Jie Zhou，Zhi－Yong Huang，Lei Lin，Shi－Yun Li

(College of Material Science and Engineering，Chongqing University，Chongqing 400044，China)

1 Introduction

As the rise of the strength and the decrease of the elongation，the formability of the steel is limited.To improve the forming quality of the high-strength steel (HSS)or the ultra high strength steel(UHSS)，a controlled manufacturing technique known as hot stamping is developed［1－2］.According to the research results of Naderi［3－4］et al.，hot stamping is a nonisothermal high-temperature forming process，in which complex ultra-high strength parts are produced，with the goal of no springback.In hot stamping process，the blanks are austenitized and subsequently formed and quenched in the die with cooling system.Hot stamping is a complex manufacturing process where the properties and quality of the final part are strongly affected by the material and the numerous parameters in the thermo-mechanical cycle，such as strain-and strain rate-paths，temperature and microstructure evolution that mutually interact during the forming and the cooling stages［5-6］.Centering on the hot stamping process of steel sheets，lots of researches have been done.Bariani and Hoffmann［7－8］presents an innovative experimental procedure，based on Nakazima test，for evaluating the formability limits in the hot stamping of high strength steel(HSS)which iscapableof generating formability data suitable for an FE modeling of the process.In the book of Formability of Metallic Materials［9］，detailed statement of the properties of workpiecesaftermetalforming，as wellas the fundamentals of the theory of plasticity and finite element simuation of metal forming processes is present.Ikeuchi and Yanagimoto［10］used hot forming simulator to study the valuation method for the effects of hot stamping process parameters on product properties.Xinga and Bergman［11－12］set up material models under hot stamping condition of quenchable steel，and made the numerical simulations to the whole hot stamping process of hot forming，quenching and spring-back of bending parts.Xiang［13］conducted a quenching experiment with B1500HS under three different cooling mediums—water，copper and iron chopping board with cooling duct，and studied the influence of specimen’s austenitizing temperature and holding time to the mechanical property and microstructure.And it concluded that different cooling medium did not show dramatic influence to the final mechanical properties of the quenched part，and better strength performance would be achieved under the austenitizing temperature of 900 to 950℃.Jiang［14］simulated the hot stamping processes of B-bearing steel sheet，and found that the boron segregation and austenization of martensite steel were accelerated with the increase of heating temperature and holding time.Yu［15］conducted basic processparameters improvementexperiments with 22MnB5， and found the influence of heating temperature，soaking time and cooling rate on mechanical properties of the quenched material.For 22MnB5 with thickness of 1.6 mm，the optimized heating temperaturerangeis 900－950 ℃，the optimized soaking time range is 2－4 min，and the optimized cooling rate range is 36－173℃/s，both high strength and good plasticity can be obtained.

He［16］got the optimization of quenching parameters for cold rolled steel B1500HS based on Response Surface Method，including austenitizing temperature and soaking time.However，it could be found that，when the explanatory variable soaking time ranged from 0 to 45 min(0 min，5 min，30 min，45 min)， the response variables tensile strength，quenching hardness and elongation all reached their best when the soaking time was 0，except in the low temperature quenching area when the soaking time was 15.95 min and tensile strength got 1658.94 MPa.Therefore，it could be concluded that maybe because the soaking time was too long that we can not get an accurate result.In this paper，the same idea is adopted as Lianfang He used—The Orthogonal Design was adopted to arrange experiment，and Response Surface Methodology(RSM)was applied to analyze the experimental results.At the same time，there are mainly three differences，for one the hot rolled steel BR1500HS was used.And for another the soaking time was set as 0，5，15，30 and 45 s.Besides，the mechanism about the influence of austenitizing temperature and soaking time to the mechanical property(tensile strength，quenching hardness and elongation)was analyzed.The steps of RSM included: 1)Establish regression models.Austenite temperature and soaking time were taken asthe explanatory variables and quenching hardness，tensile strength and elongation as the response variables.Regression analysis was carried out and regression equations were obtained.2)Test the reliability of the regression models and the statistical significance of the individual model coefficients by Analysis of Variance(ANOVA).3)Analysis the response surface.By analyzing the response surfaces， the relationship between the explanatory variables(austenite temperature，soaking time)and the response variables (quenching hardness，tensile strength and elongation)can be directly perceived through the senses.In this paper，the optimum value of each single-objective model was firstly obtained by using the response surface models.Then，Ideal Point Method was chosen to find the global optimum value of this multi-objective programming.

