Establishment of a Numerical Model and Process Optimization for the Moving Induction Hardening of a Whole-Roll Flatness Roll

Abstract

The surface of a whole-roll flatness roll is in long-term contact with the steel strip, leading to slipping and wear and placing higher demands on the performance of the roll surface. This study establishes a finite element model for moving induction quenching and a phase transformation hardness numerical model by generating multi-field simulations and hardness predictions for the flatness roll during induction quenching. First, the thermal–physical properties of the roll material, MC3, are calculated using JMatPro V13.0. The dynamic domain and moving mesh techniques are applied in COMSOL Multiphysics to simulate time-varying boundary conditions, and the JMAK and K-M phase transformation models are used for electromagnetic–thermal–microstructure field simulations. Subsequently, the Taguchi method is used to optimize the induction quenching process of the flatness roll. After optimization, the martensitic hardened layer depth along the axial direction of the roll becomes uniformly distributed near the target value of 3 mm. Finally, through the modified Maynier hardness model, the corrected formula for the Vickers hardness of MC3 is obtained. The calculated hardness value of the roll surface in the simulation model reaches 950 HV, which agrees well with the experimental hardness results, validating the ability of the numerical model to guide specific processes.

1. Introduction

Flatness is a key quality indicator for cold-rolled steel strips [1,2], and the flatness roll is the core piece of detection equipment in a flatness measurement and control system [3,4,5]. By coming into direct contact with the surface of the steel strip, the flatness roll can detect the tension distribution in the width direction of the steel strip and calculate the flatness distribution in real time. During detection, the steel strip fully covers the flatness roll and enables transmission. The roll surface not only bears periodic alternating loads from the steel strip but also experiences slipping and wear due to the different speeds between the strip and the roll [6,7].

Before leaving the factory, the whole-roll flatness roll is often subjected to surface heat treatment through electromagnetic induction quenching to strengthen the roll surface, resulting in a relatively thin martensitic hardened layer on the surface, while the original pearlitic structure in the core is retained. The depth of the hardened layer determines the service life of the flatness roll, and its uniformity affects the overall mechanical performance, while hardness is an important parameter for evaluating the wear resistance of the roll surface. Unlike the surface quenching process of work rolls [8,9], the whole-roll flatness roll has axial through-holes near the roll surface, where sensors are installed and subjected to high pre-tension, causing elastic deformation of the roll surface by tens of micrometers. This requires minimizing the depth of the hardened layer and ensuring its uniform distribution, so that the surface hardness and roll body toughness are coordinated to prevent cracking of the roll surface when the sensors are installed.

Many scholars have conducted detailed studies on electromagnetic induction quenching. Lu et al. [10] used JIS SUJ2 steel as the material for induction quenching and studied the distribution of residual stress and residual austenite. The results showed that the microstructure of the specimen before induction quenching had a significant impact on the depth of the hardened layer, and the maximum residual compressive stress increased with the rise in austenitizing temperature. Derouiche et al. [11] developed two data-driven models for the temperature–time evolution and austenite phase transformation during induction heating and created data-driven and hybrid models to predict hardness after quenching. Fang et al. [12] used cold rolling and electromagnetic induction heating to manufacture gradient structures on AISI 316L stainless steel and studied the effect of surface softening on the fatigue behavior of AISI 316L stainless steel. Jian et al. [13] successfully prepared a gradient microstructure using high-frequency induction quenching to improve the mechanical behavior of the Ti-6Al-4V alloy. Gao et al. [14] evaluated the effect of induction quenching on the damage tolerance of the EA4T axle under fly ash impact damage, finding that the induction-quenched specimens exhibited higher stability. Smalcerz et al. [15] performed induction quenching simulations and field tests on cylindrical components made of 38Mn6 steel and observed the microstructure after testing. The results showed that the components achieved appropriate performance and hardness layer thickness. Garstka et al. [16] used the Barkhausen method for induction quenching tests on samples and discovered a relationship between the hardness change of the sample surface layer and Barkhausen noise parameters. Choi et al. [17] measured the hardened depth of high-frequency induction-quenched specimens using a Vickers hardness tester and compared it with simulation results, finding that high-frequency induction quenching is beneficial for surface hardening. Slatter et al. [18] studied the impact wear resistance of compacted graphite iron (CGI) before and after induction hardening.

