Weimin Zhuang， Pengyue Wang,?， Yang Liu， Dongxuan Xie and Hongda Shi

（1. State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130022, China；2. FAW-Volkswagen Automotive Co., Ltd., Changchun 130011, China）

Abstract: The effects of forming damage are analyzed, which occur during hot stamping process,on the load-carrying capacity and failure mode of hot stamped beams. A damage-coupled pre-forming constitutive model was proposed, in which the damage during hot stamping process was introduced into the service response. The constitutive model was applied into the three-point bending simulation of a hot stamped beam, and then the influences of forming damage on the load-carrying capacity and cracks propagation were investigated. The results show that the forming damage reduces the maximum load capacity of the hot stamped beam by 7.5%. It also causes the crack to occur earlier and promotes crack to propagate along the radial direction of the punch.

Key words: constitutive modelling；hot-stamped beam；forming damage；three-point bending；crack propagation

For the enhancements of passenger safety and lightweight car body, the ultra-high strength steel hot-stamped members are widely used in car body crashworthiness components[1-2], for example A-pillars, B-pillars and bumpers. A component forming is a plastic process. In this process, not only the thickness of sheet metal is not uniform, but also the dislocation density changes with the forming process, resulting in microvoids, micro-cracks and other micro-damage in the material. These invisible forming damages are likely to lead to early failures of components.Therefore, it is very important to analyze the influence of forming damage on the component’s service performances such as load-carrying capacity, failure mode and so on.

The researches show that after cold stamping, the strength[3-4], thickness distribution[5]and residual stresses[6]in work-hardened regions change greatly, which has the significant effects on the crush behaviour, load-carrying capacity and energy absorption of automobile body components[7]. Oliveira et al.[8]studied the effect of forming process variables on crashworthiness of srail tubes. They found that the bending process resulted in an increase in the crash force by approximately 25%–30% and energy absorption by 18%. The similar result was obtained by Najafi et al.[9]who integrated the process-performance optimization using a multi-objective genetic algorithm approach. Niu et al.[10]observed that the samples formed by stamping showed a peculiar collapse mode and their energy absorption capacity was significantly reduced. Gumruk et al.[11]found that the residual forming data and the effects of spring-back caused by a deep drawing forming process significantly influenced the axial quasi-static crash behaviour of straight thinwalled top-hat section. Other investigations[12-14]indicate that cold forming history has a significant influence on the service performance of metallic structures, and the forming history should be considered to accurately evaluate the structural performance.

The mechanism of hot stamping affecting the performance of material is completely different from cold stamping. The hot stamping is a high temperature and low speed viscoplastic deformation process of metal. High temperature significantly reduces the work hardening effect of the material. Instead, the micro-voids and microcracks[15-16], which occur at the crystal boundary during hot forming process, play important roles to impact the performance of material[17]. Many researches have been conducted in the past several decades to develop predictive models for the hot forming damage. Aboutalebi et al.[18-19]predicted ductile damage in a sheet metal forming process by using the continuum damage mechanics and finite strain theory. Li et al.[20-21]proposed a set of viscoplastic constitutive equations for predicting boron steel damage under hot stamping conditions. Shutov et al.[22]suggested a phenomenological model that takes thermodynamics into consideration and incorporates damage-induced volume change and a new void nucleation rule, and they used this model to predict ductile damage in metal forming simulations. Hu et al.[23]and Shi et al.[24]presented a set of thermo-elastic-plastic constitutive equations,which can accurately predict ductile damage and material failure in hot stamping processes. In the process of hot stamping, the components are damaged locally. Because the existing constitutive model cannot introduce hot forming damage to predict the performance of components, the current general method of predicting the performance of components is to ignore the impact of hot forming damage[25-27], which leads to an inaccurate analysis of the performance of hot stamping components.

To introduce the effects of forming damage into the service performance, a damage-coupled pre-forming constitutive model for 22MnB5 steel was proposed based on Lin’s original constitutive equations[28]in this work. The sequential coupled process-performance simulations were conducted to predict the structural performance with forming damage. Firstly, both forming damage and component geometry were computed from the hot stamping simulation, and the forming damage was simultaneously cast in forming strain which was necessarily used in the damagecoupled pre-forming constitutive model. Subsequently, forming strain and component geometry were mapped to performance simulation.Finally, the proposed constitutive model was applied into the hot stamped beam three-point bending simulations, and the influences of forming damage on the load-carrying capacity and cracks propagation were investigated.