2 Experimental Materials and Procedures

2.1 Preparation of Experimental Materials

The material(BR1500HS)used in this experiment is developed by Shanghai Baosteel Group.The thickness of the experimental material is 1.8 mm.Table 1 shows the chemical composition(wt.%)of BR1500HS.Fig.1 shows the configuration and dimension of specimen.

Table 1 Chemical composition of BR1500HSwt.%

Fig.1 Configuration and dimension of specimen(Unit:mm)

In order to investigate the effects of austenite temperature and soaking time on the mechanical properties of BR1500HS，the orthogonal experiment with two factors and five levels is carried out.The experimentalfactorsand levels are presented in Table 2.

Table 2 Experimental factors and levels

2.2 Experimental Procedures

Actual experimental procedure includes:1)Set different heat treatment temperatures in the range of 800 to 950℃;2)Put the samples into the heat treated furnace to be heated for definite time at a certain temperature;3)Removethesamplesby mechanical hand and put them into the hot forming mould with water-cooling channels;4)Cool the samples at about 30℃/s in the mound;5)Analyze the mechanical properties of samples by using the Rockwell hardness test system and tensile test system.

In the hardness testing experiment，six different points of each sample are chosen.The average reading is chosen as the experimental value.An electronic tensile testing machine SY-6014 is selected for the tensile test.According to the thickness and the width of the sample，the tensile displacement as well as the tension load and the tensile speed，the value of elongation can be obtained.

According to the method of orthogonal design，twenty-five experiments must be carried out for the orthogonal experiment with two factors and five levels.The orthogonal test table and the experimental results are shown in Table 3，where θ(℃)represents austenite temperature;t(s)represents the soaking time;H(HRC)represents the quenching hardness;σ (MPa)represents the tensile strength，and δ(%) represents the elongation.

Table 3 Orthogonal test table and the experimental results

3 Mathematical Modeling

In statistics，response surface methodology［17］(RSM)is used to explore the relationships among explanatory variablesand response variables.The relationship can be founded by Polynomial functions and further displayed in the graphics.

The response variable y can be defined as follows:

where x1and x2represent the variables;ε represents the variables noise or error observed in the response y，and f is the unknown function.If the expected response is expressed as E(y)=f(x1，x2)=η，the surface represented by η=f(x1，x2)is called a response surface.

In mostcases，the relationship between the response variable and the independent variables is unknown.Thus，finding a suitable approximation to the true relationship is the first step in RSM.Generally，a lower order polynomial equation is available for a simple case.But in this research，the regression analysis results shows that a third-order polynomial approaches more to the true function between the explanatory variables(austenite temperature，soaking time)and the quenching hardness.And fourth-order polynomials approach more to the true function between the explanatory variables and the response variables (tensile strength，elongation).Hence，a third-order polynomial is used to simulate the relationship between the explanatory variables and quenching hardness，while the fourth-order polynomials are applied to fit the relationship between the explanatory variables and the response variables.

4 Results and Discussion

4.1 Regression Model of Responses

Before establishing the regression model，the singular points should be got rid of.The regression response surface models of the quenching hardness H，the strength of extension σ and the elongation δ are expressed as:

where θ(℃)represents the austenite temperature;t(s) represents the soaking time.The regression equations can show an approximate relationship between the response variables and the independent variables.The fitted formulas can be applied to predict the values of the quenching hardness H，the tensile strength σ and the elongation δ.Figs.2(a)－2(c)show the difference between the predicted and experimental values.The comparison results imply that the predicted values of H，σ and δ are close to the experimental values within a 95%confidence interval.

Fig.2 Comparisons of predicted and experimental values about H，σ and δ

4.2 Analysis of Proposed Mathematical Model on Mechanical Performance Characteristics

To evaluate the reliability of the experimental results and the credibility of the responses model，both the statistical significance of the regression models and the statistical significance on the individual model coefficients need to be tested.These tests are performed as ANOVA procedure by calculating the“F－value”，the“P－value”，the determination coefficients(R2)as well as the adjusted R－squared

Usually，the desired confidence level is set as 95%.If the value of“Prob.＞F”is smaller than 0.05，the regression model is considered to be statistically significantand the variables in the modelhave significant effects on the responses.When R2approaches to unity，the better the response model fits the actual data，the less the difference between the predicted and actual values exists.If those additional terms do not add value to the model，the adjusted R－squareddecreases as the number of terms in the model increases.Therefore，the biggerthe value ofthe adjusted R－squared is，the better the regression effects are.

