Multi-stage coupled connection structure modeling analysis method

Abstract

The invention relates to the field of mechanical analysis, in particular to a multi-stage coupled connection structure modeling analysis method. According to the method, the problems of continuity and monotonicity of a model are solved through the advantages of a wave function in the aspect of geometric authenticity and a low-order function, a wave-shaped micro-convex body is constructed through a cosine function, and the micro-convex body is compared with a spherical micro-convex body in the aspects of stress-strain distribution, deformation process and plastic expansion behavior; the analysis of the mechanical behavior of the wave-shaped micro-convex body in the stage from elastoplasticity to complete plasticity is closer to the real situation.

Description

Technical Field

[0001] This invention relates to the field of elastoplastic analysis of micro-convex bodies, and particularly to a method for modeling and analyzing multi-stage coupled connection structures. Background Technology

[0002] Rough surfaces in connection structures involve various contact mechanisms, such as friction, wear, and sealing, all of which involve complex nonlinear mechanical behaviors. Research on these phenomena is crucial in tribology, materials science, and mechanical engineering, and is currently a hot topic. Since Greenwood and Williamson proposed the landmark statistical contact model (GW model), a research approach based on the micro-protrusion assumption has been established. The GW model assumes that the rough surface is a series of spherical micro-protrusions with the same radius of curvature. The contact behavior of these micro-protrusions follows Hertz's elastic contact theory. However, this assumption does not consider elastoplastic and plastic states, and therefore its limitations have gradually become apparent when dealing with the elastoplastic deformation and multi-scale effects of practical engineering surfaces. The theoretical analysis error increases continuously with increasing contact pressure.

[0003] To more accurately describe the elastoplastic contact process, subsequent research focused on improving the mechanical models of micro-asperities. The CEB model, proposed by Chang, Etsion, and Bogy, was the first to systematically consider the continuous deformation process of micro-asperities from elastic to plastic, thus extending the GW model. This model defines the critical contact depth ω... c And the depth of fully plastic contact ω p The contact load versus area relationship in the elastoplastic stage was established. However, the CEB model at the critical yield point ω... c The contact variables are discontinuous at the point of contact, and the elastic-plastic curve jumps, which contradicts the physical laws of actual contact.

[0004] To address the discontinuity of the CEB model at the critical point, CHANG et al. proposed an improved CEB model. However, this model only considered the fully elastic and fully plastic contact states, without in-depth analysis of the elastoplastic stage. ABBOT et al. proposed that after the micro-protrusion undergoes plastic yielding, the pressure on the contact surface is linearly related to the material hardness H, the contact surface follows the assumption of equal area, and the micro-protrusion follows the assumption of volume conservation, thus obtaining the AF model for the fully plastic stage. Based on this, they proposed the ZMC model, which uses a high-order polynomial interpolation method to connect the elastic and plastic solutions, ensuring that the contact variable is constant at ω. e It maintains continuity at the point of contact, but the higher-order interpolation terms it introduces cause non-physical oscillations in the contact pressure during the elastoplastic stage.

[0005] In recent years, optimizing the geometry of micro-convexities has become a key direction for improving the predictive power of models. Several innovative approaches have been proposed to improve model continuity, abandoning high-order polynomial interpolation and employing elliptic curves and power functions. These methods allow for strictly monotonic and continuous changes of contact variables in the elastoplastic stage using low-order functions, effectively avoiding the oscillation problem of the ZMC model. However, the surface morphology of micro-convexities is still based on the simplified assumption that they are spherical. While this simplification facilitates mathematical processing, it fails to characterize the widespread multi-scale, periodic, and asymmetric features of real rough surfaces. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a multi-stage coupled connection structure modeling and analysis method. This invention leverages the geometric realism of wave functions and the use of low-order functions to solve the continuity and monotonicity problems of the model. A wave-shaped micro-protrusion is constructed using a cosine function. By comparing the stress-strain distribution, deformation process, and plastic extension behavior of this micro-protrusion with that of a spherical micro-protrusion, the analysis shows that the mechanical behavior of the wave-shaped micro-protrusion in the elastoplastic to fully plastic stage more closely resembles reality.

[0007] The technical solution of this invention is: a method for modeling and analyzing multi-stage coupled connection structures, characterized by comprising the following steps:

[0008] Step 1: Establishing elastoplastic contact between wavy micro-protrusions.

[0009] The surface profile of a single micro-convex body is constructed using a cosine function. The micro-convex body is modeled on the xoy plane, and its two-dimensional morphological features are as follows:

[0010]

[0011] h is the height of the micro-protrusion, L is the radius of the wavy micro-protrusion, and x represents the length of the base of the micro-protrusion.

