## Abstract

Stress distributions and plastic deformation zones are factors directly influencing the fatigue life of components under cyclic contact. An effective approach to improving the resistance of a steel to contact fatigue failure is surface hardening, which builds gradient yield strength from the surface of the steel to the bulk. When using the distortion energy theory as the criterion to identify failure initiation for a case-hardened steel, contact yield starts in the subsurface wherever the von Mises stress reaches the local material strength, rather than at the point of the maximum von Mises stress in the subsurface. If the yield strength changes from the surface to the bulk following a straight line, the location of yield initiation should occur at the tangency of the strength line and the von Mises stress curve. Analyses on circular, rectangular, and elliptical contacts are presented to reveal the locations of contact yield initiation for such case-hardened steels subjected to rolling contact stresses, for which the influence of friction can be ignored. A group of formulas relating contact yield initiation, in terms of the critical pressure, location of the first yield, and plasticity index (transition to plasticity) to case-hardening parameters, such as the case slope, the minimum case depth, and surface and bulk strengths, are derived to facilitate contact element designs using case-hardened materials. The results are applied to examine the rolling contact behaviors of several case-hardened steels, and the data correlation suggests that their rolling contact fatigue lives are related to a nondimensional case-hardening slope besides external loading.

## 1 Introduction

Surface case hardening is extensively and successively used to improve wear and contact fatigue resistances of steels without affecting bulk material properties. Because the combination of a harder surface and a more ductile core can effectively protect the surface from contact fatigue damage, case hardening is widely implemented in the design of bearings, crankshafts, turbine blades, and other mechanical components [1,2]. Longer rolling contact fatigue (RCF) life and more reliable RCF resistance are usually expected for components processed with surface-hardening treatments, such as carburizing, nitriding, and induction hardening [3,4]. Several types of commonly used bearing and gear steels, for example, AISI M50, M50NiL, and 52100, carburized or nitro-carburized, have been widely studied for their RCF lives and wear resistance. The recent works by Rosado et al. [5], Arakere et al. [6], Forster et al. [7], Londhe et al. [8], Guetard et al. [9], and Rydel et al. [10] on rolling contact fatigue (RCF) behaviors of several materials, as well as many other studies, suggest that good material characterization, precise identification of the location of critical stresses, and adequate evaluation of the stress field are among the keys to the RCF analysis of steels, which are also true to the understanding of RCF behaviors of case-hardened materials.

Surface-hardening treatments usually result in gradient strength reductions with depth from the surface to the bulk. The key issues are (1) the case depth that should be sufficient to avoid subsurface yielding and (2) the case-hardening parameters that are quantitatively meaningful to RCF lives of surface-treated materials. Sharma et al. [11] explored the effect of case depth on fatigue failures and indicated that if the peak subsurface shear stress exceeds the critical stress defined by a straight strength gradient, or near the surface, pitting fatigue prevailed in carburized gears tested, whereas the gears might encounter the spalling fatigue from the vicinity of the case–bulk interface, if the effective case depth was insufficient. Elkholy [12] employed the torsional fatigue strength–hardness relationship to evaluate the case depth in carburized gears and investigated the effects of the geometry-material properties, pressure angle, and gear ratio on the design of case depth. Considering the fact that a deep case depth might reduce the compressive residual stress at the surface and the fatigue resistance, Genel and Demirkol [13] conducted rotary bending fatigue tests of AISI 8620 carburized steels with different case depths and constructed empirical relationships between the case profile and fatigue performance to determine the relative case depth defined by the ratio of case depth to specimen diameter. Sun and Bell [14] investigated the wear and corrosion resistance of the carburizing layer produced on AISI 316 austenitic stainless steel through a low-temperature plasma carburizing technique and found that the layer thickness and hardness can benefit the surface wear resistance. Wang et al. [15] examined the microstructure and hardness of wheel/rail materials treated by a laser quenching approach and reported that the spalling damage and surface wear are two major failure modes caused by surface crack and subsurface repeated plastic shear due to tangential and normal forces, contact pressure, creep forces, etc. [16]. Tyfour and Beynon [17], Daves et al. [18] and Wang and coauthors [19] explored the effect of rolling direction reversal, surface roughness, and spherical dents on crack and plastic strain, which may play certain roles in the fatigue life of wheels and rails. Izciler and Tabur [20] did a group of wear tests of gas carburized AISI 8620 on a pin-on-disc test machine and suggested that a thicker case depth subjected to a long gas carburized time could enhance the wear resistance, similar to the experiment results by Sun and Bell [14]. Tobie et al. [21] investigated the influence of case depth on the local stress/strength profiles in the material subsurface and the load capacity of gears. Later, Tobie et al. [22] explored the effect of case depth on pitting resistance of case carburized gears and identified an optimum case depth, linked to contact geometries, to ensure the maximum allowable contact stress. Recently, Zhang et al. [23] studied the influences of hardness distributions (such as the convex, concave, and linear variations of hardness from the surface to the bulk) and case depth on plastic strain and contact pressure and showed that they were critical to the service durability of such a material in contact, especially when it is subjected to rolling contact fatigue (RCF).