The optimization of induction hardening process has demonstrated significant application value in industries such as automotive, aerospace, and steel production. It not only enhances the performance and production efficiency of critical components but also promotes the advancement of materials science and manufacturing technology. This paper uses numerical simulation methods to perform electromagnetic–thermal–microstructure multi-field simulations of the whole-roll flatness roll moving induction quenching process. At the same time, the induction quenching process is optimized using the Taguchi method [19,20] for specific hardened layer parameters, achieving satisfactory optimization results. Finally, a corrected formula for the Vickers hardness of MC3 high-carbon steel is established, and the accuracy of the numerical model and hardness correction formula is verified through metallographic and hardness experiments.

The dynamic mesh method employed in this study significantly enhances the realism and practicality of moving induction quenching simulations through adaptive mesh optimization, high-precision multi-field coupling, and efficient transient modeling. Its advantages are not only reflected in reducing R&D costs and cycle times but also in revealing transient physical mechanisms that are difficult to observe using traditional methods, thereby providing deeper theoretical support for process optimization. Compared to existing technologies and process optimizing approaches, this research provides a more targeted solution to the specific process requirements of a series of special industrial rolls, particularly the whole roll flatness meter rolls. Through geometric model reconstruction, dynamic mesh optimization, material property updates, phase transformation model adjustments, parametric design, and multi-physics coupling model updates, the proposed model in this study can be effectively adapted to industrial flatness rolls with varying geometric shapes or material selections.

2. Numerical Model of Moving Induction Quenching

The moving induction quenching process of the flatness roll is shown in Figure 1. The electrically energized coil surrounds the roll body and moves along the axial direction at a process-set speed v, heating the roll and causing the matrix pearlitic structure to undergo a diffusion phase transformation to austenite. After a period of heating, the cooling liquid is introduced and the sprayer, which also surrounds the roll body, moves synchronously with the coil. This process causes the transition structure in the surface austenite to undergo a non-diffusion phase transformation to martensite, forming a hardened layer on the roll surface.

The process described above is simulated using COMSOL Multiphysics 5.5. The 3D schematic diagram of the model, dimensions, observation point locations, and mesh division are shown in Figure 2. The induction heating 3D structure used for the simulation is shown in Figure 2a. The roll body specification is Φ 300 mm × 150 mm, and the coil cross-sectional dimensions are 60 mm × 20 mm. The space between the roll body and the coil is filled with air, with a gap of 12 mm between them. The coil cross-section is filled with cooling liquid. Each part of the model is made of the same material and is homogeneous. Due to the axisymmetric nature of the 3D model, it was simplified into a 2D model. A total of 18 observation points were taken along the roll surface and below it, numbered from A to R. The centerline of the model was used as the observation path. The dimensions and observation point locations of the 2D model are shown in Figure 2b. It was assumed that the internal roll body follows Ampère’s law, and an infinite element domain [21,22] (essentially a diffusion-type control equation) is introduced, surrounding the entire simulation space, to model magnetic field dissipation. The boundary of the infinite element domain is set as magnetic insulation so that the high-frequency magnetic field generated by the induction coil only propagates within this region. Due to the skin effect, phase transformation mainly occurs in the surface region during induction heating.

