A New Combined Model for considering the Plasticity Effects in Contacting Asperities

Wheel-rail contact in railway engineering is an important topic. Due to different materials and surface roughness of wheel and rail, the contact characteristics can alter significantly. +is article aims to investigate the effects of surface roughness and asperities on the contact parameters such as contact area, contact force, and contact stiffness. +e lateral contacts between asperities are assumed to be the general contact condition. Azimuthal and contact angles distributions are assumed to be spherical harmonic distribution. +is assumption is compatible with the asperity distribution on the wheel and the rail surfaces. Besides, a new combined model is developed to cover the stick-slip and the plasticity effects in contacting asperities. +e results of the presented model offer very good estimations for the asperities contact characteristics, especially at the small-contact area and separation where high-contact pressure and plastic deformation usually exist.


Introduction
Nowadays, railway engineering has enormously developed. However, many unidentified subjects regarding wheel-rail failure and maintenance cycles need to be investigated yet [1]. Contact of surfaces is a high-risk matter from the damage initiation point of view in engineering problems and the railway industry. To develop the accuracy of the contact modeling, the topography classification of the surfaces is a real need. To this end, probabilistic models are presented to describe the contact characteristics like contact load, contact area, separation of surfaces, and contact stiffness. Furthermore, the contact stiffness may alter according to the stated features.
ese features usually cause complicated and nonlinear problems in studying the contacting surfaces [2,3]. To improve the modeling of contacting rough surfaces, the effects of the asperities should be taken into account. In this regard, the following applied models are introduced.
Greenwood and Williamson [4] defined the roughness distribution by considering the spherical asperities with different heights on the surface and introduced the GW model in terms of the Hertz theory for elastic contact. To spread the GW model to a fully plastic zone, Chang et al. [5] suggested their model named as CEB. e ZMC model is introduced by Zhao et al. [6] to evaluate the deformation of asperities in the elastic-plastic regime. In this regard, they examined the mathematical fitting approach. In [7], the elastic-plastic asperity deformation in contacting surfaces by using FEM is investigated by Kogut and Etsion. eir approach is known as the KE model.
In [8], Etsion used another FE model to examine the dependence of load to deformation and force considering plasticity in contact problems. In [9], Jackson and Green consider the effects of hardness variation by a FE analysis (JG Model). ey examined elastic-plastic contact of a sphere with a rigid flat surface considering geometrical hardness variation. e mentioned models are constructed in terms of the idea of contacting an equal rough surface with a rigid one without roughness. As a result, the stick-slip phenomenon at the asperities shoulder may not be considered by these models [10]. In [11], Ciavarella used an uncomplicated model for contacting rough surfaces. However, his model does not show good accuracy because of the assumptions and simplifications in many parameters in contact problems that impose a significant effect on the final results.
Cinat et al. [12] proposed an algorithm that exploited an analogy between genetics and the multiscale characterization of roughness, where various-length scales are described in terms of rough profiles. Wen et al. [13] studied a new elliptical microcontact model considering elastoplastic deformation. ey described the relationship between the normal deformation of the asperity and the mean contact pressure.
Some other works like Abuzeid et al. [14] have used fractal methods for estimating the properties of the rough surface contact. is method is different from the statistical model and usually is adequate for accounting based on frequencies.
Lately, Megalingam and Hanumanth Ramji [15] proposed a model of a deformable spherical asperity contact with a rigid flat plate in a statistical method and combined their model with the effects of Young's modulus, Poisson's ratio, yield strength, and isotropic strain hardening rate by using the finite element method.
Abdo and Farhang [16] suggested the AF model for contacting of the rough surfaces by considering shoulder-toshoulder asperity contact and plasticity. ey considered a distinct condition among the different contact zones.
In [5], Chang et al. at the macroscopic level used an elastic-plastic model while at the microscopic scale, they considered asperities experience initially elastic deformation, and where their defined criterion had been met, they assumed purely plastic deformation.
In recent studies, the used models almost consider the elastic-plastic behavior of the materials [17,18].
In [19], Gao et al. gave an elastic-plastic contact model for contacting asperities. ey assumed the lateral contact, effects of adjacent asperities, and distribution of contact angles. Although their research is the closest work to the presented study, they did not consider the effects of the stickslip in the modeling of contacting asperities, and they used a rather complicated model for implying the effects of contact angles.
According to the performed analysis by using the conventional models, there are still many lacks in the contact modeling of asperities. For example, most of the existing models assume that an equal rough surface contact with a rigid flat one while the effects of friction and interaction between adjacent asperities are passed over. Besides, many models consider isotropic features for a rough surface that may be unrealistic.
In this task, a new combined model is constructed to study the contacting asperities. By this model, the lateral contact between asperities and the elastic-plastic regime in the contact area are considered. Using the base of the AF model [16] for the elastic-plastic regime, focus on the slipstick in the asperity contact region is done. Furthermore, a new spherical harmonic function is developed for asperity contact modeling that its usage shows more precision in comparison with the experimental results.
For a better understanding, the characteristics and main fields of application of different well-known models are shown in Table 1.

