A New Method for Measuring Stress Inside Movable Element in Contact

Subsurface stress plays an important role in the damage of a movable contact element, but subsurface stresses are usually obtained with numerical calculation according to the theory of contact mechanics. In the present paper, a new method to measure the subsurface stress of the movable element is proposed with using photoelastic technology. Although the technology has been widely used in measuring the stress in the static state, it is seldom used in a moving body because the observed point is moving. After the experimental tester is introduced in detail, the principles of the photoelastic technology are presented. The tester is designed to be able to working in three conditions, the static, the rolling, and the sliding, in the line or surface contacts. The experimental results, that is, the interference fringes, of the three states are then presented in different loads and rotational speeds. Because the fringe figure indicates the maximum shear stress distribution in the body of the moving element, we can find what the real stress distribution in the rolling or sliding element is alike.


Introduction
Subsurface stresses are always the key factors to cause a mechanical element failure in contacts. Among the stresses, the maximum shear stress is usually an important one to cause a crack to expanse [1,2]. If the stress can be correctly measured, we can accurately predict the fatigue of an element. Although the stresses have been obtained easily with a numerical method by computer, a real stress distribution inside an element, especially in a moving element, is much difficultly found yet. In tribology research, rolling friction and sliding friction are two basic contact states which have attracted the attention of many scholars. There has been a large amount of theoretical study on the subsurface stress in contact after Hertz gave the analytical solutions of the contact problems [3]. Hamilton and Goodman solved the analytical solution of the full-field stress inside the material when the sphere is in contact with the normal force and the tangential force [4]. Kalker studied the non-Hertzian contact problem with the numerical method in the rolling contact, and he et al. studied the three-dimensional rolling contact problem under dry friction [5][6][7][8][9]. Johnson and Popov et al. proposed the theoretical distribution of the contact and tangential stresses [10,11]. With increase of friction due to sliding, the maximum shear stress under the surface will move upward gradually 1 3 156 Page 2 of 8 [12]. Hariprasad et al. concluded that the stress is no longer asymmetrically distributed under friction [13]. The theoretical researches have revealed most of the contact stress field already [14]. Therefore, the subsurface stress distributions can have been precisely calculated based on the contact mechanics.
The observation of the plane stress field in the static condition mostly adopts the photoelastic experiment method, which can directly observe the distribution of the principal stress difference inside an object. Brewster discovered the phenomenon of stress optics in 1815 [15,16]. Many studies have applied this method to measure the contact stress in the static state [13,[17][18][19][20]. Zhan et al. used the numerical calculation combined with photoelastic experiment to solve the distribution of rolling contact pressure and describe the evolution of the contact stress field in the wear process [19,21]. Bryant and Lin used the photoelastic technology to measure real contact interfaces and sliding wear between the contact interfaces [22]. In the study of viscoelastic rolling contact, Yoneyama et al. used the photoviscoelastic technique of elliptically polarized light to calculate the variation of principal stress, principal strain, and their direction with time [23].
Although the experimental results of the subsurface stress in the static state have been obtained and are consistent with the theoretical results, those in the kinematic state have not been obtained yet. In the present paper, a photoelastic tester is developed for measuring the inner stress of elements not only in the static but in the kinematic (rolling or sliding). This can be achieved because we keep the moving contact area in the same position. Furthermore, the stress state in the observed region is the same even if the element moves during measurement.
The principal stress difference, which is proportional to the maximum shear stress, can be obtained by the tester, which can work in the static or kinematic situation under different loads and speeds using the photoelastic method. With the tester, the comparison between the photoelastic experiment result and the numerical calculation result will help us to verify the validity of the theoretical analysis of the subsurface stress of a moving element.

