Saturation behavior and full-field reconstruction of residual stress in quenched AISI 304 stainless steel via the contour method

Abstract

Residual stresses induced by quenching play a critical role in determining the structural integrity and service performance of engineering components. This study investigates the residual-stress distributions in 304 austenitic stainless subjected to quenching using the contour method. By integrating high-resolution surface deformation measurements with finite element-based inverse analysis, full-field residual-stress profiles were reconstructed across both transverse and longitudinal sections. A comparative analysis was conducted under water and oil quenching conditions at different temperatures to examine the effects of cooling media and thermal parameters. The results reveal that water quenching generates higher stress magnitudes and steeper stress gradients compared to oil quenching, due to its faster cooling rate. Additionally, once the quenching temperature exceeds 700 °C, the residual-stress distributions exhibit limited sensitivity to further increases in temperature. These findings provide valuable insights into the mechanisms of thermal stress formation in non-transforming steels and offer practical guidance for tailoring residual-stress distributions to enhance the structural performance of quenched components.

Introduction

Residual stresses are self-equilibrated internal stresses that persist in the absence of external loads¹. They are typically introduced during manufacturing processes involving non-uniform plastic deformation² or temperature gradients, such as welding³, machining⁴, surface treatments⁵, and most notably, heat-treatment processes such as quenching⁶. Residual stresses can significantly influence performance⁷, dimensional stability⁸, and structural integrity⁹ of engineering components. In particular, tensile residual stresses may promote crack initiation¹⁰ and fatigue failure¹¹, whereas compressive residual stresses can enhance resistance to crack propagation and improve fatigue life¹².

Among various thermal processes, quenching remains a widely employed technique to enhance mechanical properties by rapid cooling¹³. However, this process inevitably introduces complex residual-stress fields due to steep thermal gradients¹⁴. The severity and distribution of these stresses depend on cooling rate, quenching medium, and thermal history, as they control the transient thermal gradients and the extent of thermo-mechanical plasticity during cooling¹⁴.

Numerous studies have investigated quenching-induced residual stresses because of their critical role in the mechanical reliability of heat-treated components. For aluminum alloys, residual stresses are often discussed together with precipitation/aging behavior and resulting property changes. For example, Liu et al.¹⁵ examined the influence of quenching-induced residual stresses on precipitation behavior in 7085 aluminum alloy and showed that residual stresses can modify dislocation configurations and promote heterogeneous nucleation, thereby affecting aging response and the resulting mechanical properties. Similarly, Gao et al.¹⁶ studied SiCp/Al–Cu–Mg composites under different heat treatments (T6, T79, T8) and combined X-ray diffraction with mechanical testing to relate heat-treatment routes to residual-stress evolution and property changes.

For alloys in which phase transformation contributes significantly to stress evolution, thermo-mechanical modeling is frequently coupled with transformation kinetics to capture transformation-induced strain and its interaction with thermal gradients. Teixeira et al.¹⁷ investigated residual-stress formation during quenching of Ti17 and Ti–6Al–4 V using FEM with phase transformation modeling, and reported that transformation characteristics strongly affect stress evolution, with Ti17 exhibiting higher residual stresses. For steels, Esfahani et al.¹⁸ performed coupled thermo-mechanical simulations for large steel forgings and analyzed how cooling rates and temperature-dependent material behavior govern residual-stress development during quenching.

At the component scale, geometry and boundary conditions can dominate the resulting residual-stress field, motivating simulation-based assessments and targeted experimental validation. Ba et al.¹⁹ developed a three-dimensional sequentially coupled thermo-mechanical model to study quenching of large aluminum alloy forgings, emphasizing the roles of cooling conditions and part geometry in stress evolution. Li et al.²⁰ investigated 7085 aluminum alloy plates with varying dimensions and found thickness to be the most influential factor on residual-stress levels, while length and width had comparatively smaller effects. Beyond process simulation, experimental measurements are increasingly used to validate and interpret model predictions and stress evolution under different cooling conditions. For instance, Yang et al.²¹ combined experimental measurements with finite element simulations to show that appropriate pre-stretching can reduce residual stresses and improve dimensional stability in 7050 aluminum alloy die forgings, while Wang et al.²² investigated quenching-induced residual stresses in medium-carbon steel using 3D-DIC together with FEM to analyze stress evolution and provide experimentally supported guidance for process optimization.

Collectively, these studies demonstrate the strong influence of material response, boundary conditions, and component geometry on quenching-induced residual stresses, while also highlighting the need for robust full-field experimental constraints. Full-field mapping of internal residual stresses remains challenging for surface-based techniques, motivating the use of sectioning-based reconstruction methods such as the contour method.

Despite extensive work on quenching-induced residual stresses, systematic like-for-like comparisons across quenching media and temperature, together with multi-plane full-field reconstructions, remain limited for non-transforming austenitic stainless steels such as AISI 304.

This study combines full-field measurements and thermo-mechanical modeling to examine quenching-induced residual stresses in AISI 304 under controlled conditions: (i) Water- and oil-quenching are performed from 400 to 1000 °C to build a like-for-like dataset that isolates the role of cooling severity; (ii) Residual stresses are reconstructed as full-field maps using the contour method (calibrated non-contact optical profilometry and elastic FE inversion) on both transverse and longitudinal sections, offering multi-plane insight into the internal stress state; (iii) This study aims to examine the temperature-dependence of residual stress distributions, specifically focusing on the potential occurrence of a saturation-type response at elevated temperatures (700–800 °C).

Materials and methods

Material and heat treatment

Cylindrical specimens (D = 38 mm, L = 40 mm) were prepared from 304 austenitic stainless steel (AISI 304). Unless otherwise stated, the cylinder axis is referred to as the longitudinal direction, and the plane normal to the axis as the transverse section (see Fig. 4). The same geometry was used for sectioning and for the finite-element model employed in the contour-method back-calculation.

