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New insight into the mechanism of static friction: A theoretical prediction of the effect of loading history on static friction force based on the static friction model proposed by Lorenz and Persson
Author’s Accepted Manuscript New insight into the mechanism of static friction: A theoretical prediction of the effect of loading history on static friction force based on the static friction model proposed by Lorenz and Persson Satoru Maegawa, Fumihiro Itoigawa, Takashi Nakamura www.elsevier.com/locate/jtri
PII: DOI: Reference:
S0301-679X(16)30067-6 http://dx.doi.org/10.1016/j.triboint.2016.04.021 JTRI4165
To appear in: Tribiology International Received date: 5 September 2015 Revised date: 14 April 2016 Accepted date: 17 April 2016 Cite this article as: Satoru Maegawa, Fumihiro Itoigawa and Takashi Nakamura, New insight into the mechanism of static friction: A theoretical prediction of the effect of loading history on static friction force based on the static friction model proposed by Lorenz and Persson, Tribiology International, http://dx.doi.org/10.1016/j.triboint.2016.04.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
New insight into the mechanism of static friction: A theoretical prediction of the effect of loading history on static friction force based on the static friction model proposed by Lorenz and Persson Satoru Maegawa1,2*, Fumihiro Itoigawa1, Takashi Nakamura1 1
Department of Mechanical Engineering, Nagoya Institute of Technology,
Gokiso-cho Showa-ku Nagoya Aichi 466-8555, Japan 2
Present Address: Department of Mechanical and Aerospace Engineering, Tottori University,
4-101 Minami Koyama, Tottori 680-8553, Japan
*Corresponding author. Tel. & Fax: +81–857–31–5735
E-mail: [email protected]
Abstract
This study provides a new insight into the field of static friction. Based on the static friction model developed by Lorenz and Persson, in which it is assumed that a slip distance characterizes the level of frictional shear stress acting on the contact interface, we pointed out that tangential loading history is an important factor for determining the value of static friction. The stop-restart motion increased the level of static friction force, while the stop-inversion motion reduced the static friction.
Keywords: Static friction; Frictional shear stress; Loading history; Elastomer
1. Introduction
The mechanism of static friction has been an actively studied topic in the fields of tribology, geophysics, and mechanical engineering [1-3]. Following the pioneering experiments by Fineberg’s group [4, 5], in which temporal and spatial development of precursory local slip were observed in situ, a number of studies were performed to gain a complete understanding of the origin of static friction. As pointed out by David and Fineberg [6], “static friction is not a material constant”, i.e., static friction depends on several system parameters: the stiffness properties of the contact interface [7-9]; the magnitude of normal load [10-12]; the loading configuration of the external forces; and the geometry of the sliding object [13-15]. These results indicate there are a number of methods for tuning static friction force. In mechanical engineering, it is important to fully optimize the system parameters listed above to improve the ability of sliding systems to perform their design functions. Based on a simplified static friction model, Lorenz and Persson derived a general rule in characterizing the level of static friction force [14]. They stated that the static friction force decreases due to the existence of different surface regions that begin to slip at different times during the external loading period. Thus, the initiation process of sequential precursory local slip that occurs prior to the onset of the global slip characterizes the level of static friction force. This consideration explains why static friction depends on the system parameters described above. This study focused on the effect of tangential loading history on the magnitude of static friction. Based on the static friction model developed by Lorenz and Persson [14], we developed a theoretical prediction where the sequence of the external loading affects the magnitude of the static friction force. Sequences of external loadings used for this study were stop-restart and stop-inversion motions.
2. Static friction model by Lorenz and Persson
Lorenz and Persson [14] discussed how static friction depends on the elasticity and geometry of a sliding object using a simplified analysis based on a 1D slab model. The advantage to using this simplified model is that the analysis is effective in finding dominant factors for characterizing phenomena. Additionally, a simplified model allows a theoretical solution to be derived analytically. In our study, we used the Lorenz and Persson model, as illustrated in Figure 1, to discuss the effect of the loading history on static friction force. To facilitate the comparison between their results and results obtained in our paper, identical symbols were used. In this model the frictional shear stress τf acts on the contact interface between the elastic slider and the smooth surface of the rigid substrate. It is assumed τf decreases from τs to τk as the surface displacement u increases in the X direction, as shown in Figure 1(b). When u is smaller than a characteristic distance D, τf = τs – (τs – τk) u/D. In contrast, when u is larger than D, τf = τk. Thus, the amount of the local slip D is needed to drop frictional stress from τs to τk. As discussed in the original paper by Lorenz and Persson [14], the amount of D is typically characterized by the time of interfacial relaxation process (aging), which ranges from nm to mm. For example, thermally activated creep or interdiffusion of polymer chains characterizes D as approximately 1 nm. On the other hand, in the case of plastic creep, D will be under the order
of a few μm that is characterized by surface roughness, elastic modulus and the hardness of the contacting solids. In other cases, D is determined by a surface waviness, or structure in the order of a few mm. It should be noted that the local stress fluctuations resulting from the randomly distributed asperity contacts are averaged over large enough surface areas. Thus, the local distribution of shear and normal stresses are removed. The slider is made of an elastic material with Young’s modulus E. The deformation of the elastic slider is modeled by a 1D model. In addition, any kinetic effects such as the fast propagation of a crack tip were not considered. The crack-like front (at x = L in Figure 1) was assumed to move adiabatically as the loading force increased. Here, it should be noted that L0 is the slider length. L1 and L are the lengths from the left edge of the slider to the front edge of the stick and slip regions where τs and τk act on the contact interface as shown in Figure 1(a), respectively. Additionally, the width of slider in the Y direction is described as Ly, and h is the height of the slider in the Z direction, as illustrated in Figure 1(a). Lorenz and Persson derived the theoretical solution of the maximum value of the nominal (macroscopic) frictional shear stress τmax. This maximum value corresponds to the frictional shear stress required to initiate global slip as follows:
𝜏k −
𝜏k 𝜅𝐿0
𝜏
𝜏s
𝜏𝑠
𝜅𝐿0
acos ( k ) + 𝜏s
𝜏max =
𝜅𝐿0 𝜏s
{
𝜅𝐿0
sin(𝜅𝐿0 ) 𝜏
2 1/2
[1 − ( k ) ] 𝜏s
2 1/2
𝜏
[1 − ( k ) ] 𝜏s
𝜏
for 𝜅𝐿0 > acos ( k ) 𝜏𝑠
𝜏k
for 𝜅𝐿0 < acos ( ) 𝜏𝑠
(1)
𝜏
for 𝜅𝐿0 = acos ( k ) 𝜏𝑠
where,
𝜅= (
𝜏s −𝜏k 1/2 ℎ𝐸𝐷
)
(2)
From equation (1) it is clear that the maximum value of the nominal frictional shear stress τmax is not a material constant. Here, it should be noted that the maximum static friction force Fsmax is described as τmaxA, where A is the area of the apparent contact region; that is, Fsmax = τmaxA = τmaxLyL0. Thus, the Lorenz and Persson model provides a clear solution to the question of how static friction depends on the geometry of the slider, and the elastic properties of the slider. The details of the analysis, results and the methodology used to develop the solution are described in Lorenz and Persson’s [14] original paper.