Construction and Principles of the Tester
The photoelastic stress measurement method can be used in a surface contact. According to the principles of photoelasticity, the sample must be made of the transparent photoelastic material. Then, a polarized light source irradiates from one side surface of the sample in the contact area and a camera is used to observe the interference light behind the polarizer on the other side of the sample. Since the photoelastic material may change the refractive index under different stresses, the different refractive indexes will cause the light through the sample in different velocities. Therefore, as the same source light passes through one point of the sample and is divided to two different polarized lights in the directions of the two principal stresses, they will form an interference image because of the optical path difference. Then, the photoelastic fringes will appear in the camera. The brightness of the fringes shows the intensity of the principal stress difference, which is two times proportional to the maximum shear stress. Therefore, if the fringe image has been obtained, the distribution of the maximum shear stress in the element has been known.

Construction of Experimental Tester
The diagram of the experimental tester is shown in Fig. 1. Along the direction of the optical path, it consists of a collimating light source, a polarizer, two quarter-wavelength plates, a photoelastic disc, an analyzer, and a CCD camera. The contact pairs consist of a steel disc which is a rolling bearing on the top and the photoelastic disc made of epoxy resin on the bottom. The photoelastic disc is driven by a motor through a synchronous belt. The transmission ratio of the motor and the disc is equal to 1. When the speeds of the steel disc and the photoelastic disc are the same under the action of the frictional force, the rolling state is formed. The steel disc can be braked so that the steel disc will be stationary so that the two discs will slide relatively. Therefore, the tester can realize three states, static, rolling, and sliding, as shown in Fig. 2, where R is the radius of the disc and ω is the angular velocity of the disc. Different weights can be applied on the top of the steel disc so as to change the load. By controlling the speed of the motor and the load of the weight, we can form a variety of experimental working conditions.
The photoelastic disc and steel disc are placed in the middle of the photoelastic experimental platform. The monochromatic light generated by the collimating light source reaches the CCD camera through the polarizer, the quarterwavelength plate, the photoelastic disc, the other quarterwavelength plate, and the analyzer. The observed area is placed just in the optical path. The distance between the lens of the camera and the contact area can be adjusted so that the camera can capture clear photoelastic images. The fringe order of the images will give the magnitude of the stress.

Principles of Experiment
In the experiment, the contact pairs are unlubricated at the room temperature. The upper contact pair is the deep-groove ball bearing 6900-2Z. Its inner diameter is 10 mm, the outer diameter is 22 mm, and the width is 6 mm. The lower contact pair is a photoelastic disc which is assembled on a shaft to be able to rotate.
The load is applied from the top of the rolling bearing. Although the load can be changed, the material is always kept in the elastic state. The load magnitude will be measured by a sensor. The loaded photoelastic disc is placed in a circularly polarized light field.
In the current experiment, a monochromatic laser light with a wavelength of 623 nm is mainly used, which can be regarded as parallel light and used directly. Although the other light, such as white light, can be used as the light source, a convex lens is usually needed to make the light to be parallel. Finally, an industrial camera is used to capture fringe images. The camera is set to record at a speed of 120 f/s.
The photoelastic disc used in the experiment is made of epoxy resin. Its material fringe value is 14.6 N/(mm·fringe) which is determined by a four-point bending test. The outer diameter of the disc is 50 mm, the inner diameter is 10 mm, and the width is 4 mm.
The photoelastic disc is driven by a motor. There are three states shown in Fig. 2 which are as follows: (a) static, (b) rolling (U = Rω), and (c) sliding (U = Rω). For the rolling state, the upper rolling bearing rotates synchronously with the photoelastic disc. When the upper steel disc is braked, it is in the stationary so that the sliding state occurs. Adjusting the brake force the state may be between the rolling and the sliding states. In the current experiment, no lubricant is used between the contact regions.

Photoelastic Method and Principles
The photoelastic experimental method is suitably used in a plane stress field. Transparently photoelastic materials are isotropic and have no birefringence property without external forces. However, under an external force, the material will become anisotropic. This is a temporary birefringence phenomenon caused by stress. Birefringence causes the incident light to be split into two rays in the two principal stress directions. The two rays have an optical path difference so as to cause interference. The distribution of stress inside the Fig. 2 Schematic diagram of three different working states in line contact disc can be determined by collecting interference images in response to the principal stress difference (isochromatics) and its direction (isomele) in the photoelastic material.

