13
Research Article Thermomechanical-Phase Transformation Simulation of High-Strength Steel in Hot Stamping Wenhua Wu, Ping Hu, and Guozhe Shen State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle and Mechanics, Dalian University of Technology, Dalian 116024, China Correspondence should be addressed to Wenhua Wu; [email protected] Received 18 September 2014; Accepted 1 December 2014 Academic Editor: Chenfeng Li Copyright © 2015 Wenhua Wu et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e thermomechanical-phase transformation coupled relationship of high-strength steel has important significance in forming the mechanism and numerical simulation of hot stamping. In this study a new numerical simulation module of hot stamping is proposed, which considers thermomechanical-transformation multifield coupled nonlinear and large deformation analysis. In terms of the general shell finite element and 3D tetrahedral finite element analysis methods related to temperature, a coupled heat transmission model for contact interfaces between blank and tools is proposed. Meanwhile, during the hot stamping process, the phase transformation latent heat is introduced into the analysis of temperature field. Next the thermomechanical-transformation coupled constitutive models of the hot stamping are considered. Static explicit finite element formulae are adopted and implemented to perform the full numerical simulations of the hot stamping process. e hot stamping process of typical U-shaped and B- pillar steel is simulated using the KMAS soſtware, and a strong agreement comparison between temperature, equivalent stress, and fraction of martensite simulation and experimental results indicates the validity and efficiency of the hot stamping multifield coupled constitutive models and numerical simulation soſtware KMAS. e temperature simulated results also provide the basic guide for the optimization designs of cooling channels in tools. 1. Introduction In order to meet the demands for reducing weight and improving safety, the hot stamping of high-strength steel, a new modern forming technology, has attracted more and more attention in the automobile industry [1]. e steel specimen produces martensitic microstructure and exhibits strong hardening abilities aſter hot stamping. e final yield strength and ultimate tensile strength can reach up to about 1000 MPa and 1500 MPa, respectively. With comparison to the traditional cold stamping pro- cess, which has been thoroughly studied by scholars through- out the world [25], in the standard hot stamping process a steel blank is heated in a furnace to a certain high temper- ature (880–950 C). e matrix organization and mechanical behavior change with the thermal effect. e austenite trans- formation is distributed uniformly in the blank [6]. en the heated blank is transferred into the tools (stamping molds) with water-cooled channels and formed and quenched simul- taneously for a short time (less than 10 seconds). e phase transformation behavior from austenitic to martensitic state in the material causes an increase of the tensile strength up to 1200–1500 MPa. roughout the entire stamping process, coupling thermomechanical characteristics are exhibited. e accurate control of temperature and final martensite rate play an important role in the final qualities of the molded parts [7]. e difficulty of hot stamping simulation is far beyond that of traditional cold forming. e thermodynamic prop- erties of materials are affected by the changing of the microstructure, especially in the phase transform phase. e strong rate-dependent constitutive behaviors and tem- perature sensitivities are significant throughout the entire stamping process. ermal contact and heat transfer between the tools and the blank surface must be considered with accuracy. e numerical modeling of the full process requires Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 982785, 12 pages http://dx.doi.org/10.1155/2015/982785

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Research ArticleThermomechanical-Phase Transformation Simulation ofHigh-Strength Steel in Hot Stamping

Wenhua Wu Ping Hu and Guozhe Shen

State Key Laboratory of Structural Analysis for Industrial Equipment Faculty of Vehicle and MechanicsDalian University of Technology Dalian 116024 China

Correspondence should be addressed to Wenhua Wu lxyuhuadluteducn

Received 18 September 2014 Accepted 1 December 2014

Academic Editor Chenfeng Li

Copyright copy 2015 Wenhua Wu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The thermomechanical-phase transformation coupled relationship of high-strength steel has important significance in formingthe mechanism and numerical simulation of hot stamping In this study a new numerical simulation module of hot stampingis proposed which considers thermomechanical-transformation multifield coupled nonlinear and large deformation analysis Interms of the general shell finite element and 3D tetrahedral finite element analysis methods related to temperature a coupled heattransmission model for contact interfaces between blank and tools is proposed Meanwhile during the hot stamping process thephase transformation latent heat is introduced into the analysis of temperature field Next the thermomechanical-transformationcoupled constitutivemodels of the hot stamping are considered Static explicit finite element formulae are adopted and implementedto perform the full numerical simulations of the hot stamping process The hot stamping process of typical U-shaped and B-pillar steel is simulated using the KMAS software and a strong agreement comparison between temperature equivalent stressand fraction of martensite simulation and experimental results indicates the validity and efficiency of the hot stamping multifieldcoupled constitutive models and numerical simulation software KMAS The temperature simulated results also provide the basicguide for the optimization designs of cooling channels in tools

1 Introduction

In order to meet the demands for reducing weight andimproving safety the hot stamping of high-strength steel anew modern forming technology has attracted more andmore attention in the automobile industry [1] The steelspecimen produces martensitic microstructure and exhibitsstrong hardening abilities after hot stamping The final yieldstrength and ultimate tensile strength can reach up to about1000MPa and 1500MPa respectively

With comparison to the traditional cold stamping pro-cess which has been thoroughly studied by scholars through-out the world [2ndash5] in the standard hot stamping process asteel blank is heated in a furnace to a certain high temper-ature (880ndash950∘C) The matrix organization and mechanicalbehavior change with the thermal effect The austenite trans-formation is distributed uniformly in the blank [6] Then theheated blank is transferred into the tools (stamping molds)

with water-cooled channels and formed and quenched simul-taneously for a short time (less than 10 seconds) The phasetransformation behavior from austenitic to martensitic statein the material causes an increase of the tensile strength upto 1200ndash1500MPa Throughout the entire stamping processcoupling thermomechanical characteristics are exhibitedThe accurate control of temperature and final martensite rateplay an important role in the final qualities of the moldedparts [7]

The difficulty of hot stamping simulation is far beyondthat of traditional cold forming The thermodynamic prop-erties of materials are affected by the changing of themicrostructure especially in the phase transform phaseThe strong rate-dependent constitutive behaviors and tem-perature sensitivities are significant throughout the entirestamping processThermal contact and heat transfer betweenthe tools and the blank surface must be considered withaccuracyThe numerical modeling of the full process requires

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 982785 12 pageshttpdxdoiorg1011552015982785

2 Mathematical Problems in Engineering

thermomechanical coupledmaterial descriptions for the typeof steel considered

Bergman and Oldenburg implemented the numericalsimulation of hot forming [8] Eriksson et al investigatedthe temperature and strain rate dependence of hot formingmaterial [9] Naderi Bleck and Merklein researched theplastic flow criterion and material parameters under hightemperature [10 11] The theoretical analysis numericalsimulations and experimental studies of hot forming werepresented by Ma and Hu who provided a new method forhot forming die design and technical process design [12 13]

At present most of the forming software programs avail-able perform simplifications by using the rigid shell elementto simulate the 3D tools in the hot forming modeling Suchassumption causes difficulties in simulating the temperaturetransport process In 2012 Jiang Wu et al presented anumerical frame to simulate the thermal transport process byusing a thick shell element of tools [14 15] It can simulatethe simple components such as U-shaped steel with highaccuracy However for the majorities of autopanels thenumerical simulations of temperature domain with complex3D tool surfaces still face many difficulties and challenges

In this paper a fully coupled 3D thermomechanical-phase transformation finite element simulation of the hotstamping process is developed The rate-dependent thermalconstitutive model is implemented in the present softwareframework In terms of general shell finite element andthe 3D tetrahedral finite element analysis method relatedto temperature a coupled thermal transport modeling forcontact interaction between blank and tools is proposedThe hot stamping process of benchmark typical U-shapedsteel and B-pillar steel is simulated by the present numeri-cal program Strong agreements of temperature equivalentstress and fraction of martensite between the simulated andexperimental results indicate the validity and efficiency of thepresentmultifield coupled numericalmodel in simulating thehot stamping process for the automobile industry

2 Key Technologies for Thermomechanical-Phase Transformation Coupled Behavior

21 Constitutive Laws of High-Strength Steel 22MnB5 boronsteel is the most common steel used for hot stamping A briefintroduction to the thermomechanical-phase transformationcoupled constitutive relationship for high-strength steel ispresented in the following section Detailed descriptions canbe found in [13]

In order to model the thermal-elastic-plastic responseduring the hot stamping process an additive decompositionof the total strain increment ( 120576

119894119895) is utilized [13 14] The total

stain 120576119894119895can be expressed with the decomposition rate form

as follows

120576

119894119895= 120576

119890

119894119895

+ 120576

119901

119894119895

+ 120576

th119894119895

+ 120576

tr119894119895

+ 120576

tp119894119895

(1)

where 120576

119890

119894119895

is the elastic strain increment 120576

119901

119894119895

is the plasticstrain increment 120576

th119894119895

is the thermal strain increment 120576

tr119894119895

is the isotropic transformation strain increment and 120576

tp119894119895

is

00 01 02 03 04 05

True strain 120576

100

150

200

250

300

350

400

True

stre

ss120590

(MPa

)

500∘C

550∘C

600∘C

650∘C

700∘C750∘C

800∘C

ExperimentalQuadratic interpolation

22MnB5

h0 = 16mmdTdt = 50 ∘Csd120576dt = 1 sminus1

Figure 1 Stress-strain curves at different evaluated temperaturefrom 500∘C to 800∘C

the transformation induced plasticity increment 120576

tr119894119895

and 120576

tp119894119895

play an important role in hot forming and are expressed as(2) in the traditional quenching treatment

120576

tr= 120573

120585

120576

tp= 119896 (120590) 120590 (1 minus 120585)

120585

(2)

where 120585 is the fraction of martensitic transformation 120573 is thephase transformation volume coefficient and 119896 is the phasetransformation plastic coefficient related to equivalent stress120590

The sensitivity of temperature on the forming behaviorof 22MnB5 is investigated [16] The flow curves shown inFigure 1 indicate that the temperature has a strong influenceon the forming behavior of quenchable steel Temperatureincreasing leads to an appreciable reduction of stress level anda decreased work hardening exponent

For temperature variation from 500∘C to 800∘C rep-resentative true stress-strain curves are displayed for anexemplarily strain rate of 1 sminus1 Other curves under differenttemperatures are obtained by quadratic interpolation usingthe existing experimental curves such as the curves of 600∘Cand 700∘C as shown in Figure 1

22 Martensite Phase Transformation For the martensitephase transformation the relationship between temperatureand phase change is modeled using the equation proposed byMa et al [13] which can be written as follows

120585 = 1 minus exp [minus120579 (120590) (119872119904(120590) minus 119879)] (3)

where 120585 represents the fraction ofmartensitic transformation119872

119904represents the martensite transformationrsquos beginning

temperature 120579 represents the material parameter whichreflects the austenite-martensite transformation rate 120590 rep-resents equivalent stress and 119879 represents temperature

