Residual stress distribution and evolution in gear shaft journals induced by mechanical roll hardening: experimental characterization and numerical prediction

1 Introduction

As the core component for power transmission in prime movers, gears are subject to extremely stringent requirements for both manufacturing precision and service performance. Due to the periodic nature of meshing loads, the shaft journal is subjected to large-amplitude alternating impact forces over extended periods, readily inducing bending and torsional deformations. Under sustained cyclic bending stresses, the material progressively accumulates fatigue damage, eventually leading to crack initiation, which severely compromises the reliability and service life of the gear transmission system (Luo et al., 2023; Aliakbari et al., 2022a ; Singh and Sengar, 2021). With the advancement of computational capabilities, finite element simulation technology has matured considerably, providing theoretical support for the structural design of gear shaft systems under fatigue loading. Nevertheless, relying solely on structural optimization to enhance fatigue performance presents distinct limitations: on one hand, the local optimization space available at the gear shaft journal is inherently constrained; on the other hand, adjustments to structural parameters will trigger dimensional chain variations in associated mating components, resulting in broad-ranging impacts that are difficult to reconcile with practical manufacturing constraints. Consequently, under current technological conditions, achieving fatigue strengthening through the refinement of manufacturing processes and optimization of machining parameters remains a more viable technical pathway (Aliakbari et al., 2022b; Xu et al., 2025; Fu et al., 2026).

Currently, numerous surface strengthening methods have been developed for engineering practice, each possessing specific applicable operating conditions and inherent limitations (Ma et al., 2024). Surface rolling strengthening, as an effective chipless surface treatment technology, applies reciprocating rolling pressure to the component surface via specially designed rolling tools (commonly fabricated from quenched steel, cemented carbide, or ruby, with geometries encompassing rollers, balls, and wheels) (the strengthening process is illustrated in Figure 1) (Li et al., 2023; Zhou et al., 2023; Španiel et al., 2021). This process significantly reduces surface roughness and introduces beneficial residual compressive stress fields within the surface layer, thereby concurrently enhancing the corrosion resistance, wear resistance, and fatigue performance of components, demonstrating excellent engineering applicability (Kluz et al., 2021; Nguyen et al., 2022; Noronha et al., 2024).

As a surface modification technique, rolling strengthening has achieved remarkable progress in improving the fatigue performance of mechanical components. The Internal Combustion Engine Research Institute of Tianjin University employed rolling processes to strengthen the gear shaft journals of 370Q and 376Q diesel engines. Results demonstrated that the fatigue limit of the strengthened journals increased by 92.3%, with the safety factor improved to nearly twice the original value (Heikebrügge et al., 2022b ). Liuzhou Nanfang Automotive Cylinder Liner Factory and Henan Yingwei Dongfeng Machinery Manufacturing Co., Ltd. Conducted rolling strengthening experiments on automotive cylinder liners and hub axle tubes, respectively, confirming that this technology significantly improves product yield rates and enhances fatigue performance (Morrev and Gordon, 2018; Pertoll et al., 2024). The stress corrosion cracking strength of 300 M steel after strengthening treatment could be elevated to the alloy’s yield strength level was reported, with increased fracture threshold values, thereby effectively enhancing resistance to stress corrosion cracking (Fonseca and Faria, 2019). The Academy of Armored Force Engineering investigated rolling strengthening of machined surfaces on heavy-duty components following surfacing repair. Their findings revealed that rolling the surfacing layer with specific process parameters could generate nanoscale grain structures within the surface layer, improving surface hardness, wear resistance, and residual stress states (Kinner-Becker et al., 2020; Bialowas et al., 2023).

Regarding the regulation of surface microstructure and stress states, surface rolling treatment of LZ50 steel and found that this process increases surface layer hardness and introduces high residual compressive stresses was investigated (Kinner-Becker and Sölter, 2023). The synergistic interaction between hardness enhancement and compressive stress effectively suppresses microcrack initiation and propagation, attenuates the accelerating influence of short crack colony effects on crack growth, and thereby improves the local fatigue resistance of the material (Kinner-Becker et al., 2021). The very-high-cycle fatigue behavior of high-carbon chromium bearing steel was studied, demonstrating that in the high-stress, low-life regime, cracks in both treated and untreated specimens initiated from the surface (Fonseca and Faria, 2018). However, the presence of surface residual compressive stress after rolling led to increased fatigue strength.

In the low-stress, high-life regime, crack origins in untreated specimens were located in the subsurface, whereas in treated specimens they shifted to interior regions devoid of residual compressive stresses and hardened layers, resulting in significant fatigue strength improvement (Kumar et al., 2021). X-ray diffraction was employed to investigate the rolled AA6110-T6 aluminum alloy, with particular attention to the distribution of surface residual stresses and their impact on fatigue performance. The residual stress relaxation behavior under varying thermal conditions and its underlying mechanisms were analyzed (Dänekas et al., 2022; Schubnell et al., 2022). The research indicates that rolling strengthening establishes a residual compressive stress zone of appreciable depth within the specimen surface layer, effectively reducing the operational stress level during actual service, thereby suppressing fatigue crack nucleation and propagation in the surface layer and enhancing material fatigue life (Reggiani and Olmi, 2019; Heikebrügge et al., 2022a). Additionally, rolling treatment improves specimen surface quality, reduces surface roughness, renders machined surface profiles smooth and continuous, effectively mitigates stress concentration, and diminishes the detrimental influence of geometric notches on fatigue strength (Dos et al., 2021; Martins et al., 2023).

In summary, rolling strengthening exhibits significant efficacy in improving surface microstructure, optimizing stress distribution states, and enhancing surface integrity, constituting an effective fatigue strengthening methodology.