By comparing the results from coarse, baseline, and refined meshes, convergence of multi-physics fields was verified by setting specific convergence criteria for the electromagnetic field, temperature field and stress field, respectively, and ensuring overall error control through coupled iterations. Using the single-variable variation method, the sensitivity of mesh density to stress concentration areas at the coil edges and roller ends was quantified (error reduced to <2%), with results meeting industrial standards. Based on an error estimator, the mesh density in critical regions (e.g., coil and roll edges) was dynamically adjusted to balance accuracy and computational efficiency. Through a local mesh partitioning strategy, the total number of mesh elements was controlled within 30,000. A dual criterion—residual <1 × 10⁻⁵ and temperature gradient variation in critical regions <1%—was adopted to ensure result stability. A fine mesh with a size of 0.2 mm is applied to the surface layer, with a thickness of 10 mm. The infinite element domain uses a mapped mesh with seven layers of elements. The remaining regions use conventional triangular mesh sizes, as shown in Figure 2c.

The boundary conditions of the model and the time-varying mesh division within the dynamic domain are shown in Figure 3. During the moving induction quenching process, as the coil moves upwards, the lower surface area undergoes austenitization and the high-temperature region of the roll body experiences enhanced thermal radiation. After the coil passes the cooling interval, the sprayer begins spraying cooling liquid. The cooling liquid, under the effect of its own weight, causes the region of the roll body below the sprayer to undergo forced convection, while the area above it remains under natural convection with air. In the heating completion phase, the coil stops heating, but the sprayer continues to move and operate. When the sprayer reaches above the roll body, the entire roll surface undergoes forced convection with the cooling liquid. Figure 3a is a schematic diagram of the model boundary condition changes. As can be seen, the simulation model needs to comprehensively set the boundary condition changes according to the actual situation at each stage [23,24].

The dynamic mesh is introduced to simulate the coil mesh movement process [25,26], enabling the continuous movement of the coil mesh. A consistent boundary pair is added to the model to divide the dynamic domain (the coil movement range, i.e., the area in which the mesh size changes in the figure) and the static domain (the stationary region outside the dynamic domain). The mesh within the dynamic domain is re-meshed in real time as the coil moves, as shown in Figure 3b.

The initial overall temperature of the model is set to 20 °C, and a single-turn coil is used with the initial magnetic vector potential, set to 0. The incoming material is set as a 100% uniformly distributed pearlite structure.

3. Material Property Setting and Phase Transformation Model

3.1. Material Property Setting

The material used for the induction coil is copper. In the simulation software, its relative magnetic permeability is set to 1, resistivity is set to 1.71 × 10⁻⁸ Ω·m, and the dielectric constant is set to 1. The relative magnetic permeability, electrical conductivity, and dielectric constant of air and water are all set to 1. The material of the flatness roll is MC3 high-carbon alloy steel, and its main chemical composition is shown in

Table 1. Main chemical composition of MC3 (wt.%).

Element	C	Mn	Cr	Mo	V	Si	S	P	Ni
Content	0.87	0.58	2.92	0.3	0.12	0.45	0.009	0.014	0.7

.

The simulation uses temperature-related parameters. The temperature-dependent curves of resistivity, relative magnetic permeability, thermal conductivity, and the specific heat of MC3 during induction quenching are calculated using Jmatpro [27,28] and shown in Figure 4. The density of the MC3 material is calculated to change little with temperature, and is set to 7862 kg/m³. The CCT and TTT curves of MC3 are shown in Figure 5, and are used to set the model parameters during the phase transformation process.

Due to the limited research on the thermal properties related to MC3, the calculation results were verified using the alloy structural steel 12CrMoV, which has a similar composition [29].

3.2. Phase Transformation Models

The JMAK (Johnson–Mehl–Avrami–Kolmogorov) simplified diffusion equation is used in the model to describe the diffusional phase transformation from pearlite to austenite:

$${\epsilon }_{\mathrm{A}}=1-\mathrm{e}\mathrm{x}\mathrm{p}\left\{\mathrm{A}{\left\{{\displaystyle \frac{T-T_{\mathrm{S}}}{T_{\mathrm{e}}-T_{\mathrm{S}}}}\right\}}^{\mathrm{D}}\right\}$$

(1)

where ${\epsilon }_{\mathrm{A}}$ is the volume fraction of austenite generated during induction heating; $T_{\mathrm{e}}$ and $T_{\mathrm{s}}$ are the starting and ending temperatures of the pearlite-to-austenite phase transformation; and T is the heating temperature. A and D are constants, calculated using Jmatpro: A = −7; D = 2.