4.2.1 Analysis of variance(ANOVA)for quenching hardness

Tables 4－5 presents the analysis of variance (ANOVA)results of the quenching hardness model.The significance of each coefficient is determined by using T－test and P－value.It is shown that the P－value of each item is less than significance level α(α=0.05)，indicating that the terms in the model have a significant effect on the quenching hardness. The total determination coefficient(R2)is 0.9905，suggesting that the polynomial model can represents the experimental results adequately. The adjusted determination coefficient(R2)is 0.9848，which implies that 98.48%of the changes of this model are attributed to the independent variables.

The ANOVA results show that the model F－value is 174.244(F＞F0.05(9，15)= 2.59).And the model P－values is 1.58E-13 which is far less than 0.05.Both the F－value and the P－value demonstrate that the regression result is very significant.

The residual plot of quenching hardness is shown in Fig.3.The residual plot shows that all the residual plots contain“0”.All the tracing points are surrounding the zero value line.And no abnormal point does exist.Hence，it is evidenced that the regression model fits the measurements well.

4.2.2 Analysis of variance(ANOVA)for tensile strength

Tables 6－7 gives the regression analysis results of the tensile strength.It is shown that the P－value of each item is less than significance level α(α=0.05).The results suggest that there is a significant difference between the dependent variable(tensile strength)and the polynomial terms.The model value of the coefficient of multiple determination(R2= 0.9770)implies that the fitted model adequately represents the experimental results.Both the model F－test value(F=35.8942＞F0.05(13，11)=2.76)and the P－values of(P＜0.05) indicate the significance of the regression model.

Table 4 Analysis of variance of regression equation for quenching hardness

Table 5 Analysis of variace table for quenching hardness

Fig.3 Residual plot of quenching hardness

A residual plot for tensile strength is shown in Fig.4.The residual plot shows that all the tracing points are surrounding the zero value line，implying that the regression model agrees well with the measurements.In short，the analysis of variance results shows that the regression equation can not only fit the experimental data，but also can predict the results properly.

4.2.3 Analysis of variance(ANOVA)for the elongation

Tables 8－9 lists the ANOVA results of the polynomial model of elongation.For the elongation，the ANOVA results show that the associated model with a large value of the coefficient of multiple determination (R2= 0.9856)is adequate to representthe experimental results.Besides，the model F－test value (F=68.6074＞ F0.05(12，12)=2.69)and the P－values of(P=3.82E－09＜0.05)indicate the significance of the regression model.

Table 6 Analysis of variance of regression equation for tensile strength

Table 7 Analysis of variance table for tensile strength

Fig.4 Residual plot of tensile strength

A residual plot for elongation is depicted in Fig.5.It is shown that all the tracing points surround the zero value line.Thus，the regression model agrees well with the measurements.

In summary，these mathematical expressions can successfully pass the F statistics and the R2－test，implying that the current data fitting is excellent.The independent variables are significantly correlative to the elongation.

4.3 Response Surface Analysis

To visualize the effect of the variables on the required responses，the 3-D response surface plots are applied to describe the regression equations.For the three required responsesincluding the quenching hardness，the tensile strength and the elongation of BR1500HS，the corresponding 3-D response surfaces are shown in Figs.6－8.

Table 8 Analysis of variance of regression equation for elongation

Table 9 Analysis of variance table for elongation

4.3.1 Response surface analysis for quenching hardness

Fig.6 shows the status of response surface and contour plot for quenching hardness.The region where the value of the quenching hardness is greater than 50 HRC is defined as the optimum region.As shown in Fig.6，the optimum region for the quenching hardness is at the temperature of 820－930℃ and soaking time 0－15 s.In thisregion，the quenching hardness decreases significantly byincreasing the value of soaking time.By increasing the austenite temperature，the quenching hardness increases firstly to a peak and then decreases.

Fig.5 Residual plot of elongation

Fig.6 Response surface and contour plot for quenching hardness

Eq.(2)indicates the optimum values for the single-objective model of quenching hardness.The optimum value is shown in Table 10.