2 Materials and methods

2.1 Experimental materials

30CrMoA steel with dimensions of Φ 50 × 200 mm is used; its chemical composition is shown in Table 1, and its mechanical and material property parameters are shown in Table 2. In these tables, E represents the elastic modulus of the material, σ 0.2 represents the yield strength, σ b represents the tensile strength, v represents the Poisson’s ratio, ρ represents the density, u represents the coefficient of friction, and C represents the specific heat capacity.

TABLE 1. Chemical composition of 30CrMoA steel used in testing (Mass fraction, %).

TABLE 2. 30CrMoA material properties.

Taking a large, high-precision gear drive shaft as the prototype, and considering the costs and duration of the tests under the principle of material similarity, process experiments and measurements are conducted on scaled-down specimens fabricated from the same material with the same initial heat-treated condition. The tempering specifications for the mechanically rolled specimens are listed in Table 3.

TABLE 3. Modulation technical specifications.

2.2 Mechanism of mechanical roll-forming strengthening

Rolling is a typical elastic-plastic contact problem, the essence of which lies in the fact that when the maximum contact pressure between the roller and the workpiece exceeds the yield strength of the material, the material undergoes irreversible plastic deformation, thereby forming a surface-hardened layer and a residual compressive stress field. According to Hertz’s line contact theory (Łach, 2024), the contact half-width b is given by:where P is the normal rolling force, R is the roller radius, L is the effective contact length of the roller, and E * is the equivalent modulus of elasticity. Among these,where E 1 and E 2 represent the moduli of elasticity of the roller and the workpiece, respectively, while v 1 and v 2 represent the Poisson’s ratios of the roller and the workpiece, respectively.

b = 4 PR π L E * (1)

1 E * = 1 − v 1 2 E 1 + 1 − v 2 2 E 2 (2)

The maximum contact pressure is:

p 0 = 2 P π b L = P E * π R L (3)

The pressure distribution is semi-elliptical and perpendicular to the roller’s axis.where x represents the lateral distance from the center of contact.

p x = p 0 1 − x b 2 (4)

The critical condition for initial yield (based on the maximum shear stress criterion, i.e., the Tresca criterion) corresponds to a critical rolling pressure P Y of:where σ Y represents the uniaxial yield strength of the material.

P Y = π R L 1.1 σ Y 2 E * (5)

When P < P Y, the contact zone is in a fully elastic state, and Hertz’s theory can be applied; when P > P Y, the material yields.

The entire rolling process is divided into elastic, elastoplastic, and fully plastic stages. Rolling induces a certain amount of stress in the journal. To introduce residual stresses into the journal’s surface layer that can counteract the thermal stresses and centrifugal forces generated during operation, the applied stress reaches the yield limit, that is, P > P Y, and plastic deformation occurs. Elastoplastic problems fall under the category of material nonlinearity. (Caddemi, 1994).

The strain formula for plastic deformation is:where d ε e and d ε p represent the elastic strain increment and the plastic strain increment, respectively. d ε e obeys Hooke’s law, which states:where D e represents the elastic matrix of the material, and d σ represents the stress increment. d ε e follows the flow law, which is:where F represents the loading surface. The Mises yield criterion is used. This yields the stress-strain relationship in incremental form for the ideal elastoplastic model:

d ε = d ε e + d ε p (6)

d ε e = D e − 1 d σ (7)

d ε P = d λ ∂ F ∂ σ (8)

d σ = D e − D P d ε = D e p d ε (9)

Since rolling is a form of microplastic deformation, it is assumed that rolling does not alter the overall elastic properties of the workpiece. For both roller and roller-column rolling, the mechanical behavior can be simplified to a plane-strain problem. Since the contact between the roller and the workpiece is approximately linear, the rolling force is simplified to a concentrated force. By solving the Boussinesq-Flamant problem, the internal stresses in the workpiece can be determined as (Xu et al., 2025): where p s is the concentrated pressure at the centerline in the Boussinesq-Flamant problem, which is related to the rolling pressure p as follows:where a is the equivalent contact length of the rolling action of a roller or roll.

σ r = − 2 p s ⁡ cos ⁡ θ π r , σ θ = 0 (10)

σ z = ν σ r + σ θ = − 2 ν p s ⁡ cos ⁡ θ π r (11)

τ r θ = τ θ r = 0 , τ r z = τ z r = 0 , τ θ z = τ z θ = 0 (12)

p s = p a (13)

Substitute Equations 10–13 into the von Mises yield criterion:

σ s = 2 2 σ θ − σ r 2 + σ θ − σ z 2 + σ r − σ z 2 + 6 τ r θ 2 + τ θ z 2 + τ r z 2 (14)

We obtain:where σ s is the von Mises yield stress of the workpiece; v is the Poisson’s ratio of the workpiece; therefore, during the rolling process, microplastic deformation occurs within a cylinder subjected to a linear load. Setting θ = 0 yields the relationship between the depth of microplastic deformation and the rolling force in roller or roll-forming processes:

r = 2 p s 1 − ν + ν 2 cos ⁡ θ π σ s (15)

d = 2 p s 1 − ν + ν 2 π σ s = 2 p 1 − ν + ν 2 π a σ s (16)

For the ball-rolling method, since the contact between the ball and the workpiece is approximately point contact, the rolling force is similarly simplified to a concentrated force. Based on the solution to the Boussinesq problem, the stress components within the workpiece are given by:

σ ρ = p 2 π R 2 − 3 ρ 3 x R 3 + 1 − 2 ν R R + x (17)