Fig. 4.

Schematic Illustration of Sectioning Strategies for Cylindrical Specimens.

To investigate the evolution of residual stresses induced by quenching at different temperatures, a series of heating experiments were conducted at target temperatures of 400 °C, 500 °C, 600 °C, 700 °C, 800 °C, 900 °C, and 1000 °C. The experimental heating process was conducted using a laboratory furnace, as shown in Fig. 1. Once the furnace reached each specified temperature, a subset of specimens was removed and immediately subjected to the quenching process. The remaining specimens were kept inside the furnace and continuously heated until the next target temperature was attained. This procedure was repeated sequentially until all designated temperatures had been reached and corresponding quenching operations had been performed.

Fig. 1.

Heat Treatment Furnace Used for Quenching Experiments.

Both water and a commercially available quenching oil (Houghto-Quench G, Houghton, USA) were employed as quenching media to investigate their respective influences on the development and distribution of residual stresses. The selection of these two commonly used quenching media allows for a comparative analysis of cooling severity and its effect on the resulting internal stress states within the material. By systematically varying the quenching medium, this study aims to reveal the role of thermal gradients and cooling rates in shaping the residual-stress profiles.

Microstructural observations

To document the material conditions examined in this work, representative optical micrographs of 304 austenitic stainless steel are presented in Fig. 2. The three conditions are the as-received state and specimens heated to 800 °C followed by either water quenching or oil quenching. Metallographic coupons were ground with SiC papers (240, 400, 800, 1200, 2000 grit), polished to a 0.05 µm alumina finish, ultrasonically cleaned in ethanol, and chemically etched with glyceregia (glycerol:HCl:HNO₃ = 1:1:1) for 10–30 s at room temperature prior to optical microscopy (scale bar 100 µm).

Fig. 2.

Optical Micrographs of 304 Austenitic Stainless Steel.

Figure 2(a) shows equiaxed austenite grains with clean, finely etched boundaries and few intragranular features in the as-received material. After heating to 800 °C and water quenching (Fig. 2(b)), the overall grain morphology remains similar; grain boundaries are mostly clean, with only scattered etch pits or particles and no continuous boundary decoration. In contrast, oil quenching from 800 °C (Fig. 2(c)) produces a darker and more continuous network along grain boundaries, indicating enhanced boundary decoration consistent with the slower cooling rate. No stereological quantification is pursued, as these micrographs serve solely to document the three material states used in the study.

These micrographs indicate boundary decoration differences without a bulk phase transformation; 304 is therefore treated here as a non-transforming alloy in the investigated range. Qualitative implications for mechanical properties are discussed in "Implications for mechanical properties Section".

Principles of the contour method

Accurate characterization of internal residual stresses—especially across full cross-sections in non-transforming materials like austenitic stainless steels—requires robust experimental techniques capable of capturing spatial stress variations with high fidelity. To this end, the contour method was adopted in this study due to its proven ability to reconstruct two-dimensional residual-stress fields with high spatial resolution and minimal assumptions.

The contour method is a destructive but highly effective technique for evaluating residual-stress distributions in solid components²³. It is particularly suited for measuring stress normal to a planar cross-section by mapping the deformation caused by residual stress release²⁴. The method involves sectioning the component, measuring the resulting surface deformation, and reconstructing the original stress field through finite element analysis. The fundamental principle of the contour method for residual stress measurement is illustrated in Fig. 3.

Fig. 3.

Schematic of Residual-Stress Evaluation Using the Contour Method.

As shown in Step-1, residual stresses are initially present within the intact workpiece, typically introduced by thermal treatments such as quenching. In Step-2, the specimen is carefully sectioned along a predefined mid-plane using Wire Electrical Discharge Machining (EDM) to prevent the induction of additional stresses during the cutting process. Following the cut, the release of internal stresses causes deformation on the newly exposed surfaces, as depicted in Step-3. These surface contours are then precisely measured using high-resolution profilometry or laser scanning techniques. In Step-4, the measured displacement field is applied in reverse as boundary conditions to a finite element (FE) model, enabling simulation of the stresses required to generate the observed deformation. Finally, as shown in Step-5, the original residual-stress distribution normal to the cut plane is reconstructed through elastic finite element analysis. This approach provides a quantitative map of residual stresses across the cross-section of the specimen.

Experimental procedures of the contour method

Specimen sectioning and EDM process

Two sectioning strategies were considered for the cylindrical specimen: longitudinal sectioning along the longitudinal axis (Fig. 4(b)), and transverse sectioning perpendicular to the axis (Fig. 4(c)). These sectioning methods are fundamental to investigating the residual-stress distributions using the contour method.

Wire Electrical Discharge Machining (EDM) was utilized to perform precise sectioning of the cylindrical specimens for residual stress evaluation. As shown in Fig. 5(a), the machining was conducted on a CNC-controlled wire EDM system equipped with an automatic wire feeding mechanism, tension adjustment, and real-time process monitoring. During the cutting process, the specimens were fully submerged in deionized water, as illustrated in Fig. 5(b). This dielectric fluid was continuously circulated to maintain a low electrical conductivity, facilitate debris removal, stabilize the discharge channel, and provide effective cooling to minimize thermal distortion.

Fig. 5.

Wire EDM Machine and Cutting Environment.