[0012] Step Two: Steps in the Elasticity Stage Analysis

[0013] According to the mathematical formula for curvature:

[0014]

[0015] k is the curvature Denotes the first derivative; Represents the second derivative;

[0016] The formulas for the first and second derivatives of the wave function are:

[0017]

[0018]

[0019] The radius of curvature at the point of contact x=0 at the top of the cosine function is obtained. .

[0020] Contact radius of micro-protrusions in the elastic stage Contact area of ​​micro-protrusions in the elastic stage Micro-protrusion contact load in the elastic stage ; Indicates the actual contact depth;

[0021] Step 3: Steps for Elastic-Plastic Stage Analysis

[0022] Define the contact radius in the elastoplastic stage for:

[0023]

[0024] , This is the contact radius correction factor for the elastoplastic stage of the micro-protrusion;

[0025] Elastic-plastic contact area for:

[0026]

[0027] Elastic-plastic contact pressure The formula is:

[0028]

[0029] Where γ and δ are the correction parameters for the elastoplastic average contact pressure;

[0030] Step 4: Steps for the analysis of the fully plastic stage

[0031] Contact radius during fully plastic stage Its projection in the x-direction:

[0032]

[0033] Fully plastic contact area for:

[0034]

[0035] Average contact pressure p during the plastic stage p The average value is equal to the material's hardness H.

[0036]

[0037] Therefore, the contact load P can be obtained. p for:

[0038]

[0039] At the critical point between elasticity and elastoplasticity The continuity of contact radius, contact area, and average contact pressure are satisfied respectively, that is:

[0040]

[0041]

[0042]

[0043] At the critical point between elastic-plastic and fully plastic states:

[0044]

[0045]

[0046] .

[0047] The beneficial effects of the present invention are: the finite element model of the wavy micro-protrusion of the present invention can better reflect its contact behavior and make up for the defect that it is difficult to carry out experiments on a single micro-protrusion at the micro level due to the precision of the instrument and equipment. Attached Figure Description

[0048] Figure 1 shows a planar model of a wavy micro-convex body.

[0049] Figure 2 Mesh refinement in the micro-protrusion contact area.

[0050] Figure 3 Micro-protrusions contact the overall mesh. Detailed Implementation

[0051] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0052] The present invention provides a method for modeling and analyzing multi-stage coupled connection structures, comprising the following steps:

[0053] Step 1: Establishing elastoplastic contact between wavy micro-protrusions.

[0054] The surface profile of a single micro-convex body is constructed using a cosine function, where x=0 represents the highest point of the micro-convex body. Since the micro-convex body exhibits axisymmetric characteristics in its three-dimensional structure, its mechanical properties are uniformly distributed radially. For ease of analysis and calculation, this invention selects a micro-convex body on the xoy plane for model construction. The two-dimensional morphological characteristics of the micro-convex body are as follows:

[0055]

[0056] In the formula, h is the height of the micro-convexity, L is the radius of the wavy micro-convexity, which is 1 / 4 wavelength of the single-period cosine function, and x represents the base length of the micro-convexity. The micro-convexity is axisymmetric, and its elastic-elastoplastic and elastoplastic-plastic critical points are determined with reference to the CEB model. This step uses a cosine function to construct a single micro-convexity model, breaking through the traditional spherical shape in terms of morphology and making the mechanical behavior closer to reality.

[0057] Step Two: Flexible Phase Analysis steps

[0058] First, define the boundary between elasticity and elastoplasticity, and the critical contact depth. ,in , E represents the Poisson's ratio of a material. ∗ Equivalent elastic modulus Critical contact depth for perfect elasticity Critical contact depth for fully plasticity .

[0059] ω e ω represents the elastic contact depth of the micro-convexity. p This represents the depth of fully plastic contact with the micro-protrusion. Indicates the actual contact depth; ω c The critical elastic contact depth is given by E1, E2 and υ1, υ2, which are the elastic modulus and Poisson's ratio of the two materials, respectively. H is the material hardness, and R is the contact radius of the micro-protrusion.

[0060] Under the assumptions of elastic contact and small deformation, the contact area is approximated as the contact between a flat plate and a wave function. Therefore, it is necessary to solve for the radius of curvature of this function at the vertex x=0. Based on the radius of curvature... Where k is the curvature, Let w be the radius of curvature of the soil mass, defined by the mathematical formula for curvature:

[0061]

[0062] Denotes the first derivative; Represents the second derivative;

[0063] The formulas for the first and second derivatives of the wave function are:

[0064]

[0065]

[0066] The radius of curvature at the point of contact x=0 at the top of the cosine function is obtained. .