Acquisition of Interference Images
In experiments, the photoelastic disc subjected to an external force is placed between a polarized light field. Due to the temporary birefringence effect of the material, when a beam of plane-polarized light enters the loading sample disc vertically, it will be decomposed into two plane-polarized light beams along the principal stress directions at a point. Because of their different propagation velocities, the phase difference will occur after passing through the loaded disc. According to the stress-optic law, the optical path difference can be obtained by the basic formula of photoelasticity [24]: where Δ is the optical path difference, C is the optical constant of the material, h is the thickness of the photoelastic material, 1 and 2 are the two principal stresses in the plane stress field, and λ is the wavelength of the incident light. Figure 3 is the schematic diagram to show how to use the photoelastic method to obtain interference images. The tester includes a light source, a polarizer, a galvanometer, two quarter-wavelength plates, a photoelastic disc, and a CCD camera. The light source can be monochrome or white light source. Set the optical axis of the polarizer to be perpendicular to the reference axis OX. The angle between the fast axis of the first quarter-wavelength plate and OX is 45°. The photoelastic disc is placed between two quarter-wavelength plates. The angle of the second quarter-wavelength plate between its fast axis and OX is 135°. Then, set the optical axis of the analyzer horizontal. Finally, a CCD camera is used to obtain the interference images.

Relationship Between Interference Image and Stress
The light passes through the polarizer and the first quarterwavelength plate, circularly polarized light is incident on the photoelastic disc. Then, optical path difference is resulted in while the polarized light will undergo birefringence under different stresses. Therefore, the interference due to the optical path difference occurs. The intensity of the interference light can be expressed as follows.
where I 0 is the incident light intensity and I is the interference light intensity. The interference image obtained by the above optical device reflects the principal stress difference.
According to Eq. (1), the relationship between the principal stress difference and the optical path difference can be deduced as follows: According to Eq. (2), the intensity of the interference light changes periodically with the optical path difference. With variation of the principal stress difference, the interference image changes alternately between light and dark. If the light level changes for N times as it begins at I = 0, the principal stress difference will appear dark again. If Δ = 2Nπ (N = 0, 1,…), Eq. (3) can be written as: where N is the order of isochromatic fringes, d is the thickness of the disc, and f α is the stress fringe coefficient of the photoelastic material.
(2) I = I 0 sin 2 Δ 2 The stress fringe coefficient f α only depends on the type of the birefringence material and the wavelength of the incident light, but has no relation to the size and shape of the disc. If the principal stress difference is an integer multiple of f α /d, then the dark fringes of the isometric lines appear.
According to theory of elasticity, the relationship between the maximum shear stress and the principal stress difference in the stress plane can be expressed as: It can be seen from Eqs. (4) and (5) that the larger the fringe order N is, the larger the principal stress difference will be. Therefore, the position where the maximum shear stress appears can be judged according to the order of the photoelastic fringe. From the knowledge of material fatigue damage, it is known that the main cause of fatigue usually begins at the position of the maximum shear stress. Therefore, the acquisition of photoelastic images is of great significance to find the real damaged position in rolling or sliding contact. Moreover, the results can be used to check whether the theoretical analysis is accurate or not.

Experimental Results of Rolling and Sliding in Line Contact
In the process of photoelastic experiment, the load can be changed by increasing or decreasing the number of weights, and the speed can be changed by adjusting the speed of the servo motor. An industrial camera is used to capture the fringe image. The light source can be a common light focused into a parallel light by a convex lens or it can be directly a parallel light such as the laser. In Fig. 4, there is a fringe image of photoelasticity obtained with a white LED light. For convenience, the following analysis figures are always obtained with a laser used in the experiments.