σ θ = 1 − 2 ν p 2 π R 2 x R − R R + x (18)

σ x = 3 ρ x 3 2 π R 5 (19)

τ x r = τ r x = − 3 p ρ x 2 2 π R 5 (20)

τ r θ = τ θ r = 0 ; τ θ x = τ x θ = 0 (21)

R 2 = x 2 + ρ 2 = x 2 + y 2 + z 2 (22)

Substituting Equations 17–22 into the von Mises yield criterion and setting y = 0 and z = 0 yields the following relationship between the microplastic deformation depth at the axis and the rolling force:

d = p 2 ν + 5 4 π σ s (23)

According to fracture mechanics theory, a surface crack or defect propagates only when the stress intensity factor at the crack tip reaches the critical stress intensity factor of the material under the applied cyclic loading. In the presence of a mean stress, the range of the critical stress intensity factor is expressed as (Luo et al., 2023; Ma et al., 2024):where R represents the average stress under alternating loads.

Δ K t h = 1.2 Δ K 1 + 0.2 1 + R / 1 − R (24)

Since residual compressive stress counteracts the mean stress induced by external cyclic loads, it reduces the amplitude of the stress intensity factor Δ K actually experienced by the component. The reduction in R, in turn, increases the critical stress intensity factor Δ K t h at which crack propagation initiates. Consequently, under given cyclic loading conditions, a crack that would otherwise propagate may require a further increase in the applied cyclic stress to initiate surface crack propagation in the presence of residual compressive stress, owing to the elevated Δ K t h. This mechanism accounts for the beneficial effect of residual compressive stress on improving the fatigue strength of cracked materials.

2.3 Mechanical deep rolling test

The rolling experiments were conducted on a CAK3665 CNC lathe equipped with a variable-frequency stepless speed-regulated spindle, offering a rotational speed range of 20–3,000 r/min. The maximum travels of the X- and Z-axes are 220 mm and 650 mm, respectively, with corresponding rapid traverse rates of 3.8 m/min and 7.8 m/min. The positioning repeatability is 0.012 mm for the X-axis and 0.016 mm for the Z-axis, satisfying the stringent control accuracy required for the rolling depth, workpiece rotational speed, and feed rate investigated in this study.

During the roll-forming process, the forming tool is mounted on the X-axis slide of the CNC lathe via a tool holder, and the roller is pressed against the gear journal surface at a predetermined penetration depth. The workpiece rotates at a constant speed with the spindle, while the forming tool traverses axially along the journal at a preset feed rate, generating a continuous helical rolled track on the journal surface. Under the applied static load, the rolling balls exert a normal pressure on the journal surface; once this pressure exceeds the yield strength of the material, directional plastic flow is induced in the surface layer. The penetration depth is precisely controlled by the CNC system and directly governs the stress level and the extent of plastic deformation within the contact zone. The rolling test platform used in the mechanical deep rolling tests is shown in Figure 2.

X-ray diffraction (XRD) measurements are performed using a PROTO LXRD residual stress analyser to determine the residual stresses in the surface layer of the specimens. A PROTO 8818-V3 electrolytic polishing instrument is employed for layer-by-layer material removal, and, in conjunction with the XRD measurements, residual stress values at successive depths are obtained, enabling the residual stress distribution along the depth direction to be characterized. A deep-focus microscope is used to examine the microstructure of the specimen surfaces before and after rolling, from which the deformation layer thickness and grain morphology under different process parameters are captured; the characteristics of rolling-induced surface plastic deformation and grain refinement are then analysed based on these images. To investigate the effects of rolling depth, feed rate, and workpiece rotational speed on the surface residual stress and the stressed layer depth after deep rolling, the experimental design and the corresponding results are summarized in Table 4.

TABLE 4. Experimental design and results.

2.4 Mechanical deep rolling test

Deep rolling is a highly nonlinear process involving triple nonlinearities, material, geometric, and contact, coupled with frictional effects and heat conduction. Reasonable simplifications of the actual operating conditions are therefore necessary to ensure computational accuracy, improve convergence, and reduce computational cost. A block-type model is adopted without compromising the accuracy of the results, and the roller, whose stiffness is far greater than that of the workpiece, is idealized as a rigid body. The three-dimensional model is illustrated in Figures 3, 4. In the finite element model of the 30CrMoA workpiece developed in this study, the workpiece dimensions are 40 mm (length) × 16 mm (width) × 20 mm (height). To ensure consistency between the experimental and simulation conditions, the roller is defined with a diameter of 65 mm, a fillet radius of 2.0 mm, and a centre height of 30 mm. An equivalent modelling strategy is employed: the bottom surface of the workpiece is fully constrained to suppress its rotation, while the roller, treated as a rigid body, reproduces the rolling action through a coupled rotational and reciprocating translational motion along the workpiece surface. Upon completion of each reciprocating stroke, the roller is indexed laterally in a stepwise manner, thereby covering the entire target processing area.

Since the rollers are modelled as rigid bodies, they require neither meshing nor the assignment of material properties. The workpiece is discretized with hexahedral elements. To balance simulation accuracy against computational cost, local mesh refinement combined with absolute mesh control is applied: the machined region on the upper surface of the workpiece is locally refined, whereas the unmachined region at the bottom is meshed relatively coarsely. The downward pressure and feed of the roller are imposed through a prescribed translational path, while all other degrees of freedom are constrained. Time-dependent functions are defined to specify the motion parameters, thereby establishing the boundary conditions for the roller.

