# Optimization of Forging Process Parameters and Prediction Model of Residual Stress of Ti-6Al-4V Alloy

Authors: Xiurong Fang; Liu Liu; Jia Lu; Yang Gao
Journal: Advances in Materials Science and Engineering
 (2021)

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## Abstract

Nonisothermal forging is an efficient plastic forming method for titanium alloys, but at the same time, it can produce large and uneven residual stress, which seriously affects the service life of components. In order to quantitatively analyze the influence of forging process parameters on the residual stress of Ti-6Al-4V alloy forgings, a numerical model was first established and optimized in combination with experiments. Then, the effects of deformation temperature, deformation degree, and deformation speed on the residual stress of forgings were analyzed by orthogonal test, and the optimal combination of forging process parameters was obtained. Finally, the multiple regression analysis was employed to propose multivariate regression models for the prediction of the average equivalent residual stress. Results show that the prediction model can be used for predicting the residual stress of Ti-6Al-4V alloy forgings with a higher reliability.


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## Body

## 1. Introduction

Titanium and its alloys are often selected for use in high-value structural components in a number of applications due to their high specific strength, relatively low density, excellent corrosion resistance, and fatigue resistance [1–3]. Aerospace industry accounts for the majority of titanium consumption in the world, and more than 80% of titanium is used to produce a variety of alloys, among which Ti-6Al-4V is the most widespread variant [4]. In order to ensure the mechanical properties and dimensional stability of related parts, nonisothermal forging is usually used to process them [5]. Nonisothermal forging is a deformation method with high strain rate, which can greatly improve the mechanical properties of titanium alloys. However, residual stress often occurs in the forging process due to thermal conductivity, high deformation resistance, and narrow forging temperature range of Ti-6Al-4V alloy [6, 7]. In the process of machining, residual stress will cause the forging to deform and even crack, which reduces the qualified rate [8, 9]. In the service process, residual stress will affect the mechanical properties of components and reduce their service life [10, 11]. Therefore, it is necessary to reduce the residual stress of Ti-6Al-4V alloy forgings in the forging process.At present, many researchers have conducted a lot of studies on reducing residual stress. Araghchi et al. [12] studied the effect of different concentrations of quenchant on the residual stress of the alloy. The results showed that quenching in 15% polymer solution and aging at 190°C for 12 h could reduce the residual stress, but the mechanical properties of the alloy decreased. Robinson et al. [13] analyzed the influence of quenching temperature on residual stress, and the results showed that quenching had little effect on residual stress when the quenching temperature was low, but the residual stress of the alloy could be reduced when the water temperature was above 100°C. Ya Bo Dong et al. [14] analyzed the effect of heat treatment on the residual stress of forged plate and improved the balance of mechanical properties and residual stress of forged plate by changing the heat treatment process parameters. It can be seen that most researchers reduce the residual stress of forgings by adjusting the heat treatment process parameters. However, as far as the present technology is concerned, it is impossible to completely eliminate the residual stress of forgings by heat treatment [15]. In addition, it was found that the qualified rate of a certain type of blade is increased by 20%–30% after heat treatment and the adjustment of cutting process parameters, but there are still a lot of scrapped products due to deformation, and the total qualified rate is only 70%. After the analysis of the scrapped products, it was found that forging process parameters have a great influence on the residual stress of forgings. In addition, Karunathilaka et al. [16] analyzed the influence of lubrication and forging load on residual stress and found that there is a positive relationship between the forging load and residual stress. Ameli et al. [17] investigated the effects of process parameters such as workpiece and die geometries, percentage of deformation, and workpiece motions on residual stresses. Ye Zhang et al. [18] analyzed the influences of different process parameters on residual stress during turbine disc hot die forging by numerical simulation. They all found that the key forging process parameters have a considerable influence on the residual stress of forgings [19]. Therefore, it is of great practical significance to reduce the residual stress and improve the distribution of residual stress in forgings by optimizing the forging process parameters.In this paper, the residual stress of Ti-6Al-4V alloy forgings under different forging processes (deformation temperature, deformation degree, and deformation speed) were analyzed based on the finite element method, and the optimal combination of forging process parameters was obtained through the orthogonal test. Then, a mathematical model for predicting the average equivalent residual stress was established by regression analysis of the simulation data, which can provide effective guidance for the actual forging process.


