compModeOfJominyTestPaperID1396

8/11/2019 compModeOfJominyTestPaperID1396

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Paper ID: 1396

Computational Modeling of a Jominy TestDung-An Wang

1*, Wei Lo2, Wei-Yi Chien

2and Yhu-Jen Hwu

2

1Graduate Institute of Precision Engineering, National Chung-Hsing University

2

Iron & Steel Research & Development Department, China Steel CorporationCorresponding: [email protected]

AbstractA computational model of the Jominy end-quench

test is developed. Scheil s additivity rule andJohnson-Mehl-Avrami-Kolmogorov (JMAK) model areused to model the phase transformation. The model isimplemented in a finite element code. Temperaturehistory, stress, and volume fraction of microstructural phases of a S50C steel during the Jominy test are

calculated. The simulation results are verified byexperiments.

Keywords: Jominy test, Model, Phase transformation

1. IntroductionDimensional accuracy of steel parts after

quenching is required by customers of steel companies.Control of shape and deformation of the quenched steel plates is not effective by manual adjustment of the parameters of quenching processes. Throughdevelopment and usage of simulation tools for quenching, better mechanical properties,microstructures, and final dimensional accuracies of thequenched steel parts can be achieved without resortingto manual labor. Finite element analysis has beenextensively used as an effective tool to understand theeffects of the processing parameters on the properties of the quenched steel parts[1].

Mathematical modeling of quenching process of steel has been investigated by many researchers. Inorder to simulate the quenching process, accuratemodeling of phase transformation is important. Themodels implemented in finite element codes needverification before being used in the steel industry.The Jominy test has been extensively used tocharacterize the hardenability of quenched steels. Thissimple test can be utilized to obtain temperature historyand volume fraction of phase transformation duringquenching for verification of the mathematical models.

In this investigation, a finite element program for prediction of thermal stresses and distortion in quenchedsteel parts is developed. The diffusional phasetransformation model of JMAK model and diffusionless phase transformation model of Koistinen and Marburger (KM) model are introduced in the program. Theresults of this program are compared with the Jominytest results.

2. Mathematical Model

2.1 Phase Transformation ModelVarious microstructures of steel parts during

quenching evolve differently according to cooling rates,temperatures, sizes, and alloying elements. Themicrostructural evolution has a significant effect on thefinal shape and dimensions of the quenched steel parts.Phase transformations during steel quenching can bediffusional transformation or diffusionlesstransformation[2]. The formulation of the phasetransformation presented below follows the basic procedure reported by Kang and Im[1].

Volume fractions i of diffusionaltransformation, including ferrite, pearlite and bainitetransformation, can be described by the JMAK equation[1]

1 exp bi j at (1)where a and b are material parameters obtainedfrom the time-temperature-transformation (TTT)

diagram. j t is the transformation time. Fig. 1 is aschematic of a TTT diagram. As illustrated in Fig. 1,initial and final volume fractions of each phase are takenas s (=0.01) and f (=0.99), respectively,

corresponding to the transformation beginning time s

and transformation finishing time f , respectively. a

and b are determined by

ln(1 )sbs

a

(2)

ln ln(1 ) / ln(1 )

ln( / )

s f

s f

b

(3)

The transformation time t is expressed as1/1ln(1 )

b j i

j j bs

t t

(4)

where j t is the time interval as shown in Fig. 1.

We adopt the Scheils additive rule to decide if the phase transformation occurs. According to this rule, phase transformation begins when the sum of thefollowing equation equals to unity[1]


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Paper ID: 1396

1 j j j

t

(5)

where j t is the time step and j the transformation

beginning time at the jth time step.When the diffusionless transformation occurs, the

volume fraction m of martensite transformation, isdescribed by the KM equation[1]

1 1 exp 0.011m i si

M T

(6)

where i are the volume fractions except the austeniteand martensite.

