ISIJ International, Vol. 54 (2014), No. 8, pp. 1856–1865
Finite Element Analysis of Three-Dimensional Hot Bending and Direct Quench Process Considering Phase Transformation and Temperature Distribution by Induction Heating
1) Nippon Steel & Sumitomo Metal Corporation, 1-8 Fuso-cho, Amagasaki, Hyogo, 660-0891 Japan.2) Graduate School of Energy Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501 Japan.
(Received on February 14, 2014; accepted on May 18, 2014)
In this study, a coupled thermo-mechanical-metallurgical finite element analysis (FEA) method wasdeveloped to investigate the deformation behavior in the three-dimensional hot bending and direct quenchprocesses. In the developed FEA procedure, the temperature distribution was calculated by two methods.First was a three dimensional electromagnetic and heat conduction analysis that considered a non-linearityof permeability and magnetic transformation. Second was a simplified method that used an original heatsource model for induction heating. In the deformation analysis, temperature, micro structure and strainrate dependencies of flow stress were taken into consideration. As for the microstructure evolution, anexperimental formula was used to track the ferrite-austenite transformation, and Koistinen-Marburger rela-tionship was employed to describe the austenite-martensite change. To confirm the effectiveness of thedeveloped FEA method, the thickness change upon bending and the camber by inhomogeneous coolingwere simulated. The results were in good agreement with the experimental measurements.
In recent years, the automotive industry has been focusingon two issues: the development of lighter vehicles to improvefuel economy in an effort to prevent global warming, andthe improvement in crash safety. A three-dimensional hotbending and direct quench (3DQ) mass processing technol-ogy of tube has been developed1,2) as a means of satisfyingthese two conflicting needs. The technology enables auto-motive parts to be a hollow tubular structure with an ultra-high-tensile strength and a three-dimensional complexshape.
Because a high dimensional accuracy is usually requiredfor automotive parts, it is necessary to know accurately thedeformation behavior of tubes in 3DQ. However the actual3DQ is a complicated process where electromagnetic fields,heat conduction and heat transfer, latent heat of transforma-tion, microstructure change, mechanical properties change,thermal and transformation expansion, and transformationplasticity are involved.
In the past, some numerical models for tube bending pro-cesses using induction heating (Induction bending) havebeen proposed. Asao et al.3) and Kuriyama et al.4) predictedthe bending moment and wall thickness of cylindrical tubes
using elementary analyses. The models were based on theassumption of plane strain, and they stated that the cross-section shape did not change during the deformation.
A finite element method (FEM) has also been used tosimulate the induction bending process. Wang et al.,5) Huet al.6) and Tropp et al.7) calculated thickness change andflattening by a coupled temperature-deformation FEM.Additionally, Lee et al.8) tried to calculate a residual stressdistribution. However, the above studies did not take intoaccount phase transformation and an accurate temperaturedistribution generated by induction heating. Hence, thesemodels are not enough to apply to the 3DQ process.
To generate the temperature distribution for deformationanalysis, heat source model of induction heating is neces-sary. In previous studies, some heat source models9,10) forinduction heating were proposed. However, these modelscannot be used in the present study because they are for spotheating of plate.
In this study, a reliable simulation framework for 3DQ isdeveloped in which an accurate temperature distributiongenerated by induction heating and phase transformation aretaken into account. In the framework, also a simplifiedmethod to calculate temperature distribution including aheat source model for local induction heating of tube isproposed to save the computation time. Finally thesimulation accuracy is examined by comparing withexperimental results.
The characteristics of the 3DQ process and its productsare explained in the literatures.1,2) The outline is as follows.3DQ is a consecutive forming technique that allows three-dimensional complex hollow bending and quenching at thesame time. Figure 1 shows an example of a 3DQ machine.The machine consists of five components: an axial feedingdevice, guide rolls, an induction heater, a cooling device,and a bending device (Robot). 0.21 mass% carbon-boronsteel tubes are used as the material. First, the tube is fed inthe axial direction and heated to higher than Ac3
temperature locally. Just after heating, the tube is bent by anindustrial robot. At the same time, the tube is quenched toroom temperature by the cooling device. To simulate the3DQ process accurately using FEM, it is important toconsider the temperature history and the metal structurechange.