## 2. Calculation Model

### 2.1. Establishment of the Simulation Model

In the process of numerical simulation, the blank is defined as an elastic-plastic body because of the elastoplastic transformation of materials while the dies are defined as a rigid body. In addition, due to the temperature difference between the blank and the dies, as well as between the blank and the air, the simulation control parameters such as the heat conduction between the blank and the dies, the heat convection between the blank and the air, and the heat radiation of the blank should be considered when establishing the simulation model. The heat transfer of forgings is shown in Figure1 [20].Figure 1
Heat transfer of forgings.The geometric models of the upper and lower dies and the forging blank were established using three-dimensional drawing software, as shown in Figure2(a). The size of the forging blank is Φ60 mm × 100 mm. Considering the lengthwise symmetrical characteristic and the complexity of the elastic-plastic finite element analysis, one fourth of the forging billet was taken for analysis, and the finite element model is shown in Figure 2(b).Figure 2
Simulation model of forging. (a) Geometric model. (b) Finite element model.
(a)(b)The accuracy of the simulation results is affected by the control parameters. Therefore, in order to obtain more accurate control parameters, this paper combines physical experiments and numerical simulation to analyze the residual stress of Ti–6Al–4V alloy forgings after nonisothermal forging [21]. Firstly, the forging experiment and residual stress detection test were carried out to obtain the residual stress of the forging. Two sets of different forging process parameters were selected (one set is deformation temperature 975°C, deformation degree 50%, and deformation speed 10 mm/s, and the other set is deformation temperature 925°C, deformation degree 50%, and deformation speed 20 mm/s). Five experimental samples were taken under each set of process parameters, and the average value of residual stress was taken as the final result. The specific experimental process is shown in Figure 3 [20]. Then, the corresponding numerical simulation was carried out. Based on the forging residual stress measured in physical experiment, the error between experimental results and simulation results was analyzed. Finally, the simulation control parameters were adjusted until the residual stress values of the simulation are as close as possible to the experimental measurements. The final simulation control parameters are shown in Table 1.Figure 3
Physical experiment process [20].Table 1
Forging simulation parameters.
Simulation parametersValueSimulation parametersValueEnvironment temperature (°C)20Thermal radiation rate0.7Die temperature (°C)300Step (mm)0.25Friction coefficient0.3Thermal convection coefficient (N s−1 mm−1 °C−1)0.02Thermal conductivity of the bottom die (N s−1 mm−1 °C−1)1Thermal conductivity of the forging (N s−1 mm−1 °C−1)5


### 2.2. Results’ Analysis

In order to intuitively show the distribution of residual stress of Ti-6Al-4V alloy forgings under nonisothermal forging, a set of forging process parameters (deformation temperature 925°C, deformation degree 50%, and deformation speed 100 mm/s) was selected for simulation analysis according to the engineering practice, and the result is shown in Figure4.Figure 4
Distribution of equivalent residual stress.In order to further reveal the distribution characteristics of residual stress, 11 tracking points with a spacing of 5 mm were selected from the upper surface to the lower surface along the central axis of the forging. The distribution of the residual stress at the tracking points along the XYZ direction is shown in Figure5. It can be seen from Figure 5 that the distribution characteristics of the residual stress in the X and Y directions of the forging are basically similar, which are symmetrical but uneven about the center of the axis. In the direction of the central axis, the residual stress on the surface of the forging is tensile stress while the center is compressive stress. From the surface to the core, the tensile stress is transformed into compressive stress. The residual stress in Z direction is less than that in X and Y directions, and the distribution is more uniform.Figure 5
Distribution of residual stress in XYZ direction of the forging at different depths.