Mathematical Problems in Engineering 3

600

650

700

750

800

850

900

Ms(

K)

0 100 200 300 400 500

120590 (MPa)

Ms

(a) Martensite transformation temperature

0 100 200 300 400 500

120590 (MPa)

0008

0009

001

0011

0012

120579

120579

(b) Material parameter 120579

Figure 2 Relationships of the martensite start temperature119872119904

and material parameter 120579 with the variation to equivalent stress 120590

The corresponding relation between the equivalent stress120590 and starting temperature of martensite transformation119872

119904and the relationship between material parameter 120579 and

equivalent stress 120590 are shown in Figure 2In general the martensite start temperature119872

119904rises with

increasing tensile and compressive normal stresses as well asshear stresses The hydrostatic component always decreasesthe martensite start temperature In the present paper (3) isused to model the austenite to martensite reaction

3 Nonlinear Large-Deformation FEM Formulaof Hot Forming

Hot stamping is a coupled thermomechanical forming pro-cess with intended phase transformation During the solid-state phase transformations heat is released which theninfluences the thermal field Furthermore both the mechan-ical and thermal properties vary with the temperature vari-ation and deformation Consequently a realistic FE modelfor full process numerical simulationmust consider the inter-action among the mechanical thermal and microstructuralfields

31 Thermal Transport Model and FE Formulae

(i) Basic Formula of Heat Transfer in the Hot Forming ProcessThe equation of heat conduction in an isotropic solid is asfollows

119896(

120597

2

119879

120597119909

2

+

120597

2

119879

120597119910

2

+

120597

2

119879

120597119911

2

) + 119902V minus 120588119888120597119879

120597119905

= 0 (4)

where 120588 is the mass density 119888 is the specific heat 119896 is thethermal conductivity and 119902V is the internal heat generationrate per unit volume 119902V includes external heat sources aswell as transformation heat (119902tr) and heat generated by plasticdeformation (119902119901) as

119902

119901

= 119886120590

1015840

119894119895

120576

119894119895 (5)

where 1205901015840119894119895

is the deviatoric stress tensor and 120576

119894119895is the strain

increment

The latent heat 119902tr can be defined by the following equa-tion [16 17]

119902

tr= 120588119871

120597120585

120597119905

(6)

where 119871 is the latent heat of fusion (Jkg) 120585 is the volumefraction martensite transformation

The heat transfer processes between the contacted sur-faces were considered by using the following convectionboundary conditions

minus119896

120597119879

120597119899

= ℎ (119879 minus 119879

119886)

(7)

where 119879120572is the temperature at the contacted surface and ℎ is

the heat transfer coefficient

(ii) Governing Equation for 3D Transient Heat ConductionProcessThe finite element formulation for 3D transient heatconduction is as follows

119862

119890

119879

119890

+ 119870

119890

119879

119890

= 119865

119890

(8)

where119862119890 is the heat capacitymatrix119870119890 is the heat conductionmatrix and

119879

119890 and 119879119890 are the derivatives of element nodaltemperature with respect to time and element nodal temper-ature respectively 119865119890 is the nodal temperature load vectorTheir components are as follows

119862

119890

119894119895

= int

119881119890

120588119888119873

119894119873

119895119889119881

119870

119890

119894119895

= int

119881119890

119896(

120597119873

119894

120597119909

120597119873

119895

120597119909

+

120597119873

119894

120597119910

120597119873

119895

120597119910

+

120597119873

119894

120597119911

120597119873

119895

120597119911

)119889119881

119865

119890

119894

= int

119881119890

119873

119894119902V119889119881 + int

120597119881119890

119873

119894119902119889119860

(9)

where 119902 is the distributed heat flux per unit area and119873119894refers

to the shape function

4 Mathematical Problems in Engineering

X1

X2

X3

e2e3

e1

i

12058531205852

1205851

Figure 3 Quadrilateral temperature shell element

(iii) 3DThermal Shell ElementThe basic equation of temper-ature shell elements as shown in Figure 3 was developed byWang and Tang [18] The temperature shell elements adoptthe 1205851120585

2120585

3curvilinear coordinates with 120585

1 120585

2in the neutral

plane and 1205853perpendicular to the neutral plane Assume that

the inner element temperature field changes in second orderalong the 120585

3direction

119879 (120585

1 120585

2 120585

3 119905) = 119879

0(120585

1 120585

2 119905)

+ 119879

1(120585

1 120585

2 119905) 120585

3+ 119879

2(120585

1 120585

2 119905) 120585

2

3

(10)

where 1198790(120585

1 120585

2 119905) represents neutral surface temperature

and 1198791(120585

1 120585

2 119905) and 119879

2(120585

1 120585

2 119905) can be determined by the

boundary condition on 1205853= plusmn12

(iv) 3DThermal Tetrahedral Element [19]The 3D tetrahedralelement is implemented in the heat conduction simulation oftools It is assumed that temperature is a linear transformationfunction with the coordinates (119909 119910 119911)

119879 = 119886

1+ 119886

2119909 + 119886

3119910 + 119886

4119911 (11)

where 1198861 1198862 1198863 and 119886

4are the undetermined coefficients unit

four coordinates of the nodes 119894 119895119898 and 119897As shown in Figure 4 the shape function of the tetrahe-

dral element is obtained as follows

119873

120585=

1

6119881

119890

(119886

120585+ 119887

120585119909 + 119888

120585119910 + 119889

120585119911) (120585 = 119894 119895 119898 119897) (12)

where 119881119890is the volume of the tetrahedral element and 119886

120585 119887120585

119888

120585 and 119889

120585are the coefficients related to the node locations

32 FE Formula with Static Explicit Algorithm Based onthe continuum theory and virtual work principle [20] thestatic explicit finite element equation of coupled multifield isobtained by the following

[119870

119901] V119890 =

119891

119901 + 119892

119901 +

119891

119890

(13)

l (xl yl zl)

m (xm ym zm)

j (xj yj zj)

i(xi yi zi)

X

Z

YO

Figure 4 Temperature tetrahedral element

where V119890 is the nodal velocity vector and [119870119901] is the mass

matrix with the form of

[119870

119901] = int

119890

[119861]

119879

([

119863

119890119901] minus [119865]) [119861] + [119864]

119879

[119876] [119864] 119889V

119892

119901 = int

119890

[119861]

119879

(

120578

1 + 120577

119875 +

119879 120573

1015840

) 119889V

119891

119901 = int

119886120590

[119873]

119879

119901 119889119886 + int

119890

[119873]

119879

119901 119889V

(14)

where 119901119894and 119901

119894

(119894 = 1 2 3) are the body force and surfaceforce vector [119861] is the strain element and the force vector[119865] takes the form of

119865

119894119895119896119897=

1

2

(120590

119897119895120575

119896119894+ 120590

119896119895120575

119897119894+ 120590

119897119894120575

119896119895+ 120590

119896119894120575

119897119895) (15)

4 Numerical Simulations of U-Shaped Steeland B-Pillar in Hot Stamping

Based on the abovemultifieldmodels FEM and temperaturefield analysis the numerical modeling of the hot stampingprocess is developed and implemented into independentlydeveloped CAE commercial software called KMAS for metalsheet forming Two numerical examples are presented in thissection to show the strong ability and feasibility of the presentnumerical framework to simulate the hot stamping process

According to the three-dimensional temperature fieldcharacteristics of hot forming the proprocessing of 3D toolstypically follows the rules below

(1) We must divide the 3D elements into internal andboundary elements Different surfaces with differentthermal boundary conditions must be defined

(2) The contact surfaces and cooling channel surfaces onthe tools are separated to realize the contact interfacesimulation and treatment of the boundary conditions

Mathematical Problems in Engineering 5

100

200

300

400

500

600

700

120590s

(MPa

)

0 200 400 600 800 1000

T (∘C)

120590s

(a) Yield stress

0 200 400 600 800 1000

T (∘C)

100

120

140

160

180

200

220

E(G

Pa)

E

(b) Elastic modulus

Figure 5 Relationships of yield stress and elastic modulus with increasing of temperature

25

30

35

40

45

50

IW(m

∘ C)

IW

0 200 400 600 800 1000

T (∘C)

(a) Thermal conductivity

0 200 400 600 800 1000

T (∘C)

300

400

500

600

700

800

900

C

C(J

kg ∘ C)

(b) Specific heat

Figure 6 Relationships of heat conductivity and specific heat with increasing of temperature

(3) For the triangular elements on the boundary surfacesthe nodes should be organized with a certain order toguarantee the same normal direction

(4) We assume there is only one surface of each elementat the border The deal to some extent can meet theaccuracy requirements of the thermal simulation

41 Material Properties of 22MnB5 Steel The materials ofU-shaped steel and B-pillar steel are composed of 22MnB5high-strength steel and 45 steel is chosen as the material ofthe tools Figures 5 and 6 show the major mechanical andthermal related properties of 22MnB5 steel with variationof temperature We can see that the yield stress elasticmodulus and thermal conductivity decrease respectively asthe temperature increases Also as the temperature increasesthe variation of specific heat reaches the maximum at 800Kand then remains constant

Punch

Blank holder

BlankDie

X Y

ZETADYNAFORM

Figure 7 Schematic diagram of FE model

42 Numerical Simulation of U-Shaped Steel in the Hot Stamp-ing Process Figure 7 presents the finite element configura-tions of tools and blank for U-shaped steel The thicknessof blank is 119905 = 16mm Heat transfer between the contact

6 Mathematical Problems in Engineering

X Y

Z

X Y

Z

X Y

Z

Heat transfer between blankand tool contact surfaces

Heat transfer between coolingchannels and tools surfaces

Figure 8 Process of split surfaces from 3D tools

XY

Z

XY

Z

Figure 9 Simulation temperature results on different element of blank

interfaces takes place via conduction through the contact-ing spots the conduction through the interstitial gas andradiation across the gaps The convection of blank and toolsto the environment (ℎ

119890= 36W(m2K)) convection from

tools into cooling channels (ℎ119888= 4700W(m2K)) and heat

transfer fromhot blank to tools (120572119888= 5000W(m2K)) are also

considered as shown in Figure 8

The changing of the contact situation between blank andtools leads to changing of blank temperature Three periodscan be summarized to describe the contact behavior

(1) Stamping Phase (0ndash05 s) point 1 is not fully con-tacted with the tools due to being gently warped andthe temperature drops slowlyThe temperatures of theother points decrease greatly

Mathematical Problems in Engineering 7

XY

Z

XY

Z

(Die) (Water channels of die)

Figure 10 Simulation temperature results of die (119905 = 05 s)

X

Y

Z

X

Y

Z

(Punch) (Water channels of punch)

Figure 11 Simulation temperature results of punch (119905 = 05 s)

X

Y

Z

X

Y

Z

(Blank holder) (Water channels of blank holder)

Figure 12 Simulation temperature results of blank holder (119905 = 05 s)

8 Mathematical Problems in Engineering

300

400

500

600

700

800

900

1000

1100

T(K

)

