Analysis method for revolved body shape micro-convex body contact based on quadratic function

Abstract

The invention relates to the field of micro-convex body contact analysis, in particular to a rotary body shape micro-convex body contact analysis method based on a quadratic function. The micro-convex body contact model meets the following requirements: (1) the shape of a micro-convex body is a quadratic function revolving body, the height of the micro-convex body follows Gaussian distribution, and the micro-convex body has the same peak curvature; (2) the contact angle is kept unchanged in the whole contact process, namely the contact angle during initial contact; (3) the micro-convex bodies are mutually independent and do not influence each other; (4) friction of the micro-convex body is not considered; and (5) the upper and lower micro-convex bodies have the same shape and size. According to the method, the contact load, the average contact load and the contact rigidity can be analyzed more accurately.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of micro-asperity contact analysis, in particular to a method for analyzing micro-asperity contact based on a quadratic function of a rotary body shape. BACKGROUND

[0002] In the field of machinery, connecting structures are important structures for transmitting force and load between parts in various complex equipment. The surfaces of the parts in contact with each other are called joint surfaces, and the contact characteristics of the joint surfaces directly affect the working performance of the mechanical system. During the whole life cycle of the product, when subjected to vibration and impact load, the joint surface will exhibit mechanical behaviors such as friction, wear, slip, collision and separation, causing phenomena such as a decrease in interfacial preload, structural damage and a decrease in contact stiffness, which affect the contact characteristics of the joint surface and thus affect the life and reliability of the equipment. Predicting and calculating the contact characteristics of the joint surface is an important topic in the engineering field. Generally, a joint surface contact model can be established to calculate and predict the contact characteristics. How to establish a more accurate and effective contact model to study the contact stiffness is the basis for further developing the overall dynamics modeling and analysis of equipment, and has important theoretical and engineering application value for evaluating the mechanical performance of complex equipment, and is of great significance for in-depth understanding of the mechanical behavior of the joint surface and improving the overall performance and extending the service life of the equipment.

[0003] The elastic contact theory proposed by Hertz in 1882 is the origin of classical contact mechanics, which regards the height distribution of micro-asperities on a rough surface as a random function, and proposes a statistical micro-elastic contact model (GW model) based on the Hertz theory, laying the foundation of contact mechanics. Based on the GW model, most models use the traditional flat-micro-asperity assumption to establish the model. The existing BGT model, CEB model and ZMC model are combined to establish a new statistical model of rough surfaces. For example, the shape of the micro-asperity is assumed to be an ellipsoid to establish a finite element model, which first analyzes in detail the elastic-plastic deformation behavior of an ellipsoid with different ellipticities in contact with a rigid plane, filling the gap in this field. JENG studied the change of the friction coefficient of the contact surface with elliptical micro-asperities under different parameters, and found that the friction coefficient increases with the increase of the plastic index, but decreases with the increase of the effective radius ratio.

[0004] The existing rough surface measurement research found that the sine contact can better simulate the micro-convex contact of the rough surface than the hemispherical contact. Jackson et al. found that for light load contact, the behavior of the sine contact can be very similar to that of the sphere, but as the load increases and the contact approaches completion, the behavior of the sine surface is very different from that of the sphere. Domestic scholar An Qi et al. aimed at the micro-convex profile of the grinding surface, assumed the micro-convex shape as a half-period sine curve of revolution, and established a contact model for effectively simulating the micro-convex of the grinding surface. Sepehri and Farhang derived the control equation of the micro-convex oblique contact based on the KE model, and established the SF contact model. Closed-form approximate equations were developed for the contact force components and the contact area. Li and Hong et al. combined the KE model and the SF model to establish an oblique contact model of the ellipsoidal micro-convex of the double rough surface. Jamshidi and Ahmadian et al. only considered the elastic deformation, established an oblique contact model of the spherical micro-convex of the double rough surface, studied the friction contact hysteresis behavior between two planes, corrected the Mindlin spherical elastic contact equation by analyzing the partial slip and total slip of the rough surface, and developed an improved interface contact model by using the double rough surface contact theory.

