Energy Loss in Frictional Hertzian Contact Subjected to Two-Dimensional Cyclic Loadings

Abstract

We investigate the effect of three different harmonically varying loads as a function of the friction coefficient on energy loss in a three-dimensional discrete uncoupled frictional contact problem. Three loading cases include (1) a normal force is constant and a tangential force varies, (2) normal and tangential forces both vary, but the loading and unloading curves are identical, and (3) normal and tangential forces both vary, but the loading and unloading curves are different. For a higher coefficient of friction, three loading cases show different characteristics. If a normal force is constant and a tangential force varies, there is always some slip, but dissipation tends asymptotically to zero at large coefficient of friction. If normal and tangential forces both vary, but the loading and unloading curves are identical, there is no slip and no dissipation above a critical coefficient of friction. If the loading and unloading curves are different, dissipation occurs for all values of the coefficient of friction, and we expect that the dissipation is asymptotic to the relaxation damping value as the coefficient of friction approaches infinity. For lowering coefficient of friction, the three loading cases show similar behavior. Dissipation increases and reaches a maximum just before a state where gross slip is possible.

1. Introduction

If elastic bodies in engineering systems are subjected to both normal and tangential forces, the frictional contact area consists of $stick$ regions where there is no relative velocity and $microslip$ regions where there is a relative velocity. Cattaneo [1] and Mindlin [2] independently solved a frictional Hertzian contact problem, which is loaded by a constant normal force and then by a monotonically increasing tangential force. By finding the traction distribution theoretically, they showed that a microslip region develops around the edge of the contact area, while a central stick region decreases. Mindlin and Deresiewicz [3] extended the solution to consider cases in which both normal and tangential forces change with time. Ciavarella [4] and Jäger [5] independently have shown that the exact solution could be found by the Cattaneo–Mindlin superposition method for any uncoupled two-dimensional contact between two half-planes. Barber et al. [6] extended the Ciavarella–Jäger superposition to find evolution of the traction distribution for both normal and tangential loadings that vary periodically in time. They showed that there is no slip during any period of the loading cycle when the normal force is increasing and when the stick condition between the increment of the normal and tangential forces is satisfied. Otherwise, frictional energy dissipation due to a microslip region can occur, which results in the initiation of fretting fatigue cracks on the contact surface [7].

The energy dissipation due to the microslip also influences the structural damping of the engineering systems at the contact interface. Therefore, the frictional contact is used to reduce the peak response of the system by designing the contacting pairs to have an optimum friction constraint. Griffin and Meng [8] addressed how the choice of an optimum friction force and the performance of the damper with two-dimensional loading are affected in a dynamic system. They showed that when the damper is optimized, the amplitude of response in the circular motion is lower by 36 percent than in linear motion. In the quasi-static frictional system, previous studies focused on the effect of the phase between normal and tangential loads on friction dissipation. Jang and Barber [9] studied the effect of the phase in a simple frictional system subjected to sinusoidally changing normal and tangential loads on the energy dissipation. They showed that the out-of-phase loading makes the frictional energy dissipation reach its maximum. Putignano et al. [10] derived expressions for the energy dissipation per loading cycle in contact of nominally flat rough surfaces under harmonically varying loads and showed that the energy dissipation becomes larger when the loads are out of phase. Davis et al. [11] presented the global dissipation during one loading cycle, which provides information about damping when the varying normal and tangential loads are out of phase.

The previous research [3] showed that if the loading and unloading trajectory are the same and the coefficient of friction is above a critical value, energy dissipation is not possible. However, if the loading and unloading paths are different, the energy dissipation is always possible. This suggests that the loading trajectory as a function of the coefficient of friction has a significant influence on the frictional energy dissipation, and hence on the effective damping of the contact structure. This effect has not been studied so far, and its magnitude is not quantified. Therefore, we will explore the effect of the coefficient of friction on the energy dissipation for various two-dimensional cyclic loadings and investigate the characteristic of effective damping for each loading. For this purpose, we will devise a general three-dimensional finite element model to track the frictional microslip behavior and calculate the corresponding energy dissipation.

2. Finite Element Model for the Frictional Contact Problem

We consider a general three-dimensional frictional Hertz contact problem under a two-dimensional periodic loading. The Hertz contact problem consists of two identical elastic spheres pressed into each other, as shown in Figure 1. We restrict the problem using the Dundurs constant $\beta =0$, which is equivalently defined as

$${\displaystyle \frac{(1-{\nu }_1)}{{\mu }_1}}={\displaystyle \frac{(1-{\nu }_2)}{{\mu }_2}},$$

(1)

where ${\nu }_k,{\mu }_k$ and ($k=1,2$) represent Poisson’s ratio and the shear modulus for the materials of bodies 1 and 2, respectively [12]. Therefore, the contact problem is uncoupled, and the normal contact problem cannot be influence by the tangential loading. In particular, the normal indentation $v\left(P\right)$ and the extent of the contact area $\mathcal{A}\left(P\right)$ are uniquely determined by the increasing normal force of P only [13].