0 1 2 3 4 5 6 7

t (s)

T = 1073K

t = 15 s

T = 350K

Figure 13 Experimental and simulated temperature results of blank

768

0

100

200

300

400

500

600

700

800

120590(M

Pa)

400 600 800 1000

T (K)

0

20

40

60

80

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

(a)

0

100

200

300

400

500

600

700

800

120590(M

Pa)

200 400 600 800 1000

T (K)

0

20

40

60

80

100

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

906

(b)

Figure 14 Equivalent stress and fraction of martensite curve with different temperature

30120583m

Figure 15 Typical microstructure of sample

Mathematical Problems in Engineering 9

3D tools

2D surfaces

Contact surfaces

Channel surfaces

X Y

Z

X Y

Z

X Y

ZX Y

Z

Figure 16 Schematic diagram of tools model

0 2 4 6 8 10

Time (s)

200

300

400

500

600

700

800

900

Tem

pera

ture

(K)

t = 116 sMs

Start quenching at 873KX Y

Z

12

34

5

1

2

3

4

5

X Y

Z

Figure 17 Simulation temperature results of blank

(2) Quenching Phase (05ndash65 s) due to the effect of thecooling channels on the inner surface of the tools thecooling rate of the blank is high (about 300Ks) at thebeginning of the quenching phase

(3) Latent heat of phase transformations is released andthus affects the blank thermal history We observedthat the temperature curve fluctuates at around thetime of 15 s

(i) Results for Thermal Field Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiencyThe temperature distributions of the 3D toolsslices surfaces and cooling channels surfaces are shown inFigures 9ndash12

Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot formingTheexperimental and simulation results are shown in Figure 13A high consistency can be observed for the entire stampingprocess Table 1 presents the maximum temperature valueslocated at the die punch and blank holder

(ii) Results for Equivalent Stress and Fraction of Martensite Inthe process of hot stamping temperature and deformationat the shoulder (a position with large curvature) largelyaffect the steel forming performance Two character pointsare selected to present the variation of equilibrium stressand fraction of martensite with temperature as shown inFigure 14The presence of an applied or internal stress affectsboth the martensite start temperature 119872

119904and the fraction

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

thermomechanical coupledmaterial descriptions for the typeof steel considered

Bergman and Oldenburg implemented the numericalsimulation of hot forming [8] Eriksson et al investigatedthe temperature and strain rate dependence of hot formingmaterial [9] Naderi Bleck and Merklein researched theplastic flow criterion and material parameters under hightemperature [10 11] The theoretical analysis numericalsimulations and experimental studies of hot forming werepresented by Ma and Hu who provided a new method forhot forming die design and technical process design [12 13]

At present most of the forming software programs avail-able perform simplifications by using the rigid shell elementto simulate the 3D tools in the hot forming modeling Suchassumption causes difficulties in simulating the temperaturetransport process In 2012 Jiang Wu et al presented anumerical frame to simulate the thermal transport process byusing a thick shell element of tools [14 15] It can simulatethe simple components such as U-shaped steel with highaccuracy However for the majorities of autopanels thenumerical simulations of temperature domain with complex3D tool surfaces still face many difficulties and challenges

In this paper a fully coupled 3D thermomechanical-phase transformation finite element simulation of the hotstamping process is developed The rate-dependent thermalconstitutive model is implemented in the present softwareframework In terms of general shell finite element andthe 3D tetrahedral finite element analysis method relatedto temperature a coupled thermal transport modeling forcontact interaction between blank and tools is proposedThe hot stamping process of benchmark typical U-shapedsteel and B-pillar steel is simulated by the present numeri-cal program Strong agreements of temperature equivalentstress and fraction of martensite between the simulated andexperimental results indicate the validity and efficiency of thepresentmultifield coupled numericalmodel in simulating thehot stamping process for the automobile industry

2 Key Technologies for Thermomechanical-Phase Transformation Coupled Behavior

21 Constitutive Laws of High-Strength Steel 22MnB5 boronsteel is the most common steel used for hot stamping A briefintroduction to the thermomechanical-phase transformationcoupled constitutive relationship for high-strength steel ispresented in the following section Detailed descriptions canbe found in [13]

In order to model the thermal-elastic-plastic responseduring the hot stamping process an additive decompositionof the total strain increment ( 120576

119894119895) is utilized [13 14] The total

stain 120576119894119895can be expressed with the decomposition rate form

as follows

120576

119894119895= 120576

119890

119894119895

+ 120576

119901

119894119895

+ 120576

th119894119895

+ 120576

tr119894119895

+ 120576

tp119894119895

(1)

where 120576

119890

119894119895

is the elastic strain increment 120576

119901

119894119895

is the plasticstrain increment 120576

th119894119895

is the thermal strain increment 120576

tr119894119895

is the isotropic transformation strain increment and 120576

tp119894119895

is

00 01 02 03 04 05

True strain 120576

100

150

200

250

300

350

400

True

stre

ss120590

(MPa

)

500∘C

550∘C

600∘C

650∘C

700∘C750∘C

800∘C

ExperimentalQuadratic interpolation

22MnB5

h0 = 16mmdTdt = 50 ∘Csd120576dt = 1 sminus1

Figure 1 Stress-strain curves at different evaluated temperaturefrom 500∘C to 800∘C

the transformation induced plasticity increment 120576

tr119894119895

and 120576

tp119894119895

play an important role in hot forming and are expressed as(2) in the traditional quenching treatment

120576

tr= 120573

120585

120576

tp= 119896 (120590) 120590 (1 minus 120585)

120585

(2)

where 120585 is the fraction of martensitic transformation 120573 is thephase transformation volume coefficient and 119896 is the phasetransformation plastic coefficient related to equivalent stress120590

The sensitivity of temperature on the forming behaviorof 22MnB5 is investigated [16] The flow curves shown inFigure 1 indicate that the temperature has a strong influenceon the forming behavior of quenchable steel Temperatureincreasing leads to an appreciable reduction of stress level anda decreased work hardening exponent

For temperature variation from 500∘C to 800∘C rep-resentative true stress-strain curves are displayed for anexemplarily strain rate of 1 sminus1 Other curves under differenttemperatures are obtained by quadratic interpolation usingthe existing experimental curves such as the curves of 600∘Cand 700∘C as shown in Figure 1

22 Martensite Phase Transformation For the martensitephase transformation the relationship between temperatureand phase change is modeled using the equation proposed byMa et al [13] which can be written as follows

120585 = 1 minus exp [minus120579 (120590) (119872119904(120590) minus 119879)] (3)

where 120585 represents the fraction ofmartensitic transformation119872

119904represents the martensite transformationrsquos beginning

temperature 120579 represents the material parameter whichreflects the austenite-martensite transformation rate 120590 rep-resents equivalent stress and 119879 represents temperature

Mathematical Problems in Engineering 3

600

650

700

750

800

850

900

Ms(

K)

0 100 200 300 400 500

120590 (MPa)

Ms

(a) Martensite transformation temperature

0 100 200 300 400 500

120590 (MPa)

0008

0009

001

0011

0012

120579

120579

(b) Material parameter 120579

Figure 2 Relationships of the martensite start temperature119872119904

and material parameter 120579 with the variation to equivalent stress 120590

The corresponding relation between the equivalent stress120590 and starting temperature of martensite transformation119872

119904and the relationship between material parameter 120579 and

equivalent stress 120590 are shown in Figure 2In general the martensite start temperature119872

119904rises with

increasing tensile and compressive normal stresses as well asshear stresses The hydrostatic component always decreasesthe martensite start temperature In the present paper (3) isused to model the austenite to martensite reaction

3 Nonlinear Large-Deformation FEM Formulaof Hot Forming

Hot stamping is a coupled thermomechanical forming pro-cess with intended phase transformation During the solid-state phase transformations heat is released which theninfluences the thermal field Furthermore both the mechan-ical and thermal properties vary with the temperature vari-ation and deformation Consequently a realistic FE modelfor full process numerical simulationmust consider the inter-action among the mechanical thermal and microstructuralfields

31 Thermal Transport Model and FE Formulae

(i) Basic Formula of Heat Transfer in the Hot Forming ProcessThe equation of heat conduction in an isotropic solid is asfollows

119896(

120597

2

119879

120597119909

2

+

120597

2

119879

120597119910

2

+

120597

2

119879

120597119911

2

) + 119902V minus 120588119888120597119879

120597119905

= 0 (4)

where 120588 is the mass density 119888 is the specific heat 119896 is thethermal conductivity and 119902V is the internal heat generationrate per unit volume 119902V includes external heat sources aswell as transformation heat (119902tr) and heat generated by plasticdeformation (119902119901) as

119902

119901

= 119886120590

1015840

119894119895

120576

119894119895 (5)

where 1205901015840119894119895

is the deviatoric stress tensor and 120576

119894119895is the strain

increment

The latent heat 119902tr can be defined by the following equa-tion [16 17]

119902

tr= 120588119871

120597120585

120597119905

(6)

where 119871 is the latent heat of fusion (Jkg) 120585 is the volumefraction martensite transformation

The heat transfer processes between the contacted sur-faces were considered by using the following convectionboundary conditions

minus119896

120597119879

120597119899

= ℎ (119879 minus 119879

119886)

(7)

where 119879120572is the temperature at the contacted surface and ℎ is

the heat transfer coefficient

(ii) Governing Equation for 3D Transient Heat ConductionProcessThe finite element formulation for 3D transient heatconduction is as follows

119862

119890

119879

119890

+ 119870

119890

119879

119890

= 119865

119890

(8)

where119862119890 is the heat capacitymatrix119870119890 is the heat conductionmatrix and

119879

119890 and 119879119890 are the derivatives of element nodaltemperature with respect to time and element nodal temper-ature respectively 119865119890 is the nodal temperature load vectorTheir components are as follows

119862

119890

119894119895

= int

119881119890

120588119888119873

119894119873

119895119889119881

119870

119890

119894119895

= int

119881119890

119896(

120597119873

119894

120597119909

120597119873

119895

120597119909

+

120597119873

119894

120597119910

120597119873

119895

120597119910

+

120597119873

119894

120597119911

120597119873

119895

120597119911

)119889119881

119865

119890

119894

= int

119881119890

119873

119894119902V119889119881 + int

120597119881119890

119873

119894119902119889119860

(9)

where 119902 is the distributed heat flux per unit area and119873119894refers

to the shape function

4 Mathematical Problems in Engineering

X1

X2

X3

e2e3

e1

i

12058531205852

1205851

Figure 3 Quadrilateral temperature shell element

(iii) 3DThermal Shell ElementThe basic equation of temper-ature shell elements as shown in Figure 3 was developed byWang and Tang [18] The temperature shell elements adoptthe 1205851120585

2120585

3curvilinear coordinates with 120585

1 120585

2in the neutral

plane and 1205853perpendicular to the neutral plane Assume that

the inner element temperature field changes in second orderalong the 120585

3direction

119879 (120585

1 120585

2 120585

3 119905) = 119879

0(120585

1 120585

2 119905)