Material yield due to contact loading should be avoided in designing mechanical components. Greenwood and Williamson [24] proposed a criterion for the initiation of plastic flow in contacting asperities, named the plasticity index. Kogut and Etsion [25] offered a slightly different version of plasticity index, where the critical interference to cause plastic deformation for a spherical Hertz contact was involved. The definition of the critical interference was presented by Jackson and Green [26]. Chen et al. [27] calculated the cumulative plastic deformation of contact bodies due to repeated loading and investigated their shakedown and ratchetting behaviors under rolling/sliding conditions based on plastic strain volume integral (PV). Zhu et al. [28] examined the pitting life of gears under mixed elastohydrodynamic lubrication (EHL) condition and employed the fatigue life model [29,30] in terms of subsurface von Mises stress and plastic strain volume (PV) to predict the pitting lives of gear teeth, fit to the experimental results. Jackson and Green [31] improved the understanding of plasticity index and defined a closed-form solution for surface asperities in a Gaussian distribution, along with an alternative method to define a plasticity index. Xie et al. [32] investigated the plastic deformation zones beneath the surface of AISI 8620, 9310 and 4140 steels subjected to the surface hardening under the cycle loading, and the relationships between surface hardness, case depth, and RCF life. Recently, Gupta and Zaretsky [33] presented a stress-fatigue life model based on the fundamental Lundberg-Palmgren life [34], indicating the dependence of contact fatigue on subsurface maximum shear stress and the stress volume.

Most researches have been focused on the plasticity index for homogeneous and isotropic materials and on the location of maximum stresses without considering strength distribution. The location of the maximum stress, in terms of von Mises or Tresca stress without considering the effect of friction, is at a fixed location in a homogeneous material, which is 0.48 times the contact radius, *a*, (i.e., 0.48*a*), for circular contacts [35], which should also be the location of the first yield in the material if the strength is homogeneous. However, for case-hardened steels, the hardness varies in a gradient from the surface to the bulk, and so does the yield strength [36]. Therefore, the first yield in a case-hardened material may be away from the location of the maximum stress; rather, it should depend on the comparison between the subsurface stress distribution and the hardness, or the strength, profile.

The work reported in this paper is a systematic research on (1) the first yield locations of case-hardened steels that initiate contact plastic deformation for components under circular contact, rectangular contact, or elliptic contact and (2) a set of formulae to predict contact yield initiation. The research aims to quantify the location of the first yield with respect to case depth and surface hardness and to determine the critical hardness profile that helps avoid contact yielding. The results are used to explain the rolling contact fatigue behaviors of several case-hardened steels reported in our previous work [32].

## 2 Numerical Description of Case-Hardened Steel

Case hardening creates a gradient hardness that reduces from the surface to the bulk. Figure 1 shows the hardness profiles for vacuum carburized AISI 8620 steel specimens, where the hardness values were taken from cross sections and measured using a Vickers Hardness Tester with a 200-gram load. The dwell time with loading for each test was 5 s. The specimens were vacuum carburized at 940 °C for 194 min (low case depth), 287 min (medium case depth), or 445 min (high case depth), and then cooled to 815 °C, held at temperature for 20 min, quenched in agitated 80 °C oil, and tempered at 177 °C for 3 h [32].

*a*)

*b*)

In the above, *σ*_{y} (kgf/mm^{2}) is the yield strength, *HB* is for the Brinell hardness, and *HV* is for the Vickers hardness (kgf/mm^{2}). The yield strength–hardness relationships may be material specific.