Table 10 Optimum values for quenching hardness

4.3.2 Response surface analysis for tensile strength

Fig.7 shows the effects of austenite temperature and soaking time on the tensile strength of the ultrahigh strength steel BR1500HS.The region where the value of the tensile strength is greater than 1600 MPa is defined as the optimum region.It is shown that there are two optimum regions forthe tensile strength atthe temperature of 800－950℃ and soaking time 0－45 s.One is at the temperature of 800－840℃ and soaking time 3－30 s.In this region，the value of the tensile strength reaches a maximum and then drops down with the increase of the soaking time.The tensile strength decreasessignificantly by increasing the value of austenite temperature.The other one is at the temperature of 870－950℃ and soaking time 0－10 s.In this region，the tensile strength increases firstly with the increase of the austenite temperature and then decreases.The tensile strength decreases by increasing the soaking time.

The function Eq.(3)is calculated to find out the optimum values in the optimum region.The maximum value is chosen as the optimum value for the singleobjective model of tensile strength.The results are shown in Table 11.

Table 11 Optimum values for tensile strength

4.3.3 Response surface analysis the elongation

Fig.8 shows the status of response surface and contour plot for the values of the elongation in relation to the austenite temperature and soaking time.The region where the value of the elongation is greater than 8%is defined as the optimum region.It is shown that there are also two peak regions for the elongation at the temperature of 800－950℃ and soaking time 0－45 s.One of them is at the temperature of 800－830℃ and soaking time 0－20 s.In this region，the elongation increases firstly and then decreases with the increase of soaking time，and decreases by increasing the austenite temperature.The other one is at the temperature of 910－950℃ and soaking time 0－10 s.In the region，the value of the elongation reaches a maximum and then drops down with the increase of the austenite temperature.The elongation decreases significantly by increasing the value of soaking time.

Fig.7 Response surface and contour plot for tensile strength

Fig.8 Response surface and contour plot for elongation

The function Eq.(4)is calculated to find out the optimum values in these peak areas.The maximum value is regarded as the optimum value for the singleobjective model of elongation.The results are presented in Table 12.

Table 12 Optimum values for elongation

4.3.4 Mechanism about the influence of austenitizing temperature and soaking time to mechanical property

As we all know，in all the factors influencing the formation ofaustenite， temperature is the most remarkable.With the elevating of the temperature，the atomic diffusion speed gets faster，so the austenite formation rate gets higher and austenite grain grows faster.Thus，we can get austenite which is under the same situation by giving more soaking time under low temperature or giving less socking time under high temperature.However，when the temperature gets too high or the soaking time gets too long，the austenite grain will grow up to the size which significantly decreases tensile strength，quenching hardness and elongation.Therefore，it is reasonable to explain the difference of the result in this paper from what Lianfang He［16］got.In He’s paper，the maximum of the tensile strength was got when the austenite temperature was 820℃;however，in this paper we got the maximum value when it was 916℃.It is mainly because in Lianfang He’paper the soaking time took about 16 min which was far beyond the maximum soaking time(15 s)set in this paper.

4.4 Multi-Objective Optimization

Generally，the parts that made of UHSS serve as reinforcement in the car body.Hence，the higher the value of the quenching hardness，the tensile strength and the elongation are，the better the quality of product is.In the present study，the ideal point method is chosen to find the solution to this multi-objective programming.The quenching hardness，the tensile strength as well as the elongation are taken as target variables.

According to the ideal point method，the multiobjective problem must be transformed into singleobjectives.The first step of the ideal point method is to solve the single-objectives.The simultaneous equations of the single-objectives H(θ，t)，σ(θ，t)and δ(θ，t)are written as follows:

The solution of H*，σ*，δ*can be gained by solving the above-mentioned equations.Vector= (H*，σ*，δ*)Tis seen as the ideal point.In fact the ideal point is difficult to realize because of complexity among objective.An evaluation function φ(z)must be defined to find a point that approaches the ideal point mostly.

According to the optimization procedures，the evaluation function φ(z)should be minimized.

By solving theaboveequation，the global optimum values were obtained，which are presented in Table 13.

Table 13 Global optimum values

5 Conclusions

In the current research， Response Surface Methodology(RSM)and Ideal Point Method are applied to optimize the quenching parameters for ultra high strength steel BR1500HS.Quenching hardness，tensile strength and elongation are taken asthe response variables.Finally，the global optimum values of austenite temperature and soaking time are obtained.The main results are as follows:

1)The results of ANOVA and comparisons of experimental data show that the mathematical models of the value of the quenching hardness，tensile strength and elongation are fairly well fitted with the experimental values with a 95%confidence interval.Both the austenite temperature and soaking time have a significant effect on the quenching hardness，tensile strength and elongation.Besides，the residual plots show that all the tracing points are surrounding the zero value line without abnormal points.

