# A Normal Contact Stiffness Statistical Model of Joint Interface considering Hardness Changes

Authors: Donghua Yin; Xueliang Zhang; Shuhua Wen; Guosheng Lan; Yonghui Chen
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

Based on Kogut and Etsion’s finite element contact model (KE model) and citing the hardness geometric limit proposed by Jackson et al., the deformation stage of an asperity is re-divided, and a normal contact stiffness model of joint interface is established by using a statistical theory. The simulation results show that the normal contact stiffness of joint interface is a nonlinear function of the mean surface separation and decreases with the increase of the mean surface separation under different plasticity indexes. By comparing the normal contact stiffness model of joint interface considering hardness changes with the normal contact stiffness model of joint interface without considering hardness changes, it is found that, with the increase of the plasticity index, the difference between calculated results becomes larger.


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## Body

## 1. Introduction

There are various interfaces in contact with each other in mechanical equipment, such as the bolt connection interface of machine tools and the tooth meshing interface of gears. However, the common feature of all kinds of complicated contact interfaces is that surfaces in contact with each other have rough topography. And the contact stiffness is one of the most important parameters describing interface features, whose changes directly affect interfaces and static characteristics and dynamic characteristics of mechanical equipment system, including fatigue and wear, vibration noise response, contact pressure distribution, and work stability [1–3]. So, the accurate modeling of the normal contact stiffness of joint interface is very important for the performance analysis of mechanical equipment. At present, the research on the contact stiffness characteristics of joint interface has received extensive attention. Based on the statistical model description and fractal model description of rough surface topography [4–6] and considering the elastic, elastoplastic, or plastic deformation characteristics of interfaces, different contact models were proposed to analyze the normal contact stiffness characteristics of joint interface [7–11]. Tian et al. [12] established a new normal contact stiffness equation of joint interface. However, the normal contact stiffness of the above joint interfaces did not consider hardness changes caused by the change of contact geometry topography during the elastoplastic deformation stage of asperities. Tian et al. [13] fitted a power function considering the variation of material hardness with surface depth to represent the limiting average pressure by using a method of natural transition equality under boundary conditions. However, the limiting average pressure in this study only considered the elastoplastic transition stage and did not consider the intermediate stage of elastoplastic deformation. Jackson et al. [14] used the finite element method to analyze contact problems and fitted a hardness geometric limit varying with spherical geometry topography. However, they simply unified the elastoplastic deformation stage and plastic deformation stage of an asperity into one stage. Jackson et al. [14–16] also did not propose a contact stiffness model.In this study, based on Kogut and Etsion’s finite element contact model [17] (KE model) and citing the hardness geometric limit proposed by Jackson et al. [14], the deformation stage of an asperity was re-divided, and a normal contact stiffness model of joint interface was established by using a statistical theory. The influence of the mean surface separation of joint interface on the normal contact stiffness was studied by model simulation. The characteristic of this study is that a more complete deformation theory system was given considering hardness changes, so it is more consistent with the actual contact situation of joint interface.


## 2. Contact Model of an Asperity

Joint interface refers to rough surfaces in contact with each other produced by the combination of various parts according to certain requirements. Generally, the contact between two rough surfaces is simplified to the contact between a rigid plane and a rough surface, and for a single asperity on a rough surface, it is usually equivalent to a half sphere. It is assumed that the displacement of the rigid planeω is the deformation of the asperity under the applied of the normal load p, which is also called interference. Interference at the initial point of yield is called critical interference ωc. Jackson et al. [14] derived the critical interference by using the analysis method of von Mises yield criterion, and the resulting equation is(1)ωc=πCSy2E2R,where C is the yield strength coefficient and satisfies the relationship C=1.295exp0.736υ, where υ is Poisson’s ratio of the softer material, Sy is the yield strength of the softer material, R is the radius of curvature at the peak of an asperity, and E is the comprehensive elastic modulus, satisfying 1/E=1−υ12/E1+1−υ22/E2, where υ1, υ2 and E1, E2 are Poisson’s ratios and elastic modulus of the two materials, respectively. In this study, the rigid plane is smooth by E2⟶∞, so the comprehensive elastic modulus can be simplified as E=E1/1−υ12.


