CN108613922B - Bonding surface static friction factor three-dimensional fractal prediction method considering adhesive force - Google Patents

Bonding surface static friction factor three-dimensional fractal prediction method considering adhesive force Download PDF

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CN108613922B
CN108613922B CN201810381922.XA CN201810381922A CN108613922B CN 108613922 B CN108613922 B CN 108613922B CN 201810381922 A CN201810381922 A CN 201810381922A CN 108613922 B CN108613922 B CN 108613922B
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薛棚升
朱立达
刘阔
阚德广
张海权
黄绪杰
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Abstract

本发明提供一种考虑粘着力的结合面静摩擦因数三维分形预测方法，涉及机械结合面技术领域。该方法首先将描述结合面表面形貌的二维分形函数改进为三维分形函数，并用此函数波谷与波峰的差值表示接触变形量。再分别计算弹性变形阶段、弹塑性变形阶段的结合面表面微凸体的变形量与临界接触面积；同时计算结合面所受总法向载荷、总粘着力、总切向载荷与结合面接触面积的关系。最后建立结合面静摩擦因数与法向载荷、切向载荷及粘着力之间的关系。本发明提供的考虑粘着力的结合面静摩擦因数三维分形预测方法，得到的结合面静摩擦因数可靠性强，更贴近实际情况，可为预测、控制精密机械结合面的静摩擦因数提供理论依据。

Figure 201810381922

The invention provides a three-dimensional fractal prediction method for the static friction coefficient of a joint surface considering the adhesive force, and relates to the technical field of mechanical joint surfaces. In this method, the two-dimensional fractal function describing the surface topography of the joint surface is first improved into a three-dimensional fractal function, and the difference between the trough and the peak of this function is used to represent the contact deformation. Then calculate the deformation amount and the critical contact area of the surface asperities on the joint surface in the elastic deformation stage and the elastic-plastic deformation stage respectively; at the same time, calculate the total normal load, total adhesive force, total tangential load and the joint surface contact area on the joint surface. Relationship. Finally, the relationship between the static friction factor of the joint surface and the normal load, tangential load and adhesive force is established. The three-dimensional fractal prediction method for the static friction coefficient of the joint surface provided by the invention considering the adhesion force has strong reliability and is closer to the actual situation, and can provide a theoretical basis for predicting and controlling the static friction coefficient of the precision mechanical joint surface.

Figure 201810381922

Description

一种考虑粘着力的结合面静摩擦因数三维分形预测方法A three-dimensional fractal prediction method for the static friction factor of the joint surface considering the adhesive force

技术领域technical field

本发明涉及机械结合面技术领域，尤其涉及一种考虑粘着力的结合面静摩擦因数三维分形预测方法。The invention relates to the technical field of mechanical joint surfaces, in particular to a three-dimensional fractal prediction method for the static friction coefficient of joint surfaces considering adhesive force.

背景技术Background technique

机床及大部分复杂的机械工程设备除了结构本体之外，还包含大量的零部件，装配零部件间的接触面称为结合面，或称为结合部。机械结合面由于外部载荷的作用，不可避免的产生摩擦现象，从而影响机械结构的整体性能。因此从理论上研究结合面的接触行为，在设计阶段预测结合面的摩擦特性，对于提高机械加工设备的工作效率、稳定性和加工精度显得尤为重要。In addition to the structural body, machine tools and most complex mechanical engineering equipment also contain a large number of components. The contact surface between the assembled components is called the joint surface, or joint part. Due to the external load, the mechanical joint surface inevitably produces friction, which affects the overall performance of the mechanical structure. Therefore, theoretically studying the contact behavior of the joint surface and predicting the friction characteristics of the joint surface in the design stage are particularly important for improving the working efficiency, stability and machining accuracy of the machining equipment.

对于结合面静摩擦因数的研究，一部分学者在经典赫兹接触理论的基础上，假设结合面微凸体的高度分布近似高斯分布并从统计学的角度出发，建立了机械结合面的静摩擦因数微观统计接触模型；另一部分学者基于分形理论，利用表征粗糙表面轮廓曲线的分形函数及岛屿面积分布函数建立了机械结合面的静摩擦因数分形接触模型，避免了微观统计接触模型受表面形貌测量仪器分辨率和采样长度影响的缺点。For the study of the static friction factor of the joint surface, some scholars, on the basis of the classical Hertzian contact theory, assumed that the height distribution of the asperities on the joint surface was approximately Gaussian distribution, and from a statistical point of view, established the static friction coefficient of the mechanical joint surface. Based on the fractal theory, some scholars established the static friction factor fractal contact model of the mechanical joint surface by using the fractal function representing the rough surface profile curve and the island area distribution function, avoiding the micro-statistical contact model being affected by the resolution and the resolution of the surface topography measuring instrument. Disadvantages affected by sampling length.

虽然在国内外学者的不懈努力下，机械结合面的静摩擦因数模型在不断地发展、完善，但是在重载条件下利用现有静摩擦因数模型计算结合面间的静摩擦因数仍有较大误差。其主要原因在于：在重载条件下，结合面间的粘着力对静摩擦因数计算的影响较大，不能忽略；同时，现有机械结合面静摩擦因数计算模型忽略了微凸体三维表面分形分布的特点。因此在重载条件下，利用现有理论方法并不能准确地计算出结合面的静摩擦因数。With the unremitting efforts of scholars at home and abroad, the static friction factor model of the mechanical joint surface is continuously developed and improved, but there is still a large error in the calculation of the static friction factor between the joint surfaces by using the existing static friction factor model under heavy load conditions. The main reasons are: under heavy load conditions, the adhesive force between the joint surfaces has a great influence on the calculation of the static friction factor and cannot be ignored; at the same time, the existing calculation model of the static friction factor of the mechanical joint surface ignores the fractal distribution of the three-dimensional surface of the asperities. Features. Therefore, under heavy load conditions, the static friction factor of the joint surface cannot be accurately calculated by using the existing theoretical methods.

