An elastic-plastic stepwise probabilistic integral contact thermal resistance calculation method based on heat transfer
Patent Information
- Application Number
- CN202610965927.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-07-01
- Publication Date
- 2026-10-30
AI Technical Summary
[0004]针对现有研究不足，以赫兹接触与经典传热学为理论基础，摒弃了传统分形理论用简化的几何模型来代替真实微观接触状态，引入多阶段接触力学模型，解决了传统单一变形假设模型无法覆盖全载荷工况的问题
[0056]以赫兹接触为理论基础，建立弹塑性分阶段接触热阻模型。本模型根据材料变形的不同程度，明确量化了弹性、弹塑性与塑性三个接触阶段，使接触状态的划分标准更贴合力学实际。摒弃了传统分形理论用简化的几何模型来代替真实微观接触状态，引入多阶段接触力学模型，解决了传统单一变形假设模型无法覆盖全载荷工况的问题。创新性地将基于传热学理论的接触热阻公式与粗糙峰高度概率密度积分相结合，构建了弹塑性分阶段接触热阻模型。该方法通过逐点统计不同高度粗糙峰的接触热传导贡献，更直接地还原了从微观单峰热传导到宏观界面总接触热传导的物理过程，有效降低了统计误差。
Smart Images
Figure CN123013397A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a method for calculating the elastoplastic stepwise probability integral contact thermal resistance based on heat transfer, belonging to the field of mechanical interface technology. Background Technology
[0002] The contact thermal resistance at assembly mating surfaces is a key factor affecting the overall thermal characteristics and thermal error control of high-end CNC machine tools and precision equipment. At the mating surfaces of critical machine tool components, contact thermal resistance directly determines the efficiency of heat transfer between components. Accurate modeling and calculation of contact thermal resistance is not only the core foundation for constructing a comprehensive thermal network model of the machine tool and predicting temperature field distribution, but also a prerequisite for achieving real-time thermal error compensation and improving equipment machining accuracy and operational stability, thus possessing significant engineering value for enhancing the performance of high-end equipment.
[0003] Current methods for calculating contact thermal resistance are mostly based on fractal theory. They establish geometric fractal models of rough surfaces to statistically map the microscopic contact state to the macroscopic thermal resistance. However, these methods simplify the description of microscopic contact mechanics and fail to accurately reflect the true contact state under different load conditions, including elastic, elastoplastic, and plastic deformation stages. This results in significant discrepancies between calculated contact thermal resistance and actual operating conditions. Furthermore, most existing methods do not deeply couple the contact mechanics model with classical heat transfer theory, failing to fully reconstruct the physical process from microscopic single-roughness peak contact to macroscopic interface heat transfer, thus limiting the model's engineering applicability and computational accuracy. Summary of the Invention
[0004] To address the shortcomings of existing research, this study, based on Hertzian contact and classical heat transfer theory, abandons the traditional fractal theory's use of simplified geometric models to represent the true microscopic contact state. Instead, it introduces a multi-stage contact mechanics model, solving the problem that traditional single-deformation assumption models cannot cover all load conditions. Innovatively, it combines the contact thermal resistance formula based on heat transfer theory with the probability density integral of roughness peak height to construct an elasto-plastic multi-stage contact thermal resistance model.
[0005] A method for calculating elastoplastic stepwise probability integral contact thermal resistance based on heat transfer includes the following steps:
[0006] Step 1: Establish the heat transfer formula for the contact thermal resistance of the interface based on heat transfer, the law of conservation of energy and the Fourier formula for heat conduction.
[0007] According to the law of conservation of energy and Fourier's formula for heat conduction, the contact thermal conductivity... It can be represented as:
[0008] (1)
[0009] In the formula Contact thermal conductivity; Lg The thickness of the gap between the mating surfaces; A C Actual contact area of the mating surface gap; A V The area of non-contact surfaces in the gap between mating surfaces; A is the nominal contact area; The thermal conductivity of the medium filling the uncontacted gaps in the bonding surface; k A k B These are the thermal conductivity coefficients of part A and part B, respectively.
