Fractal Model of Normal Contact Stiffness between Two Spheres of Joint Interfaces with Simulation

The contact stiffness of joint surface plays a significant role in the overall static and dynamic characteristics of mechanical systems. When considering joint interfaces with two rough surfaces, the traditional model based on Hertz contact theory between a sphere and a plane is difficult to use, especially for increasingly complex engineering surfaces. In order to overcome the weakness, here we propose a new contact stiffness model in view of the influence of domain extension factor between two rough surfaces. We study the deformation mechanism and the critical contact parameters. Subsequently, we analyze evolution of the elastic-plastic contact involving three distinct stages ranged from complete elastic through elastic-plastic to fully plastic deformation. Our fractal model is more universal than the traditional model based on some strict assumptions which simplify the contact of two rough surfaces to a rough surface and a rigid plane. In fact, the traditional model could be regarded as a special case in our new model. Simulations show that non-dimensional normal contact stiffness increases with dimensionless contact total load under the mechanism of elastic-plastic transition when the fractal dimension is between 1.1 to 1.5. The results indicate that contact stiffness of our fractal model is appropriate and the theoretical contact stiffness is consistent with the experiment data. DOI: http://dx.doi.org/10.5755/j01.mech.23.5.19356


Introduction
Normal contact stiffness of joint interfaces is vital to processes ranging from adhesion to friction, wear and lubrication [1,2].It is the fact that the rough surface morphology affects the contact stiffness profoundly, and contact between surfaces is the interaction of asperities.Therefore, the interaction of asperities has been extensively studied.
According to the Hertz solution for a single elastic sphere, Greenwood and Williamson [1] proposed that interaction between contacting asperities was equivalent to contact between a rough surface comprising spherical asperities of constant radius and an ultra-smooth flat surface.Assuming asperity heights follow Gaussian distribution, they gave a statistical model called GW model, a popular method to simulate contact stiffness of joint surface.
Due to the limitation of statistical approach, the GW model relies heavily on statistical roughness parameters of sampling length and resolution of measuring instrument.It means that unbiased information from surface topography is difficult to obtain [2].Consequently, Majumdar and Tien [2] gave a fractal characterization and simulated rough surfaces to remedy these deficiencies which depended on the scale in the GW model.In addition, Majumdar and Bhushan [3] characterized the multi-scaled self-affine topography by scale-independent parameters, then they proposed a novel mathematical model by using fractal geometry.In their theory, the model (MB model) mainly considered the contact between rough surface and ultra-smooth plane.Additionally, Wang and Komvopoulos [4] analysed the irrationality of MB model and introduced a domain extension of factor associated with fractal dimension to better match real situation.As a result, they established WK contact fractal model with more reasonable and impersonal asperity size distribution function.
Recently, some researchers managed to compute normal contact stiffness by using microscope property of asperities.For example, Zhang [5] proposed a fractal model for normal contact stiffness of machine joint surfaces based on Hertz contact theory between a sphere and a plane under some given hypotheses.It overcame the shortcoming of statistical analysis in the MB model to avoid excessive dependence upon scale.Subsequently, Wen [6] discussed a modified model for normal contact stiffness by using micro-contact size distribution function, as well as a mathematical relationship between normal contact stiffness and characteristic parameters of joint surfaces.Generally, they concluded that normal contact stiffness increased with the normal load and decreased with the fractal characteristic length scale parameter G. Nevertheless, the variation becomes complicated when the fractal dimension D is involved.
Meanwhile, numerous researchers studied the stringency of parameters.For example, Ganti and Bushan [7] doubted the scale-independence of fractal parameters, especially D and G, when they concentrated on the fractal model.It leaded more discussion about fractal parameters [8].He and Zhu [9] investigated the fractal parameters D and G of two typical kinds of engineering surfaces.The scale-independence of fractal parameters was studied by changing the sampling spacing t corresponding to spatial measuring resolution and the sampling length L. Jiang et al. [10] employed structure function method to calculate the rough surface fractal dimension and scale coefficient, and they indicated that the theoretical contact stiffness of diverse samples is consistent with the experiment data under the different contact load.
Recently, researchers turned their main interests into contact stiffness from different points of views.For example, Ghafoor et al. [11] established a contact stiffness model by using the finite element analysis and gave a novel probability concept in the case of mechanical soft contact.For the most popular GW model, Wu [12] discussed the statistical method of micro-bulges on the one dimensional case.But the prerequisite of their research is that all the microbulges are half spheres and curvature radiuses are equal, which is rather idealized for real situations.Additionally, Robert and Jeffrey [13] adopted multi scale methods to study the contact load between rough surfaces.Persson [14] studied the mechanical properties of the random rough surface.Daniel and Dickrell [15] obtained the contact rigidity from the transverse contact stiffness model, which is consistent with the experiment data under the condition of small load and small contact stress.Filippi and Kartal [16][17] measured the contact stiffness of the joint surfaces.
The research mentioned above involving the fractal model of normal contact stiffness is applicable for a multitude of mechanical joint interface.The relationships between the parameters of fractal dimensions and roughness amplitude are indicated in the mathematical formulas and simulation.Furthermore, another benefit is the convenience of calculating.
However, from the literature, these models have a shared problem that they are all based on MB fractal model.This means that the contact of two rough surfaces is simplified to a rigid plane and a single micro-bulge which is equivalent to half a sphere [18][19][20][21].Obviously, the model has its limitation.For example, some of these works are confined to pure elastic deformation of the contacting sphere [22][23][24][25][26][27][28][29][30][31].An attempt to solve this problem is given by L. Kogut and I. Etsion [32].They presented general solutions for the elastic-plastic contact, but it still aimed at a deformable sphere and a rigid flat.
It is well-known that joint interfaces are constituted by two rough surfaces.For the single micro-bulge of the rough surfaces, they could be regarded as equivalent spheres with curvature radius R. Thus, the two micro-bulge of the rough surfaces are equivalent to two different half spheres.In this paper, the main objective of our research is to offer a new fractal model of normal contact stiffness which is more universal than before.Meanwhile, the evolution of the elastic-plastic contact is analysed involving three distinct stages which range from complete elastic through elastic-plastic to fully plastic deformation.Moreover, we could find that the previous model is only one of the special cases.

