Natural Science Foundation of Shanxi Province201901D111248Shanxi Provincial “1331” Engineering Key Discipline Construction Project of ChinaShanxi Provincial Graduate Education Innovation Project of China2020BY112
1. Introduction

The machine tool is the mother of manufacturing industry, which is assembled by various components. There are some parts that contact each other, namely joint surface. Among them, the fixed joint surface is one of the widely existing joint surfaces. Research showed that  the ratio of contact stiffness of fixed joint surface to total stiffness of the whole machine was more than 60%. Therefore, it is of great significance to establish a more accurate contact stiffness model for the analysis of static and dynamic characteristics of the machine tool structure.

2. Model of Normal Contact Stiffness of a Single Asperity

Figure 1 presents the deformation of a single asperity before and after contact with a rigid plane, where the dashed lines show the situation before deformation. The displacement of the rigid plane ω is the deformation of the asperity under the applied of the normal load p. R is the curvature radius of the asperity summit.

Contact deformation diagram of a single asperity with a rigid plane.

2.1. Model of Normal Contact Stiffness of a Single Asperity during Loading

The critical deformation of a single asperity ωc, when it transforms from the elastic to the elastic-plastic deformation regime, is given by (1)ωc=πKH2E2R,where H is the hardness of the softer material, which is related to its yield strength by H=2.8σs , K is the hardness coefficient in the form K=0.454+0.41ν, where υ is Poisson’s ratio of the softer material, And E, the comprehensive elastic modulus, is given in 1/E=1υ12/E1+1υ12/E2, where E1 and E2 and υ1 and υ2 are the elastic modulus and Poisson’s ratios of the two materials, respectively. In this model, the rigid plane is smooth by E2, so the comprehensive elastic modulus can be simplified as E=E1/1υ12.

When ωωc, a single asperity deforms elastically. According to Hertz theory , the normal contact load during loading can be expressed as follows:(2)pel=43ER1/2ω3/2.

Therefore, the critical contact load of a single asperity pc that marks the transition from the elastic to the elastic-plastic deformation regime can be expressed as follows:(3)pc=43ER1/2ωc3/2.

According to equation (2), the normal contact stiffness of a single asperity during loading kel is given by(4)kel=dpeldω=2ER1/2ω1/2.

When ωcω110ωc, elastic-plastic deformation of a single asperity occurs. The normal contact load during loading can be expressed as follows:(5)pepl1=4.123ER1/2ωc3/2ωωc1.425,ωcω6ωc,(6)pepl2=5.63ER1/2ωc3/2ωωc1.263,6ωcω110ωc.

Similarly, the normal contact stiffness of a single asperity during this loading is obtained as follows:(7)kepl1=1.957ER1/2ωc1/2ωωc0.425,ωcω6ωc,(8)kepl2=2.3576ER1/2ωc1/2ωωc0.263,6ωcω110ωc.

When ω110ωc, a single asperity has a completely plastic deformation, in which there is no stiffness.

2.2. Model of Normal Contact Stiffness of a Single Asperity during Unloading

When ωmaxωc, the normal contact load peu and the normal contact stiffness keu of the asperity during unloading can be expressed as follows:(9)peu=43ER1/2ω3/2,(10)keu=2ER1/2ω1/2.

According to , the relationship between normal contact load pepu and unloading deformation of asperity ω is(11)pepu=pmaxωωresωmaxωresnp,where the index of the plastic stage is given by np=1.5ωmax/ωc0.0331.

The residual deformation ωres and the radius of residual nonuniform curvature Rres depend on the maximum load at the beginning of the unloading pmax. According to relations (5) and (6) between contact load and deformation, the maximum load pmax can be deduced as follows:(12)pmax1=4.123ER1/2ωc3/2ωmaxωc1.425,ωcωmax6ωc,(13)pmax2=5.63ER1/2ωc3/2ωmaxωc1.263,6ωcωmax110ωc.

By substituting equations (12), (13) into equation , respectively, and making further differentiation, the unloading stiffness of an elastic-plastic deformable asperity can be deduced as follows:(14)kepu1=4.12ER1/2ωc3/2np3ωmaxωresωmaxωc11.425ωωresωmaxωresnp1,ωcωmax6ωc,(15)kepu2=5.6ER1/2ωc3/2np3ωmaxωresωmaxωc11.263ωωresωmaxωresnp1,6ωcωmax110ωc.

In the same way, the plastically deformable asperity does not recover. There is no contact stiffness during unloading.

In , the ratio relation ωres/ωmax was given as follows:(16)ωresωmax=11ωmax/ωc0.2811ωmax/ωc0.69.

3. Statistical Model of Normal Contact Stiffness of Joint Surface3.1. Statistical Model of Normal Contact Stiffness of Joint Surface during Loading

Based on Greenwood and Williamson’s model (GW model), this paper assumes that there is no interaction between asperities, and all deformation is limited to the contacting asperities. So, the fixed joint surface is equivalent to the contact between a rough surface and a smooth rigid plane, as shown in Figure 3. Z is the height of asperities, d is the distance between the mean of asperity heights and the rigid plane, h is the distance between the mean of surface heights and the rigid plane, and ys is the distance between the mean of asperity heights and the mean of surface heights, satisfying the relation d=hys. The rough surface is isotropic, and its morphology is defined by three independent parameters: the area density of asperities η, the ratio of the standard deviation of asperity heights to the standard deviation of surface heights σs/σ, and the radius of curvature of asperity summit R.