3. Two types of different loading history
Figure 2 illustrates the two types of different loading sequences that are the focus of this study. Figure 2(a) shows the basic configuration of the sliding model. As in the case of the Lorenz and Persson model, the normal and tangential loads F and FN are applied on the top and side face of the slider. In our model, it is assumed that F is applied via a spring with a stiffness K that moves in the X direction with a driving speed of V. In this analysis, the kinetic effects are not considered. Therefore, the value of K does not affect the conclusions
obtained in this study. The changes in driving speed V and nominal frictional shear stress τ of the stop-restart motion are schematically drawn in Figure 2(b). As illustrated, the loading sequence is divided into three periods, i.e., first loading period, stopping period, and second loading period. In the first loading period, the left edge of the spring moves in the positive X direction at a constant speed of V0. The nominal stress τ increases with time until τ reaches the maximum value τmax, at which time slip occurs over the entire contact region, i.e., global slip. The value of τmax corresponds to the maximum static friction force. Subsequently, τ decreases from τmax to τk as the frictional phase changes from static friction to kinetic friction. The decrease in τ occurs instantaneously since this model does not account for kinetic effects such as crack tip propagation. In this study, it is assumed that during the total slip period, the sliding speed of the contacting surface of the slider is equal in all of the regions. Additionally, it is also assumed that the transition from the slip to re-static friction occurs when the sliding speed reaches zero. Therefore, the transition from the total slip to the total stick at each contact region occurs simultaneously at the moment of the onset of the stopping period, because during the stopping period, the increase in tangential load stopped as the motion of the spring stopped, i.e., V = 0. Furthermore, it should be noted that during the stopping period, tangential and normal loads are continuously applied. In the stopping period, the applied tangential force is equal to the static friction force. Therefore, the internal strain during the total slip period is stored before and after the transition. After a certain waiting time, the increase in tangential loading restarts under a constant speed V0. This study focused on the value of the maximum shear stress at the second loading period, i.e., τmax’. As described in following sections, it is found that τmax’ is larger than τmax. Thus, the existence of the stopping motion of the tangential loading increases the maximum static friction. Figure 2(c) illustrates the changes in V and τ for the stop-inversion motion. First, as with the stop-restart motion, a constant speed V0 is applied; τ increases with time during the first loading period. Subsequently, the increase in the tangential load comes to a standstill during the stopping period. In contrast to the stop-restart scenario, during the second loading period, the moving direction of the spring edge is reversed, i.e., V = –V0. The development of the local slip for the contact interface during the stop-restart motion is illustrated in Figure 3, where times t0 to t6 correspond to those times denoted in Figure 2(b). At the initial condition, the entire region of the contact interface was under a static friction condition as illustrated in Figure 3(a). As the tangential load increased, the slip region extended from the trailing to the leading edge of the slider as shown in Figure 3(b). Before the transition from static to kinetic friction, the leading edge of the elastic block does not slide since the shear stress does not act (or relatively small) on the contact interface at the leading edge. At the moment of the transition, in which the local slip develops over the entire region of the contact interface, global slip occurs. During the total slip condition, i.e., Figure 3(c), the slider deforms due to the tangential load and kinetic friction force; the value of tangential load equals the total value of kinetic friction force over each contact region. Subsequently, during the stopping period, i.e. Figure 3(d), the amount of the slider deformation is stored; thus, ΔL(t2) = ΔL(t3) = ΔL(t4), where ΔL means the deformation amount of the slider as illustrated in Figure 3. The increase in tangential load restarts after the end of the stopping period as illustrated in Figure 2(b). The deformation of the elastic slider at the initial moment of the second loading period is illustrated in Figure 3 (d). In contrast to the first loading period, the internal deformation of the slider is already formed at the initial
moment of the second tangential loading. Thus, the initial value of the stress distribution is nonzero value. The total deformation of ΔL(t4) is not zero. Similar to the first loading period, friction shear stress increases until the initiation of global slip, i.e., the transition from static to kinetic friction, at which point the friction shear stress decreased to τk. Similar to the case of the stop-restart scenario, the stop-and-inversion motion as illustrated in Figure 2(c) has a non-zero deformation (initial stress distribution) of the slider at the initiation of the second loading period. In this study, we theoretically predict that the existence of the inversion motion reduces the nominal value of static friction.
4. Theoretical analysis
4.1. Maximum value of nominal frictional shear stress in stop-restart motion
Figure 4 illustrates the method for obtaining the analytical solution of τmax’, which is the maximum value of the nominal frictional shear stress in the second loading period after the stop-restart motion. The time change of τ during the second loading period is highlighted by the solid dashed circle, as seen in Figure 4(a). This study focuses on that the value of τmax’ can be obtained as the sum of τ1’ and τ2’, i.e., τmax’ = τ1’ + τ2’, where τ1’ and τ2’ means the kinetic and maximum static shear stress illustrated in the time changes of τ, which are shown in the right side of Figure 4(a). Therefore, we can obtain the analytical solution of τmax’ from the calculation of the theoretical solutions of τ1’ and τ2’. Figure 4(b) shows the illustration of the analytical model for obtaining τ1’, in which shear stress τk uniformly acts on the entire region of the contact interface. Thus, the relationships between τf and u, and τ and t are described in the center and right graphs in Figure 4(b), respectively. In this case, τ1’ can be easily described as following equation, because a uniform kinetic shear stress τk acts on the contact region, τ1 ′ = 𝜏k
(3)