## 2.1. Establishment of the Simulation Model

In the process of numerical simulation, the blank is defined as an elastic-plastic body because of the elastoplastic transformation of materials while the dies are defined as a rigid body. In addition, due to the temperature difference between the blank and the dies, as well as between the blank and the air, the simulation control parameters such as the heat conduction between the blank and the dies, the heat convection between the blank and the air, and the heat radiation of the blank should be considered when establishing the simulation model. The heat transfer of forgings is shown in Figure1 [20].Figure 1
Heat transfer of forgings.The geometric models of the upper and lower dies and the forging blank were established using three-dimensional drawing software, as shown in Figure2(a). The size of the forging blank is Φ60 mm × 100 mm. Considering the lengthwise symmetrical characteristic and the complexity of the elastic-plastic finite element analysis, one fourth of the forging billet was taken for analysis, and the finite element model is shown in Figure 2(b).Figure 2
Simulation model of forging. (a) Geometric model. (b) Finite element model.
(a)(b)The accuracy of the simulation results is affected by the control parameters. Therefore, in order to obtain more accurate control parameters, this paper combines physical experiments and numerical simulation to analyze the residual stress of Ti–6Al–4V alloy forgings after nonisothermal forging [21]. Firstly, the forging experiment and residual stress detection test were carried out to obtain the residual stress of the forging. Two sets of different forging process parameters were selected (one set is deformation temperature 975°C, deformation degree 50%, and deformation speed 10 mm/s, and the other set is deformation temperature 925°C, deformation degree 50%, and deformation speed 20 mm/s). Five experimental samples were taken under each set of process parameters, and the average value of residual stress was taken as the final result. The specific experimental process is shown in Figure 3 [20]. Then, the corresponding numerical simulation was carried out. Based on the forging residual stress measured in physical experiment, the error between experimental results and simulation results was analyzed. Finally, the simulation control parameters were adjusted until the residual stress values of the simulation are as close as possible to the experimental measurements. The final simulation control parameters are shown in Table 1.Figure 3
Physical experiment process [20].Table 1
Forging simulation parameters.
Simulation parametersValueSimulation parametersValueEnvironment temperature (°C)20Thermal radiation rate0.7Die temperature (°C)300Step (mm)0.25Friction coefficient0.3Thermal convection coefficient (N s−1 mm−1 °C−1)0.02Thermal conductivity of the bottom die (N s−1 mm−1 °C−1)1Thermal conductivity of the forging (N s−1 mm−1 °C−1)5


## 2.2. Results’ Analysis

In order to intuitively show the distribution of residual stress of Ti-6Al-4V alloy forgings under nonisothermal forging, a set of forging process parameters (deformation temperature 925°C, deformation degree 50%, and deformation speed 100 mm/s) was selected for simulation analysis according to the engineering practice, and the result is shown in Figure4.Figure 4
Distribution of equivalent residual stress.In order to further reveal the distribution characteristics of residual stress, 11 tracking points with a spacing of 5 mm were selected from the upper surface to the lower surface along the central axis of the forging. The distribution of the residual stress at the tracking points along the XYZ direction is shown in Figure5. It can be seen from Figure 5 that the distribution characteristics of the residual stress in the X and Y directions of the forging are basically similar, which are symmetrical but uneven about the center of the axis. In the direction of the central axis, the residual stress on the surface of the forging is tensile stress while the center is compressive stress. From the surface to the core, the tensile stress is transformed into compressive stress. The residual stress in Z direction is less than that in X and Y directions, and the distribution is more uniform.Figure 5
Distribution of residual stress in XYZ direction of the forging at different depths.


## 3. Optimization of Forging Process Parameters Based on Orthogonal Test

### 3.1. Orthogonal Test Factors

In nonisothermal forging, it is necessary to select the appropriate forging process parameters for analysis to correctly guide the actual processing. According to the characteristics of Ti-6Al-4V alloy, such as transformation temperature (985°C), forging method, allowable forging deformation degree, and speed, the deformation temperature, deformation degree, and deformation speed were determined as the key process parameters. Through further research, it was found that these parameters have a great influence on the residual stress of forgings [20].


### 3.2. Orthogonal Test Schemes

Taking deformation temperature, deformation degree, and deformation speed as test factors, five levels are set for each factor, as shown in Table2.Table 2
Level table of orthogonal test factors.
LevelsFactorsT (deformation temperature (°C))D (deformation degree (%))S (deformation speed (mm·s−1))1925301029504050397550100410006050051025701000In accordance with theL25 (53) orthogonal table, 25 sets of orthogonal test schemes are established, as shown in Table 3, where T is deformation temperature, D is deformation degree, S is deformation speed, and σ is the average equivalent residual stress.Table 3
Orthogonal test schemes.
Test no.FactorsTest indexTDSσ (MPa)11 (925）1 (30）1 (10）14.521 (925）2 (40）2 (50）18.331 (925）3 (50）3 (100）18.141 (925）4 (60）4 (500）17.451 (925）5 (70）5 (1000）18.962 (950）1 (30）2 (50）12.172 (950）2 (40）3 (100）15.382 (950）3 (50）4 (500）15.492 (950）4 (60）5 (1000）14.3102 (950）5 (70）1 (10）20.8113 (975）1 (30）3 (100）11.8123 (975）2 (40）4 (500）11.4133 (975）3 (50）5 (1000）11.8143 (975）4 (60）1 (10）19.7153 (975）5 (70）2 (50）18.5164 (1000）1 (30）4 (500）9.79174 (1000）2 (40）5 (1000）9.04184 (1000）3 (50）1 (10）10.7194 (1000）4 (60）2 (50）10.7204 (1000）5 (70）3 (100）11215 (1025）1 (30）5 (1000）8.04225 (1025）2 (40）1 (10）8.58235 (1025）3 (50）2 (50）9.22245 (1025）4 (60）3 (100）8.83255 (1025）5 (70）4 (500）9.2


