On the normal contact stiffness and contact resonance frequency of rough surface contact based on asperity micro-contact statistical models

European Journal of Mechanics / A Solids 75 (2019) 450–460

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On the normal contact stiffness and contact resonance frequency of rough surface contact based on asperity micro-contact statistical models

T

Huifang Xiaoa,∗, Yunyun Sunb a b

School of Mechanical Engineering, University of Science and Technology Beijing, Beijing, 100083, PR China Collaborative Innovation Center of Steel Technology, University of Science and Technology Beijing, Beijing, 100083, PR China

ARTICLE INFO

ABSTRACT

Keywords: Normal contact stiffness Asperity micro-contact statistical model Rough surface Contact resonance frequency

Contact stiffness is an important parameter for describing the interface characteristics in many engineering applications. In this paper, five different statistical micro-models including the Greenwood-Williamson (GW), Zhao-Maietta-Chang (ZMC), Kogut-Etsion (KE), Jackson-Green (JG) and Brake are employed to predict the normal contact stiffness for rough surface contact. It is found that the expressions of contact stiffness obtained using the statistical micro-models are very complex and the direct relationship between the contact stiffness and normal load is not available. Accordingly, an explicit approximated expression for contact stiffness is established in terms of normal load based on the results of numerical simulations. The normal contact stiffness as a function of normal load can be approximated using a power law, in which the coefficient and power are related to surface roughness parameters, material properties as well as nominal contact area. The close agreement between the predicted results and full numerical simulations verify the accuracy of the established explicit expression. The contact stiffness calculated using the predictive expressions are also compared with available experimental results from both ultrasonic method and contact resonance method. Further, the explicit expression of contact resonance frequency for rough surface contact with respect to the normal load is also provided, which can be used to evaluate the contact resonance frequency. The predicted contact resonance frequency is also validated through comparing with experimental results.

1. Introduction Contact interfaces are indispensable in machine tools include bolted joints, rolling bearings and gear meshes, which can transmit coupling forces and moments between structures, as well as dissipate energy to provide damping. Contact stiffness is an essential parameter for describing the interface characteristics and exhibits significant effect on both the static and dynamic behavior of the machine system, such as contact pressure distribution, real contact area, vibration, fatigue as well as interface modeling (Andres and Catalin, 2016; Zou and Wang, 2015; Oskar et al., 2016; Akarapu et al., 2011; Xiao et al., 2018; Sun et al., 2018). Investigation of contact stiffness would suffice for determining the whole class of properties. Therefore, it is of fundamental importance to effectively model the contact stiffness for engineering contact interfaces to characterize and understand the interfacial behavior of practical machine tools more accurately. In practice, engineering surfaces are inherently rough and great attention has been expended on the calculation and measurement of the normal contact stiffness at rough contact interfaces. On the contact

∗

modeling of rough surface, the statistical approach is one of the most popular methods and different asperity micro-contact statistical models have been proposed. The pioneering contribution to rough surface contact was made by Greenwood and Williamson (1966), who developed the basic elastic contact model (the GW model). In the GW model, the rough surface topography is described by asperities possessing spherical summits with identical radius of curvature and the height following a Gaussian distribution. The GW model assumes purely elastic contact, but the elastoplastic and plastic deformation regimes of asperities were neglected. Based on the GW model, Chang et al. (1987) proposed the CEB model, which considers both the elastic and plastic deformation regimes for rough surfaces on the basis of volume conservation of plastically deformed asperities. However, the transit from fully elastic to plastic deformation is not modeled and the contact load is discontinuous at the critical point of the initial yielding. To model the transition from elastic deformation to fully plastic flow, Zhao et al. (2000) proposed an elastic-elastoplastic-plastic asperity contact model (ZMC model) based on a mathematical function combining with the continuity of variables across different deformation regimes. Later,

Corresponding author. E-mail address: [email protected] (H. Xiao).

https://doi.org/10.1016/j.euromechsol.2019.03.004 Received 18 April 2018; Received in revised form 18 February 2019; Accepted 4 March 2019 Available online 08 March 2019 0997-7538/ © 2019 Elsevier Masson SAS. All rights reserved.