[0010] In general, the medium in the gap between the contact surfaces of parts in a machine tool is air, and air has a low thermal conductivity. It is much smaller than the thermal conductivity of metals. , The second term in equation (1) can be ignored. Equation (1) can be written as equation (2):
[0011] (2)
[0012] If the pressure at the joint surface between the contacting parts is small and the thermal conductivity of the filling medium in the gap is large, the heat conduction at the joint surface mainly relies on the filling medium, then the second term in equation (1) is the main term and cannot be ignored.
[0013] Step 2: Construct a three-stage deformation contact model of the assembly interface based on Hertzian contact theory, encompassing elastic, elastoplastic, and plastic stages.
[0014] This method, based on the GW and KE models, divides the contact behavior of rough surfaces between parts into three stages: elastic stage, elastoplastic stage, and plastic stage. When the deformation is less than the critical elastic deformation ω... e At this point, the roughness is in the elastic stage. When the deformation ω is greater than the critical elastic deformation ω... e At this point, the roughness is in the elastic and elastoplastic deformation stage. When the deformation ω is greater than 110×ω e At this point, the roughness is in the fully plastic stage. The critical elastic deformation ω e for:
[0015] (3)
[0016] (4)
[0017] In the formula ω e The critical deformation value in the elastic deformation-elastoplastic deformation stage; k u Average contact pressure coefficient; R: average radius of all micro-convex spheres before deformation; H: sphere hardness, the relationship between the sphere hardness H and its yield strength Y is H=2.8Y; E: *Hertzian modulus; Young's modulus of part 1 (E1); Young's modulus of part 2 (E2); Poisson's ratio of part 1 (V1); Poisson's ratio of part 2 (V2).
[0018] Step 2.1: Construct a contact model for the elastic deformation stage.
[0019] When 0 < < At this time, the micro-protrusion is in the elastic deformation stage, and the contact area and contact pressure of the micro-protrusion are respectively expressed by equations (5) and (6):
[0020] (5)
[0021] (6)
[0022] In the formula Contact area of the sphere during the elastic deformation stage; Contact pressure between the two parts during the elastic deformation stage; ω roughness peak elastic deformation amount;
[0023] Step 2.2: Constructing a contact model for the elastoplastic deformation stage
[0024] when < < hour( The micro-protrusion is in the elastoplastic deformation stage, and the contact area and contact pressure of the micro-protrusion are respectively Equations (7) and (8).
[0025] (7)
[0026] (8)
[0027] Here, k1(ω) and g(ω) are polynomial expressions that can be constructed as functions of the independent variable ω, as shown in the following equation:
[0028] (9)
[0029] (10)
[0030] In the formula Contact area of the sphere during the elastoplastic deformation stage; Contact pressure between two parts during the plastic deformation stage; Critical deformation values between the elastic-plastic deformation stage and the plastic deformation stage. ;
[0031] Step 2.3: Constructing a contact model for the plastic deformation stage
[0032] when < hour( The micro-protrusion is in the plastic deformation stage, and the contact area and contact pressure of the micro-protrusion are respectively Equation (11) and Equation (12).
[0033] (11)
[0034] (12)
[0035] Where H is the hardness of the softer material in the two contact surfaces; Contact area of the sphere during the plastic deformation stage; Contact pressure between two parts during the plastic deformation stage.
[0036] Step 3: Substitute the elastic-elastoplastic-plastic contact model into the contact thermal resistance heat transfer formula to construct an elastoplastic stepwise probability integral contact thermal resistance model based on heat transfer.
[0037] Step 3.1: Thermal conductivity model of micro-convexity contact in the elastic stage, as shown in equation (13).
[0038] Right now
[0039] (13)
[0040] In the formula, Z represents the height of the roughness peak relative to the average line of the contact surface; The elastic deformation of the rough peak; d is the average gap height between the two contact surfaces.
[0041] Step 3.2: Model of thermal conductivity of micro-protrusion contact in the elastoplastic stage, as shown in equation (14).