During wire-EDM sectioning, the cylindrical specimens were not mechanically clamped. To maintain positional stability, each specimen was temporarily bonded at one end to a rigid base using a room-temperature, low-shrinkage adhesive, with the assembly fully submerged in a deionized-water dielectric. This arrangement prevented clamp-induced stresses or local plastic deformation in the gauge region while ensuring accurate alignment of the cut. The only body force acting during cutting was the specimen’s self-weight; for the present geometry, a worst-case cantilever estimate yielded a bending stress of approximately 0.01–0.03 MPa, which was negligible relative to the MPa-level residual-stress fields measured. After sectioning, the halves were released from the temporary bond and prepared for surface profilometry. The cutting parameters were carefully selected to ensure minimal introduction of additional stresses. A brass wire with a diameter of 0.25 mm was used as the electrode. The pulse current was set to 3 A, the pulse duration to 20 μs, and the pulse interval to 30 μs. The wire feed rate was maintained at 8 mm/s, and the table feed speed was set to 0.5 mm/min. These parameters were optimized to achieve a balance between material removal rate and surface integrity, ensuring that the cut surfaces could be accurately measured for contour deformation and subsequent residual stress reconstruction.

Surface contour measurement via optical profilometry

The cut-surface relief was measured with a ZEISS O-DETECT optical measuring system in camera-based focus-variation mode (CALYPSO), as shown in Fig. 6. Prior to scanning, the system was calibrated with a traceable step-height standard; accuracy follows the manufacturer’s ISO 10,360 specification. This non-contact method is well suited for capturing the small out-of-plane surface relief released during sectioning and avoids probe-induced deformation.

Fig. 6.

Optical Surface Profiling of the Cut Specimen Using a Non-contact 3D Measurement System.

Surface height maps were acquired with a lateral sampling of 0.10 mm. The raw maps were leveled by least-squares plane removal and lightly low-pass filtered to suppress high-frequency noise before inversion. The measured relief fields then served as inputs to an elastic finite-element back-calculation to reconstruct the original residual-stress component normal to the cut surface using the contour method.

Data processing and numerical inversion

Transverse section displacement processing

Before reconstructing the residual-stress field on the transverse section, the raw displacement data obtained via non-contact profilometry must be preprocessed to improve accuracy and numerical stability. As illustrated in Fig. 7(a), the original out-of-plane displacement measurements often contain noise and high-frequency surface roughness due to machining and scanning artifacts. To address this, a two-dimensional Gaussian filter is applied to smooth the measured data, preserving the main deformation characteristics while eliminating noise components. The Gaussian smoothing function is defined as²⁵:

1

where is the standard deviation determining the extent of smoothing. The filtered height field is obtained by convolving the raw measured field with the Gaussian kernel:

2

where ∗ denotes the 2D convolution operator. Figure 7(b) displays the smoothed surface profile, which preserves the overall deformation pattern while eliminating high-frequency fluctuations.

Fig. 7.

Gaussian Smoothing of Measured Displacement Field for Transverse Sectioning.

Both cut surfaces were profiled. Both height maps H1(x, y) and H₂(x, y) were leveled by least-squares plane removal. The second surface was mirrored and registered to the first surface’s coordinate frame, and its sign convention was aligned (flipped where necessary). A narrow margin near the cut edge was masked. The final relief used for inversion was the average:

3

where denotes the mirrored, registered, and sign-aligned map. Averaging reduces random noise and cancels small tilts and misalignments from individual scans, providing a more robust boundary condition for the elastic finite-element back-calculation.

The residual-stress field must satisfy static equilibrium either globally or locally within the material domain²⁶. In the contour method, the component is sectioned, and the deformation caused by the release of residual stress is measured. The measured data corresponds to the normal stress component (σ_n) across the cutting plane.

To ensure that the reconstructed residual-stress field satisfies internal equilibrium, it is necessary to eliminate any rigid-body offset present in the measured displacement data. This is achieved by identifying an appropriate reference plane on the measured surface and correcting the displacement field such that the net force across the section becomes zero.

With the sign convention adopted here, the averaged height/relief map is identified with the measured out-of-plane displacement of the cut surface as:

4

Let u_meas (x, y) represent the out-of-plane displacement measured over the cutting plane. The corrected displacement field u_coor (x, y) is defined as:

5

where Δu is the mean displacement over the area _A of the section, given by:

6

By subtracting this average displacement, we ensure that the resulting corrected displacement field satisfies the following force equilibrium condition on the section:

7

Instead of a simple analytical solution, the relationship between the measured surface deformation and the original residual stress field is governed by the theory of linear elasticity for a continuum solid. Since the cross-section of the cylindrical specimen involves a triaxial stress state where Poisson effects are non-negligible, the corrected displacement field serves as a Dirichlet boundary condition for an elastic Finite Element (FE) analysis. The residual stress distribution is then reconstructed numerically by solving the equilibrium equations subject to these boundary conditions, as detailed in the subsequent modeling section.

In Fig. 8(a), the highlighted grid plane represents the reference surface selected as the datum for contour alignment. All surface profiles were adjusted relative to this plane to ensure that the reconstructed stress field on the section remains balanced. The presented data correspond to the specimen quenched in water at 1000 °C. Figure 8(b) shows the corrected displacement field after aligning the data to the selected reference plane, ensuring the net force and moment balance on the cross-section. Subsequently, Fig. 8(c) illustrates the reverse displacement field imposed on the finite element model in the direction normal to the cut plane, which is used to reconstruct the residual-stress distribution via elastic back-calculation.

Fig. 8.

Displacement Correction and Application Process in the Contour Method for Transverse Sectioning.

Longitudinal section displacement processing

Similarly, the displacement field measured on the longitudinal cutting plane underwent the same data processing workflow as applied to the transverse section. This process included noise reduction using Gaussian smoothing, alignment with a reference plane to ensure force equilibrium, and preparation of the displacement data for input into the finite element model. The results shown in Fig. 9 correspond to the specimen quenched in water at 1000 °C, illustrating the final corrected displacement field used for stress reconstruction.

Fig. 9.

Sequential Processing of Displacement Data in the Contour Method for Longitudinal Sectioning.