+ 119879

1(120585

1 120585

2 119905) 120585

3+ 119879

2(120585

1 120585

2 119905) 120585

2

3

(10)

where 1198790(120585

1 120585

2 119905) represents neutral surface temperature

and 1198791(120585

1 120585

2 119905) and 119879

2(120585

1 120585

2 119905) can be determined by the

boundary condition on 1205853= plusmn12

(iv) 3DThermal Tetrahedral Element [19]The 3D tetrahedralelement is implemented in the heat conduction simulation oftools It is assumed that temperature is a linear transformationfunction with the coordinates (119909 119910 119911)

119879 = 119886

1+ 119886

2119909 + 119886

3119910 + 119886

4119911 (11)

where 1198861 1198862 1198863 and 119886

4are the undetermined coefficients unit

four coordinates of the nodes 119894 119895119898 and 119897As shown in Figure 4 the shape function of the tetrahe-

dral element is obtained as follows

119873

120585=

1

6119881

119890

(119886

120585+ 119887

120585119909 + 119888

120585119910 + 119889

120585119911) (120585 = 119894 119895 119898 119897) (12)

where 119881119890is the volume of the tetrahedral element and 119886

120585 119887120585

119888

120585 and 119889

120585are the coefficients related to the node locations

32 FE Formula with Static Explicit Algorithm Based onthe continuum theory and virtual work principle [20] thestatic explicit finite element equation of coupled multifield isobtained by the following

[119870

119901] V119890 =

119891

119901 + 119892

119901 +

119891

119890

(13)

l (xl yl zl)

m (xm ym zm)

j (xj yj zj)

i(xi yi zi)

X

Z

YO

Figure 4 Temperature tetrahedral element

where V119890 is the nodal velocity vector and [119870119901] is the mass

matrix with the form of

[119870

119901] = int

119890

[119861]

119879

([

119863

119890119901] minus [119865]) [119861] + [119864]

119879

[119876] [119864] 119889V

119892

119901 = int

119890

[119861]

119879

(

120578

1 + 120577

119875 +

119879 120573

1015840

) 119889V

119891

119901 = int

119886120590

[119873]

119879

119901 119889119886 + int

119890

[119873]

119879

119901 119889V

(14)

where 119901119894and 119901

119894

(119894 = 1 2 3) are the body force and surfaceforce vector [119861] is the strain element and the force vector[119865] takes the form of

119865

119894119895119896119897=

1

2

(120590

119897119895120575

119896119894+ 120590

119896119895120575

119897119894+ 120590

119897119894120575

119896119895+ 120590

119896119894120575

119897119895) (15)

4 Numerical Simulations of U-Shaped Steeland B-Pillar in Hot Stamping

Based on the abovemultifieldmodels FEM and temperaturefield analysis the numerical modeling of the hot stampingprocess is developed and implemented into independentlydeveloped CAE commercial software called KMAS for metalsheet forming Two numerical examples are presented in thissection to show the strong ability and feasibility of the presentnumerical framework to simulate the hot stamping process

According to the three-dimensional temperature fieldcharacteristics of hot forming the proprocessing of 3D toolstypically follows the rules below

(1) We must divide the 3D elements into internal andboundary elements Different surfaces with differentthermal boundary conditions must be defined

(2) The contact surfaces and cooling channel surfaces onthe tools are separated to realize the contact interfacesimulation and treatment of the boundary conditions

Mathematical Problems in Engineering 5

100

200

300

400

500

600

700

120590s

(MPa

)

0 200 400 600 800 1000

T (∘C)

120590s

(a) Yield stress

0 200 400 600 800 1000

T (∘C)

100

120

140

160

180

200

220

E(G

Pa)

E

(b) Elastic modulus

Figure 5 Relationships of yield stress and elastic modulus with increasing of temperature

25

30

35

40

45

50

IW(m

∘ C)

IW

0 200 400 600 800 1000

T (∘C)

(a) Thermal conductivity

0 200 400 600 800 1000

T (∘C)

300

400

500

600

700

800

900

C

C(J

kg ∘ C)

(b) Specific heat

Figure 6 Relationships of heat conductivity and specific heat with increasing of temperature

(3) For the triangular elements on the boundary surfacesthe nodes should be organized with a certain order toguarantee the same normal direction

(4) We assume there is only one surface of each elementat the border The deal to some extent can meet theaccuracy requirements of the thermal simulation

41 Material Properties of 22MnB5 Steel The materials ofU-shaped steel and B-pillar steel are composed of 22MnB5high-strength steel and 45 steel is chosen as the material ofthe tools Figures 5 and 6 show the major mechanical andthermal related properties of 22MnB5 steel with variationof temperature We can see that the yield stress elasticmodulus and thermal conductivity decrease respectively asthe temperature increases Also as the temperature increasesthe variation of specific heat reaches the maximum at 800Kand then remains constant

Punch

Blank holder

BlankDie

X Y

ZETADYNAFORM

Figure 7 Schematic diagram of FE model

42 Numerical Simulation of U-Shaped Steel in the Hot Stamp-ing Process Figure 7 presents the finite element configura-tions of tools and blank for U-shaped steel The thicknessof blank is 119905 = 16mm Heat transfer between the contact

6 Mathematical Problems in Engineering

X Y

Z

X Y

Z

X Y

Z

Heat transfer between blankand tool contact surfaces

Heat transfer between coolingchannels and tools surfaces

Figure 8 Process of split surfaces from 3D tools

XY

Z

XY

Z

Figure 9 Simulation temperature results on different element of blank

interfaces takes place via conduction through the contact-ing spots the conduction through the interstitial gas andradiation across the gaps The convection of blank and toolsto the environment (ℎ

119890= 36W(m2K)) convection from

tools into cooling channels (ℎ119888= 4700W(m2K)) and heat

transfer fromhot blank to tools (120572119888= 5000W(m2K)) are also

considered as shown in Figure 8

The changing of the contact situation between blank andtools leads to changing of blank temperature Three periodscan be summarized to describe the contact behavior

(1) Stamping Phase (0ndash05 s) point 1 is not fully con-tacted with the tools due to being gently warped andthe temperature drops slowlyThe temperatures of theother points decrease greatly

Mathematical Problems in Engineering 7

XY

Z

XY

Z

(Die) (Water channels of die)

Figure 10 Simulation temperature results of die (119905 = 05 s)

X

Y

Z

X

Y

Z

(Punch) (Water channels of punch)

Figure 11 Simulation temperature results of punch (119905 = 05 s)

X

Y

Z

X

Y

Z

(Blank holder) (Water channels of blank holder)

Figure 12 Simulation temperature results of blank holder (119905 = 05 s)

8 Mathematical Problems in Engineering

300

400

500

600

700

800

900

1000

1100

T(K

)

0 1 2 3 4 5 6 7

t (s)

T = 1073K

t = 15 s

T = 350K

Figure 13 Experimental and simulated temperature results of blank

768

0

100

200

300

400

500

600

700

800

120590(M

Pa)

400 600 800 1000

T (K)

0

20

40

60

80

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

(a)

0

100

200

300

400

500

600

700

800

120590(M

Pa)

200 400 600 800 1000

T (K)

0

20

40

60

80

100

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

906

(b)

Figure 14 Equivalent stress and fraction of martensite curve with different temperature

30120583m

Figure 15 Typical microstructure of sample

Mathematical Problems in Engineering 9

3D tools

2D surfaces

Contact surfaces

Channel surfaces

X Y

Z

X Y

Z

X Y

ZX Y

Z

Figure 16 Schematic diagram of tools model

0 2 4 6 8 10

Time (s)

200

300

400

500

600

700

800

900

Tem

pera

ture

(K)

t = 116 sMs

Start quenching at 873KX Y

Z

12

34

5

1

2

3

4

5

X Y

Z

Figure 17 Simulation temperature results of blank

(2) Quenching Phase (05ndash65 s) due to the effect of thecooling channels on the inner surface of the tools thecooling rate of the blank is high (about 300Ks) at thebeginning of the quenching phase

(3) Latent heat of phase transformations is released andthus affects the blank thermal history We observedthat the temperature curve fluctuates at around thetime of 15 s

(i) Results for Thermal Field Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiencyThe temperature distributions of the 3D toolsslices surfaces and cooling channels surfaces are shown inFigures 9ndash12

Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot formingTheexperimental and simulation results are shown in Figure 13A high consistency can be observed for the entire stampingprocess Table 1 presents the maximum temperature valueslocated at the die punch and blank holder

(ii) Results for Equivalent Stress and Fraction of Martensite Inthe process of hot stamping temperature and deformationat the shoulder (a position with large curvature) largelyaffect the steel forming performance Two character pointsare selected to present the variation of equilibrium stressand fraction of martensite with temperature as shown inFigure 14The presence of an applied or internal stress affectsboth the martensite start temperature 119872

119904and the fraction

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

600

650

700

750

800

850

900

Ms(

K)

0 100 200 300 400 500

120590 (MPa)

Ms

(a) Martensite transformation temperature

0 100 200 300 400 500

120590 (MPa)

0008

0009

001

0011

0012

120579

120579

(b) Material parameter 120579

Figure 2 Relationships of the martensite start temperature119872119904

and material parameter 120579 with the variation to equivalent stress 120590

The corresponding relation between the equivalent stress120590 and starting temperature of martensite transformation119872

119904and the relationship between material parameter 120579 and

equivalent stress 120590 are shown in Figure 2In general the martensite start temperature119872

119904rises with

increasing tensile and compressive normal stresses as well asshear stresses The hydrostatic component always decreasesthe martensite start temperature In the present paper (3) isused to model the austenite to martensite reaction

3 Nonlinear Large-Deformation FEM Formulaof Hot Forming

Hot stamping is a coupled thermomechanical forming pro-cess with intended phase transformation During the solid-state phase transformations heat is released which theninfluences the thermal field Furthermore both the mechan-ical and thermal properties vary with the temperature vari-ation and deformation Consequently a realistic FE modelfor full process numerical simulationmust consider the inter-action among the mechanical thermal and microstructuralfields

31 Thermal Transport Model and FE Formulae

(i) Basic Formula of Heat Transfer in the Hot Forming ProcessThe equation of heat conduction in an isotropic solid is asfollows

119896(

120597

2

119879

120597119909

2

+

120597

2

119879

120597119910

2

+

120597

2

119879

120597119911

2

) + 119902V minus 120588119888120597119879

120597119905

= 0 (4)

where 120588 is the mass density 119888 is the specific heat 119896 is thethermal conductivity and 119902V is the internal heat generationrate per unit volume 119902V includes external heat sources aswell as transformation heat (119902tr) and heat generated by plasticdeformation (119902119901) as

119902

119901

= 119886120590

1015840

119894119895

120576

119894119895 (5)

where 1205901015840119894119895

is the deviatoric stress tensor and 120576

119894119895is the strain

increment

The latent heat 119902tr can be defined by the following equa-tion [16 17]