### 2.1. Elastic Contact Model

Jackson and Green [14] showed that Hertz’s theory is still applicable when the deformation of an asperity ω≤1.9ωc. Therefore, this stage can be considered as the elastic deformation stage of an asperity. Then, the normal contact load of a single asperity can be expressed as follows:(2)pe=43ER1/2ω3/2.Therefore, the critical contact load of a single asperity during the transition from elastic to elastoplastic can be expressed as follows:(3)pc=43ER1/2ωc3/2.According to (2), the normal contact stiffness of a single asperity is given by(4)ke=dpedω=2ER1/2ω1/2.


### 2.2. Elastoplastic Contact Model

According to [14], when the deformation of an asperity exceeds 1.9ωc, hardness is no longer a constant material property and is defined as the uniform pressure at the time of completely yielding contact, which varies with the change of contact geometry topography of an asperity. At this point, the hardness geometric limit HG can be expressed as follows:(5)HG=2.84Sy1−exp−0.82πCSy2EωωcωωcB/2−0.7.According to [17], the elastoplastic deformation stage of an asperity is defined as 1.9ωc≤ω≤110ωc in this study. Then, the contact load of a single asperity is expressed as follows:(6)pep=pcexp−0.25ωωc5/12ωωc3/2+4HGCSy1−exp−0.04ωωc5/9ωωc=43ER1/2ωc3/2A1ωωc3/2+11.36C1−A2−A3+A4ωωc,where,(7)A1=exp−0.25ωωc5/12,A2=exp−0.04ωωc5/9,A3=exp−0.82πCSy2E−0.711.9−0.35Bωωc−0.35B+1,A4=exp−0.82πCSy2E−0.711.9−0.35Bωωc−0.35B+1−0.04ωωc5/9,B=0.14exp23SyE.According to (6), the normal contact stiffness of a single asperity is given by(8)kep=dpepdω,=43ER1/2ωc3/2A1−548ωc5/12ω−7/12ωωc3/2+A132ωc3/2ω1/2+11.36Cωc1−A2−A3+A4+11.36ωCωcω−4/945ωc5/9A2−0.287B+1πCSy2E−0.711.9−0.35B1ωc−0.35B+1ω−0.35B−1.35A3+A40.287B+1πCSy2E−0.711.9−0.35B1ωc−0.35B+1ω−0.35B−1.35−ω−4/945ωc5/9.


### 2.3. Fully Plastic Contact Model

Whenω≥110ωc, an asperity has a fully plastic deformation, in which there is no stiffness.


## 2.1. Elastic Contact Model

Jackson and Green [14] showed that Hertz’s theory is still applicable when the deformation of an asperity ω≤1.9ωc. Therefore, this stage can be considered as the elastic deformation stage of an asperity. Then, the normal contact load of a single asperity can be expressed as follows:(2)pe=43ER1/2ω3/2.Therefore, the critical contact load of a single asperity during the transition from elastic to elastoplastic can be expressed as follows:(3)pc=43ER1/2ωc3/2.According to (2), the normal contact stiffness of a single asperity is given by(4)ke=dpedω=2ER1/2ω1/2.