The selection of an appropriate time increment is critical: an excessively large increment may cause severe mesh distortion and convergence difficulties, whereas an excessively small one substantially increases the computation time. For regions requiring high local accuracy, the time increment should be smaller than one-tenth of the characteristic element size. In the present simulation, the time increment is set to 0.0001 s, and the total number of increments is determined from the simulation duration. To limit the storage size of the output files without compromising the post-processing analysis, results are saved every 50 increments. The fidelity of the dynamic simulation of the machining process is governed by the constitutive model of the material, which must therefore be selected with care. Constitutive models for a wide range of workpiece materials have been established through experimental, analytical, and numerical studies. Given that heat-assisted surface rolling shares the same fundamental strengthening mechanism as conventional cold plastic rolling, and that the Johnson–Cook model accommodates a broad range of strain rates while accounting for strain hardening, strain-rate hardening, and thermal softening, including material softening induced by adiabatic heating arising from plastic deformation, this model is adopted to define the plasticity parameters (Jiang et al., 2024; Neuvonen et al., 2020). The constitutive equation for the material flow stress is expressed as follows:where A is the yield stress coefficient, B is the strain hardening coefficient, n is the strain hardening exponent, C is the strain rate coefficient, m is the thermal softening exponent, ε is the equivalent plastic strain, ε * is the normalized effective plastic strain rate, and T * is the corresponding temperature. Among these,where T is the current temperature, T r o o m is the room temperature, T m e l t is the melting temperature.

σ = A + B ε n 1 + C ⁡ ln ⁡ ε * 1 − T * m (25)

T * = T − T room T melt − T room (26)

Deep rolling is a cold plastic deformation process, and the effect of temperature during the process can be neglected. The simplified form of this model is shown below. (Zhong et al., 2022). The constitutive parameters are shown in Table 5.

σ = A + B ε n 1 + C ⁡ ln ⁡ ε * (27)

TABLE 5. Johnson-Cook model material constitutive parameters for 30CrMoA medium-carbon low-alloy structural steel.

Select a sparse matrix solver and use the Newton-Larsen method for the iterative procedure.

3 Results and analysis

3.1 Typical characteristics of residual stress distribution

The stress field exhibits a typical layered distribution pattern, characterized by moderate compressive stress in the surface layer, peak compressive stress in the subsurface layer, a gradient-decaying transition zone, and near-zero stress in the matrix (Zhou et al., 2023). This pattern is consistent with the formation mechanism of residual stresses induced by elastoplastic contact deformation. The strengthening effect of deep rolling is concentrated only in the near-surface region of the workpiece is shown in Figure 5. The compressive stress level in the near-surface stress zone is slightly lower than the peak value in the subsurface layer. The surface acts as a free boundary with no material constraint on the outer side; moreover, during the rolling process, the surface metal undergoes intense plastic flow, and part of the internal stress is released through dislocation slip and grain rotation. Consequently, the compressive stress amplitude does not reach its maximum value. At a certain depth below the surface, a continuous, straight band of high-amplitude compressive stress forms. At this depth, the material undergoes the most complete plastic deformation while being strongly constrained by the surrounding elastic matrix; during the unloading phase, the elastic matrix attempts to recover from deformation but is most significantly impeded by the plastic deformation layer at the surface, thereby forming a peak in residual compressive stress.

This region serves as the core strengthening layer that suppresses the steady-state propagation of fatigue cracks and can effectively reduce the effective stress intensity factor at the crack tip. As the depth increases further, the amplitude of the residual compressive stress decreases approximately monotonically with depth; after reaching a certain depth, the compressive stress disappears completely, and the stress in the deep matrix approaches zero, remaining largely unaffected by the rolling process. This distribution conforms to the principle of mechanical self-equilibrium: the residual compressive stress introduced in the surface layer is balanced by weak tensile stress within the interior. Since the thickness of the strengthened layer is much smaller than the overall dimensions of the workpiece, the magnitude of the tensile stress in the matrix is extremely low and does not adversely affect the material’s overall mechanical properties.

The typical distribution characteristics described above confirm that deep rolling can introduce a residual compressive stress field with high amplitude and great depth into the surface layer of the journal, with the peak compressive stress located in the subsurface layer, which precisely corresponds to the primary region of steady-state propagation following fatigue crack initiation, thereby directly enhancing the journal’s bending fatigue resistance at the stress level. At the same time, the uniform distribution of axial stress ensures consistency in the strengthening effect across the entire length of the journal, providing a stress foundation for the long-term operational reliability of the gear. This typical distribution pattern also serves as a reference benchmark for subsequent analyses of the influence of process parameters and parametric modeling.

3.2 Effect of individual process parameters on residual stress

To isolate the individual effect of each process parameter, single-factor experiments were conducted using the controlled-variable method, in which the remaining parameters are held constant at their intermediate reference values. Three key parameters, rolling depth, feed rate, and workpiece rotational speed, are examined with respect to two evaluation metrics, namely, the surface residual compressive stress and the compressive stress layer depth, thereby identifying the effective range of each parameter and the associated plastic saturation effect.

3.2.1 Analysis of the effects of rolling depth

With the workpiece rotational speed held at 600 r/min and the feed rate at 0.05 mm/r, five rolling depths, 0.05, 0.07, 0.10, 0.15, and 0.20 mm, are selected to examine the influence of rolling depth on the residual stress field. The depth-wise distributions of residual stress under different rolling depths, together with the relationships of rolling depth to surface residual stress and compressive stress layer depth, are presented in Figures 6–8, respectively.