3. Modeling of 3DQ
We propose a simulation framework for 3DQ includingcoupled thermo-mechanical-metallurgical FEM in thisstudy. Figure 2 shows the procedure of the framework. Thisprocedure consists of the following two stages.
At the first stage, a temperature distribution is calculated.In this study, two methods are employed. One is the three-dimensional electromagnetic and heat conduction analysis14)
(Section 3.1.1). We call this model the three-dimensionalmodel in this paper. This model is used in the case wherethe detail consideration of the effect of three dimensionaldistribution is needed (Section 4.3). Another is a simplified
method that includes original heat source model and onedimensional heat conduction analysis (Section 3.1.2). Thismethod is called the simplified method in this paper. Whenthe temperature distribution is already known by an experi-ment or the three dimensional model, the simplified methodis used to save time to prepare simulation models and thecomputation time (Sections 4.1 and 4.2).
At the second stage, the temperature distributioncalculated in the first stage is assigned to integration pointsin finite elements. Then, the deformation analysis is carriedout considering temperature, micro structure and strain ratedependencies of flow stress using the coupled thermo-mechanical-metallurgical FEM. The static implicit methodin ABAQUS Standard is used in this stage.
To take into account the microstructure change duringdeformation, the Kunitake’s equations16) as a function ofmaterial composition are used to describe the transformationtemperatures, the experimental formula11) is used to track theferrite-austenite transformation, and a Koistinen-Marburgerrelationship12) is employed to describe the austenite-martensitechange. In order to facilitate the model when the materialcomposition is changed, not the dilatometry test result spe-cific to the material but the above-mentioned generalizedequations are used in the present study. Then, the flow stressis determined using these transformation models and a lin-ear mixture rule.13) These models were implemented intoABAQUS through user subroutines.
The above mentioned models are described in detail inthe following sections.
3.1. Temperature Analysis3.1.1. Three-dimensional Model
In this analysis, coupled electromagnetic and heat con-duction analysis is carried out. A joule heat density due toinduction heating is calculated by an electromagnetic anal-ysis, and a temperature distribution is calculated by a heatconduction analysis. It is difficult to obtain an accurate threedimensional result if the actual quenching process is mod-eled as the tube is very long and moves in the longitudinaldirection. To solve such a problem, the modeling area is nar-rowed down to locally near the induction coil. Further, thefeeding of material is considered as the flow field using anadvective term in the equation of heat conduction. Theseanalysis algorithms are described in the literature.14) Thisanalysis considers the non-linearity of permeability, themagnetic transformation and the temperature dependent ofthermal and electric property.
When the temperature distribution reaches a steady state,it is assigned to integration points in finite elements used inthe deformation analysis.
3.1.2. Simplified MethodIn the simplified method, to generate the temperature dis-
tribution, the heat conduction analysis using the heat sourcemodel is carried out.
The heat source model is based on a simple theory andan empirical equation. An electric current density of high-frequency induction heating decreases exponentially with adistance from outer surface as follows15)
........................... (1)
Fig. 1. Schematic illustration of 3DQ machine using industrialrobot. (Online version in color.)
Fig. 2. Procedure of 3DQ analysis.i i x= −0 exp( / ),δ
where i is an electric current density, i0 is an electric currentdensity of outer surface, x is a distance from outer surface,and δ is a skin depth. The distribution of the electric currentdensity calculated from Eq. (1) is schematically shown inFig. 3(a). When δ is small, the electric current concentratesin the outer surface of a tube. Next, a longitudinal distribu-tion of outer surface electric current density is approximatedby a normal distribution function as follows
........................ (2)
where m is a parameter for adjusting the position of thepeak, a is a half width of heating area in the longitudinaldirection, and z is a longitudinal position. The longitudinaldistribution of electric current density calculated from Eq.(2) is schematically shown in Fig. 3(b). The value of i0 inEq. (2) is normalized to a range between 0 and 1.
Here, the relationship between joule heat density andelectric current density is given in the form
.................................. (3)
where R is an electric resistance and Q is a joule heat den-sity. From Eqs. (1) to (3), a two-dimensional joule heat den-sity is approximated by the following equation
........ (4)
where Qmax is a maximum joule heat density. The distribu-tion of the joule heat density calculated from Eq. (4) is sche-matically shown in Fig. 4, where the origins of x-axis andz-axis are the outer surface of tube and the point of centerof induction coil, respectively. τ is a thickness of tube.