Next, we will obtain the analytical solution of τ2’, which corresponds to the increment amount of τ during the second loading period. Figure 4(c) shows the equivalent model for obtaining τ2’. In the model, as shown in the right graph in Figure 4(c), τ gradually increases until τ reaches τ2’, and then it drops to zero. Considering that as shown in the right graph in Figure 4(c), the kinetic friction does not act on the equivalent model, kinetic shear stress should be set to zero. Therefore, in this case, the displacement dependence of τf can be described as, τf_model 2’ = τs – τk – (τs – τk) u/D, when u < D. Considering that in the Lorenz and Persson model, τf = τs – (τs – τk) u/D when u < D, the model in Figure 4(c) is identical to the model in Figure 1 when τs and τk in Equation (1) is replaced with τs – τk and 0, respectively. Therefore, using the replacements τs → τs – τk and τk → 0 in Equation (1), τ2’ is derived as
𝜏s −𝜏k 𝜏s −𝜏k
𝜏2 ′ =
𝜅𝐿0
{
for 𝜅𝐿0 > acos(0)
𝜅𝐿0
sin(𝜅𝐿0 )
𝜏s −𝜏k
for 𝜅𝐿0 < acos(0)
(4)
for 𝜅𝐿0 = acos(0)
𝜅𝐿0
Thus, from τmax’ = τ1’ + τ2’, we obtain τmax’ that is required to initiate global slip in the second loading period,
𝜏k + 𝜏max ′ = 𝜏k + {
𝜏s −𝜏k
𝜏s −𝜏k 𝜅𝐿0
𝜏k +
for 𝜅𝐿0 > acos(0)
𝜅𝐿0
sin(𝜅𝐿0 )
𝜏s −𝜏k
for 𝜅𝐿0 < acos(0)
(5)
for 𝜅𝐿0 = acos(0)
𝜅𝐿0
4.2. Maximum value of nominal frictional shear stress in stop-inversion motion
As with the case of the stop-restart motion, the method for obtaining the analytical solution of τmax” in the stop-inversion scenario is illustrated in Figure 5. The meaning of τ1” and τ2” are illustrated in Figure 2(c), and also Figure 5(a). The stress acting on the contact interface during the first loading period is identical for that in the stop-restart motion; therefore, τ1” = τ1’ = τk. As illustrated in Figure 2(c), for the case of the stop-inversion motion, the moving direction of the spring in the second loading period is opposite compared to that in the first loading period. Thus, in the second loading period, V = –V0. As is the case in the stop-restart motion, we should find the equivalent analytical model for obtaining τ2”, which can be illustrated in Figure 5(c). As shown in the right graph in Figure 5(c), during the second loading period, τ increases from -τk to τ2”, and then it reaches τk. In this situation, the maximum frictional shear stress and kinetic shear stress acting on the contact interface are τs + τk and 2τk, respectively. Thus, the relationship between τf and u in the model 2” can be described as τf_model 2’ = τs + τk – (τs – τk) u/D, when u
2𝜏k −
2𝜏k 𝜅𝐿0
acos (
𝜏2 ′′ =
2𝜏k
𝜏𝑠 +𝜏𝑘 𝜏s +𝜏k 𝜅𝐿0
{
)+
𝜏s +𝜏k 𝜅𝐿0
𝜏s +𝜏k 𝜅𝐿0
[1 − (
sin(𝜅𝐿0 )
[1 − (
2𝜏k 𝜏𝑠 +𝜏𝑘
2 1/2
) ]
2𝜏k 𝜏𝑠 +𝜏𝑘
2 1/2
) ]
for 𝜅𝐿0 < acos (
for 𝜅𝐿0 > acos ( 2𝜏k 𝜏𝑠 +𝜏𝑘
for 𝜅𝐿0 = acos (
) 2𝜏k
𝜏𝑠 +𝜏𝑘
2𝜏k 𝜏𝑠 +𝜏𝑘
) (6)
)
Considering that the loading direction during the second loading period is different from that in the first loading period, τmax” = – τ1” + τ2”. Therefore, using Equation (6), we obtain the τmax’’ that is required to initiate global slip in the second loading period:
𝜏k −
2𝜏k 𝜅𝐿0
acos (
2𝜏k 𝜏𝑠 +𝜏𝑘
−𝜏k +
𝜏max ′′ = {
−𝜏k +
)+
𝜏s +𝜏k
𝜏s +𝜏k 𝜅𝐿0
𝜏s +𝜏k
𝜅𝐿0
𝜅𝐿0
[1 − (
2𝜏k 𝜏𝑠 +𝜏𝑘
sin(𝜅𝐿0 )
[1 − (
2𝜏k 𝜏𝑠 +𝜏𝑘
2 1/2
) ]
for 𝜅𝐿0 > acos (
for 𝜅𝐿0 < acos (
2 1/2
) ]
2𝜏k 𝜏𝑠 +𝜏𝑘
for 𝜅𝐿0 = acos (
2𝜏k 𝜏𝑠 +𝜏𝑘
)
) 2𝜏k
𝜏𝑠 +𝜏𝑘
(7) )
4.3. Effects of loading history on nominal frictional shear stress
From the analysis above, it was found that the maximum static friction after the stop-restart or the stop-inversion motion, i.e., Equations (5) or (7), differs from the maximum static friction before them, i.e., Equation (1). In order to investigate the effect of loading history on the maximum value of the nominal frictional shear stress, we define the ratio of the maximum value for nominal shear stress over τs, as below 𝜆I =
𝜏max 𝜏s
, 𝜆II =
𝜏max ′ 𝜏s
, 𝜆III =
𝜏max ′′ 𝜏s
(8)
The value of λI expresses the difference between the static frictional shear stress acting on the contact interface and the nominal (or macroscopic) frictional stress. Similarly, λII and λIII indicate the same effect after the stop-restart motion and the stop-inversion motion, respectively. Figure 6 shows calculation results of three kinds of λ as a function of κL0. As pointed out in the study by Lorentz and Persson [14], the value of λ depends on only two parameters, i.e., κL0 and τk/τs. In all conditions, λ decreased as κL0 increased. It indicated that static friction is characterized by the parameter κL0. For example, for a slider with a relatively larger length, the value of the static friction will be smaller than that in the slider with a small length. The value of λ asymptotically approached the value of τk/τs. It indicates that as κL0 increased, the static friction force approached the kinetic friction force. In addition, from the figures, it was found that the value of λII is always larger than λI. This indicates that the static friction force after the stop-restart motion is always greater than that before the motion. In contrast, λIII is always smaller than λI. Thus, the static friction force after the stop-inversion motion is always smaller than that before the motion. The effect of the loading history on the static friction is explained as follows. Figure 7 shows calculation results in L1/L0 as a function of κL0 at τk/τs = 0.3. This is the moment of transition from static to kinetic friction. Here, L1/L0 indicates the ratio of the length of the kinetic friction region to the slider length, as shown in Figure 1(a). The analytical solution of L1/L0 can be obtained from Equation (26) in the original paper by Lorenz and Persson [14] with L = L0, i.e., cos[κ(L0 – L1) = τk/τs. As a general trend, it is found that L1/L0 increases with increasing κL0. Thus, for example, for a slider with a relatively large length, the value of L1/L0 will be smaller than that in the slider with a small length. From Figure 7, it can be also seen that after the stop-restart motion, L1/L0 has a relatively small value. In contrast, after the stop-inversion motion, L1/L0 has a relatively large value. Considering that the macroscopic static friction force after the stop-restart motion is relatively large as seen in Figure 6, a small L1/L0 corresponds to a large static friction force. While, as seen in the stop-inversion motion, a large L1/L0 corresponds to a small static friction force. This effect is the essence of the loading history dependence of static friction force. The amounts of the
displacement from the initial state to that moment of transition from static to kinetic friction under different loading sequences are illustrated schematically in Figure 8. In the first loading period shown in the left side of Figure 8, the deformation is not applied at the initial state. Alternately during the second loading period after the stop-restart motion, i.e., the middle column in the figure, the initial displacement has a nonzero value since the tangential load is applied during the stopping period. In the case of the stop-inversion motion, the initial displacement is also non zero as illustrated in the right column of Figure 8. Consequently, as denoted in the gray region in the figure, the displacement from initial state to time at τ = τmax (or τ = τmax’ or τ = τmax”) depends on the loading method. The displacement u at the moment of the transition from static to kinetic friction determines the frictional shear stress acting on the contact interface, because the frictional stress τ was defined as a function of u. Therefore, the maximum value of the nominal (macroscopic) frictional shear stress depends on the type of loading history.
