Thermomechanical-Phase Transformation Simulation of High-Strength Steel in Hot Stamping

The 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 constitutivemodels 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. The hot stamping process of typical U-shaped and Bpillar 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 multifield coupled constitutive models and numerical simulation software KMAS. The temperature simulated results also provide the basic guide for the optimization designs of cooling channels in tools.


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].The steel specimen produces martensitic microstructure and exhibits strong hardening abilities after hot stamping.The 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 process, which has been thoroughly studied by scholars throughout the world [2][3][4][5], in the standard hot stamping process a steel blank is heated in a furnace to a certain high temperature (880-950 ∘ C).The matrix organization and mechanical behavior change with the thermal effect.The austenite transformation is distributed uniformly in the blank [6].Then the heated blank is transferred into the tools (stamping molds) with water-cooled channels and formed and quenched simultaneously for a short time (less than 10 seconds).The phase transformation behavior from austenitic to martensitic state in the material causes an increase of the tensile strength up to 1200-1500 MPa.Throughout the entire stamping process, coupling thermomechanical characteristics are exhibited.The accurate control of temperature and final martensite rate play an important role in the final qualities of the molded parts [7].
The difficulty of hot stamping simulation is far beyond that of traditional cold forming.The thermodynamic properties of materials are affected by the changing of the microstructure, especially in the phase transform phase.The strong rate-dependent constitutive behaviors and temperature sensitivities are significant throughout the entire stamping process.Thermal contact and heat transfer between the tools and the blank surface must be considered with accuracy.The numerical modeling of the full process requires thermomechanical coupled material descriptions for the type of steel considered.
Bergman and Oldenburg implemented the numerical simulation of hot forming [8].Eriksson et al. investigated the temperature and strain rate dependence of hot forming material [9].Naderi, Bleck, and Merklein researched the plastic flow criterion and material parameters under high temperature [10,11].The theoretical analysis, numerical simulations, and experimental studies of hot forming were presented by Ma and Hu, who provided a new method for hot forming die design and technical process design [12,13].
At present, most of the forming software programs available perform simplifications by using the rigid shell element to simulate the 3D tools in the hot forming modeling.Such assumption causes difficulties in simulating the temperature transport process.In 2012, Jiang, Wu et al. presented a numerical frame to simulate the thermal transport process by using a thick shell element of tools [14,15].It can simulate the simple components, such as U-shaped steel, with high accuracy.However, for the majorities of autopanels, the numerical simulations of temperature domain with complex 3D tool surfaces still face many difficulties and challenges.
In this paper, a fully coupled 3D thermomechanicalphase transformation finite element simulation of the hot stamping process is developed.The rate-dependent thermal constitutive model is implemented in the present software framework.In terms of general shell finite element and the 3D tetrahedral finite element analysis method related to temperature, a coupled thermal transport modeling for contact interaction between blank and tools is proposed.The hot stamping process of benchmark typical U-shaped steel and B-pillar steel is simulated by the present numerical program.Strong agreements of temperature, equivalent stress, and fraction of martensite between the simulated and experimental results indicate the validity and efficiency of the present multifield coupled numerical model in simulating the hot stamping process for the automobile industry.

Key Technologies for Thermomechanical-Phase Transformation Coupled Behavior
2.1.Constitutive Laws of High-Strength Steel.22MnB5 boron steel is the most common steel used for hot stamping.A brief introduction to the thermomechanical-phase transformation coupled constitutive relationship for high-strength steel is presented in the following section.Detailed descriptions can be found in [13].
In order to model the thermal-elastic-plastic response during the hot stamping process, an additive decomposition of the total strain increment ( ε  ) is utilized [13,14].The total stain   can be expressed with the decomposition rate form as follows: where ε   is the elastic strain increment, ε   is the plastic strain increment, ε th  is the thermal strain increment, ε tr  is the isotropic transformation strain increment, and ε  play an important role in hot forming and are expressed as (2) in the traditional quenching treatment: where  is the fraction of martensitic transformation,  is the phase transformation volume coefficient, and  is the phase transformation plastic coefficient related to equivalent stress .
The sensitivity of temperature on the forming behavior of 22MnB5 is investigated [16].The flow curves shown in Figure 1 indicate that the temperature has a strong influence on the forming behavior of quenchable steel.Temperature increasing leads to an appreciable reduction of stress level and a decreased work hardening exponent.
For temperature variation from 500 ∘ C to 800 ∘ C, representative true stress-strain curves are displayed for an exemplarily strain rate of 1 s −1 .Other curves under different temperatures are obtained by quadratic interpolation using the existing experimental curves, such as the curves of 600 ∘ C and 700 ∘ C, as shown in Figure 1.