Influence of Load on Distribution of Principal Stress Difference
As shown in Table 1, the load w is equal to 94.8, 114.5, or 142.7 N in the static contact (the rotational speed n = 0) and in the rolling contact or the sliding contact (n = 25 rpm).
Corresponding the different loads, the maximum Hertzian stress σ H is shown in Table 1. Using Eq. (3), it can be determined that when the fringe order of the photoelastic principal stress difference distribution diagram is equal to 1, that is N = 1, the principal stress difference is 3.6 MPa. It can be seen from Table 1 that the principal stress difference is symmetrical in the static state and almost symmetrical in the rolling state. The number of stripes near the surface is the largest, which is quite similar to the solutions of Hertzian contact theory. At the same time, it can be seen that with the increase of the load, the fringe order in the photoelastic principal stress difference distribution diagram increases significantly. The more the fringe order, the greater the stress on the contact pair. This is also consistent with the theoretical results.
For the sliding state, however, if the frictional coefficient is assumed to be a constant, the friction force between the contact pairs will increases as the load increases. Since the photoelastic disc rotates counterclockwise, it can be seen that the photoelastic principal stress difference fringes distribution under the sliding state deflect toward the direction of rotation, and the degree of deflection of the fringes in the sliding state is more obvious than that in the rolling state, which shows that the friction force in the sliding state has more significant influence than that in the rolling state.
In the experiments, four times average sliding friction coefficients corresponding to loads are listed in the following table, while the rolling friction coefficient μ is about 0.0010-0.0015.
Since the rolling frictional force on the surface is very small, the stress is nearly symmetric after a symmetric load is applied on the disc during experiment. However, because the sliding frictional force on the surface is larger enough, it brings about a transverse action to distort the distribution of stress. Therefore, the sliding imagines become asymmetric ( Table 2).

Influence of Rotational Speed on Principal Stress Difference in Rolling and Sliding
When considering the influence of the speed on the subsurface stress, we adjust the motor speed to make the contact pair roll or slide under a certain load. Table 3 shows the photoelastic interference fringes at different rotational speeds to keep the load as a constant, that is, w = 142.7 ± 1 N (σ H = 59.81 MPa).  It can be seen from Table 3 that without rotation, that is, no frictional force, the images have no deflection. As the disc rotates, there is no obvious change in the rolling, but there is some deflection of the photoelastic fringes in the sliding. The difference between the rolling and sliding images depends on the degrees of deflection, which indicates that under the action of friction, the photoelastic fringes will deflect and the stress field will deflect along the direction of motion. The effect of sliding friction on fringe deflection is greater than that of rolling friction because the sliding frictional force is much larger than that of rolling friction so that the shear stress in sliding is larger than that in rolling. Therefore, the effect of the frictional force in sliding must be considered on fatigue of elements, but need not in rolling.

Relative Sliding Results in Surface Contact
The tester can also be changed slightly to be used for measuring the subsurface stress of relative sliding elements in surface contact. However, since the study region may move away from the scope, the movement can only be limited in a short distance. The relative sliding model in surface contact is shown in Fig. 5, where the fixed block is made of steel and the movable block is made of the photoelastic material. The observed region is at the upper side of the movable block. And, it can move while the transverse (frictional) force is gradually increased largely enough.
The results of the interference images corresponding to the distribution of the principal stress difference in the movable block are shown in Fig. 6. It can be seen that the fringes change with time as the transverse force driving the movable block increases.
We can see that without transverse force (that is, t = 0 s), the fringes (the maximum shear stresses) are nearly symmetrical and concentrated at the two edges because there are a large number of interference fringes. As the transverse force increases, the fringe images gradually change. The stresses are severely distorted, that is, the stress at the front (left) edge increases while the stress at the back (right) edge decreases. In fact, the experiment is quasi-dynamic. The pictures of Fig. 6 show the movement trend. Only the images in the last column show that the movable block begins to slide.

Conclusion
In the present paper, a method for measuring the subsurface stress of a movable element is presented by means of photoelasticity. The images of the principal stress difference, that is, equal to twice of the maximum shear stress, are obtained under the rolling and sliding contacts. The main conclusions are as follows.
(1) The photoelastic method for measuring the shear stress inside the element can be used in the kinematic states, rolling or sliding, as well as the static one for the plane stress field. (2) The interference fringes obtained during photoelastic experiment indicate the maximum shear stress of a