Thermal contact resistance modeling prediction method based on L-L contact model and modified three-dimensional fractal

Abstract

The invention discloses a contact thermal resistance modeling prediction method based on an L-L contact model and a modified three-dimensional fractal, and belongs to the technical field of mechanical joint surfaces. Comprising the following three steps: firstly, considering the deformation factor of the micro-convex body on the joint surface, expressing the contact deformation of the micro-convex body according to a corrected three-dimensional fractal theory, obtaining the critical cross sectional areas of the micro-convex body in three stages of elasticity, elastoplasticity and plasticity, and determining a size distribution function of micro-contact points; then determining an integral interval according to the maximum cross section area of the single micro-bulge, expressing the relation between the cross section areas of the three deformation stages of the single micro-bulge and the real area by using an L-L contact model, and then performing integration to obtain the total real contact area and the total normal load between the joint surfaces; and finally, establishing a relationship between the total normal load of the junction surface and the contact area and a mathematical expression of the total contact thermal resistance of the junction surface, deducing the contact thermal conductance between the junction surfaces of the three stages of the micro-convex body, and obtaining a contact thermal resistance result by obtaining a reciprocal of the result.

Description

Technical Field

[0001] The present invention belongs to the technical field of mechanical joint surfaces, and particularly relates to a modeling method for theoretical calculation of contact thermal resistance. Background Art

[0002] During the manufacturing process of machine tools or various mechanical equipment, a large number of components are assembled through bolts or other connecting parts, and the contact surface between the assembled parts is called the joint surface. During the working process of mechanical equipment, each component, each part, and the entire mechanical equipment will be under the action of various heat sources, and the whole machine will generate a temperature field and thermal deformation, thereby affecting the overall performance of the mechanical structure. During the design stage, the contact thermal resistance will affect the temperature transfer between joint surfaces. For example Figure 1 As shown, there will be a temperature step between joint surfaces. Accurate thermal characteristic analysis between joint surfaces is very crucial for product design, and it is particularly important for improving the stability and machining accuracy of machine tools.

[0003] In the research on the contact thermal resistance of joint surfaces, for the expression of the contact state of micro-protrusions, the currently more commonly used ones are the W-A contact model based on statistical parameters and the M-B model based on the fractal theory of rough surfaces (two-dimensional fractal theory) and Hertz contact theory. Currently, there are still large errors in calculating the contact thermal resistance between joint surfaces using existing contact thermal resistance models. The main reasons are as follows: The existing calculation models for the contact thermal resistance of joint surfaces do not consider the influence of the deformation of the micro-protrusion matrix. The mutual interaction of micro-protrusions between joint surfaces generates a large influence on the calculation of contact thermal resistance due to matrix deformation, which cannot be ignored. The relationship between the cross-sectional area and the actual contact area cannot be accurately expressed, resulting in inaccurate actual area results obtained from the final calculation. These factors all lead to a large difference between the calculated value of the contact thermal resistance of joint surfaces and the actual data. This method uses the modified three-dimensional fractal theory instead of the two-dimensional fractal theory to express the surface topography of micro-protrusions, which is closer to the actual situation, and uses the L-L contact model to better express the area relationship of micro-protrusions, improving the accuracy of theoretical calculation. Summary of the Invention

[0004] The present invention proposes a contact thermal resistance modeling and prediction method based on the L-L contact model and the modified three-dimensional fractal theory to solve the problem that the theoretical calculation value of the contact thermal resistance between joint surfaces differs greatly from the actual value, so as to improve the accuracy of the theoretical calculation value of the contact thermal resistance. This method is based on the modified three-dimensional fractal theory, considering the accurate relationship between the actual contact area and the cross-sectional area of micro-protrusions, and proposes a method to improve the theoretical calculation effect of the contact thermal resistance between joint surfaces. The flow chart is as Figure 2 shown, and the specific implementation steps are as follows:

[0005] Step 1: Considering the deformation factors of asperities on the joint surface, express the contact deformation of asperities according to the modified three-dimensional fractal theory, obtain the critical cross-sectional areas of asperities in the elastic, elastoplastic, and plastic stages, and determine the size distribution function of micro-contact points.

[0006] Step 1.1: Analyze the contact deformation of asperities using the modified three-dimensional fractal theory. The contact state of asperities is as Figure 3 shown, and the contact deformation is expressed by the amplitude difference between the wave peaks and wave valleys of the three-dimensional function.

[0007] Step 1.2: By analyzing the critical deformation of asperities in the three stages (elastic, elastoplastic, and plastic), obtain the critical cross-sectional areas of the three stages and determine the size distribution function of micro-contact points.