Methods
In this research, based on Gao et al.'s works [19], contact of two single asperities under shoulder-to-shoulder interaction is considered. e model of contacting asperity is demonstrated in Figure 1, where the mean separation d is the gap between mean planes 1 and 2. e mean plane represents the mean value of asperity height distribution on each surface.
In this task, to avoid the complexity in formulations, the shape of the asperity's tip was assumed to be spherical.
Abdo and Farhang [16] defined a fictitious asperity with a fictitious surface which is related to the plastic deformation of asperity ( Figure 1). As is shown in Figure 1, ω c represents the critical deformation at the beginning of plastic deformation and R represents the radius of the spherical tip of asperity. By analyzing the contact at the tip of the asperities, according to Figure 2, the contact force relationships can be found. e total contact force decomposes into three components as follows: where φ and θ are contact and azimuthal angles, respectively. e resultant force F i is calculated as follows: e normal component force F in makes a relating normal deformation δ b in the Z′-axis direction which includes the deformation δ bf (as shown in Figure 2) produced due to the lateral contact of lower and upper asperities and the deformation δ bw created due to contact of the nearby asperities.
e normal component of deformation δ b is shown as follows: e tangential component of force F iτ creates a tangential displacement in the X′-axis direction, and the horizontal component of force F is generates a horizontal displacement in the Y′-axis direction.
Considering the elastic-plastic model of material, Gao et al [19] introduced the relations of force, contact area, and contact stiffness between contacting asperities. ey considered interactional deformation effects of adjacent asperities and anisotropic spreading of asperities. In this work based on [7], [16], and [19], we analysed the elastic-plastic contact regime by considering a new combined model for the asperities contact problem. e presented model is accurately compatible with the physics of the asperities contact problem and the results of the experiment. To examine the validity of the presented method, the obtained results are compared with the results of the other related works.

