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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 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. This 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.

    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 stress,and 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 [2–5], in the standard hot stamping process asteel blank is heated in a furnace to a certain high temper-ature (880–950∘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 1200–1500MPa. Throughout the entire stamping process,coupling thermomechanical characteristics are exhibited.The 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 phase.The strong rate-dependent constitutive behaviors and tem-perature sensitivities are significant throughout the entirestamping process.Thermal contact and heat transfer betweenthe tools and the blank surface must be considered withaccuracy.The numerical modeling of the full process requires

    Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 982785, 12 pageshttp://dx.doi.org/10.1155/2015/982785

  • 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 proposed.The 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

    2.1. 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 ( ̇𝜀

    𝑖𝑗) is utilized [13, 14]. The total

    stain 𝜀𝑖𝑗can be expressed with the decomposition rate form

    as follows:

    ̇𝜀

    𝑖𝑗= ̇𝜀

    𝑒

    𝑖𝑗

    + ̇𝜀

    𝑝

    𝑖𝑗

    + ̇𝜀

    th𝑖𝑗

    + ̇𝜀

    tr𝑖𝑗

    + ̇𝜀

    tp𝑖𝑗

    , (1)

    where ̇𝜀𝑒𝑖𝑗

    is the elastic strain increment, ̇𝜀𝑝𝑖𝑗

    is the plasticstrain increment, ̇𝜀th

    𝑖𝑗

    is the thermal strain increment, ̇𝜀tr𝑖𝑗

    is the isotropic transformation strain increment, and ̇𝜀tp𝑖𝑗

    is

    0.0 0.1 0.2 0.3 0.4 0.5

    True strain 𝜀

    100

    150

    200

    250

    300

    350

    400

    True

    stre

    ss𝜎

    (MPa

    )

    500∘C

    550∘C

    600∘C

    650∘C

    700∘C750∘C

    800∘C

    ExperimentalQuadratic interpolation

    22MnB5h0 = 1.6mm

    dT/dt = 50 ∘C/sd𝜀/dt = 1 s−1

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

    the transformation induced plasticity increment. ̇𝜀tr𝑖𝑗

    and ̇𝜀tp𝑖𝑗

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

    ̇𝜀

    tr= 𝛽

    ̇

    𝜉,

    ̇𝜀

    tp= 𝑘 (𝜎) 𝜎 (1 − 𝜉)

    ̇

    𝜉,

    (2)

    where 𝜉 is the fraction of martensitic transformation, 𝛽 is thephase transformation volume coefficient, and 𝑘 is the phasetransformation plastic coefficient related to equivalent stress𝜎.

    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 s−1. 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.

    2.2. 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:

    𝜉 = 1 − exp [−𝜃 (𝜎) (𝑀𝑠(𝜎) − 𝑇)] , (3)

    where 𝜉 represents the fraction ofmartensitic transformation,𝑀

    𝑠represents the martensite transformation’s beginning

    temperature, 𝜃 represents the material parameter whichreflects the austenite-martensite transformation rate, 𝜎 rep-resents equivalent stress, and 𝑇 represents temperature.

  • Mathematical Problems in Engineering 3

    600

    650

    700

    750

    800

    850

    900

    Ms(

    K)

    0 100 200 300 400 500

    𝜎 (MPa)

    Ms

    (a) Martensite transformation temperature

    0 100 200 300 400 500

    𝜎 (MPa)

    0.008

    0.009

    0.01

    0.011

    0.012

    𝜃

    𝜃

    (b) Material parameter 𝜃

    Figure 2: Relationships of the martensite start temperature𝑀𝑠

    and material parameter 𝜃 with the variation to equivalent stress 𝜎.

    The corresponding relation between the equivalent stress𝜎 and starting temperature of martensite transformation𝑀

    𝑠and the relationship between material parameter 𝜃 and

    equivalent stress 𝜎 are shown in Figure 2.In general, the martensite start temperature𝑀

    𝑠rises 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.

    3.1. Thermal Transport Model and FE Formulae

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

    𝑘(

    𝜕

    2

    𝑇

    𝜕𝑥

    2

    +

    𝜕

    2

    𝑇

    𝜕𝑦

    2

    +

    𝜕

    2

    𝑇

    𝜕𝑧

    2

    ) + 𝑞V − 𝜌𝑐𝜕𝑇

    𝜕𝑡

    = 0, (4)

    where 𝜌 is the mass density, 𝑐 is the specific heat, 𝑘 is thethermal conductivity, and 𝑞V is the internal heat generationrate per unit volume. 𝑞V includes external heat sources aswell as transformation heat (𝑞tr) and heat generated by plasticdeformation (𝑞𝑝) as:

    𝑞

    𝑝

    = 𝑎𝜎

    𝑖𝑗

    ̇𝜀

    𝑖𝑗, (5)

    where 𝜎𝑖𝑗

    is the deviatoric stress tensor and ̇𝜀𝑖𝑗is the strain

    increment.