Contour data interpolation for FE modeling

After preprocessing, the corrected displacement data are interpolated onto the FE mesh nodes, since displacements in the finite element model must be defined at discrete nodal points. Let the preprocessed data be denoted by scattered sample points , where (x_i, y_i) are spatial coordinates on the cutting plane, and is the measured displacement normal to the plane. The interpolated displacement u (x,y) at any point (x,y) on the FE mesh is obtained by constructing a continuous function u (x,y) satisfying:

8

where w_i (x,y) are interpolation weights determined by a scattered data interpolation scheme. In this study, the interpolation is performed using natural neighbor interpolation, which ensures smooth and spatially coherent displacement mapping over irregularly spaced mesh nodes. This method is particularly suitable for transferring laser-scanned data to nonuniform FE meshes without introducing spurious oscillations.

The resulting nodal displacement field is then applied as boundary conditions on the cutting plane of the FE model, enabling inverse stress reconstruction via elastic simulation.

To verify the effectiveness of applying processed contour data as boundary conditions, the interpolated displacement fields obtained from 1000 °C water-quenched specimens were mapped onto the finite element mesh. The imposed displacements were used to drive a static elastic simulation, where the resulting residual-stress field was recovered through reverse deformation analysis. The simulation results, as shown in Fig. 10, clearly demonstrate the reconstructed normal residual-stress distributions on both the transverse and longitudinal sections. A displacement amplification factor of 100 was used for better visualization of the deformed mesh.

Fig. 10.

Finite Element Reconstruction of Residual-Stress Fields Using Processed Contour Data From 1000 °C Water-Quenched Specimens.

Finite element modeling setup

To validate the residual-stress distributions measured by the contour method, a finite-element simulation of the quenching process was carried out in Abaqus using a fully coupled temperature–displacement step (COUPLED TEMPERATURE–DISPLACEMENT), in which the transient heat transfer and mechanical response were solved simultaneously. The simulation aimed to predict the thermal and mechanical responses during quenching, focusing on the evolution of residual stresses in the cylindrical AISI 304 stainless steel specimens (diameter 38 mm, length 40 mm). This numerical approach complements the experiments and enables a direct, quantitative comparison with the measured residual-stress profiles. Given the computational demands, simulations were conducted for representative conditions: water quenching at 400 °C, 700 °C, and 1000 °C, and oil quenching at 700 °C and 1000 °C. These selections capture the low-temperature behavior, the onset of stress saturation above 700 °C, and the effects of cooling media.

Material properties and model setup

To reduce computational cost, a three-dimensional half-model of the cylindrical specimen was constructed by exploiting mirror symmetry about a longitudinal plane passing through the cylinder axis. The mesh was uniform, comprising approximately 112,400 eight-node coupled temperature–displacement brick elements (C3D8T), with a nominal element size of ≈0.8 mm throughout the domain. For consistency and direct identification of the stress fields, the stress components S11 and S33 in the Abaqus simulation legend correspond to the normal stresses and , respectively. The quenching simulation was conducted in a single analysis step, with the transient heat transfer and stress evolution computed simultaneously.

Temperature-dependent material properties for AISI 304 were implemented, as summarized in Table 1²⁷. Phase transformations were not modeled, consistent with the non-transforming behavior of austenitic 304 within 400–1000 °C.

Table 1.

Temperature-Dependent Properties of AISI 304 Stainless Steel.

Temperature, T /℃	Density, ρ /kg·m-3	Specific Heat, c /J·(kg-1·K-1)	Thermal Conductivity, k /W·(m-1·K-1)	Thermal expansion, α /10−6·K-1	Young’s Modulus , E /GPa	Poisson Ratio, ν
20	7910	456	16.2	15.5	200	0.29
100	7876	494	16.6	16.3	191.4	0.285
200	7840	532	17.45	16.7	183.5	0.289
300	7798	553	18.9	17.1	177.5	0.307
400	7754	569	20.35	17.6	168.2	0.309
500	7708	582	20.64	18	158.2	0.282
600	7663	593	20.93	18.3	149.3	0.289
700	7617	609	–	19	140.3	0.28
800	7572	629	–	20	127.5	0.255

The temperature dependence of the 0.2% proof (yield) stress of annealed AISI 304 can be represented by retention factors k_0.2,θ (T) recommended in the Design Manual for Structural Stainless Steel (Euro Inox/SCI, 4th ed.). The yield stress at temperature _T is obtained by scaling the room-temperature value:

9

where (20℃) is the adopted room-temperature 0.2% proof stress. In this study, a representative room-temperature 0.2% proof stress () of 210 MPa was adopted based on the standard range for annealed AISI 304, and k_0.2,θ (T) is taken from the manual and linearly interpolated between tabulated temperatures:

10

The recommended tabulated pairs {T /°C , k_0.2,θ } for AISI 304 are²⁸:

11

These factors are derived from elevated-temperature stress–strain models and test compilations and are intended for short-term tensile properties of annealed austenitic stainless steels.

