Research of Residual Stresses and Deformations for Parts After Shot Peening Using Finite Element Analysis

Abstract

Despite significant advancements in understanding the effects of hardening modes, particularly shot peening, on the stress–strain state of materials, the determination of residual stress remains a considerable challenge, especially when temporal factors are involved. Conventional methods for measuring residual stress are often laborious and costly. In this study, the modeling of residual stress formation and the calculation of manufacturing-induced residual strains are performed. This article’s objective is to investigate the redistribution and formation of residual stresses in the surface layer of components, along with the resulting deformations using shot-peening samples. This objective is realized through the incorporation of the concept of initial stresses. The findings demonstrate the effectiveness of finite element analysis in predicting residual deformations in complex components such as blades, disks, and other geometrically intricate aircraft engine parts. The proposed approach offers a faster and more cost-effective alternative to conventional residual stress evaluation techniques.

1. Introduction

In materials science, the nonlinear strain-hardening ability of metallic materials is a critical mechanical property that is indicative of their strength, hardness, and plastic deformation capacity [1]. This phenomenon is also referred to as work hardening or, given that the temperature at which deformation occurs is “cold” relative to the absolute melting temperature of the metal, as cold working. It is noteworthy that the majority of metals undergo strain hardening at ambient temperatures [2]. The most effective methods of hardening parts are based on Surface Plastic Deformation (SPD).

The employment of SPD has gained widespread popularity as a finishing and strengthening treatment due to its cost-effectiveness, labor efficiency, and minimal chip generation. The underlying principle of all SPD methods is the exploitation of metals’ plastic properties, which enable the acceptance of residual stresses without compromising the integrity and volume of the workpiece [3]. Notable methods within this category include pneumatic and hydroblasting hardening, as well as running-in with a ball or roller. In the context of hardening thin-walled, low-rigid parts for aviation applications, the thickness of the surface layer that undergoes plastic deformation is directly proportional to the thickness of the part itself.

Shot peening is a well-known cold-working surface treatment characterized by the chaotic contact of shots with the hardened surface of a part and is widely used to increase the fatigue resistance of metallic components, since it introduces compressive residual stresses on the surface of metals [4]. This is particularly true in the aircraft industry, where structural components made of high-strength aluminum alloys are shot-peened as a matter of course [5]. The effect of shot peening is generally neglected in the fatigue calculations (a choice in favor of component safety) or included in the form of empirical coefficients [6]. Residual stress is defined as the stress that is generated when the material is loaded beyond its elastic limit, causing the stresses to remain after the external load is removed [7].

It has been observed that the residual deformations that emerge during the manufacturing process frequently exceed the acceptable values that have been established subsequent to the redistribution of the hardening residual stresses. It is evident that the forecast of residual deformations caused by residual stresses is of significant importance in order to meet the strict tolerances of the components for gas turbine engines (GTEs). Consequently, studies have been conducted to investigate the deformation of thin components [8,9] and even new methods to manufacture them in order to improve parameters that are critical for these parts [10].

It is important to note that several experimental techniques are available for measuring residual stress. As stated in [11], the evaluation of residual stress is typically performed through the application of two primary methodologies.

The initial category comprises relaxation-based methods, also known as destructive techniques. The implementation of these methodologies requires the removal of material, a process that results in deformations. The pre-existing residual stress state can be deduced from these deformations. This methodology is generally applicable to linear-elastic materials. It only requires knowledge of the elastic constants, without further material-specific calibrations. However, it necessarily damages the specimen. Common techniques in this category include the sectioning technique and the contour method.

The second category comprises non-destructive testing methods, which rely on the measurement of physical phenomena that are influenced by the stress field. Such phenomena include interatomic spacing, as well as other indirectly correlated effects such as alterations in magnetic properties or vibrational spectra. The interpretation of these alterations can be used to determine the residual stress distribution. Representative methods in this category include Barkhausen noise analysis, X-ray diffraction, neutron diffraction, and ultrasonic techniques.

Additionally, a third category, frequently designated as semi-destructive methods, includes techniques such as hole-drilling, ring-core, and deep-hole. These methods involve the localized removal of material and are less invasive than fully destructive approaches. However, they still provide valuable stress profile information.