### 3.3. Orthogonal Test Results

Range analysis is usually used to analyze the results of an orthogonal test [22]. It can be described as shown in the following equation [23]:(1)Ki=1n∑j=1nEj,(2)R=Kimax−Kimin,where Ej is the value of a certain factor, Ki is the average of each factor, and R is the range and used to estimate design variable sensitivities [24]. In general, the greater the Ki value, the higher the indicator values under this level, and if the value of R is larger, the factor value would be more influential [25]. The range analysis of the orthogonal test results was calculated according to equations (1) and (2), as shown in Table 4Table 4
Influence analysis of parameters on the residual stress.
ParametersTDSK117.44011.24614.856K215.58012.52413.764K314.64013.04413.006K410.24614.18612.638K58.77415.68012.416R8.6664.4322.440Sorting123It can be clearly found from Table4 that the primary and secondary relationship of factors affecting the residual stress of the forging is deformation temperature, deformation degree, and deformation speed. And, their contribution rates to the residual stress of forgings are 56%, 28%, and 16%, respectively. Comparing the minimum values of K1∼K5 under each factor, the optimal combination of process parameters with the smallest residual stress can be determined as deformation temperature 1025 °C, deformation degree 30%, and deformation speed 1000 mm/s.


### 3.4. Numerical Simulation of Optimized Process Parameters

The forging process of Ti-6Al-4V alloy was simulated under the optimal process parameters of deformation temperature 1025 °C, deformation degree 30%, and deformation speed 1000 mm/s. The results are shown in Figure 6. It can be seen from Figure 6 that the residual stress of the forgings under the optimal combination of process parameters basically belongs to normal distribution, without obvious stress concentration, and the distribution is relatively uniform. This is because when the deformation temperature (1025 °C) far exceeds the transformation temperature of Ti-6Al-4V alloy (985°C), the forging is completed in the β single-phase region. The plastic forming ability of the forgings is significantly improved and the deformation resistance is greatly reduced, which greatly improves the deformation uniformity. Furthermore, when the deformation degree is 30%, the main softening mechanism of Ti-6Al-4V alloy forgings is dynamic recovery. The residual stress of the forgings decreases sharply during the recovery stage. In addition, the deformation temperature is high and the degree of deformation is small at this time, which reduces the heat converted and weakens the thermal effect of metal deformation, thereby greatly reducing the residual stress of the forging and improving the distribution of the residual stress.Figure 6
Distribution of forging residual stress under the optimal process parameters. (a) Equivalent residual stress. (b) Residual stress in theX direction. (c) Residual stress in the Y direction. (d) Residual stress in the Z direction.
(a)(b)(c)(d)However, when the forging temperature is above theβ-transition temperature, there are a lot of coarse β-grains in the forgings. The coarser the grain, the more uneven the plastic deformation and the greater the internal stress concentration. Moreover, the coarse grains reduce the chance of crisscross between grains, which will be conducive to the propagation and development of cracks, and the strength and toughness will become worse. So, the mechanical properties will be reduced. Therefore, for the parts that require high-dimensional stability but not too much mechanical properties, this combination of forging parameters can be used as an effective guide.


## 3.1. Orthogonal Test Factors

In nonisothermal forging, it is necessary to select the appropriate forging process parameters for analysis to correctly guide the actual processing. According to the characteristics of Ti-6Al-4V alloy, such as transformation temperature (985°C), forging method, allowable forging deformation degree, and speed, the deformation temperature, deformation degree, and deformation speed were determined as the key process parameters. Through further research, it was found that these parameters have a great influence on the residual stress of forgings [20].