5. Discussion
The theoretical prediction discussed in this study is based on the Lorenz and Persson static friction. In that case, the kinetic effects e.g., the propagation of a crack-like front or dynamic precursor [4, 5], are not considered. Therefore, for complete understanding of the effect of loading history on static friction, the effects of these dynamic behaviors on the stress field of an elastic sliding object should be considered. However, we believed that the qualitative conclusion of this study is not modified, because the essence of the findings presented in this study is to indicate that the difference of the initial stress field characterized by a pre-loading strongly affects the level of static friction. From the other perspective, the effect of initial stress distribution on macroscopic static friction was discussed using a finite element analysis. Ozaki et al. [12] pointed out that the initial stress field resulting from the Poisson effects affected the local slip development process. Similar to the present study, they concluded that the initial stress field of the elastic slider (or substrate) is an important factor for determining the macroscopic frictional behaviors. Additionally, Rubinstein et al. [16] discussed the normal loading history dependence of static friction force, which is similar to the tangential loading history dependence of static friction force presented in this study. Furthermore, recently, Maegawa et al. experimentally confirmed that the existence of the stop-restart motion works to increase static friction force, using a different experimental setup [17]. The essence of the current study is identical to these previous studies; the development process of precursory local slip characterizes the macroscopic static friction force. It should be noted that the increase (or reduction) effect of static friction force can be seen only for the first stop-restart motion (or stop-inversion motion). It means that the second and subsequent values of static friction force have a same value, i.e., τmax’ = τmax’2 = τmax’3 = τmax’n, where n means the number of stop-restart motion. Similarly, τmax” = τmax”2 = τmax”3 = τmax”m, where m is the number of stop-inversion motion. The reason is that the initial stress distribution (the deformation of the slider) at the subsequent loading period is identical to that at the second loading period. Equations (1), (5), and (7) includes the kinetic shear stress τk. This is a thought-provoking conclusion; it implies that the macroscopic static friction force may depend on not only local static friction but also local
kinetic friction force. We are not yet able to provide an experimental proof. However, we believe that it is one of important keys for designating static friction force.
6. Conclusion
We present a new insight into static friction force based on a theoretical analysis using the static friction model developed by Lorenz and Persson. The results showed that loading history is an important factor for characterizing the magnitude static friction force. It was found that the existence of the stop-restart motion acts to increase the static friction force. In contrast, the existence of the stop-inversion motion acts to reduce the magnitude of static friction force. The findings of this study provide a novel insight for improving the positioning accuracy and stability of sliding systems included in many mechanical components, because the stop-restart motion and stop-inversion motion are basic types of motion used in many mechanical systems.
FIGURE CAPTIONS
Figure 1: Static friction model developed by Lorenz and Persson [14]. (a) A rectangular elastic block with a nominally flat surface slides on a rigid substrate. Normal and tangential loads F and FN are applied at the side and top faces of the slider. (b) Frictional stress τf acts on the contact interface between the elastic block and substrate and depends on displacement u. When the displacement u is smaller than a characteristic distance D, τf = τs – (τs – τk) u/D. When u is larger than D, τf = τk.
Figure 2: Two types of loading sequences. (a) Basic configuration of a sliding system, (b) Time changes in driving velocity V and the predicted value of frictional shear stress τ for the stop-restart scenario. (c) Time changes in driving velocity V and the predicted value of frictional shear stress τ for the stop-inversion scenario.
Figure 3: Snapshots of deformation of an elastic slider from t0 to t6. These correspond to them denoted in Figure 2(b). ΔL indicates the total deformation from the initial shape of the slider to the final shape. The initial shape is shown by the dashed lines.
Figure 4: Analytical method for obtaining the maximum frictional shear stress at the second loading period, τmax’ for stop-restart motion, τmax’ = τ1’ + τ2’. Figure 5: Analytical method for obtaining the maximum frictional shear stress at the second loading period, τmax” for stop-restart motion, τmax” = –τ1” + τ2”. Figure 6: Calculation results: λ as a function of κL0. Figure 7: Calculation results: L1/L0 as a function of κL0.
Figure 8: Effect of loading history on the sliding distance (displacement u).
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
References [1]
Bowden FP, Tabor D. The friction and lubrication of solids. Oxford: Oxford University Press; 1950.
[2]
Persson BNJ. Sliding friction: physical principles and applications. 2nd ed. Heidelberg: Springer;
2000. [3]
Popov VL. Contact mechanics and friction: Physical principals and applications. Heidelberg:
Springer; 2010. [4]
Rubinstein SM., Cohen G., Fineberg J. Detachment fronts and the onset of dynamic friction. Nature
2004;430:1005–1009. [5]
Rubinstein SM, Cohen G, Fineberg J. Dynamics of precursors to frictional sliding. Phys Rev Lett
2007;98:226103. [6]
David OB, Fineberg J. Static friction coefficient is not a material constant. Phys Rev Lett
2011;106:254301. [7]
Varenberg M, Gorb S. Hexagonal surface micropattern for dry and wet friction. Adv Mater
2009;21:483–486. [8]
Murarash B, Itovich Y, Varenberg M. Tuning elastomer friction by hexagonal surface patterning. Soft
Matter 2011;7:5553–5557. [9]
Nakano K, Maegawa S. Stick-slip in sliding systems with tangential contact compliance. Tribol Int
2009;42:1771–1780. [10]
Otsuki M, Matsukawa H. Systematic breakdown of Amonton’s law of friction for an elastic object
locally obeying Amonton’s law. Sci Rep 2013;3:1586. [11]
Katano Y, Nakano K, Otsuki M, Matsukawa H. Novel friction law for the static friction force based
on local precursor slipping. Sci Rep 2015;4:06324. [12]
Ozaki S, Inanobe C, Nakano K. Finite element analysis of precursors to macroscopic stick-slip motion
in elastic materials: analysis of friction test as a boundary value problem. Tribol Lett 2014;55:151–163. [13]
Scheibert J, Dysthe DK. Role of friction-induced torque in stick-slip motion. Europhys Lett
2010;92:54001. [14]
Lorenz B, Persson BNJ. On the origin of why static or breakloose friction is larger than kinetic friction,
and how to reduce it: the role of aging, elasticity and sequential interfacial slip. J Phys Condens Matter 2012;24:225008. [15]
Maegawa S, Suzuki A, Nakano K. Precursors of global slip in a longitudinal line contact under
non-uniform normal loading. Tribol Lett 2010;38:313–323. [16]
Rubinstein SM, Cohen G, Fineberg J. Contact area measurements reveal loading-history dependence
of static friction. Phys Rev Lett 2006:96:256103. [17]
Maegawa S, Itoigawa F, Nakamura T. Effects of stress distribution at the contact interface on static
friction force: numerical simulation and model experiment. Tribol Lett 2016;62:15.