发明内容SUMMARY OF THE INVENTION

针对现有技术的缺陷，本发明提供一种考虑粘着力的结合面静摩擦因数三维分形预测方法，实现重载条件下机械结合面静摩擦因数的计算。In view of the defects of the prior art, the present invention provides a three-dimensional fractal prediction method for the static friction coefficient of the joint surface considering the adhesive force, so as to realize the calculation of the static friction coefficient of the mechanical joint surface under heavy load conditions.

一种考虑粘着力的结合面静摩擦因数三维分形预测方法，包括以下步骤：A three-dimensional fractal prediction method for the static friction factor of the joint surface considering the adhesion force, including the following steps:

步骤1、模拟结合面表面微凸体的三维形貌：将描述结合面表面形貌的二维分形函数改进为三维分形函数，并用此函数的波谷与波峰的差值表示结合面单个微凸体的接触变形量δ，如下公式所示：Step 1. Simulate the three-dimensional topography of the asperities on the surface of the junction surface: improve the two-dimensional fractal function describing the surface topography of the junction surface into a three-dimensional fractal function, and use the difference between the trough and the peak of this function to represent a single asperity on the junction surface. The contact deformation δ of , is shown in the following formula:

δ＝2(11-3D)/2GD-2(lnγ)1/2π(D-3)/2a(3-D)/2 δ=2 (11-3D)/2 G D-2 (lnγ) 1/2 π (D-3)/2 a (3-D)/2

其中，G为结合面表面的分形尺度系数，D为结合面表面的分形维数，2＜D＜3，γ为与结合面表面形貌频率密度有关的参数，γ>1，一般取1.5，a为结合面表面单个微凸体实际弹性接触面积；Among them, G is the fractal scale coefficient of the surface of the bonding surface, D is the fractal dimension of the surface of the bonding surface, 2<D<3, γ is a parameter related to the frequency density of the surface topography of the bonding surface, γ>1, generally take 1.5, a is the actual elastic contact area of a single microprotrusion on the surface of the joint surface;

步骤2、分别计算弹性变形阶段、弹塑性变形阶段，结合面表面单个微凸体的变形量与临界接触面积，具体方法为：Step 2. Calculate the elastic deformation stage and the elastic-plastic deformation stage respectively, and the deformation amount and the critical contact area of a single asperity on the surface of the joint surface. The specific method is as follows:

步骤2.1、根据赫兹接触理论，结合面两粗糙表面相互接触时，将两粗糙表面转化为一等效粗糙表面与一刚性光滑平面相互接触，则结合面表面单个微凸体实际弹性接触面积a为等效粗糙表面上变形的单个微凸体与刚性光滑平面相交的截面面积，如下公式所示：Step 2.1. According to the Hertzian contact theory, when the two rough surfaces of the joint surface are in contact with each other, the two rough surfaces are transformed into an equivalent rough surface and a rigid smooth plane is in contact with each other, then the actual elastic contact area a of a single asperity on the joint surface is: The cross-sectional area of a deformed single asperity on an equivalent rough surface intersecting a rigid smooth plane is given by:

a＝πRδa=πRδ

其中，R为单个微凸体的曲率半径，δ为单个微凸体的接触变形量；where R is the radius of curvature of a single asperity, and δ is the contact deformation of a single asperity;

步骤2.2、计算单个微凸体的弹性临界变形量和弹性临界接触面积；Step 2.2. Calculate the elastic critical deformation and elastic critical contact area of a single asperity;

所述单个微凸体的弹性临界变形量δc的计算如下公式所示：The calculation of the elastic critical deformation amount δ c of the single asperity is shown in the following formula:

Figure BDA0001641260870000021
Figure BDA0001641260870000021

其中，kμ为摩擦力的修正因子，φ为材料特性系数；Among them, k μ is the correction factor of the friction force, and φ is the material characteristic coefficient;

所述单个微凸体的弹性临界接触面积ac的计算如下公式所示：The calculation of the elastic critical contact area a c of the single asperity is shown in the following formula:

Figure BDA0001641260870000022
Figure BDA0001641260870000022

进一步求出，单个微凸体的弹塑性一区临界接触面积apt1如下公式所示：Further, the critical contact area a pt1 of the elastic-plastic first region of a single asperity is given by the following formula:

apt1＝1101/(2-D)ac；a pt1 =110 1/(2-D) a c ;

单个微凸体的弹塑性二区临界接触面积apt2如下公式所示：The critical contact area a pt2 of the elastic-plastic second region of a single asperity is shown in the following formula:

apt2＝61/(2-D)ac；a pt2 =6 1/(2-D) a c ;

步骤3、计算结合面所受总法向载荷与结合面的接触面积的关系；Step 3. Calculate the relationship between the total normal load on the joint surface and the contact area of the joint surface;

根据分形理论，当al＞ac且D≠2.5时，结合面所受总法向载荷P与结合面的接触面积的关系如下公式所示：According to the fractal theory, when a l > a c and D≠2.5, the relationship between the total normal load P on the joint surface and the contact area of the joint surface is shown in the following formula:

Figure BDA0001641260870000031
Figure BDA0001641260870000031

当al＞ac且D＝2.5时，结合面所受总法向载荷P与结合面的接触面积关系如下公式所示：When a l > a c and D=2.5, the relationship between the total normal load P on the joint surface and the contact area of the joint surface is shown in the following formula:

Figure BDA0001641260870000032
Figure BDA0001641260870000032

其中，al为单个微凸体最大接触点面积，k为与结合面两接触材料中较软材料的屈服强度σy和硬度H相关的系数，三者之间的关系为：H＝kσy；E为结合面两接触材料的等效弹性模量，