## 2.2. Elastoplastic Contact Model

According to [14], when the deformation of an asperity exceeds 1.9ωc, hardness is no longer a constant material property and is defined as the uniform pressure at the time of completely yielding contact, which varies with the change of contact geometry topography of an asperity. At this point, the hardness geometric limit HG can be expressed as follows:(5)HG=2.84Sy1−exp−0.82πCSy2EωωcωωcB/2−0.7.According to [17], the elastoplastic deformation stage of an asperity is defined as 1.9ωc≤ω≤110ωc in this study. Then, the contact load of a single asperity is expressed as follows:(6)pep=pcexp−0.25ωωc5/12ωωc3/2+4HGCSy1−exp−0.04ωωc5/9ωωc=43ER1/2ωc3/2A1ωωc3/2+11.36C1−A2−A3+A4ωωc,where,(7)A1=exp−0.25ωωc5/12,A2=exp−0.04ωωc5/9,A3=exp−0.82πCSy2E−0.711.9−0.35Bωωc−0.35B+1,A4=exp−0.82πCSy2E−0.711.9−0.35Bωωc−0.35B+1−0.04ωωc5/9,B=0.14exp23SyE.According to (6), the normal contact stiffness of a single asperity is given by(8)kep=dpepdω,=43ER1/2ωc3/2A1−548ωc5/12ω−7/12ωωc3/2+A132ωc3/2ω1/2+11.36Cωc1−A2−A3+A4+11.36ωCωcω−4/945ωc5/9A2−0.287B+1πCSy2E−0.711.9−0.35B1ωc−0.35B+1ω−0.35B−1.35A3+A40.287B+1πCSy2E−0.711.9−0.35B1ωc−0.35B+1ω−0.35B−1.35−ω−4/945ωc5/9.


## 2.3. Fully Plastic Contact Model

Whenω≥110ωc, an asperity has a fully plastic deformation, in which there is no stiffness.


## 3. Contact Model of Joint Interface

### 3.1. Fundamental Theory of Statistical Model

Greenwood and Williamson model (GW model) equates the contact between two rough surfaces to the contact between an ideal smooth surface and an equivalent rough surface and establishes a mechanical model of normal contact between a rough surface and a smooth surface. The model is based on the following assumptions. (1) The rough surface is isotropic. (2) All asperities are spherical before contact and have the same radius of curvature, but their heights are randomly varying and follow the Gaussian distribution. (3) The deformation of substrate material can be ignored. (4) In the contact process, there is no volumetric deformation of the contact asperity. It can be seen from Figure1 that the parameters in the contact model include the height of asperities Z, the distance h between the rigid plane and the mean of surface heights, the distance d between the rigid plane and the mean of asperity heights, and the distance ys between the mean of surface heights and the mean of asperity heights. h, d, and ys satisfy the relationship d=h−ys. The three independent parameters that express the morphology of isotropic rough surface include the area density of asperities η, the radius of curvature of asperity summit R, and the ratio of the standard deviation of asperity heights to the standard deviation of surface heights σs/σ.Figure 1
Schematic diagram of contact between a rough surface and a smooth rigid surface.Figure1 is reproduced from Donghua Yin et al. [18] (under the Creative Commons Attribution License/public domain).In this study, Gaussian distribution suitable for the height distribution of asperities is used in the dimensionless form:(9)ϕ∗z∗=12πσσsexp−12σσs2z∗2.Assuming that there areN asperities on the nominal contact area An, the expected number of contact asperities on joint interface n can be expressed as(10)n=N∫d∞ϕzdz=ηAn∫d∞ϕzdz.The relation of the ratioσs/σ is given by [19](11)σsσ=1−3.717×10−4β2,where β is a dimensionless surface roughness parameter in the form β=ηRσ.GW defines a plasticity index in terms of surface properties and critical interference, which is given as [5](12)ψ=ωc∗σσs−1/2.The plasticity index relates the critical interference to the plastic deformation of a surface. A higher plasticity index indicates that asperities on a surface are more likely to yield. Therefore, the asperities on a rough surface with lower critical interference value are more likely to undergo plastic deformation.The dimensionless separation between the dimensionless mean of asperity heightsd∗ and the dimensionless mean of surface heights h∗ is given by [19](13)ys∗=h∗−d∗=1.5108πβ.The random dimensionless interference of a single asperity can be expressed as follows:(14)ω∗=z∗−d∗.