2)The results of response surface analysis show that the optimum region for the quenching hardness is at the temperature of 820－930℃ and soaking time 0－15 s.Both the tensile strength and elongation have two optimum regions.One of the optimum regions for the tensile strength is at the temperature of 800－840℃and soaking time 3－30 s.The other one is at the temperature of 870－950℃ and soaking time 0－10 s.One of the optimum regions for the elongation is at the temperature of 800－830℃ and soaking time 0－20 s.The other one is at the temperature of 910－950℃ and soaking time 0－10 s.

3)The results of three single objective optimizations show that the peak value of quenching hardness 52.29 HRC is achieved at the temperature of 875.40℃ and soaking time 0 s.The peak value of the tensile strength 1671.2 MPa isobtained atthe temperature of 916.01℃ and soaking time 0 s.The peak value of the elongation 9.027%is gained at the temperature of 935.39℃ and soaking time 0 s.

4) The results of three multi-objective optimizations show that the global optimum value is gained when the austenite temperature is 914.17℃and the soaking time is 0 s.The predicted values of quenching hardness，tensile strength and elongation are not less than 51.03 HRC，1671 MPa and 8.75%，respectively.

［1］Geiger M，Merklein M，Hoff C.Basic investigation on the hot stamping steel 22MnB5.Advanced Materials Research.Erlangen:Trans Tech Publications，2005.795－802.

［2］Mori K，Maki S，Tanaka Y.Warm and hot stamping of ultra high tensile strength steel sheets using resistance heating.Manufacturing Technology，2005，54(1):209－212.

［3］Naderi M，Saeed-Akbari A，Bleck W.The effects of nonisothermal deformation on martensitic transformation in 22MnB5 steel.Materials Science and Engineering A，2008，487(1/2):445－455.

［4］Naderi M，Durrenberger L，Molinari A，et al.Constitutive relationships for 22MnB5 boron steel deformed isothermally at high temperatures.Materials Science and Engineering A，2008，478(1/2):130－139.

［5］Hein P.A global approach of the finite element simulation of hot stamping.Advanced Materials Research.Erlangen: Trans Tech Publications，2005.763－770.

［6］Turetta A，Bruschi S，Ghiotti A.Investigation of 22MnB5 formability in hot stamping operations.Journal of Materials Processing Technology，2006，177(1/2/3):396－400.

［7］ Bariani P F，Bruschi S，Ghiotti A，et al.Testing formability in the hot stamping of HSS.Manufacturing Technology，2008，57(1):265－268.

［8］Hoffmann H，So H，Steinbeiss H.Design of hot stamping tools with cooling system.Manufacturing Technology，2007，56(1):262—272.

［9］Tekkaya A E，Banabic D，Bunge H J.Formability of Metallic Materials:Plastic Anisotropy， Formability Testing，F(xiàn)orming Limits.Verlag，Berlin:Springer，2010.1－175.

［10］Ikeuchi K，Yanagimoto J.Valuation method for effects of hot stamping process parameters on product properties using hot forming simulator.Journal of Materials Processing Technology，2011，211(8):1441－1447.

［11］Xinga Z W，Baoa J，Yang Y Y.Numerical simulation of hot stamping of quenchable boron steel.Materials Science and Engineering，2009，499(1/2):28－31.

［12］Bergman G，Oldenburg MA.A finite element model for thermomechanical analysis of sheet metal forming.International Journal for Numerical Methods in Engineering，2004，59(9):1167－1186.

［13］Xiang N.Research on properties after quenching and flow behavior at elevated temperature of B1500HS boron sheet steel.Shandong:Shandong University，2012.

［14］Jiang Haitao，Tang Di，Mi Zhenli.Influence of processing parameters of hot stamping to mechanical properties of martensite steeland segregation ofboron.Material Engineering，2010(2):69－73.

［15］Yu Yuanhong.Research on hotforming process improvement of automobile high strength steel and its application.Da Lian:Dalian University of Technology，2013.

［16］He Lianfang，Zhao Guoqun，Li Huiping.Optimization of quenching parameters for hot stamping boron steel B1500HS based on response surface methodology.Journal of Mechanical Engineering，2011，47(8):77－82.

［17］KhuriA I， Mukhopadhyay S.Response surface methodology. Wiley Interdisciplinary Reviews: Computational Statistics，2010，2(2):128－149.