The work in [12] provides a comprehensive classification of the different residual stress measurement methods and offers an overview of some of the recent advances in this area. This overview aims to assist researchers in selecting the most suitable techniques for their specific applications, taking into account the availability of those techniques. The work also summarizes the scope, physical limitations, advantages, and disadvantages of the techniques.

Talking specifically about the shot-peening process, there are studies that employ different methodologies; for example, in [13], a prediction of the plastic deformation and residual stresses induced in metallic parts was made, with a predictive model as a function of the process parameters, resulting in good correlation with the residual stresses obtained by X-ray diffraction.

In the field of shot peening, a considerable body of literature has emerged on the subject of predicting residual stress. For instance, ref. [4] employed Moiré interferometry, a high-resolution optical technique that boasts a unique combination of high sensitivity and excellent contrast, range, and spatial resolution. This technique was demonstrated to be relatively accurate for measuring residual stress distributions.

The investigation [14] developed a mathematical model of the shape and stress–strain state of a steel plate subjected to shot peening. The investigation also performed experimental verification and numerically reconstructed the profiles of residual stresses and strains with reasonable accuracy using the experimental data. Furthermore, the investigation provided an explicit formula for the dependence of the residual plastic deformation along the thickness of the plate and also analyzed the sources of errors for such mathematical expressions.

The research in [15] focused on a thin-walled fan blade composed of titanium alloy (Ti–6Al–4V). Utilizing the Abaqus finite element analysis (FEA) software, the study simulated the deformation resulting from residual stress after shot peening, analyzing the correlation between residual stress and deformation. Subsequently, the FEA was employed to optimize the shot-peening parameters, with the objective of controlling the deformation of the blade, thereby ensuring the preservation of dimensional and shape accuracies. A comparison was then conducted between the measured and simulated values of residual stress, with the purpose of verifying the validity of the FEA model of shot peening. An experimental study demonstrated that the optimized parameters of shot peening led to a reduction in blade deformation, thereby confirming the suitability of the optimization method. This research is of great significance for the control of deformation in thin-walled structures post-shot peening.

The investigation in [16] predicts the deformation of a fan blade made of titanium alloy after machining. The residual stresses on the surface and subsurface of the blade after milling and shot-peening processes were measured and analyzed by experimental testing. The distribution of residual stress was analyzed and discretized according to separation layers, and the discrete results were applied to the blade finite element model (FEM) created in Abaqus software. This model was employed to analyze deformation by considering the effect of the established residual stress. The simulation deformation was then compared with the measured deformation, and the feasibility of the finite element analysis (FEA) model was verified. The study provides an effective analysis method for the deformation prediction of fan blades after milling and shot-peening processes.

The research in [17] mentions that the residual stresses in the machining of thin-walled parts can cause serious distortions and considers that the finite element cutting models for entire parts are time-consuming and redundant. Consequently, a method was proposed to map residual stress profiles to complicated surfaces of the entire model. This method involves a transformation from the local coordinate system to the global coordinate system. Typical residual stress profiles are added to the elements at surface layers of the model, according to their depths and directions. The simulation results demonstrate a strong agreement with the basic distortion magnitudes and linear superposition principles, thereby validating the feasibility of the proposed method.

The work in [18] proposes a mathematical model for predicting surface roughness and compressive residual stress distribution for abrasive waterjet peening. The authors of this work mention that the energy of the particles is partially transformed into workpiece plastic deformation, further implanting compressive residual stress. The model was validated by comparing the predicted and experimental compressive residual stresses. The findings indicate that the maximum discrepancy is 8.3%.

In the investigation made in [19], they made a computational model of bending deformation of peened plate samples to determine the deformation energy of the shot-peened plate by means of the total kinetic energy input due to shot impacts.

The work in [20] presented a numerical simulation of the shot-peening process using FEA performed by the software ABAQUS version 6.4.1. The feasibility of the model was demonstrated, as the resulting shot-peening residual stress and plastic deformation profiles were in good agreement with the experimental observations. It is important to note that, in order to simplify the application of the coupled elastic–plastic damage analysis, static analysis using energy equivalence was adopted. It is noteworthy that the dynamic analysis necessitates substantial computing time and often yields unstable results.