Using the joule heat density distribution of Eq. (4), onedimensional heat conduction analysis in thickness directionis then carried out. The governing equation of one dimen-sional heat conduction is of the form
......................... (5)
where T is a temperature, t is a time, λ is a heat conductioncoefficient, ρ is a density, and c is a specific heat. As for theboundary condition, convection heat transfer by water jet onthe outer surface of tube is considered by
.................... (6)
where h is a heat transfer coefficient, Ts is an outer surfacetemperature and Tw is a water temperature. On the inner sideof tube, thermal heat insulation is assumed by the followingequation
................................ (7)
From Eqs. (4) to (7) the heat conduction analysis is carriedout using a finite difference method with Crank-Nicolsonmethod. Figure 5 shows the schematic illustration of finitedifference analysis. The finite difference lattice is movedalong the z-axis with the tube feeding speed V. Then thejoule heat density Q is given by Eq. (4). As a result, a two-dimensional steady state temperature distribution in thethickness and longitudinal directions is obtained.
The advantages of the simplified method are as follows:it is not necessary to model the coil and tube geometries, thecomputation time is very short and an experimental temper-ature distribution can be reproduced easily by adjusting theparameters in Eq. (4).
3.1.3. Comparison between Three-dimensional Model andSimplified Method
The effectiveness of simplified method is examined bythe comparison with the three-dimensional model.
The analysis using the three dimensional model was car-ried out considering a circular cross-section tube with 1.6 mmin thickness and 52.56 mm in outer diameter. The feedingspeed was 80 mm/s. The center of coil was z=0 mm and thecooling area was z >25.33 mm. The frequency of inductionheating was 10 kHz. The magnetic transformation tempera-ture (737°C) was also taken into account. The distributionof joule heat density obtained by this analysis is shown inFig. 6. The skin depth rapidly increased at around z=0. Fig-ure 7 shows the temperature distribution in the longitudinal
Fig. 3. Electric current distributions in thickness and longitudinaldirections.
Fig. 4. Heat source model for local induction heating of tube.(Online version in color.)
iz m
a0
2
22 9= −
−( )⎡
⎣⎢⎢
⎤
⎦⎥⎥
exp( / )
,
Q i R= 2 ,
Q Qz m
a
x= −
−( )( )
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥max exp
/exp ,
2
2
2
2 9 δ
Fig. 5. One dimensional heat conduction analysis using movingfinite difference lattice.
direction on the inner and outer surfaces. The temperatureexceeds the magnetic transformation temperature (737°C) ataround z=0. Then the skin depth rapidly increased here.
In the simplified method, the thickness, feeding speedand cooling area were the same as those used in the three-dimensional model. The parameters of Eq. (4) were as fol-lows: m=–5.67 mm, a=53 mm, Qmax=2.39×1010 W/m3, andδ =2.8 mm. The thermal properties were assumed constantand the values at 600°C were used: λ =35.6 W/mK, ρ =7 830 kg/m3 and c=731 J/kgK. The distribution obtained byEq. (4) is shown in Fig. 8. The complex joule heat densitydistribution that was predicted in the three-dimensionalmodel (Fig. 6) could not be achieved in the simplified meth-od. However, the simplified method could describe the bellshape distribution in the longitudinal direction and the skineffect in the distribution in thickness direction as those pre-dicted using the three-dimensional model.
As shown in Fig. 7, the temperature distributions predict-ed using the simplified method are also consistent withthose of the three-dimensional model with a sufficient accu-racy for practical use. There is a little discrepancy at a tem-perature of around 200°C or less, but this is not a practicalproblem since it is a low temperature range with small
deformation. The above results confirmed the effectivenessof the simplified method.
3.2. Modeling of Microstructural EvolutionFor the analysis of the phase transformations, a mathe-
matical model based on physical and empirical approachesis used. The 3DQ deformation model has both a heating areaand a cooling area. In the heating area, an above-Ac3 tem-perature is mandatory to obtain the full austenite structure.In the cooling area, the full martensite structure is obtainedwithout bainite or ferrite transformation as the cooling ratein 3DQ is very high.
The transformation temperatures of Ac1°C, Ac3°C andMs°C are described using Kunitake’s equations (Eqs. (8) to(10)16)) and a regression formula for chemical compositionto provide a versatile model. Xi is a weight percentage of acomposition i
... (8)
... (9)
... (10)
The approximation Eq. (11) is used to simulate the for-mation of austenite in the heating area11)
.........................................(11)
where ξγ is a volume fraction of austenite and cA and da areconstants.