The K-M (Koistinen–Marburger) model is used to describe the non-diffusional phase transformation from austenite to martensite:

$${\epsilon }_{\mathrm{M}}=1-\mathrm{e}\mathrm{x}\mathrm{p}\left\{\alpha \left(M_{\mathrm{S}}-T\right)\right\}$$

(2)

where ${\epsilon }_{\mathrm{M}}$ is the volume fraction of martensite; $M_{\mathrm{S}}$ is the starting temperature of the martensite transformation; $\alpha$ is the martensite transformation rate constant; and T is the cooling temperature. $M_{\mathrm{S}}$ and $\alpha$ are calculated using Jmatpro: $M_{\mathrm{S}}$ = 150 °C; $\alpha$ = 0.014.

To ensure the accuracy of the models, the experimental data of 12CrMoV were referenced, and the simulation results were in good agreement with them. Through sensitivity analysis, it was found that when the coefficient changed slightly, the models had little impact on the results. Therefore, the current coefficient values were considered reasonable.

4. Simulation Results Analysis

Based on the operator experience, the simulation parameters are selected as shown in

Table 2. Selected simulation parameters.

Current Frequency (Hz)	Current Density (A/m2)	Air Gap (mm)	Moving Speed (mm/s)	Cooling Interval (s)	Initial Coil Position (mm)	Heating Time (s)	Total Time (s)
7000	7 × 106	12	1	50	y = 0	150	800

. To better describe the time-varying characteristics of each parameter in the moving induction quenching process, the analysis is conducted with respect to three aspects: the electromagnetic field, temperature field, and microstructure field.

Figure 6 and Figure 7 show the time-varying electromagnetic field and temperature field at the roll body cross-section, respectively. During processing, the magnetic flux lines initially concentrate at the surface layer of the roll body and spread outward, causing the surface temperature to rise rapidly. Once the temperature reaches the Curie point, the surface undergoes demagnetization, and the magnetic flux lines gradually move inward. During spray cooling, the temperature in the demagnetized surface layer rapidly decreases below the Curie point, and the magnetic flux lines re-concentrate at the surface, forming a “bump-like” distribution of magnetic flux lines at the cross-section. As induction quenching progresses, this distribution continuously moves upward. Since the surface temperature rise rate is slower than the coil movement speed, relative to the coil position, the high-temperature region in the temperature field shows a certain delay in the “bump” position.

During the starting and ending stages of induction heating, the magnetic flux lines concentrate at the upper and lower ends of the roll body, leading to a high density of induced eddy currents and a rapid temperature rise rate at these locations. Therefore, the starting and stopping positions of the coil should not exceed the upper and lower ends of the roll body in order to prevent overheating of the surface or the formation of a deep hardened layer at the end faces, which could cause large residual stresses and lead to roll surface cracking. Figure 8 shows the time-varying temperature curves at two sets of observation points, surface and internal, providing a more intuitive observation of the temperature rise at the upper and lower ends.

Figure 9 shows the martensite distribution within the cross-section after quenching and the time-varying martensite volume fraction at the internal observation points. At the end of heating, a noticeable “depression” in martensite distribution is observed near the upper end face, consistent with the temperature field distribution at that time. Since the Q-point is close to this “depressed” area, its martensite content is much lower than that of the other points, with the generated martensite volume being only 6% at the end of quenching. The data from multiple fields show high consistency. In conclusion, the peak temperature at the upper and lower ends of the roll body is too high under the current process, and the martensite distribution along roll axial direction is uneven. To ensure consistent mechanical properties of the roll body, process optimization for moving induction quenching will be carried out.