It should be mentioned that (1) in many cases, hardness values in different scales are convertible; (2) different yield strength–hardness relationships may be elected for different materials or different considerations; and (3) the yield strength–hardness relation may vary at different stages of plastic deformation; however, at the surface yield initiation, Eq. (1*a*) is largely true based on Refs. [26,36,37], which is exactly the need for this work; and (4) compared with the hardness variation from the surface to the bulk of a non-case-hardened material, which reflects the difference of the very top surface layer, the surface hardness of a case-hardened material is significantly higher than that of the bulk and the hardening layer is much thicker; these two cases should never be mingled. In fact, the use of yield strength–hardness relationship simply means that yield strength and hardness can be mutually converted, and the choice of a relationship does not affect the following derivation work.

*n*, the strain hardening exponent.

*n*= 0.140 for AISI 8620 and

*n*= 0.113 for AISI 9310 can be found from the AISI Bar Steel Fatigue database [38].

*σ*

_{y}of a case-hardened steel can be defined in Eq. (3) as a function of depth

*z*.

*K*=

*σ*

_{yb}−

*σ*

_{ys}/

*d*is the case slope of yield strength,

*σ*

_{yb}and

*σ*

_{ys}are the yield strengths of the bulk and the surface, respectively, and

*d*is the case depth as Fig. 1 shows. The yield strength

*σ*

_{y}of a case-hardened steel can be normalized in terms of $K\xaf=Ka/Ph$ (nondimensional case slope),

*Z*=

*z*/

*a*(nondimensional depth from the surface), $\sigma \xafy=\sigma y/Ph$ (nondimensional yield strength), and $\sigma \xafys=\sigma ys/Ph$ (nondimensional surface yield strength), with

*a*and

*P*for the Hertz contact radius (or half width, if rectangular contact) and maximum Hertz pressure.

_{h}## 3 Contact Yield Initiation and Critical Hertz Pressure, Their Expressions

Based on Eq. (1), the yield strength and hardness of a case-hardened steel with a linearly distributed hardness profile should have a similar gradient. Figure 2 depicts the nondimensional von Mises stress profile and the yield strength line along the *z*-axis in the depth direction, and shows that the first yield beneath the surface initiates at point B (*z*/*a* = 0.62), namely the tangent point of the von Mises stress profile and the yield strength line, instead of point A (*z*/*a* = 0.48) where the von Mises stress is maximum for point contact. Therefore, the yield initiation in a case-hardened steel should occur at the depth where the stress distribution profile is tangent to the yield strength line. It should be mentioned that residual stress is an important factor of a case-hardened steel, which can be considered through stress distribution modification. However, the residual stress distribution depends on many issues and should be considered case by case. In the case residual stresses have to be taken into account, the stress curve should be the total stress distribution profile.

### 3.1 Circular Contact.

*a*, and maximum Hertz pressure,

*P*, for circular contact under the applied normal load,

_{h}*W*, are given as follows [35]:

*R*, and the equivalent Young’s modulus, $E*$, defined as (1/

_{e}*R*

_{e}) = (1/

*R*

_{1}) + (1/

*R*

_{2}), $(1/E*)=(1\u2212\nu 12/E1)+(1\u2212\nu 22/E2)$, where and

*R*

_{1}

*R*

_{2}are the radii of curvature of the two contact bodies, and

*E*

_{1},

*E*

_{2}and

*ν*

_{1},

*ν*

_{2}are the Young’s moduli and Poisson ratios of the two bodies, respectively. In fact, there is an effect of case hardening on Young’s modulus and Poisson’s; based on the findings of Londhe et al. [8] and Klecka et al. [39], the change in modulus can be up to 10%. However, this may only result in a small change in

*P*for a set of typical steel in contact, and ignoring this effect should not make a significant difference in the solutions. Then, substituting Eqs. (4) into (3) results in a nondimensional case slope, $K\xaf$, of the yield strength

_{h}*P*), or in connection to the yield strengths of the surface and bulk, as well as the case depth which is the depth when the yield strength or hardness drops to the bulk value. Considering the symmetry of the stress field about the contact origin, the nondimensional von Mises stress along the

_{h}*z*-axis can be written as follows [35,40]:

Although Gupta and Zaretsky [33] suggested the dependence of contact fatigue on subsurface maximum shear stress, the authors consider that the von Mises stress is also a reasonable choice for the yield analysis because both stresses reflect the distortion energy responsible for ductile failure. In addition, Eqs. (6) and (7) indicate that the nondimensional von Mises stress along the *z*-axis, as well as case slope $K\xaf$, depends on depth and material properties, but not explicitly on external loading.