The material response is modeled using von Mises (J2) elastoplasticity under small-strain and isotropic assumptions, with temperature-dependent isotropic hardening defined via tabulated data. The total strain tensor ε is additively decomposed into elastic (ε_e ), plastic (ε^p), and thermal (ε^th) components:

12

The thermal strain tensor is defined using the instantaneous coefficient of linear thermal expansion α(T) /K⁻¹ and a reference stress-free temperature T_ref /K:

13

where T /K is the absolute temperature and I is the second-order identity tensor. The Cauchy stress tensor σ /Pa is related to the elastic strain via temperature-dependent isotropic linear elasticity:

14

where C(T) /Pa is the fourth-order stiffness tensor, expressed as:

15

Here, G(T) /Pa is the shear modulus, λ(T) /Pa is the first Lamé parameter, and I_sym is the fourth-order symmetric identity tensor. The shear modulus and Lamé parameter are related to Young’s modulus E(T) /Pa and Poisson ratio υ (T) by:

16

Plastic yielding follows the von Mises (J2) criterion with a temperature- and plastic-strain-dependent yield stress /Pa:

17

The equivalent von Mises stress /Pa is:

18

where /Pa is the deviatoric stress tensor, and tr(·) denotes the trace. An associated flow rule is adopted, with the plastic strain rate tensor (s^-1) and equivalent plastic strain rate (s⁻¹) given by:

19

where (s^-1) is the plastic multiplier rate. The plastic flow satisfies the Kuhn–Tucker conditions and consistency condition:

20

Isotropic hardening is defined by tabulated true stress–plastic strain curves at discrete temperatures. The yield stress is obtained via linear interpolation in plastic strain and temperature, without extrapolation. The tangent hardening modulus H′ /Pa is:

21

Each hardening curve is monotonically increasing in stress, with the first point typically at (σ_y (T), 0). The elastic properties E(T) and Poisson ratio υ(T) are provided as tabulated data or analytical functions, with linear interpolation applied if tabulated. For temperatures or plastic strains outside the tabulated range, the solver uses the nearest tabulated values to avoid extrapolation. This simplification is justified as the thermo-physical properties of AISI 304, such as thermal conductivity and specific heat, exhibit relative stability at temperatures exceeding 800 ℃. Moreover, the early stage of quenching, where the highest thermal gradients occur and the bulk of plastic deformation is triggered, is predominantly influenced by the material response within the 600–800 ℃ range²⁷.

Boundary conditions and simulation procedure

The simulation was initialized by assigning a uniform temperature to the specimen as the initial condition. Quenching was then applied to all external surfaces using temperature-dependent heat-transfer coefficients h(T) specified by the curves in Fig. 11^29,30. Two boundary-condition sets were used to represent the cooling media—water-quench h(T) and oil-quench h(T)—both supplied to the solver as tabulated, piecewise-linear data with linear interpolation in temperature.

Fig. 11.

Heat Transfer Coefficient (HTC) Profiles for Water and Oil Quenching of AISI 304 Stainless Steel.

As expected, the water-quench curve attains a higher peak h(T) and broader high-HTC region than the oil-quench curve, reflecting more intense cooling. Both curves exhibit a pronounced rise through the nucleate/transition-boiling regime and a subsequent decline toward lower values at high and low temperatures. In the simulations, these h(T) profiles primarily control near-surface cooling rates and thereby the magnitude and depth of the compressive residual-stress layer; sensitivity analyses indicate that variations around the peak of h(T) have the greatest influence on the final stress profile.

The quenchant temperature was set to 25 °C, and the simulation ran until the specimen reached thermal equilibrium. Mechanical boundary conditions included symmetry constraints along the axis and free expansion/contraction elsewhere. Residual stresses were extracted at the end of the cooling step, representing the self-equilibrated state.

Results and discussion

Residual stress analysis on the transverse section

Corrected displacement profiles

To investigate the effect of quenching temperature on residual stress evolution, surface contour data were measured on the transverse cutting plane after wire-cut electrical discharge machining (WEDM). The deformation data were corrected to satisfy force equilibrium conditions across the cross-section, enabling accurate residual stress recovery using the contour method.

Figure 12 presents the corrected displacement fields obtained under oil quenching conditions at different temperatures. These corrected fields reflect the distortion caused by the release of internal stresses during cutting and are directly related to the magnitude of residual stress initially present in the material. It is worth noting that no significant surface contour deformation was detected at oil quenching temperatures below 700 °C. Therefore, only results starting from 700 °C are presented in this figure.

Fig. 12.

Corrected Displacement Fields on the Transverse Cutting Plane Under Different Quenching Temperatures in Oil Quenching.

As shown in Fig. 12, the corrected displacement magnitude generally increases with oil quenching temperature, indicating a rising level of residual stress induced by higher thermal gradients. However, the increase becomes less pronounced above 800 °C, suggesting that the rate of residual stress development tends to plateau at higher quenching temperatures. The displacement profiles also exhibit a generally symmetric distribution about the specimen axis, particularly at 1000 °C, reflecting the uniformity of thermal contraction in oil-quenched samples.

Following the oil quenching results, further analysis was conducted under water quenching conditions, which are characterized by a significantly higher cooling rate. This rapid cooling is expected to introduce larger thermal gradients and more intense residual-stress distributions. The corrected displacement fields on the transverse cutting plane obtained at different water quenching temperatures are shown in Fig. 13.

Fig. 13.

Corrected Displacement Fields on the Transverse Cutting Plane Under Different Quenching Temperatures in Water Quenching.

As shown in Fig. 13, water quenching resulted in significantly greater displacement magnitudes than oil quenching, especially at elevated temperatures. The deformation fields exhibit a more pronounced bowl-shaped profile, with sharper gradients near the edge regions.

Notably, when the quenching temperature exceeds 700 °C, the increase in displacement magnitude becomes less pronounced, suggesting a saturation effect in stress development under severe quenching conditions.

These results reflect the intense internal stress release associated with high thermal shock in water quenching. The displacement fields provide critical input for calculating residual-stress distributions using the contour method and will be further examined in the following sections.

Reconstructed stress distributions

Based on the corrected displacement fields obtained on the transverse cutting planes, Fig. 14 illustrates the reconstructed normal residual-stress distributions under various oil quenching temperatures ranging from 700 to 1000 °C.

Fig. 14.

Reconstructed Normal Residual-Stress Distributions (σ_zz) on the Transverse Section Under Different Oil Quenching Temperatures.

The inverse analysis was performed using a finite element model, where the reverse displacement fields were imposed on the cross-section to recover the self-equilibrated residual-stress field normal to the cutting plane.