Analysis of the New Combined
Method. In [7], Kogut and Etsion solved the problem of elastic-plastic contacting asperities and noted that the complete elastic-plastic state Considering shoulder-to-shoulder asperity contact e elastic-plastic contact in rough surfaces Mathematical Problems in Engineering spreads over interference values in the range of 1≤δ bf / ω c < 110 with a marked variation in the mean contact pressure at δ bf /ω c � 6. Up to the δ bf /ω c � 6, which is a shifting value; a plastic zone grows under the contact boundary while the whole contact area is elastic. Above δ bf /ω c � 6, the contact area includes an internal elastic circular core that is enclosed by an outer plastic annulus.
Gao et al. [19] reported that the stick area is a circular core at the elastic-plastic state and is bounded by an outer plastic zone that is based on Cattaneo [20], Mindlin [21], and Eriten et al.'s [22] works; we can mention this region as the partial-slip area ( Figure 3).
According to Kogut and Etsion's [7] model, the whole area of the contact is plastic, but the mean contact pressure stays growing until it turns out to be unvarying and equal to the hardness at δ bf /ω c � 110, where the fully plastic contact arises.
Considering the three general regimes elastic, elasticplastic, and entirely plastic deformations for asperities under contact, Gao et al. [19] showed that the total force, contact area, and stiffness are obtained as follows (if we assume that Q is a property like a force or stiffness of asperities in contact): where Q TOT , Q e , Q ep , and Q p are the total property of asperities in contact, the property related to elastic regime, the property related to elastic-plastic regime, and the property related to the fully plastic regime, respectively. erefore, based on the Kogut and Etsion's [7] and Abdo and Farhang's [16] models and Gao et al.'s [19] works, we can rewrite equation (4) as follows: where Q ep1 is the elastic-plastic first component of the property that is correlated to the stick area, and in terms of the Abdo and Farhang's [16] model, it should be subtracted from the elastic property and Q ep2 is the elastic-plastic second component of the property that is related to the slip area, and based on the Gao et al. [19], the formulation should be added with the other regimes. In this analysis, we can consider both slip and stick effects in the elastic-plastic zone and modify the elastic-plastic formulations in the contact problem of asperities. e effects of fictitious asperity (AF model) may be considered in relations as follows: where R is the equivalent radius of asperities, R p is the equivalent plastic radius based on the AF model [16], and ω c is the critical deformation at the beginning of plastic state.
According to the CEB model [5] the relations for these parameters can be expressed as follows: where R 1 and R 2 are the radii of the asperities near the contact zone and S is the maximum contact pressure factor, which is based on the CEB model [5] obtained by S � 0.454 + 0.41], ] and H are Poisson's ratio and the hardness of the softer material, respectively, and E is the equivalent elastic modulus which is attained as follows:

Harmonic Spherical Distribution Function for Effects of Contact Angles.
In this work, we have chosen a new harmonic spherical distribution function to reveal the effects of shoulder-to-shoulder contact angles. Poorsolhjouy and Gonzalez [23] introduced a new distribution function ψ(φ, θ) for asperities contact as follows: where φ and θ are contact and azimuthal angles, respectively. e summation over even k is represented by Σ ' . P k (cosφ) is the k th order Legendre polynomial with respect to cosφ,  Mathematical Problems in Engineering polynomials P 2 and P 1 2 are matched to the single asperity contact problem which we assumed at the beginning to simplify equation (9). According to Poorsolhjouy and Gonzalez [23], parameter a20 is equal to −0.5 in a wide range of density values. Furthermore, we assume parameters a 21 and b 21 are equal to 1 because the shape of the distribution function for the azimuthal angles is harmonic in the asperities contact zone. Equation (9) can be simplified with these assumptions as follows: e distribution function ψ, used in asperities contact formulation, states the effects of the contact angle distribution in the final formulation. e variations of φ and θ during integration are 0 to π and 0 to 2π, respectively.