*ν*= 0.3) with the bulk yield strength of $\sigma \xafyb$ and the depth profile of von Mises stress, $\sigma \xafVM$, is plotted in Fig. 3(a) based on Eqs. (3) and (6). At the point of tangency, the nondimensional von Mises stress reaches the yield strength given by the depth–yield strength relationship from a case-hardening process. The case slope, $K\xaf$, is for both the curve of the depth–von Mises stress relationship at the point of yield initiation and the line for the depth–yield strength relationship. Therefore, the case slope, $K\xaf$, is linked to the depth of yield initiation, $Z\xafc$, which is shown in Eq. (7), and then $Z\xafc$ can be expressed in terms of $K\xaf$.

*a*)

*b*)

*a*)

*b*)

It is noted that for untreated materials of an isotropic strength with $K\xaf=0$, the critical Hertz pressure $Ph_c$ is about 0.62*P*_{h} from Eq. (9) at $Z\xafc=0.48$ from Eq. (8).

*Case*) in a circular contact (subscript:

*P*) and can be defined as follows:

### 3.2 Rectangular Contact.

*a*and maximum Hertz pressure

*P*[35]

_{h}*z*-axis for rectangular contact can be obtained as a function of

*Z*= z/

*a*, referring to Johnson’s book [35] and Green’s paper [40], for materials of Poisson’s ratio larger than 0.2 [40,41], which is true for case-hardened steels having

*ν*= 0.3. Furthermore, the analytical solution to $K\xaf$ in terms of $Z\xafc$ can be expressed in Eq. (14)

*a*)

*b*)

*a*)

*b*)

Similar to the circular-contact situation, knowing the heat treatment results, such as hardness of the surface and bulk and case depth, one can obtain the nondimensional case slope and then determine the depth of yield initiation and critical Hertz pressure. For a untreated material of the homogeneous strength with $K\xaf=0$, the critical Hertz pressure is about 0.558*P*_{h} from Eq. (16) at a depth of $Z\xafc=0.7$ from Eq. (15).

*Case*) in a rectangular contact (subscript:

*L*) can be defined as follows:

Yield and residual stresses thus induced may appear if hardening is not well controlled, and Fig. 6 shows three such cases. Figure 6(a) is for the case of contact yield initiation close to the location of the maximum von Mises stress, which leads to the residual von Mises stress near the surface (the inserted contour plot). This yield can cause pitting failure. Figure 6(b) shows the case of contact yield initiation at the case–bulk interface, together with the residual von Mises stress. The case depth is insufficient, and spalling failure would occur [11,12]. Figure 6(c) presents a combined case, in which the material yields concurrently near the surface and at the case–bulk interface, which can result in both pitting and spalling failures. These failure mechanism arguments should hold for both circular (Sec. 3.2) and elliptical contacts (Sec. 3.4) as well.

### 3.3 Elliptic Contact.

Circular and rectangular contacts are special variants of elliptical contact. The stress field in the elliptic contact can be analyzed, similar to the previous two cases but with respect to the ellipticity ratio, *k*, the contact radius *a*, and maximum Hertz pressure *P _{h}* for nondimensionalization [35]. The nondimensional stresses are related to the location and ellipticity ratio

*k*. Figure 7 shows the nondimensional case slope $K\xaf$ of the yield strength and depth of yield initiation, affected by ellipticity ratio

*k*. Therefore, if the elliptic ratio is given, the relationship between nondimensional case slope $K\xaf$ and yield initiation depth $Z\xafc$ can be determined through the same approach as that for the circular-contact problem.