As shown in Figure 6, the residual stress distributions exhibit a consistent gradient across all rolling depths: the residual compressive stress at the outermost surface is moderate; as the depth increases, the magnitude of the compressive stress rises continuously, reaching a peak in the subsurface layer at 0.2–0.4 mm; beyond the peak depth, the compressive stress decays gradually, approaching zero at a depth of approximately 0.8 mm; and at depths greater than 1.5 mm, it transforms into a balanced residual tensile stress, stabilizing in the range of 100–180 MPa to equilibrate the compressive stress field induced by plastic deformation in the surface layer. With increasing rolling depth, the entire stress profile shifts towards more negative values, the peak subsurface compressive stress increases accordingly, and the compressive stress layer deepens continuously, indicating that a greater rolling depth effectively enhances the degree of plastic deformation in the surface layer.

When the rolling depth is excessive, severe plastic flow occurs in the surface metal, leading to localized tearing and metal bulging in the surface microstructure. The stress-relaxation effect associated with this excessive surface plastic deformation partially offsets the gain in compressive stress; meanwhile, the increased heat accumulation in the surface layer promotes slight dynamic recovery, which further weakens the residual compressive stress.

As shown in Figure 7, at a rolling depth of 0.05 mm, the surface residual compressive stress is only −430.96 MPa. In this case, the shallow radial penetration of the roller results in insufficient plastic flow of the surface metal, limited dislocation multiplication and grain refinement, and a weak confinement effect upon unloading and spring back, yielding only a low-magnitude residual compressive stress. As the rolling depth increases from 0.05 to 0.10 mm, the surface residual compressive stress rises continuously, reaching a maximum of −592.93 MPa at 0.10 mm. At this stage, the increased rolling depth directly enlarges the contact pressure and contact area between the roller and the workpiece; extensive irreversible plastic slip occurs in the surface metal, the degree of lattice distortion increases markedly, and the constraining effect of the subsurface elastic matrix on the deformed layer is progressively strengthened, macroscopically manifested as a continuous increase in the magnitude of the surface residual compressive stress. When the rolling depth exceeds 0.10 mm, the magnitude of the surface residual compressive stress declines slightly, measuring −563.60 MPa at 0.15 mm and −515.36 MPa at 0.20 mm.

As shown by the compressive stress layer depth curve in Figure 8, the depth of the stress-strengthened layer increases rapidly at first, then grows more slowly, and eventually approaches saturation with increasing rolling depth. At a rolling depth of 0.05 mm, the effective compressive stress layer depth is only 0.5594 mm; within the range of 0.05–0.15 mm, the layer depth increases approximately linearly, reaching 0.8591 mm at 0.15 mm; when the rolling depth is further increased to 0.20 mm, the layer depth rises only marginally to 0.8764 mm, an increment of less than 0.02 mm, indicating that the plastic deformation has reached saturation. Owing to the high constraint of the deep-layer matrix material, the rolling load cannot be transmitted effectively further into the bulk; consequently, further increasing the radial indentation depth cannot continuously extend the influence zone of the residual compressive stress.

3.2.2 Analysis of the effect of workpiece rotational speed

Single-factor experiments were performed at five workpiece rotational speeds of 120, 360, 600, 720, and 840 r/min, with the rolling depth and feed rate held constant at 0.10 mm and 0.05 mm/r, respectively, to examine the influence of rotational speed on the residual stress distribution, surface stress magnitude, and hardened layer depth, as shown in Figures 9–11.

Figure 9 presents the residual stress distributions along the depth direction at different rotational speeds. The overall distribution trend remains essentially consistent across all speeds, with a pronounced compressive stress peak occurring in the subsurface layer. As the rotational speed increases, however, the entire stress profile shifts toward more negative values, accompanied by a corresponding increase in the magnitude of the peak compressive stress.

As shown in Figure 10, the magnitude of the surface residual compressive stress is only 560.62 MPa at a workpiece rotational speed of 120 r/min; it increases slightly to 578.46 MPa at 360 r/min, reaches its maximum of 609.73 MPa at 720 r/min, and then decreases slightly to 607.70 MPa as the speed is further increased to 840 r/min. At low rotational speeds, the residual compressive stress generated by plastic deformation in the subsurface layer partially relaxes, which reduces the thickness of the effective strengthened layer. A moderate increase in rotational speed alleviates the local temperature rise, enabling the rolling load to be transmitted more effectively into the subsurface layer and thereby expanding the plastic deformation zone. At excessively high rotational speeds, however, the contact between the roller and the workpiece becomes insufficient; consequently, the plastic deformation per rolling pass is inadequate, and the strengthened layer thickness decreases slightly.

As shown in Figure 11, the depth of the compressive stress layer is only 0.67417 mm at 120 r/min; it increases with rotational speed, reaching a maximum of 0.74384 mm at 720 r/min, beyond which it decreases slightly to 0.73417 mm at 840 r/min. A comparison of the variations in residual stress induced by rolling depth and by workpiece rotational speed reveals that the latter has a weaker influence than the former, as rotational speed regulates plastic deformation only indirectly, by altering the contact duration and the process-induced temperature rise. Considering both the overall strengthening effectiveness and processing stability, the optimal workpiece rotational speed range is 600–720 r/min, within which both a higher surface residual compressive stress and a thicker strengthened layer can be achieved. At lower rotational speeds, the roller dwells at a given circumferential position for a longer duration, resulting in greater localized heat accumulation from plastic deformation, which promotes thermal relaxation of the residual compressive stress in the surface layer. As the rotational speed increases, the duration of each contact is shortened and heat accumulation is significantly suppressed, allowing the residual compressive stress generated by plastic deformation to be more fully retained. In the low-speed range, the rotational speed has a relatively minor influence on the surface stress, with a difference of less than 20 MPa; in the medium-to-high-speed range, the magnitude of the surface compressive stress increases markedly, as the suppression of thermal relaxation by the elevated rotational speed becomes dominant. When the rotational speed exceeds 720 r/min, however, the relative sliding speed between the workpiece and the roller becomes excessive, intensifying localized micro-slip and causing slight frictional damage to the surface layer, which leads to a marginal decay of the residual compressive stress.