As shown, all cases exhibit a characteristic compressive–tensile stress profile from the surface to the core.

At 700 °C, compressive stresses are dominant in the outer region with a relatively moderate tensile peak near the core. As the quenching temperature increases, the compressive stress band near the surface becomes narrower and sharper, while the central tensile zone expands and intensifies. At 1000 °C, the peak compressive stress near the surface reaches approximately − 50 MPa, suggesting that higher quenching temperatures promote stronger thermal gradients, which in turn lead to more severe compressive stress buildup at the surface.

To further investigate the effect of water quenching temperature on the residual-stress distribution, the reconstructed normal residual-stress fields (σ_zz) on the transverse section were analyzed under different water quenching temperatures.

Figure 15 presents the reconstructed residual-stress distributions on the transverse section under different water quenching temperatures, derived from finite element analysis using the corrected displacement fields obtained via the contour method. This figure illustrates how the residual-stress profile evolves with increasing water quenching temperature, providing insight into the role of thermal gradients and cooling severity in shaping the stress distribution.

Fig. 15.

Reconstructed Normal Residual-Stress Distributions (σ_zz) on the Transverse Section Under Different Water Quenching Temperatures.

As shown in Fig. 15, the residual-stress distribution patterns exhibit clear dependence on the quenching temperature. At lower quenching temperatures such as 400 °C and 500 °C, a relatively steep stress gradient is observed near the surface, with compressive stress gradually transitioning to tensile stress toward the core. As the quenching temperature increases, the compressive zone becomes more pronounced and penetrates deeper into the section, indicating a higher thermal gradient and stronger thermo-mechanically induced stress.

To facilitate a comparative analysis of the residual-stress behavior under different quenching media, the reconstructed normal residual-stress distributions along the diameter of the transverse section are plotted together in Fig. 16. The comparison includes results from oil quenching and water quenching over a range of temperatures. This visualization enables a more intuitive evaluation of the influence of quenching severity and thermal gradients on the internal stress field.

Fig. 16.

Distributions of Residual Stress () Normal to the Transverse Section Along the Diameter Under Various Quenching Conditions.

As shown in Fig. 16, the surface compressive residual stresses induced by water quenching (Fig. 16(b)) are significantly higher than those observed in oil quenching (Fig. 16(a)), especially at elevated temperatures. While the overall stress magnitude increases with temperature up to a certain point, the width of the tensile region in water-quenched samples remains relatively stable beyond 700 °C, without exhibiting a clear broadening trend.

Moreover, for both oil and water quenching, the residual-stress profiles show minimal change once the quenching temperature exceeds 700 °C. This convergence may be attributed to the saturation of thermal gradients, which dominate stress development at lower temperatures but become less influential at higher ones. These observations indicate that increasing quenching temperature beyond 700 °C may have a limited impact on further altering residual-stress distributions.

Residual stress analysis on the longitudinal section

Corrected displacement profiles

In addition to the transverse section, displacement measurements and corrections were also performed on the longitudinal cutting plane to provide a more comprehensive understanding of residual stress development along the longitudinal direction. The longitudinal section captures the longitudinal deformation behavior primarily resulting from thermal gradients during quenching.

Figure 17 presents the corrected displacement fields obtained from the longitudinal section at different oil quenching temperatures ranging from 700 to 1000 °C. These results were processed through Gaussian smoothing and reference plane alignment to eliminate noise and rigid-body motion, ensuring that the displacement fields are suitable for accurate residual stress reconstruction.

Fig. 17.

Corrected Displacement Fields on the Longitudinal Cutting Plane Under Different Quenching Temperatures in Oil Quenching.

As illustrated in Fig. 17, the displacement fields on the longitudinal cutting plane exhibit a distinct saddle-shaped curvature, with peak displacements observed near the specimen ends and a trough near the midsection. The displacement magnitude increases with quenching temperature, indicating more severe thermally induced distortion at higher temperatures.

To further investigate the impact of quenching medium on deformation behavior, the corrected displacement fields on the longitudinal cutting plane under water quenching at various temperatures (400 °C to 1000 °C) are presented in Fig. 18.

Fig. 18.

Corrected Displacement Fields on the Longitudinal Cutting Plane Under Different Quenching Temperatures in Water Quenching.

Compared to oil quenching, water quenching induces significantly larger displacement amplitudes across the longitudinal section, particularly at higher temperatures. The central region exhibits a deeper valley, while the edge zones show more pronounced upward deformation, forming a stronger "saddle-shaped" profile.

These intensified displacement patterns reflect the elevated thermal gradients and thermo-mechanically induced strain under rapid cooling conditions. The corrected fields will be subsequently used to reconstruct the residual-stress distributions along the longitudinal direction and evaluated against the transverse results for comparative analysis.

Reconstructed stress distributions

To provide a more comprehensive understanding of the internal stress state, the spatial evolution of residual stresses was further analyzed across the longitudinal section. Figure 19 illustrates the reconstructed normal residual stress (σ_xx) fields under various oil quenching temperatures. A consistent distribution pattern is observed, characterized by a predominantly tensile stress region situated near the center of the section, balanced by compressive zones closer to the outer surface. As the quenching temperature is increased from 700 to 1000 °C, the magnitude of the central tensile peak becomes more pronounced, and the overall stress gradient across the section intensifies, indicating that elevated temperatures significantly enhance longitudinal stress concentrations.

Fig. 19.

Reconstructed Normal Residual-Stress Distributions (σ_xx) on the Longitudinal Section at Different Oil Quenching Temperatures.