[0042] Right now ( )
[0043] (14)
[0044] Step 3.3: Model of thermal conductivity of micro-protrusion contact during the plastic stage, as shown in equation (15).
[0045] Right now
[0046] (15)
[0047] Step 3.4: Construct an overall model of elastoplastic stepwise probability integral contact thermal resistance based on heat transfer.
[0048] The contact thermal conductivity of a single micro-protrusion under different contact conditions is as follows:
[0049] (16)
[0050] Since the height z of the rough peaks at the interface is randomly distributed and the entire contact surface conforms to a Gaussian distribution, the total contact thermal conductivity per unit area of the interface is obtained by summing the contact heat transfer coefficients of all rough peaks through probability density integration, as shown in equation (17):
[0051] (17)
[0052] The contact thermal resistance R is given by the following formula (18).
[0053] (18)
[0054] Step 4: Input the relevant data to calculate the contact thermal resistance based on the elastoplastic fractional probability integral of heat transfer.
[0055] This invention provides a method for calculating elasto-plastic stepwise probability integral contact thermal resistance based on heat transfer, which has the following advantages:
[0056] Based on Hertzian contact theory, an elastoplastic staged contact thermal resistance model is established. This model clearly quantifies the three contact stages—elastic, elastoplastic, and plastic—according to different degrees of material deformation, making the classification criteria for contact states more closely aligned with mechanical reality. It abandons the traditional fractal theory's use of simplified geometric models to replace the true microscopic contact state, introducing a multi-stage contact mechanics model to solve the problem that traditional single-deformation assumption models cannot cover all load conditions. Innovatively, the contact thermal resistance formula based on heat transfer theory is combined with the probability density integral of roughness peak height to construct an elastoplastic staged contact thermal resistance model. This method, by statistically analyzing the contact heat conduction contribution of roughness peaks at different heights point by point, more directly reconstructs the physical process from microscopic single-peak heat conduction to macroscopic total interface contact heat conduction, effectively reducing statistical errors. Attached Figure Description
[0057] Figure 1 This is a flowchart of an elastoplastic stepwise probability integral contact thermal resistance method based on heat transfer.
[0058] Figure 2 This is a schematic diagram of the contact thermal resistance of the mechanical assembly mating surface.
[0059] Figure 3 This is a schematic diagram of heat flowing through the interface.
[0060] Figure 4 This is a schematic diagram of the contact surfaces of the mating surfaces. Detailed Implementation
[0061] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings:
[0062] Step 1: Establish the heat transfer formula for the contact thermal resistance of the interface based on heat transfer, the law of conservation of energy and the Fourier formula for heat conduction.
[0063] like Figure 2 As shown, based on the law of conservation of energy and the Fourier formula for heat conduction, we can obtain equations (1) and (2).
[0064]
[0065]
[0066] Where h c Contact heat transfer coefficient; Nominal contact area; k A k B The thermal conductivity of parts A and B are respectively; T1 is the temperature at position 1 of part A; T 3A Temperature at the mating surface of part A; T 3B Temperature at the mating surface of part B; Temperature at position 5 of part B (T5); 𝛥, 𝑥- 𝐴 The distance between positions 1 and 3 of part A; 𝛥, 𝑥- B The distance between positions 3 and 5 of part B;
[0067] In equations (1) and (2) That is, contact thermal resistance, using This gives the expression (3):
[0068]
[0069] For contact problems involving metallic materials, radiative heat transfer weakens below 900K, accounting for less than 2% of the total heat transfer. When calculating heat conduction between the mating surfaces of metallic parts, radiative heat transfer between the gaps in the mating surfaces is neglected. Equations (1) and (2) can be written as follows:
[0070]
[0071] In the formula, Q represents heat conduction; L represents heat transfer. g The thickness of the gap between the mating surfaces; A C Actual contact area of the mating surface gap; A V The area of non-contact surfaces in the gap between mating surfaces; The thermal conductivity of the medium filling the uncontacted gaps in the bonding surface;
[0072] Contact thermal conductivity It can be represented as:
[0073]
[0074] In general, the medium in the gap between the contact surfaces of parts in a machine tool is air, and air has a low thermal conductivity. It is much smaller than the thermal conductivity of metals. , The second term in equation (5) can be ignored. Equation (5) can be written as equation (6):
[0075]
[0076] If the pressure at the joint surface between the contacting parts is small and the thermal conductivity of the filling medium in the gap is large, the heat conduction at the joint surface mainly relies on the filling medium, then the second term in equation (5) is the main term and cannot be ignored.