[0067] Although the wavy micro-protrusion deviates from a spherical shape, its deformation is still relatively small during initial contact. Referring to the deformation of a spherical micro-protrusion, the overall deformation of the wavy micro-protrusion in the elastic stage is considered not to exceed the small deformation range. Considering the assumptions and applicability of Hertz contact theory, the contact radius, contact area, and contact load in the elastic stage of this model also adopt Hertz contact theory. Therefore, using the formulas from Hertz contact theory, the contact radius of the micro-protrusion in the elastic stage is obtained. Contact area of ​​micro-protrusions in the elastic stage Micro-protrusion contact load in the elastic stage .

[0068] Step 3: Elastic-plastic stage Analysis steps

[0069] Existing elastic-plastic curves suffer from oscillations and runge phenomena. This invention employs an overly smooth, monotonic low-order function for correction. Based on the description of the mechanical behavior in the elastic-plastic stage and finite element analysis results, the expansion of the plastic zone inside the wavy micro-protrusion leads to a nonlinear increase in the contact area, with the growth rate gradually decreasing as the contact depth ω increases. Simultaneously, the contact load in the elastic-plastic stage exhibits a nonlinear increasing trend. Furthermore, the function needs to be strictly monotonic and continuous at the elastic-elastic-plastic and elastic-plastic-plastic critical points. Therefore, this invention uses a simple, smooth, and monotonically increasing hyperbolic tangent function as the functional description for this stage, by introducing a correction term. , elastic solution Extend to the range of elastoplastic deformation. Based on the correction term, redefine the contact radius in the elastoplastic stage. for:

[0070]

[0071] , This is the contact radius correction factor for the elastoplastic stage of the micro-convex body. The parameter value is determined based on the finite element analysis results.

[0072] Meanwhile, based on the axisymmetric properties of the wavy micro-protrusion, the contact area is circular when the plate contacts the micro-protrusion, therefore the elastoplastic contact area is... for:

[0073]

[0074] Where α and β are elastoplastic nonlinear expansion parameters used to control the expansion rate of the plastic region.

[0075] Elastic-plastic contact pressure Similarly, in elastic contact pressure Introducing correction terms based on The formula is:

[0076]

[0077] Wherein γ and δ are correction parameters for the elastoplastic average contact pressure, the values ​​of which are determined based on the finite element analysis results, and are used to describe the influence of plastic deformation on the nonlinear growth of the load.

[0078] Step 4: Full Plasticity Stage Analysis steps

[0079] When the contact depth is large, the contact area of ​​the micro-protrusion undergoes significant deformation. However, compared to the overall structure of the micro-protrusion, assuming the deformation range still falls within the small deformation theory, and therefore adhering to the volume conservation assumption, the micro-protrusion in the contact area, after being compressed under plastic deformation, becomes highly deformable. Substitute the point set into the surface profile equation:

[0080]

[0081] Simplified inequalities:

[0082]

[0083] make Then the inequality simplifies to:

[0084]

[0085] Where s∈[0,1], s is an intermediate parameter.

[0086] The above cosine function exist The function is even in the interval [-L, L], and has a maximum value at x=0 and a minimum value at x=±L. The inequality is... The solution corresponds to:

[0087]

[0088] Substitution Solving for:

[0089]

[0090] Then the contact radius in the fully plastic stage Its projection in the x-direction:

[0091]

[0092] The present invention employs the projected area theory during the plastic stage of a single micro-protrusion, correcting the defect of overestimation of area caused by the volume conservation theory.

[0093] When ω << h, expand the above formula using the Taylor series and simplify to obtain the following formula:

[0094]

[0095] Therefore, the fully plastic contact area can be obtained as :

[0096]

[0097] The average contact pressure p in the plastic stage p , when the asperities enter the fully plastic deformation, the pressure distribution in the contact area tends to be uniform, and the average value is equal to the hardness H of the material.