At the cooling area, the transformation from austenite tomartensite is described by Koistinen-Marburger’s formula12)
.... (12)
where ξM is a volume fraction of martensite. An austenitevolume fraction ξγ in the cooling area is defined as follows.
Fig. 6. Joule heat density by three dimensional model. (Online ver-sion in color.)
Fig. 7. Comparison between three-dimensional model and simpli-fied method.
Fig. 8. Joule heat density by heat source model (Eq. (4)). (Onlineversion in color.)
3.3. Temperature, Metal-structure, and Strain RateDependencies of Flow Stress
To carry out an elastic-plastic deformation analysis con-sidering temperature, metal-structure and strain rate depen-dencies of flow stress, the flow stresses were measuredexperimentally at various conditions under uniaxial tensionusing electrical resistance heating and inert gas flow coolingas follows.
To measure the flow stress of ferrite-pearlite structure, thetemperature was increased by heating rate of 500°C/s. Whenthe temperature reached the testing temperature, the tensiletest was immediately carried out. The tests were carried outin the temperature range between room temperature and800°C.
To measure the flow stress of austenite structure, the tem-perature was first increased to over 1 000°C to obtain fullaustenite structure. After that, the specimen was cooleddown in 100°C/s until the temperature reached a preset val-ue and the tensile test was immediately carried out. Theupper critical cooling rate of this material is 30°C/s;17) hencethe tensile test can be carried out in full supercooled austen-ite state. The test was conducted in the temperature rangebetween 400 and 1 100°C.
To measure the flow stress of the martensite structure,quenched full martensite material was used. The tensile testwas carried out after the temperature was increased to a pre-set value. The test was conducted in the temperature rangebetween room temperature and 400°C.
In this deformation analysis, mechanical properties suchas Young’s modulus and flow stress of mixture are definedby the linear mixture rule.13) For example, the flow stress ofmixture is given as follows
..................... (14)
where σ yΙ(T) is an experimental value of flow stress of phaseI at the temperature, ξ Ι is a predicted value of the volumefraction of the phase, and σ y(T) is a calculated flow stress. Inthis analysis, N is 3 because the three kinds of phases appear.
Strain rate dependency of flow stress is defined by thefollowing method. The tensile test was carried out in theconditions: temperature of 800°C to 1 000°C and strain rateof less than 0.6 s–1.
Figure 9 shows the obtained flow stresses at a strain of0.03: the medium value of this simulation. Here, the exper-imental result is approximated by the following Cowper-Symonds law18)
...................... (15)
where D and P are material constants, is a strain rate, andσ0 is a flow stress at = 0. The lines in Fig. 9 are theapproximations using Eq. (15). In this case, the increase offlow stress from 800°C to 1 000°C is described by a pair ofparameters: D=1.53 s–1, and P=2.57.
3.4. Thermal and Transformation StrainThermal and transformation strain is defined as follows.
First, a density of each metal structure is calculated byMiettinen’s equations (Eqs. (16) to (22)19)).
........................................ (16)
........................................ (17)
........................................ (18)
................. (19)
.... (20)
.... (21)
........................................ (22)
where ργ(T) is a density of austenite structure, ρα+C(T) is adensity of ferrite-pearlite structure and ρM(T) is a density ofmartensite structure. (T) is a density of austenite struc-ture of pure iron, (T) is a density of ferrite structure ofpure iron and ρC(T) is a density of cementite structure.ρsub(T) is a member to express the effect of compositionexcept carbon.
Using the above definitions, the thermal and transforma-tion strain dεT,Tr is given in the form11)
.......... (23)
where l(T) is a length of line element in the temperature T.Fig. 9. Flow stress fitting by Cowper-Symonds law.
ξ ξγ = − ≤1 M sT M( ).
σ σ ξy yI II
N
T T( ) = ( )=∑
1
,
σ σε
= + ⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪0
1
1D
P/
,
εε
ρ ργ γT T X X
X X
Fe C Si
Mn M
( ) = + − + ×( ) −
− +
−118 26 7 39 10 68 24
6 01 12 45
3. . .
. . oo
Cr Cr
T
T X X
+ − + ×(− × − × )+
−
− −
7 59 3 422 10
5 388 10 1 4271 10
1 54
3
7 2 2
. .
