the horizontal deflection of the springs is exactly equal to the rigid

Fig. 3. Rigid body displacement u in the case of a shakedown with a fixed tangential force ft = 0.4

Fig. 4. Rigid body displacement u in the case of a shakedown with a fixed rotational amplitude 0 = 0.4

body displacement U. In the region c < x < b, the springs slide during the beginning of the shakedown process. After several few rotation periods, a fixed displacement is reached. According to Eq. (6), the horizontal displacement in this area is a function of the rotational amplitude:

Ux, c-b = I~T U kt

x +1 R3DÎ>

d( x +1/2 R3D p)2

At the position x = c, this displacement must match the rigid body displacement:

d (c +1/2 R3D p)

from which the stick radius c can be derived:

- 2 R3D p

d Ft = ux d x t Ax x

over the entire contact region -a < x < a:

Ft = J A. ux (x)dx. t _ A x

Taking into account the different contact radii and the symmetry yields

Ft = 2^4G_judx+2^4G_|

d ( x +12 R3D cp)2

dx + 2

Inserting the norms defined in Eqs. (13)-(15) into the expression for the tangential force (22) yields

ft = 1 - -u 0-(1 - u )32.

Equation (23) specifies the relation between the tangential force, the rigid-body displacement and the rotational amplitude in the case of a safe shakedown. This makes it possible to predict the tangential displacement for known tangential forces and rotational amplitudes. According to Eqs. (13) and (14), the dimensionless tangential force as well as the dimensionless tangential displacement of the rigid body can achieve a maximum value of 1. The maximum of the rotational amplitude is 0 = 2. In this case complete slippage occurs, even if the tangential force vanishes. Altogether, the range of the three dimensionless variables is

Without any rotation of the elastic sphere (0 = 0), we get the well-known relationship between the tangential force and the tangential displacement in a non-dimensional form [12]:

ft = 1 - (1 - u )32. (25)

In this case, the maximum rigid body displacement u = 1 is only achieved if the tangential force corresponds to the

//h ? ? W A? } ) ? ? ? ? ¿y N ? J ? h /

-a-b-c c b a

-a - b -c

z, uz, Ux b

"N . x

Fig. 5. Three different contact radii after a safe shakedown (a), the dotted curve shows the deflection in the z-direction uz, whereas the solid line shows the displacement of the springs in the x-direction ux (b)

Fig. 6. Dimensionless tangential rigid body displacement as a function of the tangential force for different rotational amplitudes in the case of a safe shakedown

maximum holding power /t = 1. Figure 6 shows the rigid body displacement as a function of the tangential force for different rotational amplitudes.

The displacement u corresponds to the time-independent shakedown displacement defined by Klarbring et al. [9, 10]. If the actual rigid body deflection reaches this certain value, no further frictional slip occurs and all springs remain in a state of stick. As Klarbring et al. place no restriction on the number of load cycles or the time required to achieve the equilibrium, it is also possible that the final displacement of the system might fall below the theoretical shakedown displacement. In our system this would occur if the oscillating rotation ends before the number of rotational periods necessary to achieve the shakedown displacement was reached.

The different displacements in Fig. 6 all have their own certain maximums, shown by the dotted u lim -curve. This line indicates which maximum displacement can be achieved before complete slippage occurs. In the limiting case, all of the springs in the contact region are slipping and the stick radius c, described in Eq. (20), is exactly zero:

This yields a relation between the dimensionless maximum possible rigid body displacement ulim and the associated dimensionless rotational amplitude 0lim :

Rotational amplitude < Fig. 8. Tangential force limit vs. rotational amplitude for shakedown

^lim = 2V! - Ulim • (27)

Inserting (27) into (23) gives the relation between the tangential force limit and the maximum displacement ulim, the ulim - curve shown in Fig. 6:

ft,lim ~ 1 2 U lim

y&2+i(i lim

Alternatively, the tangential force and the displacement in the limiting case can be formulated as a function of the corresponding rotational amplitude as depicted in Figs. 7 and 8:

=1 - 4 ^

and ft,lim = 1 - 4 ^lim +16 ^lim- (29)

If the maximum displacement is reached, a slight increase in the tangential force or the rotational amplitude leads to an induced rigid body motion. If this is the case, the theorem of Klarbring et al. does not hold for our system anymore, as there exists no time-independent shakedown displacement. The rigid body displacement is, therefore, not restricted to the shakedown value, even if our system is not from a normal-tangential coupling type.