[0098]

[0099] Therefore, the contact load P can be obtained as p :

[0100]

[0101] To sum up, the theoretical formulas for the wavy asperities in the elastic, elastic-plastic, and fully plastic stages are shown in the following table:

[0102] Table Elastic-Plastic Theoretical Model of Wavy Asperities

[0103] stage Contact radius Contact area A Contact load P Average contact pressure p elasticity Elasticity Fully plastic

[0104] The above formulas must satisfy the following conditions, that is, at the critical point between elastic and elastic-plastic respectively satisfy the continuity of the contact radius, contact area, and average contact pressure, that is:

[0105]

[0106]

[0107]

[0108] At the critical point between elastic-plastic and fully plastic also satisfy the continuity conditions, that is:

[0109]

[0110]

[0111]

[0112] This invention overcomes the limitations of the spherical assumption, capturing stress concentration and plastic flow phenomena. Results show that the contact pressure in the elastic stage is 22% higher than that of a sphere, while the plastic strain in the elastoplastic stage is 52% lower. The contact area error is corrected by 20% in the fully plastic stage. This invention employs a hyperbolic tangent function in the elastoplastic stage of a single micro-protrusion, and four correction parameters were determined through finite element analysis. The hyperbolic tangent function eliminates non-physical oscillations in the elastoplastic stage, ensuring the continuity and monotonicity of the contact variables, with an error of <5% compared to finite element analysis. Simultaneously, a method for rough surface contact was established based on this model, verifying that the wavy micro-protrusion can better match the mechanical properties of real rough surfaces and exhibits a gradual decrease in stiffness during plastic flow. The rough surface contact analysis method of this invention provides a theoretical basis for predicting stress distribution, plastic evolution, and multi-scale mechanical behavior in interface connections.

[0113] The present invention may also include step five, an analysis method for rough surface contact:

[0114] Based on the elastoplastic contact model of a wavy micro-protrusion, and combined with a Gaussian distribution, the correspondence between the normal load on the rough surface and the average contact depth was established. To analyze the influence of the wavy micro-protrusion on the overall contact mechanics model of the rough surface, and comparing it with the spherical micro-protrusion, a Gaussian height distribution was adopted, the function expression of which is:

[0115]

[0116] In the formula, Let Variance be the height distribution. The mean of the height distribution.

[0117] When contact occurs on a rough surface, from a macroscopic perspective, as long as the contact surface is not completely smooth, some stress concentration will inevitably occur. Therefore, the micro-protrusions will undergo elastic and plastic deformation. By adding up the contact loads of all the micro-protrusions, we can obtain the change in the contact state of the entire rough surface in the normal direction, that is:

[0118]

[0119]

[0120]

[0121] In the formula: A is the actual contact area of ​​the rough surface, P is the total contact load on the rough surface, and p is the average contact pressure on the rough surface. d represents the number of micro-protrusions that actually contact the rough surface, and d is the distance between the contact surface and the reference plane for the average height of the micro-protrusions.

[0122] This invention includes dispersed stress concentration regions and plastic region expansion; for non-monotonic, oscillatory, and fully plastic deformation problems in the elastoplastic stage, based on the contact mechanics behavior of wavy micro-protrusions, a suitable low-order function is selected to modify the analytical expression of the contact mechanics of the micro-protrusions, forming an elastoplastic contact theory model of cosine function wavy micro-protrusions. Combined with refined finite element analysis to identify the parameters of the low-order function, an optimized elastoplastic contact model of a single wavy micro-protrusion is obtained; based on the height distribution of the micro-protrusions, a rough surface contact model composed of wavy micro-protrusions is constructed, and the influence of wavy micro-protrusions on the morphology of rough surfaces is compared, providing support for the development of a more accurate rough surface contact theory.

[0123] This invention also provides a practical example of establishing elastoplastic contact of wavy micro-protrusions. Wavy and spherical micro-protrusion models are established in numerical analysis software. Since both types of micro-protrusions exhibit axisymmetry in the three-dimensional structure, they can be considered as being formed by sweeping a single plane 180°. To simplify calculations, micro-protrusions are constructed on the two-dimensional plane xoy. The material parameters of the micro-convex contact model are as follows: As shown, the micro-protrusion is made of low-carbon steel, and the contact plate is made of structural steel with a yield strength slightly greater than that of the micro-protrusion. Therefore, the contact pressure is mainly reflected in the micro-protrusion.

[0124] surface Micro-protrusion material parameters

[0125] Parameter type Elastic modulus E Poisson's ratio ν Yield strength Y Hardness H Micro-convexity 200 GPa 0.3 400MPa 2.8Y flat 200 GPa 0.3 460MPa 2.8Y

[0126] To compare and analyze the influence of micro-convexity morphology on the elastoplastic model, a cosine-shaped micro-convexity model was established by assuming that the volume of the wavy function and the micro-convexity constructed by the semicircle are equal. and hemispherical micro-convex bodies, the base length of the two types of micro-convex bodies mm, and maintains equal area on a two-dimensional plane.