1 Forming Simulation

1.1 Forming constitutive model

The hot stamping process was simulated by the unified visco-plastic-damage constitutive equation for boron steel under hot stamping conditions[29]. The constitutive model considers the effect of temperature and strain rate on the mechanical properties of the material during hot stamping, which provides an accurate failure limit prediction for the sheet. This model is applied for the failure prediction of 22MnB5 steel.

The forming constitutive model comprises stiff ordinary differential equations(ODEs), and it cannot normally be solved analytically. For such problems, a more convenient and versatile solution is to use genetic algorithm-based optimization methods to “draw” the “distance” between the test data and the model calculation data to obtain a set of optimal material constants. The constants in the equations were determined by fitting the experimental data[15]using optimization methods[30]. Fig.1 shows a comparison between the computed (solid curves) and experimental (symbols) stress-strain curves for 22MnB5 steel, and the minimum average error was 3.23% for this case. The determined constants are listed in Tab. 1.

Fig.1 Comparison between the computed (solid curves) and experimental (symbols) stress-strain curves

Tab. 1 Determined material constants for boron steel for use in the visco-plastic-damage constitutive model

1.2 Forming finite element model

In the forming simulation of hat beam, tools including punch, binders and die were set as the rigid bodies, as shown in Fig.2. The blank has a dimension of 320 mm×80 mm×1 mm, and it was meshed using C3D8R element with the size of 1.6 mm×0.8 mm×0.25 mm. The contact between the tools and the blank were maintained as automatic surface to surface contact and the coefficient of friction was assumed to be 0.2. The forming constitutive model and material constants were used to simulate the deformation behavior of the blank. Other blank material properties including the density, Poisson’s ratio, coefficient of thermal conductivity and the specific heat capacity have been set to 7.85×103kg/m3, 0.3,24 W·m–1·°C–1and 460 J·kg–1·°C–1, respectively.

While the binders and die were fixed, the punch applied a displacement load of 15.5 mm.The initial temperatures of the blank and the molds were 850 °C and 700 °C, respectively. The effective heat transfer coefficient between the

blank and environment was set as a function of the temperature as shown in Tab. 2, combining both convection and radiation effects[31].

Fig.2 Hat section of finite element model for hot stamping

Tab. 2 Convection and radiation heat transfer coefficients for 22MnB5 boron steel[31]

1.3 Forming result

The results of forming simulation and performance simulation(in Section 2.5) can be intuitively analyzed by defining the middle section A and the local top view B of the hat beam, as shown in Fig.3. In section A, Path 1 and Path 2 are the node paths of the upper and lower surfaces of the hat beam, respectively. The main deformation areas of the hat beam are defined as top corner, sidewall and flange corner in section A.

Fig.3 Section A and the local top view B of the hat beam

Fig.4 shows the thickness distribution of hat section beam after the hot stamping process. The thickness mainly decreases at the top corner,sidewall and flange corner. The thickness is 0.93 mm at the top corner where the thinning is the largest. The thicknesses of flange corner and sidewall are 0.95 mm and 0.96 mm, respectively.The thicknesses of top plate and flange basically unchanged. The similar trend can be seen from the damage contour of hat section shown in Fig.5.The more the thinning is, the more serious the damage is. The forming damage is also distributed along the top corner, sidewall and flange corner, in which the damage values are 0.12, 0.05 and 0.07, respectively.

The damage described by the forming constitutive model is a nonlinear variable. Nonlinear damage is difficult to get in the experiment and inconvenient to characterize the impact of damage on service performance. Therefore, the nonlinear damage was transformed into linear damage(defined as forming strain α=εp/εf) as in Eq.(1) in the previous work[32]. Eq. (1) makes it easy to establish the service constitutive model and determine the material constants.

Fig.4 Thickness distribution of hat section beam

Fig.5 Damage contour of hat section

The forming strains in Path 1 and Path 2 were calculated by forming damage through Eq.(1), and the results are shown in Fig.6, in which the largest variation in the forming strain occurs at top corner of Path 1. The probable reasons are summarized and given here: (i) small values of radius of curvature at the corners increase deformation at this region; (ii) the frictional forces at the top and bottom surfaces give rise to an increase in the deformation of this region; (iii) the temperature reduction effected by the punch decrease the formability of this region.