### 3.2. Statistical Model of the Normal Contact Stiffness

According to (1), the normal contact stiffness of joint interface can be expressed as follows:(15)K=ηAn∫dd+1.9ωckeϕzdz+ηAn∫d+1.9ωcd+110ωckepϕzdz.The dimensionless form of (15) is(16)K∗=KEAn/σ=2βσR1/2∫h∗−ys∗h∗−ys∗+1.9ωc∗ω∗1/2ϕ∗z∗dz∗+43βπCSy2E3σR−1∫h∗−ys∗+1.9ωc∗h∗−ys∗+110ωc∗A1∗−5ω∗−7/1248ωc∗5/12ω∗ωc∗3/2+A1∗3ω∗1/22ωc∗3/2+11.36Cωc∗1−A2∗−A3∗+A4∗+11.36ω∗Cωc∗ω∗−4/945ωc∗5/9A2∗−0.287B+1πCSy2E−0.711.9−0.35Bω∗−0.35B−1.35ωc∗−0.35B+1A3∗+A4∗0.287B+1πCSy2E−0.711.9−0.35Bω∗−0.35B−1.35ωc∗−0.35B+1−ω∗−4/945ωc∗5/9ϕ∗z∗dz∗,where(17)A1∗=exp−0.25ω∗ωc∗5/12,A2∗=exp−0.04ω∗ωc∗5/9,A3∗=exp−0.82πCSy2E−0.711.9−0.35Bω∗ωc∗−0.35B+1,A4∗=exp−0.82πCSy2E−0.711.9−0.35Bω∗ωc∗−0.35B+1−0.04ω∗ωc∗5/9.


## 3.1. Fundamental Theory of Statistical Model

Greenwood and Williamson model (GW model) equates the contact between two rough surfaces to the contact between an ideal smooth surface and an equivalent rough surface and establishes a mechanical model of normal contact between a rough surface and a smooth surface. The model is based on the following assumptions. (1) The rough surface is isotropic. (2) All asperities are spherical before contact and have the same radius of curvature, but their heights are randomly varying and follow the Gaussian distribution. (3) The deformation of substrate material can be ignored. (4) In the contact process, there is no volumetric deformation of the contact asperity. It can be seen from Figure1 that the parameters in the contact model include the height of asperities Z, the distance h between the rigid plane and the mean of surface heights, the distance d between the rigid plane and the mean of asperity heights, and the distance ys between the mean of surface heights and the mean of asperity heights. h, d, and ys satisfy the relationship d=h−ys. The three independent parameters that express the morphology of isotropic rough surface include the area density of asperities η, the radius of curvature of asperity summit R, and the ratio of the standard deviation of asperity heights to the standard deviation of surface heights σs/σ.Figure 1
Schematic diagram of contact between a rough surface and a smooth rigid surface.Figure1 is reproduced from Donghua Yin et al. [18] (under the Creative Commons Attribution License/public domain).In this study, Gaussian distribution suitable for the height distribution of asperities is used in the dimensionless form:(9)ϕ∗z∗=12πσσsexp−12σσs2z∗2.Assuming that there areN asperities on the nominal contact area An, the expected number of contact asperities on joint interface n can be expressed as(10)n=N∫d∞ϕzdz=ηAn∫d∞ϕzdz.The relation of the ratioσs/σ is given by [19](11)σsσ=1−3.717×10−4β2,where β is a dimensionless surface roughness parameter in the form β=ηRσ.GW defines a plasticity index in terms of surface properties and critical interference, which is given as [5](12)ψ=ωc∗σσs−1/2.The plasticity index relates the critical interference to the plastic deformation of a surface. A higher plasticity index indicates that asperities on a surface are more likely to yield. Therefore, the asperities on a rough surface with lower critical interference value are more likely to undergo plastic deformation.The dimensionless separation between the dimensionless mean of asperity heightsd∗ and the dimensionless mean of surface heights h∗ is given by [19](13)ys∗=h∗−d∗=1.5108πβ.The random dimensionless interference of a single asperity can be expressed as follows:(14)ω∗=z∗−d∗.