Figure BDA0001641260870000033
E1、E2分别表示两接触材料的弹性模量，ν1、ν2分别表示两接触材料的泊松比；Among them, a l is the maximum contact point area of a single asperity, k is the coefficient related to the yield strength σ y and the hardness H of the softer material in the two contact materials of the joint surface, and the relationship between the three is: H = kσ y ; E is the equivalent elastic modulus of the two contact materials on the joint surface,
Figure BDA0001641260870000033
E 1 and E 2 respectively represent the elastic moduli of the two contact materials, and ν 1 and ν 2 respectively represent the Poisson's ratio of the two contact materials;

步骤4、计算结合面所受总粘着力与结合面接触面积的关系，具体方法为：Step 4. Calculate the relationship between the total adhesive force on the joint surface and the contact area of the joint surface. The specific method is:

步骤4.1、建立不同变形区间单个微凸体的粘着力与其接触面积的关系；Step 4.1. Establish the relationship between the adhesion force of a single asperity and its contact area in different deformation intervals;

根据分形理论，不同变形区间单个微凸体的粘着力与其接触面积的关系为：According to the fractal theory, the relationship between the adhesion force of a single asperity and its contact area in different deformation intervals is:

(1)在非接触区，δ/δc＜0，单个微凸体的粘着力与其接触面积的关系如下公式所示：(1) In the non-contact area, δ/δ c <0, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:

Figure BDA0001641260870000034
Figure BDA0001641260870000034

(2)在完全弹性区，0＜δ/δc＜1，单个微凸体的粘着力与其接触面积的关系如下公式所示：(2) In the fully elastic region, 0 < δ/δ c < 1, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:

Figure BDA0001641260870000035
Figure BDA0001641260870000035

(3)在弹塑性一区，1＜δ/δc＜6，单个微凸体的粘着力与其接触面积的关系如下公式所示：(3) In the elastic-plastic region, 1 < δ/δ c < 6, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:

Figure BDA0001641260870000041
Figure BDA0001641260870000041

(4)在弹塑性二区，6＜δ/δc＜110，单个微凸体的粘着力与其接触面积的关系如下公式所示：(4) In the second elastic-plastic region, 6 < δ/δ c < 110, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:

Figure BDA0001641260870000042
Figure BDA0001641260870000042

其中，Fsi为不同变形区间下的单个微凸体的粘着力，i＝1、…、4，Δγ′为粘着力能，ε为分子间距；Among them, F si is the adhesive force of a single asperity under different deformation intervals, i=1,...,4, Δγ' is the adhesive force energy, and ε is the molecular distance;

步骤4.2、建立结合面总粘着力与结合面接触面积的关系；Step 4.2, establish the relationship between the total adhesion of the joint surface and the contact area of the joint surface;

根据分形理论，结合面总粘着力与结合面接触面积的关系如下公式所示：According to the fractal theory, the relationship between the total adhesion force of the joint surface and the contact area of the joint surface is shown in the following formula:

Figure BDA0001641260870000043
Figure BDA0001641260870000043

其中，in,

Figure BDA0001641260870000044
Figure BDA0001641260870000044

Figure BDA0001641260870000045
Figure BDA0001641260870000045

Figure BDA0001641260870000046
Figure BDA0001641260870000046

Figure BDA0001641260870000051
Figure BDA0001641260870000051

其中，An为结合面名义接触面积，

Figure BDA0001641260870000052
为结合面接触点的面积分布函数，Fsi为不同变形区间下的单个微凸体的粘着力，Δγ′为粘着力能；Among them, An is the nominal contact area of the junction surface,
Figure BDA0001641260870000052
is the area distribution function of the contact point of the bonding surface, F si is the adhesion force of a single asperity under different deformation intervals, and Δγ′ is the adhesion force energy;

步骤5、计算结合面所受总切向载荷与结合面接触面积的关系；Step 5. Calculate the relationship between the total tangential load on the joint surface and the contact area of the joint surface;

根据分形理论，结合面所受总切向载荷与结合面接触面积的关系为：According to the fractal theory, the relationship between the total tangential load on the joint surface and the contact area of the joint surface is:

当al＞ac且D≠2.5时，结合面所受总切向载荷与结合面接触面积关系式为：When a l > a c and D≠2.5, the relationship between the total tangential load on the joint surface and the contact area of the joint surface is:

Figure BDA0001641260870000053
Figure BDA0001641260870000053

当al＞ac且D＝2.5时，结合面所受总切向载荷与结合面接触面积关系式为：When a l > a c and D=2.5, the relationship between the total tangential load on the joint surface and the contact area of the joint surface is:

Figure BDA0001641260870000054
Figure BDA0001641260870000054

步骤6、建立结合面静摩擦因数μ与结合面的总法向载荷P、总切向载荷Q及总粘着力Fs之间的关系，如下公式所示：Step 6. Establish the relationship between the static friction factor μ of the joint surface and the total normal load P, the total tangential load Q and the total adhesion force F s of the joint surface, as shown in the following formula:

Figure BDA0001641260870000055
Figure BDA0001641260870000055

由上述技术方案可知，本发明的有益效果在于：本发明提供的一种考虑粘着力的结合面静摩擦因数三维分形预测方法,与传统的基于有限元和二维分形理论的方法相比，采用三维分形函数表征结合面表面形貌，更贴近实际情况，同时使预测结果更准确。同时，本发明方法充分考虑了重载条件下结合面粘着力对静摩擦因数的影响，克服了现有基于分形理论的方法计算结合面静摩擦因数不准确的缺点。As can be seen from the above technical solutions, the beneficial effects of the present invention are: a three-dimensional fractal prediction method for the static friction factor of a joint surface provided by the present invention considering the adhesion force, compared with the traditional method based on finite element and two-dimensional fractal theory, adopts three-dimensional fractal method. The fractal function characterizes the surface topography of the joint surface, which is closer to the actual situation and makes the prediction result more accurate. At the same time, the method of the invention fully considers the influence of the adhesive force of the joint surface on the static friction factor under heavy load conditions, and overcomes the disadvantage of inaccurate calculation of the static friction factor of the joint surface by the existing method based on fractal theory.