## 3.2. Orthogonal Test Schemes

Taking deformation temperature, deformation degree, and deformation speed as test factors, five levels are set for each factor, as shown in Table2.Table 2
Level table of orthogonal test factors.
LevelsFactorsT (deformation temperature (°C))D (deformation degree (%))S (deformation speed (mm·s−1))1925301029504050397550100410006050051025701000In accordance with theL25 (53) orthogonal table, 25 sets of orthogonal test schemes are established, as shown in Table 3, where T is deformation temperature, D is deformation degree, S is deformation speed, and σ is the average equivalent residual stress.Table 3
Orthogonal test schemes.
Test no.FactorsTest indexTDSσ (MPa)11 (925）1 (30）1 (10）14.521 (925）2 (40）2 (50）18.331 (925）3 (50）3 (100）18.141 (925）4 (60）4 (500）17.451 (925）5 (70）5 (1000）18.962 (950）1 (30）2 (50）12.172 (950）2 (40）3 (100）15.382 (950）3 (50）4 (500）15.492 (950）4 (60）5 (1000）14.3102 (950）5 (70）1 (10）20.8113 (975）1 (30）3 (100）11.8123 (975）2 (40）4 (500）11.4133 (975）3 (50）5 (1000）11.8143 (975）4 (60）1 (10）19.7153 (975）5 (70）2 (50）18.5164 (1000）1 (30）4 (500）9.79174 (1000）2 (40）5 (1000）9.04184 (1000）3 (50）1 (10）10.7194 (1000）4 (60）2 (50）10.7204 (1000）5 (70）3 (100）11215 (1025）1 (30）5 (1000）8.04225 (1025）2 (40）1 (10）8.58235 (1025）3 (50）2 (50）9.22245 (1025）4 (60）3 (100）8.83255 (1025）5 (70）4 (500）9.2


## 3.3. Orthogonal Test Results

Range analysis is usually used to analyze the results of an orthogonal test [22]. It can be described as shown in the following equation [23]:(1)Ki=1n∑j=1nEj,(2)R=Kimax−Kimin,where Ej is the value of a certain factor, Ki is the average of each factor, and R is the range and used to estimate design variable sensitivities [24]. In general, the greater the Ki value, the higher the indicator values under this level, and if the value of R is larger, the factor value would be more influential [25]. The range analysis of the orthogonal test results was calculated according to equations (1) and (2), as shown in Table 4Table 4
Influence analysis of parameters on the residual stress.
ParametersTDSK117.44011.24614.856K215.58012.52413.764K314.64013.04413.006K410.24614.18612.638K58.77415.68012.416R8.6664.4322.440Sorting123It can be clearly found from Table4 that the primary and secondary relationship of factors affecting the residual stress of the forging is deformation temperature, deformation degree, and deformation speed. And, their contribution rates to the residual stress of forgings are 56%, 28%, and 16%, respectively. Comparing the minimum values of K1∼K5 under each factor, the optimal combination of process parameters with the smallest residual stress can be determined as deformation temperature 1025 °C, deformation degree 30%, and deformation speed 1000 mm/s.


## 3.4. Numerical Simulation of Optimized Process Parameters

The forging process of Ti-6Al-4V alloy was simulated under the optimal process parameters of deformation temperature 1025 °C, deformation degree 30%, and deformation speed 1000 mm/s. The results are shown in Figure 6. It can be seen from Figure 6 that the residual stress of the forgings under the optimal combination of process parameters basically belongs to normal distribution, without obvious stress concentration, and the distribution is relatively uniform. This is because when the deformation temperature (1025 °C) far exceeds the transformation temperature of Ti-6Al-4V alloy (985°C), the forging is completed in the β single-phase region. The plastic forming ability of the forgings is significantly improved and the deformation resistance is greatly reduced, which greatly improves the deformation uniformity. Furthermore, when the deformation degree is 30%, the main softening mechanism of Ti-6Al-4V alloy forgings is dynamic recovery. The residual stress of the forgings decreases sharply during the recovery stage. In addition, the deformation temperature is high and the degree of deformation is small at this time, which reduces the heat converted and weakens the thermal effect of metal deformation, thereby greatly reducing the residual stress of the forging and improving the distribution of the residual stress.Figure 6
Distribution of forging residual stress under the optimal process parameters. (a) Equivalent residual stress. (b) Residual stress in theX direction. (c) Residual stress in the Y direction. (d) Residual stress in the Z direction.
(a)(b)(c)(d)However, when the forging temperature is above theβ-transition temperature, there are a lot of coarse β-grains in the forgings. The coarser the grain, the more uneven the plastic deformation and the greater the internal stress concentration. Moreover, the coarse grains reduce the chance of crisscross between grains, which will be conducive to the propagation and development of cracks, and the strength and toughness will become worse. So, the mechanical properties will be reduced. Therefore, for the parts that require high-dimensional stability but not too much mechanical properties, this combination of forging parameters can be used as an effective guide.