European Journal of Mechanics / A Solids 75 (2019) 450–460

H. Xiao and Y. Sun

Kogut and Estion (Kogut and Etsion, 2002), Jackson and Green (2005) developed convenient empirical elastoplastic model (KE model and JG model) of a single asperity using finite element analysis based on realistic physical consideration. More recently, Brake (2012) developed an elastic-elastoplastic-plastic contact model (Brake model) for the normal contact of two rough surfaces by including the three separate domains of elastic regime, mixed elastoplastic regime, and fully plastic flow. The force-displacement relationship of the Brake model in the elastoplastic deformation regime is approximated by the cubic Hermite polynomials. Regarding the experimental measurements of normal contact stiffness at rough interfaces, the ultrasonic reflection method and the contact resonance method are the two most popular measurement methods. In the ultrasonic reflection method, the interface is modeled as a single spring and the contact stiffness is related to the reflection coefficient of an incident ultrasonic wave (Kendall and Tabor, 1971; Biwa et al., 2005), which has been widely used to measure the contact behavior in different practical applications, such as rolling bearings, threaded contacts, wheel-rail interaction (Dwyer-Joyce, 2005; Du et al., 2015; Dwyer-Joyce et al., 2009). The limitation of the ultrasonic reflection method is that the wavelength of the incident ultrasound should be large compared to the interface separation. In the contact resonance method, the contact resonant frequency of the dynamic system, which is related to the contact stiffness, is measured through an impact experiment and the contact stiffness is then extracted based on the known dynamic system model (Zhao et al., 2016; Shi and Polycarpou, 2005). Recently, finite element method has also been used to investigate contact characteristics between rough surfaces in real engineering applications (Tian et al., 2011). The normal contact stiffness is investigated by introducing the experimental results of the resonance frequency into finite element models (Mao et al., 2010). In this work, the above mentioned five different statistical micromodels, including the GW model, ZMC model, KE model, JG model and the Brake model are employed to predict the normal contact stiffness for rough surface contact. It is found that the expressions of contact stiffness obtained are very complex and the direct relationship between the contact stiffness and normal load is not available. Accordingly, an explicit approximated expression of power law for normal contact stiffness is established in terms of normal load for rough interface contact based on asperity micro-contact statistical models. The coefficient and power are also determined which are functions of surface roughness parameters, material properties and nominal contact area. The accuracy of the established explicit expression is verified through the close agreement between the predicted results and the full numerical simulations, as well as the available experimental results from both ultrasonic method and contact resonance method. The second purpose of this work is to establish the explicit expression of contact resonance frequency as function of normal load and contact surface property to bridge the theoretical analysis and experimental measurements. Based on the established expression for normal contact stiffness, the explicit expression of contact resonance frequency for rough surface contact with respect to the normal load is also provided, which can be used to evaluate the contact resonance frequency.

Fig. 2. The equivalent contact of a rough plate against a flat surface for the two rough surfaces at the contact region.

initial roughness profiles for the two surfaces are z1(x) and z2(x). The contact of two rough plates can be represented by the equivalent contact of a rough plate and a flat rigid plate with effective elastic modulus 1/E=(1-v12)/E1+(1-v22)/E2, as shown in Fig. 2. E1, E2, v1, and v2 are the Young's modulus and Poisson's ratios of the contacting surfaces, respectively. z represents the asperity height measured from the mean line of summit heights. The separation between the mean lines of the rough surface and the flat surface is h. d denotes the distance between the mean of summit heights and the flat rigid surface. The distance between the mean of surface heights and mean of summit heights is ys, (i.e. ys = h-d). The distances h and d is correspondingly reduced as a normal force F is applied on the contact interface. The five different statistical micro-contact models of the GW model, ZMC model, KE model, JG model and Brake model are utilized to investigate the normal contact stiffness between the rough surfaces. The differences between the five statistical micro-contact models are that the contact mechanism of a single asperity is different. The GW model is a purely elastic statistical contact model between rough surfaces, whilst the other four statistical models consider the multiple regimes of contact deformation. However, the contact behaviors of a rough surface are all obtained by integrating the single asperity contact parameters with the number of contact asperities according to the statistical distribution of asperities, generally assumed to be Gaussian distribution, in the five different statistical micro-contact models. The other assumptions in the statistical micro-contact model include: the rough surface is isotropic; all asperities possess spherical summits with identical radius of curvature; each individual asperity deforms separately and there is no interaction between neighboring asperities; there is no bulk deformation and adhesion. 2.1. Asperity micro-contact statistical models 2.1.1. Greenwood and Williamson (GW) model Greenwood and Williamson (1966) proposed the basic purely elastic statistical micro-contact model for a rough surface in contact with a smooth flat surface, based on the elastic Hertzian solution of a single asperity contact and the extension to the whole surface with the asperity heights following Gaussian distribution. Accordingly, the normal force and contact stiffness of the rough interface can be obtained as

F (h ) = N

2. Modeling of normal contact stiffness The contact interface between rough surfaces is shown in Fig. 1. The

K (h) = N

f h ys a

h ys

(z ) dz

ka (z ) dz

(1) (2)

where N is the total number of asperities with N = nAn, n denotes the asperity distribution density, An denotes the nominal contact area, (z ) is the probability density function of height following the Gaussian distribution, fa and ka are the contact force and contact stiffness of a single asperity with

fa (w ) =

Fig. 1. Contact between rough surfaces. 451

4 E 3

1 3 2 w2

(3)

European Journal of Mechanics / A Solids 75 (2019) 450–460

H. Xiao and Y. Sun

k a (w ) =

dfa dw

1 1 2 w2

= 2E

wcn1 = [ qH /(2E )]2 ( / ), wcn2 = 54wcn1 (Zhao et al., 2000).