附图说明Description of drawings

图1为本发明实施例提供的一种考虑粘着力的结合面静摩擦因数三维分形预测方法的流程图；Fig. 1 is a flow chart of a three-dimensional fractal method for predicting the static friction factor of a joint surface considering adhesive force provided by an embodiment of the present invention;

图2为本发明实施例提供的结合面表面形貌三维分形示意图；2 is a schematic diagram of a three-dimensional fractal surface topography of a joint surface provided by an embodiment of the present invention;

图3为本发明实施例提供的结合面受力示意图；FIG. 3 is a schematic diagram of force on a joint surface provided by an embodiment of the present invention;

图4为本发明实施例提供的三维表面分形维数取不同值时静摩擦因数与无量纲法向载荷的关系图；Fig. 4 is the relation diagram of static friction factor and dimensionless normal load when the three-dimensional surface fractal dimension provided by the embodiment of the present invention takes different values;

图5为本发明实施例提供的考虑粘着力、不考虑粘着力和静摩擦因数试验值三种情况下静摩擦因数与总法向载荷的关系图。FIG. 5 is a graph of the relationship between the static friction factor and the total normal load under three conditions of considering the adhesion force, not considering the adhesion force and the static friction factor test value provided by the embodiment of the present invention.

其中，1、刚性光滑平面；2、等效粗糙表面。Among them, 1. Rigid smooth surface; 2. Equivalent rough surface.

具体实施方式Detailed ways

下面结合附图和实施例，对本发明的具体实施方式作进一步详细描述。以下实施例用于说明本发明，但不用来限制本发明的范围。The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and embodiments. The following examples are intended to illustrate the present invention, but not to limit the scope of the present invention.

本实施例以某机械结合面为例，使用本发明的考虑粘着力的结合面静摩擦因数三维分形预测方法计算该结合面的静摩擦因数。In this embodiment, a mechanical joint surface is taken as an example, and the static friction coefficient of the joint surface is calculated by using the three-dimensional fractal prediction method of the joint surface static friction coefficient considering the adhesion force of the present invention.

一种考虑粘着力的结合面静摩擦因数三维分形预测方法，如图1所示，包括以下步骤：A three-dimensional fractal prediction method for the static friction coefficient of the joint surface considering the adhesion force, as shown in Figure 1, includes the following steps:

步骤1、模拟结合面表面微凸体的三维形貌：将描述结合面表面形貌的二维分形函数改进为三维分形函数，并用此函数的波谷与波峰的差值表示结合面单个微凸体的接触变形量δ，如下公式所示：Step 1. Simulate the three-dimensional topography of the asperities on the surface of the junction surface: improve the two-dimensional fractal function describing the surface topography of the junction surface into a three-dimensional fractal function, and use the difference between the trough and the peak of this function to represent a single asperity on the junction surface. The contact deformation δ of , is shown in the following formula:

δ＝2(11-3D)/2GD-2(lnγ)1/2π(D-3)/2a(3-D)/2 δ=2 (11-3D)/2 G D-2 (lnγ) 1/2 π (D-3)/2 a (3-D)/2

其中，G为结合面表面的分形尺度系数，D为结合面表面的分形维数，2＜D＜3，γ为与结合面表面形貌频率密度有关的参数，γ>1，一般取1.5，a为结合面表面单个微凸体实际弹性接触面积；Among them, G is the fractal scale coefficient of the surface of the bonding surface, D is the fractal dimension of the surface of the bonding surface, 2<D<3, γ is a parameter related to the frequency density of the surface topography of the bonding surface, γ>1, generally take 1.5, a is the actual elastic contact area of a single microprotrusion on the surface of the joint surface;

本实施例中，将描述结合面表面轮廓曲线的二维分形W-M函数改进为描述结合面表面分形曲面的三维分形W-M函数。给定等效粗糙面的参数为：结合面表面的分形尺度系数G＝2.01×10-9m，分形维数D＝2.501，与结合面表面形貌频率密度有关参数γ＝1.5，得到的结合面表面三维形貌如图2所示。In this embodiment, the two-dimensional fractal WM function describing the surface profile curve of the joint surface is improved into a three-dimensional fractal WM function describing the fractal curved surface of the joint surface. The parameters of the given equivalent rough surface are: the fractal scale coefficient G=2.01×10 -9 m, the fractal dimension D=2.501, and the parameter γ=1.5 related to the frequency density of the surface topography of the joint surface. The three-dimensional topography of the surface is shown in Fig.

步骤2、分别计算弹性变形阶段、弹塑性变形阶段，结合面表面单个微凸体的变形量与临界接触面积，具体方法为：Step 2. Calculate the elastic deformation stage and the elastic-plastic deformation stage respectively, and the deformation amount and the critical contact area of a single asperity on the surface of the joint surface. The specific method is as follows:

步骤2.1、根据赫兹接触理论，结合面两粗糙表面相互接触时，如图3所示，将两粗糙表面转化为一等效粗糙表面2与一刚性光滑平面1相互接触，则结合面表面单个微凸体实际弹性接触面积a为等效粗糙表面上变形的单个微凸体与刚性光滑平面相交的截面面积，如下公式所示：Step 2.1. According to the Hertzian contact theory, when the two rough surfaces of the joint surface are in contact with each other, as shown in Figure 3, the two rough surfaces are transformed into an equivalent rough surface 2 and a rigid smooth plane 1 is in contact with each other. The actual elastic contact area a of the convex body is the cross-sectional area of the single deformed asperity on the equivalent rough surface intersecting the rigid smooth plane, as shown in the following formula:

a＝πRδa=πRδ

其中，R为单个微凸体的曲率半径，δ为单个微凸体的接触变形量；where R is the radius of curvature of a single asperity, and δ is the contact deformation of a single asperity;