Once the relations formulated in Eqs. (27) and (28) are not satisfied in the sense that the maximum rotational amplitude or the maximum tangential force are exceeded, the reciprocating rotation of the sphere causes an increasing,

Fig. 7. Maximum rigid body displacement vs. rotational amplitude for shakedown

Fig. 9. Rigid body displacement for complete slippage with a fixed tangential force / = 0.4

Fig. 10. Rigid body displacement for complete slippage with a fixed rotational amplitude 0 = 1.3

Fig. 12. Coefficient k vs. static displacement computed through numerical experiments and the approximation K(ustat) = 2.39u1{2t

in principle unlimited, displacement of the rigid base, as can be seen in Figs. 9 and 10.

In this case, the first few rotational cycles cause a loosening of the springs at the boundary regions. Therefore, the slip region grows and after a certain number of periods, none of the springs stick to the rigid base anymore. Now, any additional rigid-body displacement no longer increases the force of the springs. Consequently, the rigid base begins to slide with a constant mean velocity, which depends on both the rotational amplitude and the tangential force, as depicted in Figs. 9 and 10.

A higher rotational amplitude leads to a higher rigid body deflection per period. If the tangential force is increased, the springs are deflected more during a rotational cycle. Again this leads to a higher rigid body displacement and velocity. Using the non-dimensional representation stated in Eqs. (14) and (15) as well as the static displacement of the rigid body (the displacement without any rotation of the sphere) as a measure for the tangential force, we can write the dimensionless mean velocity:

In the case of a fixed static displacement, the mean velocity is proportional to the difference between the rotational amplitude 0 and rotational amplitude limit 0lim before complete slippage occurs, as shown in Fig. 11:

^ 0ustat=const =K(0-0iim). (31)

A0 " 0.0053 .............. A0 = 0.086 ----A0 = 0.18 -----A<j) = 0.26

..........................-J"*.....

^ ..J................

Fig. 11. Rigid body displacement vs. time for a fixed static displacement ustat = 037 and varying A0 = 0-0iim

Numerical experiments using the stepwise scheme and linear regression show that the constant of proportionality k is an unknown function of the static displacement, as shown in Fig. 12.

We approximate this function as

K(ustat) = 2 •39ustat, (32)

which yields the dimensionless mean velocity:

For static displacements in the region of 0.2 < ustat < < 0.8, this analytical expression differs ±5 % from the results computed with the previously described stepwise scheme.

Two different scenarios are possible for the tangentially loaded frictional contact under the influence of an oscillating rotation. If the rotational amplitude does not exceed the limiting value, the rigid body will shake down in the sense that the frictional slip ceases after the first few rotational cycles. If the limiting value is exceeded, the rigid body starts to slip with a constant mean velocity. This velocity depends on the static displacement and the difference of the rotational amplitude and the maximum amplitude.

The author would like to thank V.L. Popov for the idea for this work, many valuable discussions, and for a critical review of the manuscript.

References

Shrink-Fit Holding Torque Using Probabilistic, Micromechanical and Experimental Approaches // Proc. Inst. Mech. Engineers, Part B: J. Eng. Manufacture. - 2004. - V. 218. - P. 175-187.

Experimental study of n non-linear effects in a typical shear lap joint configuration // J. Sound Vibration. - 2004. - V. 277. - P. 327-351.

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Wetter Robbin, Ing., Berlin University of Technology, Germany, r.wetter@tu-berlin.de