## 4. Development of Prediction Model

### 4.1. Regression Model and Calculation

In order to further quantitatively analyze the relationship between the key forging process parameters and the average equivalent residual stress, regression analysis was carried out on the simulation data to find the mapping relationship between them, so as to determine a mathematical model that can effectively predict the average equivalent residual stress of the forgings and provide effective guidance for the actual forging process [26].Based on Table2 and the analysis results of Table 4, the change curve of the average equivalent residual stress under each factor is shown in Figure 7.Figure 7
Variation curve of average equivalent residual stress under each factor.It can be seen from Figure7 that when the deformation temperature exceeds the transformation temperature (985°C) of Ti-6Al-4V alloy, the average equivalent residual stress of forgings decreases sharply. However, there is a linear relationship between the deformation temperature and the average equivalent residual stress in the two different temperature ranges above and below the transformation temperature. In addition, when the deformation degree is between 40% and 70%, there is a linear relationship between the deformation degree and the average equivalent residual stress. And, when the deformation velocity is between 100 mm/s and 1000 mm/s, the relationship between the deformation velocity and the average equivalent residual stress is basically linear.Based on the above characteristics, the deformation temperature was divided into two intervals. Then, regression model I was established by taking deformation temperature, deformation degree, and deformation velocity as independent variables and average equivalent residual stress as dependent variable. At the same time, considering the difference in the order of magnitude of the selected parameters, the logarithmic transformation was carried out to establish regression model II.Set the average equivalent residual stress asσ (MPA), deformation temperature as t (°C), deformation degree as d (%), and deformation speed as s (mm/s), among which t∈[925, 980]∪[ 990, 1025 ], d∈[40, 70], and s∈[100, 1000].The two regression models were as follows:(3)ModelI:σ=a10t+a11d+a12s+C1925≤t≤980σ=a20t+a21d+a22s+C2990≤t≤1025,ModelII:σ=ea10t+a11d+a12s+C1925≤t≤980σ=ea20t+a21d+a22s+C2990≤t≤1025.Fifty sets of sample data were obtained using the modified finite element model of residual stress of Ti-6Al-4V alloy forgings under nonisothermal forging. Among them, 25 sets of sample data with deformation temperature between 925°C and 980°C are shown in Table5, and the other 25 sets of sample data with deformation temperature between 990°C and 1025°C are shown in Table 6.Table 5
Regression analysis of sample data (925°C–980°C).
SampleTDSσ (MPa)19254010018.529256050017.43925501001849256010018.559505010015.3695060100014.379507050015.589505030014.999506050014.61095040100014.4119506030015.11295050100014.4139504010015.3149754050011.4159754010012.2169755010012.6179757010013.51897540100011.11997550100011.8209757030012.6219804030011.4229805010012.12398040100010.5249805050011.5259807010012.4Table 6
Regression analysis of sample data (990°C–1025°C).
SampleTDSσ (MPa)19904010010.229905030010.739906050010.649904030010.459904010009.6169905050010.871000403009.63810006010009.79910007050010.61010004010009.041110007010011121010405008.82131010403008.99141010601009.51510107030010161010505009.3171010603009.33181025603008.35191025701009.19201025601008.83211025705009.2221025401008.2231025501008.93241025503008.3251025405008.1Based on the sample data in Tables5 and 6, the mathematical analysis software was used to analyze and solve the two regression models. The results are shown in Figures 8 and 9.Figure 8
Analysis results of model I. (a) Normal probability plot (925 ≤ (t) ≤ 980). (b) Histogram (925 ≤ (t) ≤ 980). (c) Normal probability plot (990 ≤ (t) ≤ 1025). (d) Histogram (990 ≤ (t) ≤ 1025).
(a)(b)(c)(d)Figure 9
Analysis results of model II. (a) Normal probability plot (925 ≤ (t) ≤ 980). (b) Histogram (925 ≤ (t) ≤ 980). (c) Normal probability plot (990 ≤ (t) ≤ 1025). (d) Histogram (990 ≤ (t) ≤ 1025).
(a)(b)(c)(d)The regression models obtained after calculation are as follows:(4)ModelI:σ=−0.11233t+0.02085d−0.001276s+121.16925≤t≤980σ=−0.06263t+0.0336d−0.000721s+71.13990≤t≤1025,ModelII:σ=e−0.00773t+0.00181d−0.000087s+9.98925≤t≤980σ=e−0.00661t+0.00358d−0.000073s+8.75990≤t≤1025,where t∈[925, 980]∪[990, 1025], d∈[ 40, 70], and s∈[100, 1000].By comparing Figures8 and 9, it can be seen that the normalized residual of regression model II is closer to the normal distribution and has higher accuracy. Therefore, regression model II was selected as the prediction model of the average equivalent residual stress.