Martensite Phase Transformation.
For the martensite phase transformation, the relationship between temperature and phase change is modeled using the equation proposed by Ma et al. [13], which can be written as follows: where  represents the fraction of martensitic transformation,   represents the martensite transformation's beginning temperature,  represents the material parameter which reflects the austenite-martensite transformation rate,  represents equivalent stress, and  represents temperature.The corresponding relation between the equivalent stress  and starting temperature of martensite transformation   and the relationship between material parameter  and equivalent stress  are shown in Figure 2.
In general, the martensite start temperature   rises with increasing tensile and compressive normal stresses as well as shear stresses.The hydrostatic component always decreases the martensite start temperature.In the present paper, ( 3) is used to model the austenite to martensite reaction.

Nonlinear Large-Deformation FEM Formula of Hot Forming
Hot stamping is a coupled thermomechanical forming process with intended phase transformation.During the solidstate phase transformations, heat is released, which then influences the thermal field.Furthermore, both the mechanical and thermal properties vary with the temperature variation and deformation.Consequently, a realistic FE model for full process numerical simulation must consider the interaction among the mechanical, thermal, and microstructural fields.

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 as follows: where  is the mass density,  is the specific heat,  is the thermal conductivity, and  V is the internal heat generation rate per unit volume. V includes external heat sources as well as transformation heat ( tr ) and heat generated by plastic deformation (  ) as: where    is the deviatoric stress tensor and ε  is the strain increment.
The latent heat  tr can be defined by the following equation [16,17]: where  is the latent heat of fusion (J/kg). is the volume fraction martensite transformation.
The heat transfer processes between the contacted surfaces were considered by using the following convection boundary conditions: where   is the temperature at the contacted surface and ℎ is the heat transfer coefficient.
(ii) Governing Equation for 3D Transient Heat Conduction Process.The finite element formulation for 3D transient heat conduction is as follows: where   is the heat capacity matrix,   is the heat conduction matrix, and Ṫ  and   are the derivatives of element nodal temperature with respect to time and element nodal temperature, respectively.  is the nodal temperature load vector.Their components are as follows: where  is the distributed heat flux per unit area and   refers to the shape function.(iii) 3D Thermal Shell Element.The basic equation of temperature shell elements, as shown in Figure 3, was developed by Wang and Tang [18].The temperature shell elements adopt the  1  2  3 curvilinear coordinates, with  1 ,  2 in the neutral plane and  3 perpendicular to the neutral plane.Assume that the inner element temperature field changes in second order along the  3 direction: where  0 ( 1 ,  2 , ) represents neutral surface temperature and  1 ( 1 ,  2 , ) and  2 ( 1 ,  2 , ) can be determined by the boundary condition on  3 = ±1/2.
(iv) 3D Thermal Tetrahedral Element [19].The 3D tetrahedral element is implemented in the heat conduction simulation of tools.It is assumed that temperature is a linear transformation function with the coordinates (, , ): where  1 ,  2 ,  3 , and  4 are the undetermined coefficients, unit four coordinates of the nodes , , , and .
As shown in Figure 4, the shape function of the tetrahedral element is obtained as follows: where   is the volume of the tetrahedral element and   ,   ,   , and   are the coefficients related to the node locations.

FE Formula with Static
Explicit Algorithm.Based on the continuum theory and virtual work principle [20], the static explicit finite element equation of coupled multifield is obtained by the following: where {V}  is the nodal velocity vector and [  ] is the mass matrix with the form of where   and   ( = 1, 2, 3) are the body force and surface force vector, [] is the strain element, and the force vector [] takes the form of

Numerical Simulations of U-Shaped Steel and B-Pillar in Hot Stamping
Based on the above multifield models, FEM, and temperature field analysis, the numerical modeling of the hot stamping process is developed and implemented into independently developed CAE commercial software called KMAS for metal sheet forming.Two numerical examples are presented in this section to show the strong ability and feasibility of the present numerical framework to simulate the hot stamping process.
According to the three-dimensional temperature field characteristics of hot forming, the proprocessing of 3D tools typically follows the rules below.
(1) We must divide the 3D elements into internal and boundary elements.Different surfaces with different thermal boundary conditions must be defined.
(2) The contact surfaces and cooling channel surfaces on the tools are separated to realize the contact interface simulation and treatment of the boundary conditions.(3) For the triangular elements on the boundary surfaces, the nodes should be organized with a certain order to guarantee the same normal direction.
(4) We assume there is only one surface of each element at the border.The deal, to some extent, can meet the accuracy requirements of the thermal simulation.