(4)

where β is the asperity radius, w is the deflection with

w=z

d=z

2.1.3. Kogut and Etsion (KE) model Kogut and Etsion (2002) presented the empirical expressions between contact load and deformation for a single asperity using finite element analysis considering the elastic-elastoplastic-fully elastic contact behavior. They also formulated the expressions for the whole rough surface contact assuming the Gaussian distribution of asperity heights. Given the contact force relationships for a single asperity contact in different deformation regimes, the corresponding normal force and contact stiffness of the whole surface can be determined as

(5)

h + ys

Substituting Eqs. (3)–(5) into Eqs. (1) and (2) and introducing the following non-dimensional parameters

zn =

z

,wn =

w

, hn =

h

ys

, ysn =

,

n (z n )

1 2

=

exp s

1 2

2 s

zn2

The normal contact force and stiffness of the rough surface can be expressed as

F (h n ) =

4 nAn E 3

3 2

1 2

hn ysn

K (hn ) = 2nAn E

3

(z n + ysn

hn ) 2

F (h n ) = N

(6)

1 hn) 2 n (z n) dzn

(zn + ysn

hn ysn

n (z n ) dz n

2 1.03 qHwcn0.425 3 E 2 1.4 qHwcn0.263 + 3 E

(7)

+

2.1.2. Zhao, Maietta and Chang (ZMC) model Zhao et al. (2000) provided an elastic-elastoplastic-fully plastic contact model based on contact-mechanics theories combining with the continuity of variables across different deformation regimes. The mean contact pressure-interference and contact area-interference relations in the elastoplastic deformation regime are modeled by logarithmic and third-order polynomial functions, respectively. The corresponding stiffness is obtained by evaluating the differential quotient of force and interference. The normal force and normal contact stiffness of the whole surface can be obtained as

F (hn ) = N + +

E

2 H E hn ysn + wcn2

H hn ysn + wcn2 E hn ysn + wcn1

2

(

3 hn ysn + wcn1 wn2 hn ysn

)

wn wcn1 3 wcn2 wcn1

)

hn ysn + wcn2

H

+3

wn wcn2

hn ysn + wcn1

+ 6wn 1

wcn1 wcn1

(1

n (z n ) dz n + 2

1+3

(

H

n (z n ) dz n

wn1.263

n (z n ) dz n

wn (z n) dzn

1

wn2

qHwcn0.425 qHwcn0.263

hn ysn + 110wcn

wn1.425

(10)

n (z n ) dz n hn ysn + 6wcn hn ysn + wcn

wn0.425

hn ysn + 110wcn hn ysn + 6wcn

n (z n ) dz n

wn0.263

n (z n ) dz n

n (z n ) dz n

(11)

where wcn is dimensionless critical yield interference demarcating the transition between elastic and elastoplastic deformation, i.e. wcn = wc/ σ = wcn1.

)

2

1

ln wcn2

n (z n ) dz n

(1

2 q 3

) (1

ln wcn2

wn wcn2

) (ln w

2.1.4. Jackson and Green (JG) model Jackson and Green (2005) presented a micro-contact model based on finite element analysis. Considering the effects of material properties and geometry as well as hardness variation during the deformation, they also formulated the empirical expressions between contact load and deformation for a single asperity and showed that the elastic deformation for the asperity contact is valid up to 1.9 times the critical interface. However, the critical interface at which fully plastic deformation begins was not determined. The normal force and contact stiffness of the whole surface can be written as

(8)

wn2

1+

2

2 q 3

H

+2

hn ysn + 110wcn hn ysn + 6wcn

hn ysn + wcn

2 3 2 + 1.7682 × 3

wn wcn1 2 wcn2 wcn1

wn (zn) dz n

hn ysn

+

hn ysn + 6wcn

hn ysn

+ 1.4677 ×

n (z n ) dz n

hn ysn + wcn

hn ysn + 110wcn

K (hn ) = N 2E

(z n) dzn

2 ln w ln w q ln w cn2 ln w n 3 cn2 cn1

hn ysn + wcn1

K (hn ) = N 2E

2 H E

3

wn2

wn (zn ) dz n

(1

1

hn ysn + wcn hn ysn

+

where σ is the standard deviation of the surface heights distribution, i.e. surface roughness, σs is the standard deviation of asperity heights distribution, n (z n) is the normalized probability density function of height distribution.

4 3

4 3

E

wcn1 wcn1

ln wcn2 + ln wn) ln wcn1

1

3

F (h n ) = N

ln wn) (w wn )(wn wcn1) cn2 ln wcn1 (wcn2 wcn1)3

+

cn2

hn ysn + wcn2

n (z n ) dz n

+

(9)

where H is the hardness related to the yield strength Sy satisfying H = 2.8Sy, q is the hardness coefficient related to the Poisson's ratio ν satisfies q = 0.454 + 0.41ν (Chang et al., 1988). The dimensionless critical interferences of transition from elastic to elastoplastic and from elastoplastic to plastic deformation regime are respectively defined as

E

4 3

hn ysn + 1.9w cn hn ysn

4 C Sy × 3 2E

hn ysn + 1.9w cn

2.84 × 4wn × 1 C

e

1 wn 25 w cn

e

0.82

3

wn2

n (z n ) dz n

(w cn )

wn

1 wn 1 3 2 wn2 e 4 w cn

wn 1.9w cn

D /2

5/12

0.7

1

5/9

(zn) dz n (12)