### 4.2. Model Validation

Since the simulation model has been modified by experiments, it can be used to verify the accuracy of the prediction model. Another 5 sets of forging process parameters were selected as test samples for simulation and model prediction, as shown in Table7. The comparison result is shown in Figure 10, from which it can be seen that the simulation data are very close to the predicted data and the error is small. This indicates that the prediction model can accurately reflect the mapping relationship between the forging process parameters and the average equivalent residual stress of Ti-6Al-4V alloy forgings. It also indicates that the prediction model can effectively guide the prediction of the residual stress of Ti-6Al-4V alloy forgings in practical engineering.Table 7
Process parameters of test samples.
SampleTDS192570100029504030039756010041000501005102550500Figure 10
Comparison between simulated data and predicted data.


## 4.1. Regression Model and Calculation

In order to further quantitatively analyze the relationship between the key forging process parameters and the average equivalent residual stress, regression analysis was carried out on the simulation data to find the mapping relationship between them, so as to determine a mathematical model that can effectively predict the average equivalent residual stress of the forgings and provide effective guidance for the actual forging process [26].Based on Table2 and the analysis results of Table 4, the change curve of the average equivalent residual stress under each factor is shown in Figure 7.Figure 7
Variation curve of average equivalent residual stress under each factor.It can be seen from Figure7 that when the deformation temperature exceeds the transformation temperature (985°C) of Ti-6Al-4V alloy, the average equivalent residual stress of forgings decreases sharply. However, there is a linear relationship between the deformation temperature and the average equivalent residual stress in the two different temperature ranges above and below the transformation temperature. In addition, when the deformation degree is between 40% and 70%, there is a linear relationship between the deformation degree and the average equivalent residual stress. And, when the deformation velocity is between 100 mm/s and 1000 mm/s, the relationship between the deformation velocity and the average equivalent residual stress is basically linear.Based on the above characteristics, the deformation temperature was divided into two intervals. Then, regression model I was established by taking deformation temperature, deformation degree, and deformation velocity as independent variables and average equivalent residual stress as dependent variable. At the same time, considering the difference in the order of magnitude of the selected parameters, the logarithmic transformation was carried out to establish regression model II.Set the average equivalent residual stress asσ (MPA), deformation temperature as t (°C), deformation degree as d (%), and deformation speed as s (mm/s), among which t∈[925, 980]∪[ 990, 1025 ], d∈[40, 70], and s∈[100, 1000].The two regression models were as follows:(3)ModelI:σ=a10t+a11d+a12s+C1925≤t≤980σ=a20t+a21d+a22s+C2990≤t≤1025,ModelII:σ=ea10t+a11d+a12s+C1925≤t≤980σ=ea20t+a21d+a22s+C2990≤t≤1025.Fifty sets of sample data were obtained using the modified finite element model of residual stress of Ti-6Al-4V alloy forgings under nonisothermal forging. Among them, 25 sets of sample data with deformation temperature between 925°C and 980°C are shown in Table5, and the other 25 sets of sample data with deformation temperature between 990°C and 1025°C are shown in Table 6.Table 5
Regression analysis of sample data (925°C–980°C).
SampleTDSσ (MPa)19254010018.529256050017.43925501001849256010018.559505010015.3695060100014.379507050015.589505030014.999506050014.61095040100014.4119506030015.11295050100014.4139504010015.3149754050011.4159754010012.2169755010012.6179757010013.51897540100011.11997550100011.8209757030012.6219804030011.4229805010012.12398040100010.5249805050011.5259807010012.4Table 6
Regression analysis of sample data (990°C–1025°C).