Final Contact Formulations.
According to the stated theories and combining them for elastic-plastic interaction zone and based on [19], the AF model [16], CEB model [5], and KE model [7], the relations for contact force, contact area, and contact stiffness may be computed as follows: where e nondimensional deformation parameter δ * bc is introduced as δ * bc � δ bf /ω c . e distribution function of Gauss in nondimensional form [9] may be attained as follows: in equations (12)-(16) η, A n , z, d, σ, σ z , and * are the area density of asperities, nominal contact area, height of asperity calculated from the mean asperity height, mean separation of two rough surfaces, standard deviation of the surface heights distribution, standard deviation of asperity heights distribution, and nondimensional parameters, respectively. Gao et al. [19] obtained the relations for F * ie , F * iep2 , and F * ip which are used in equations (12)- (15). In this task, considering the same relations for F * ie , F * iep2 , and F * ip , we have obtained a new expression for F * iep1 based on Eriten's model [22] and combined it with the equation (11). e main formulations are as follows: where δ * vf is the nondimensional normal component of deformation in the Z-axis direction that is produced by the normal component of force, ζ * v and ξ * v are nondimensional tangential and horizontal component of displacement in the Z-axis direction, respectively, and r * ie is the nondimensional elastic radius of the contact area between the asperities in contact and is defined as r * ie � ���� � δ * bc /π. e relation between δ * vf and δ * bc is δ * bc � δ * vf cos φ. For elastic-plastic and fully plastic regimes according to Section 3 and the AF model [16], the nondimensional radius of the contact area can be considered as follows: where r * i(p)AT is the nondimensional new radius of contact area at plastic zones, r i(p) is the plastic or elastic-plastic radius of contact area, r ic is the critical radius at the beginning of plastic deformation, and r ic � ��� � ω c R. e Mathematical Problems in Engineering subscript (p) can be "ep" or "p" for elastic-plastic or plastic deformations, respectively. en, we have where and the relations for nondimensional tangential and horizontal displacement are where G is the equivalent shear modulus, G � (2 − v 1 / G 1 + 2 − v 2 /G 2 ) − 1 , and F inc is the critical value at the beginning of the plastic deformation that may be found as F inc � 4ER 1/2 ω 3/2 c /3. According to the above corrections in the radius of the contact area, we can write where, according to Etsion's model [12], F * in−ep � 1.32 (δ * vf cos φ − 1) 1.27 + 1 and μ ep is the friction factor at elasticplastic contact deformations, and based on Jaeger's model [24] and CEB model [5], it can be written as According to the JG model [6] and radius corrections in this work, and B � 0.14 exp(23Sy/E), and Sy � (H/2.8).
Similarly, the formulations for total contact area can be written as follows: where 6

Mathematical Problems in Engineering
where β and β p are elastic and plastic roughness parameters defined as β � πRσ z and β p � π(R − ω c )σ z . e relations for the contact area are as follows: Similarly, for the total contact stiffness, we can write where Mathematical Problems in Engineering

Results and Discussion
In this paper, a MATLAB program is employed to assess the outcomes of the suggested theoretical model to determine the contact problem of the rough surfaces considering asperities effects. e rough surfaces' properties assume to be constant, and the contact properties like the contact stiffness, which is very important in many applied problems such as wheel and rail interaction, are studied.
In this paper, in order to compare the obtained results with the results of the other works, the contact properties of the Gao model [19] are used.
Considering the presented model, Figure 4 shows the total contact force variation versus mean separation for different surface parameters σ z .
is figure demonstrates that the total contact force diminishes with growing separation, which is acceptable in real contact situations. Furthermore, whenever the surface parameter σ lessens, the contact force decreases with a higher rate. is fact shows that, in the smooth surfaces, the contact force reduces at a higher rate in comparison to a rough one with rising separation, which is logical since the figures of asperities in contact on a smooth surface decrease at a higher rate than a rough one.
In Figure 5, the total contact force variation versus separation for different models is shown (the presented model is called the "AT model").
As is seen in Figure 5, the contact force of the AT model is slightly lesser than the other models. However, its trend of variation shows suitable concurrence with the other models. In the AT model, the effects of the slip-stick area are considered in the formulation, and in this way, the slight diversity between the AT model and the other models may be explained.
Considering the AT model, the real contact area variations versus mean separation for different surface parameters are shown in Figure 6. e results indicate that the real contact area has a reducing trend by rising mean separation, which is logical because the figure of asperities in contact diminishes with growing separation.
In Figure 6, considering a smoother surface (i.e., with smaller σ z ), the real contact area decreases at a higher rate with increasing separation in comparison to a rougher one, because the figures of asperities in contact on a smooth surface reduce at a higher rate with increasing separation in comparison to a rough one. e real contact area variations versus mean separation for different models are shown in Figure 7. e effects of considered fictitious asperity in the elastic-plastic zone and its subtraction from the total results and some considered corrections in the AT model like r iepAT and β p cause the results of the real contact area in this model to be lesser than the other models. At bigger separations, where the plastic region diminishes, the results are closer. e variations of the contact stiffness versus the mean separation and the effects of the surface parameter σ z on the contact stiffness are shown in Figure 8. As is seen from this figure in lower separation, the contact stiffness rises.
is trend can be explained by the hardening process, which occurs in contacting asperities or the connection which exists between the mean separation and the real contact area. Figure 8 demonstrates that the contact stiffness of smoother surfaces rises at a higher rate than the rougher one with diminishing separation. is is logical since the real contact area of the smoother surface rises at a higher rate than the rougher one with diminishing separation.
Contact stiffness variations versus mean separation for different models are shown in Figure 9. As can be seen from  Mathematical Problems in Engineering this figure, the variation trend of the AT model results shows good agreement with other recognized models. In Figure 10 the results of the contact area variations versus mean separation considering the experimental results, based on Kucharski et al [17] and other models, are compared. e material properties of the steel specimens considered in the experimental work are as follows: Young's modulus E � 200 GPa, Poisson's ratio v � 0.3, and tensile yield strength Y � 400 MPa. e specimens are made of carbon steel with 0.45% of carbon.
In [17], measuring of contact load and the contact area is performed by the test, and sand-blasting is used to produce the surface roughness of specimens. e loadings of the surfaces are performed incrementally in a few steps up to 300 MPa of nominal load. After each step of loading, the 3D topography of the surface is mapped and areas of plastic deformation are identified in the isometric view. e real contact area is measured by summarizing the areas of individual contact spots.
According to Figure 10, the results of the AT model rather than other analytical models are very close to the experimental results. Since in employed formulations of this model, the slip-stick effects are considered in elastic-plastic   zones and corrections on the modified plastic radius and plastic roughness parameters are performed.
In comparison to the experimental results, the error of the presented model at low mean separation and where the effects of plasticity are maximized is about 14%. However, for the other models, this error is too high, and for bigger mean separation, where the plastic region diminishes, all the results are closed to each other. On the other hand, the AT model offers a worthy assessment for the asperities contact problem, especially at the low contact area and separation where high contact pressure and plastic deformation usually exist.