Both the magnitude of the compressive residual stress and the depth of the compressive layer increase monotonically with the workpiece rotational speed: the former rises from 560.6 MPa to 609.7 MPa (+8.8%; Figure 10), and the latter from 0.674 mm to 0.744 mm (+10.4%; Figure 11). These trends are physically reasonable and can be attributed to three mechanisms: (i) strain-rate hardening, whereby the flow stress of the rate-sensitive material increases with the equivalent strain rate, enlarging the plastically deformed zone and thereby deepening the compressive layer retained after unloading; (ii) enhanced accumulation of plastic strain, at a constant feed per revolution, a higher rotational speed increases the number of rolling passes per unit time, which intensifies strain accumulation and drives the zero-stress boundary deeper into the substrate; and (iii) the higher normal contact load and the associated hydrostatic pressure component at elevated speeds, which shift the residual stress field toward greater compression. Accordingly, the minimum layer depth of 0.674 mm at 120 r/min corresponds to the lowest strain rate and the weakest strain accumulation, which is fully consistent with the observed monotonic trends. Moreover, the moderate variations (<11%) over the entire speed range indicate a wide process window, within which the workpiece rotational speed can serve as an auxiliary parameter for regulating the residual stress field.

3.2.3 Analysis of the effects of feed rate

With the rolling depth fixed at 0.10 mm and the workpiece rotational speed at 360 r/min, single-factor experiments are performed at five feed rates of 0.01, 0.03, 0.05, 0.07, and 0.10 mm/r to investigate the influence of the axial feed rate on the residual stress field. The resulting residual stress distributions along the depth direction, together with the variations in surface stress and stress layer depth, are presented in Figures 12–14, respectively.

As shown in Figure 12, the feed rate directly determines the overlap ratio of adjacent axial rolling tracks, thereby governing the extent of plastic deformation accumulation in the surface layer. At lower feed rates, the overlap between adjacent axial rolling tracks increases, so that each location undergoes repeated rolling, which promotes sufficient accumulation of plastic deformation and shifts the stress profile further towards negative values. As the feed rate increases, the degree of track overlap diminishes; the number of rolling passes over a given local region becomes insufficient, the extent of plastic deformation is reduced, and both the peak compressive stress and the surface compressive stress decrease accordingly. At low feed rates, the roller repeatedly passes over the same region, causing continuous accumulation of process heat in the surface layer; the resulting thermal relaxation of the residual compressive stress leads to a lower compressive stress magnitude. At feed rates of 0.05–0.07 mm/r, the track overlap is moderate, the process temperature rise remains controllable, and the accumulation of plastic deformation is optimal, so that the surface residual compressive stress reaches its peak.

When the feed rate exceeds 0.07 mm/r, pronounced gaps emerge between adjacent axial rolling tracks, leaving certain regions subjected to only a single rolling pass or incompletely compressed; the resulting insufficient plastic deformation leads to a marked attenuation of the surface residual compressive stress. The quantitative variation of the surface residual stress with feed rate is presented in Figure 13. At a feed rate of 0.01 mm/r, the surface residual compressive stress is only −523.72 MPa; as the feed rate increases from 0.01 to 0.07 mm/r, the stress increases continuously in magnitude, attaining its maximum of −613.29 MPa at 0.07 mm/r; beyond this critical feed rate, however, the stress magnitude decreases rapidly, dropping to −540.59 MPa at 0.10 mm/r. This non-monotonic evolution can be attributed to the distinct deformation mechanisms operating in the different feed-rate regimes. At extremely low feed rates, the temperature rise induced by repeated rolling relaxes the subsurface compressive stress, thereby limiting the thickness of the strengthened layer.

At moderate feed rates, plastic deformation accumulates uniformly and the rolling load is fully transmitted into the substrate, yielding the maximum strengthened-layer depth. At high feed rates, the axial rolling coverage becomes non-uniform, and plastic deformation is confined to an extremely thin surface layer, which significantly reduces the effective strengthened-layer depth. As shown in Figure 14, the strengthened-layer depth is only 0.63091 mm at a feed rate of 0.01 mm/r; it increases to a peak of 0.71923 mm at 0.05 mm/r and then gradually decreases to 0.68035 mm as the feed rate further rises to 0.10 mm/r. Considering both the surface residual stress magnitude and the strengthened-layer depth, the optimal feed rate lies in the range of 0.05–0.07 mm/r: an excessively low feed rate prolongs the processing time and exacerbates thermal relaxation, whereas an excessively high feed rate induces non-uniform axial work hardening and substantially degrades the overall level of residual compressive stress.

3.2.4 Comprehensive comparison of individual process parameters and summary of patterns

A comparison of the maximum variations in surface residual stress and strengthened layer depth across the three single-factor experiments indicates that the significance of the three process parameters in influencing the residual stress field decreases in the following order: rolling depth > feed rate > workpiece rotational speed.

The single-factor experiments have quantified the independent control mechanism of each parameter and identified the corresponding processing thresholds, thereby avoiding two types of failure scenarios: first, insufficient parameter levels, which lead to inadequate plastic deformation and a low residual compressive stress; and second, excessive parameter levels, which induce defects such as surface over-extrusion, thermal relaxation, and non-uniform rolling, all of which weaken the strengthening effect on the gear journal. Moreover, the parameter ranges determined here provide a rational basis for the subsequent Central Composite Design (CCD) and response surface methodology used to develop multi-factor coupling and predictive models for residual stress, effectively narrowing the sample space required for the multi-factor experiments and improving the fitting accuracy of the resulting parametric models.

