Research on stress through photoelastic experiment and finite element method considering sliding wear

. Sliding contact on the contact interface of friction pairs is a common type of contact. The sliding wear caused by sliding contact has an obvious influence on the stress in the contact area. In this study, the photoelastic experiment and finite element method are adopted to study variation laws of stress in the contact area. The results show that the stress in the contact region is very concentrated, and the contact half-width gradually ascends with the increase of sliding wear. The stress intensity in the contact region and von Mises stress at the contact centre decrease with the increase of wear depth. In the case of a wear depth of less than 0.3 mm, the stress intensity and the contact stress decrease rapidly with the growth of wear depth. When the wear depth exceeds 0.3 mm, the influence of wear depth on the stress intensity and contact stress is small. The results of this research clarify the effect of sliding wear on the stress in the contact area, and provide a reference for studying the contact issues.


Introduction
Sliding wear, also known as adhesive wear, is a complex frictional process occurring when two surfaces experience relative motion.This wear mechanism is omnipresent in industrial engineering, affecting the efficiency and longevity of critical components such as gears, bearings, and other friction pairs.The mechanism that causes sliding wear is the adhesion of the contact surface.When relative motion occurs, this adhesion will cause deformation, material transfer, and eventually wear.
Sliding wear has profound influences on diverse industries.Taking wheel/rail contact as an example, the sliding contact between wheel and rail will occur during the braking process of a train.The train braking will cause serious wear to the wheel surface (Fig. 1).Wear changes the geometry of the contact surface, directly affecting the stress distribution, which also affects the train braking frictional heat, thermal fatigue and material damage on the contact surface [1]- [4].In this paper, the effect of sliding wear on stress are studied through experiment and numerical calculation.
The wear on the interface is significant when the friction pair is in the sliding contact.Bowden and Taber conducted extensive studies on contact and friction between friction pairs [5].And a large number of research results are widely used in a variety of fields.Archard established the Archard wear theory, which can be adopted to analyze the wear of various contact pairs during contact [6].In addition, scholars conducted many fundamental studies on the contact and friction of friction pairs [7]- [11].Compared to other wear theories, Archard's theory is a simple and practical model, and can be applicable to various wear conditions and different materials, exhibiting broad applicability [12]- [14].
The main theories for calculating the contact issues include Hertz contact theory [15], Carter two-dimensional contact theory [16], Vermeulen-Johnson contact theory [17], and Kalker theory [18].The above theories usually assume the contact stress on the interface is semi-ellipsoidal or quadratic parabolic.However, sliding wear significantly changes the contact surface geometry and also affects the distribution form of contact stress when the friction pair is in sliding contact.In actual contact components, it is almost impossible to directly observe the changes in contact stress.Photoelastic technology provides a non-intrusive, highly sensitive stress analysis method.By measuring optical distortions on material surfaces, it precisely reveals stress distribution caused by wear, providing insightful understanding of material performance and wear mechanisms [19]- [21].In this paper, photoelastic experiment and finite element method are employed to research the stress during friction pair sliding contact.The research results can clarify the effect of sliding wear on the stress, and provide a reference for studying the other contact issues.

Photoelastic experiment
The photoelastic method is an experimental method that can provide information about the full-field stress distribution.The photoelastic streak map contains two important physical quantities, namely isobars and isotropes [22], and the photoelastic experiment can present the stress distribution law in the contact region very well [23]- [26].Fig. 2 is the sketch of the circularly polarized light field arrangement.During the photoelastic experiment, the incident light will be decomposed into two beams of polarised light along the two principal stresses when it passes through the birefringent material.Due to the different propagation velocities of the two beams of polarised light, a phase difference of ∆ will generate when it is ejected from the specimen.The birefringence is proportional to the applied stress according to the law of planar stresses-optics.That is: where  is the material optical constant. and  are the first and third principal stress.ℎ is the thickness of the specimen. is the wavelength of the incident light.When incident light with an intensity of  passes through the detector mirror, the intensity of light is: From Eqs. ( 1) and ( 2), it can be seen that two beams of polarised light with a phase difference will produce interference fringes.While there is no phase difference when the principal stress are equal, in which case dark fringes will appear.The more concentrated the stress on the specimen, the greater the change in the principal stress difference, and therefore the denser the interference fringes, and vice versa.The information from the interference fringes can be used to characterize the stress distribution in the specimen.The relationship between stress intensity and principal stresses is:

Numerical model
In this paper, a sliding wear finite element model is built, and is shown in Fig. 3.In the model, the friction pair is simulated with PLANE182 element, and the contact behaviour is simulated with the CONTA171 and TARGE169 elements.The model has a total of 13411 elements and 13350 nodes.After trial and error, the minimum element size is set to 0.15 mm under the conditions of meeting calculation speed and accuracy.Meanwhile, the wear elements are created based on the Archard wear theory.The wear can be calculated by: where  is the amount of wear per unit area per unit time, or is called the wear rate. is the wear coefficient. is the sliding speed. is the hardness of the material.And  ,  is the contact stress.In the finite element model, the extended Lagrangian algorithm is used to calculate the contact stress [27].The expression is: where  is the normal contact stiffness,  is the contact gap, Ω is: where Θ is the intrusion tolerance, Ω is the Lagrange multiplier component of the -th iteration step.