119902

tr= 120588119871

120597120585

120597119905

(6)

where 119871 is the latent heat of fusion (Jkg) 120585 is the volumefraction martensite transformation

The heat transfer processes between the contacted sur-faces were considered by using the following convectionboundary conditions

minus119896

120597119879

120597119899

= ℎ (119879 minus 119879

119886)

(7)

where 119879120572is the temperature at the contacted surface and ℎ is

the heat transfer coefficient

(ii) Governing Equation for 3D Transient Heat ConductionProcessThe finite element formulation for 3D transient heatconduction is as follows

119862

119890

119879

119890

+ 119870

119890

119879

119890

= 119865

119890

(8)

where119862119890 is the heat capacitymatrix119870119890 is the heat conductionmatrix and

119879

119890 and 119879119890 are the derivatives of element nodaltemperature with respect to time and element nodal temper-ature respectively 119865119890 is the nodal temperature load vectorTheir components are as follows

119862

119890

119894119895

= int

119881119890

120588119888119873

119894119873

119895119889119881

119870

119890

119894119895

= int

119881119890

119896(

120597119873

119894

120597119909

120597119873

119895

120597119909

+

120597119873

119894

120597119910

120597119873

119895

120597119910

+

120597119873

119894

120597119911

120597119873

119895

120597119911

)119889119881

119865

119890

119894

= int

119881119890

119873

119894119902V119889119881 + int

120597119881119890

119873

119894119902119889119860

(9)

where 119902 is the distributed heat flux per unit area and119873119894refers

to the shape function

4 Mathematical Problems in Engineering

X1

X2

X3

e2e3

e1

i

12058531205852

1205851

Figure 3 Quadrilateral temperature shell element

(iii) 3DThermal Shell ElementThe basic equation of temper-ature shell elements as shown in Figure 3 was developed byWang and Tang [18] The temperature shell elements adoptthe 1205851120585

2120585

3curvilinear coordinates with 120585

1 120585

2in the neutral

plane and 1205853perpendicular to the neutral plane Assume that

the inner element temperature field changes in second orderalong the 120585

3direction

119879 (120585

1 120585

2 120585

3 119905) = 119879

0(120585

1 120585

2 119905)

+ 119879

1(120585

1 120585

2 119905) 120585

3+ 119879

2(120585

1 120585

2 119905) 120585

2

3

(10)

where 1198790(120585

1 120585

2 119905) represents neutral surface temperature

and 1198791(120585

1 120585

2 119905) and 119879

2(120585

1 120585

2 119905) can be determined by the

boundary condition on 1205853= plusmn12

(iv) 3DThermal Tetrahedral Element [19]The 3D tetrahedralelement is implemented in the heat conduction simulation oftools It is assumed that temperature is a linear transformationfunction with the coordinates (119909 119910 119911)

119879 = 119886

1+ 119886

2119909 + 119886

3119910 + 119886

4119911 (11)

where 1198861 1198862 1198863 and 119886

4are the undetermined coefficients unit

four coordinates of the nodes 119894 119895119898 and 119897As shown in Figure 4 the shape function of the tetrahe-

dral element is obtained as follows

119873

120585=

1

6119881

119890

(119886

120585+ 119887

120585119909 + 119888

120585119910 + 119889

120585119911) (120585 = 119894 119895 119898 119897) (12)

where 119881119890is the volume of the tetrahedral element and 119886

120585 119887120585

119888

120585 and 119889

120585are the coefficients related to the node locations

32 FE Formula with Static Explicit Algorithm Based onthe continuum theory and virtual work principle [20] thestatic explicit finite element equation of coupled multifield isobtained by the following

[119870

119901] V119890 =

119891

119901 + 119892

119901 +

119891

119890

(13)

l (xl yl zl)

m (xm ym zm)

j (xj yj zj)

i(xi yi zi)

X

Z

YO

Figure 4 Temperature tetrahedral element

where V119890 is the nodal velocity vector and [119870119901] is the mass

matrix with the form of

[119870

119901] = int

119890

[119861]

119879

([

119863

119890119901] minus [119865]) [119861] + [119864]

119879

[119876] [119864] 119889V

119892

119901 = int

119890

[119861]

119879

(

120578

1 + 120577

119875 +

119879 120573

1015840

) 119889V

119891

119901 = int

119886120590

[119873]

119879

119901 119889119886 + int

119890

[119873]

119879

119901 119889V

(14)

where 119901119894and 119901

119894

(119894 = 1 2 3) are the body force and surfaceforce vector [119861] is the strain element and the force vector[119865] takes the form of

119865

119894119895119896119897=

1

2

(120590

119897119895120575

119896119894+ 120590

119896119895120575

119897119894+ 120590

119897119894120575

119896119895+ 120590

119896119894120575

119897119895) (15)

4 Numerical Simulations of U-Shaped Steeland B-Pillar in Hot Stamping

Based on the abovemultifieldmodels FEM and temperaturefield analysis the numerical modeling of the hot stampingprocess is developed and implemented into independentlydeveloped CAE commercial software called KMAS for metalsheet forming Two numerical examples are presented in thissection to show the strong ability and feasibility of the presentnumerical framework to simulate the hot stamping process

According to the three-dimensional temperature fieldcharacteristics of hot forming the proprocessing of 3D toolstypically follows the rules below

(1) We must divide the 3D elements into internal andboundary elements Different surfaces with differentthermal boundary conditions must be defined

(2) The contact surfaces and cooling channel surfaces onthe tools are separated to realize the contact interfacesimulation and treatment of the boundary conditions

Mathematical Problems in Engineering 5

100

200

300

400

500

600

700

120590s

(MPa

)

0 200 400 600 800 1000

T (∘C)

120590s

(a) Yield stress

0 200 400 600 800 1000

T (∘C)

100

120

140

160

180

200

220

E(G

Pa)

E

(b) Elastic modulus

Figure 5 Relationships of yield stress and elastic modulus with increasing of temperature

25

30

35

40

45

50

IW(m

∘ C)

IW

0 200 400 600 800 1000

T (∘C)

(a) Thermal conductivity

0 200 400 600 800 1000

T (∘C)

300

400

500

600

700

800

900

C

C(J

kg ∘ C)

(b) Specific heat

Figure 6 Relationships of heat conductivity and specific heat with increasing of temperature

(3) For the triangular elements on the boundary surfacesthe nodes should be organized with a certain order toguarantee the same normal direction

(4) We assume there is only one surface of each elementat the border The deal to some extent can meet theaccuracy requirements of the thermal simulation

41 Material Properties of 22MnB5 Steel The materials ofU-shaped steel and B-pillar steel are composed of 22MnB5high-strength steel and 45 steel is chosen as the material ofthe tools Figures 5 and 6 show the major mechanical andthermal related properties of 22MnB5 steel with variationof temperature We can see that the yield stress elasticmodulus and thermal conductivity decrease respectively asthe temperature increases Also as the temperature increasesthe variation of specific heat reaches the maximum at 800Kand then remains constant

Punch

Blank holder

BlankDie

X Y

ZETADYNAFORM

Figure 7 Schematic diagram of FE model

42 Numerical Simulation of U-Shaped Steel in the Hot Stamp-ing Process Figure 7 presents the finite element configura-tions of tools and blank for U-shaped steel The thicknessof blank is 119905 = 16mm Heat transfer between the contact

6 Mathematical Problems in Engineering

X Y

Z

X Y

Z

X Y

Z

Heat transfer between blankand tool contact surfaces

Heat transfer between coolingchannels and tools surfaces

Figure 8 Process of split surfaces from 3D tools

XY

Z

XY

Z

Figure 9 Simulation temperature results on different element of blank

interfaces takes place via conduction through the contact-ing spots the conduction through the interstitial gas andradiation across the gaps The convection of blank and toolsto the environment (ℎ

119890= 36W(m2K)) convection from

tools into cooling channels (ℎ119888= 4700W(m2K)) and heat

transfer fromhot blank to tools (120572119888= 5000W(m2K)) are also

considered as shown in Figure 8

The changing of the contact situation between blank andtools leads to changing of blank temperature Three periodscan be summarized to describe the contact behavior

(1) Stamping Phase (0ndash05 s) point 1 is not fully con-tacted with the tools due to being gently warped andthe temperature drops slowlyThe temperatures of theother points decrease greatly

Mathematical Problems in Engineering 7

XY

Z

XY

Z

(Die) (Water channels of die)

Figure 10 Simulation temperature results of die (119905 = 05 s)

X

Y

Z

X

Y

Z

(Punch) (Water channels of punch)

Figure 11 Simulation temperature results of punch (119905 = 05 s)

X

Y

Z

X

Y

Z

(Blank holder) (Water channels of blank holder)

Figure 12 Simulation temperature results of blank holder (119905 = 05 s)

8 Mathematical Problems in Engineering

300

400

500

600

700

800

900

1000

1100

T(K

)

0 1 2 3 4 5 6 7

t (s)

T = 1073K

t = 15 s

T = 350K

Figure 13 Experimental and simulated temperature results of blank

768

0

100

200

300

400

500

600

700

800

120590(M

Pa)

400 600 800 1000

T (K)

0

20

40

60

80

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

(a)

0

100

200

300

400

500

600

700

800

120590(M

Pa)

200 400 600 800 1000

T (K)

0

20

40

60

80

100

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

906

(b)

Figure 14 Equivalent stress and fraction of martensite curve with different temperature

30120583m

Figure 15 Typical microstructure of sample

Mathematical Problems in Engineering 9

3D tools

2D surfaces

Contact surfaces

Channel surfaces

X Y

Z

X Y

Z

X Y

ZX Y

Z

Figure 16 Schematic diagram of tools model

0 2 4 6 8 10

Time (s)

200

300

400

500

600

700

800

900

Tem

pera

ture

(K)

t = 116 sMs

Start quenching at 873KX Y

Z

12

34

5

1

2

3

4

5

X Y

Z

Figure 17 Simulation temperature results of blank

(2) Quenching Phase (05ndash65 s) due to the effect of thecooling channels on the inner surface of the tools thecooling rate of the blank is high (about 300Ks) at thebeginning of the quenching phase

(3) Latent heat of phase transformations is released andthus affects the blank thermal history We observedthat the temperature curve fluctuates at around thetime of 15 s

(i) Results for Thermal Field Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiencyThe temperature distributions of the 3D toolsslices surfaces and cooling channels surfaces are shown inFigures 9ndash12

Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot formingTheexperimental and simulation results are shown in Figure 13A high consistency can be observed for the entire stampingprocess Table 1 presents the maximum temperature valueslocated at the die punch and blank holder

(ii) Results for Equivalent Stress and Fraction of Martensite Inthe process of hot stamping temperature and deformationat the shoulder (a position with large curvature) largelyaffect the steel forming performance Two character pointsare selected to present the variation of equilibrium stressand fraction of martensite with temperature as shown inFigure 14The presence of an applied or internal stress affectsboth the martensite start temperature 119872