452

European Journal of Mechanics / A Solids 75 (2019) 450–460

H. Xiao and Y. Sun

+

1

hn ysn + 1.9wcn

K (hn ) = N 2E

wn2

hn ysn

4E 3

3 12 wn e 2

hn ysn + 1.9wcn 5 11 1 wn 12 w 12 e 4 w cn n

5 wcn 48

1

11.36wcn2 1 + C

K (hn ) = N 2E

+

H

wn 1.9wcn

0.7

D /2

1

e

1 wn 25 wcn

5/9

45.44E

+ D) 3C

× 1+

1

0.35 (1.9w )0.35Dw 2 cn cn

× 1

+

e

0.35D

0.7

D /2

hn ysn + 1.9wcn

wn 1.9w cn

D /2

+

0.7

e

f2 (wn) =

1

1 wn 25 w cn

5/9

5

wn9

n (z n ) dz n

+

H E

hn ysn + w1n hn ysn

hn ysn + w 2n hn ysn + w1n

× wn + w 2n )

1 3 2 2

wn w2n

n (z n ) dz n

3

wn2

n (z n ) dz n

q (wn w1n) 2 (wn + w1n) + f (wn) 3w1n 2w1n (w2n w1n )3 1

w1n w1n +

2f2 (wn) w2n

w1n

+

wn w2n

w1n w1n

2

w2n + w1n w1n w2n

n (z n ) dz n

2

w2n 2 H E

H

hn ysn + w 2n

n (z n ) dz n

(15)

w2n + w1n (wn w2n w1n hn ysn + w 2n

wn

n (z n ) dz n

4w1n

wn w2n

4 2 2 qw2n + q 3 3 2 + 2qw1n + qw2n wn 3

2q) w1n w2n

w1n w1n

w2n

w2n + w1n (wn w2n w1n

2 w12n

w2n)

The contact stiffness of a single asperity contact is first examined by the models described above. Fig. 3 shows the variation of normal contact stiffness with non-dimensional interference for a single asperity contact using different micro-contact models. The parameters used in calculation are β = 0.32 mm, E = 113 GPa, v = 0.29, H = 1.96 GPa. The GW model is a purely elastic contact model and thus the contact stiffness increases monotonically and smoothly with interference. However, the contact stiffness values across different regimes of deformation based on the ZMC model are non-monotonic and not continuous for a single asperity contact. This is due to that the contact force is not smooth at the transit points from elastic to elastoplastic and from elastoplastic to fully plastic regime, which results in the discontinuous contact stiffness at the transit points. For the KE model and the JG model, the contact stiffness increases monotonically with interference. The KE and JG models are both established based on finite element analysis and describe the physical behavior of the materials in detail. For the Brake model, the contact stiffness increases non-monotonic with interference in the elastoplastic deformation regime due to the higher order of cubic Hermite polynomials employed in this model. In addition, unrealistic negative stiffness exists in the elastoplastic deformation regime. This is presumably an artefact of the fitting method. It can be also seen in Fig. 3 that the amplitudes of stiffness predicted using different micro-contact models are quite different. The Brake model predicts the largest contact stiffness values in the elastoplastic deformation regime. The purely elastic GW model predicts smaller stiffness values than the Brake model; however, much larger stiffness values than the

2.1.5. Brake model Brake (2012) developed an elastic-elastoplastic-plastic contact model for the normal contact of two rough surfaces by including the three separate domains of elastic regime, elastoplastic regime, and fully plastic flow. The force-displacement relationship in the mixed elastoplastic regime is approximated by the cubic Hermite polynomials to enforce continuity between the end of the elastic regime and the beginning of the plastic regime. Accordingly, the normal force and normal contact stiffness of the whole surface can be given as

4 3

w1n f (wn ) w1n 2

3. Results and discussion

where w cn is the non-dimensional critical interference and given by w cn = w c / = [ CSy /(2E )] / , constants C and D can be expressed as C = 1.295e0.736v and D = 0.14e23Sy/E, respectively (Jackson and Green, 2005).

E

)

where w1n and w2n are critical interferences of transition from elastic to elastoplastic and from elastoplastic to plastic and given as w1n = wcn, w2n = 110wcn. (13)

F (h n ) = N

wn w2n

w1n

2qw 2n 3

)3

where

n (z n ) dz n

1 18

wn

0.82

wn 1.9w cn

f1 (wn) = (6

135C

e

wn

0.82

wn

4w1n + 2qw1n +

hn ysn + 1.9wcn

5/9

1 wn 25 w cn

45.44Ewcn

e

(

n (z n ) dz n

(wn w1n)2f1 (wn) q (wn + w1n) + 3w1n 2w1n (w2n w1n )3

+2

0.287wn 1.35

w1n)2

1

wn2

(wn w1n) f1 (wn ) q + 3w1n w1n (w2n w1n)3

hn ysn + w 2n

× wn +

n (z n ) dz n

+

hn ysn

2w1n (w2n

+

1 0.15 (1 2

hn ysn + w1n

hn ysn + w1n

(wn

+ wn

1 1 2 2

5/12

1 wn 4 w cn

5/12

0.82

e

n (z n ) dz n

(14)