Analysis parameter
In the experiment, the photoelastic specimen consists of a hollow circle and a rectangle, and they are made of epoxy resin.The inner and outer diameter of the hollow circle is 10 mm and 50 mm, respectively.The height and length of the rectangle is 50 mm and 130 mm, respectively.The thickness of the specimen is 6 mm (Fig. 4).In the finite element wear model, the dimensions of the contact pair are the same as the photoelastic specimen.The vertical displacement at the bottom of the rectangle is constrained [28].The modulus and Poisson's ratio of the material is 3 GPa and 0.38, respectively, and the density is 1200 kg/m 3 .The wear coefficient is 4×10 -4 [29], and the hardness is 85 HD.The vertical load applied on the top of the hollow circle is the same as that of the photoelastic experiment, both are 100 N. The sliding speed is 1 m/s.

Experiment results
The stress intensity at different wear depths are illustrated as Fig. 5. Fig. 5(a) shows the photoelastic stripes are symmetrically distributed, with a closed-curve distribution at the position of the contact centre.And the number of stripes is the most intensive, in other words the stress is very concentrated.Fig. 5(a) also shows the stripe gap gradually increases from the contact centre outwards, and it indicates the stress in the contact centre is the highest, and decreases gradually from the contact centre outwards.

Finite element calculation results
Stress intensity contours obtained by finite element model are shown as Fig. 6.The contact half-widths, contact stress and von Mises stress at the contact centre under the condition of different wear depths are respectively shown as Fig. 7, Fig. 8  Fig. 6 shows the maximum stress intensity for the four cases of wear depth of 0 mm, 0.15 mm, 0.3 mm and 0.45 mm is 8.65 MPa, 4.67 MPa, 2.78 MPa and 2.3 MPa, respectively.The results indicate the maximum value of stress intensity decreases rapidly with the growth of the wear depth.Fig. 6 also shows the shape of the contour line near the contact surface is gradually flat with the increase of wear depth.Meanwhile, the results show the stress distribution laws obtained from numerical calculation are consistent with the experimental results (Fig. 5 and Fig. 6).7 shows with the growth of wear depth (ℎ), the contact half-width () on the contact surface gradually increases.Moreover, Fig. 7 shows the results obtained from the finite element wear model fit well with the experimental results.Fig. 8 shows as the depth of wear increases, the contact stress on the contact surface is gradually transformed from the initial sharp convex distribution to a flat shape, the contact size is gradually increasing, and the maximum contact stress is gradually decreasing.In other words, the distribution of contact stress gradually becomes uniform.Fig. 9 shows the von Mises stress in the contact centre descends with the increase of the wear depth.And when the wear depth is more than 0.3 mm, the influence of wear on the von Mises stress is small.

Conclusions
Sliding contact is a common contact phenomenon in friction devices.In the sliding contact process, the wear and contact stress on the contact interface will change significantly.The change rule of stress on the contact surface at different wear depths is analyzed by photoelastic experiment and numerical calculation, and the conclusions are as follows.
1) At different wear depths, the photoelastic stripes are symmetrically distributed and are very dense in the contact area, indicating that the stress is concentrated in the contact region.As the wear depth increases, the number of photoelastic stripes decreases and the distance between the stripes gradually increases.
2) As the wear depth increases, the contact size increases, the stress intensity in the contact region and the von Mises stress of the contact centre rapidly decrease, and the contact stress distribution gradually become uniform.
3) When the wear depth is less than 0.3 mm, the contact stress on the contact surface will decrease rapidly as the wear depth increases.If the wear depth is greater than 0.3 mm, the effect of the wear depth on the stress intensity and contact stress gradually decreases.

Discussion
The effect of sliding wear on the stress in the contact area is investigated in this article by experimental and numerical methods.The small-scale models are used, and how to apply the research results to actual components is a challenge.Therefore, a mathematical model of contact stress considering wear is hopefully established to reflect the changes in contact stress during sliding wear, and research in this aspect is currently underway.

1 .
a) High-speed railway b) Normal-speed railway Fig. Damage to wheel tread

Fig. 5 ( 5 .
b), Fig. 5(c) and Fig. 5(d) show that as the depth of wear increases, the number of stripes near the contact surface decreases, and the gap between the stripes increases.a) Unworn (0 mm ) b) 0.15 mm c) 0.3 mm d) 0.45 mm Fig. Stress intensity diagram at different wear depths

Fig. 7 .
Fig. 7. Contact half-width at different wear depths Fig.7shows with the growth of wear depth (ℎ), the contact half-width () on the contact surface gradually increases.Moreover, Fig.7shows the results obtained from the finite element wear model fit well with the experimental results.Fig.8shows as the depth of wear increases, the contact stress on the contact surface is gradually transformed from the initial sharp convex