119904and the fraction

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

X1

X2

X3

e2e3

e1

i

12058531205852

1205851

Figure 3 Quadrilateral temperature shell element

(iii) 3DThermal Shell ElementThe basic equation of temper-ature shell elements as shown in Figure 3 was developed byWang and Tang [18] The temperature shell elements adoptthe 1205851120585

2120585

3curvilinear coordinates with 120585

1 120585

2in the neutral

plane and 1205853perpendicular to the neutral plane Assume that

the inner element temperature field changes in second orderalong the 120585

3direction

119879 (120585

1 120585

2 120585

3 119905) = 119879

0(120585

1 120585

2 119905)

+ 119879

1(120585

1 120585

2 119905) 120585

3+ 119879

2(120585

1 120585

2 119905) 120585

2

3

(10)

where 1198790(120585

1 120585

2 119905) represents neutral surface temperature

and 1198791(120585

1 120585

2 119905) and 119879

2(120585

1 120585

2 119905) can be determined by the

boundary condition on 1205853= plusmn12

(iv) 3DThermal Tetrahedral Element [19]The 3D tetrahedralelement is implemented in the heat conduction simulation oftools It is assumed that temperature is a linear transformationfunction with the coordinates (119909 119910 119911)

119879 = 119886

1+ 119886

2119909 + 119886

3119910 + 119886

4119911 (11)

where 1198861 1198862 1198863 and 119886

4are the undetermined coefficients unit

four coordinates of the nodes 119894 119895119898 and 119897As shown in Figure 4 the shape function of the tetrahe-

dral element is obtained as follows

119873

120585=

1

6119881

119890

(119886

120585+ 119887

120585119909 + 119888

120585119910 + 119889

120585119911) (120585 = 119894 119895 119898 119897) (12)

where 119881119890is the volume of the tetrahedral element and 119886

120585 119887120585

119888

120585 and 119889

120585are the coefficients related to the node locations

32 FE Formula with Static Explicit Algorithm Based onthe continuum theory and virtual work principle [20] thestatic explicit finite element equation of coupled multifield isobtained by the following

[119870

119901] V119890 =

119891

119901 + 119892

119901 +

119891

119890

(13)

l (xl yl zl)

m (xm ym zm)

j (xj yj zj)

i(xi yi zi)

X

Z

YO

Figure 4 Temperature tetrahedral element

where V119890 is the nodal velocity vector and [119870119901] is the mass

matrix with the form of

[119870

119901] = int

119890

[119861]

119879

([

119863

119890119901] minus [119865]) [119861] + [119864]

119879

[119876] [119864] 119889V

119892

119901 = int

119890

[119861]

119879

(

120578

1 + 120577

119875 +

119879 120573

1015840

) 119889V

119891

119901 = int

119886120590

[119873]

119879

119901 119889119886 + int

119890

[119873]

119879

119901 119889V

(14)

where 119901119894and 119901

119894

(119894 = 1 2 3) are the body force and surfaceforce vector [119861] is the strain element and the force vector[119865] takes the form of

119865

119894119895119896119897=

1

2

(120590

119897119895120575

119896119894+ 120590

119896119895120575

119897119894+ 120590

119897119894120575

119896119895+ 120590

119896119894120575

119897119895) (15)

4 Numerical Simulations of U-Shaped Steeland B-Pillar in Hot Stamping

Based on the abovemultifieldmodels FEM and temperaturefield analysis the numerical modeling of the hot stampingprocess is developed and implemented into independentlydeveloped CAE commercial software called KMAS for metalsheet forming Two numerical examples are presented in thissection to show the strong ability and feasibility of the presentnumerical framework to simulate the hot stamping process

According to the three-dimensional temperature fieldcharacteristics of hot forming the proprocessing of 3D toolstypically follows the rules below

(1) We must divide the 3D elements into internal andboundary elements Different surfaces with differentthermal boundary conditions must be defined

(2) The contact surfaces and cooling channel surfaces onthe tools are separated to realize the contact interfacesimulation and treatment of the boundary conditions

Mathematical Problems in Engineering 5

100

200

300

400

500

600

700

120590s

(MPa

)

0 200 400 600 800 1000

T (∘C)

120590s

(a) Yield stress

0 200 400 600 800 1000

T (∘C)

100

120

140

160

180

200

220

E(G

Pa)

E

(b) Elastic modulus

Figure 5 Relationships of yield stress and elastic modulus with increasing of temperature

25

30

35

40

45

50

IW(m

∘ C)

IW

0 200 400 600 800 1000

T (∘C)

(a) Thermal conductivity

0 200 400 600 800 1000

T (∘C)

300

400

500

600

700

800

900

C

C(J

kg ∘ C)

(b) Specific heat

Figure 6 Relationships of heat conductivity and specific heat with increasing of temperature

(3) For the triangular elements on the boundary surfacesthe nodes should be organized with a certain order toguarantee the same normal direction

(4) We assume there is only one surface of each elementat the border The deal to some extent can meet theaccuracy requirements of the thermal simulation

41 Material Properties of 22MnB5 Steel The materials ofU-shaped steel and B-pillar steel are composed of 22MnB5high-strength steel and 45 steel is chosen as the material ofthe tools Figures 5 and 6 show the major mechanical andthermal related properties of 22MnB5 steel with variationof temperature We can see that the yield stress elasticmodulus and thermal conductivity decrease respectively asthe temperature increases Also as the temperature increasesthe variation of specific heat reaches the maximum at 800Kand then remains constant

Punch

Blank holder

BlankDie

X Y

ZETADYNAFORM

Figure 7 Schematic diagram of FE model

42 Numerical Simulation of U-Shaped Steel in the Hot Stamp-ing Process Figure 7 presents the finite element configura-tions of tools and blank for U-shaped steel The thicknessof blank is 119905 = 16mm Heat transfer between the contact

6 Mathematical Problems in Engineering

X Y

Z

X Y

Z

X Y

Z

Heat transfer between blankand tool contact surfaces

Heat transfer between coolingchannels and tools surfaces

Figure 8 Process of split surfaces from 3D tools

XY

Z

XY

Z

Figure 9 Simulation temperature results on different element of blank

interfaces takes place via conduction through the contact-ing spots the conduction through the interstitial gas andradiation across the gaps The convection of blank and toolsto the environment (ℎ

119890= 36W(m2K)) convection from

tools into cooling channels (ℎ119888= 4700W(m2K)) and heat

transfer fromhot blank to tools (120572119888= 5000W(m2K)) are also

considered as shown in Figure 8

The changing of the contact situation between blank andtools leads to changing of blank temperature Three periodscan be summarized to describe the contact behavior

(1) Stamping Phase (0ndash05 s) point 1 is not fully con-tacted with the tools due to being gently warped andthe temperature drops slowlyThe temperatures of theother points decrease greatly

Mathematical Problems in Engineering 7

XY

Z

XY

Z

(Die) (Water channels of die)

Figure 10 Simulation temperature results of die (119905 = 05 s)

X

Y

Z

X

Y

Z

(Punch) (Water channels of punch)

Figure 11 Simulation temperature results of punch (119905 = 05 s)

X

Y

Z

X

Y

Z

(Blank holder) (Water channels of blank holder)

Figure 12 Simulation temperature results of blank holder (119905 = 05 s)

8 Mathematical Problems in Engineering

300

400

500

600

700

800

900

1000

1100

T(K

)

0 1 2 3 4 5 6 7

t (s)

T = 1073K

t = 15 s

T = 350K

Figure 13 Experimental and simulated temperature results of blank

768

0

100

200

300

400

500

600

700

800

120590(M

Pa)

400 600 800 1000

T (K)

0

20

40

60

80

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

(a)

0

100

200

300

400

500

600

700

800

120590(M

Pa)

200 400 600 800 1000

T (K)

0

20

40

60

80

100

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

906

(b)

Figure 14 Equivalent stress and fraction of martensite curve with different temperature

30120583m

Figure 15 Typical microstructure of sample

Mathematical Problems in Engineering 9

3D tools

2D surfaces

Contact surfaces

Channel surfaces

X Y

Z

X Y

Z

X Y

ZX Y

Z

Figure 16 Schematic diagram of tools model

0 2 4 6 8 10

Time (s)

200

300

400

500

600

700

800

900

Tem

pera

ture

(K)

t = 116 sMs

Start quenching at 873KX Y

Z

12

34

5

1

2

3

4

5

X Y

Z

Figure 17 Simulation temperature results of blank

(2) Quenching Phase (05ndash65 s) due to the effect of thecooling channels on the inner surface of the tools thecooling rate of the blank is high (about 300Ks) at thebeginning of the quenching phase

(3) Latent heat of phase transformations is released andthus affects the blank thermal history We observedthat the temperature curve fluctuates at around thetime of 15 s

(i) Results for Thermal Field Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiencyThe temperature distributions of the 3D toolsslices surfaces and cooling channels surfaces are shown inFigures 9ndash12

Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot formingTheexperimental and simulation results are shown in Figure 13A high consistency can be observed for the entire stampingprocess Table 1 presents the maximum temperature valueslocated at the die punch and blank holder

(ii) Results for Equivalent Stress and Fraction of Martensite Inthe process of hot stamping temperature and deformationat the shoulder (a position with large curvature) largelyaffect the steel forming performance Two character pointsare selected to present the variation of equilibrium stressand fraction of martensite with temperature as shown inFigure 14The presence of an applied or internal stress affectsboth the martensite start temperature 119872

119904and the fraction

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

100

200

300

400

500

600

700

120590s

(MPa

)

0 200 400 600 800 1000

T (∘C)

120590s

(a) Yield stress

0 200 400 600 800 1000

T (∘C)

100

120

140

160

180

200

220

E(G

Pa)

E

(b) Elastic modulus

Figure 5 Relationships of yield stress and elastic modulus with increasing of temperature

25

30

35

40

45

50

IW(m

∘ C)

IW

0 200 400 600 800 1000

T (∘C)

(a) Thermal conductivity

0 200 400 600 800 1000

T (∘C)

300

400

500

600

700

800

900

C

C(J

kg ∘ C)

(b) Specific heat

Figure 6 Relationships of heat conductivity and specific heat with increasing of temperature

(3) For the triangular elements on the boundary surfacesthe nodes should be organized with a certain order toguarantee the same normal direction

(4) We assume there is only one surface of each elementat the border The deal to some extent can meet theaccuracy requirements of the thermal simulation

41 Material Properties of 22MnB5 Steel The materials ofU-shaped steel and B-pillar steel are composed of 22MnB5high-strength steel and 45 steel is chosen as the material ofthe tools Figures 5 and 6 show the major mechanical andthermal related properties of 22MnB5 steel with variationof temperature We can see that the yield stress elasticmodulus and thermal conductivity decrease respectively asthe temperature increases Also as the temperature increasesthe variation of specific heat reaches the maximum at 800Kand then remains constant