## 3.2. Statistical Model of the Normal Contact Stiffness

According to (1), the normal contact stiffness of joint interface can be expressed as follows:(15)K=ηAn∫dd+1.9ωckeϕzdz+ηAn∫d+1.9ωcd+110ωckepϕzdz.The dimensionless form of (15) is(16)K∗=KEAn/σ=2βσR1/2∫h∗−ys∗h∗−ys∗+1.9ωc∗ω∗1/2ϕ∗z∗dz∗+43βπCSy2E3σR−1∫h∗−ys∗+1.9ωc∗h∗−ys∗+110ωc∗A1∗−5ω∗−7/1248ωc∗5/12ω∗ωc∗3/2+A1∗3ω∗1/22ωc∗3/2+11.36Cωc∗1−A2∗−A3∗+A4∗+11.36ω∗Cωc∗ω∗−4/945ωc∗5/9A2∗−0.287B+1πCSy2E−0.711.9−0.35Bω∗−0.35B−1.35ωc∗−0.35B+1A3∗+A4∗0.287B+1πCSy2E−0.711.9−0.35Bω∗−0.35B−1.35ωc∗−0.35B+1−ω∗−4/945ωc∗5/9ϕ∗z∗dz∗,where(17)A1∗=exp−0.25ω∗ωc∗5/12,A2∗=exp−0.04ω∗ωc∗5/9,A3∗=exp−0.82πCSy2E−0.711.9−0.35Bω∗ωc∗−0.35B+1,A4∗=exp−0.82πCSy2E−0.711.9−0.35Bω∗ωc∗−0.35B+1−0.04ω∗ωc∗5/9.


## 4. Simulation and Result Analysis of the Model

There are many factors that affect the normal contact stiffness of joint interface. This study mainly analyzes the influence law of the mean separation on the normal contact stiffness of joint interface. As shown in (16), the dimensionless normal contact stiffness K∗ is related to the dimensionless surface roughness parameter β, the radius of curvature at the peak of an asperity R, the standard deviation of surface heights σ, the dimensionless surface mean separation h∗, and other parameters. For the simulation analysis, the parameters given are the elastic modulus E1=E2=2.07×1011Pa, the yield strength of the material Sy=0.7×109, and Poisson’s ratio υ1=υ2=0.29. In addition, the dimensionless surface roughness parameters β and σ/R are shown in Table 1. The data of each variable are used to simulate (16), and the corresponding results are shown in Figure 2.Table 1
Dimensionless surface roughness parameters [20].
Numberβσ/R10.03391.600×10−420.04766.576×10−430.05411.144×10−340.06011.770×10−3Figure 2
Influence ofh∗ on lgK∗.Figure2 shows relationship between the mean separation and the normal contact stiffness. As shown in Figure 2, the normal contact stiffness of joint interface is a nonlinear function of the mean surface separation and decreases with the increase of the mean surface separation under different plasticity indexes.In order to verify the validity of the statistical model of normal contact stiffness of joint interface established in this study, the model in this study is compared with the model in [21], and the results are shown in Figure 3.Figure 3
A comparison between the model in this study and that in [21]. (a) ψ=0.73498. (b) ψ=1.571. (c) ψ=2.0946. (d) ψ=2.6232.
(a)(b)(c)(d)As can be seen from Figure3, when the plasticity index is lower, the normal contact stiffness of joint interface considering hardness changes is higher than that without considering hardness changes. When the plasticity index is higher, the normal contact stiffness of joint interface considering hardness changes is lower than that without considering hardness changes. This is in line with the reality. Because when the plastic index is lower, the proportion of plastic deformation is smaller, so the normal contact stiffness of joint interface is higher. When the plasticity index is higher, the proportion of plastic deformation is larger, so the normal contact stiffness of joint interface is lower. The above studies indicate that considering hardness changes cannot be ignored in the study of the normal contact stiffness of joint interface. The above research provides a theoretical basis for further and accurate research on the normal contact stiffness of joint interface.


## 5. Conclusions

Based on the contact statistical model and considering hardness changes, a statistical model of the normal contact stiffness of joint interface is established in this study. The findings are as follows:(1)
The normal contact stiffness of joint interface decreases with the increase of the mean surface separation under different plasticity indexes(2)
When the plasticity index is lower, the normal contact stiffness of joint interface considering hardness changes is higher than that without considering hardness changes(3)
When the plasticity index is higher, the normal contact stiffness of joint interface considering hardness changes is lower than that without considering hardness changes



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