[0077] Step 2: Construct a three-stage deformation contact model of the assembly interface based on Hertzian contact theory, encompassing elastic, elastoplastic, and plastic stages.
[0078] This method, based on the GW and KE models, divides the contact behavior of rough surfaces between parts into three stages: elastic stage, elastoplastic stage, and plastic stage. When the deformation is less than the critical elastic deformation ω... e At this point, the roughness is in the elastic stage. When the deformation ω is greater than the critical elastic deformation ω... e At this point, the roughness is in the elastic and elastoplastic deformation stage. When the deformation ω is greater than 110×ω e At this point, the roughness is in the fully plastic stage. The critical elastic deformation ω e for:
[0079]
[0080]
[0081] In the formula ω e The critical deformation value in the elastic deformation-elastoplastic deformation stage; k u Average contact pressure coefficient; R: average radius of all micro-convex spheres before deformation; H: sphere hardness, the relationship between the sphere hardness H and its yield strength Y is H=2.8Y; E: * Hertzian modulus; Young's modulus of part 1 (E1); Young's modulus of part 2 (E2); Poisson's ratio of part 1 (V1); Poisson's ratio of part 2 (V2);
[0082] Where, k u The average contact pressure coefficient is related to the Poisson's ratio ν of the softer material of the two contacting parts, and satisfies the formula... .
[0083] Based on Hertzian contact theory and the Kogut-Etsion model, the relationship between the elastic contact area, elastic contact load, and deformation ω can be established as follows:
[0084] Step 2.1: Construct a contact model for the elastic deformation stage.
[0085] When 0 < < At this time, the micro-protrusion is in the elastic deformation stage, and the contact area and contact pressure of the micro-protrusion are respectively expressed by equations (9) and (10):
[0086]
[0087]
[0088] In the formula Contact area of the sphere during the elastic deformation stage; Contact pressure between the two parts during the elastic deformation stage; ω roughness peak elastic deformation amount;
[0089] Step 2.2: Constructing a contact model for the elastoplastic deformation stage
[0090] when < < hour( The micro-protrusion is in the elastoplastic deformation stage, and the contact area and contact pressure of the micro-protrusion are respectively Equation (11) and Equation (12).
[0091]
[0092]
[0093] Here, k1(ω) and g(ω) are polynomial expressions that can be constructed as functions of the independent variable ω, as shown in the following equation:
[0094]
[0095]
[0096] In the formula Contact area of the sphere during the elastoplastic deformation stage; Contact pressure between two parts during the plastic deformation stage; Critical deformation values between the elastic-plastic deformation stage and the plastic deformation stage. ;
[0097] Step 2.3: Constructing a contact model for the plastic deformation stage
[0098] when < hour( The micro-protrusion is in the plastic deformation stage, and the contact area and contact pressure of the micro-protrusion are respectively Equation (15) and Equation (16).
[0099]
[0100]
[0101] Where H is the hardness of the softer material in the two contact surfaces; Contact area of the sphere during the plastic deformation stage; Contact pressure between two parts during the plastic deformation stage.
[0102] Step 3: Substitute the elastic-elastoplastic-plastic contact model into the contact thermal resistance heat transfer formula to construct an elastoplastic stepwise probability integral contact thermal resistance model based on heat transfer.