Conclusions
In this task, the effects of the surface asperities on the contact force, contact area, and contact stiffness are studied. e lateral contact between asperities is assumed to be the general condition for contacting surfaces. e Gao model is used for the fundamental of formulations, and corrections for the elastic-plastic contact zone, fictitious asperity theory, and slip-stick effects are carried out. Also, a new harmonic spherical distribution function for applying the contact angle effects is introduced, and a corrected radius for plastic deformation is obtained. From the achieved results and performed discussion, the following major outcomes are derived: (1) In the contact of rough surfaces, total contact force, real contact area, and contact stiffness decrease with increasing the mean separation. ese parameters increase at a higher rate with decreasing mean separation on a smoother surface.
(2) e outcomes of the AT model are close to the other available analytical models. e deviations in some results, especially where the plastic deformation is greater, are due to the effects of the considered parameters like the effects of the slip-stick in the elastic-plastic formulation and corrections in some important parameters like plastic radiuses and plastic roughness. (3) Because of considering the effects of slip-stick in the elastic-plastic regime, performing some modifications in plastic region parameters such as plastic radius, roughness parameter, formulations of F ep1 , A ep1 , and K ep1 , and using a new harmonic spherical distribution function for contact angles effects, the results of the AT model, in comparison with the other considered models with high accuracy, are near to the experimental results, especially in a lower separation where the plastic deformation is dominated. (4) In comparison to the experimental results, the error of the AT model considering low mean separation is about 14%. While for the other models, this error is too high. For greater mean separation, where the plastic region diminishes, all results are approximately close to each other. On the other hand, the results of the presented model offer very good estimations for the asperities contact characteristics, especially at the small-contact area and separation where high-contact pressure and plastic deformation usually exist.
In future works, the presented model can be used to simulate the applied problems with high accuracy. For example, in the wheel-rail contact problem, various failures such as fracture and wear are prevalent that need to be expressed and analyzed with an accurate model.