As previously evidenced by the methodologies employed to determine deformation caused by residual stress, all of them depend on experimental data to validate analytical models and/or finite element methods.

Despite significant progress in understanding the effects of hardening modes, particularly shot peening, on the stress–strain state of materials, the determination of residual stress remains a considerable challenge, especially when considering temporal factors. As stated in [21], conventional methods for measuring residual stress are often laborious and costly.

In response to the aforementioned challenges, the present study proposes an experimental wet etching technique, based on the principle of layer-by-layer material removal, integrated with a compliance function. This approach aims to determine changes in the size and shape of parts, which are consequence of stress redistribution during the shot-peening process.

According to the classification presented in [22], this wet etching technique is considered a relaxation method. In such methods, the relationship between the measured deformations and the residual stresses is expressed as an integral equation, which requires the application of an inverse method to evaluate the residual stress solution.

The proposed technique enables an expedite and more cost-effective evaluation of residual stress in comparison to traditional methods. Notably, this method allows the accurate determination of subsurface residual stress profiles, and a spatial resolution as fine as 1 μm can be accurately determined by this method. This makes it especially well suited for thin-walled, low-rigid parts. Furthermore, this manuscript undertakes an evaluation of the impact of initial stress on the stress–strain state of components, employing the finite element method as a methodological framework.

2. Materials and Methods

In order to accomplish the objective of determining the residual stresses and deformations induced by shot peening on a component, a series of procedures was executed. These procedures involved experimental activities, analytical calculations, and the utilization of a finite element model (FEM). This section outlines the methodology employed and provides an example of its application. Figure 1 illustrates the primary activities involved in the development of this methodology. These activities are described in brief in the subsequent paragraphs and in greater detail in their corresponding sections.

The initial step in the procedure involves the fabrication of the test specimens using the same manufacturing process as the final components. The manufacturing process entails the utilization of electrical discharge machining (EDM) to cut the raw material blanks, followed by milling and grinding of the workpiece. This replication is imperative to preserve the process inheritance, which exerts a direct influence on the stress–strain state of the specimens prior to the hardening treatment. Subsequent to the specimens’ fabrication, shot peening was applied.

The second step of the procedure involves experimental activities aimed at determining the residual stresses induced by the hardening process. The methodology employed in this study consisted in measuring the deformations that occurred during the layer-by-layer etching of the hardened surface layer of the specimens, resulting in a deformation curve. This curve is indicative of the redistribution of internal stresses that occurs during the removal of successive layers.

The third step involves the analytical determination of the initial stress distribution based on the residual stresses obtained from the surface-hardened layer. The residual stress profile, derived from the layer-by-layer etching experiment, is recalculated by processing the measured deformation curve.

The fourth step in the process is the finite element analysis (FEA). In this step, the geometry of the specimens is meshed, and the surface-hardened layer is simulated within the model. The model is configured with appropriate boundary conditions, and the initial stress distribution obtained in the previous step is applied to the surface layer.

Finally, a comparison of the deflection results obtained from the experimental, analytical, and numerical analyses is conducted to establish a correlation.

2.1. Characteristics and Creation of the Samples

To illustrate the developed method for modeling stress redistribution during shot peening based on initial stresses, physical samples were produced. In this study, both flat and circular specimens were selected for analysis. The utilization of flat specimens in this study facilitates a straightforward analysis of surface deformation and residual stress distribution. Conversely, circular specimens are more accurate representations of components with axial symmetry. This dual approach enhances the relevance and applicability of the findings to real-world engineering contexts. Furthermore, the samples are regarded as representative due to the existence of analytical solutions for both sample types. Consequently, a comparison of the results of these analytical solutions with finite element calculations can be made. This method can be employed to ascertain the precision of calculations in a FEA software such as ANSYS. The characteristics of the samples are presented below.

2.1.1. Characteristics of Flat Samples

The material selected for the flat specimens is a heat-resistant nickel-based superalloy EI 698 VD, which is similar to Nimonic^® 90 [23]. The EI 698 VD is used for turbine disks, in particular for the NK-8 family of engines for civil aviation [24], and the chemical composition of this material is presented in

Table 1. Chemical composition as percentages of EI 698 VD nickel-based superalloy.