[0005] However, the prediction accuracy of the existing model is low, and the general prediction accuracy is about 20%. SUMMARY

[0006] In view of the deficiencies of the prior art, the present application provides an analysis method based on the contact of the micro-convex of the quadratic function of revolution. Based on the fitting results of the rough topography measurement data, the parabolic body of revolution is used as the micro-convex shape, and the quadratic function of revolution micro-convex side contact model is established by considering the substrate deformation factor, so that the contact behavior of the micro-convex in the whole process of elasticity, elastoplasticity and plasticity can be accurately described.

[0007] The technical scheme of the present application is: an analysis method based on the contact of the micro-convex of the quadratic function of revolution, comprising the following steps: characterized in that: step 1, the step of establishing a micro-convex contact model, the micro-convex contact model meets the following requirements: (1) the micro-convex shape is a quadratic function of revolution, the height obeys Gaussian distribution, and has the same peak curvature; (2) the contact angle remains unchanged in the whole contact process, that is, the contact angle at the initial contact; (3) the micro-convexes are independent of each other and do not affect each other; (4) the friction of the micro-convex is not considered; (5) the upper and lower micro-convex shapes are the same size;

[0008] Step 2, the step of analyzing the contact characteristics in the elastic stage;

[0009] Step 3, the step of analyzing the contact characteristics in the plastic stage;

[0010] Step 4, the step of analyzing the contact characteristics in the elastoplastic stage;

[0011] Elastic stage, plastic stage, elastic-plastic stage output contact load, average contact load, contact stiffness three parameter results.

[0012] The beneficial effects of the present application are: the present application can more accurately analyze the contact load, the average contact load, the contact stiffness. BRIEF DESCRIPTION OF DRAWINGS

[0013] Figure 1 It is a two-dimensional geometric modeling schematic diagram.

[0014] Figure 2 It is a force analysis diagram. DETAILED DESCRIPTION

[0015] The technical solutions of the present application are further described below in combination with the drawings.

[0016] As Figure 1 and Figure 2 The inventors found through multiple test results that the fitting mean square deviation of the parabola to the micro-convex shape is smaller than that of the circular arc, so it can be preliminarily inferred that the parabola is more accurate for fitting the micro-convex shape, in order to avoid contingency, a longer two-dimensional topography data is randomly selected along the X-axis direction to perform the same operation. Through rough topography data analysis, the quadratic function fitting micro-convex shape is more accurate than the circular arc, so the present application will establish a quadratic function rotary body micro-convex contact model based on rough topography measurement.

[0017] The analysis method of the present application based on the shape of the quadratic function rotary body micro-convex contact includes the following steps:

[0018] Step 1, the step of establishing a micro-convex contact model

[0019] The micro-convex geometry model is constructed as Figure 1 shown, the micro-convex contact model meets the following requirements: (1) the micro-convex shape is a quadratic function rotary body, the height obeys Gaussian distribution, and has the same peak curvature; (2) the contact angle remains unchanged throughout the contact process, that is, the contact angle when initially contacting; (3) the micro-convex bodies are independent of each other and do not affect each other; (4) the friction of the micro-convex body is not considered; (5) the upper and lower micro-convex shapes are the same size.

[0020] Step 2, analysis method of contact characteristics in the elastic stage

[0021] Step 21, derivation of the contact angle in the elastic contact stage analysis

[0022] The relationship between the critical deformation when the micro-convex body begins to deform completely plastically and the critical deformation of the initial yield point is: The relationship between the critical deformation when the micro-convex body begins to deform completely plastically and the critical deformation of the initial yield point is:​

[0023]

[0024] When the micro-convex body deformation According to the Hertz contact theory, the micro-convex body is in the elastic contact stage, The actual deformation is

[0025] The derivation of the contact angle in the elastic contact stage The relationship between the parabola fat and thin, the distance between the micro-convex bodies, and the contact angle of the entire contact process.

[0026] The equation of the cross-section of the micro-convex body contact pair in the XZ plane is as follows:

[0027] 1. Left side opening upward parabola

[0028] 2. Right side opening downward parabola

[0029] Where, , is the opening coefficient of the parabola; , are the horizontal coordinates of the vertexes of the two parabolas respectively; , are the vertical coordinates of the vertexes of the two parabolas respectively. The parabolas and do not intersect at the beginning, and the central axis distance is , that is, When moves downward until it is tangent to , let the vertex of drop by at this time, then The new equation of .