Fig.6 Distribution forming strains in Path 1 and Path 2 of hat section A

2 Performance Simulations

2.1 Performance constitutive model

To introduce the forming strain into service performance, a damage-coupled pre-forming constitutive model was proposed. This model is able to consider the effect of forming strain on plasticity and failure of quenched 22MnB5 steel. This model is formed as

2.2 Determination of material constants

To determine the material constants in the constitutive equation, the stress-strain curves for materials with different forming strains are necessary. In our previous work, a two-steps tensile test was conducted to obtain the stress-strain curve of quenched 22MnB5 steel with different forming strains[17]. The two-steps tensile test consists of the following steps:

① Using the MMS-200 multifunction thermal-mechanical simulator, five hot tensile samples were tested at 850 °C and strain rate of 0.1. Then the failure strain was calculated by tensile results.

② Calculated the displacement, which is corresponding to hot tensile samples with different forming strains, by using the failure strain and a modified method.

③ Stretched hot tensile samples to forming strains of 0%, 10%, 30%, 50% and 70%.

④ After grinding and cutting samples, service tensile tests were conducted by using a 3D digital image correlation(DIC).

The dimensions and deformations of samples are shown in Fig.7. The stress-strain curves of the samples with different forming strains are shown in Fig.8. The material constants of the performance constitutive model are determined from the experimental true stress-strain curves using the same optimization method as Section 1.1. A close agreement was obtained in this case(the average error isless than 2.65%). The determined values of the constants are listed in Tab. 3.

Fig.7 Dimensions and deformations of samples

Fig.8 Stress-strain curves of the samples with different forming strains

Tab. 3 Determined material constants for the damage-coupled pre-forming constitutive model

To investigate the applicability of material constants, two-steps tensile tests were conducted using the conditions shown in Tab. 4. the failure strain and ultimate stress with different conditions were extracted and summarized in Fig.9.The failure strain and ultimate stress decrease nonlinearly with increasing forming strain, but are not sensitive to temperature and strain rate.Therefore, the material constants of the performance constitutive model are suitable for service performance prediction of hot stamping steel within the temperature range of 650°C–850°C and the strain rate range of 0.01–1.00.

Tab. 4 Test conditions for hot tensile tests of 22MnB5 steel

Fig.9 Failure strain(FS) and ultimate stress(US) of quenched 22MnB5 steel under different pre-forming conditions

2.3 Verified constitutive models

The performance constitutive model was implemented into the commercial FE solver ABAQUS via the user-defined subroutine VUMAT. To verify the validity of the constitutive models, a two-steps simulation was conducted.The dimension of sample and boundary conditions were the same as the experiment in the previous work[31]. A hot forming finite element(FE)model and a service FE model were established.The hot forming FE model was characterized by C3D8R elements. A displacement load was applied to one side of sample, while the other side was fixed. The forming constitutive model and material constants were used as the user material input. In the service FE model, the sample was read from the result of the hot forming model.The grid size and element type of the hot forming FE model were simultaneously delivered to the service model. One side of the sample was applied to the displacement load, and the other side was fixed. The proposed performance constitutive model with the material constants were used as user material inputs. The service stressstrain curves from the experiments and simulations are shown in Fig.10. The maximum error between the simulation and experimental test is 2%, which verified that the established constitutive model can predict the service performance of 22MnB5 steel with different forming strains.

Fig.10 Comparison of the service stress-strain curves between the experiment and simulation

2.4 Performance finite element model

The performance simulations(three-point bending simulation) include two models. The geometry and forming damage from the previous forming simulation are used to establish the initial state for the performance simulation of model 1. To compare and analyze the influence of the forming damage, the model 2 only mapped the geometry from the forming simulation. The grid size and element type of hat beam in two models are obtained from the forming simulation.

Fig.11 shows the full size of three-point bending finite element model. The radii of the punch and the supporting rollers are 20 mm and 10 mm, respectively, and the lengths are 100 mm.The distance between the supporting rollers is 240 mm. During the three-point bending process,the supporting rollers were treated as rigid bodies, and remained stationary. The punch was taken as the rigid body with a constant velocity of 300 mm/s. The surface-to-surface contact was used between the punch, supporting rollers and the hat beam, and the coefficient of friction is assumed to be 0.2. A damage degree fds=0.5 is assumed as a threshold value to estimate if failure occurs.