Figure 10 shows the distribution of austenite at the centerline of the roll cross-section after quenching. At the end of quenching, a significant amount of austenite still exists from the core of the roll body to several millimeters below the roll surface. It decreases abruptly at approximately 3 mm below the roll surface, forming a transition layer. Within the range of 3 mm below the roll surface, 15.2–16.6% of residual austenite still remains.

5. Optimization of Moving Induction Quenching Process

The Taguchi method was used to optimize the induction quenching process, with the target value for the hardened layer depth set to 3 mm. An orthogonal experiment using a five-factor, four-level table was conducted. The five factors for evaluating the depth and uniformity of the hardened layer were the coil current density (P₁), moving speed (P₂), coil starting position (P₃), spray interval (P₄), and coil position at the end of heating (P₅). Based on the simulation model results, corresponding levels were selected for each factor, and the other levels were adjusted with small disturbances. The four levels of five factors are shown in

Table 3. Four levels of five factors of moving induction quenching processes.

Experimental Factors	P1 (A/m2)	P2 (mm/s)	P3 (mm)	P4 (s)	P5 (mm)
Level 1	6.1 × 106	0.8	−25	36	130
Level 2	6.15 × 106	0.9	−27.5	38	131
Level 3	6.26	1	−30	40	132
Level 4	6.25 × 106	1.1	−32.5	42	133

, and the sixteen sets of orthogonal experimental processes are shown in

Table 4. Sixteen sets of orthogonal experimental processes.

Number of Experiments	P1 (A/m2)	P2 (mm/s)	P3 (mm)	P4 (s)	P5 (mm)
1	6100	0.8	−25	36	130
2	6100	0.9	−27.5	38	131
3	6100	1	−30	40	132
4	6100	1.1	−32.5	42	133
5	6150	0.8	−27.5	40	133
6	6150	0.9	−25	42	132
7	6150	1	−32.5	36	131
8	6150	1.1	−30	38	130
9	6200	0.8	−30	42	131
10	6200	0.9	−25	40	130
11	6200	1	−32.5	38	133
12	6200	1.1	−27.5	36	132
13	6250	0.8	−32.5	38	132
14	6250	0.9	−30	36	133
15	6250	1	−27.5	42	130
16	6250	1.1	−25	40	131

.

The model was recalculated according to the 16 sets of process parameters in the orthogonal table. The resulting martensite volume fraction at the centerline of the roll body, the cooling rate, and the surface hardness for the 16 sets are shown in Figure 11. The surface hardness remained largely consistent under different process parameters, although significant differences were noted in the cooling rate and phase proportions at the center. It was therefore concluded that the post-quenching surface hardness primarily depends on the cooling capacity of the cooling liquid.

The designed orthogonal experimental table was substituted into the model, and the martensite depth at the lower end line, centerline, dimpled line, and upper end line positions was obtained after calculation, with the values denoted as H₁, H₂, H₃, and H₄, respectively, and with the line positions shown in Figure 12. The influence weights of each parameter to be optimized on the current evaluation index were calculated, and the results are shown in

Table 5. Influence weights of each optimization parameter.

	H1		H2		H3		H4
	SSD	Weight (%)	SSD	Weight (%)	SSD	Weight (%)	SSD	Weight (%)
P1	1.5734	7.94	0.0721	2.8	0.6409	6.32	0.4826	39.27
P2	12.5	63.11	2.4262	94.64	10.1048	67.91	0.2152	17.89
P3	1.5727	7.94	0.0109	0.4	0.9267	6.23	0.0067	0.54
P4	1.5727	7.94	0.0316	1.23	1.95	13.11	0.2936	23.86
P5	2.5877	13.07	0.0227	0.93	1.2564	12.43	0.5258	18.44
Total	19.807	1	2.5635	1	14.879	1	1.231	1

.