Schematic diagram of contact between a rough surface and a smooth rigid surface.

The relation of the ratio σs/σ can be expressed as(17)σsσ=13.717×104β2,where β is a dimensionless surface roughness parameter in the form: (18)β=ηRσ.

Assuming that there are N asperities on the nominal contact area An, the expected number of contact asperities on the joint surface n is given by(19)n=Ndϕzdz=ηAndϕzdz,where ϕz is the probability density function of the normal distribution of asperity heights.

The distribution function ϕz of dimensionless asperity heights z is described by a dimensionless Gaussian standard probability density function in the form:(20)ϕz=12πσσsexp12σσs2z2.

The dimensionless distance ys between the mean of asperity heights and the mean of surface heights is given by (21)ys=hd=1.5108πβ.

The random dimensionless interference of a single asperity can be expressed as follows: (22)ω=zd.

In this paper, the plastic index form proposed by GW is adopted:(23)ψ=ωcσσs1/2.

According to equation (1), the normal contact stiffness of the joint surface during loading is(24)Kl=ηAndd+ωckelϕzdz+ηAnd+ωcd+6ωckepl1ϕzdz+ηAnd+6ωcd+110ωckepl2ϕzdz=2ER1/2ηAndd+ωcω1/2ϕzdz+0.9785πKHRηAnd+ωcd+6ωcωωc0.425ϕzdz+1.1788πKHRηAnd+6ωcd+110ωcωωc0.263ϕzdz.

The dimensionless form of equation (24) is(25)Kl=KlEAn/σ=2βσR1/2hyshys+ωcω1/2ϕzdz+0.9785πKHβEhys+ωchys+6ωcωωc0.425ϕzdz+1.1788πKHβEhys+6ωchys+110ωcωωc0.263ϕzdz.

The normal contact stiffness of joint surface during unloading can be expressed as(26)Ku=2ER1/2ηAndmindmin+ωcω1/2ϕzdz+1.03πKH3R2ηAn6E2dmin+ωcdmin+6ωcnpωmaxωresωmaxωc1.425ωωresωmaxωresnp1ϕzdz+0.7πKH3R2ηAn3E2dmin+6ωcdmin+110ωcnpωmaxωresωmaxωc1.263ωωresωmaxωresnp1ϕzdz.

The dimensionless form of the above equation is(27)Ku=KuEAn/σ=2βσR1/2hminyshminys+ωcω1/2ϕzdz+1.03πKH3Rβ6E3hminys+ωchminys+6ωcnpωmaxωresωmaxωc1.425ωωresωmaxωresnp1ϕzdz+0.7πKH3Rβ3E3hminys+6ωchminys+110ωcnpωmaxωresωmaxωc1.263ωωresωmaxωresnp1ϕzdz,where np=1.5ωmax/ωc0.0331.

4. Simulation and Result Analysis of the Model

It can be seen from equation (25) that the dimensionless loading normal contact stiffness Kl is a function of the standard deviation of surface heights σ, the radius of curvature at the initial summit of asperity R, the dimensionless surface roughness parameter β, the dimensionless surface mean separation h, and so on. It can be seen from equation (27) that the dimensionless unloading normal contact stiffness Ku is a function of the standard deviation of surface heights σ, the radius of curvature at the initial summit of asperity R, the dimensionless surface roughness parameter β, the dimensionless surface mean separation h, the dimensionless residual deformation ωres, and so on. And, it is not affected by residual nonuniform curvature radius Rres. In the simulation analysis, the parameters are given such as the elastic modulus E1=E2=2.07×1011Pa, Poisson’s ratio υ1=υ2=0.29, the hardness H=1.96×109Pa, the radius of curvature at the initial summit of asperity R=6.89×104mm, and the dimensionless surface roughness parameters β and σ/R  (shown in Table 1). Equations (25) and (27) are simulated by using the data of each variable, and the corresponding results are shown in Figure 4.

Dimensionless surface roughness parameters .

Numberβσ/R
10.03391.600×104
20.04766.576×104
30.05411.144×103
40.06011.770×103

Influence of h on lgK. (a) ψ=0.73498. (b) ψ=1.571. (c) ψ=2.0946. (d) ψ=2.6232.

It can be seen from Figure 4 that the normal contact stiffness of joint surface during loading and unloading is a nonlinear function of the mean surface separation and decreases with the increase of the mean surface separation. When the plastic index is smaller, the contact between asperities is more elastic, so the normal contact stiffness curves of joint surface during loading and unloading are close. When the plastic index is larger, the plastic deformation cannot recover due to the large proportion of the plastically deformed asperities, so the normal contact stiffness decreases rapidly during unloading.

A comparison between the unloading model in this paper and that in . (a) ψ=0.73498. (b) ψ=1.571.

5. Conclusions

In this paper, a statistical model of the normal contact stiffness of fixed joint surface during unloading after the first load is established, and a simulation analysis is carried out on the model to study the influence of the mean surface separation on the normal contact stiffness. The findings are as follows:

However, the larger the plastic index is, the slower the loading normal contact stiffness decreases, while the faster the unloading normal contact stiffness decreases.

Unfortunately, our paper did have some limitations and shortcomings, which will be verified by supplementary experiments in the future.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by Shanxi Provincial Natural Science Foundation of China (Grant no. 201901D111248), Shanxi Provincial “1331” Engineering Key Discipline Construction Project of China, and Shanxi Provincial Graduate Education Innovation Project of China (Grant No. 2020BY112).