. .
. ++ × − ×(+ × )
− −
−
2 267 10 11 26 10
6 2642 10
3 7 2
2
. .
. ,
T T
X XNi Ni
ρρ ρ
ραα
+
−
=−
( )+
( )⎛
⎝⎜⎜
⎞
⎠⎟⎟ + ( )C C
Fe
CC subT
X
T
X
TT( )
.
. .,
6 69
6 69 6 69
1
ρ ρ ραMFe C C subT T X X T( ) . . ,= ( ) + −( ) + ( )7 92 168 2
ρk(T) and ρκ+1(T) are densities at the beginning and end ofincrement, respectively. ρκ+1(T) is calculated using the lin-ear mixture rule with volume fraction ξi(T) and the densityρi(T) at the end of increment as follows.
................ (24)
Figure 10 shows the temperature-strain curves obtainedfrom the simulation using Eqs. (8) to (13) and Eqs. (16) to(24) and the experiment. In the experimental dilatometrytest, the heating rate of 10°C/s and the cooling rate of60°C/s were employed. An external force was not applied.Although the simulated result is in agreement with theexperimental one qualitatively, this does not match quanti-tatively. Especially, a discrepancy is observed in the trans-formation curve from austenite to martensite. A discrepancyis also observed in the slope of the cooling curve. The effectof the discrepancies on the simulation of 3DQ will be dis-cussed in sections 4.1 and 4.2.
4. Results
4.1. ThicknessTo confirm the analysis accuracy, an experiment and an
analysis of bending processes were carried out under thethree conditions shown in Table 1. Because of the validationof large deformation, the thermal and transformation strainsare neglected in these analyses. The temperature distributionwas calculated by the simplified method to save time for thepreparation of simulation models. The parameters in heatsource model in section 3.1.2 were identified as m=–5.67mm, a=53 mm and δ =2.8 mm. Qmax was adjusted to reachthe outer surface temperature 1 000°C. The followingparameters are the same among the three conditions. Thefeeding speed is 80 mm/s, and a heat transfer coefficient ofthe cooling area is 25 000 W/m2K. Figure 11 shows thegeometry and boundary conditions of FE model. A four-node shell element with reduced integration was used. Fiveintegration points were given through thickness. The num-ber of element is about 20 000. The movement of robot wasmodeled and reproduced as displacement boundary condi-tions given at the top of tube.
Figure 12 shows the thickness strain distribution in the
longitudinal direction at the center of inner and outer sideof tubes (points a and b). The simulation results are in goodagreement with the experimental ones.
Figure 13 shows the comparison of maximum and mini-mum thickness strains between the experiment and the sim-ulation for all the conditions. Clearly the simulation resultsare in good agreement with the experiment results regard-less of the bending conditions.
In these analyses, the plastic deformation occurs in theportions of the ferrite-pearlite structure, the austenite structureand the mixture of ferrite-pearlite and austenite structure.The transformation model describes well the transformationfrom ferrite-pearlite structure to austenite structure as shownin Fig. 10, leading to the accurate prediction of flow stresschange. On the other hand, because the plastic deformationdoes not occur after the martensite transformation starts, itis also considered that the martensite transformation inwhich the discrepancy was observed in Fig. 10 does not
Fig. 10. Temperature-strain curves obtained from simulation andexperiment.
ρ ξ ρki i
i
n
T T T+
=( ) = ( ) ( ){ }∑1
1
.
Table 1. Analytical and experimental conditions.
Tube geometry Bending radius
Condition A 40 H × 46 W × 1.6 t 1 000 mm
Condition B 24 H × 56 W × 2.3 t 1 120 mm
Condition C 24 H × 56 W × 2.3 t 250 mm
Fig. 11. Geometry and boundary condition of FE model (Condi-tion C).
Fig. 12. Comparison of longitudinal thickness strain distributionsbetween experiment and FEA (Condition C).
4.2. Camber by Inhomogeneous CoolingThe above bending analysis was good enough to confirm
the analysis accuracy of large deformation. However, thismodel was not suitable to confirm the thermal and transfor-mation deformation because the deformation due to bendingis much larger than thermal deformation. Therefore,quenching analyses with various inhomogeneous coolingconditions were carried out to confirm the accuracy of thesimplified method and the thermal and transformation strainmodels.