It can be seen that the martensite depth at the upper end face H₄ was most influenced by P₁; H₁, H₂, and H₃ were mainly influenced by the coil moving speed P₂, indicating that P₂ played a decisive role in the depth of the hardened layer after quenching. Moreover, H₁ was also influenced by P₅. H₂ was almost only affected by P₂. H₃ was also influenced by P₄, P₅. H₄ was also influenced by P₄, P₅, and P₂, and hardly affected by P₃. After repeated simulations and comprehensive consideration, P₁ was selected at level 1, P₂ at level 1, P₃ at level 2, P₄ at level 3, and P₅ at level 2.

The continuous induction quenching simulation was carried out again with the selected parameters, and the temperature variation at each point on the surface and the martensite depth (H₁~H₄) at each position after optimization are shown in Figure 13.

It can be seen that compared to the original process, the maximum temperature near the upper and lower ends of the roll body had significantly decreased, effectively preventing material overburning caused by excessively high temperatures. The temperature variation trends at each point show good consistency, which resulted in a more uniform distribution of the hardened layer. Although not completely consistent, the dimpled area had largely been filled. H₁~H₄ were all close to the target value of 3 mm, effectively reducing the locally high residual stresses and significantly lowering the risk of cracking when the roll surface is subjected to loading.

6. Metallographic and Hardness Experiments on the Microstructure After Quenching

6.1. Hardness Calculation Numerical Model

In addition to the hardened layer indicators, the surface hardness of the roll can be used to more intuitively measure the wear resistance of the tested roll. To reasonably assess the post-quenching surface hardness of the MC3 high-carbon steel material in the simulation model, a mixed-phase weighted hardness calculation method is proposed considering the influence of alloy element content and actual cooling rate on hardness, specifically for the residual austenite that has not fully transformed after the phase change.

Using a large amount of experimental data, Maynier [30] established the relationship between the hardness of martensite and pearlite in low-carbon steel, the actual cooling rate, and material composition. The expression is as follows:

$${HV}_{\mathrm{M}}=127+949C+27Si+11Mn+16Cr+8Ni+21 lg{\nu }_{\mathrm{M}}$$

(3)

$${HV}_{\mathrm{P}}=42+223C+53Si+30Mn+12.6Ni+7Cr+19Mo+[10-19Si+4Ni+8Cr+130{\nu }_P ]\cdot lg{\nu }_{\mathrm{P}}$$

(4)

where ${HV}_{\mathrm{M}}$ and ${HV}_{\mathrm{P}}$ are the Vickers hardness values of martensite and pearlite at the end of quenching; and $v_{\mathrm{M}}$ and $v_{\mathrm{P}}$ are the actual cooling rates of martensite and pearlite.

G. Krauss [31] proposed the formula for the effect of residual austenite formation on the Vickers hardness of martensite:

$$∆HV\approx {\displaystyle \frac{{\epsilon }_{\mathrm{A}}}{0.1+0.015{\epsilon }_{\mathrm{A}}}}$$

(5)

$$H{V}_{\mathrm{M}}^{\prime }=H{V}_{\mathrm{M}}-∆HV$$

(6)

where ${\epsilon }_{\mathrm{A}}$ is the volume fraction of residual austenite. When the carbon content is consistent throughout the material, the hardness of the residual austenite has a constant value.

Through experimental methods, Zhang [32] obtained an empirical formula for the hardness value of residual austenite as a function of carbon content.

$$H{V}_{\mathrm{A}}={\displaystyle \frac{351\cdot C^2+166C-122}{1.3\cdot C-0.64}}$$

(7)

By combining Equations (4), (6) and (7), the mixed-phase weighted hardness calculation formula after induction quenching is obtained:

$$HV=H{V}_{\mathrm{M}}^{\prime }\cdot {\epsilon }_{\mathrm{M}}+H{V}_{\mathrm{A}}\cdot {\epsilon }_{\mathrm{A}}+H{V}_{\mathrm{P}}\cdot {\epsilon }_{\mathrm{P}}$$