In this study, ABAQUS/Standard [14] is used to track the frictional contact behavior numerically. We first set up an MPC (Multi-Point Constraint) BEAM to constrain the motion of the top and bottom surface region of each half-sphere to the motion of a rigid body reference point. The motion of the bottom reference point is constrained to be fixed, and the motion of the other upper reference point is only allowed to move in three translational directions. The upper reference point is used to apply the external loading to avoid the stress concentration. The following periodic normal and tangential forces are loaded at the upper reference point,

$$P\left(t\right)=P_0+P_1\cos \left(wt\right),$$

(2)

$$Q\left(t\right)=Q_0+Q_1\cos (wt-\varphi ),$$

(3)

where t is time, and w is frequency. We assume that the loading rate is sufficiently slow for the acceleration terms to be negligible. Static stress analysis is used based on the quasi-static assumption, and a period is assigned to the static analysis to represent the variation of external loads with the amplitude option in ABAQUS/Standard. The frictional contact between the contacting surfaces is governed by the Coulomb friction law, which is implemented by a finite sliding contact model in the tangential direction and a hard contact model in the normal direction, respectively.

Mesh Convergence

We consider that the contact area is relatively small compared to the radius of the spheres. The relation between the semi-contact width a and normal load P is expressed as

$$a\left(P\right)={\left({\displaystyle \frac{3P{R}_e}{4E^{*}}}\right)}^{1/3},$$

(4)

where $R_e$ is the effective radius between the spheres, and $E^{*}$ is the composite elastic contact modulus [15]. The half-sphere is modeled with C3D8 linear 8-node hex elements. The meshes near the expected contact region predicted by Equation (4) are refined, and the other region is coarsely meshed to reduce the computational time. The influence of the mesh density on the contact area is investigated to ensure that the contact pressure and contact area are adequate. After the mesh sizes are refined, the corresponding contact pressure and contact areas are compared with the exact solution. The constant normal load P is applied to the upper sphere with the bottom sphere fixed, and contact is assumed to be smooth (no friction). Figure 2 shows the normalized pressure values concerning the normalized position, and the contact area is almost the same as the the exact solution. The numerical solution at the center of the contact area shows a slightly higher pressure value (4.5 %) than the exact solution, and it is less accurate near the contact edge where the contact pressure shows a strong gradient. The number of total refined meshes is 210,472 elements, and twenty contacting nodes are located inside the contact semi-width.

3. Results

The sphere in this example has a radius of 50 mm and is elastic, with a Young’s modulus of 1 MPa and a $Poisson$’s ratio of 0.3. The tangential mean value in Equation (3) has no influence on the energy loss, so it is set to zero, and Equations (2) and (3) become

$$P\left(t\right)=P_0+P_1\cos \left(wt\right),$$

(5)

$$Q\left(t\right)=Q_1\cos (wt-\varphi ).$$

(6)

Three different loading scenarios concerning the coefficient of friction f were studied by assigning the parameters as $P_0=3$, $P_1=-1$, and $Q_1=-1$. We assumed that the path for each loading satisfies the condition that $-fP\left(t\right)<Q\left(t\right)<fP\left(t\right)$ so that gross slip would not occur. Also, the maximum step time was limited so that the loading per cycle was input by eighty discretization points for a period, which was enough to identify contact status for external forces. A static analysis algorithm in ABAQUS/Standard was performed with an HP-Z8 workstation, and each case took around ten hours.