Cr	Mo	Ti	Fe	Nb
13–16	2.8–3.2	2.35–2.75	2 max.	1.9–2.2
Si	C	P	S	Ni
0.5 max.	0.03 max.	0.015 max.	0.007	Base

as percentages.

The material properties employed in the analysis were extracted from [25], and for a temperature of 20 °C the material properties are the following:

• Density, 8320 kg/m³.
• Young’s modulus, 200 GPa.
• Poisson’s ratio, 0.3.

The dimensions of the samples are the following:

• Length l = 60 mm.
• Width b = 8 mm.
• Height h = 3 mm.

2.1.2. Creation of Flat Samples

Following the fabrication of the flat samples using the previously mentioned manufacturing processes and the final dimensions being achieved, a hardening process was initiated. The shot-peening application process involved the utilization of a robotic pneumatic blasting machine (Rosler) with the following operational parameters:

• Material of the shot: steel.
• Shot diameter ds = 315 μm.
• Air pressure P_air = 0.3 MPa.
• Nozzle movement speed vnm = 30 mm/min.
• Processing time t = 8 min.

2.1.3. Characteristics of Ring Samples

The ring specimens were created from carbon steel 45 (SM45C). Carbon steel 45 is typically employed in gears that are not subjected to thermal stress. The material properties utilized in the analysis were extracted from [26]. The most significant properties are listed below:

• Density, 7600 kg/m³.
• Young’s modulus, 207 GPa.
• Poisson’s ratio, 0.3.

The dimensions of the samples are the following:

• Average radius R = 43.6 mm.
• Thickness h = 3.7 mm.
• Width b = 10 mm.

2.1.4. Creation of Ring Samples

The ring samples were fabricated in accordance with the same methodology employed for the flat samples. Subsequent to the manufacturing process, a hardening procedure was executed. This procedure involved the utilization of the same blasting machine that was employed for the flat samples. However, the operational parameters of this machine were adjusted as follows:

• Material of the shot: steel.
• Shot diameter ds = 1.4–1.8 μm.
• Air pressure P_air = 0.35 MPa.
• Nozzle movement speed vnm = 30 mm/min.
• Processing time t = 8 min.

The hardening process was applied to the outer surface, which had a width of 5 mm (see Figure 2). Following the completion of the shot-peening process, the ring was cut into three sections of 120° each to facilitate the etching procedure.

2.2. Experimental Procedures

To evaluate the deflection of the samples, they were subjected to a layer-by-layer electrolytic etching process applied to the surface-hardened layer. This procedure leads to a redistribution of residual stresses, resulting in measurable deformation of the specimen. The residual stress profile was subsequently obtained by recalculating the deformation curve recorded during the etching process.

The deflection, defined as the displacement of the free end in a cantilever mounting configuration, was determined based on sensor readings collected throughout the etching sequence.

The etching process was completed in 1.5 h with a solution of sulfuric acid, phosphoric acid, and water (H₂SO₄ + H₃PO₄ + H₂O). The etching process was carried out with an automated system for determining residual stresses in complex specimens (ASB-1) with the following parameters:

• Current density = 20 A/dm².
• Electric current = 1.6 A.
• Voltage = 30 W.

As illustrated in Figure 3, a flat sample is mounted in the ASB-1 equipment prior to the initiation of the etching process. The ASB-1 equipment facilitates precise regulation of the aforementioned etching parameters, thereby ensuring repeatability and reliability in the experimental procedure. This is a critical component for the subsequent analysis of residual stress distribution.

2.3. Analytical Methods to Determine the Initial Stresses

In order to evaluate the deformations induced by residual stresses in a surface-hardened component, it is first necessary to determine the initial stresses. These stresses function as the mechanical baseline prior to the onset of plastic deformation, which is caused by the hardening process. This section presents the analytical approaches that were employed to estimate the initial stress field. These approaches consist in classical methods of elasticity theory and mechanics of deformed solids. These approaches will subsequently be incorporated into a finite element model described in the following section. This model will simulate stress redistribution and resulting strain fields.

As a preliminary step, the initial stresses are employed to estimate the stress–strain state of the surface-hardened layer, neglecting both stress redistribution and the presence of residual stresses [27,28,29,30]. The distinction between initial and final stresses within the hardened surface enables the identification of the stress redistribution mechanism that occurs during the hardening process.