Fig.11 Three-point bending finite element model

2.5 Result analysis

Fig.12 shows the stress and damage contours for two performance models at the displacement of 10.2 mm. Fig.12 shows that the high stress areas of both models are “X-shaped”. The radii created by hot stamping are gradually flattened, and new radii from the upper part of the sidewall are produced. It is worth mentioning that the stress level in model 1 is less than that in model 2. Instead, in the damage contours shown in Fig.12 , the damage areas in model 1 are significantly larger than those in model 2.These phenomena indicate that the performance of the hat beam was weakened by the forming damage.

Fig.12 Stress and damage contours of model 1 and the stress and damage of model 2

The failure fraction r=d/dfof the hat section beam was defined to analyze the damage changes during three-point bending process. The variable d is the punch displacement, and dfis the failure displacement. The failure displacements of models 1 and 2 are all 10.2 mm. Fig.13 shows the forcedisplacement curves for two models and the local damage evolution of the hat section beam with different failure fractions. The peak load is 7.9 kN when the forming damage is not considered and the peak load decreases by 7.5% when the forming damage is considered. In addition, the displacement corresponding to the peak load also decreases by 4.7% due to forming damage.

Before r=0.3, there are the same reaction forces between model 1 and model 2. When r=0.3, no crack occurs in model 2, but a crack occurs in model 1 at the top corner. After that,the reaction force of the model 1 gradually decreases below the model 2 due to the forming damage. When r=0.4, a crack occurs in model 2 at the top plate. As loading increases, two models show a series of force fluctuations until the failure fraction reached to 0.75. In this stage, the radii created by hot-stamping are gradually flattened, and new radii located in the upper part of the sidewall are produced. This phenomenon leads to local structural instability and force fluctuations. As the failure fraction changed from 0.4 to 0.1, the crack in model 2 mainly propagate along the axial direction of the punch, whereas the crack in model 1 not only propagates along the axial direction of the punch but also propagates along the radial direction.

Fig.13 Force-displacement curve for models 1 and 2 and the local damage evolution of the hat beam under different failure fractions

Fig.14 Comparison of performance damage in two models’Path 1 under different failure fractions

To analyze the effect of forming damage on crack propagation along the axial direction of the punch, the comparisons of performance damages in two models’ Path 1 were shown in Fig.14. The failure of two models mainly occurs at nodes 37 and 65, and gradually propagates to both sides.But there are significant differences in degree of damage between the two models. Due to the formation damage, the performance damage in model 1 is always greater than that in model 2,especially in the nodes with larger forming damage. Thus, the failure of model 1 occurs earlier than that in model 2. Therefore, the forming damage not only decreases the load carrying capacity of the hat section beam but also changes the damage evolution and crack propagation in a hat beam.

Fig.15 shows the stress-strain and damagestrain curves of node 21 in Path 1 of the hot stamping model and the stress-strain and damage-strain curves in two models. During the hot stamping process, the damage and the forming strain of node 21 are 0.12 and 75%, respectively.This damage dramatically decreases the fracture toughness. As an example, the fracture toughness at node 21 of model 1 is 2.0×107J/m3,whereas this node’s fracture toughness is 9.9×107J/m3in model 2. The 75% forming strain decreases the fracture toughness by 79.6%. In addition, the forming damage increases the slope of the damage-strain curve and decreases the effect of work hardening in model 1. As a result, the damage at node 21 of model 1 rapidly increases to 0.5 and the stress is less than that in model 2.

Fig.15 Stress-strain and damage-strain curves of node 21 in Path 1 of the hot stamping model and its stress-strain and damage-strain curves in models 1 and 2

3 Conclusions

① A damage-coupled pre-forming constitutive model was proposed for 22MnB5 steel.This model can characterize the effect of forming strains on the service performance of 22MnB5 steel. By the experimental verification, the constitutive model with the determined material constants is valid.

② The constitutive equations for boron steel under hot stamping conditions were implemented to predict the thickness change and material damage during the hot stamping process. The forming damage and thinning were mainly distributed along top corner, sidewall and flange corner of the hat beam.

③ The peak load of the hat beam was reduced by approximately 7.5%, when the forming damage was considered into the three-point bending simulation of hat section beam. The displacement corresponding to the peak load decreases by 4.7% due to the forming damage. In addition, the forming damage dramatically decreased the fracture toughness of 22MnB5 steel.The 75% forming strain decreased the fracture toughness by 79.6%.

④ The forming damage strongly influence the failure behavior of hot stamped beam. It not only caused the crack to occur earlier but also promoted the crack to propagate along the radial direction of the punch.