3.1. A Periodic Tangential Force with a Constant Normal Force

We first considered a Hertzian contact problem where a constant normal force $P(=P_0)$ was loaded and then held constant, while a tangential force $Q\left(t\right)$ was varied sinusoidally with the period for one complete cycle, where $T(=2\pi /w)$. Since the problem is uncoupled, the contact semi-width a was determined by the constant normal load $P_0$ and remained constant, even though the tangential loading varied. But, slip regions appear in an annulus of the contact edge, and the stick regions at the center shrink with radius c. The evolution of semi-contact traction with the coefficient of friction $f=0.35$ for one complete cycle is illustrated in Figure 3. The second cycle was used to exclude the effect of the initial loading trajectory, which had faded out after the first cycle. As the tangential loading increased, all contact areas were stuck, and a backward microslip region began at the edge of the circular contact area, as shown in Figure 3a. As the tangential loading increased to the extreme point, $Q\left(t\right)=Q_1$, as shown in Figure 3b, the microslip region became the maximum extent. The inside boundary in a slip annulus represents the permanent stuck region c. As the tangential loading was about to change direction and decrease, all contact areas instantaneously were stuck, as shown in Figure 3c. As the magnitude of the tangential loading approached zero, a reverse forward slip region developed at the edge of the contact area, as shown in Figure 3d. Passing through the developing microslip region, as shown in Figure 3e, again the maximum extent reached the opposite extreme point, $Q\left(t\right)=-Q_1$, as shown in Figure 3f, in which the same extent of the permanent stuck region appeared. As the applied tangential loading changed its direction, all contact areas instantaneously were stuck, and it maintained its state until the reverse microslip region develops. Therefore, if the loading cycles repeat, we could predict that the severe damage region would occur in a slip annulus bounded by the maximum slip edge, as in Figure 3b,f.

3.2. Out-of-Phase Loading

We considered a Hertzian contact problem in which both normal and tangential forces ($P_1,Q_1$) varied periodically with a relative phase angle $\varphi$ in Equation (6). In general, the trajectory would be elliptical. In this study, we assumed a special case with $\left|P_1\right|=\left|Q_1\right|$ and $\varphi =\pi /2$ in which the maximum energy dissipation would occur for the given loads. The resulting trajectory became circular, and the loading and unloading paths were different. The development of the traction distribution was evolved by the increments $\Delta P$, $\Delta Q$ as the normal and tangential loads P, Q changed during time increment $\Delta t$. We also assumed that the transient trajectory joined at a point of the periodic cycle, ($P,Q$) = ($P_0+P_1,0$), where the contact area became minimal, because the normal force was the smallest over the entire loading cycle. The evolution of the traction distribution for the complete cycle with the period T and the coefficient of friction $f=0.4$ are illustrated in Figure 4. When the second periodic loading started, the traction distribution shows that the edge of the contact area already belonged to a backward microslip region, and it continued to develop further in the same direction and then reached the maximum extent of slip region, as shown in Figure 4a. As loading proceeded, the entire contact area was instantaneously stuck when the normal force was increasing ($dP/dt>0$), $d\left|Q\right|/dP$ was of magnitude f, as shown in Figure 4b, and there was no energy dissipation. When the normal force increased ($dP/dt>0$) and $d\left|Q\right|/dP$ was greater than f, a reverse microslip region began at the edge of the contact area, as shown in Figure 4c. As the forward microslip region developed, the normal force reached a maximum when ($P,Q$) = ($P_0-P_1,0$), at which point the extent of the contact area was a maximum, as shown in Figure 4d. A forward microslip region continued to develop until the normal force was decreasing ($dP/dt<0$) and the derivative $d\left|Q\right|/dP$ was of magnitude f, as shown in Figure 4e. At this point, a complete stick region appeared everywhere, and a reverse microslip region started to develop from the edges of the contact area, as shown in Figure 4f.

3.3. In-Phase Loading

We considered a Hertzian contact problem in which both varying normal and tangential forces had the same magnitude ($\left|P_1\right|=\left|Q_1\right|$) with a relative phase angle $\varphi =0$ in Equation (6). The normal and tangential components were in phase, and the loading trajectory collapsed in a straight line from an ellipsoidal. During the cycle, therefore, the loading moved back and forth along the straight line. The evolution of the traction distribution per loading cycle with the period, T, and the coefficient of friction $f=0.55$ are illustrated in Figure 5. The loading started at ($P=P_0+P_1,Q=Q_1$), and a backward microslip region began at the edge of the contact area, as shown in Figure 5a, and continued to develop during the loading phase, as shown in Figure 5b, and the maximum extent of the slip zones reached the extreme point ($P=P_0-P_1,Q=-Q_1$), as shown in Figure 5c. As the loading changed its direction, a complete stick instantaneously occurred everywhere inside the contact area, as shown in Figure 5d, so there was no energy dissipation during the period. As the tangential force changed its applied direction and the normal force decreased, a reverse microslip region developed at the contact edge, as shown in Figure 5e, and reached a maximum at the other extreme point ($P=P_0+P_1,Q=Q_1$), as shown in Figure 5f. For the given coefficient of friction f, a microslip region occurred during both the loading and unloading phases only when the condition $Q_1>f{P}_1$ was satisfied.