[0030] When the two parabolas are tangent, they satisfy the following two conditions at a certain point

[0031] 1. The function values are equal:

[0032]

[0033] 2. The derivative value (slope) is equal:

[0034]

[0035] The slope at the tangent point is ​Therefore, the angle between the tangent and the x-axis satisfy:

[0036]

[0037] Will , Substituting into the above equation, we get:

[0038]

[0039] Therefore, the included angle The relationship with the shape parameters of the micro-convexity is as follows:

[0040]

[0041] When the shape and size of the micro-protrusions are consistent, that is... included angle satisfy:

[0042]

[0043] Step 22: Solve for the combined radius of curvature of the two micro-convex bodies at the contact point.

[0044] The shape equations of the two micro-convex bodies in three-dimensional space are as follows:

[0045]

[0046]

[0047] x and y represent the x-coordinate and y-coordinate, respectively. Solve for the contact point of the two parabolic surfaces. The combined radius of curvature at that point Gradually move down until it is in line with Tangent at point Then the following conditions apply:

[0048]

[0049] Normal vector condition:

[0050] for Its gradient (normal vector) is:

[0051]

[0052] for Its gradient (normal vector) is:

[0053]

[0054] Since the normal vectors are the same at the contact point:

[0055]

[0056]

[0057] Substitute , we have :

[0058]

[0059]

[0060] Because , we have , substitute the above equation, we have

[0061]

[0062] For At the point , i.e. and , the principal radii of curvature are respectively

[0063] The principal radius of curvature along the direction :

[0064]

[0065] Substitute , we have

[0066]

[0067] The principal radius of curvature along the direction :

[0068]

[0069] For

[0070] Similarly, at the point , i.e. and , the principal radii of curvature are respectively

[0071] The principal radius of curvature along the direction :

[0072]

[0073] The principal radius of curvature along the direction :

[0074]

[0075] The integrated curvature is calculated for each principal direction separately:

[0076]

[0077] The integrated radius of curvature at the contact point is calculated using the harmonic mean method (closed-form equations for three dimensional elastic-plastic contact of nominally flat rough surfaces) :

[0078]

[0079] where

[0080] When , the integrated radius of curvature is:

[0081]

[0082] Step 23, solve the normal deformation of the asperity contact plane

[0083] According to Sepehri's theory, let , be the radii of curvature of the two asperity vertices, then we have:

[0084]

[0085] According to the geometric relationship, we have: ,

[0086] Let: , using the quadratic approximation near the asperity vertices, we can get

[0087]

[0088] denotes the distance between the two plates, denotes the normal deformation without considering the deformation of the substrate

[0089] Consider the deformation of the substrate: when the maximum pressure on the substrate is constantly changing, the relationship between the normal displacement of different positions on the substrate surface and the horizontal distance from the central axis can be expressed as

[0090]

[0091] where F is the load on the asperity, Eeff and E are the effective and elastic modulus of the substrate, respectively, ν is the Poisson's ratio of the substrate material; R is the radius of the contact area at the top of the substrate. Thus, the deformation of the substrate at the contact center can be expressed as

[0092]

[0093] The contact stiffness of the substrate is

[0094]

[0095] According to the Hertz contact theory, the contact stiffness of the asperity is

[0096]

[0097] Due to the force balance, we have Thus, we have , δs, δa, and δ are the substrate deformation, asperity deformation, and total deformation, respectively;

[0098] From we have:

[0099]

[0100] Assume that the elastic modulus of the asperity is the same as that of the substrate, i.e. Ea represents the elastic modulus of the asperity.

[0101] Substitute and into we have:

[0102]

[0103] Substitute into we have:

[0104]

[0105] Let and substitute into we have: is a characteristic parameter, which has no physical meaning in order to make the formula look more concise;

[0106] ​​​​​

[0107] Using iterative method, we can get:

[0108]

[0109] where , , ,

[0110] Step 24 Derivation of contact load formula

[0111] According to Hertz law, the contact load in elastic stage is proportional to the deformation of the contact area.

[0112]

[0113] where is the combined Young's modulus of the two surfaces, The formula for calculating is where , is the Poisson's ratio of the upper and lower micro-convex. is the equivalent curvature radius at the contact point, is the deformation along the normal direction of the contact area.