Material Properties of 22MnB5
Steel.The materials of U-shaped steel and B-pillar steel are composed of 22MnB5 high-strength steel, and #45 steel is chosen as the material of the tools.Figures 5 and 6 show the major mechanical and thermal related properties of 22MnB5 steel with variation of temperature.We can see that the yield stress, elastic modulus, and thermal conductivity decrease, respectively, as the temperature increases.Also, as the temperature increases, the variation of specific heat reaches the maximum at 800 K and then remains constant.interfaces takes place via conduction through the contacting spots, the conduction through the interstitial gas, and radiation across the gaps.The convection of blank and tools to the environment (ℎ  = 3.6 W/(m 2 K)), convection from tools into cooling channels (ℎ  = 4700 W/(m 2 K)), and heat transfer from hot blank to tools (  = 5000 W/(m 2 K)) are also considered, as shown in Figure 8.

Numerical Simulation of U-Shaped Steel in the Hot Stamping Process.
The changing of the contact situation between blank and tools leads to changing of blank temperature.Three periods can be summarized to describe the contact behavior.
(1) Stamping Phase (0-0.5 s): point 1 is not fully contacted with the tools due to being gently warped, and the temperature drops slowly.The temperatures of the other points decrease greatly.(2) Quenching Phase (0.5-6.5 s): due to the effect of the cooling channels on the inner surface of the tools, the cooling rate of the blank is high (about 300 K/s) at the beginning of the quenching phase.
(3) Latent heat of phase transformations is released and thus affects the blank thermal history.We observed that the temperature curve fluctuates at around the time of 1.5 s.
(i) Results for Thermal Field.Tools with water channels were designed to make the temperature distribution homogeneous with efficiency.The temperature distributions of the 3D tools, slices surfaces, and cooling channels surfaces are shown in Figures 9-12.
Thermocouples were installed and utilized to measure the temperature variations during the prototype hot forming.The experimental and simulation results are shown in Figure 13.A high consistency can be observed for the entire stamping process.Table 1 presents the maximum temperature values located at the die, punch, and blank holder.
(ii) Results for Equivalent Stress and Fraction of Martensite.In the process of hot stamping, temperature and deformation at the shoulder (a position with large curvature) largely affect the steel forming performance.Two character points are selected to present the variation of equilibrium stress and fraction of martensite with temperature, as shown in Figure 14.The presence of an applied or internal stress affects both the martensite start temperature   and the fraction  It can be seen from Figure 14 that, when quenching begins, there is no martensite transformation.The amount of martensite transformation increases as time increases (Figure 14(a)).6 s later, the martensite volume fraction of the blank reaches above 90% (Figure 14(b)).
The optical microscopy images of the blank related to the same position as that in Figure 14(a) are presented in Figure 15 after quenching in the final condition.Ferrite appeared as white, bainite appeared as light gray, and martensite appeared as black lath-shape.The average martensite volume fraction was 75.9%.

Numerical Simulation Results of B-Pillar
Steel.B-pillar steel stamping is chosen as the second example.The tool model design for B-pillar is shown in Figure 16 and the 3D finite element model in Figure 17.The quenching stages of Bpillar are also simulated by KMAS software.
The beginning temperature of the quenching of the blank is 873 K. Five character points are selected (Figure 18(a)) in the surface of the die tool.The temperature curves shown in Figure 18(b) present the temperature changing process of each point.We can see from the temperature curve that point 2 does not fully contact with the tools, due to a gentle warp, and the temperature drops slowly.The temperatures of the other points decrease greatly.Similar with the die, as shown in Figure 19, four character points on the punch surface are selected.
The maximum temperatures of the tools and cooling channel appear within a short time after quenching begins.Then, due to the continuing cooling effect of the water channel, the temperature of the tools decreases at a rapid speed.Figure 20

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

Mathematical Problems in Engineering
At the end of this phase, the tools reach the maximum temperature at the shoulder.
(2) The temperature of a cooling duct placed closer to the tool surface in a convex area is higher than that in a concave area.Most of the surfaces of the cooling channels remain at a low temperature in the quenching process.
(3) Due to the ignorance of the flow velocity and pressure of water, the temperature distributions on the different slices surfaces are almost the same.
It can be concluded that the efficient cooling is most desired in convex areas, that the geometry of cooling ducts is restricted due to constraints in drilling, and that the ducts should be placed but sufficiently far from the tool contour to avoid any deformation.A more accurate contacted condition and flow analysis for hot stamping remains to be studied.

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

Figure 2 :
Figure 2: Relationships of the martensite start temperature   and material parameter  with the variation to equivalent stress .

Figure 5 :Figure 6 :
Figure 5: Relationships of yield stress and elastic modulus with increasing of temperature.
Figure 7  presents the finite element configurations of tools and blank for U-shaped steel.The thickness of blank is  = 1.6 mm.Heat transfer between the contact
presents the final B-pillar products by the hot stamping process.The martensite volume fraction of the blank reaches above 90%, as shown by the optical microscopy image test.

Figure 20 :
Figure 20: High-strength B-pillar steel with hot stamping technique.

Table 1 :
Maximum temperature of tools.
of martensite formed as a function of the cooling below   .When  changes within a certain range,   increases with the increase of .