3.3 Effect of individual process parameters on residual stress

The rolling depth, feed rate, and workpiece rotational speed are taken as the input factors, each at five levels. Based on the single-factor experimental results, the effective ranges of the three parameters are determined as 0.05–0.20 mm for rolling depth, 120–840 r/min for workpiece rotational speed, and 0.01–0.10 mm/r for feed rate. The corresponding coded levels are listed in Table 6. The surface residual compressive stress and the depth of the compressive stress-affected layer are selected as the response variables.

TABLE 6. Response surface experiment factors and levels coding.

A CCD is employed in this study, comprising 14 factorial runs and 3 centre-point replicates. A finite element model is established for each parameter combination to perform the corresponding numerical simulations. The factor settings for each experimental run and the corresponding response values are listed in Table 7.

TABLE 7. Design matrix and response values.

Based on the data in Table 7, regression models are developed to relate the two responses, the surface residual compressive stress and the compressive stress layer depth, to the rolling depth, feed rate, and workpiece rotational speed. Three candidate models are considered: a linear model, a two-factor interaction (2FI) model, and a quadratic model. The fitting accuracy of the three models is evaluated, and the results are summarized in Table 8.

TABLE 8. Comparison of regression model fitting accuracy.

A comparison of the model statistics in Table 8 indicated that the quadratic model provides the best fit; it is therefore adopted for the regression analysis of the experimental data, yielding a mathematical model relating each response variable to the coded values of the process parameters. The general form of the model is expressed as:where Y represents the response; β 0 is the constant term; β i is the coefficient of the linear term; β i i is the coefficient of the quadratic term; β i j is the coefficient of the interaction term; and ε is the random error. The coefficients of each term were estimated using the least squares method, and insignificant terms were eliminated through stepwise regression, ultimately yielding the simplified regression equations for each response:

Y = β 0 + ∑ i = 1 3 β i X i + ∑ i = 1 3 β i i X i 2 + ∑ i = 1 3 ∑ j = i + 1 3 β i j X i X j + ε (28)

σ = − 592.86 − 49.11 A − 6.61 B − 13.54 C − 2.49 AB + 2.70 AC − 4.36 BC + 93.64 A 2 + 15.28 B 2 + 31.89 C 2 (29)

d = 0.8246 + 0.0929 A + 0.0060 B + 0.0069 AB + 0.0350 AC − 0.0151 BC − 0.0440 A 2 + 0.0029 B 2 − 0.0410 C 2 (30)

As can be seen from the regression coefficients, the absolute value of coefficient A for the linear term is the largest, indicating that rolling depth exerts the dominant influence on the response value. The interaction terms AB and AC are significant, demonstrating pronounced coupling effects between rolling depth and workpiece rotational speed, as well as between rolling depth and feed rate. The presence of the quadratic term indicates that the response metric exhibits nonlinear behavior with varying process parameters, which is consistent with the plastic saturation pattern observed in the single-factor experiments.

Analysis of variance (ANOVA) is performed on the regression models, and the results are summarized in Table 9. As shown in the table, the F-values of the models are 29.70 and 22.49, respectively, with corresponding p-values all below 0.01, indicating that the regression models are highly significant and that the differences among the process parameters are statistically significant. The coefficients of variation are all below 5%, well within the 10% acceptance criterion, indicating good data stability. Moreover, the signal-to-noise ratios are all well above 4, confirming that the models possess adequate precision. These results demonstrate that the models are statistically reliable.

TABLE 9. Analysis of variance for response surface regression models.

To comprehensively evaluate the predictive reliability of the parametric model under practical operating conditions, three sets of process parameter combinations are randomly selected within the CCD design space, distinct from those used in the regression modeling, and employed in supplementary validation experiments. The relative errors between the predicted and measured values are calculated, and the validation results are presented in Table 10.

TABLE 10. Model accuracy validation results.

The results show that the maximum relative error between the values predicted by the surface residual compressive stress model and the experimentally measured values is 5.71%, while the maximum relative error for the depth of the residual compressive stress strengthened layer is 6.01%. Both errors are well below 10%, satisfying the prediction accuracy requirement and confirming that the parametric model can reliably predict the residual stress characteristics of 30CrMoA gear journals under various rolling process conditions.

3.4 Multi-objective process optimization and experimental validation

To achieve superior surface finishing quality in the deep rolling of journal surfaces, a multi-objective optimization is performed based on the aforementioned response surface models to identify the optimal combination of process parameters, and the reliability of the optimization results is verified experimentally. Taking the maximization of the surface residual compressive stress and the depth of the strengthened layer as the optimization objectives, the desirability function approach (DFA) is employed for multi-objective optimization to obtain the Pareto-optimal process parameter combination.

Using the built-in optimization module of Design-Expert software, the overall desirability value D is calculated for each candidate combination, and the ten solutions with the highest D values are selected. To avoid the arbitrariness associated with relying on a single optimal solution, the feasibility of the process parameters and the capabilities of the equipment are comprehensively considered. The final optimal combination of process parameters is presented in Table 11. The respective roles of the physical experiments and FE simulations are clarified in Table 12.

TABLE 11. Candidate solutions and overall expected value in multi-objective optimization.