[0114] Step 25 Derivation of contact stiffness formula

[0115] According to the definition of contact stiffness, the derivative of the elastic contact load is obtained:

[0116]

[0117] Step 26 Derivation of contact area formula

[0118] According to Hertz law, the contact area in the elastic stage is

[0119]

[0120] Therefore, the normal average contact load is

[0121]

[0122] Step 3, analysis method of plastic stage contact characteristics

[0123] Step 31, derivation of contact area formula

[0124] When the micro-convex deformation At this point, the micro-forehead will undergo complete plastic contact deformation. Based on the principle of volume conservation, the contact area model for the plastic contact stage of the micro-forehead is derived. According to geometric conditions, assume that the two vertices of the micro-forehead are initially at the same height, and the lateral distance is... The downward displacement of the upper micro-convex body is The volume enclosed by the intersection of the contours at this point is:

[0125]

[0126] in:

[0127]

[0128] therefore

[0129]

[0130] It represents the distance between two planes.

[0131] Substituting the values:

[0132]

[0133] Based on the assumption of volume conservation, the shape of the plastic flow region can be considered as a cylinder, whose volume is the product of the contact area and the deformation. Therefore:

[0134]

[0135] From geometric relations, we can obtain Deformation in direction for:

[0136]

[0137] Substituting the values, we can obtain the result. The contact area of ​​the plastic flow region in the direction is

[0138]

[0139] The contact area in the z-direction is

[0140]

[0141] The actual contact area is the sum of the areas of the plastic flow region and the central region.

[0142]

[0143] Step 32, Contact Load Formula Derivation

[0144] The contact load of the plastic contact stage of the micro asperity is equal to the product of the contact area and the average contact pressure, where, Therefore, the contact load of the micro asperity in the normal direction is

[0145]

[0146] represents the material hardness of the micro asperity, represents the average contact load of the micro asperity, represents the deformation amount of the micro asperity in the plastic stage (the symbols and subscripts are explained: is the deformation amount, a refers to the micro asperity, which is distinguished from the base b, and p represents the plastic stage in the contact stage, which is distinguished from the elastic stage e and the elastic-plastic stage ep)

[0147] Step 33, Contact Stiffness Formula Derivation

[0148] Then the normal contact stiffness of the micro asperity is:

[0149]

[0150] Step 4, Analysis Method of Contact Characteristics in Elastic-Plastic Stage

[0151] When δc≤ ≤110δc, that is, the micro asperity is in the elastic-plastic deformation stage, the contact deformation is relatively complex, both elastic deformation and plastic deformation exist, which is a "mix" of the two. When the contact deformation is near the critical deformation δc, the elastic deformation plays a major role at this time; and when the contact deformation is close to 110δc, the plastic deformation plays a major role at this time. Therefore, the relationship between the contact parameters and the deformation in this stage is determined by the "mix" of the elastic deformation and the plastic deformation.

[0152] Step 41, Average Contact Load

[0153] Before discussing the mixed elastic-plastic stage, the following assumptions are made: the pressure on the contact area is uniform.

[0154] Initial Yielding Critical Point

[0155]

[0156] represents the maximum contact pressure coefficient, which is a constant parameter related to the material Poisson's ratio.

[0157] Critical deformation at which the asperity begins to deform plastically The relationship between the critical deformation at which the asperity begins to deform plastically and the critical deformation at which the asperity begins to deform elastically is:

[0158]

[0159] According to the theory of contact mechanics, when the normal approach increases to the critical point of initial yielding , the asperity begins to deform plastically, and at this time the plastic deformation only occurs inside the asperity, which is surrounded by a large elastic deformation area; with the increase of the normal contact load, the plastic deformation area gradually evolves from the inside to the surface, and finally when the critical point of complete plastic deformation is reached , the entire asperity is in a complete plastic deformation state, and the contact pressure on the contact surface is everywhere the same, all H. From the perspective of continuum mechanics, this process is continuous, smooth and monotonic. Therefore, the relationship between the actual contact area and the average contact pressure and the normal approach should satisfy the following conditions at the two critical points, respectively

[0160]

[0161]

[0162] (Symbols and subscripts are explained: A represents the contact area, P represents the average contact load; subscript e represents the elastic stage, ep represents the elastic-plastic stage, and p represents the plastic stage; the above all represent the corresponding function values when )

[0163] The ZMC model uses an interpolation method to connect the complete elastic deformation stage and the complete plastic deformation stage, but the average contact pressure given by it is in logarithmic relationship with the normal approach, which cannot guarantee the smoothness at the critical points. This is improved to realize the continuity and smoothness of the change of the contact state variable, but the change of the average contact pressure of the elastic-plastic deformation stage derived by using a high-order interpolation function is not monotonic.