In contrast, the σ_xx distributions resulting from water quenching exhibit a much higher sensitivity to temperature variations (see Fig. 20). While the stress profile remains relatively uniform across the central region at a lower temperature of 400 °C, the distributions at higher temperatures (700–1000 °C) transition into a markedly non-uniform state. This non-uniformity is particularly evident near the edges of the section, where the fields exhibit substantially sharper stress gradients.

Fig. 20.

Reconstructed Normal Residual-Stress Distributions (σ_xx) on the Longitudinal Section at Different Water Quenching Temperatures.

For a more quantitative assessment, residual-stress profiles were extracted along the diameter and axial paths defined in the schematic in Fig. 21.

Fig. 21.

Schematic Representation of the Axial and Diameter Paths Within the Specimen Used for Stress Path Extraction.

The comparisons between oil and water quenching conditions along these paths are summarized in Figs. 22 and 23. These results further demonstrate the decisive role of the cooling medium in shaping the longitudinal stress profile. In oil quenching, the stress distributions along both the diameter path (Fig. 22a) and the axial path (Fig. 23a) appear relatively mild and uniform. Conversely, water quenching induces significantly higher tensile stress magnitudes and steeper gradients, which underscores the intense thermal shock effects experienced during the rapid cooling process. Both the transverse and axial profiles (Figs. 22b and 23b, respectively) reveal a markedly more non-uniform stress state in water-quenched specimens. In contrast, oil quenching maintains a more stable and less intense distribution due to its comparatively slower cooling rate.

Fig. 22.

Distributions of Residual Stress (σ_xx) Normal to the Longitudinal Section Along the Diameter Path Under Various Quenching Conditions.

Fig. 23.

Distributions of Residual Stress (σ_xx) Normal to the Longitudinal Section Along the Axial Path Under Various Quenching Conditions.

Numerical simulation and validation

Simulated residual-stress distributions

Figure 24 presents representative finite-element contour plots of the residual-stress field in the cylinder after water quenching from 1000 °C. The plots show normal stress components on two orthogonal sections—σ_zz on a transverse section and σ_xx on a longitudinal section—extracted at the end of cooling (ambient temperature). The fields show a predominantly axisymmetric distribution, with near-surface compression balanced by tensile stresses in the interior.

Fig. 24.

Simulated Residual-Stress Contours After Water Quenching From 1000 °C.

The contour is included to visualize the simulated residual-stress field at the end of quenching. Comprehensive quantitative comparisons with measurements are presented in the subsequent sections. Given the geometric symmetry and the symmetry boundary conditions imposed on the longitudinal symmetry plane, the stresses obtained on the symmetry section are representative of the interior residual-stress distribution in the full cylindrical specimen.

Comparison with Experimental Results

To assess the predictive capability of the model, we compare measured and simulated residual-stress profiles for both oil and water quenching. Stresses are extracted along two representative sections: the transverse section for the axial component (along the diameter), and the longitudinal section for (along the diameter path and the axial path, respectively). Figure 25 reports oil-quench results at selected temperatures (e.g., 700 and 1000 °C), while Fig. 26 summarizes water-quench results over multiple temperatures (e.g., 400/700/1000 °C). The comparison focuses on (i) profile shape—characteristic compressive rims and tensile cores, (ii) locations and magnitudes of extrema, and (iii) consistency of trends with quench medium and temperature. Legends denote “Experiment” vs. “Simulation.” Any localized discrepancies near surface minima or axial peak tension are analyzed later with respect to heat-transfer boundary conditions and constitutive assumptions.

Fig. 25.

Comparison of Experimental and Simulated Residual-Stress Profiles After Oil Quenching.

Fig. 26.

Comparison of Experimental and Simulated Residual-Stress Profiles After Water Quenching.

As shown in Figs. 25 and 26, the profiles resulting from the simulation are in good agreement with the experimental data. For both quench media, the profiles exhibit the compressive surface rim and tensile core, symmetric about the mid-planes. With increasing temperature (700 and 1000 °C for oil quenching; 400, 700, and 1000 °C for water quenching), both experiment and simulation show stronger compressive rims and higher core tension, indicating that the model responds correctly to steeper thermal gradients and enlarged plastic zones.

Water quenching yields substantially larger magnitudes than oil quenching, and the simulation reproduces this ranking of the quenching media. The consistency between the experimental results on the transverse section (σ_zz /MPa) and the longitudinal section (σ_xx /MPa) supports the assumption of circumferentially uniform cooling and lends confidence to the calibrated constitutive parameters.

Residual discrepancies are localized: (i) a slight overprediction of the near-surface compressive extremum in oil quenching; and (ii) a mild underprediction of the axial peak tension for severe water quenching. These discrepancies can be attributed to: (i) experimental uncertainty (instrumentation, specimen-to-specimen variability, cooling-condition fluctuations, and alignment/registration); (ii) model-parameter uncertainty (simplified/calibrated heat-transfer coefficients, temperature-dependent constitutive parameters, and idealized boundary constraints); and (iii) the inherent limitation of the contour method (CM)—it reconstructs the normal residual stress from a single-component displacement field after cut-and-release, making it sensitive to cut quality and measurement noise and less capable of fully recovering the 3D stress state, especially near high gradients.

It is worth noting that the experimental profile for water quenching at intermediate temperatures (e.g., 500 ℃) exhibits a localized irregularity or ‘dent’ in the central region. This feature is attributed to the self-equilibrating nature of residual stresses rather than experimental error. During quenching, plastic deformation is concentrated in the outer layers due to steep thermal gradients, while the core material cools more slowly and remains largely elastic. Consequently, the stress distribution in the central region acts as a passive reaction that must adjust to balance the plastic strain gradients generated in the outer shell. Verification simulations performed for this condition confirm this non-parabolic behavior, demonstrating that the observed ‘dent’ is a deterministic thermo-mechanical response to the specific cooling history. Overall consistency in sign, extrema locations, and medium/temperature trends supports the model’s physical fidelity.