Highlights
A theoretical prediction of the effect of loading history on static friction force was
derived based on the Lorenz and Persson static friction model.
The stop-restart motion acts to increase the static friction force.
The stop-inversion motion acts to decrease the static friction force.
PII: DOI: Reference:
S0301-679X(16)30067-6 http://dx.doi.org/10.1016/j.triboint.2016.04.021 JTRI4165
To appear in: Tribiology International Received date: 5 September 2015 Revised date: 14 April 2016 Accepted date: 17 April 2016 Cite this article as: Satoru Maegawa, Fumihiro Itoigawa and Takashi Nakamura, New insight into the mechanism of static friction: A theoretical prediction of the effect of loading history on static friction force based on the static friction model proposed by Lorenz and Persson, Tribiology International, http://dx.doi.org/10.1016/j.triboint.2016.04.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
New insight into the mechanism of static friction: A theoretical prediction of the effect of loading history on static friction force based on the static friction model proposed by Lorenz and Persson Satoru Maegawa1,2*, Fumihiro Itoigawa1, Takashi Nakamura1 1
Department of Mechanical Engineering, Nagoya Institute of Technology,
Gokiso-cho Showa-ku Nagoya Aichi 466-8555, Japan 2
Present Address: Department of Mechanical and Aerospace Engineering, Tottori University,
4-101 Minami Koyama, Tottori 680-8553, Japan
*Corresponding author. Tel. & Fax: +81–857–31–5735
E-mail: [email protected]
Abstract
This study provides a new insight into the field of static friction. Based on the static friction model developed by Lorenz and Persson, in which it is assumed that a slip distance characterizes the level of frictional shear stress acting on the contact interface, we pointed out that tangential loading history is an important factor for determining the value of static friction. The stop-restart motion increased the level of static friction force, while the stop-inversion motion reduced the static friction.
Keywords: Static friction; Frictional shear stress; Loading history; Elastomer
1. Introduction
The mechanism of static friction has been an actively studied topic in the fields of tribology, geophysics, and mechanical engineering [1-3]. Following the pioneering experiments by Fineberg’s group [4, 5], in which temporal and spatial development of precursory local slip were observed in situ, a number of studies were performed to gain a complete understanding of the origin of static friction. As pointed out by David and Fineberg [6], “static friction is not a material constant”, i.e., static friction depends on several system parameters: the stiffness properties of the contact interface [7-9]; the magnitude of normal load [10-12]; the loading configuration of the external forces; and the geometry of the sliding object [13-15]. These results indicate there are a number of methods for tuning static friction force. In mechanical engineering, it is important to fully optimize the system parameters listed above to improve the ability of sliding systems to perform their design functions. Based on a simplified static friction model, Lorenz and Persson derived a general rule in characterizing the level of static friction force [14]. They stated that the static friction force decreases due to the existence of different surface regions that begin to slip at different times during the external loading period. Thus, the initiation process of sequential precursory local slip that occurs prior to the onset of the global slip characterizes the level of static friction force. This consideration explains why static friction depends on the system parameters described above. This study focused on the effect of tangential loading history on the magnitude of static friction. Based on the static friction model developed by Lorenz and Persson [14], we developed a theoretical prediction where the sequence of the external loading affects the magnitude of the static friction force. Sequences of external loadings used for this study were stop-restart and stop-inversion motions.
2. Static friction model by Lorenz and Persson
Lorenz and Persson [14] discussed how static friction depends on the elasticity and geometry of a sliding object using a simplified analysis based on a 1D slab model. The advantage to using this simplified model is that the analysis is effective in finding dominant factors for characterizing phenomena. Additionally, a simplified model allows a theoretical solution to be derived analytically. In our study, we used the Lorenz and Persson model, as illustrated in Figure 1, to discuss the effect of the loading history on static friction force. To facilitate the comparison between their results and results obtained in our paper, identical symbols were used. In this model the frictional shear stress τf acts on the contact interface between the elastic slider and the smooth surface of the rigid substrate. It is assumed τf decreases from τs to τk as the surface displacement u increases in the X direction, as shown in Figure 1(b). When u is smaller than a characteristic distance D, τf = τs – (τs – τk) u/D. In contrast, when u is larger than D, τf = τk. Thus, the amount of the local slip D is needed to drop frictional stress from τs to τk. As discussed in the original paper by Lorenz and Persson [14], the amount of D is typically characterized by the time of interfacial relaxation process (aging), which ranges from nm to mm. For example, thermally activated creep or interdiffusion of polymer chains characterizes D as approximately 1 nm. On the other hand, in the case of plastic creep, D will be under the order
of a few μm that is characterized by surface roughness, elastic modulus and the hardness of the contacting solids. In other cases, D is determined by a surface waviness, or structure in the order of a few mm. It should be noted that the local stress fluctuations resulting from the randomly distributed asperity contacts are averaged over large enough surface areas. Thus, the local distribution of shear and normal stresses are removed. The slider is made of an elastic material with Young’s modulus E. The deformation of the elastic slider is modeled by a 1D model. In addition, any kinetic effects such as the fast propagation of a crack tip were not considered. The crack-like front (at x = L in Figure 1) was assumed to move adiabatically as the loading force increased. Here, it should be noted that L0 is the slider length. L1 and L are the lengths from the left edge of the slider to the front edge of the stick and slip regions where τs and τk act on the contact interface as shown in Figure 1(a), respectively. Additionally, the width of slider in the Y direction is described as Ly, and h is the height of the slider in the Z direction, as illustrated in Figure 1(a). Lorenz and Persson derived the theoretical solution of the maximum value of the nominal (macroscopic) frictional shear stress τmax. This maximum value corresponds to the frictional shear stress required to initiate global slip as follows:
𝜏k −
𝜏k 𝜅𝐿0
𝜏
𝜏s
𝜏𝑠
𝜅𝐿0
acos ( k ) + 𝜏s
𝜏max =
𝜅𝐿0 𝜏s
{
𝜅𝐿0
sin(𝜅𝐿0 ) 𝜏
2 1/2
[1 − ( k ) ] 𝜏s
2 1/2
𝜏
[1 − ( k ) ] 𝜏s
𝜏
for 𝜅𝐿0 > acos ( k ) 𝜏𝑠
𝜏k
for 𝜅𝐿0 < acos ( ) 𝜏𝑠
(1)
𝜏
for 𝜅𝐿0 = acos ( k ) 𝜏𝑠
where,
𝜅= (
𝜏s −𝜏k 1/2 ℎ𝐸𝐷
)
(2)
From equation (1) it is clear that the maximum value of the nominal frictional shear stress τmax is not a material constant. Here, it should be noted that the maximum static friction force Fsmax is described as τmaxA, where A is the area of the apparent contact region; that is, Fsmax = τmaxA = τmaxLyL0. Thus, the Lorenz and Persson model provides a clear solution to the question of how static friction depends on the geometry of the slider, and the elastic properties of the slider. The details of the analysis, results and the methodology used to develop the solution are described in Lorenz and Persson’s [14] original paper.