[0164] An elliptical curve is used, with its short semi-axis coinciding with the vertical line passing through , and the elliptical curve is selected as:

[0165]

[0166] where A is the length of the long semi-axis of the elliptical curve; (H-B) is the length of the short semi-axis of the elliptical curve; and B is the offset of the center of the elliptical curve relative to the axis.

[0167] Therefore, the normal average contact load is​

[0168]

[0169] According to the formula , can be rewritten as

[0170]

[0171] According to the formula, the contact load at and the slope is

[0172]

[0173] According to the smooth and continuous condition

[0174]

[0175] The expression of A, B is solved

[0176]

[0177] , are all intermediate variables.

[0178] Substitute The average contact load in the elastic-plastic stage can be solved as:

[0179]

[0180] The whole process:

[0181]

[0182] Step 42 contact load

[0183] The hermite interpolation method is used to deduce the contact load formula in the elastic-plastic stage, and the contact load function in the elastic-plastic stage is set as a cubic hermite interpolation function form:

[0184]

[0185] According to the ZMC model, the following boundary conditions are set:

[0186] (1) The elastic contact load and the elastic-plastic contact load at

[0187] (2) The elastic-plastic contact load and the plastic contact load are equal (equivalence condition).

[0188] (3) The elastic contact stiffness and the elastoplastic contact stiffness are equal (equal slope condition).

[0189] (4) The elastoplastic contact stiffness and plastic contact stiffness are equal (equal slope condition).

[0190]

[0191] Elastic phase contact load:

[0192]

[0193] Contact load during the plastic stage:

[0194]

[0195]

[0196]

[0197] Solving for interpolation coefficients

[0198]

[0199] because

[0200]

[0201] Substitution :

[0202]

[0203] make ,but

[0204]

[0205] Substitute the known

[0206]

[0207] make

[0208]

[0209] Then in Place Simplify to:

[0210]

[0211] From Solve:

[0212]

[0213] Substitute Simplify to:

[0214]

[0215] Solve

[0216]

[0217] Solve the coefficient:

[0218]

[0219] Therefore, the final elastic-plastic stage solution load formula is

[0220]

[0221] Where .

[0222] The whole process of micro convex contact load calculation formula is as follows:

[0223]

[0224] Step 43 contact stiffness

[0225] According to the contact stiffness formula of elastic stage and plastic stage, , (constant), similar to the initial condition of average contact load, so the same can be fitted with elliptical curve.

[0226] Let

[0227] Select elliptical curve:

[0228]

[0229] Where is the length of the major axis of the elliptical curve; This is the length of the minor semi-axis of the elliptic curve; relative to the center of the ellipse The offset of the shaft. The contact stiffness in the elastic stage is: ,Will Substituting the values, we can obtain the contact stiffness in the elastic stage as:

[0230]

[0231] According to the formula ,exist The contact stiffness and slope at the point are

[0232]

[0233] Based on the conditions of smoothness and continuity, we have

[0234]

[0235] Solving for the expressions of A and B yields the following results.

[0236]

[0237] Substitution The contact stiffness in the elastoplastic stage can be solved as follows:

[0238]

[0239] Therefore, the contact stiffness throughout the entire contact process of the micro-protrusion is:

[0240]

[0241] The final outputs of all three stages are the results of three parameters: contact load, average contact load, and contact stiffness.

[0242] Elastic phase:

[0243] Contact load:

[0244] Average contact load:

[0245] Contact stiffness:

[0246] Plastic stage:

[0247] Contact load:

[0248] Average contact load:

[0249] Contact stiffness:

[0250] Elastic-plastic stage:

[0251] Contact load:

[0252]

[0253] Average contact load:

[0254]

[0255] Contact stiffness:

[0256]

[0257] The contact load, average contact load and contact stiffness formulas of the whole process can be obtained by connecting the three stages in series

[0258] The contact load calculation formula of the whole process is:

[0259]

[0260] The average contact load calculation formula of the whole process is:

[0261]

[0262] The contact stiffness calculation formula of the whole process is:

[0263]

[0264] The present application can more accurately contact load, average contact load, contact stiffness. Through simulation calculation, the contact load precision can reach 8%, the average contact load precision can reach 6%, the contact stiffness precision can reach 10%, thereby effectively improving the analysis precision.