453

European Journal of Mechanics / A Solids 75 (2019) 450–460

H. Xiao and Y. Sun

Fig. 3. Contact stiffness of a single asperity across multiple deformation regimes for different contact models (a) GW (b) ZMC (c) KE (d) JG (e) Brake. Solid line: elastic; dashed line: elasto-plastic; dotted line: plastic.

other three models. The magnitudes of contact stiffness calculated using the KE and JG models are similar. The three elastic-elastoplastic-plastic models of ZMC, KE and Brake predict the same stiffness values in the fully plastic regime due to the same plastic deformation regime employed. The normal force and contact stiffness are both functions of dimensionless distance hn, so the variation of the contact stiffness with

the force can be obtained. Fig. 4 shows the variation of contact stiffness versus the normal load for the rough interface contact with different surface topographies using the five asperity micro-contact models. The parameters of surfaces with different roughness are listed in Table 1 (Beheshti and Khonsari, 2012). The material parameters used in calculations are E = 113 GPa, v = 0.29, H = 1.96 GPa, and the nominal contact area is An = 100 mm2. As can be seen, the contact stiffness of 454

European Journal of Mechanics / A Solids 75 (2019) 450–460

H. Xiao and Y. Sun

Fig. 4. Plots of the normal contact stiffness as a function of normal load for different surface topographies (a) smooth (b) medium (c) rough.

consideration. Meanwhile, the KE and JG models yield the monotonic evolution of stiffness at all different regimes of deformation in comparison to ZMC and Brake models. The GW model has been widely used as a statistical contact model between rough surfaces, because the stiffness expression of the whole rough surface is relatively simple compared to other contact models. Therefore, the GW, KE and JG models are employed for the analysis of contact stiffness in the following sections of this paper.

Table 1 Parameters of rough surfaces with different topographies (Beheshti and Khonsari, 2012). Cases 1 Smooth 2 Medium 3 Rough

σ (m)

n (m−2)

β (m) −6

0.08 × 10 0.3 × 10−6 1 × 10−6

−3

0.32 × 10 0.17 × 10−3 0.055 × 10−3

ξ = nβσ 9

1.55 × 10 1.18 × 109 1.15 × 109

0.040 0.060 0.064

4. Predictive expressions

the rough surfaces all increases monotonically and smoothly with normal load for different contact models. This is due to that the contact stiffness of the whole rough surface is obtained by integrating all asperities involved in contact. The values of normal stiffness decrease as the roughness of the contact interface increases. The surface roughness also has significant effect on the results of contact stiffness between different asperity micro-contact models. For smooth surface, i.e. σ = 0.08 μm, different micro-contact models yield similar values of contact stiffness, especially for low loads, as shown in Fig. 3 (a). As the surface roughness increases, the differences between contact models become obviously. In the cases studied, the KE and JG models predict close contact stiffness profiles even at high loads, whilst the GW model predicts higher contact stiffness values than KE and JG models since it assumes that all asperities deform elastically. From the above discussion, the KE and JG models are found to be appropriate statistical contact models for analyzing the contact stiffness behavior since the two models are based on realistic physical

The derivation process for contact stiffness require considerable time and effort, and the obtained expressions of contact stiffness are very complex for different micro-asperity contact models, as demonstrated in section 2. Moreover, the numerical calculations are required to solve for the contact stiffness due to the lack of explicit expression of stiffness with respect to the load. The convenient equation of contact stiffness based on the micro-contact models should be of interest to the tribology community. In this section, the direct relationship between normal contact stiffness and normal load are developed according to the numerical results from the GW, KE and JG models. Fig. 5 illustrates the dependence of dimensionless contact stiffness on the dimensionless normal force for three different surface topographies. The roughness parameters used for calculation are listed in Table 1. It can be observed that the logarithm of dimensionless stiffness is proportional to the logarithm of dimensionless force, especially for low to medium forces. The normal contact stiffness in terms of normal 455

European Journal of Mechanics / A Solids 75 (2019) 450–460

H. Xiao and Y. Sun

expression of Eq. (17) is much more convenient in representing the contact stiffness behavior compared with the full numerical calculations of Eqs. (6) and (7), Eqs. (10) and (11), Eqs. (12) and (13), which are very complex and involve integral operation. Accordingly, the contact stiffness can be predicted directly at a given load according to the simple analytical approximation. In order to compare the approximate predictions with the numerical calculations, the predicted contact stiffness K using Eq. (17) is plotted against the full numerical contact stiffness obtained using different micro-contact models, as shown in Fig. 6. The surface roughness parameters listed in Table 1 are used for the calculations. The solid lines in Fig. 6 indicate that the predictive values are exactly equal to the full numerical results of the statistical models. It can be seen that the contact stiffness obtained using the established predictive expression agree well with the full numerical calculations, showing the good accuracy of the approximated expression. For the GW, KE and JG models, the maximum errors due to curve fitting for the three cases are 8.2%, 9.4% and 11.2%, respectively. The exponent of the predictive expression, i.e. = 1 ( / ) 2 3 4 based on the GW, KE, and JG model are in the range of α = [0.6293, 0.7615], α = [0.4813, 0.7468], α = [0.4499, 0.7598] for the varying surface topographies, respectively. A similar power law dependence of normal contact stiffness on normal load has also been presented by Pohrt and Popov (2012) for the normal contact between an elastic solid with rough self-affine surface and a flat rigid body. The exponent is in the range of α = [0.51,0.77] for different fractal surfaces studied. However, their results are obtained based on the fractal theory description of rough surface, in which the surface roughness are characterized by fractal dimensions and is different from the statistical micro-contact models and statistical roughness parameters employed in this work.

