Numerical simulation of contact surface stress distribution based on stress-magnetization effect surface

Taking the rough surface of C45 steel grinding as the research object, we established a two-dimensional �nite element model of the rough surface and its surrounding air by using ANSYS, and studied the contact properties between the rough surface and the rigid plane under normal load ( ≤ 10MPa). On this basis, ANSYS Apdl language was used to establish a force-magnetic coupling model, and further simulated the impact of different load conditions on the leakage magnetic �eld on the contact surface. The results showed that according to the zero-crossing point of the normal component of the leakage magnetic �eld or the extreme point of the tangential component, the number of stress concentrations on the contact surface and the stress level of the corresponding area can be effectively determined. This method has certain reference value for studying the surface contact stress distribution between parts.


Introduction
In mechanical assemblies, the surface contact between parts was not smooth surface-to-surface contact, but consists of mutual contacts between many micro-convex bodies [1][2][3].The real contact area will be much smaller than the nominal Contact area.When the equipment was running, the contact surfaces between parts were prone to vibration, reducing the stiffness of the equipment.According to research, the stiffness of the joint surface in a machine tool accounted for approximately 60-80% of the total stiffness of the machine tool, and the deformation caused by the joint surface accounts for approximately 85-90% of the total static deformation of the machine tool [4].When two surfaces squeezed each other due to the action of load, the actual contact area will be subject to a large contact load, and phenomena such as contact surface wear and local crushing were prone to occur [5].Therefore, conveniently and accurately obtaining reliable surface contact stress distribution is a key theoretical and technical issue in the design and development of general mechanical products, and it is also a research hotspot in domestic and foreign academic circles.on the proposed concept of fractal smoothness, and derived the relationship between the deformation of a given asperity and the contact stiffness [9].In recent years, magnetic non-destructive testing technology based on the magnetization theory of ferromagnetic materials has developed rapidly [10].This technology can not only quickly detect the stress state of materials, but also evaluate the damage state of materials [11][12][13].Roskosz et al. applied the magnetic memory method to evaluate the stress distribution of specimens containing holes under tension [14].The research results showed that there is a clear correlation between the residual magnetic eld and the stress state.Lopez et al. studied the impact of the stamping process on residual stress and magnetic domain structure [15].The results showed that the residual compressive stress caused by the stamping process will change the shape and width of the magnetic domain structure near the edge of the tested sample.Yang et al. used the high magnetic permeability characteristics of ferromagnetic materials to establish an evaluation system for measuring defect characteristics through magnetic ux leakage signals [16].

Under certain assumptions
Numerical simulation was an effective method to analyze the quantitative relationship between leakage eld distribution and damage or other physical properties of ferromagnetic materials, and was an important part of theoretical research on ferromagnetic materials [17].Sorabh  to conduct nite element static simulation of a pipeline magnetic ux leakage detector and studied the effects of defect size and sensor lift-off value on the magnetic ux leakage signal [21].Numerical simulation methods can more intuitively describe the magnetic eld and stress distribution on the contact surface through nite element models, and can more accurately present the simulation results of magnetization [22][23][24].
Our research goal was to use the force-magnetic effect to characterize the stress distribution characteristics of two rough surfaces under a certain load.We assumed that the stress distribution level in the contact area can be judged based on the magnetic eld distribution pattern of the asperities on the rough contact surface.Using ANSYS nite element software, a plane model of the rough surface was rst constructed, and the contact properties between the rough surface and the rigid plane under normal load (≤ 10MPa) were studied.On this basis, ANSYS APDL language was used to establish a force-magnetic coupling model to further simulate the impact of different load conditions on the leakage magnetic eld on the contact surface.

Experimental methods
This article used ANSYS nite element software to establish a nite element plane strain model of a twodimensional rough surface.We rst performed component mechanics calculations in the static eld to obtain the contact characteristics under different loads.Then, the stress values of each unit under different loads were brought into the force-magnetic coupling model equation, and the magnetic permeability of each unit under different load intensities was calculated to prepare for magnetostatic analysis.Figure 1 was the surface pro le curve of the C45 steel surface grinding sample obtained by the surface pro le measuring instrument.The sampling spacing was 1 µm, and the number of sampling points was 5,000.The measured data points were processed using the least squares method and connected with straight lines to form a rough surface pro le.
As shown in Fig. 2, the model gradually densi ed the mesh in layers along the vertical direction, making the mesh density transition appropriately.In order to ensure the calculation accuracy and the convergence of the contact analysis, a uniform quadrilateral grid was used near the contact surface to evenly divide the surface contour in the horizontal direction, ensuring the uniformity of the grid in the horizontal direction.The number of units in the entire model was 38647, the number of nodes was 96024, and the unit type was PLANE183.The material properties of the model were shown in Table I, using a multilinear elastic-plastic strain hardening material model obtained from uniaxial tensile experiments.
Add vertical constraints to the bottom of the model, fully constrain the lower left corner, and add horizontal constraints to the rigid surface.Finally, a normal pressure of 10 MPa was applied to the rigid surface, and the load was carried out according to 100 load steps with equal step length.3 Results and discussion