At present, two well-established methodologies are employed to determine the initial stress from the residual stress data. The first approach involves the numerical solution of an integral equation, as outlined in [28]. The second approach employs the method of successive approximations, detailed in [29,30], which allows the use of data obtained through experimental means.

For the method of successive approximations, the initial stress value is determined by the technique of layer-by-layer etching applied to the hardened surface of rectangular samples. This process leads to the redistribution of residual stresses and deformation of the sample under study. The residual stress profile is determined by recalculating the deformation curve, which is obtained during the etching process.

It is essential that the thickness of the sample, designated as “h”, be as proximate as possible to the thickness of the hardened component, denoted as “h_p”. In the event that a/h_p ≤ 0.1 (where “a” represents the thickness of the hardened layer), the minimum value of “h” must be at least 2.5 ÷ 3 mm.

2.3.1. Initial Stresses for Flat Samples

In the case of flat samples, the initial stress σ_I(ξ) is dependent on the deformation ξ, which is based on residual stress σ_R(ξ). This dependence is determined by the method of successive approximations [30] and can be expressed as follows:

$${\sigma }_{I,i}={\sigma }_R\left(\xi \right)+f_{p,i-1}\left(\xi \right){\int }_0^{a_{i-1}}{\sigma }_{I,i-1}\left(\xi \right)d\xi , i=1, 2,\dots$$

(1)

where $f_{\mathrm{p}}\left(\xi \right)$ is a coefficient and is given by:

$$f_{\mathrm{p}}(\xi )={\displaystyle \frac{3(h-2\xi )(h-2{\xi }_{\mathrm{c}})+h^2}{h^3}}$$

(2)

${\xi }_L$ is the deformation of the layer:

$${\xi }_L\approx {\displaystyle \frac{{\int }_0^a{\sigma }_R\mathsf{\xi }\mathrm{d}\xi }{{\int }_0^a{\sigma }_R\mathrm{d}\mathsf{\xi }}}$$

(3)

For Equation (1), under the condition that i = 1, i − 1 = 0, the result is σ_I(ξ) = σ_R(ξ).

The thickness “a” of the hardened surface layer can be determined in two ways. The first method involves the use of the initial stress diagram σ_I(ξ) to identify its intersection with the deformation axis ξ. The second method involves measuring microhardness on oblique sections of a layer that have undergone plastic deformation.

2.3.2. Initial Stresses for Ring Samples

In the case of ring components, the determination of residual stress typically involves the utilization of rectangular cross-section ring samples. Given the absence of bending deformations in the radial direction of the ring components, the initial stress is unloaded exclusively by tensile force factors. In such instances, the expression for f_p is therefore as follows:

$$f_p={\displaystyle \frac{1}{h-\xi }}$$

(4)

where “h” represents the thickness of the ring in the radial direction.

Based on Equations (1) through (4), an initial stress program was developed, which was presented in [31]. The calculations have demonstrated that the number of necessary approximations is significantly dependent on the a/h ratio. In instances where the ratio a/h < 0.05, the difference between residual stress and initial stress can be disregarded. It is anticipated that the deviations will be within the range of 3 to 5%. In such a scenario, residual stress can be employed for the calculation of residual deformations due to manufacturing.

2.4. The Use of Initial Stress in the FEA

The methodology for modeling stress redistribution during shot peening, developed in this study, is based on the concept of initial stresses. This approach requires the construction of an FEM to calculate the deformations in a given part, caused by residual stresses in the surface-hardened layer. To exemplify the method, an FEM was constructed for the two types of samples: a flat specimen and a ring specimen. The FEM was created through the utilization of ANSYS Mechanical APDL 2021 R1.

The construction of the models was based on the actual dimensions previously mentioned and the corresponding material properties assigned. The volume of the models was divided in order to allocate the hardened layer into a separate volume. The dimensions of the hardened layer correspond to the intersection of the initial stresses with the deformation axis in the residual stress distribution diagram, as presented in the Results and Discussion section.

With regard to the mesh of the volumes, the surface layer was meshed with 3-D eight-node structural solid shell elements (SOLSH190), while the remaining body was meshed with 3-D structural solid elements (SOLID45).