Fig. 5. Plots of the dimensionless contact stiffness versus the dimensionless normal force in logarithmic coordinate.

force follows a power law relationship. The power law relationship between normal contact stiffness and normal force has also been presented in published references for experimental measurements (Buzio et al., 2003; Pohrt and Popov, 2012; Pohrt et al., 2012). The general form of a power function connecting the dimensionless contact stiffness and dimensionless normal force can be approximated by

K F =c EAn E An

(16)

where c and α are dimensionless coefficient and power, respectively, which are related to the rough surface topography. Introducing the following dimensionless topographical parameters

K¯ =

K F , F¯ = , EAn E An

n

=

, 2

An

4.1. Experimental verification In this section, the experimental results of two popular measurement methods including the ultrasonic reflection method and the contact resonance method are compared with the predictive values to further verify the accuracy of the proposed analytical expression of contact stiffness.

=n

and after considerable numerical simulations, the expression of dimensionless contact stiffness can be finally determined as c2

K = c1

2

c 3 c4 n

F EAn

1

3+

4.2. Ultrasonic reflection method

4

The ultrasonic reflection method is based on the principle that the proportion of the incident wave reflected from the contact interface between two solids is depending on the stiffness of the rough interface. The proportion of the wave amplitude reflected, i.e., ultrasonic reflection coefficient can be measured and further used to evaluate the contact stiffness by means of the acoustic model of interface. Gonzales-Valadez et al. (Gonzalez-Valadez et al., 2010) measured the normal stiffness of rough surface contact using the ultrasound technique based on the principle that the reflection coefficient of ultrasound is related to the interfacial stiffness by means of a spring model. The experimental material is steel with equivalent material parameters E = 114 GPa, ν = 0.29 and nominal contact area An = 95 mm2. The roughness parameters of the tested specimens are listed in Table 4. The contact stiffness expression of equation (17) shows that the surface roughness values σ, as well as the asperity distribution density n and asperity radius β are all needed to determine the stiffness. In many published experimental results on normal contact stiffness of rough surface contact, the surface roughness values σ of tested specimen are reported while the values for asperity distribution density n and asperity radius β are not (Mulvihill et al., 2013; Du et al., 2014). Gonzales-Valadez et al. (Gonzalez-Valadez et al., 2010) provided the normal contact stiffness experimental results for rough contacting interfaces and, fortunately, they also provided the experimental spectral

(17)

where ci and αi (i = 1,2,3,4) are dimensionless constants. Considering the topography of real engineering surfaces, the ranges of dimensionless parameters used for the curve fitting are presented in Table 2 (Beheshti and Khonsari, 2012). Since the normalized probability density function of height distribution following the standard Gaussian distribution, the probability for asperity height in the range of zn = [-3, 3] is 99.73%. Therefore, the dimensionless distance dn, which is directly related to the real contact area, ranges from dn = [0,3] to cover all levels of real contacts even for condition when the applied load is very high. Equation (17) shows that the contact stiffness is related to the normal force, surface topographies, material properties as well as nominal contact area. The dimensionless coefficients in Eq. (17) for different micro-contact models are listed in Table 3. The approximated Table 2 Ranges of dimensionless parameters used for the curve fitting (Beheshti and Khonsari, 2012).

minimum maximum

σ/β

ξ = nβσ

σn=σ/(An1/2)

5.0 × 10−4 2.0 × 10−1

3.0 × 10−2 1.0 × 10−1

1.0 × 10−7 1.0 × 10−3

456

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Table 3 Dimensionless constants in Eq. (17) for different micro-contact models. c1 GW KE JG

c2

10.2378 4.9587 5.2561

c3

0.1445 −0.0058 −0.0652

c4

0.7838 0.9296 0.9577

α1

−0.9393 −0.9280 −0.8968

−6.1896 × 10 −0.0029 −0.0126

−4

α2

α3

α4

0 0.0877 0.0867

−1.5761 −1.3642 −1.0081

0.7848 0.7811 0.8263

Fig. 6. Plots of contact stiffness predictions against the numerical calculations (a) smooth (b) medium (c) rough. The solid lines represent the predictions are exactly equal to the numerical results.

n=

Table 4 Experimental statistical parameters of the rough surface samples (GonzalezValadez et al., 2010). Sample

σ (μm)

m0 (μm2)

m2

m4 (μm−2)

n (m−2)

β (μm)