### 3.4 Contact Yield Initiation in General.

*P*

_{h}_{_c}, as well as the plasticity index $\Psi TypeCase$, can be expressed as follows:

*a*)

*b*)

*c*)

*d*)

*P*

_{h}_{_c}to cause the first contact yield, where

*q*

_{1}= 0.482,

*q*

_{1}= −0.426, and

*q*

_{3}= 0.62 are for circular-contact problems, and

*q*

_{1}= 1.147,

*q*

_{1}= −0.673, and

*q*

_{3}= 0.558 are for rectangular contact problems. From the critical Hertz pressure, one can obtain plasticity index $\Psi TypeCase$, where $\Psi PCase$ represents the circular contact, and $\Psi LCase$ is for rectangular contact. Equation group (19) states that the value of $K\xaf$ is negative, and as the equivalent radius of the contacting element,

*R*, increases, the nondimensional case slope $K\xaf$ decreases. In addition, the minimum case depth $d\xafmin$ for circular contact, Eq. (11) and that for rectangular contact, Eq. (18), can be generalized as

_{e}*p*

_{1}= 2.081,

*p*

_{2}= −0.369, and

*p*

_{3}= −1.773 for circular contact, and

*p*

_{1}= 0.877,

*p*

_{2}= −1.003, and

*p*

_{3}= 0 for rectangular contact. It is emphasized that for a given bulk material, if the case depth is smaller than the minimum of case depth, the contact element should experience plastic deformation at the case–bulk interface and would encounter the risk of spalling fatigue.

Taking the rectangular contacts of case-hardened steels of rollers with the bulk hardness of 400 HV as examples, Eqs. (19) and (20) render the conditions of contact yield initiations, and the results are given in Table 1. For the rollers with a large equivalent radius, *R _{e}* = 300 mm, if the surface hardness is H_surface = 700 HV, one can find that the depth of yield initiation is at about

*Z*= 1.28

_{c}*a*. For a smaller roller with

*R*= 20 mm and H_surface = 700 HV, the contact first yield would be at the depth of

_{e}*Z*

_{c}= 0.73

*a*. As Table 1 shows, Eqs. (19

*a*)–(19

*d*) can be utilized to obtain the critical Hertz pressure,

*P*

_{h}_{_c}, and the depth of yield initiation,

*Z*, from the yield strength of the surface and bulk, case depth, and the contact geometry. It is noted that the equivalent radius significantly influences the contact yield initiation depth. Table 1 also lists several other cases of rollers with a different surface hardness and their corresponding contact yield initiation conditions.

_{c}Referring to Fig. 1, if the nondimensional case depth *d/a* is larger than $d\xafmin$, a higher surface strength and deeper case depth are preferred for the case of Fig. 1(a), a smaller absolute slope is preferred for the case of Fig. 1(b), and a higher surface strength is preferred for the case of Fig. 1(c).

It should be mentioned that although the above results are obtained with reference to steels, they are not limited to ferrous materials only. Equations (19) and (20) for the case slope, minimum case depth, critical load, and plasticity index for each contact type are applicable to other materials graded from the surface, as long as the yield strength varies linearly from the surface to the bulk and the Poisson’s ratio is about 0.3. The same approach can be implemented to derive similar formulas for other materials.

## 4 Rolling Contact Fatigue of Case-Hardened Steels in an Elliptical Rolling Contact

The rolling contact fatigue (RCF) lives of vacuum degassed aircraft-quality steel (AISI 8620 and 9310) from the RCF tester, reported by Xie et al. [32], are further analyzed. The specimens were exposed to different heat treatments: AISI 8620 with atmosphere carburizing (8620-A) and vacuum carburizing (8620-V), as shown in Fig. 1, AISI 9310 with vacuum carburizing (9310-V). Employing the elastoplastic contact model developed by Wang et al. [42], with the solution methods reported by Liu et al. [43] and Liu and Wang [44], and the power-law strain hardening in Eq. (2), but for the elliptical contact between a rod specimen and a ball with the ellipticity ratio of *k* = 0.57, the stress fields of the vacuum carburized 9310 steel (9310-V) and the AISI 9310 steel without the heat treatment, in terms of the total von Mises stress and residual von Mises stress distributions along the *z*-axis, were calculated, where the computational domain −2*a* ≤ *x* ≤ 2*a*, −2*a* ≤ *y* ≤ 2*a*, and 0 ≤ *z* ≤ 3*a* composed of 64 × 64 × 128 elements, the semi-minor axis *a* = 0.432 mm, and the Hertz pressure *P _{h}* = 5930 MPa, and the results are given in Figs. 8(a) and 8(b), respectively. Apparently, case hardening significantly reduces the residual von Mises stress by reducing the stress due to plastic strain. Figure 9 shows the worn groove depth profiles of the vacuum carburized 9310 (9310-V) steel with different case depths after RCF. The total stress of the case-hardened graded 9310-V steel than that of the AISI 9310 steel without heat treatment, as shown in Fig. 8(a), is related to the higher yield strength as a result of the heat treatment, which reduces the plastic stress in the subsurface, as shown in Fig. 8(b), and improves the RCF life and the surface wear resistance, as shown in Fig. 10. Steels hardened with deeper case depths are advantageous against rolling contact fatigue.