3. Two types of different loading history
Figure 2 illustrates the two types of different loading sequences that are the focus of this study. Figure 2(a) shows the basic configuration of the sliding model. As in the case of the Lorenz and Persson model, the normal and tangential loads F and FN are applied on the top and side face of the slider. In our model, it is assumed that F is applied via a spring with a stiffness K that moves in the X direction with a driving speed of V. In this analysis, the kinetic effects are not considered. Therefore, the value of K does not affect the conclusions
obtained in this study. The changes in driving speed V and nominal frictional shear stress τ of the stop-restart motion are schematically drawn in Figure 2(b). As illustrated, the loading sequence is divided into three periods, i.e., first loading period, stopping period, and second loading period. In the first loading period, the left edge of the spring moves in the positive X direction at a constant speed of V0. The nominal stress τ increases with time until τ reaches the maximum value τmax, at which time slip occurs over the entire contact region, i.e., global slip. The value of τmax corresponds to the maximum static friction force. Subsequently, τ decreases from τmax to τk as the frictional phase changes from static friction to kinetic friction. The decrease in τ occurs instantaneously since this model does not account for kinetic effects such as crack tip propagation. In this study, it is assumed that during the total slip period, the sliding speed of the contacting surface of the slider is equal in all of the regions. Additionally, it is also assumed that the transition from the slip to re-static friction occurs when the sliding speed reaches zero. Therefore, the transition from the total slip to the total stick at each contact region occurs simultaneously at the moment of the onset of the stopping period, because during the stopping period, the increase in tangential load stopped as the motion of the spring stopped, i.e., V = 0. Furthermore, it should be noted that during the stopping period, tangential and normal loads are continuously applied. In the stopping period, the applied tangential force is equal to the static friction force. Therefore, the internal strain during the total slip period is stored before and after the transition. After a certain waiting time, the increase in tangential loading restarts under a constant speed V0. This study focused on the value of the maximum shear stress at the second loading period, i.e., τmax’. As described in following sections, it is found that τmax’ is larger than τmax. Thus, the existence of the stopping motion of the tangential loading increases the maximum static friction. Figure 2(c) illustrates the changes in V and τ for the stop-inversion motion. First, as with the stop-restart motion, a constant speed V0 is applied; τ increases with time during the first loading period. Subsequently, the increase in the tangential load comes to a standstill during the stopping period. In contrast to the stop-restart scenario, during the second loading period, the moving direction of the spring edge is reversed, i.e., V = –V0. The development of the local slip for the contact interface during the stop-restart motion is illustrated in Figure 3, where times t0 to t6 correspond to those times denoted in Figure 2(b). At the initial condition, the entire region of the contact interface was under a static friction condition as illustrated in Figure 3(a). As the tangential load increased, the slip region extended from the trailing to the leading edge of the slider as shown in Figure 3(b). Before the transition from static to kinetic friction, the leading edge of the elastic block does not slide since the shear stress does not act (or relatively small) on the contact interface at the leading edge. At the moment of the transition, in which the local slip develops over the entire region of the contact interface, global slip occurs. During the total slip condition, i.e., Figure 3(c), the slider deforms due to the tangential load and kinetic friction force; the value of tangential load equals the total value of kinetic friction force over each contact region. Subsequently, during the stopping period, i.e. Figure 3(d), the amount of the slider deformation is stored; thus, ΔL(t2) = ΔL(t3) = ΔL(t4), where ΔL means the deformation amount of the slider as illustrated in Figure 3. The increase in tangential load restarts after the end of the stopping period as illustrated in Figure 2(b). The deformation of the elastic slider at the initial moment of the second loading period is illustrated in Figure 3 (d). In contrast to the first loading period, the internal deformation of the slider is already formed at the initial
moment of the second tangential loading. Thus, the initial value of the stress distribution is nonzero value. The total deformation of ΔL(t4) is not zero. Similar to the first loading period, friction shear stress increases until the initiation of global slip, i.e., the transition from static to kinetic friction, at which point the friction shear stress decreased to τk. Similar to the case of the stop-restart scenario, the stop-and-inversion motion as illustrated in Figure 2(c) has a non-zero deformation (initial stress distribution) of the slider at the initiation of the second loading period. In this study, we theoretically predict that the existence of the inversion motion reduces the nominal value of static friction.
4. Theoretical analysis
4.1. Maximum value of nominal frictional shear stress in stop-restart motion
Figure 4 illustrates the method for obtaining the analytical solution of τmax’, which is the maximum value of the nominal frictional shear stress in the second loading period after the stop-restart motion. The time change of τ during the second loading period is highlighted by the solid dashed circle, as seen in Figure 4(a). This study focuses on that the value of τmax’ can be obtained as the sum of τ1’ and τ2’, i.e., τmax’ = τ1’ + τ2’, where τ1’ and τ2’ means the kinetic and maximum static shear stress illustrated in the time changes of τ, which are shown in the right side of Figure 4(a). Therefore, we can obtain the analytical solution of τmax’ from the calculation of the theoretical solutions of τ1’ and τ2’. Figure 4(b) shows the illustration of the analytical model for obtaining τ1’, in which shear stress τk uniformly acts on the entire region of the contact interface. Thus, the relationships between τf and u, and τ and t are described in the center and right graphs in Figure 4(b), respectively. In this case, τ1’ can be easily described as following equation, because a uniform kinetic shear stress τk acts on the contact region, τ1 ′ = 𝜏k
(3)