Punch

Blank holder

BlankDie

X Y

ZETADYNAFORM

Figure 7 Schematic diagram of FE model

42 Numerical Simulation of U-Shaped Steel in the Hot Stamp-ing Process Figure 7 presents the finite element configura-tions of tools and blank for U-shaped steel The thicknessof blank is 119905 = 16mm Heat transfer between the contact

6 Mathematical Problems in Engineering

X Y

Z

X Y

Z

X Y

Z

Heat transfer between blankand tool contact surfaces

Heat transfer between coolingchannels and tools surfaces

Figure 8 Process of split surfaces from 3D tools

XY

Z

XY

Z

Figure 9 Simulation temperature results on different element of blank

interfaces takes place via conduction through the contact-ing spots the conduction through the interstitial gas andradiation across the gaps The convection of blank and toolsto the environment (ℎ

119890= 36W(m2K)) convection from

tools into cooling channels (ℎ119888= 4700W(m2K)) and heat

transfer fromhot blank to tools (120572119888= 5000W(m2K)) are also

considered as shown in Figure 8

The changing of the contact situation between blank andtools leads to changing of blank temperature Three periodscan be summarized to describe the contact behavior

(1) Stamping Phase (0ndash05 s) point 1 is not fully con-tacted with the tools due to being gently warped andthe temperature drops slowlyThe temperatures of theother points decrease greatly

Mathematical Problems in Engineering 7

XY

Z

XY

Z

(Die) (Water channels of die)

Figure 10 Simulation temperature results of die (119905 = 05 s)

X

Y

Z

X

Y

Z

(Punch) (Water channels of punch)

Figure 11 Simulation temperature results of punch (119905 = 05 s)

X

Y

Z

X

Y

Z

(Blank holder) (Water channels of blank holder)

Figure 12 Simulation temperature results of blank holder (119905 = 05 s)

8 Mathematical Problems in Engineering

300

400

500

600

700

800

900

1000

1100

T(K

)

0 1 2 3 4 5 6 7

t (s)

T = 1073K

t = 15 s

T = 350K

Figure 13 Experimental and simulated temperature results of blank

768

0

100

200

300

400

500

600

700

800

120590(M

Pa)

400 600 800 1000

T (K)

0

20

40

60

80

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

(a)

0

100

200

300

400

500

600

700

800

120590(M

Pa)

200 400 600 800 1000

T (K)

0

20

40

60

80

100

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

906

(b)

Figure 14 Equivalent stress and fraction of martensite curve with different temperature

30120583m

Figure 15 Typical microstructure of sample

Mathematical Problems in Engineering 9

3D tools

2D surfaces

Contact surfaces

Channel surfaces

X Y

Z

X Y

Z

X Y

ZX Y

Z

Figure 16 Schematic diagram of tools model

0 2 4 6 8 10

Time (s)

200

300

400

500

600

700

800

900

Tem

pera

ture

(K)

t = 116 sMs

Start quenching at 873KX Y

Z

12

34

5

1

2

3

4

5

X Y

Z

Figure 17 Simulation temperature results of blank

(2) Quenching Phase (05ndash65 s) due to the effect of thecooling channels on the inner surface of the tools thecooling rate of the blank is high (about 300Ks) at thebeginning of the quenching phase

(3) Latent heat of phase transformations is released andthus affects the blank thermal history We observedthat the temperature curve fluctuates at around thetime of 15 s

(i) Results for Thermal Field Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiencyThe temperature distributions of the 3D toolsslices surfaces and cooling channels surfaces are shown inFigures 9ndash12

Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot formingTheexperimental and simulation results are shown in Figure 13A high consistency can be observed for the entire stampingprocess Table 1 presents the maximum temperature valueslocated at the die punch and blank holder

(ii) Results for Equivalent Stress and Fraction of Martensite Inthe process of hot stamping temperature and deformationat the shoulder (a position with large curvature) largelyaffect the steel forming performance Two character pointsare selected to present the variation of equilibrium stressand fraction of martensite with temperature as shown inFigure 14The presence of an applied or internal stress affectsboth the martensite start temperature 119872

119904and the fraction

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

X Y

Z

X Y

Z

X Y

Z

Heat transfer between blankand tool contact surfaces

Heat transfer between coolingchannels and tools surfaces

Figure 8 Process of split surfaces from 3D tools

XY

Z

XY

Z

Figure 9 Simulation temperature results on different element of blank

interfaces takes place via conduction through the contact-ing spots the conduction through the interstitial gas andradiation across the gaps The convection of blank and toolsto the environment (ℎ

119890= 36W(m2K)) convection from

tools into cooling channels (ℎ119888= 4700W(m2K)) and heat

transfer fromhot blank to tools (120572119888= 5000W(m2K)) are also

considered as shown in Figure 8

The changing of the contact situation between blank andtools leads to changing of blank temperature Three periodscan be summarized to describe the contact behavior

(1) Stamping Phase (0ndash05 s) point 1 is not fully con-tacted with the tools due to being gently warped andthe temperature drops slowlyThe temperatures of theother points decrease greatly

Mathematical Problems in Engineering 7

XY

Z

XY

Z

(Die) (Water channels of die)

Figure 10 Simulation temperature results of die (119905 = 05 s)

X

Y

Z

X

Y

Z

(Punch) (Water channels of punch)

Figure 11 Simulation temperature results of punch (119905 = 05 s)

X

Y

Z

X

Y

Z

(Blank holder) (Water channels of blank holder)

Figure 12 Simulation temperature results of blank holder (119905 = 05 s)

8 Mathematical Problems in Engineering

300

400

500

600

700

800

900

1000

1100

T(K

)

0 1 2 3 4 5 6 7

t (s)

T = 1073K

t = 15 s

T = 350K

Figure 13 Experimental and simulated temperature results of blank

768

0

100

200

300

400

500

600

700

800

120590(M

Pa)

400 600 800 1000

T (K)

0

20

40

60

80

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

(a)

0

100

200

300

400

500

600

700

800

120590(M

Pa)

200 400 600 800 1000

T (K)

0

20

40

60

80

100

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

906

(b)

Figure 14 Equivalent stress and fraction of martensite curve with different temperature

30120583m

Figure 15 Typical microstructure of sample

Mathematical Problems in Engineering 9

3D tools

2D surfaces

Contact surfaces

Channel surfaces

X Y

Z

X Y

Z

X Y

ZX Y

Z

Figure 16 Schematic diagram of tools model

0 2 4 6 8 10

Time (s)

200

300

400

500

600

700

800

900

Tem

pera

ture

(K)

t = 116 sMs

Start quenching at 873KX Y

Z

12

34

5

1

2

3

4

5

X Y

Z

Figure 17 Simulation temperature results of blank

(2) Quenching Phase (05ndash65 s) due to the effect of thecooling channels on the inner surface of the tools thecooling rate of the blank is high (about 300Ks) at thebeginning of the quenching phase

(3) Latent heat of phase transformations is released andthus affects the blank thermal history We observedthat the temperature curve fluctuates at around thetime of 15 s

(i) Results for Thermal Field Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiencyThe temperature distributions of the 3D toolsslices surfaces and cooling channels surfaces are shown inFigures 9ndash12

Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot formingTheexperimental and simulation results are shown in Figure 13A high consistency can be observed for the entire stampingprocess Table 1 presents the maximum temperature valueslocated at the die punch and blank holder

(ii) Results for Equivalent Stress and Fraction of Martensite Inthe process of hot stamping temperature and deformationat the shoulder (a position with large curvature) largelyaffect the steel forming performance Two character pointsare selected to present the variation of equilibrium stressand fraction of martensite with temperature as shown inFigure 14The presence of an applied or internal stress affectsboth the martensite start temperature 119872

119904and the fraction

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

XY

Z

XY

Z

(Die) (Water channels of die)

Figure 10 Simulation temperature results of die (119905 = 05 s)

X

Y

Z

X

Y

Z

(Punch) (Water channels of punch)

Figure 11 Simulation temperature results of punch (119905 = 05 s)

X

Y

Z

X

Y

Z

(Blank holder) (Water channels of blank holder)

Figure 12 Simulation temperature results of blank holder (119905 = 05 s)

8 Mathematical Problems in Engineering

300

400

500

600

700

800

900

1000

1100

T(K

)

0 1 2 3 4 5 6 7

t (s)

T = 1073K

t = 15 s

T = 350K

Figure 13 Experimental and simulated temperature results of blank

768

0

100

200

300

400

500

600

700

800

120590(M

Pa)

400 600 800 1000

T (K)

0

20

40

60

80

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

(a)

0

100

200

300

400

500

600

700

800

120590(M

Pa)

200 400 600 800 1000

T (K)

0

20

40

60

80

100

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

906

(b)

Figure 14 Equivalent stress and fraction of martensite curve with different temperature

30120583m

Figure 15 Typical microstructure of sample

Mathematical Problems in Engineering 9

3D tools

2D surfaces

Contact surfaces

Channel surfaces

X Y

Z

X Y

Z

X Y

ZX Y

Z

Figure 16 Schematic diagram of tools model

0 2 4 6 8 10

Time (s)

200

300

400

500

600

700

800

900

Tem

pera

ture

(K)

t = 116 sMs

Start quenching at 873KX Y

Z

12

34

5

1

2

3

4

5

X Y

Z

Figure 17 Simulation temperature results of blank

(2) Quenching Phase (05ndash65 s) due to the effect of thecooling channels on the inner surface of the tools thecooling rate of the blank is high (about 300Ks) at thebeginning of the quenching phase

(3) Latent heat of phase transformations is released andthus affects the blank thermal history We observedthat the temperature curve fluctuates at around thetime of 15 s

(i) Results for Thermal Field Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiencyThe temperature distributions of the 3D toolsslices surfaces and cooling channels surfaces are shown inFigures 9ndash12

Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot formingTheexperimental and simulation results are shown in Figure 13A high consistency can be observed for the entire stampingprocess Table 1 presents the maximum temperature valueslocated at the die punch and blank holder

(ii) Results for Equivalent Stress and Fraction of Martensite Inthe process of hot stamping temperature and deformationat the shoulder (a position with large curvature) largelyaffect the steel forming performance Two character pointsare selected to present the variation of equilibrium stressand fraction of martensite with temperature as shown inFigure 14The presence of an applied or internal stress affectsboth the martensite start temperature 119872

119904and the fraction

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

300

400

500

600

700

800

900

1000

1100

T(K

)

0 1 2 3 4 5 6 7

t (s)

T = 1073K

t = 15 s

T = 350K

Figure 13 Experimental and simulated temperature results of blank

768

0

100

200

300

400

500

600

700

800

120590(M

Pa)

400 600 800 1000

T (K)

0

20

40

60

80

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

(a)

0

100

200

300

400

500

600

700

800

120590(M

Pa)

200 400 600 800 1000

T (K)

0

20

40

60

80

100

120585 m(

)