1 Ra = 1.58 μm 2 Ra = 2.42 μm 3 Ra = 3.09 μm

2.04 3.10 3.90

4.22 9.70 15.28

0.046 0.074 0.100

0.011 0.015 0.018

7.324 × 109 6.209 × 109 5.513 × 109

6.337 5.427 4.954

6

m4 3 m2

= 0.375

m4

(18)

where m2, m4 are the spectral moments of the rough surface. The calculated statistical parameters n and β are also listed in Table 4. Fig. 7 shows the predicted normal contact stiffness as a function of the pressure using the established analytical approximation Eq. (17) for different micro-contact models, as well as the experimental results in the first loading process reported by Gonzales-Valadez et al. (GonzalezValadez et al., 2010). As seen, the predicted values of contact stiffness exhibit the same increasing trend with load as the experimental results. However, the measured stiffness based on the spring model is much higher than the predicted results for different statistical micro-contact models.

moments, i.e. m0, m2, m4 of the samples studied, which can be used to calculate the statistical parameters, i.e. n, β required for the contact stiffness models according to (Bush et al., 1976) 457

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H. Xiao and Y. Sun

Fig. 7. Contact stiffness of different specimens predicted by statistical models compared with the ultrasound results based on the spring model (a) sample 1 (b) sample 2 (c) sample 3.

The large discrepancy between ultrasound experimental measured results and theoretical contact models has been observed for different contact interfaces such as the aluminum-aluminum contact (Drinkwater et al., 1996), the pad-flat plate contact (Mulvihill et al., 2013) and the steel ball-flat steel disk contact (Dwyer-Joyce et al., 2011). These results show that at a contact pressure of 50 MPa, ultrasound was around 2.4 times stiffer and the deviation increases with normal contact pressure. The same condition in these ultrasound experimental measurements of normal contact stiffness is that the spring acoustic model is employed to describe the dependence of interface stiffness on ultrasound reflection. However, the spring model assumes the interface is ideally elastic and neglects the ultrasound attenuation generated at the interface, and thus results in the large deviation. Kim et al. (2004) have also presented that in the elastoplastic contact regime the ultrasonic interfacial stiffness is much higher than the static loading interfacial stiffness—as predicted using the micro-contact models. The physical implication provided is that the ultrasonically determined interfacial stiffness is the local unloading stiffness and it does not include the plastic softening effect of the asperities in the elastoplastic regime. Recently, Du et al. (2014) proposed the series spring-damper acoustic model to incorporate the effect of ultrasonic attenuation at the interface. The comparison results show that the maximum relative deviation between ultrasound and model prediction reduces to 16% for pressure smaller than 100 MPa with the modified stiffness. However, since the surface roughness parameters and the relaxation time of Maxwell material required in the series spring-damper acoustic model are not presented in (Du et al.,

2014), the predicted normal contact stiffness Eq. (17) is unable to be compared with the contact stiffness calculated using the series springdamper acoustic model. Fig. 7 also shows that the predictions of the GW statistical model, which considers only the contact stiffness arising from elastic deformation, provides much higher results of contact stiffness than that of the KE and JG statistical models, which include the elasto-plastic and plastic deformation regimes. 4.3. Contact resonance method The contact resonant frequency method is based on the principle that the resonant frequency of the contact system is related to the contact stiffness based on the system dynamic model. Accordingly, the contact stiffness can be extracted based on the measurements of the contact resonant frequency through an impact experiment. Shi and Polycarpou (2005) measured the contact stiffness of realistic rough surfaces for both spherical contacts and flat rough surfaces in contact under lightly loaded conditions using the contact resonance method. The experimental results for flat rough surface contact are employed for comparison in this work. The stainless steel flat samples of type 17/4 PH were employed for tests. The samples were attached on the top mass and bottom mass using rigid connections to avoid introducing additional compliance in the system. Two stainless steel soft tube springs were used to isolate the contact part from the tester frame. The impact was applied on the top 458

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H. Xiao and Y. Sun

Fig. 10. Comparison of the contact resonance frequency between analytical predictions and experimental results obtained using contact resonance method.

Fig. 8. Comparision of contact stiffness between analytical approximations and the experimental results based on resonance method.

schematic of the lumped parameter model for the rough surface in contact with a rigid surface as shown in Fig. 2. The rough contact interface is modeled as a nonlinear spring in parallel with a damping term. The stiffness of the spring is a function of applied load F. The system equation of motion is given by

mass using a small impactor with a diameter of 0.8 mm and a weight of 2 mN. Accelerometers were attached on the lower mass to measure the accelerations of the mass block. The maximum acceleration was kept to be smaller than 1 g (g = 9.81 m/s2) to avoid loss of contact at the interface. The contact force was measured using strain gauges attached to the lower tube spring. The resonance frequencies were extracted based on the acceleration signals from impact experiments. The material parameters and surface roughness of the flat rough surface contact were E = 105.3 GPa, v = 0.29, H = 2.96 GPa, σ = 0.189 μm, β = 2.402 μm, n = 0.1256 μm−2, and the nominal contact area was An = 1.824 mm2 (Shi and Polycarpou, 2005). Fig. 8 shows the predicted contact stiffness for different normal loads using the established expression Eq. (17) and the experimental results from resonance measurements. It can be observed that the there is a close agreement between them, especially for lower forces. However, the experimental values are lower than the theoretical predictions and the difference increases with the normal force. This can be attributed to the reason that the nonlinearity of the experimental contact system increases as the normal load increases. The strong nonlinearity will cause a large distortion of contact resonance frequency, resulting in the contact stiffness obtained from the experimental measurements lower than the predictive values.