The RCF cases for AISI 8620 with vacuum carburizing (8620-V), AISI 9310 with vacuum carburizing (9310-V), and AISI 8620 with atmosphere carburizing (8620-A) are revisited (Fig. 10), because their hardness profiles as a function of depth are approximately linear (Fig. 11). If the hardness at the location of the maximum von Mises stress (Fig. 10(a)) is correlated with the RCF life, as shown in Fig. 10(b), no general agreement can be found overall or for two of these three materials, especially for the case of 8620-A. The discussion in Sec. 3 has indicated that the yield initiation location is at the tangency of the stress curve and the line of the yield-strength variation. It is necessary to examine how sensitive the RCF of these case-hardened steels is on the gradient-hardening-related mechanics characteristics, such as (1) the nondimensional case slope of the yield strength, $K\xaf$, and (2) hardness at the first yield point, which is the location identified by the tangency of the depth profiles of the nondimensional von Mises stress and the yield-strength. The present work can examine (1) because the size of the samples in the tests for the data in Fig. 10 is small, making it difficult to tell the difference between the hardness at the locations of the maximum von Mises stress and at the point of tangency mentioned in (2).

Figure 11(a) shows the measured microhardness profiles for specimens of 8620-V, 9310-V, and 8620-A as a function of depth, which are approximately linear. The yield strength profiles, based on the hardness variations, are fit into straight lines, and Fig. 11(b) plots one such an example for the linear hardness profile of 8620-A nondimensionalized as *H*/3.0*P*_{h} with depth *Z* = *z*/*a*, where the contact radius is *a* = 0.3 mm and the Hertz pressure is *P _{h}* = 4124.6 MPa for the contact yield initiation. From Fig. 11(b), the nondimensional case slope $K\xaf$ of the yield strength can be fitted as $K\xaf=\u22120.0625$. As discussed in Sec. 3 (Eqs. (5) and (13)), the nondimensional case slope $K\xaf$ depends on contact geometry and material property, and changing load leads to parallel lines of yield strength, as Fig. 12(a) shows. The normal distance between the parallel strength lines reflects margining to the external loading levels, and the distance

*S*to the stress curve suggests the difference in RCF lives if other conditions are the same. Nondimensional case slope $K\xaf$, together with

*S*, can be utilized to characterize the RCF lives of AISI 8620-A, AISI 8620-V, and AISI 9310-V, plotted in Fig. 12(b) as a function of $(Sn|K\xaf|)m$. It is noted that $(Sn|K\xaf|)m$ corresponds to the area of the shaded triangle in proportion to the area under the yield strength as shown in Fig. 12(a), which indicates the relationship of fatigue life with stress volume, and

*n*= 1.3 from the analogy to Weibull exponent [33],

*m*= ±1. The negative sign is for the case tested at accelerated conditions where the von Mises stress is higher than the yield strength, shown by the stress curve interfering with the yield strength line (the dashed line). The positive sign is for the cases at the conditions where the von Mises stress is lower than the yield strength. It should be mentioned that the fatigue life shown by the solid line is considered as RCF life baseline, where the yield strength line is tangent to stress curve. When

*m*= −1, decreasing

*S*means increasing the RCF life to that corresponding to the solid line. On the other hand, when

*m*= +1, increasing

*S*means increasing the RCF life. Figure 12(b) shows that the RCF life appears to increase with the value of $(Sn|K\xaf|)m$, and the tendency is clear for all. This is especially true for 8620-A, as compared to the data discrepancy in Fig. 10(b). Figure 12(b) shows nearly the same trends of the RCF lives for all these three case-hardened steels, suggesting that the nondimensional case slope of the yield strength and distance between the yield strength line and the stress curve have significant influences on the RCF lives of these steels although not all data fall closely to this correlation. Some other factors may also influence the RCF lives. Ideally, as the value of $K\xaf$ equals 0, the nondimensional value of the case depth becomes infinity, indicating that for an isotropically treated material, distance

*S*becomes the major dominant factor for RCF lives of these materials.