[0103] Step 3.1: Thermal conductivity model of micro-convexity contact in the elastic stage. Right now
[0104] The formula for the elastic deformation stage Substituting into the contact thermal conductivity formula (6) based on heat transfer, we obtain the following equation: :
[0105]
[0106] In the formula, z represents the height of the roughness peak relative to the average line of the contact surface; The elastic deformation of the rough peak; d is the average gap height between the two contact surfaces.
[0107] Step 3.2: Model of thermal conductivity of micro-astigmas in the elastoplastic stage. Right now ( )
[0108] The formula for the elastic-plastic deformation stage Substituting into the contact thermal conductivity formula (6) based on heat transfer, we obtain equation (18):
[0109]
[0110] Step 3.3: Model of thermal conductivity of micro-asperities in the plastic stage. Right now
[0111] Formulas for the plastic deformation stage Substituting into the contact thermal conductivity formula (6) based on heat transfer, we obtain equation (19).
[0112]
[0113] Step 3.4: Construct an overall model of elastoplastic stepwise probability integral contact thermal resistance based on heat transfer.
[0114] The contact thermal conductivity of a single micro-protrusion under different contact conditions is as follows:
[0115]
[0116] Since the height z of the rough peaks at the interface is randomly distributed and the entire contact surface follows a Gaussian distribution, the total contact thermal conductivity per unit area of the interface is obtained by summing the contact heat transfer coefficients of all rough peaks through probability density integration, as shown in the equation. :
[0117]
[0118] The contact thermal resistance R is given by the following formula (22).
[0119]
[0120] Step 4: Input the relevant data to calculate the contact thermal resistance based on the elastoplastic fractional probability integral of heat transfer.
Claims
1. A method for calculating elastoplastic stepwise probability integral contact thermal resistance based on heat transfer, characterized in that, Includes the following steps: Step 1: Establish the heat transfer formula for the contact thermal resistance of the interface based on heat transfer, the law of conservation of energy, and the Fourier formula for heat conduction; According to the law of conservation of energy and Fourier's formula for heat conduction, the contact thermal conductivity... It can be represented as: （1） In the formula Contact thermal conductivity; L g The thickness of the gap between the mating surfaces; A C Actual contact area of the mating surface gap; A V The area of non-contact surfaces in the gap between mating surfaces; A is the nominal contact area; The thermal conductivity of the medium filling the uncontacted gaps in the bonding surface; k A k B These are the thermal conductivity coefficients of part A and part B, respectively.
2. If the second term in equation (1) is ignored, then equation (1) can be written as equation (2): （2） Step 2: Construct a three-stage deformation contact model of the assembly interface based on Hertzian contact theory, encompassing elastic, elastoplastic, and plastic phases; Based on the GW model and the KE model, the contact behavior of rough surfaces between parts is divided into three stages: elastic stage, elastoplastic stage and plastic stage. When the deformation is less than the critical elastic deformation ω e At this point, the roughness is in the elastic stage; when the deformation ω is greater than the critical elastic deformation ω... e At this point, the roughness is in the elastic and elastoplastic deformation stage; when the deformation ω is greater than 110×ω e At this point, the roughness is in the fully plastic stage; the critical elastic deformation ω e for: (3) (4) In the formula ω e The critical deformation value in the elastic deformation-elastoplastic deformation stage; k u Average contact pressure coefficient; R: average radius of all micro-convex spheres before deformation; H: sphere hardness, the relationship between the sphere hardness H and its yield strength Y is H=2.8Y; E: * Hertzian modulus; Young's modulus of part 1 (E1); Young's modulus of part 2 (E2); Poisson's ratio of part 1 (V1); Poisson's ratio of part 2 (V2); Step 3: Substitute the elastic-elastoplastic-plastic contact model into the contact thermal resistance heat transfer formula to construct an elastoplastic step-probability integral contact thermal resistance model based on heat transfer. Step 4: Measure the relevant micro-morphological parameters using a 3D profilometer, consult relevant manuals to obtain the relevant performance parameters of the material, and input the relevant data into the contact thermal resistance model based on the elastoplastic fractional probability integral of heat transfer to obtain the relevant values.