mz¨ + cm z + k (F ) z = F

(19)

where k(F) is the contact stiffness of the rough surfaces, and cm denotes the contact damping representing energy dissipation. Combining the characteristic equation of Eq. (19) and analytical approximations of contact stiffness Eq. (17), the contact resonance frequency of the contact interface with respect to the normal force can be obtained as

fr =

1 2

1

cE1 An2 m

F

(20)

where coefficient c and power α has been described in Section 3 and given as c = c1 ( / )c 2 c3 nc4, = 1 ( / ) 2 3 + 4 , respectively. Eq. (20) indicates that the contact resonance frequency varies nonlinearly with normal force, and is affected by surface topography, material properties, equivalent mass and nominal contact area. Fig. 10 shows the predicted contact resonance frequency using Eq. (20) for different statistical micro-models as well as the experimental results measured by contact resonant method in (Shi and Polycarpou, 2005). The equivalent mass is m = 0.05375 kg. As expected, the contact resonance frequency increases nonlinearly with the normal force. Predictive values of the contact resonance frequency are higher than the experimental results, and the differences increases with the normal force. As mentioned in Section 4.2, the nonlinearity of the experimental contact system increases with the applied force, resulting a distortion of the contact resonance frequency. Accordingly, the contact resonance frequency measured from the experimental measurements will be lower than the predictive value as the normal force increases. Fig. 10 also shows that the contact resonance frequency predicted by analytical expressions based on GW, KE and JG models agrees well, especially at lower forces. The predicted value based on the GW model is slightly larger than that of the other two models. Eq. (20) can be used directly to evaluate the contact resonance frequency for rough surfaces contact.

5. Prediction of contact resonance frequency (F∼K∼ fr) Contact resonance frequency can be related to the normal contact stiffness through a known dynamic system model. Fig. 9 depicts a

6. Conclusions Fig. 9. Lumped parameter model for the rough surface in contact with a rigid surface as shown in Fig. 2.

In this work, five different statistical micro-contact models, including the GW model, ZMC model, KE model, JG model and the Brake 459

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model have been employed to predict the normal contact stiffness for rough surface contact. Based on the numerical simulation results of different micro-contact models, an explicit approximated expression of power law for contact stiffness has been established in terms of normal load. The coefficient and power have also been determined which are functions of surface roughness parameters, material properties and nominal contact area. The accuracy of the established explicit expression has been verified through the close agreement between the predicted results and the full numerical simulations, as well as the available experimental results from both ultrasonic method and contact resonance method. Further, a direct connection between contact resonance frequency and normal force for rough surface contact is provided based on the explicit expression of contact stiffness. It has been shown that the contact resonance frequency is also functions of surface topography, material properties, equivalent mass and nominal contact area.

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Nomenclature An: nominal contact area between two rough flat surfaces, m2 ci (i = 1,2,3,4): dimensionless coefficient of the predictive expressions d: separation between rigid surface and mean of summit heights, m dn: dimensionless separation between rigid surface and mean of summit heights, dn = d/σ E: effective modulus of elasticity, GPa fa: contact force of a single asperity, N fr: contact resonance frequency, Hz F: normal force, N h: distance between the mean line of the rough surface and the flat surface, m hn: dimensionless distance between the mean line of the rough surface and the flat surface, hn = h/σ H: material hardness, GPa ka: contact stiffness of single asperity, N/m K: contact stiffness, GPa/μm K : dimensionless contact stiffness based on approximate expression m: effective mass of contact bodies, kg n: asperity distribution density, m−2 N: total number of asperities P: normal pressure, MPa q: hardness coefficient Sy: yield strength, GPa v: Poisson's ratio w: asperity interference, m wn: dimensionless asperity deflection, wn = w/σ wc: critical interference according to ZMC, KE and Brake models, m wcn: dimensionless critical interference according to ZMC, KE and Brake models, wcn = wc/σ wc΄: critical interference according to JG model, m wcn΄: dimensionless critical interference according to JG model, wcn’ = wc’/σ x: displacement of the equivalent contact body, m ys: distance between mean of summit heights and surface heights, m ysn: dimensionless distance between mean of summit heights and surface heights, ysn = ys/σ z: asperity height measured from the mean line of summit heights, m zn: dimensionless asperity height, zn = z/σ αi (i = 1,2,3,4): dimensionless exponents of the predictive expressions β: asperity radius, m σ: standard deviation of the surface heights distribution, m σn: dimensionless standard deviation of the surface heights, σn = σ/An1/2 σs: standard deviation of the summit heights distribution, m (z ) : probability density function of Gaussian distribution n (z n ) : dimensionless standard normal distribution function ξ: Non-dimensional roughness parameter, nβσ

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