Subsequent to the meshing of the surface layer, the shell section is utilized to establish the required number of layers and the number of integration points within it. The thickness of each layer in the shell section is set in accordance with the total thickness of the sample surface layer.

The initial stress diagram was obtained by converting the residual stress diagram in accordance with the program presented in the referenced work [31] and subsequently loading it into the surface layer of the model, with the command “Inistate”. The number of points defining the initial stress diagram is determined based on the number of layers and integration points within each layer. To define the initial stress diagram, a total of nine points with four layers and three integration points per layer were required. The first point was located at the base of the surface layer, while the final point is on the surface. The initial stress diagram is then subdivided into nine points along the thickness “a”, as is illustrated in Figure 4. This approach facilitates the determination of the initial stress values at each point.

In the event that the surface layer is not continuous and becomes detached, for instance, in a rectangular-section sample along a longitudinal axis, the corresponding residual stresses on the surface may be absent. In such cases, it is necessary to account for the edge effect. This effect is referred to as the localized disturbance in the stress distribution near the boundaries of a component. It is attributed by the absence of surrounding material and altered boundary conditions, resulting in stress relaxation or concentration that deviates from the interior stress distribution. These effects can significantly influence the accuracy of experimental measurements, as the neglect of edge effects may result in unrealistic predictions. In thin-walled or geometrically complex components, particularly in aerospace applications, properly accounting for the edge effect is essential to ensure reliable residual stress evaluation and to avoid underestimating structurally critical zones.

The extent of the edge effect zone can be assumed to be approximately equal to the layer thickness [31]. Consequently, when meshing the surface layer for the given sample, it is imperative to select one row of stress-free Sx elements on each side.

Once the stresses have been loaded into the surface layer, the boundary conditions are specified in the model, which involves rigid fixation. For instance, in the case of calculating a disk, the hub should be fixed; similarly, when calculating a blade, the attachment should be fixed.

Upon completion of the calculation, the output data set comprises the movements of the nodes.

2.4.1. FEA for Flat Samples

The resulting model of the flat specimens is a parallelepiped with a separated volume that simulates the hardened surface layer, with its corresponding thickness. This thickness is obtained from the residual stress diagram, resulting from the experimental testing, that is presented in the Results and Discussion section in Figure 8 and is equal to a = 169.6 μm. Within the surface layer, the edge effect zones are designated as separate volumes, with a width equal to the thickness of the surface layer, as presented in the work in [32].

With regard to the mesh of the flat specimen, particular emphasis was placed on adjusting the element spacing across the model’s thickness. This adjustment was implemented to ensure that the elements near the surface layer maintained a comparable aspect ratio. Consequently, the element size gradually increased. The mesh configuration is illustrated in Figure 5, in which the elements corresponding to the surface-hardened layer are highlighted in red. It is also evident that the hardened layer does not extend across the entire top surface. This was intentional, as the simulation was designed to closely replicate the actual sample. In this sample, the region without the hardened layer corresponds to the area where the sensor is placed to measure deflection during the etching process.

The initial stress values were preliminarily calculated based on the same residual stress diagram presented in Figure 8 and loaded onto the surface layer.

Two variants were created for the flat specimen FEM. The first variant incorporates the edge effect; meanwhile, the second variant does not. In the case that the edge effect is considered, the initial stress diagram is not loaded into the inclusion zone.

The boundary conditions for the flat samples were defined to simulate a cantilever beam configuration, with one end fixed and the rest of the body free. This configuration was selected to replicate the actual experimental conditions, as previously mentioned. As illustrated in Figure 6, all six degrees of freedom—translations (Ux, Uy, Uz) and rotations (ROTx, ROTy, ROTz)—were constrained to zero at the front face of the flat sample.

2.4.2. FEA for Ring Samples

For the ring specimen, the model was constructed based on a cylindrical coordinate system. The corresponding thickness for the hardened surface layer, according to the residual stress diagram, resulting from the experimental testing, that is presented in Figure 9 in the Results and Discussion section, is a = 276 μm. The initial stress values were preliminarily calculated based on the same residual stress diagram presented in Figure 9 and loaded onto the surface layer along the Y axis, which represents the circumferential direction in the cylindrical coordinate system.