3. The method for calculating elastoplastic stepwise probability integral contact thermal resistance based on heat transfer as described in claim 1, characterized in that, Step 2 includes, Step 2.1: Constructing a contact model for the elastic deformation stage; When 0 < < At this time, the micro-protrusion is in the elastic deformation stage, and the contact area and contact pressure of the micro-protrusion are respectively expressed by equations (5) and (6): (5) (6) In the formula Contact area of the sphere during the elastic deformation stage; Contact pressure between the two parts during the elastic deformation stage; ω roughness peak elastic deformation amount; Step 2.2: Construct a contact model for the elastoplastic deformation stage; when < < hour( The micro-protrusion is in the elastoplastic deformation stage, and the contact area and contact pressure of the micro-protrusion are respectively expressed by equations (7) and (8): (7) (8) Here, k1(ω) and g(ω) are constructed as polynomial expressions of functions of the independent variable ω, as shown in the following equation: (9) (10) In the formula, Contact area of the sphere during the elastoplastic deformation stage; Contact pressure between two parts during the plastic deformation stage; Critical deformation values between the elastic-plastic deformation stage and the plastic deformation stage. ; Step 2.3: Construct a contact model for the plastic deformation stage; when < hour( The micro-protrusion is in the plastic deformation stage, and the contact area and contact pressure of the micro-protrusion are respectively Equation (11) and Equation (12). (11) (12) Where H is the hardness of the softer material in the two contact surfaces; Contact area of the sphere during the plastic deformation stage; Contact pressure between two parts during the plastic deformation stage.
4. The method for calculating elastoplastic stepwise probability integral contact thermal resistance based on heat transfer as described in claim 2, characterized in that, Step 3 includes, Step 3.1: Thermal conductivity model of micro-convex contact in the elastic stage, as shown in Equation (13). Right now ; (13) In the formula, z represents the height of the roughness peak relative to the average line of the contact surface; Roughness peak deformation; d: average gap height between the two contact surfaces; Step 3.2: Model of thermal conductivity of micro-convexity contact in the elastoplastic stage, as shown in equation (14). Right now ( ) (14) Step 3.3: Model of thermal conductivity of micro-asperities in the plastic stage, as shown in Equation (15); Right now ; (15) Step 3.4: Construct an overall model of elastoplastic fractional probability integral contact thermal resistance based on heat transfer; The contact thermal conductivity of a single micro-protrusion under different contact conditions is as follows: (16) By summing the contact heat transfer coefficients of all rough peaks through probability density integration, the total contact thermal conductivity per unit area of the interface is obtained, as shown in equation (17): (17) The contact thermal resistance R is given by the following formula (18). (18)。 5. The method for calculating elastoplastic stepwise probability integral contact thermal resistance based on heat transfer as described in claim 1, characterized in that, Step 2 is to construct a three-stage deformation contact model of the assembly interface based on Hertzian contact theory, encompassing elastic, elastoplastic, and plastic stages. First, the critical deformation value ω for the elastic-elastoplastic stage is derived. e The contact mechanical parameters are determined by combining the equivalent elastic modulus correction formula; then, ω is used as... e With 110ω e Using these as boundaries, we construct calculation formulas for the contact area and contact pressure of micro-protrusions in the elastic, elastoplastic, and plastic stages, respectively, to achieve a precise, step-by-step description of the contact mechanical behavior of rough surfaces.
6. The method for calculating elastoplastic stepwise probability integral contact thermal resistance based on heat transfer as described in claim 1, characterized in that, Step 3 involves substituting the elastic-elastoplastic-plastic contact model into the contact thermal resistance heat transfer formula, and constructing a calculation model of unit area thermal conductivity and macroscopic contact thermal resistance by integrating the probability density of the Gaussian distribution of the rough peak height. This achieves a cross-scale mapping from the microscopic rough peak contact state to the macroscopic interface thermal resistance. By combining the staged contact model with the probability integration method, the mapping from the microscopic rough peak contact state to the macroscopic interface contact thermal resistance is realized.