[0008] Step 2: Determine the integration interval based on the maximum cross-sectional area of a single asperity. Use the L-L contact model to express the relationship between the cross-sectional areas and the true area of a single asperity in the three stages (elastic, elastoplastic, and plastic), and then integrate to obtain the total true contact area and total normal load between the joint surfaces.

[0009] Step 2.1: Determine the maximum cross-sectional area of a single asperity, judge which stages the asperity deformation between the joint surfaces includes, and determine the subsequent integration interval.

[0010] Step 2.2: Establish the relationship expression between the cross-sectional areas of asperities and the true contact area in the three stages.

[0011] Step 2.3: Integrate to obtain the total true contact area and total normal load between the joint surfaces.

[0012] Step 3: Establish the relationship between the total normal load and the contact area of the joint surface and the mathematical expression of the total contact thermal resistance of the joint surface. Derive the contact thermal conductance between the joint surfaces in the three stages of asperities, and obtain the reciprocal of the result to get the contact thermal resistance result.

[0013] Step 3.1: Use the mathematical expression to establish the relationship between the total normal load and the contact area of the joint surface.

[0014] Step 3.2: Integrate with the integration interval determined in Step 2 to obtain the mathematical expression of the contact thermal conductance between the joint surfaces.

[0015] Step 3.3: Take the reciprocal of the contact thermal conductance between the joint surfaces to obtain the mathematical expression of the contact thermal resistance.

[0016] The beneficial effects of the present invention are as follows: For the theoretical calculation problem of the contact thermal resistance between joint surfaces, considering the asperity deformation and the L-L contact model, by establishing a theoretical calculation model of the contact thermal resistance, the accuracy of the theoretical calculation of the contact thermal resistance is improved, providing a basis for subsequent accurate finite element simulation. Description of the Drawings

[0017] Figure 1 It is a flow chart of a contact thermal resistance modeling prediction method based on the L-L contact model and the modified three-dimensional contact theory.

[0018] Figure 2 It is the temperature step diagram of the joint surface after adding the contact thermal resistance.

[0019] Figure 3 It is a schematic diagram of a single asperity contact model.

[0020] Figure 4 It is the difference in the expression of the real contact area between the L-L contact model and the common template function.

[0021] Figure 5 It is a flow chart of the derivation of the contact thermal resistance. Specific implementation manners

[0022] The technical solution of the present invention will be described in detail below in conjunction with the accompanying drawings of the specification:

[0023] Step 1: Considering the deformation factors of the asperities on the joint surface, the contact deformation of the asperities is characterized according to the modified three-dimensional fractal theory, and the critical cross-sectional areas of the asperities in the elastic, elastoplastic, and plastic stages are obtained, and the size distribution function of the micro contact points is determined.

[0024] Step 1.1: Analyze the contact deformation of the asperities using the modified three-dimensional fractal theory. The contact state of the asperities is as Figure 3 shown, and the contact deformation is expressed by the amplitude difference between the wave peaks and wave valleys of the three-dimensional function;

[0025] The contact deformation of the asperities is expressed as

[0026]

[0027] where δ is the contact deformation of a single asperity on the surface of the joint surface, D is the fractal dimension of the surface of the joint surface, G is the fractal scale coefficient of the surface of the joint surface, is a parameter related to the frequency density, and r' is the radius of the cross-sectional area of a single asperity.

[0028] Step 1.2: By analyzing the critical deformations of the three stages (elastic, elastoplastic, and plastic) of the asperity deformation, the critical cross-sectional areas of the three stages are obtained and the size distribution function of the micro contact points is determined.

[0029] For the deformation of a single asperity that satisfies the geometric expression

[0030] (r′) 2 = 2Rδ

[0031] The expression of the curvature R of a single asperity is obtained as

[0032]

[0033] The deformation magnitude δ at the boundary point from the elastic stage to the elastoplastic stage of a single asperity 1c and the cross-sectional area a 1c ′

[0034]

[0035] The deformation magnitude δ at the boundary point from the elastoplastic stage to the plastic stage of a single asperity 2c and the cross-sectional area a 2c ′

[0036]

[0037] where H---the hardness of the softer material, k----a parameter related to the Poisson's ratio, k = 0.454 + 0.41v, E----a parameter related to the Poisson's ratio, E = ((1 - v1 2 ) / E1+(1 - v2 2 ) / E2) -1 , v1, v2 and E1, E2 are the Poisson's ratios and elastic moduli of the two materials respectively.

[0038] The size distribution function of the micro-contact points introducing the expansion factor can be expressed as

[0039]

[0040] where D---the three-dimensional fractal dimension; a l ′----the maximum cross-sectional area of a single asperity; ψ----the domain expansion factor, which can be obtained from the transcendental equation

[0041]

[0042] Step 2: Determine the integration interval based on the maximum cross-sectional area of a single asperity, and use the L-L contact model to express the relationship between the cross-sectional area and the real area in the three deformation stages (elastic, elastoplastic, and plastic) of a single asperity. Then, integrate to obtain the total real contact area and the total normal load between the joint surfaces.