120585m120590

120590 = 223MPaMs = 703K

906

(b)

Figure 14 Equivalent stress and fraction of martensite curve with different temperature

30120583m

Figure 15 Typical microstructure of sample

Mathematical Problems in Engineering 9

3D tools

2D surfaces

Contact surfaces

Channel surfaces

X Y

Z

X Y

Z

X Y

ZX Y

Z

Figure 16 Schematic diagram of tools model

0 2 4 6 8 10

Time (s)

200

300

400

500

600

700

800

900

Tem

pera

ture

(K)

t = 116 sMs

Start quenching at 873KX Y

Z

12

34

5

1

2

3

4

5

X Y

Z

Figure 17 Simulation temperature results of blank

(2) Quenching Phase (05ndash65 s) due to the effect of thecooling channels on the inner surface of the tools thecooling rate of the blank is high (about 300Ks) at thebeginning of the quenching phase

(3) Latent heat of phase transformations is released andthus affects the blank thermal history We observedthat the temperature curve fluctuates at around thetime of 15 s

(i) Results for Thermal Field Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiencyThe temperature distributions of the 3D toolsslices surfaces and cooling channels surfaces are shown inFigures 9ndash12

Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot formingTheexperimental and simulation results are shown in Figure 13A high consistency can be observed for the entire stampingprocess Table 1 presents the maximum temperature valueslocated at the die punch and blank holder

(ii) Results for Equivalent Stress and Fraction of Martensite Inthe process of hot stamping temperature and deformationat the shoulder (a position with large curvature) largelyaffect the steel forming performance Two character pointsare selected to present the variation of equilibrium stressand fraction of martensite with temperature as shown inFigure 14The presence of an applied or internal stress affectsboth the martensite start temperature 119872

119904and the fraction

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

3D tools

2D surfaces

Contact surfaces

Channel surfaces

X Y

Z

X Y

Z

X Y

ZX Y

Z

Figure 16 Schematic diagram of tools model

0 2 4 6 8 10

Time (s)

200

300

400

500

600

700

800

900

Tem

pera

ture

(K)

t = 116 sMs

Start quenching at 873KX Y

Z

12

34

5

1

2

3

4

5

X Y

Z

Figure 17 Simulation temperature results of blank

(2) Quenching Phase (05ndash65 s) due to the effect of thecooling channels on the inner surface of the tools thecooling rate of the blank is high (about 300Ks) at thebeginning of the quenching phase

(3) Latent heat of phase transformations is released andthus affects the blank thermal history We observedthat the temperature curve fluctuates at around thetime of 15 s

(i) Results for Thermal Field Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiencyThe temperature distributions of the 3D toolsslices surfaces and cooling channels surfaces are shown inFigures 9ndash12

Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot formingTheexperimental and simulation results are shown in Figure 13A high consistency can be observed for the entire stampingprocess Table 1 presents the maximum temperature valueslocated at the die punch and blank holder

(ii) Results for Equivalent Stress and Fraction of Martensite Inthe process of hot stamping temperature and deformationat the shoulder (a position with large curvature) largelyaffect the steel forming performance Two character pointsare selected to present the variation of equilibrium stressand fraction of martensite with temperature as shown inFigure 14The presence of an applied or internal stress affectsboth the martensite start temperature 119872

119904and the fraction

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

XY

Z

(a) Temperature distributions of die

0 2 4 6 8 10

Time (s)

1

2

3

4

5

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

(b) Temperature curves of die

XY

Z

XY

Z

(c) Temperature distributions of die with water channels

Figure 18 Simulation temperature results of die (119905 = 056 s)

Table 1 Maximum temperature of tools

Maximum temperature of toolsTools Time Position Value Channel valueDie 05 s Point 2 460K 316KPunch 05 s Point 2 370K 312KBlank holder 05 s Point 1 390K 312K

of martensite formed as a function of the cooling below119872119904

When120590 changes within a certain range119872119904increases with the

increase of 120590It can be seen from Figure 14 that when quenching

begins there is no martensite transformation The amountof martensite transformation increases as time increases

(Figure 14(a)) 6 s later the martensite volume fraction of theblank reaches above 90 (Figure 14(b))

The optical microscopy images of the blank related to thesame position as that in Figure 14(a) are presented in Figure 15after quenching in the final condition Ferrite appeared aswhite bainite appeared as light gray andmartensite appearedas black lath-shape The average martensite volume fractionwas 759

43 Numerical Simulation Results of B-Pillar Steel B-pillarsteel stamping is chosen as the second example The toolmodel design for B-pillar is shown in Figure 16 and the 3Dfinite element model in Figure 17The quenching stages of B-pillar are also simulated by KMAS software

The beginning temperature of the quenching of the blankis 873K Five character points are selected (Figure 18(a)) in

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

X Y

Z

(a) Temperature curves of punch

0 2 4 6 8 10

Time (s)

1

234

Tem

pera

ture

(K)

290

300

310

320

330

340

350

360

370

(b) Temperature curves of die

X Y

Z

X Y

Z

(c) Temperature distributions of punch with water channels

Figure 19 Simulation temperature results of punch (119905 = 056 s)

the surface of the die tool The temperature curves shownin Figure 18(b) present the temperature changing process ofeach pointWe can see from the temperature curve that point2 does not fully contact with the tools due to a gentle warpand the temperature drops slowly The temperatures of theother points decrease greatly Similar with the die as shownin Figure 19 four character points on the punch surface areselected

The maximum temperatures of the tools and coolingchannel appear within a short time after quenching beginsThen due to the continuing cooling effect of the waterchannel the temperature of the tools decreases at a rapidspeed Figure 20 presents the final B-pillar products by thehot stamping process The martensite volume fraction of theblank reaches above 90 as shown by the optical microscopyimage test

Figure 20 High-strength B-pillar steel with hot stamping tech-nique

5 Conclusion

(1) During the stamping phase the blank slides along thetool wall lead to an increase in the tool temperature

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

At the end of this phase the tools reach themaximumtemperature at the shoulder

(2) The temperature of a cooling duct placed closer to thetool surface in a convex area is higher than that in aconcave areaMost of the surfaces of the cooling chan-nels remain at a low temperature in the quenchingprocess

(3) Due to the ignorance of the flow velocity and pressureof water the temperature distributions on the differ-ent slices surfaces are almost the same

It can be concluded that the efficient cooling is mostdesired in convex areas that the geometry of cooling ductsis restricted due to constraints in drilling and that the ductsshould be placed but sufficiently far from the tool contour toavoid any deformation A more accurate contacted conditionand flow analysis for hot stamping remains to be studied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are pleased to acknowledge the support of thiswork by ldquothe Fundamental Research Funds for the CentralUniversities DUT13ZD208rdquo the National Key Basic Researchand Development Program through contract Grant nos2014CB046803 and 2011ZX05026-002-02 and The NationalScienceGroup of China through contract Grant no 51221961

References

[1] N Ma P Hu and W Guo ldquoTechnology and equipment of hotforming for ultra high strength steelrdquo Automobile amp Parts no45 pp 28ndash30 2009

[2] A Makinouchi ldquoSheet metal forming simulation in industryrdquoJournal of Materials Processing Technology vol 60 no 1ndash4 pp19ndash26 1996

[3] P Hu ldquoRate-dependent quasi-flow corner theory for elasticisco-plastic materialsrdquo International Journal of Solids and Struc-tures vol 41 no 5-6 pp 1263ndash1284 2004

[4] Y Lin D Wen J Deng G Liu and J Chen ldquoConstitutivemodels for high-temperature flow behaviors of a Ni-basedsuperalloyrdquoMaterials amp Design vol 59 pp 115ndash123 2014

[5] Z-X Gui W-K Liang Y Liu and Y-S Zhang ldquoThermo-mechanical behavior of the Al-Si alloy coated hot stampingboron steelrdquoMaterials and Design vol 60 pp 26ndash33 2014

[6] E J F R Caron K J Daun and M A Wells ldquoExperimentalheat transfer coefficient measurements during hot forming diequenching of boron steel at high temperaturesrdquo InternationalJournal of Heat and Mass Transfer vol 71 pp 396ndash404 2014

[7] MRaugeiO El Fakir LWang J Lin andDMorrey ldquoLife cycleassessment of the potential environmental benefits of a novelhot forming process in automotive manufacturingrdquo Journal ofCleaner Production vol 83 pp 80ndash86 2014

[8] G Bergman and M Oldenburg ldquoA finite element model forthermomechanical analysis of sheet metal formingrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no9 pp 1167ndash1186 2004

[9] M Eriksson M Oldenburg M C Somani and L P Kar-jalainen ldquoTesting and evaluation of material data for analysis offorming and hardening of boron steel componentsrdquo Modellingand Simulation inMaterials Science and Engineering vol 10 no3 pp 277ndash294 2002

[10] M Naderi V Uthaisangsuk U Prahl and W Bleck ldquoA numer-ical and experimental investigation into hot stamping of boronalloyed heat treated steelsrdquo Steel Research International vol 79no 2 pp 77ndash84 2008

[11] M Merklein and J Lechler ldquoInvestigation of the thermo-mechanical properties of hot stamping steelsrdquo Journal of Mate-rials Processing Technology vol 177 no 1ndash3 pp 452ndash455 2006

[12] Y Chang X H Tang K M Zhao P Hu and Y C Wu ldquoInves-tigation of the factors influencing the interfacial heat transfercoefficient in hot stampingrdquo Journal of Materials ProcessingTechnology 2014

[13] N Ma P Hu W Guo et al ldquoFeasible methods applied tothe design and manufacturing process of hot formingrdquo inProceedings of the International Deep Drawing Research GroupConference (IDDRG 09) pp 835ndash843 Golden Colo USA2009

[14] N Ma W-HWu G-Z Shen and P Hu ldquoStudy of hot formingfor high strength steel numerical simulation-static explicitalgorithmrdquo Chinese Journal of Computational Mechanics vol28 no 3 pp 371ndash376 2011

[15] D-X Jiang W-H Wu and P Hu ldquoResearch on temperaturenumerical simulation of hot stamping process of high strengthsteelrdquo Engineering Mechanics vol 30 no 1 pp 419ndash424 2013

[16] N Ma P Hu and K K Yan ldquoResearch on boron steel forhot forming and its applicationrdquo Chinese Journal of MechanicalEngineering no 46 pp 68ndash72 2010

[17] P Hu H P Liu and Y Q Liu ldquoNumerical study on flangingand spring-back of stamped thick metal platerdquo Acta MechanicaSolida Sinica vol 14 no 4 pp 290ndash298 2001

[18] X C Wang and Y J Tang ldquoA finite element analysis of tem-perature fields in general shellsrdquo Journal of Tsinghua UniversityScience and Technology vol 29 no 5 pp 103ndash112 1989

[19] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[20] P Hu and Y Tomita ldquoDeformation localization for plane straintension of polymersrdquo Acta Mechanica Sinica vol 12 no 5 pp564ndash574 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of