Figures 14 and 15 show the boundary conditions andgeometry of the FE model, respectively. They were deter-mined based on practical 3DQ conditions.
A rectangular tube of 46 mm in height, 40 mm in width,1.6 mm in thickness, and 1 000 mm in length was used.Feeding speed was 80 mm/s.
Table 2 shows the simulation conditions. To clarify thetransformation effect, the analysis was carried out with orwithout considering the transformation. When the transfor-mation was not considered, metal structure remained in theferrite-pearlite structure (Cases 7 to 12 in Table 2). In thesecases, the flow stress of ferrite-pearlite structure at 800°C orover was substituted by the flow stress of austenite becausethe flow stress of ferrite-pearlite structure is not be able toobtain at 800°C or over.
To examine the analysis accuracy under different bound-ary conditions two kinds of displacement boundary condi-tions were employed as shown in Fig. 14 and Table 2, i.e.the edge of tube was constrained by the robot or was free.
When the front edge was constrained by the robot motion(Cases 1 to 3 and 7 to 9 in Table 2), the longitudinal dis-placement Ux was set to free and the other degrees of free-dom were fixed. On the other hand, in Cases 4 to 6 and 10to 12, all degrees of freedom were set to be free.
To change the curvature of camber, three kinds of coolingconditions were employed.
Condition 1 is a basic condition where homogeneouscooling is assumed. In conditions 2 and 3, inhomogeneouscooling is assumed in which the cooling rate in plane A (Fig.15) is lower than the other planes. The cooling rate in planeA of condition 3 is lower than condition 2.
In the experiment, the amount of water flow onto planeA was reduced to decrease the cooling rate in condition 2
and 3. It should be noted that this inhomogeneous coolingcondition is only for the verification analysis and is not inpractical use of 3DQ. In the analyses, a cooling curve wasfitted to experimental data by try-and-error adjustment usingthe simplified method. In such cases, the simplified methodis effective to save the computation time.
Figure 16 shows the temperature evolutions at the innersurface of the tube in planes A, B, C and D (Fig. 15)obtained by the experiment and the simulation. The data wasevaluated along the lines LA and LC (Fig. 14). The simulationresults agree well with the experimental results.
Curvature after cooling was evaluated by three-point cur-vature in a length of 800 mm.
Figure 17(a) shows the experimental results of curvatureafter quenching. A positive curvature means plane A is theinner side of camber and a negative curvature the outer side.These results have the following features. The absolute cur-vature becomes larger when the cooling rate of plane A
Fig. 13. Comparison of thickness strain between experiment andFEA.
Fig. 14. Geometry and boundary condition of FE model (Temper-ature distribution for cooling condition 3).
becomes lower. And the curvature variation with the robotconstraint condition is opposite from those of the free con-dition. The absolute values of curvature obtained with therobot constraint conditions are smaller than those of freecondition.
Figure 17(b) shows the simulated results in which phasetransformation was taken into account (Cases 1 to 6 in Table2). These results are in good agreement with the experiment.
On the other hand, the simulated results in which phasetransformation was not taken into account (Fig. 17(c)) donot agree with the experimental results. In the free conditionFig. 16. Temperature evolution of each cooling condition.
Fig. 17. Experimental and simulated results of camber for inhomo-geneous cooling.
(Cases 10 to 12 Table 2), the direction of camber was oppo-site from that observed in Fig. 17(b). In the robot constraintcondition (Case 7 to 9 of Table 2), the curvature becamemuch larger than that observed in Fig. 17(b).
The above results confirm the effectiveness of this anal-ysis framework. Furthermore, it turned out that an accuratethermal deformation analysis is necessary by considering atemperature distribution, a metal-structure change and amechanical boundary condition.
Because the results shown in Fig. 17(b) are in good agree-ment with the experiment, the effect of the discrepancies inthe temperature-strain curve (Fig. 10) on the simulationresults would be small as far as we tested in the presentstudy. However, there is a room to improve the accuracy ofthe transformation model. For example, Bok et al.20) pro-posed a model which can approximate the volume fractionof martensite using some fitting parameters. Using this mod-el, it is expected that the simulation accuracy with the vari-ety of forming conditions may be improved. This will be ourfuture work.