[0043] Step 2.1: Determine the maximum cross-sectional area of a single asperity, judge which stages the asperity deformation between the joint surfaces includes, and determine the subsequent integration interval;

[0044] Step 2.2: Establish the relationship expression between the cross-sectional area of the asperity in the three stages and the real contact area;

[0045]

[0046] where a is the real contact area, a′ is the cross-sectional area, H G1is the coefficient related to material properties and the fractal parameters of the joint surface. The L-L contact model can more accurately express the relationship between the true contact area and the cross-sectional area after asperity contact compared to the commonly used template function. The difference between the two is as Figure 4 shown.

[0047] Step 2.3: Integrate to obtain the total true contact area and the total normal load between the joint surfaces.

[0048] (1) The total true contact area between the joint surfaces is as follows

[0049] a) When the maximum asperity deformation is in the plastic stage, i.e., 0 ≤ a l ′ ≤ a 2c ′, that is, the maximum cross-sectional area a l ′ of a single asperity is less than the cross-sectional area a 2c ′ at the boundary point between the elastic-plastic stage and the plastic stage of a single asperity.

[0050]

[0051] b) When the maximum asperity deformation is in the elastic-plastic stage, i.e., a 2c ′ ≤ a l ′ ≤ a 1c ′, the maximum cross-sectional area a l ′ of a single asperity is between the cross-sectional area a 2c ′ at the boundary point between the elastic-plastic stage and the plastic stage of a single asperity and the cross-sectional area a 1c ′ at the boundary point between the elastic stage and the elastic-plastic stage of a single asperity.

[0052]

[0053] c) When the maximum asperity deformation is in the elastic stage, i.e., a l ′ ≥ a 1c ′, the maximum cross-sectional area a l ′ of a single asperity is greater than the cross-sectional area a 1c ′ at the boundary point between the elastic stage and the elastic-plastic stage of a single asperity.

[0054]

[0055] (2) The total normal load between the joint surfaces is expressed as follows

[0056] For the analysis of a single asperity, the normal loads f e , f ep and f p of a single asperity in the three stages can be respectively expressed as

[0057]

[0058] where

[0059]

[0060] a) When the maximum asperity deformation is in the plastic stage, i.e., 0 ≤ a l ′ ≤ a 2c ′

[0061]

[0062] b) When the maximum asperity deformation is in the elastic - plastic stage, i.e., a 2c ′ ≤ a l ′ ≤ a 1c ′

[0063]

[0064] c) When the maximum asperity deformation is in the elastic stage, i.e., a l ′ ≥ a 1c ′

[0065]

[0066] Step 3: Establish the relationship between the total normal load on the joint surface and the contact area, and the mathematical expression of the total contact thermal resistance of the joint surface. Derive the contact thermal conductance between the joint surfaces in the three stages of asperities, and obtain the reciprocal of the result to get the contact thermal resistance result.

[0067] Step 3.1: Use the mathematical expression to establish the relationship between the total normal load on the joint surface and the contact area;

[0068] The contact pressure P of the joint surface is expressed as

[0069]

[0070] F is the total normal load on the joint surface

[0071] Step 3.2: Integrate within the integration interval determined in Step 2 to obtain the mathematical expression of the contact thermal conductance between the joint surfaces;

[0072] The process of modeling the contact thermal conductance of the joint surface is as follows

[0073]

[0074] where

[0075]

[0076] λ A λ B The heat transfer coefficient between the two joint surfaces, represents the dimensionless actual contact area, A ais the nominal contact area, h c is the contact thermal conductance of a single asperity.

[0077] The calculation process of the contact thermal conductance of the joint surface is as Figure 5 shown. Analyzing a single asperity, the contact thermal conductance in its three different stages is as follows:

[0078]

[0079] a) When the maximum asperity deformation is in the plastic deformation stage, i.e., 0 ≤ a l ′ ≤ a 2c ′

[0080]

[0081] b) When the maximum asperity deformation is in the elastoplastic stage, i.e., a 2c ′ ≤ a l ′ ≤ a 1c ′

[0082]

[0083] c) When the maximum asperity deformation is in the elastic deformation stage, i.e., a l ′ ≥ a 1c ′

[0084]

[0085]

[0086] Step 3.3: Take the reciprocal of the contact thermal conductance between the joint surfaces to obtain the mathematical expression of the contact thermal resistance. The contact thermal resistance R c is

[0087]

[0088] Measure the roughness of the joint surface through a 3D profilometer. After calculation, the D and G parameters of the joint surface can be obtained. Substitute them into the above formula to obtain the numerical value of the joint surface thermal resistance.