步骤2.2、计算单个微凸体的弹性临界变形量和弹性临界接触面积；Step 2.2. Calculate the elastic critical deformation and elastic critical contact area of a single asperity;

所述单个微凸体的弹性临界变形量δc的计算如下公式所示：The calculation of the elastic critical deformation amount δ c of the single asperity is shown in the following formula:

Figure BDA0001641260870000071
Figure BDA0001641260870000071

其中，kμ为摩擦力的修正因子，φ为材料特性系数；Among them, k μ is the correction factor of the friction force, and φ is the material characteristic coefficient;

所述单个微凸体的弹性临界接触面积ac的计算如下公式所示：The calculation of the elastic critical contact area a c of the single asperity is shown in the following formula:

Figure BDA0001641260870000072
Figure BDA0001641260870000072

进一步求出，单个微凸体的弹塑性一区临界接触面积apt1如下公式所示：Further, the critical contact area a pt1 of the elastic-plastic first region of a single asperity is given by the following formula:

apt1＝1101/(2-D)ac；a pt1 =110 1/(2-D) a c ;

单个微凸体的弹塑性二区临界接触面积apt2如下公式所示：The critical contact area a pt2 of the elastic-plastic second region of a single asperity is shown in the following formula:

apt2＝61/(2-D)ac；a pt2 =6 1/(2-D) a c ;

本实施例中，材料特性系数φ＝σy/E，结合面两接触面材料参数分别为：一个接触面的材料参数为：弹性模量E1＝197GPa，泊松比ν1＝0.3，屈服强度σy1＝346MPa，硬度H1＝478MPa；另一个接触面的材料参数为：弹性模量E2＝205GPa，泊松比ν2＝0.3，屈服强度σy2＝353MPa，硬度H2＝500MPa，计算出的单个微凸体的弹性临界接触面积ac＝5.986319×10- 10m2。In this embodiment, the material characteristic coefficient φ=σ y /E, the material parameters of the two contact surfaces of the joint surface are respectively: the material parameters of one contact surface are: elastic modulus E 1 =197GPa, Poisson’s ratio ν 1 =0.3, yield Strength σ y1 =346MPa, hardness H 1 =478MPa; material parameters of the other contact surface are: elastic modulus E 2 =205GPa, Poisson’s ratio ν 2 =0.3, yield strength σ y2 =353MPa, hardness H 2 =500MPa, The calculated elastic critical contact area of a single asperity is a c =5.986319×10 - 10 m 2 .

步骤3、计算结合面所受总法向载荷与结合面的接触面积的关系；Step 3. Calculate the relationship between the total normal load on the joint surface and the contact area of the joint surface;

根据分形理论，当al＞ac且D≠2.5时，结合面所受总法向载荷P与结合面的接触面积的关系如下公式所示：According to the fractal theory, when a l > a c and D≠2.5, the relationship between the total normal load P on the joint surface and the contact area of the joint surface is shown in the following formula:

Figure BDA0001641260870000081
Figure BDA0001641260870000081

当al＞ac且D＝2.5时，结合面所受总法向载荷P与结合面的接触面积关系如下公式所示：When a l > a c and D=2.5, the relationship between the total normal load P on the joint surface and the contact area of the joint surface is shown in the following formula:

Figure BDA0001641260870000082
Figure BDA0001641260870000082

其中，al为单个微凸体最大接触点面积，k为与结合面两接触材料中较软材料的屈服强度σy和硬度H相关的系数，三者之间的关系为：H＝kσy；E为结合面两接触材料的等效弹性模量，

Figure BDA0001641260870000083
E1、E2分别表示两接触材料的弹性模量，ν1、ν2分别表示两接触材料的泊松比；Among them, a l is the maximum contact point area of a single asperity, k is the coefficient related to the yield strength σ y and the hardness H of the softer material in the two contact materials of the joint surface, and the relationship between the three is: H = kσ y ; E is the equivalent elastic modulus of the two contact materials on the joint surface,
Figure BDA0001641260870000083
E 1 and E 2 respectively represent the elastic moduli of the two contact materials, and ν 1 and ν 2 respectively represent the Poisson's ratio of the two contact materials;

本实施例中，施加在结合面的法向载荷P＝100kN，计算出单个微凸体最大接触点面积al＝2.293740×10-4m2。In this embodiment, the normal load applied to the joint surface is P=100kN, and the maximum contact point area a l of a single asperity is calculated to be 2.293740×10 -4 m 2 .

步骤4、计算结合面所受总粘着力与结合面接触面积的关系，具体方法为：Step 4. Calculate the relationship between the total adhesive force on the joint surface and the contact area of the joint surface. The specific method is:

步骤4.1、建立不同变形区间单个微凸体的粘着力与其接触面积的关系；Step 4.1. Establish the relationship between the adhesion force of a single asperity and its contact area in different deformation intervals;

根据分形理论，不同变形区间单个微凸体的粘着力与其接触面积的关系为：According to the fractal theory, the relationship between the adhesion force of a single asperity and its contact area in different deformation intervals is:

(1)在非接触区，δ/δc＜0，单个微凸体的粘着力与其接触面积的关系如下公式所示：(1) In the non-contact area, δ/δ c <0, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:

Figure BDA0001641260870000091
Figure BDA0001641260870000091

(2)在完全弹性区，0＜δ/δc＜1，单个微凸体的粘着力与其接触面积的关系如下公式所示：(2) In the fully elastic region, 0 < δ/δ c < 1, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:

Figure BDA0001641260870000092
Figure BDA0001641260870000092

(3)在弹塑性一区，1＜δ/δc＜6，单个微凸体的粘着力与其接触面积的关系如下公式所示：(3) In the elastic-plastic region, 1 < δ/δ c < 6, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:

Figure BDA0001641260870000093
Figure BDA0001641260870000093

(4)在弹塑性二区，6＜δ/δc＜110，单个微凸体的粘着力与其接触面积的关系如下公式所示：(4) In the second elastic-plastic region, 6 < δ/δ c < 110, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:

Figure BDA0001641260870000094
Figure BDA0001641260870000094

其中，Fsi为不同变形区间下的单个微凸体的粘着力，i＝1、…、4，Δγ′为粘着力能，ε为分子间距；Among them, F si is the adhesive force of a single asperity under different deformation intervals, i=1,...,4, Δγ' is the adhesive force energy, and ε is the molecular distance;

步骤4.2、建立结合面总粘着力与结合面接触面积的关系；Step 4.2, establish the relationship between the total adhesion of the joint surface and the contact area of the joint surface;

根据分形理论，结合面总粘着力与结合面接触面积的关系如下公式所示：According to the fractal theory, the relationship between the total adhesion force of the joint surface and the contact area of the joint surface is shown in the following formula:

Figure BDA0001641260870000095
Figure BDA0001641260870000095

其中，in,

Figure BDA0001641260870000096
Figure BDA0001641260870000096

Figure BDA0001641260870000101
Figure BDA0001641260870000101

Figure BDA0001641260870000102
Figure BDA0001641260870000102

Figure BDA0001641260870000103
Figure BDA0001641260870000103

其中，An为结合面名义接触面积，

Figure BDA0001641260870000104
为结合面接触点的面积分布函数，Fsi为不同变形区间下的单个微凸体的粘着力，Δγ′为粘着力能；Among them, An is the nominal contact area of the junction surface,
Figure BDA0001641260870000104
is the area distribution function of the contact point of the bonding surface, F si is the adhesion force of a single asperity under different deformation intervals, and Δγ′ is the adhesion force energy;

本实施例中，取结合面名义接触面积An＝3.440610×10-3m2，粘着力能Δγ′＝3.2erg/cm2，分子间距ε＝4×10-10m，计算出结合面总粘着力Fs＝2169.500842N。In this embodiment, taking the nominal contact area of the bonding surface An = 3.440610 ×10 -3 m 2 , the adhesive force energy Δγ′=3.2 erg/cm 2 , and the molecular distance ε = 4×10 -10 m, the total bonding surface is calculated. Adhesion force F s =2169.500842N.

步骤5、计算结合面所受总切向载荷与结合面接触面积的关系；Step 5. Calculate the relationship between the total tangential load on the joint surface and the contact area of the joint surface;

根据分形理论，结合面所受总切向载荷与结合面接触面积的关系为：According to the fractal theory, the relationship between the total tangential load on the joint surface and the contact area of the joint surface is:

当al＞ac且D≠2.5时，结合面所受总切向载荷与结合面接触面积关系式为：When a l > a c and D≠2.5, the relationship between the total tangential load on the joint surface and the contact area of the joint surface is:

Figure BDA0001641260870000105
Figure BDA0001641260870000105

当al＞ac且D＝2.5时，结合面所受总切向载荷与结合面接触面积关系式为：When a l > a c and D=2.5, the relationship between the total tangential load on the joint surface and the contact area of the joint surface is:

Figure BDA0001641260870000106
Figure BDA0001641260870000106

本实施例中，计算出结合面总切向载荷Q＝45318.214N。In this embodiment, the total tangential load Q=45318.214N of the joint surface is calculated.

步骤6、建立结合面静摩擦因数μ与结合面的总法向载荷P、总切向载荷Q及总粘着力Fs之间的关系，如下公式所示：Step 6. Establish the relationship between the static friction factor μ of the joint surface and the total normal load P, the total tangential load Q and the total adhesion force F s of the joint surface, as shown in the following formula:

Figure BDA0001641260870000111
Figure BDA0001641260870000111

本实施例中结合面静摩擦因数μ＝0.463232，静摩擦因数与无量纲法向载荷的关系如图4所示，由图可知，在不同的结合面表面分形维数下，随着无量纲法向载荷的增大，结合面的静摩擦因数随之增大，在无量纲法向载荷较小时，粘着力对结合面的静摩擦因数影响较小，当无量纲法向载荷较大时，粘着力对结合面的静摩擦因数影响较大。In this example, the joint surface static friction factor μ=0.463232, and the relationship between the static friction factor and the dimensionless normal load is shown in Figure 4. It can be seen from the figure that under different surface fractal dimensions of the joint surface, with the dimensionless normal load increases, the static friction coefficient of the joint surface increases. When the dimensionless normal load is small, the adhesion force has little effect on the static friction coefficient of the joint surface. The static friction factor has a great influence.

本实施例还将考虑粘着力和忽略粘着力情况下静摩擦因数的计算结果与总法向载荷的关系与文献“Tian Hong-liang,Liu Fong,Zhao Chun-hua,et al.Predicationinvestigation on static tribological performance of metallic materialsurfaces-theoretical model[J].Journal of Vibration and Shock,2014,3(1):209-220.”中静摩擦因数的试验值与总法向载荷的关系进行对比，如图5所示，从图中可以看出，在重载条件下，粘着力对结合面静摩擦因数的影响较大，同时本发明的理论计算结果与试验值的差值也较小。This embodiment will also consider the adhesion force and the relationship between the calculation results of the static friction factor and the total normal load in the case of ignoring the adhesion force and the literature "Tian Hong-liang, Liu Fong, Zhao Chun-hua, et al. of metallic materialsurfaces-theoretical model[J].Journal of Vibration and Shock,2014,3(1):209-220.” The relationship between the experimental value of static friction factor and the total normal load is compared, as shown in Figure 5, It can be seen from the figure that under heavy load conditions, the adhesion force has a great influence on the static friction coefficient of the joint surface, and at the same time, the difference between the theoretical calculation result of the present invention and the experimental value is also small.