Based on the D value and process feasibility, Group 1 is selected as the optimal process condition: a rolling depth of 0.147 mm, a workpiece rotational speed of 590 r/min, and a feed rate of 0.066 mm/r. Under this parameter combination, the model predicted a surface residual compressive stress of −601.59 MPa, and a strengthened layer depth of 0.822 mm. All individual objectives satisfied the constraint requirements, and the overall desirability reaches 0.884.

TABLE 12. Roles of physical experiments versus FE simulations in residual stress characterization and RSM modeling.

To verify the reliability of the multi-objective optimization results under practical operating conditions, experiments are conducted using the optimal process parameters, and the measured data are compared with the model predictions. The comparison between the predicted and experimental results is presented in Figures 15, 16.

The simulation predictions exhibit excellent agreement with the experimental measurements across all investigated process parameters. For the three single-factor experimental sets, rolling depth, workpiece rotational speed, and feed rate, the mean relative deviations in predicted surface residual stress are 6.42%, 7.67%, and 6.06%, respectively, while those for compressive stress layer depth are 4.33%, 7.57%, and 4.94%, respectively. Since all relative deviations remain below 8.0%, the model demonstrates satisfactory predictive accuracy for both surface residual stress and compressive stress layer depth. Moreover, the simulation results faithfully reproduce the evolution trends of residual stress with varying process parameters, showing good consistency with the experimental observations.

To comprehensively assess the improvement in journal surface integrity achieved through process optimization and to elucidate the microstructural evolution at various rolling depths under the optimal parameters, the corresponding microstructures are presented in Figure 17. With increasing rolling depth, the surface grains undergo pronounced plastic flow along the rolling direction. At a rolling depth of 0.05 mm, the surface grains are only slightly elongated, the deformation layer remains thin, and the grain boundaries are still clearly discernible.

As the rolling depth increases to 0.10 mm, grain deformation intensifies: the initially equiaxed grains transform into a fibrous microstructure, accompanied by a marked thickening of the deformation layer. Upon a further increase to 0.15 mm, the surface grains become markedly flattened and elongated, the flow lines grow denser, and the partial breakdown of grain boundaries gives rise to numerous subgrains and dislocation cells. At 0.20 mm, the deformation layer continues to thicken; however, noticeable material pile-up and micro-wrinkles appear on the surface, and although grain refinement is further enhanced, the deformation uniformity is somewhat compromised.

Figure 17f presents the surface microstructure obtained under the optimal combination of process parameters. It can be observed that the grains are finer and the deformation is more uniform under these conditions, while the deformation layer is also thicker; both the degree of grain refinement and the deformation uniformity are superior to those of the single-factor experimental groups. Compared with Figure 17d, the deformation layer of the optimal group is denser and more uniform, with no evident localized accumulation or non-uniform deformation. This indicates that, under the synergistic effect of a moderately large rolling depth combined with a moderate rotational speed and feed rate, the surface layer of the material undergoes sufficient and uniform plastic deformation, thereby achieving the most desirable grain refinement.

4 Conclusion

• Deep rolling induces a stable gradient residual compressive stress field in the surface layer of the 30CrMoA gear journal. The peak residual compressive stress is located approximately 0.20 mm beneath the surface, reaching a magnitude of over −800 MPa. The compressive stress in the surface layer ranges from 400 to 600 MPa, slightly below the peak value, whereas the deeper matrix exhibits a weak equilibrium tensile stress of approximately 100–180 MPa. This stress configuration provides an effective barrier against fatigue crack initiation and steady-state propagation.
• Single-factor experiments reveal that the significance of the three process parameters in influencing the residual stress field follows the order: rolling depth > feed rate > workpiece rotational speed. Increasing the rolling depth significantly enhances both the compressive stress magnitude and the depth of the strengthened layer; however, once the rolling depth exceeds 0.15 mm, plastic saturation and thermal relaxation occur in the surface layer, resulting in a slight decline in the stress magnitude. The workpiece rotational speed and feed rate exhibit optimal ranges of 600–720 r/min and 0.05–0.07 mm/r, respectively; values outside these ranges weaken the strengthening effect owing to heat accumulation or non-uniform rolling.
• A second-order regression model for residual stress and stress layer depth, established using the central composite design (CCD) based response surface methodology, is demonstrated to be highly significant (p < 0.01) by analysis of variance, with a signal-to-noise ratio greater than 4 and a coefficient of variation below 5%. Validation tests show that the maximum relative errors in predicting surface residual stress and stress layer depth are 5.71% and 6.01%, respectively, both well below 8.0%, confirming that the parametric model can accurately predict the residual stress characteristics under various rolling conditions.
• The optimal combination of process parameters, determined through multi-objective optimization, consists of a rolling depth of 0.147 mm, a workpiece rotational speed of 590 r/min, and a feed rate of 0.066 mm/r. Under these conditions, the model predicts a surface residual compressive stress of −601.589 MPa and a stress layer depth of 0.822 mm. Microstructural observations reveal that the surface layer exhibits uniform grain refinement and a dense deformation layer under the optimal process parameters, with an overall strengthening effect superior to that obtained under any single-factor condition, thereby validating the reliability and effectiveness of the optimization results.

It should be noted that no direct fatigue tests were performed in this study. The fatigue-related benefits described above are inferred from the measured compressive residual stress field in conjunction with well-established mechanistic understanding reported in the literature and therefore represent anticipated improvements rather than experimentally validated conclusions. Dedicated rolling contact fatigue (RCF) tests will be carried out in future work to quantitatively verify this expectation.

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Footnotes

Edited by: Deepak Kumar, Indian Institute of Petroleum and Energy (IIPE), India

Reviewed by: Narendra Varma Dantuluri, Raghu Engineering College, India

Parth Desai, Applied Materials, United States