Claims

1. A contact thermal resistance modeling and prediction method based on the LL contact model and modified three-dimensional fractal, characterized in that: include: Step 1: Considering the deformation factor of the micro-convex body on the bonding surface, the contact deformation of the micro-convex body is characterized according to the modified three-dimensional fractal theory, the critical cross-sectional area of ​​the micro-convex body in the elastic, elastoplastic and plastic stages is obtained, and the size distribution function of the micro-contact point is determined; Step 2: Determine the integration interval with the maximum cross-sectional area of ​​a single micro-convex body, and use the LL contact model to express the relationship between the cross-sectional area and the true area of ​​a single micro-convex body in three stages of deformation. The three stages are elastic, elastoplastic and plastic. Then integrate to obtain the total true contact area and total normal load between the bonding surfaces. Step 3: Establish the relationship between the total normal load of the bonding surface and the contact area and the mathematical expression of the total contact thermal resistance of the bonding surface. Derivate the contact thermal conductivity between the bonding surfaces of the three stages of the micro-convex body, and obtain the inverse of the result to obtain the contact thermal resistance result.

2. The contact thermal resistance modeling prediction method based on LL contact model and modified three-dimensional fractal according to claim 1 is characterized in that: In step 2, after determining the integral interval with the maximum cross-sectional area of ​​a single micro-convex body, the LL contact model is used to express the relationship between the cross-sectional area of ​​the micro-convex body and the true contact area, and the total true contact area and the total normal load between the bonding surfaces are calculated; The LL contact model accurately expresses the relationship between the cross-sectional area of ​​the micro-convex body in the elastic-plastic stage and the actual contact area. The LL contact model is expressed as follows: elasticity: Elasticity and plasticity: Plasticity: a = a′ Where a is the real contact area, a′ is the cross-sectional area; D is the fractal dimension of the bonding surface; H G1 is the coefficient related to the material properties and fractal parameters of the bonding surface; (1) Calculate the total real contact area of ​​the joint surface; a) When the maximum convex deformation is in the plastic stage, that is, 0≤a l ′≤a 2c ′, the maximum cross-sectional area of ​​a single microconvex body l ′ is smaller than the cross-sectional area a of the boundary between the elastic-plastic stage and the plastic stage of a single micro-convex body 2c ' b) When the maximum asperity deformation is in the elastic-plastic stage, that is, a 2c ′≤a l ′≤a 1c ′; Maximum cross-sectional area of ​​a single microconvex body a l ′ is at the cross-sectional area a of the boundary between the elastic-plastic stage and the plastic stage of a single micro-convex body 2c ′ and the cross-sectional area a at the boundary between the elastic stage and the elastoplastic stage 1c 'between c) When the maximum asperity deformation is in the elastic stage, that is, a l ′≥a 1c ′; Maximum cross-sectional area of ​​a single microconvex body a l ′ is greater than the cross-sectional area a of the boundary between the elastic stage and the elastoplastic stage of a single microconvex body 1c ′; Where D is the fractal dimension of the surface of the binding surface; ψ is the domain expansion factor. (2) Solve the total normal load on the joint surface; a) When the maximum convex deformation is in the plastic stage, that is, 0≤a l ′≤a 2c ′; b) When the maximum asperity deformation is in the elastic-plastic stage, that is, a 2c ′≤a l ′≤a 1c ′; c) When the maximum asperity deformation is in the elastic stage, that is, a l ′≥a 1c ′; Where δ is the contact deformation of a single micro-convex body on the bonding surface; R is the radius of curvature of a single micro-convex body; H is the hardness of the softer material; E is a parameter related to Poisson's ratio, E = ((1-v1 2 ) / E1+(1-v2 2 ) / E2) -1 , v1, v2 and E1, E2 are the Poisson’s ratio and elastic modulus of the two materials respectively.

3. The contact thermal resistance modeling prediction method based on LL contact model and modified three-dimensional fractal according to claim 1, characterized in that: In step 3, the contact thermal conductance between the bonding surfaces of the micro-convex body at three stages is derived based on the total real contact area and total normal load between the bonding surfaces obtained in step 2; Solution for contact thermal conductivity of the joint surface: a) When the maximum convex deformation is in plastic deformation, that is, 0≤a l ′≤a 2c ′; b) When the maximum asperity deformation is in the elastic-plastic stage, that is, a 2c ′≤a l ′≤a 1c ′; c) When the maximum convex deformation is in the elastic deformation stage, that is, a l ′≥a 1c ′; in ----Dimensionless actual contact area.

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