The mesh of the ring model can be observed in Figure 7, where the surface layer elements are highlighted in red. Two variants were created for the ring specimen FEM. The first variant incorporates the edge effect; meanwhile, the second variant does not. In the case that the edge effect is considered, the initial stress diagram is not loaded into the inclusion zone.

With respect to the boundary conditions of the ring model, two nodes of the model were constrained with displacements Ux, Uy, and Uz being equal to zero; these nodes are located at the top and bottom faces of the ring, which simulate a fix grip the same as the experimental procedure. This is illustrated in Figure 7.

3. Results and Discussion

A comparative analysis of experimental, analytical, and FEA results is conducted to facilitate a comprehensive understanding of the impact of residual stresses and initial stresses from diagrams on the geometric parameters of flat and ring samples.

3.1. Results for Flat Specimens

The experiment measurements were conducted on four samples with the characteristics previously mentioned. The residual stresses in the surface layer of the hardened samples were determined. The diagram depicting the average residual stress resulting from hardening with the aforementioned parameters is presented in Figure 8.

The resulting measurement deflection from the experiment was found to be f ≈ 251 μm, as presented in

Table 2. Resulting calculated deflections for the flat specimens.

Category	Method	Stresses Integral, MPa·m	f, mm	Difference vs. Experimental Results %
Experimental	Deflection value fexp	-	0.251	-
Analytical	Per residual stress (ΦR)	−0.0667	0.234	6.8
Analytical	Per initial stress (ΦI)	−0.0843	0.248	1.2
FEM	Considering edge effect (ΦI)	−0.0843	0.264	5.2
FEM	Not considering edge effect (ΦI)	−0.0843	0.275	9.6

.

The analytical calculation of the integral values for residual stress and initial stress was completed in accordance with the methodology described in [28,29] using the program provided in [31]. The results of the analytical calculation for different methods are presented in Table 2.

The results of the specimen deflection calculations from the FEA are presented in Table 2.

The difference between the calculated and experimental values of displacement is less than 5%, which can be attributed to potential errors in the measurements of experimental deflections, the determination of the residual stress, and the layer thickness.

3.2. Results for Ring Specimens

To determine the residual stress, three samples with the characteristics mentioned previously were simultaneously subjected to a hardening process. The residual stress diagram is presented in Figure 9. The results of the measurements of ring deformations after shot peening, along with the results of the analytical calculations, are shown in

Table 3. Calculated deformation results for a ring-shaped specimen after hardening treatment.

Category	Method	Stresses Integral, MPa·m	Diameter Difference ΔD, mm (δ, %)
Experimental	Deflection value fexp		0.005
Analytical	Per residual stress (ΦR)	−0.0785	0.0048 (4%)
Analytical	Per initial stress (ΦI)	−0.0848	0.0050 (0%)
FEM	Considering edge effect (ΦI)	−0.0848	0.0052 (4%)
FEM	Not considering edge effect (ΦI)	−0.0848	0.0054 (10.8%)

Note 1. The term “δ”, which is defined in brackets, signifies the discrepancy between the calculated values and the experimental results.

.

The residual deformations calculation for the FEM of ring-shaped parts was based on both experimental data and analytical calculations as mentioned in the work in [30].

4. Conclusions

The convergence of analytical calculation and FEA results validates the suitability of the process of modeling the initial stress redistribution and residual stress formation using an FEM.

The preceding evidence substantiates the viability of incorporating initial stress into the calculations of the stress–strain state and residual deformations of parts with an arbitrary geometric shape, including thin-walled low-rigid components with marked dimensional and shape variations.

The employment of calculations founded on the finite element method has been demonstrated to markedly improve the precision of stress–strain state analysis for components subjected to a variety of conditions and processing methods.

The initial stress redistribution and residual stress formation are assessed for simplification without consideration of the time factor (not instantaneously), a simplification that significantly facilitates the hardening calculations and modeling process.

The findings indicate that incorporating the edge effect in the FEM substantially reduces the discrepancy between the calculated results and the experimental values.

Subsequent research will concentrate on improving the convergence of experimental data and calculations in ANSYS. Furthermore, analysis of other aviation materials, such as titanium alloy VT-22 and aluminum alloys B93 and B95, will be conducted.