(8)

where ${\epsilon }_{\mathrm{M}}$ and ${\epsilon }_{\mathrm{P}}$ are the volume fractions of martensite and pearlite, respectively; and ${\epsilon }_{\mathrm{M}}+{\epsilon }_{\mathrm{A}}+{\epsilon }_{\mathrm{P}}=1$, $H{V}_{\mathrm{M}}$, $H{V}_{\mathrm{A}}$, and $H{V}_{\mathrm{P}}$ are the hardness values of martensite, austenite, and pearlite.

Figure 14a shows the cooling curves at different thicknesses along the centerline of the roll body obtained from the simulation. The volume fractions of martensite, pearlite, and austenite are determined from the cooling curves. Using the cooling rates and volume fraction data of each phase in Equation (8), the temperature and hardness distribution across the section of the roll body are obtained. The temperature and hardness distribution curves along the roll centerline are shown in Figure 14b.

The region in which the martensite volume fraction exceeds 50% is defined as the hardened layer. The model calculation results show that the depth distribution of the hardened layer is around 3 mm below the roll surface. The post-quenching surface hardness can reach approximately 950 HV, meeting the required specifications.

6.2. Post-Quenching Metallographic and Hardness Experiments

Induction quenching experiments were conducted according to the simulation conditions. Samples were obtained via wire cutting, and the metallographic distribution and hardness distribution of the samples were analyzed to verify the hardness parameters of the simulation model. The experimental subject was an MC3 test roll after quenching and tempering treatment, with billet sections and coil dimensions consistent with those used in the simulation. The tempering process is designed as a low-temperature tempering with parameters set at 200 °C for 4 h, followed by furnace cooling. Figure 15 shows the vertical high-frequency induction quenching equipment used for the experiment. The induction quenching process parameters are the same as those in Table 2.

The post-quenching roll surface-hardened microstructure at 100× and 500× magnification is shown in Figure 16. The main structure after quenching is formed of martensite with a dense grain structure, and the carbide particles are evenly distributed and fine.

Vickers hardness measurements were performed on the samples both axially and radially, with the sampling paths shown in Figure 17. On the cross-section of the flatness roll, a sample was taken every 120° along the circumference using wire cutting. For radial hardness measurements, at a radial position 0.5 mm from the surface of the roll, a point was taken every 0.5 mm along the radial direction, resulting in radial hardness gradient distribution at ten points, as shown in

Table 6. Radial hardness distribution.

Position	1	2	3	4	5	6	7	8	9	10
${HV}_{\mathrm{m}}$	890.3	885.2	881.6	876.7	872.6	741.5	460.3	346.3	344.7	342.3
${HV}_{\mathrm{c}}$	957.2	951.8	950.5	947.5	945.0	783.3	456.1	343.5	343.6	343.5
Error	−7.5	−7.6	−7.8	−8.1	−8.3	−5.6	−2.9	0.08	0.03	−0.04

. For axial hardness measurements, at the same radial position, a point was taken every 0.5 mm along the axial direction, resulting in the axial hardness gradient distribution at ten points, as shown in

Table 7. Axial hardness distribution.

Position	1	2	3	4	5	6	7	8	9	10	Average
${HV}_{\mathrm{m}}$	894.4	886.4	890.2	892.1	889.0	898.7	893.4	900.2	897.6	898.2	893.7
${HV}_{\mathrm{c}}$	957.4	957.2	958.1	957.1	957.6	957.4	958.2	958.8	957.3	957.4	957.3
Error	7.0	8.0	7.6	7.3	7.6	6.6	7.3	6.4	6.3	6.6	7.1

.

The maximum error between the numerical calculation and the measured hardness, ${HV}_{\mathrm{c}}$ and ${HV}_{\mathrm{m}}$, at radial and axial direction is −8.1% and 8.0%, respectively. The measured case-hardening depth is also distributed around 3 mm. In conclusion, the simulation model can be used to simulate the induction hardening process of flatness rolls, and the simulation results can provide data support for process optimization.