SampleTDSσ (MPa)19904010010.229905030010.739906050010.649904030010.459904010009.6169905050010.871000403009.63810006010009.79910007050010.61010004010009.041110007010011121010405008.82131010403008.99141010601009.51510107030010161010505009.3171010603009.33181025603008.35191025701009.19201025601008.83211025705009.2221025401008.2231025501008.93241025503008.3251025405008.1Based on the sample data in Tables5 and 6, the mathematical analysis software was used to analyze and solve the two regression models. The results are shown in Figures 8 and 9.Figure 8
Analysis results of model I. (a) Normal probability plot (925 ≤ (t) ≤ 980). (b) Histogram (925 ≤ (t) ≤ 980). (c) Normal probability plot (990 ≤ (t) ≤ 1025). (d) Histogram (990 ≤ (t) ≤ 1025).
(a)(b)(c)(d)Figure 9
Analysis results of model II. (a) Normal probability plot (925 ≤ (t) ≤ 980). (b) Histogram (925 ≤ (t) ≤ 980). (c) Normal probability plot (990 ≤ (t) ≤ 1025). (d) Histogram (990 ≤ (t) ≤ 1025).
(a)(b)(c)(d)The regression models obtained after calculation are as follows:(4)ModelI:σ=−0.11233t+0.02085d−0.001276s+121.16925≤t≤980σ=−0.06263t+0.0336d−0.000721s+71.13990≤t≤1025,ModelII:σ=e−0.00773t+0.00181d−0.000087s+9.98925≤t≤980σ=e−0.00661t+0.00358d−0.000073s+8.75990≤t≤1025,where t∈[925, 980]∪[990, 1025], d∈[ 40, 70], and s∈[100, 1000].By comparing Figures8 and 9, it can be seen that the normalized residual of regression model II is closer to the normal distribution and has higher accuracy. Therefore, regression model II was selected as the prediction model of the average equivalent residual stress.


## 4.2. Model Validation

Since the simulation model has been modified by experiments, it can be used to verify the accuracy of the prediction model. Another 5 sets of forging process parameters were selected as test samples for simulation and model prediction, as shown in Table7. The comparison result is shown in Figure 10, from which it can be seen that the simulation data are very close to the predicted data and the error is small. This indicates that the prediction model can accurately reflect the mapping relationship between the forging process parameters and the average equivalent residual stress of Ti-6Al-4V alloy forgings. It also indicates that the prediction model can effectively guide the prediction of the residual stress of Ti-6Al-4V alloy forgings in practical engineering.Table 7
Process parameters of test samples.
SampleTDS192570100029504030039756010041000501005102550500Figure 10
Comparison between simulated data and predicted data.


## 5. Conclusions

By using the modified finite element model, the primary and secondary relationship of key factors affecting the average equivalent residual stress of Ti-6Al-4V alloy forgings was analyzed and the optimal process parameters with minimum residual stress were obtained. Then, the multiple linear regression analysis was carried out based on the simulation data, and the mathematical model which can effectively predict the average equivalent residual stress was determined. The significant conclusions have been drawn below:(1)
In nonisothermal forging, the deformation temperature is the main factor affecting the residual stress of the forging, followed by the degree of deformation, and the deformation speed has the least influence on the residual stress of the forging.(2)
The optimal combination of process parameters for the minimum residual stress of forgings without considering mechanical properties is deformation temperature 1025°C, deformation degree 30%, and deformation speed 1000 mm/s. For the parts that require high-dimensional stability but not too much mechanical properties, this optimal combination of process parameters has certain reference significance.(3)
In the regression analysis, according to the distribution characteristics of the residual stress of Ti-6Al-4V alloy forgings in the upper and lower regions of the transformation point (985°C), the regression model was divided into two temperature regions, which can accurately reflect the mapping relationship between the key process parameters and the average equivalent residual stress.



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*Source: 3105470-2021-09-15.xml*