Discussion on residual stress formation mechanisms

Thermal gradient and plasticity

The origins of the observed residual-stress patterns, specifically the consistent formation of compressive stresses at the specimen surface, are deeply rooted in the thermo-mechanical behavior during quenching. For non-transforming materials like AISI 304 austenitic stainless steel, these compressive surface stresses typically develop even in the absence of martensitic phase transformation.

The process can be divided into two distinct stages:

1. Initial cooling stage: When the specimen is submerged in the quenchant, the surface cools and contracts much faster than the core. The still-hot interior constrains this surface contraction, thereby imposing a transient tensile stress on the outer layers. Once this induced tensile stress exceeds the material’s temperature-dependent yield strength, the surface undergoes significant plastic tensile deformation;

2. Final cooling stage: As the interior eventually begins to cool and contract, it exerts a pulling force on the now-hardened and plastically deformed surface. Since the surface can no longer accommodate this additional deformation, a mismatch occurs. This ultimately results in the generation of compressive residual stresses at the surface, balanced by compensating tensile stresses within the core.

This effect is notably intensified by the intrinsic material properties of AISI 304, such as its low thermal conductivity and high ductility. These properties facilitate the formation of sharp temperature gradients and allow for extensive plastic flow under transient thermal conditions, leading to the pronounced stress profiles observed in the experimental results.

Stress saturation effect

A significant finding in this study is the diminishing sensitivity of residual-stress distributions to the initial quenching temperature once it exceeds a certain threshold. Specifically, for both oil and water quenching, the residual-stress profiles exhibit minimal changes in magnitude or shape when the quenching temperature is increased from 700 ℃ up to 1000 ℃.

This "saturation-type" response can be attributed to the stabilization of thermal gradients at elevated temperatures. In the lower temperature range, the increase in initial temperature significantly heightens the thermal shock and resulting gradients, which dominate stress development. However, as the temperature continues to rise beyond 700–800 ℃, the heat transfer mechanisms (particularly the transition and peak boiling regimes) become the primary limiting factors. Consequently, the thermal gradients driving the thermo-mechanical plasticity begin to stabilize, thereby diminishing their influence on further stress accumulation. These observations suggest that for the investigated geometry, increasing the quenching temperature beyond this saturation point offers limited impact on further altering the internal stress state.

Implications for mechanical properties

For austenitic 304 stainless steel in the 400–1000 °C range, quenching does not induce a martensitic transformation. Accordingly, after solution heat treatment the bulk hardness and tensile strength are expected to fall within literature-reported ranges for 304 and show limited sensitivity to the quenching medium compared with the effects of cold work or precipitation hardening³¹. In contrast, the residual-stress field is strongly governed by the cooling rate: faster cooling (water) generates steeper thermal gradients and larger near-surface compressive stresses, whereas slower cooling (oil) yields a more moderate profile. The stress states reported here therefore provide qualitative insight into potential performance differences (e.g., crack initiation/fatigue), although quantitative mechanical-property testing lies outside the scope of this work. Systematic coupling of full-field residual stresses with hardness/strength measurements across all thermal conditions is identified as future work.

Conclusions

This work combined the contour method with finite element reconstruction to obtain full-field residual-stress maps on both transverse and longitudinal sections of quenched AISI 304 cylinders and then validated a temperature-dependent J2 plasticity model (tabulated hardening) across representative quench conditions. Together, the measurements and simulations establish a consistent picture of residual-stress formation in austenitic 304 over 400–1000 °C and isolate medium and temperature effects in the absence of phase transformation.

1. Parsimonious constitutive model suffices. A temperature-dependent J2 plasticity with tabulated isotropic hardening—without rate terms or transformation kinetics—accurately reproduces the measured residual-stress profiles for non-transforming AISI 304 across the investigated quench conditions. This establishes a robust baseline for process simulation and validation.

2. Heat transfer coefficient peak region controls the outcome. Sensitivity analyses indicate that the transition/peak portion of the temperature-dependent heat-transfer coefficient governs the surface compression, penetration depth, and zero-crossing location of the final residual-stress profile; calibrating this segment yields the largest accuracy gains relative to modifications of the low- and high-temperature tails.

3. Diminishing returns with higher temperatures. Above roughly 700–800 °C, further increases in initial temperature produce only second-order changes in residual-stress magnitude and shape, revealing a saturation-type response for non-transforming 304 under the present geometry and scales.

4. Symmetry sections faithfully represent the interior field. Consistency between transverse and longitudinal residual-stress profiles confirms that, for this axisymmetric geometry and boundary conditions, stresses reported on symmetry sections are representative of the true interior distribution—supporting their use for model–experiment comparison and interpretation.

Author contributions

L.M. conceived the study and developed the main research goals. A.M.K. designed the methodology and contributed to visualization and resource provision. Y.S. developed the software and conducted the investigation. K.A.A. performed data validation and prepared the original draft. L.M. carried out the formal analysis, curated the data, and led the review and editing of the manuscript. L.M. also managed project administration. A.M.K. secured funding for the project. All authors reviewed and approved the final manuscript.

Funding

The authors would like to acknowledge with thanks the support of King Abdulaziz University to this study. This research work is also supported by NSFC Research Fund for International Young Scientists RFIS-I [Grant No. GIA24002]. The computational resources generously provided by the High-Performance Computing Center of Nanjing Tech University are greatly appreciated. In the preparation of this manuscript, artificial intelligence (AI) tools were employed for linguistic refinement and enhancement of English expression, with the aim of improving the clarity and accessibility of the academic research presented. The authors have thoroughly reviewed and validated all AI-assisted modifications to ensure fidelity to the original meaning and accuracy in representation. It is important to note that the substantive work, including all research findings and innovations, is solely attributable to the authors.

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.