    The latent heat 𝑞tr can be defined by the following equa-tion [16, 17]:

    𝑞

    tr= 𝜌𝐿

    𝜕𝜉

    𝜕𝑡

    ,(6)

    where 𝐿 is the latent heat of fusion (J/kg). 𝜉 is the volumefraction martensite transformation.

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

    −𝑘

    𝜕𝑇

    𝜕𝑛

    = ℎ (𝑇 − 𝑇

    𝑎) ,

    (7)

    where 𝑇𝛼is the temperature at the contacted surface and ℎ is

    the heat transfer coefficient.

    (ii) Governing Equation for 3D Transient Heat ConductionProcess.The finite element formulation for 3D transient heatconduction is as follows:

    𝐶

    𝑒̇

    𝑇

    𝑒

    + 𝐾

    𝑒

    𝑇

    𝑒

    = 𝐹

    𝑒

    , (8)

    where𝐶𝑒 is the heat capacitymatrix,𝐾𝑒 is the heat conductionmatrix, and ̇𝑇𝑒 and 𝑇𝑒 are the derivatives of element nodaltemperature with respect to time and element nodal temper-ature, respectively. 𝐹𝑒 is the nodal temperature load vector.Their components are as follows:

    𝐶

    𝑒

    𝑖𝑗

    = ∫

    𝑉𝑒

    𝜌𝑐𝑁

    𝑖𝑁

    𝑗𝑑𝑉,

    𝐾

    𝑒

    𝑖𝑗

    = ∫

    𝑉𝑒

    𝑘(

    𝜕𝑁

    𝑖

    𝜕𝑥

    𝜕𝑁

    𝑗

    𝜕𝑥

    +

    𝜕𝑁

    𝑖

    𝜕𝑦

    𝜕𝑁

    𝑗

    𝜕𝑦

    +

    𝜕𝑁

    𝑖

    𝜕𝑧

    𝜕𝑁

    𝑗

    𝜕𝑧

    )𝑑𝑉,

    𝐹

    𝑒

    𝑖

    = ∫

    𝑉𝑒

    𝑁

    𝑖𝑞V𝑑𝑉 + ∫

    𝜕𝑉𝑒

    𝑁

    𝑖𝑞𝑑𝐴,

    (9)

    where 𝑞 is the distributed heat flux per unit area and𝑁𝑖refers

    to the shape function.

  • 4 Mathematical Problems in Engineering

    X1

    X2

    X3

    e2e3

    e1

    i

    𝜉3𝜉2

    𝜉1

    Figure 3: Quadrilateral temperature shell element.

    (iii) 3DThermal Shell Element.The basic equation of temper-ature shell elements, as shown in Figure 3, was developed byWang and Tang [18]. The temperature shell elements adoptthe 𝜉1𝜉

    2𝜉

    3curvilinear coordinates, with 𝜉

    1, 𝜉

    2in the neutral

    plane and 𝜉3perpendicular to the neutral plane. Assume that

    the inner element temperature field changes in second orderalong the 𝜉

    3direction:

    𝑇 (𝜉

    1, 𝜉

    2, 𝜉

    3, 𝑡) = 𝑇

    0(𝜉

    1, 𝜉

    2, 𝑡)

    + 𝑇

    1(𝜉

    1, 𝜉

    2, 𝑡) 𝜉

    3+ 𝑇

    2(𝜉

    1, 𝜉

    2, 𝑡) 𝜉

    2

    3

    ,

    (10)

    where 𝑇0(𝜉

    1, 𝜉

    2, 𝑡) represents neutral surface temperature

    and 𝑇1(𝜉

    1, 𝜉

    2, 𝑡) and 𝑇

    2(𝜉

    1, 𝜉

    2, 𝑡) can be determined by the

    boundary condition on 𝜉3= ±1/2.

    (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 (𝑥, 𝑦, 𝑧):

    𝑇 = 𝑎

    1+ 𝑎

    2𝑥 + 𝑎

    3𝑦 + 𝑎

    4𝑧, (11)

    where 𝑎1, 𝑎2, 𝑎3, and 𝑎

    4are the undetermined coefficients, unit

    four coordinates of the nodes 𝑖, 𝑗,𝑚, and 𝑙.As shown in Figure 4, the shape function of the tetrahe-

    dral element is obtained as follows:

    𝑁

    𝜉=

    1

    6𝑉

    𝑒

    (𝑎

    𝜉+ 𝑏

    𝜉𝑥 + 𝑐

    𝜉𝑦 + 𝑑

    𝜉𝑧) (𝜉 = 𝑖, 𝑗, 𝑚, 𝑙) , (12)

    where 𝑉𝑒is the volume of the tetrahedral element and 𝑎

    𝜉, 𝑏𝜉,

    𝑐

    𝜉, and 𝑑

    𝜉are the coefficients related to the node locations.