Claims

1. An analysis method based on quadratic function revolved body shape micro-asperity contact, comprising the following steps: characterized in that: Step 1, the step of establishing the micro asperity contact model, the micro asperity contact model meets the following requirements: (1) the micro asperity shape is a quadratic function of a rotary body, the height obeys Gaussian distribution, and has the same peak curvature; (2) the contact angle remains unchanged throughout the contact process, that is, the contact angle at the initial contact; (3) the micro asperities are independent of each other and do not affect each other; (4) the friction of the micro asperities is not considered; (5) the upper and lower micro asperities have the same shape and size; Step 2, the step of analyzing the contact characteristics in the elastic stage; Step 3, the step of analyzing the contact characteristics in the plastic stage; Step 4, the step of analyzing the contact characteristics in the elastic-plastic stage; The three parameter results of contact load, average contact load and contact stiffness in the elastic stage, plastic stage and elastic-plastic stage are outputted; The equation of the section line of the micro asperity contact pair in the XZ plane is as follows: Left side opening upward parabolic Right side opening downward parabolic wherein , is the opening coefficient of the parabola; , are the abscissas of the vertices of the two parabolas, respectively; , are the ordinates of the vertices of the two parabolas, respectively.

2. The method of claim 1, wherein the micro-asperity contact is based on a quadratic function of a body of revolution shape. The analysis result in the elastic stage is: Contact load: Average contact load: Contact stiffness: ; is the normal deformation, is the equivalent radius of curvature at the contact point, is the combined Young's modulus of the two surfaces.

3. The analytical method for contact of micro-convex bodies based on the shape of a quadratic function rotating body according to claim 1, characterized in that: The analysis result in the plastic stage is: Contact load: Average contact load: Contact stiffness: the representative micro asperity deformation amount in the plastic phase, the material hardness of the representative micro asperity, , is the opening coefficient of the parabola; is the distance from the center axis.

4. The analytical method for contact of micro-convex bodies based on the shape of a quadratic function rotating body according to claim 1, characterized in that: The analysis result in the elastic-plastic stage is: Contact load: Average contact load: Contact stiffness: ； δc is the critical deformation, A, B, C, D are intermediate variables, is the combined Young's modulus of the two surfaces, is the angle of the tangent with the x-axis, is the actual deformation, is the integrated radius of curvature, , , is an intermediate variable , , is the critical deformation, K represents the maximum contact pressure coefficient, and H is the material hardness; 。 5. The analytical method for contact of micro-convex bodies based on the shape of a quadratic function rotating body according to claim 2, characterized in that: The analysis process of step 2 includes the following steps: Step 21, derivation of the contact angle of the elastic contact phase analysis parabola and They do not intersect at the beginning. The distance is the centerline distance, when Move down until you reach When tangent, let's assume that at this time... The peak descended , The angle between the tangent and the x-axis is , and the angle between the tangent and the y-axis is Step 22, solving the comprehensive curvature radius of the two micro asperities at the contact point radius of curvature : wherein ; Step 23, solving the normal deformation of the micro-convex body contact plane wherein , , , denotes the distance between the two plates, denotes the normal deformation without considering the substrate deformation ； F represents the magnitude of the load on the microprotrusion, and Eeff and E are the effective elastic modulus and the elastic modulus of the substrate, respectively, ν is the Poisson's ratio of the substrate material; Rtop represents the radius of the contact area at the top of the substrate. Step 24, derivation of the contact load formula According to Hertz's law, the contact load in the elastic stage is inversely proportional to the deformation wherein is the combined Young's modulus of the two surfaces, The formula for the calculation of wherein , is the Poisson's ratio of the material of the upper and lower asperities; is the equivalent radius of curvature at the contact point, is the deformation along the normal to the contact area; Step 25, derivation of the contact stiffness formula According to the definition of contact stiffness, the normal contact stiffness is obtained by deriving the elastic contact load: Step 26, derivation of the contact area formula According to Hertz's law, the contact area in the elastic stage is The normal average contact load is 。

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