最后应说明的是：以上实施例仅用以说明本发明的技术方案，而非对其限制；尽管参照前述实施例对本发明进行了详细的说明，本领域的普通技术人员应当理解：其依然可以对前述实施例所记载的技术方案进行修改，或者对其中部分或者全部技术特征进行等同替换；而这些修改或者替换，并不使相应技术方案的本质脱离本发明权利要求所限定的范围。Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, but not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that it can still be The technical solutions described in the foregoing embodiments are modified, or some or all of the technical features thereof are equivalently replaced; and these modifications or replacements do not make the essence of the corresponding technical solutions depart from the scope defined by the claims of the present invention.

Claims (1)

1.一种考虑粘着力的结合面静摩擦因数三维分形预测方法，其特征在于：包括以下步骤：1. a three-dimensional fractal prediction method for the static friction factor of a joint surface considering adhesive force, is characterized in that: comprise the following steps: 步骤1、模拟结合面表面微凸体的三维形貌：将描述结合面表面形貌的二维分形函数改进为三维分形函数，并用此函数的波谷与波峰的差值表示结合面单个微凸体的接触变形量δ，如下公式所示：Step 1. Simulate the three-dimensional topography of the asperities on the surface of the junction surface: improve the two-dimensional fractal function describing the surface topography of the junction surface into a three-dimensional fractal function, and use the difference between the trough and the peak of this function to represent a single asperity on the junction surface. The contact deformation δ of , is shown in the following formula: δ＝2(11-3D)/2GD-2(lnγ)1/2π(D-3)/2a(3-D)/2 δ=2 (11-3D)/2 G D-2 (lnγ) 1/2 π (D-3)/2 a (3-D)/2 其中，G为结合面表面的分形尺度系数，D为结合面表面的分形维数，2＜D＜3，γ为与结合面表面形貌频率密度有关的参数，γ>1，一般取1.5，a为结合面表面单个微凸体实际弹性接触面积；Among them, G is the fractal scale coefficient of the surface of the bonding surface, D is the fractal dimension of the surface of the bonding surface, 2<D<3, γ is a parameter related to the frequency density of the surface topography of the bonding surface, γ>1, generally take 1.5, a is the actual elastic contact area of a single microprotrusion on the surface of the joint surface; 步骤2、分别计算弹性变形阶段、弹塑性变形阶段，结合面表面单个微凸体的变形量与临界接触面积；Step 2. Calculate the elastic deformation stage and the elastic-plastic deformation stage respectively, the deformation amount and the critical contact area of a single asperity on the surface of the joint surface; 步骤3、计算结合面所受总法向载荷与结合面的接触面积的关系；Step 3. Calculate the relationship between the total normal load on the joint surface and the contact area of the joint surface; 步骤4、计算结合面所受总粘着力与结合面接触面积的关系；Step 4. Calculate the relationship between the total adhesive force on the joint surface and the contact area of the joint surface; 步骤5、计算结合面所受总切向载荷与结合面接触面积的关系；Step 5. Calculate the relationship between the total tangential load on the joint surface and the contact area of the joint surface; 步骤6、建立结合面静摩擦因数与结合面的总法向载荷、总切向载荷及总粘着力之间的关系；Step 6. Establish the relationship between the static friction factor of the joint surface and the total normal load, total tangential load and total adhesion force of the joint surface; 所述步骤2的具体方法为：The specific method of the step 2 is: 步骤2.1、根据赫兹接触理论，结合面两粗糙表面相互接触时，将两粗糙表面转化为一等效粗糙表面与一刚性光滑平面相互接触，则结合面表面单个微凸体实际弹性接触面积a为等效粗糙表面上变形的单个微凸体与刚性光滑平面相交的截面面积，如下公式所示：Step 2.1. According to the Hertzian contact theory, when the two rough surfaces of the joint surface are in contact with each other, the two rough surfaces are transformed into an equivalent rough surface and a rigid smooth plane is in contact with each other, then the actual elastic contact area a of a single asperity on the joint surface is: The cross-sectional area of a deformed single asperity on an equivalent rough surface intersecting a rigid smooth plane is given by: a＝πRδa=πRδ 其中，R为单个微凸体的曲率半径，δ为单个微凸体的接触变形量；where R is the radius of curvature of a single asperity, and δ is the contact deformation of a single asperity; 步骤2.2、计算单个微凸体的弹性临界变形量和弹性临界接触面积；Step 2.2. Calculate the elastic critical deformation and elastic critical contact area of a single asperity; 所述单个微凸体的弹性临界变形量δc的计算如下公式所示：The calculation of the elastic critical deformation amount δ c of the single asperity is shown in the following formula:
Figure FDA0002723908040000011
Figure FDA0002723908040000011
 其中，kμ为摩擦力的修正因子，φ为材料特性系数；Among them, k μ is the correction factor of the friction force, and φ is the material characteristic coefficient; 所述单个微凸体的弹性临界接触面积ac的计算如下公式所示：The calculation of the elastic critical contact area a c of the single asperity is shown in the following formula:
Figure FDA0002723908040000021
Figure FDA0002723908040000021
 进一步求出，单个微凸体的弹塑性一区临界接触面积apt1如下公式所示：Further, the critical contact area a pt1 of the elastic-plastic first region of a single asperity is given by the following formula: apt1＝1101/(2-D)ac；a pt1 =110 1/(2-D) a c ; 单个微凸体的弹塑性二区临界接触面积apt2如下公式所示：The critical contact area a pt2 of the elastic-plastic second region of a single asperity is shown in the following formula: apt2＝61/(2-D)ac a pt2 = 6 1/(2-D) a c 所述步骤3的具体方法为：The specific method of the step 3 is: 根据分形理论，当al＞ac且D≠2.5时，结合面所受总法向载荷P与结合面的接触面积的关系如下公式所示：According to the fractal theory, when a l > a c and D≠2.5, the relationship between the total normal load P on the joint surface and the contact area of the joint surface is shown in the following formula:
Figure DEST_PATH_BDA0001641260870000031
Figure DEST_PATH_BDA0001641260870000031
 当al＞ac且D＝2.5时，结合面所受总法向载荷P与结合面的接触面积关系如下公式所示：When a l > a c and D=2.5, the relationship between the total normal load P on the joint surface and the contact area of the joint surface is shown in the following formula:
Figure FDA0002723908040000023
Figure FDA0002723908040000023
 其中，al为单个微凸体最大接触点面积，k为与结合面两接触材料中较软材料的屈服强度σy和硬度H相关的系数，三者之间的关系为：H＝kσy；E为结合面两接触材料的等效弹性模量，
Figure FDA0002723908040000024
E1、E2分别表示两接触材料的弹性模量，ν1、ν2分别表示两接触材料的泊松比；Among them, a l is the maximum contact point area of a single asperity, k is the coefficient related to the yield strength σ y and hardness H of the softer material in the two contact materials of the joint surface, and the relationship between the three is: H = kσ y ; E is the equivalent elastic modulus of the two contact materials on the joint surface,
Figure FDA0002723908040000024
E 1 and E 2 respectively represent the elastic moduli of the two contact materials, and ν 1 and ν 2 respectively represent the Poisson's ratio of the two contact materials; 所述步骤4的具体方法为：The specific method of the step 4 is: 步骤4.1、建立不同变形区间单个微凸体的粘着力与其接触面积的关系；Step 4.1. Establish the relationship between the adhesion force of a single asperity and its contact area in different deformation intervals; 根据分形理论，不同变形区间单个微凸体的粘着力与其接触面积的关系为：According to the fractal theory, the relationship between the adhesion force of a single asperity and its contact area in different deformation intervals is: (1)在非接触区，δ/δc＜0，单个微凸体的粘着力与其接触面积的关系如下公式所示：(1) In the non-contact area, δ/δ c <0, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:
Figure FDA0002723908040000031
Figure FDA0002723908040000031
 (2)在完全弹性区，0＜δ/δc＜1，单个微凸体的粘着力与其接触面积的关系如下公式所示：(2) In the fully elastic region, 0 < δ/δ c < 1, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:
Figure 139080DEST_PATH_BDA0001641260870000035
Figure 139080DEST_PATH_BDA0001641260870000035
 (3)在弹塑性一区，1＜δ/δc＜6，单个微凸体的粘着力与其接触面积的关系如下公式所示：(3) In the elastic-plastic region, 1 < δ/δ c < 6, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:
Figure 953453DEST_PATH_BDA0001641260870000041
Figure 953453DEST_PATH_BDA0001641260870000041
 (4)在弹塑性二区，6＜δ/δc＜110，单个微凸体的粘着力与其接触面积的关系如下公式所示：(4) In the second elastic-plastic region, 6 < δ/δ c < 110, the relationship between the adhesion force of a single asperity and its contact area is shown in the following formula:
Figure FDA0002723908040000034
Figure FDA0002723908040000034
 其中，Fsi为不同变形区间下的单个微凸体的粘着力，i＝1、…、4，Δγ′为粘着力能，ε为分子间距；Among them, F si is the adhesive force of a single asperity under different deformation intervals, i=1,..., 4, Δγ' is the adhesive force energy, and ε is the molecular distance; 步骤4.2、建立结合面总粘着力与结合面接触面积的关系；Step 4.2, establish the relationship between the total adhesion of the joint surface and the contact area of the joint surface; 根据分形理论，结合面总粘着力与结合面接触面积的关系如下公式所示：According to the fractal theory, the relationship between the total adhesion force of the joint surface and the contact area of the joint surface is shown in the following formula:
Figure FDA0002723908040000035
Figure FDA0002723908040000035
 其中，in,
Figure FDA0002723908040000036
Figure FDA0002723908040000036
Figure FDA0002723908040000041
Figure FDA0002723908040000041
Figure FDA0002723908040000042
Figure FDA0002723908040000042
Figure FDA0002723908040000043
Figure FDA0002723908040000043
 其中，An为结合面名义接触面积，
Figure FDA0002723908040000044
为结合面接触点的面积分布函数，Fsi为不同变形区间下的单个微凸体的粘着力，Δγ′为粘着力能；Among them, An is the nominal contact area of the junction surface,
Figure FDA0002723908040000044
is the area distribution function of the contact point of the bonding surface, F si is the adhesive force of a single asperity under different deformation intervals, and Δγ′ is the adhesive force energy; 所述步骤5的具体方法为：The specific method of the step 5 is: 根据分形理论，结合面所受总切向载荷与结合面接触面积的关系为：According to the fractal theory, the relationship between the total tangential load on the joint surface and the contact area of the joint surface is: 当al＞ac且D≠2.5时，结合面所受总切向载荷与结合面接触面积关系式为：When a l > a c and D≠2.5, the relationship between the total tangential load on the joint surface and the contact area of the joint surface is:
Figure FDA0002723908040000045
Figure FDA0002723908040000045
 当al＞ac且D＝2.5时，结合面所受总切向载荷与结合面接触面积关系式为：When a l > a c and D=2.5, the relationship between the total tangential load on the joint surface and the contact area of the joint surface is:
Figure FDA0002723908040000046
Figure FDA0002723908040000046
 步骤6所述结合面静摩擦因数μ与结合面的总法向载荷P、总切向载荷Q及总粘着力Fs之间的关系如下公式所示：The relationship between the static friction factor μ of the joint surface described in step 6 and the total normal load P, the total tangential load Q and the total adhesion force F s of the joint surface is shown in the following formula:
Figure FDA0002723908040000051
Figure FDA0002723908040000051
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