Further experiments should be conducted, using much larger specimens, for the other examination mentioned earlier, which is for the effect of the hardness at the first yield point, or on the sensitivity of the RCF on the location identified by the tangency of the depth profiles of the nondimensional von Mises stress and the yield strength. The hardness at this location of the first yield can also be an important factor for anti-RCF design.

## 5 Conclusions

The work reported in this paper investigates the relationships between contact yield initiation and case-hardening parameters for circular, rectangular, and elliptical contacts. The von Mises stress is used as the characteristic stress for yield consideration. The results are used to correlate the case slope with the rolling contact fatigue (RCF) lives of several case-hardened steels whose hardnesses, and thus the yield strengths, linearly vary in depth. The results suggest the following for the contacts of linearly gradient case-hardened materials subjected to negligible friction:

The location of the first yield is at the tangency of the depth profiles of the von Mises stress curve and the gradient yield strength of the material.

A set of formula, Eqs. (19) and (20), for nondimensional case slope $K\xaf$, yield initiation $Z\xafc$, critical Hertz pressure $Ph_c$, and plasticity index $\Psi TypeCase$ are derived for circular and rectangular contacts. The location of the first yield for an elliptical contact can be analyzed with the knowledge of ellipticity ratio.

The nondimensional case slope $K\xaf$ and the distance between the yield strength profile and stress distribution curve,

*S*, have significant influences on RCF lives of case-hardened steels. For the materials whose hardness-depth profile, as well as the yield strength profile, can be expressed by a straight line, their RCF lives can be inversely correlated to $(Sn|K\xaf|)m$.Although the above conclusions are obtained with reference to steels, they are not limited to ferrous materials only. Equations (19) and (20) for the case slope, minimum case depth, critical load, and plasticity index for each contact type are applicable to other materials graded from the surface, as long as the yield strength varies linearly from the surface to the bulk and the Poisson’s ratio is about 0.3. The same approach can be implemented to derive similar formulas for other materials with different hardening profiles.

The hardness at the location of the first yield can also be an important factor for RCF. However, it was not studied in the present work because the size of the specimens in the cited tests was too small to show the effect of such a depth.

## Acknowledgment

Q. Wang would like to acknowledge the support from the US National Science Foundation (Grant No. CMMI-1434834). Z. Wang would like to acknowledge the support from the National Science Foundation of China (Grant No. 51775457). The authors would like to thank the support from the Center for Surface Engineering and Tribology, Northwestern University and fellowship supports from China Scholar Council for D. Li (Grant No. 201706050141) and M. Zhang. The authors would also like to thank Mr. B. Enriquez at Northwestern University for his help in preparing this manuscript.

## Nomenclature

*a*=Hertzian contact radius

*d*=case depth

*H*=hardness

*K*=case slope of yield strength

- $K\xaf$=
nondimensional case slope of yield strength

- $Z\xafc$=
nondimensional yield initiation location

*d*_{min}=minimum case depth

*P*=_{h}maximum Hertz pressure

*P*_{h}_{_c}=critical Hertz pressure

*E**=equivalent Young’s modulus

*R*_{e}=equivalent radius of curvature

*E*_{1},*E*_{2}=Young’s modulus for body 1 and 2

*R*_{1},*R*_{2}=radius of curvature for body 1 and 2

*σ*_{VM}=von Mises stress

*σ*_{ys}=yield strength of the surface

*σ*_{yb}=yield strength of the bulk

- $\sigma \xafys$=
nondimensional yield strength of the surface

- $\sigma \xafyb$=
nondimensional yield strength of the bulk

- $\sigma \xafVM$=
nondimensional von Mises stress

*ν*_{1},*υ*_{2}=Poisson ratio for body 1 and 2

- Ψ
^{Case}= plasticity index for case-hardened materials