4. Discussion

4.1. Energy Dissipation Comparison for the Three Different Loadings

The frictional energy dissipation per cycle (W) is normalized in the following dimensionless form:

$$\widehat{W}=W/\left\{{\displaystyle \frac{{P_0}^2}{a}}\left({\displaystyle \frac{2-{\nu }_1}{{\mu }_1}}+{\displaystyle \frac{2-{\nu }_2}{{\mu }_2}}\right)\right\}.$$

(7)

For the three different loadings, the total energy dissipation per cycle as a function of the coefficient of friction f has been compared in Figure 6. In all three cases, a maximum dissipation occurred when we arrived at a state where gross slip (sliding) was possible, and the out-of-phase loading always showed the biggest energy dissipation compared to the other two loadings for all values of f. At a large coefficient of friction, when P and Q both varied, but the loading and unloading curves were identical, energy dissipation did not occur. The numerical solution shows that there is no slip as long as $dQ/dP<f$. Thus, there is a critical f above which there is no dissipation. This limit value of the coefficient of friction can be determined by finding the maximum slope of the given P, Q curve. For the given loading, the critical coefficient of friction f was 1, so there was no energy dissipation above the critical coefficient f. The Cattaneo–Mindlin-type loading always shows some slip, and the energy dissipation is larger than the in-phase loading. However, the numerical solution shows that energy dissipation asymptotically to zero at large coefficient of friction.

4.2. Energy Dissipation for the Varying Normal Load

For the out-of-phase loading in which loading and unloading curves are different, the numerical result shows that energy dissipation occurs for all values of f and even for a large coefficient of friction, as shown in Figure 7.

This is one of extreme cases, since energy losses can be possible in the limit f → ∞. We usually expect no energy dissipation in elastic contact problems because there is no slip. Popov et al. [16] refer to this process as ’relaxation damping’ and showed that a non-dissipative damping could appear if an asymmetric elastic contact with an infinite coefficient of friction is subjected to superimposed periodic normal and tangential forces. They used the dimensionality reduction method (MDR), which applies to asymmetric elastic contact problems. Ahn [17] gave a generalized proof of the relaxation damping with an infinite coefficient of friction and demonstrated the effect of the varying normal load on energy dissipation. He also calculated the amount of dissipation per cycle by comparing the work between loading and unloading, and the results also show that a significant variation in a normal force is required for the relaxation damping to occur. For the given loading, the dissipation will be asymptotic to the limit value represented by the dotted line, as shown in Figure 7, as f → ∞, which is calculated by the formulation in the paper [17]. Ciavarella and Ahn [18] used the crack analogue model to establish an analytic solution for a full stick contact problem with periodic loading both in normal and tangential directions by assuming that the coefficient of friction is arbitrarily large. From the tangential traction in the contact interface, a square-root singularity occurs at the edge of the contact area during the unloading process, and its magnitude depends on the tangential force difference between the loading and unloading curve at a constant normal force, as well as the instantaneous value of the contact semi-width. Therefore, we assume that the moving singularity causes infinitely rapid movements even though the contact problem assumes to be quasi-static, which eventually leads to finite energy dissipation. The exception to this is the limiting case where the normal force is constant as in the Mindlin’s problem. Therefore, energy dissipation of the Mindlin solution asymptotically goes to zero in the limit.

We expect that these findings could be used for beneficial purposes in numerous industrial applications such as friction dampers in turbomachinery, which are used to increase energy dissipation and reduce vibratory response. Coating the surface of the contacting components with different coefficient of friction can effectively change the structural damping to prevent potential risks. Therefore, the findings can be used to design friction dampers subjected to complex loading conditions.

5. Conclusions

This investigation shows that frictional energy dissipation is significantly affected by the three different loading cases: (1) a normal force is constant and a tangential force varies, (2) normal and tangential forces both vary, but the loading and unloading curves are identical, and (3) normal and tangential forces both vary, but the loading and unloading curves are different. For lowering the coefficient of friction, three loading cases showed similar behavior, and the energy dissipation in friction was greatest when we arrived at a state where gross slip was possible. For a higher coefficient of friction, the three loading cases showed different characteristics. When a normal force was constant and a tangential force varied, there was always some slip, but dissipation tended asymptotically to zero at a large coefficient of friction. When the normal and tangential forces both varied, but the loading and unloading curves were identical, there was no slip and no dissipation above a critical coefficient of friction. When the loading and unloading curves were different, dissipation occurred for all values of the coefficient of friction, and we expect that the dissipation is asymptotic to the relaxation damping value as the coefficient of friction approaches to infinity.

In this study, the energy dissipation per cycle as a function of the friction coefficient has been quantified and shown to have distinct characteristics depending on harmonically varying loads. Especially, the out-of-phase loading could be useful in designing a friction damper, since it shows energy dissipation even at a large coefficient of friction. The present research only considers the energy dissipation due to a microslip. As a future work, we need to extend the numerical method to consider a gross slip, as well as a microslip, so that the maximum energy dissipation could be found over whole range of the coefficient of friction.