Next, we will obtain the analytical solution of τ2’, which corresponds to the increment amount of τ during the second loading period. Figure 4(c) shows the equivalent model for obtaining τ2’. In the model, as shown in the right graph in Figure 4(c), τ gradually increases until τ reaches τ2’, and then it drops to zero. Considering that as shown in the right graph in Figure 4(c), the kinetic friction does not act on the equivalent model, kinetic shear stress should be set to zero. Therefore, in this case, the displacement dependence of τf can be described as, τf_model 2’ = τs – τk – (τs – τk) u/D, when u < D. Considering that in the Lorenz and Persson model, τf = τs – (τs – τk) u/D when u < D, the model in Figure 4(c) is identical to the model in Figure 1 when τs and τk in Equation (1) is replaced with τs – τk and 0, respectively. Therefore, using the replacements τs → τs – τk and τk → 0 in Equation (1), τ2’ is derived as
𝜏s −𝜏k 𝜏s −𝜏k
𝜏2 ′ =
𝜅𝐿0
{
for 𝜅𝐿0 > acos(0)
𝜅𝐿0
sin(𝜅𝐿0 )
𝜏s −𝜏k
for 𝜅𝐿0 < acos(0)
(4)
for 𝜅𝐿0 = acos(0)
𝜅𝐿0
Thus, from τmax’ = τ1’ + τ2’, we obtain τmax’ that is required to initiate global slip in the second loading period,
𝜏k + 𝜏max ′ = 𝜏k + {
𝜏s −𝜏k
𝜏s −𝜏k 𝜅𝐿0
𝜏k +
for 𝜅𝐿0 > acos(0)
𝜅𝐿0
sin(𝜅𝐿0 )
𝜏s −𝜏k
for 𝜅𝐿0 < acos(0)
(5)
for 𝜅𝐿0 = acos(0)
𝜅𝐿0
4.2. Maximum value of nominal frictional shear stress in stop-inversion motion
As with the case of the stop-restart motion, the method for obtaining the analytical solution of τmax” in the stop-inversion scenario is illustrated in Figure 5. The meaning of τ1” and τ2” are illustrated in Figure 2(c), and also Figure 5(a). The stress acting on the contact interface during the first loading period is identical for that in the stop-restart motion; therefore, τ1” = τ1’ = τk. As illustrated in Figure 2(c), for the case of the stop-inversion motion, the moving direction of the spring in the second loading period is opposite compared to that in the first loading period. Thus, in the second loading period, V = –V0. As is the case in the stop-restart motion, we should find the equivalent analytical model for obtaining τ2”, which can be illustrated in Figure 5(c). As shown in the right graph in Figure 5(c), during the second loading period, τ increases from -τk to τ2”, and then it reaches τk. In this situation, the maximum frictional shear stress and kinetic shear stress acting on the contact interface are τs + τk and 2τk, respectively. Thus, the relationship between τf and u in the model 2” can be described as τf_model 2’ = τs + τk – (τs – τk) u/D, when u
2𝜏k −
2𝜏k 𝜅𝐿0
acos (
𝜏2 ′′ =
2𝜏k
𝜏𝑠 +𝜏𝑘 𝜏s +𝜏k 𝜅𝐿0
{
)+
𝜏s +𝜏k 𝜅𝐿0
𝜏s +𝜏k 𝜅𝐿0
[1 − (
sin(𝜅𝐿0 )
[1 − (
2𝜏k 𝜏𝑠 +𝜏𝑘
2 1/2
) ]
2𝜏k 𝜏𝑠 +𝜏𝑘
2 1/2
) ]
for 𝜅𝐿0 < acos (
for 𝜅𝐿0 > acos ( 2𝜏k 𝜏𝑠 +𝜏𝑘
for 𝜅𝐿0 = acos (
) 2𝜏k
𝜏𝑠 +𝜏𝑘
2𝜏k 𝜏𝑠 +𝜏𝑘
) (6)
)
Considering that the loading direction during the second loading period is different from that in the first loading period, τmax” = – τ1” + τ2”. Therefore, using Equation (6), we obtain the τmax’’ that is required to initiate global slip in the second loading period:
𝜏k −
2𝜏k 𝜅𝐿0
acos (
2𝜏k 𝜏𝑠 +𝜏𝑘
−𝜏k +
𝜏max ′′ = {
−𝜏k +
)+
𝜏s +𝜏k
𝜏s +𝜏k 𝜅𝐿0
𝜏s +𝜏k
𝜅𝐿0
𝜅𝐿0
[1 − (
2𝜏k 𝜏𝑠 +𝜏𝑘
sin(𝜅𝐿0 )
[1 − (
2𝜏k 𝜏𝑠 +𝜏𝑘
2 1/2
) ]
for 𝜅𝐿0 > acos (
for 𝜅𝐿0 < acos (
2 1/2
) ]
2𝜏k 𝜏𝑠 +𝜏𝑘
for 𝜅𝐿0 = acos (
2𝜏k 𝜏𝑠 +𝜏𝑘
)
) 2𝜏k
𝜏𝑠 +𝜏𝑘
(7) )
4.3. Effects of loading history on nominal frictional shear stress
From the analysis above, it was found that the maximum static friction after the stop-restart or the stop-inversion motion, i.e., Equations (5) or (7), differs from the maximum static friction before them, i.e., Equation (1). In order to investigate the effect of loading history on the maximum value of the nominal frictional shear stress, we define the ratio of the maximum value for nominal shear stress over τs, as below 𝜆I =
𝜏max 𝜏s
, 𝜆II =
𝜏max ′ 𝜏s
, 𝜆III =
𝜏max ′′ 𝜏s
(8)
The value of λI expresses the difference between the static frictional shear stress acting on the contact interface and the nominal (or macroscopic) frictional stress. Similarly, λII and λIII indicate the same effect after the stop-restart motion and the stop-inversion motion, respectively. Figure 6 shows calculation results of three kinds of λ as a function of κL0. As pointed out in the study by Lorentz and Persson [14], the value of λ depends on only two parameters, i.e., κL0 and τk/τs. In all conditions, λ decreased as κL0 increased. It indicated that static friction is characterized by the parameter κL0. For example, for a slider with a relatively larger length, the value of the static friction will be smaller than that in the slider with a small length. The value of λ asymptotically approached the value of τk/τs. It indicates that as κL0 increased, the static friction force approached the kinetic friction force. In addition, from the figures, it was found that the value of λII is always larger than λI. This indicates that the static friction force after the stop-restart motion is always greater than that before the motion. In contrast, λIII is always smaller than λI. Thus, the static friction force after the stop-inversion motion is always smaller than that before the motion. The effect of the loading history on the static friction is explained as follows. Figure 7 shows calculation results in L1/L0 as a function of κL0 at τk/τs = 0.3. This is the moment of transition from static to kinetic friction. Here, L1/L0 indicates the ratio of the length of the kinetic friction region to the slider length, as shown in Figure 1(a). The analytical solution of L1/L0 can be obtained from Equation (26) in the original paper by Lorenz and Persson [14] with L = L0, i.e., cos[κ(L0 – L1) = τk/τs. As a general trend, it is found that L1/L0 increases with increasing κL0. Thus, for example, for a slider with a relatively large length, the value of L1/L0 will be smaller than that in the slider with a small length. From Figure 7, it can be also seen that after the stop-restart motion, L1/L0 has a relatively small value. In contrast, after the stop-inversion motion, L1/L0 has a relatively large value. Considering that the macroscopic static friction force after the stop-restart motion is relatively large as seen in Figure 6, a small L1/L0 corresponds to a large static friction force. While, as seen in the stop-inversion motion, a large L1/L0 corresponds to a small static friction force. This effect is the essence of the loading history dependence of static friction force. The amounts of the
displacement from the initial state to that moment of transition from static to kinetic friction under different loading sequences are illustrated schematically in Figure 8. In the first loading period shown in the left side of Figure 8, the deformation is not applied at the initial state. Alternately during the second loading period after the stop-restart motion, i.e., the middle column in the figure, the initial displacement has a nonzero value since the tangential load is applied during the stopping period. In the case of the stop-inversion motion, the initial displacement is also non zero as illustrated in the right column of Figure 8. Consequently, as denoted in the gray region in the figure, the displacement from initial state to time at τ = τmax (or τ = τmax’ or τ = τmax”) depends on the loading method. The displacement u at the moment of the transition from static to kinetic friction determines the frictional shear stress acting on the contact interface, because the frictional stress τ was defined as a function of u. Therefore, the maximum value of the nominal (macroscopic) frictional shear stress depends on the type of loading history.