Normal contact stiffness model of the joint surface
2.1.The contact of two spheres 2.1.1.Perfectly elastic deformation stage Joint surface essentially composed of two rough surfaces could be simplified to a rough surface and a true plane [33][34][35].This paper considering system of contact model of two rough surface fractal characteristics is in order to make contact more general.Two convex bodies of the rough surface contact are equivalent to two hemispheres.
Let 1 R and 2 R be the radius of the two half spheres.The two hemispheres just contact on the point O when no load is applied.The state is as shown in Fig. 1.  .The contact state of the two hemispheres is as shown in Fig. 2.
In engineering practice, it is a common phenomenon that two curved objects contact each other to transfer pressure.For example, the contact between the ball of rolling bearing and its seat, wheel and rail, etc.Before loading, these contacts are seemed to be point contacts or line contacts.Due to the material deformation after loading, these point or line contacts become contact surfaces.There are large pressures on the small convex surface and area nearby.Considering contact problem, it is often assumed that surface is rigid smooth.Based on the analysis of deformation, subjecting to the contact distribution of the deformation, the maximum compressive stress on contact surface is proposed.
According to Hertz theory, the relationship between the normal contact deformation  and the normal load p [36] can be expressed as:   Then the relation between normal load p and normal contact deformation  can be given as: where E is the equivalent elastic modulus of the two materials and it satisfies: .
Substituting from equation (3) into equation ( 2), we obtain: We will discuss two cases as following, one is , and the other is is similar to the contact between an asperity and a flat plane [37].
, because the deformation of the smaller hemisphere is stronger than the other during contacting, we choose the sectional area of the smaller one.Based on the geometrical relationship of asperities before deformation, we consider , we can get: where a  is sectional area of micro-contact with curvature radius 1 R .So the normal contact deformation can be given as: Using  from equation ( 8) into equation ( 6), we obtain:  is the critical deformation when micro contact turns from perfectly elastic stage to elastic-plastic transition deformation.Depending on the fully plastic deformation mechanism, we can express the contact load as [37]: The critical deformation c  for the inception of plastic contact is given by [33,34]: where R is the equivalent radius of curvature, H is the hardness of soft material, both of H and Y satisfy Now substituting equation ( 17) and ( 12) into equation (5), the normal contact stiffness can be expressed as:

The elastic-plastic fractal model of normal contact stiffness of joint surface
The normal deformation of asperity on joint surface will be in a fully plastic deformation stage when . That is, there exists normal contact stiffness in the other stages, complete elastic and elastic-plastic deformation.
The curvature radius of the asperity can be given as: where G is the characteristic length scale parameter and D is fractal dimension.
From equation ( 6), the  can be obtained as: R from equation ( 14) into equation (15) , yields: Comparing the equation ( 11), ( 14), ( 15), the relation of  and c  can be rewrote as: where ' c a is the critical truncated area of single asperity turns from perfectly elastic deformation into elastic-plastic transition deformation stage.And the expression can be shown as: In order to calculate cross-sectional area of normal load from the critical contact area to the largest area accurately, the size distribution function was given as: ( ') 2 where  is the domain extension factor and The normal contact stiffness of the whole joint surface can be shown as:  The relation between cross-sectional area of the elastic contact ' a and the actual contact area a is given as [18]: where ' l a is cross-sectional area of the largest micro contact, a is actual contact area of the largest contact point, r A is the total actual contact area.
Using the equation ( 21) into the equation ( 20) we obtain: . , where * G is dimensionless characteristic length scale parameter, * r A is dimensionless real contact area, * c a is dimensionless critical contact area.a A is nominal contact area and  is plasticity index.They are written as:

The elastic-plastic fractal model of normal contact load of joint surface
Using e p from Eq. ( 10), the completely plastic contact load on joint surface can be given as following: Then we combine the formula (21) into the formula (27) and get the equation as:  32 ) Using the total normal load in a non-dimensional form * P as:    ' Giving the total normal load in a non-dimensional form as:   22),( 29),(35),we build the implicit function relationship of  n K and  P ,which is a complicated non-linear relationship.
II. When: Put equation ( 36) into the equation ( 1) to get the relation of  : (37) Through the deformation of equation ( 37), the relation between normal contact deformation  and normal load p can be expressed as: ( The normal contact stiffness on elastic-plastic stage can be given as: The normal contact stiffness of the total joint surface can be obtained as: Finally, we can have the final form of the total normal contact stiffness: where Give the total normal contact stiffness in a non-dimensional form as: where , substituting for e p from equation(39) in equation where the completely elastic contact load of the joint surface can be given as following: Substituting for p p from equation (10) in equation where the completely plastic contact load of the joint surface can be obtained as:  12) in equa-tion where the elastic-plastic contact load of the joint surface can be expressed as: . 3 (2.38 1.88 ) Give the total normal contact load in a non-dimensional form as:  , the elastic, elastic-plastic, plastic normal contact load can be given as: We can have the expression of the total normal contact load as following: Put the equation (56) into the equation ( 57), so that we can get the final form:  P , which is a complicated non-linear relationship.

The digital simulation of joint surface's normal contact stiffness fractal model
In this thesis, we give the simulation for the case when

The analysis of simulation results
1.It can be seen from Fig. 3, under the consideration of elastic-plastic transition mechanism , when the fractal dimension 9 .

Fig. 1
Fig. 1 Two convex bodies of the rough surface

Fig. 2 1  2 
Fig. 2 Two spheres normal contact deformation When the two hemispheres are under the action of the normal load p, two spheres will produce normal contact deformation, the normal contact deformation of the first hemisphere is 1  and the second one is

2 E 2 v
are the elastic modulus of two contact materials, 1 v and are the corresponding Poisson ratio.
asperities entry a range of complete plastic deformation, where c

ep
from equation(9) into equation where the completely elastic contact load of the joint surface can be given as:

1 R
on equation(5) and equation (39), we obtain: geometrical relationship of microbluge before deformation and the typical value of fractal roughness parameters G, we consider   42) and equation (40), the relation of normal contact stiffness of two micro convex bodies equation (47),(53),(59), we build the implicit function relationship of  n K and  settled, with fractal dimension D and fractal roughness parameters G are given.

K 2 .K
increases with the increasing of the total dimensionless contact load  P it is non-linear relationship.According to Fig.3, total dimensionless normal contact stiffness  n K changes with the total dimensionless contact load  P and the fractal dimension's influence on total dimensionless normal contact stiffness ,we select fracparameters  G and plasticity index  's impact on the total dimensionless normal contact stiffness.The total dimensionless normal contact stiffness  n K increases with the increasing of fractal roughness parameters  G , in Fig.5, a.The total dimensionless normal contact stiffness increases with the plasticity index  's in-creasing, in Fig.5, a.But when 9 . 1  D , the total dimensionless normal contact stiffness  n K decreases with the increasing of fractal roughness parameters in Fig.5, b; the total dimensionless normal contact stiffness decreases with the plasticity index  's increasing.3.According to Fig.6, when the fractal increases with the increasing of total dimensionless contact load  P it is non-linear relationship, which gains the same result with Figs.3 and 4.4.According to Fig.7, the total dimensionless normal contact stiffness  n K increases with the increasing of fractal roughness parameters  G ,which gains the same results with Fig.5, a and b.