4.3. Deformation Behavior in 3DQTo clarify the deformation and the stress evolution during
3DQ, a simulation of a simple bending process was carriedout. First, the features of temperature distribution generatedfor the bending analysis are described in section 4.3.1. Afterthat the bending analysis conditions and the results aredescribed in section 4.3.2.
4.3.1. Temperature Distribution for Bending AnalysisTo generate the three dimensional temperature distribu-
tion for the bending analysis, the three-dimensional modeldescribed in section 3.1.1 was used. The analysis was car-ried out under the following conditions: a rectangular tube
of 35 mm in height, 45 mm in width, 1.6 mm in thickness,80 mm/s in feeding speed, and maximum temperature of1 000°C. The heat transfer coefficient of the cooling areawas assumed to be 20 000 W/m2K. Figures 18 and 19 showthe results of joule heat density and temperature distribu-tions, respectively. The broken lines show the coil portionand cooling area. The peak of joule heat density appears atthe upper stream of center of coil (point A in Fig. 18). Thisis because of the decreasing of heat efficiency owing to themagnetic transformation which occurs at 737°C duringheating. On the other hand, the temperature is distributedhomogeneously in the circumferential direction. The peaktemperature arises just before the cooling area (point B inFig. 19).
4.3.2. Deformation AnalysisFigure 20 shows the geometry and boundary conditions
of FE model. The length of the tube was 1 000 mm, and thebending radius was 1 000 mm at the center line of tube. Theelement size was 1.8 mm each side. To simplify the simu-lation, thermal and transformation strains were neglected.The temperature distribution shown in Fig. 19 was applied.
Figure 21 shows the result at the center of thickness ofthe evaluation nodes in Fig. 20. The longitudinal distribu-tions of the temperature, equivalent stress, subsequent yieldstress, and longitudinal strain rate are shown. The subse-quent yield stress decreases rapidly from 400 N/mm2 to 40N/mm2 as the temperature increases from 600°C to 800°C.
Fig. 18. Joule heat density at outer surface obtained by threedimensional model.
Fig. 19. Temperature distribution at outer surface obtained bythree dimensional model.
Fig. 20. Geometry and boundary condition of FE model.
Fig. 21. Distributions of temperature, strain rate, and stress at cen-ter of thickness.
The plastic deformation occurs at the portion where the tem-perature is higher than 800°C. There after the plastic defor-mation does not occur because the material is cooled. As aresult, the plastic deformation occurs only within the narrowarea with a longitudinal length of about 20 mm. Such a char-acteristic of the plastic deformation is effective to suppressthe change of cross sectional shape. The yield stress isincreased by martensite transformation.
The equivalent stress is relatively small in the wholeregion regardless of the temperature. This shows that themagnitude of stress in the whole region is determined by theflow stress in the hot region, yielding small springback inthe induction bending.
5. Conclusions
In this study, a simulation framework of the three-dimen-sional hot bending and direct quench (3DQ) process is pro-posed to investigate the deformation behavior. The featuresof this framework and the simulation results are as follows.
(1) A coupled thermo-mechanical-metallurgical FEManalysis framework was developed. In the electromagneticand temperature analysis, a non-linearity of permeability,magnetic transformation and temperature dependent ofthermal and electric property were considered. The threedimensional temperature distribution can be obtained bythis analysis. In the deformation analysis, temperature,micro structure and strain rate dependencies of flow stresswere taken into consideration. The phase transformation ratewas calculated using Kunitake’s equation and Koistinen-Marburger relationship. The flow stress of the mixture wascalculated by a linear mixture rule. A strain rate dependentof flow stress was defined by Cowper-Symonds law. To cal-culate the thermal and transformation strain, Miettinen’sdensity law was employed.
(2) A heat source model for local induction heating oftube and the simplified method to generate the temperaturedistribution was also proposed. The proposed method iseffective in terms of the computation time.
(3) The analysis accuracy of large deformation was con-firmed through a bending analysis. And the thermal defor-mation accuracy was confirmed through an analysis ofquenching process with inhomogeneous cooling.
(4) The detail of deformation mechanism in 3DQ wasrevealed by this analysis. For example it was found that theplastic deformation area was narrow of about 20 mm, andthe deformation stress was about 40 N/mm2.
(5) Although the simulated results of 3DQ are in agree-ment with the experimental ones, the discrepancies wereobserved in the dilatometry test. The improvement of thethermal and transformation models will be our future work.
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