5. Discussion
The theoretical prediction discussed in this study is based on the Lorenz and Persson static friction. In that case, the kinetic effects e.g., the propagation of a crack-like front or dynamic precursor [4, 5], are not considered. Therefore, for complete understanding of the effect of loading history on static friction, the effects of these dynamic behaviors on the stress field of an elastic sliding object should be considered. However, we believed that the qualitative conclusion of this study is not modified, because the essence of the findings presented in this study is to indicate that the difference of the initial stress field characterized by a pre-loading strongly affects the level of static friction. From the other perspective, the effect of initial stress distribution on macroscopic static friction was discussed using a finite element analysis. Ozaki et al. [12] pointed out that the initial stress field resulting from the Poisson effects affected the local slip development process. Similar to the present study, they concluded that the initial stress field of the elastic slider (or substrate) is an important factor for determining the macroscopic frictional behaviors. Additionally, Rubinstein et al. [16] discussed the normal loading history dependence of static friction force, which is similar to the tangential loading history dependence of static friction force presented in this study. Furthermore, recently, Maegawa et al. experimentally confirmed that the existence of the stop-restart motion works to increase static friction force, using a different experimental setup [17]. The essence of the current study is identical to these previous studies; the development process of precursory local slip characterizes the macroscopic static friction force. It should be noted that the increase (or reduction) effect of static friction force can be seen only for the first stop-restart motion (or stop-inversion motion). It means that the second and subsequent values of static friction force have a same value, i.e., τmax’ = τmax’2 = τmax’3 = τmax’n, where n means the number of stop-restart motion. Similarly, τmax” = τmax”2 = τmax”3 = τmax”m, where m is the number of stop-inversion motion. The reason is that the initial stress distribution (the deformation of the slider) at the subsequent loading period is identical to that at the second loading period. Equations (1), (5), and (7) includes the kinetic shear stress τk. This is a thought-provoking conclusion; it implies that the macroscopic static friction force may depend on not only local static friction but also local
kinetic friction force. We are not yet able to provide an experimental proof. However, we believe that it is one of important keys for designating static friction force.
6. Conclusion
We present a new insight into static friction force based on a theoretical analysis using the static friction model developed by Lorenz and Persson. The results showed that loading history is an important factor for characterizing the magnitude static friction force. It was found that the existence of the stop-restart motion acts to increase the static friction force. In contrast, the existence of the stop-inversion motion acts to reduce the magnitude of static friction force. The findings of this study provide a novel insight for improving the positioning accuracy and stability of sliding systems included in many mechanical components, because the stop-restart motion and stop-inversion motion are basic types of motion used in many mechanical systems.
FIGURE CAPTIONS
Figure 1: Static friction model developed by Lorenz and Persson [14]. (a) A rectangular elastic block with a nominally flat surface slides on a rigid substrate. Normal and tangential loads F and FN are applied at the side and top faces of the slider. (b) Frictional stress τf acts on the contact interface between the elastic block and substrate and depends on displacement u. When the displacement u is smaller than a characteristic distance D, τf = τs – (τs – τk) u/D. When u is larger than D, τf = τk.
Figure 2: Two types of loading sequences. (a) Basic configuration of a sliding system, (b) Time changes in driving velocity V and the predicted value of frictional shear stress τ for the stop-restart scenario. (c) Time changes in driving velocity V and the predicted value of frictional shear stress τ for the stop-inversion scenario.
Figure 3: Snapshots of deformation of an elastic slider from t0 to t6. These correspond to them denoted in Figure 2(b). ΔL indicates the total deformation from the initial shape of the slider to the final shape. The initial shape is shown by the dashed lines.
Figure 4: Analytical method for obtaining the maximum frictional shear stress at the second loading period, τmax’ for stop-restart motion, τmax’ = τ1’ + τ2’. Figure 5: Analytical method for obtaining the maximum frictional shear stress at the second loading period, τmax” for stop-restart motion, τmax” = –τ1” + τ2”. Figure 6: Calculation results: λ as a function of κL0. Figure 7: Calculation results: L1/L0 as a function of κL0.
Figure 8: Effect of loading history on the sliding distance (displacement u).
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
References [1]
Bowden FP, Tabor D. The friction and lubrication of solids. Oxford: Oxford University Press; 1950.
[2]
Persson BNJ. Sliding friction: physical principles and applications. 2nd ed. Heidelberg: Springer;
2000. [3]
Popov VL. Contact mechanics and friction: Physical principals and applications. Heidelberg:
Springer; 2010. [4]
Rubinstein SM., Cohen G., Fineberg J. Detachment fronts and the onset of dynamic friction. Nature
2004;430:1005–1009. [5]
Rubinstein SM, Cohen G, Fineberg J. Dynamics of precursors to frictional sliding. Phys Rev Lett
2007;98:226103. [6]
David OB, Fineberg J. Static friction coefficient is not a material constant. Phys Rev Lett
2011;106:254301. [7]
Varenberg M, Gorb S. Hexagonal surface micropattern for dry and wet friction. Adv Mater
2009;21:483–486. [8]
Murarash B, Itovich Y, Varenberg M. Tuning elastomer friction by hexagonal surface patterning. Soft
Matter 2011;7:5553–5557. [9]
Nakano K, Maegawa S. Stick-slip in sliding systems with tangential contact compliance. Tribol Int
2009;42:1771–1780. [10]
Otsuki M, Matsukawa H. Systematic breakdown of Amonton’s law of friction for an elastic object
locally obeying Amonton’s law. Sci Rep 2013;3:1586. [11]
Katano Y, Nakano K, Otsuki M, Matsukawa H. Novel friction law for the static friction force based
on local precursor slipping. Sci Rep 2015;4:06324. [12]
Ozaki S, Inanobe C, Nakano K. Finite element analysis of precursors to macroscopic stick-slip motion
in elastic materials: analysis of friction test as a boundary value problem. Tribol Lett 2014;55:151–163. [13]
Scheibert J, Dysthe DK. Role of friction-induced torque in stick-slip motion. Europhys Lett
2010;92:54001. [14]
Lorenz B, Persson BNJ. On the origin of why static or breakloose friction is larger than kinetic friction,
and how to reduce it: the role of aging, elasticity and sequential interfacial slip. J Phys Condens Matter 2012;24:225008. [15]
Maegawa S, Suzuki A, Nakano K. Precursors of global slip in a longitudinal line contact under
non-uniform normal loading. Tribol Lett 2010;38:313–323. [16]
Rubinstein SM, Cohen G, Fineberg J. Contact area measurements reveal loading-history dependence
of static friction. Phys Rev Lett 2006:96:256103. [17]
Maegawa S, Itoigawa F, Nakamura T. Effects of stress distribution at the contact interface on static
friction force: numerical simulation and model experiment. Tribol Lett 2016;62:15.
Highlights
A theoretical prediction of the effect of loading history on static friction force was
derived based on the Lorenz and Persson static friction model.
The stop-restart motion acts to increase the static friction force.
The stop-inversion motion acts to decrease the static friction force.