Contact surface analysis of the model
Figure 3 showed the stress cloud diagram of 100 load steps.From the stress cloud diagram, it can be observed that the stress distribution of the contact layer, especially the surface stress close to the contact point, is very complex.The stress started from the contact point and gradually decreased inward.The accelerated splitting near the contact layer itself caused freezing caused by the contact, and also included the extrusion and pu ng caused by other contact points.
Figure 4 showed the relationship curve between normal deformation and normal load, in which the abscissa was the normal load value and the ordinate is the displacement in the normal direction of the midpoint of the rigid surface.It can be seen from the gure that the deformation displacement gradually increases with the increase of normal load, and the relationship between the two was generally linear.By comparing with the experiment in Reference 5, the test result curves of the two were consistent, indicating that the model calculation results were reliable.
Figure 5 showed the relationship curve between contact area and normal load, where the abscissa was the size of the normal load and the ordinate was the contact area (contact line length).It can be seen from the gure that when the load was less than 1Mpa, the contact area increased generally linearly with the increase of the normal load.When the load exceeded 1Mpa, the increase in the contact area became smaller and smaller, approaching saturation.
3.2 Effects of different loading conditions on the leakage magnetic eld at the contact surface Figure 6 showed the geometric model and grid model of the two-dimensional magnetic eld.The meshing of the ferromagnetic region was the same as the mesh model in the contact analysis.The air mesh in contact with the contact surface was denser, and the air mesh became sparser farther away from the contact surface.The relative magnetic permeability of the ferromagnetic material area was 285, and the relative magnetic permeability of the air area was 1.The model used the PLANE53 unit, which had 8 nodes and each node has 4 degrees of freedom.The boundary conditions of the z component (Az) of the magnetic vector potential were the upper air boundary Az = 39.8A/m and the lower air boundary Az = 0 A/m.
By writing the APDL program, the stress values of each unit from the contact analysis were brought into the force-magnetic coupling mathematical model, the magnetic permeability of each unit was calculated, the magnetic permeability value of each unit was corrected, the boundary conditions were loaded, and ANSYS magnetostatics was performed Solve, nally.Figure 7 was the result curve of magnetostatic analysis, in which gures (a)-(c) were respectively the tangential and normal leakage magnetic eld intensity curves of air 1mm above the rough surface under various loading conditions.In the leakage magnetic eld under various normal loads from 1MPa to 10MPa, and the differences in the leakage magnetic eld under various load conditions can be clearly compared.From Fig. 7, we found that the stress value in the contact area was greater than the stress value in the uncontact area, resulting in the magnetic permeability of the contact area being smaller than the uncontact area, which caused magnetic leakage in the contact area, thereby increasing the normal component of the magnetic eld intensity in the leakage magnetic eld.All cross zero, and the tangential components all have the maximum value of the curve shape.As the load increases, each curve in the gure had more tangential magnetic eld components showing maximum values, and more normal components showing zero crossing points.This is because more contact areas appear as the load increases.
As shown in Fig. 8, the stress cloud diagram of the contact area under 10MPa normal load, it can be observed that there were 4 obvious contact areas, which were related to the number of extreme points of the tangential leakage magnetic eld intensity and the normal leakage magnetic eld intensity under 10MPa.The number of zero points was the same.
As shown in Fig. 9, under the normal load of 10Mpa, the leakage magnetic eld intensity of air layered with different heights from the contact surface.It can be seen from the gure that as the height increases, the values of the tangential and normal leakage magnetic eld strengths of the air layer above the , Fu et al. revealed the nonlinear relationship between the normal contact stiffness of the joint surface and the parameters of the joint surface based on the contact fractal theory[6].Wang et al. proposed that the wavelet decomposition method can be used to identify the contour characteristic length scale parameters in the fractal parameters of rough surfaces, and veri ed the effectiveness by comparing with other methods[7].Zhao et al. proposed a new elastic-plastic micro contact model of rough surfaces based on the theory of contact mechanics to study the elasticity, elasticplasticity, plastic contact deformation, contact area and contact load of asperities on rough surfaces[8].Wang et al. established a fractal contact stiffness model considering the interaction of asperities based et al. used ANSYS nite element software to simulate the leakage ux distribution on the surface of the pipe after the residual stress and large concave deformation of the ferromagnetic pipe under the action of spherical pressure load[18].Ren et al. simulated the force-magnetic coupling process of 18CrNi4A steel and obtained the change rules of the magnetic memory signal under different loads[19].Li et al. used numerical simulation software to analyze the relationship between defects in pipeline steel and leakage ux density, which provided a basis for quantitative analysis of pipeline steel defects[20].Shen et al. used Magnet software Figure (a), the abscissa was the coordinate value in the actual length direction, and the ordinate was the tangential leakage magnetic eld intensity of the air layer above the corresponding position.In Figure (b), the abscissa was the coordinate value in the actual length direction, and the ordinate was the normal leakage magnetic eld intensity of the magnetic eld in the air layer above the corresponding position.Figure (c) synthesized the leakage magnetic eld under normal loads from 0.1MPa to 1MPa. Figure (d) combined

Figure 1 Rough surface contour curve Figure 2 Finite 3 Figure 4
Figure 1

Figure 5 Relationship
Figure 5

Figure 7 Distribution
Figure 7