7. Discussion

• Due to the peak temperature at the roll end, the local martensitic transformation amount increases. The volume expansion caused by the phase transformation leads to an increase in local residual stress. Considering the current quenching environment, the TRIP (Transformation-Induced Plasticity) effect is relatively weak and cannot effectively alleviate the local residual stress. During sensor installation, higher residual stress may pose a risk of cracking at the thinnest part of the roll end, potentially resulting in the scrapping of the entire roll. In addition, when the roll surface is subjected to strip load, the residual stress will superimpose with the external force, which may either reduce the material fatigue life or increase the sensitivity to crack propagation. To address this issue, in addition to the process optimization methods proposed in this paper, the following measures can also be considered: Use contoured induction coils (e.g., conical end design), auxiliary coils (reverse compensation coils), or auxiliary magnetic shielding (e.g., copper ring sleeves) to reduce the eddy current density at the roll end. Employ dual-frequency induction heating (e.g., a combination of medium frequency and high frequency) to achieve rapid heat dissipation from the roll end and suppress overheating. Set up a power gradient attenuation zone in the roll end area to gradually reduce the power density within a certain roll length at the end. Implement segmented control of the induction coil movement speed (accelerate the roller end area to 1.2–1.5 times the base speed) to reduce heat accumulation at the end.
• When adapting the framework of induction hardening multi-field simulation to detection rolls composed of larger sizes or different materials, it is necessary to consider the adaptation of different material properties, such as electromagnetic parameters, thermodynamic parameters, and phase transformation mechanics models. A coupled algorithm of boundary element method (BEM) and finite element method (FEM) can be adopted to reduce computational costs. In an industrial-scale production environment, the large-sized roll can be divided into multiple heating sections to avoid energy redundancy caused by overall heating. By combining metallographic experiments (microstructure), micro-area EBSD/nanoindentation (meso-defects), and the joint measurement of hardness and residual stress (macro-performance), a three-tier bench-marking mechanism with simulation results can be established, forming a closed-loop optimization system of multi-field simulation, process optimization and data feedback
• Based on the previous analysis, the hardness after quenching depends on the cooling effect. As a direction for future research, different quenching media, such as oil and polymer solutions, as well as different cooling schemes, will be considered to study the microstructure evolution and hardness distribution during induction hardening. Additionally, the Taguchi method was adopted in this study to prioritize the analysis of main effects. Although the orthogonal array inherently captures some interaction effects through its combinatorial design, explicitly discussing parameter coupling (such as the relationship between current density and moving speed) will help deepen the analysis. Meanwhile, the verification of the optimized model parameters in this paper was solely based on numerical methods, without experimental feedback for validation. These limitations will also be a focus of future work.

8. Conclusions

• For the moving induction hardening of a whole-roll flatness roll, a finite element dynamic mesh method was used to simulate the time-varying boundary conditions and the overall temperature change process. The transient temperature field was input into the JMAK and K-M phase transformation models, and phase transformation simulations during the quenching process were carried out, resulting in the distribution of the microstructure after the quenching of the flatness roll.
• Since the uneven axial depth of the hardened layer affects the mechanical properties of the flatness roll, the Taguchi method was used to optimize the induction-hardening process. After optimization, the maximum temperature at the top and bottom ends of the roll body decreased significantly, and the martensitic hardened layer depth along the axial direction of the roll was evenly distributed around the target value of 3 mm.
• Using the Maynier hardness model, the G. Krauss modification model, and the Yufang Zhang empirical formula, a corrected formula for the Vickers hardness of MC3 high-carbon steel was obtained. The surface hardness calculated using the model reached 950 HV, meeting the required specifications. Combined with metallographic and hardness experiments, the accuracy of the simulation model was verified in terms of both the hardened layer depth and hardness.