    3.2. 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:

    [𝐾

    𝑝] {V}𝑒 = { ̇𝑓

    𝑝} + { ̇𝑔

    𝑝} + {

    ̇

    𝑓}

    𝑒

    , (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 {V}𝑒 is the nodal velocity vector and [𝐾𝑝] is the mass

    matrix with the form of

    [𝐾

    𝑝] = ∫

    𝑒

    {[𝐵]

    𝑇

    ([

    ̃

    𝐷

    𝑒𝑝] − [𝐹]) [𝐵] + [𝐸]

    𝑇

    [𝑄] [𝐸]} 𝑑V,

    { ̇𝑔

    𝑝} = ∫

    𝑒

    [𝐵]

    𝑇

    (

    𝜂

    1 + 𝜁

    {𝑃} +

    ̇

    𝑇 {𝛽

    }) 𝑑V,

    {

    ̇

    𝑓

    𝑝} = ∫

    𝑎𝜎

    [𝑁]

    𝑇

    {

    ̇

    𝑝} 𝑑𝑎 + ∫

    𝑒

    [𝑁]

    𝑇

    {̇𝑝} 𝑑V,

    (14)

    where 𝑝𝑖and 𝑝

    𝑖

    (𝑖 = 1, 2, 3) are the body force and surfaceforce vector, [𝐵] is the strain element, and the force vector[𝐹] takes the form of

    𝐹

    𝑖𝑗𝑘𝑙=

    1

    2

    (𝜎

    𝑙𝑗𝛿

    𝑘𝑖+ 𝜎

    𝑘𝑗𝛿

    𝑙𝑖+ 𝜎

    𝑙𝑖𝛿

    𝑘𝑗+ 𝜎

    𝑘𝑖𝛿

    𝑙𝑗) . (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

    𝜎s

    (MPa

    )

    0 200 400 600 800 1000

    T (∘C)

    𝜎s

    (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 surfaces,the 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.

    4.1. 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 increases,the variation of specific heat reaches the maximum at 800Kand then remains constant.

    Punch

    Blank holder

    BlankDie

    X Y

    ZETA/DYNAFORM

    Figure 7: Schematic diagram of FE model.

    4.2. 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 𝑡 = 1.6mm. 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

    𝑒= 3.6W/(m2K)), convection from

    tools into cooling channels (ℎ𝑐= 4700W/(m2K)), and heat

    transfer fromhot blank to tools (𝛼𝑐= 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 (0–0.5 s): point 1 is not fully con-tacted with the tools due to being gently warped, andthe temperature drops slowly.The 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 (𝑡 = 0.5 s).

    X

    Y

    Z

    X

    Y

    Z

    (Punch) (Water channels of punch)

    Figure 11: Simulation temperature results of punch (𝑡 = 0.5 s).

    X

    Y

    Z

    X

    Y

    Z

    (Blank holder) (Water channels of blank holder)

    Figure 12: Simulation temperature results of blank holder (𝑡 = 0.5 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 = 1.5 s

    T = 350K

    Figure 13: Experimental and simulated temperature results of blank.

    76.8%

    0

    100

    200

    300

    400

    500

    600

    700

    800

    𝜎(M

    Pa)

    400 600 800 1000

    T (K)

    0

    20

    40

    60

    80

    𝜉 m(%

    )

    𝜉m𝜎

    𝜎 = 223MPaMs = 703K

    (a)

    0

    100

    200

    300

    400

    500

    600

    700

    800

    𝜎(M

    Pa)

    200 400 600 800 1000

    T (K)

    0

    20

    40

    60

    80

    100

    𝜉 m(%

    )

    𝜉m𝜎

    𝜎 = 223MPaMs = 703K

    90.6%

    (b)

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

    30𝜇m

    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 = 1.16 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 (0.5–6.5 s): due to the effect of thecooling channels on the inner surface of the tools, thecooling rate of the blank is high (about 300K/s) 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 1.5 s.

    (i) Results for Thermal Field. Tools with water channels weredesigned tomake the temperature distribution homogeneouswith efficiency.The temperature distributions of the 3D tools,slices surfaces, and cooling channels surfaces are shown inFigures 9–12.

    Thermocouples were installed and utilized tomeasure thetemperature variations during the prototype hot forming.Theexperimental and simulation results are shown in Figure 13.A 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 14.The presence of an applied or internal stress affectsboth the martensite start temperature 𝑀

    𝑠and 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 (𝑡 = 0.56 s).

    Table 1: Maximum temperature of tools.

    Maximum temperature of toolsTools Time Position Value Channel valueDie 0.5 s Point 2 460K 316KPunch 0.5 s Point 2 370K 312KBlank holder 0.5 s Point 1 390K 312K

    of martensite formed as a function of the cooling below𝑀𝑠.

    When𝜎 changes within a certain range,𝑀𝑠increases with the

    increase of 𝜎.It 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 75.9%.

    4.3. 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 17.The 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 (𝑡 = 0.56 s).

    the surface of the die tool. The temperature curves shownin Figure 18(b) present the temperature changing process ofeach point.We can see from the temperature curve that point2 does not fully contact with the tools, due to a gentle warp,and 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 begins.Then, 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 area.Most 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 “the Fundamental Research Funds for the CentralUniversities DUT13ZD208”, the National Key Basic Researchand Development Program through contract Grant nos.2014CB046803 and 2011ZX05026-002-02, and The NationalScienceGroup of China through contract Grant no. 51221961.

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