[0127] surface Two types of micro-protrusion interface morphology

[0128] like Figure 2 and Figure 3As shown, a high-density mesh is used in the contact area (near the vertex), gradually coarsening away from the contact area. Planar constraints are applied to the normal displacement, and symmetrical boundaries constrain radial displacement. The bottom of the micro-protrusion is fixed, and a normal displacement (0.01 mm) is applied to the plate. Frictionless contact is established for the contact pair, and an enhanced Lagrangian algorithm is selected to avoid errors caused by penetration. An adaptive algorithm is used for the mesh, with the minimum element size less than 1 / 20 of the base length (0.03 mm). Mesh coupling and mesh refinement are applied to the contact surface between the plate and the micro-protrusion, the number of mesh expansion layers is set to 5, and large deformation is enabled.

Claims

1. A method for modeling and analyzing multi-stage coupled connection structures, characterized in that: Includes the following steps: Step 1: Establishing elastoplastic contact between wavy micro-protrusions. The surface profile of a single micro-convex body is constructed using a cosine function. The micro-convex body is modeled on the xoy plane, and its two-dimensional morphological features are as follows: h is the height of the micro-protrusion, L is the radius of the wavy micro-protrusion, and x represents the length of the base of the micro-protrusion. Step Two: Steps in the Elasticity Stage Analysis According to the mathematical formula for curvature: k is the curvature Denotes the first derivative; Represents the second derivative; The formulas for the first and second derivatives of the wave function are: The radius of curvature at the point of contact x=0 at the top of the cosine function is obtained. , Contact radius of micro-protrusions in the elastic stage Contact area of ​​micro-protrusions in the elastic stage Micro-protrusion contact load in the elastic stage ; Indicates the actual contact depth; Step 3: Steps for Elastic-Plastic Stage Analysis Define the contact radius in the elastoplastic stage for: , This is the contact radius correction factor for the elastoplastic stage of the micro-protrusion; Elastic-plastic contact area for: Elastic-plastic contact pressure The formula is: Where γ and δ are the correction parameters for the elastoplastic average contact pressure; Step 4: Steps for the analysis of the fully plastic stage Contact radius during fully plastic stage Its projection in the x-direction: Fully plastic contact area for: Average contact pressure p during the plastic stage p The average value is equal to the material's hardness H. Therefore, the contact load P can be obtained. p for: At the critical point between elasticity and elastoplasticity The continuity of contact radius, contact area, and average contact pressure are satisfied respectively, that is: At the critical point between elastic-plastic and fully plastic states: 。 2. The method for modeling and analyzing multi-stage coupled connection structures according to claim 1, characterized in that: Step five also includes establishing the correspondence between the normal load on the rough surface and the average contact depth, the functional expression of which is: In the formula, Let Variance be the height distribution. The mean of the height distribution.

3. The method for modeling and analyzing multi-stage coupled connection structures according to claim 2, characterized in that: Without considering the interactions between micro-protrusions, the contact mechanical properties of all micro-protrusions are integrated to obtain the change in the contact state along the normal of the entire rough surface: In the formula: A is the actual contact area of ​​the rough surface, P is the total contact load on the rough surface, and p is the average contact pressure on the rough surface. d represents the number of micro-protrusions that actually contact the rough surface, and d is the distance between the contact surface and the reference plane for the average height of the micro-protrusions.

4. A method for modeling and analyzing multi-stage coupled connection structures according to any one of claims 1 to 3, characterized in that: The dividing point between elasticity and elastoplasticity is the critical contact depth. ,in K is the plasticity correction factor, E ∗ Equivalent elastic modulus ; ω c The critical elastic contact depth is given by E1, E2 and υ1, υ2, which are the elastic modulus and Poisson's ratio of the two materials, respectively. H is the material hardness, and R is the contact radius of the micro-protrusion.

5. A method for modeling and analyzing multi-stage coupled connection structures according to any one of claims 1 to 3, characterized in that: The micro-protrusions in the contact area are compressed under plastic deformation to a height Substitute the point set into the surface profile equation: Simplified inequalities: make Then the inequality simplifies to: Where s∈[0,1], s is an intermediate parameter. The above cosine function exist The function is even in the interval [-L, L], and has a maximum value at x=0 and a minimum value at x=±L. The inequality is... The solution corresponds to: Substitution Solving for: 。 6. A method for modeling and analyzing multi-stage coupled connection structures according to any one of claims 1 to 3, characterized in that: E ∗ Equivalent elastic modulus E1, E2 and υ1, υ2 are the elastic modulus and Poisson's ratio of the two materials, respectively.

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