Heat generated due to phase transformation, latentheat, results in temperature increase of the quenched parts. The change in latent heat can be calculated

based on ehthalpy change. Enthalpy change of diffusional transformation of pearlite and ferrite, pH

and f H , respectively, can be calculated by[3]5 7 20.122 5.221 10 1.532 10 pH T T (7)

4 7 2 10 30.226 7.309 10 8.043 10 2.940 10 f H T T T (8)

Enthalpy change of diffusionless transformation of martensite, mH , is given by[4]

640mH (9)

where pH , f H and mH are the enthalpy change,

J/kg, of austenite-pearlite, austenite-ferrite andaustenite-martensite transformation. Enthalpy changeof austenite-bainite transformation, BH , is taken as the

same as pH in this investigation.

2.2 Procedure and FormulationThe phase transformation model and a

thermo-elastic-plastic constitutive equation areconsidered in the thermo-mechanical finite element program. The procedure for prediction of thetemperature history and volume fraction of themicrostructural phases is illustrated in Fig. 2. The

Newton-Raphson method is implemented for theelasto-plastic analysis. The flowchart for obtaining thestress and displacement solutions due to the changes intemperature and phase transformation is shown in Fig.3.

3. The Jominy Test3.1 Experiments

A schematic of the Jominy test considered in thisinvestigation is shown in Fig. 4. Dimensions of aS50C cylindrical specimen are indicated in the figure.The specimen is heated to austenitizing temperature,

then placed on a holding fixture. The lower end of thespecimen is quenched by a standard water jet of 28 oC.

Three thermocouples are inserted in the specimen andtheir locations are shown in Fig. 4. Temperaturehistories of the thermocouples are recorded.Micrographs of the cross section at the same locationsas the thermocouples are taken after the end-quench

testing.3.2 Finite Element Simulations

A finite element program is developed in order toobtain accurate temperature and stress solutions for theJominy test specimen,. Due to symmetry, anaxisymmetric slice of the specimen is considered. Fig.5 shows a mesh of an axisymmetric model of thespecimen. The model has the height h (=100 mm)and the radius r (=12.7 mm) according to thedimensions of the specimen. The finite element modelhas 530 4-node elements. An axisymmetric coordinatesystem is also shown in the figure. Neumann

boundary condition is applied at the axis of symmetry of the specimen and a heat flux 0q is introduced.Fourier boundary condition with heat convection andradiation is applied at the water spraying surface and atthe air cooled surfaces of specimen. The displacementin the r direction of the symmetry axis, the z axis, isconstrained to represent the symmetry condition due tothe loading conditions and the geometry of the specimen.According to the experiments, the initial temperature of the specimen is 830 oC, and the temperatures of thewater jet and air are 28 oC and 30 oC, respectively.

In this investigation, the material of the specimenis assumed to be elasto-plastic, isotropic material.

Typical values of t he Youngs modulus , Poissons ratio ,yield strength, strain hardening exponent, coefficient of thermal expansion, density, thermal conductivity andspecific heat of various phases of steel as functions of temperature are considered in the simulations. Table 1lists the free convection heat transfer coefficients of water and air as functions of temperature. Thecommercial finite element program ABAQUS isemployed to perform the computations. Axisymmetric bilinear displacement and temperature element, CAX4T,is used. The user subroutines UMAT, UEXPAN andUMATHT are used to define the material thermal andelasto-plastic behaviors.

4. Results and DiscussionsFig. 6 shows three computed temperature

histories at the distance of 2 mm, 12 mm and 22 mmfrom the quenched end. The computations are basedon the free convection of water at the quenched end.The computed results show a progressive decrease inthe rate of cooling from the quenched end. Measuredtemperature histories are also plotted in the figure.The computed temperatures drop less steeply than themeasurements at the beginning of the quenching. Asthe quenching time approaches 60 s, the computedtemperatures are lower than the measurements. Thecomputed history at the distance of 2 mm from the


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Paper ID: 1396

quenched end is in good agreement with themeasurement after the quenching time passes 20 sec.The discrepancy between the computations and theexperiments can be attributed to the different convectiveheat transfer mechanisms at the quenched end,

measurement errors, and material properties used incomputations.

Fig. 7 shows the temperature history calculatedfor various distance from the quenched end based on theforced convection heat transfer coefficients, which aremultiplied by a factor of 1.3 of the free convection heattransfer coefficients of water listed in Table 1. Thecomputed result at a distance of 2 mm from thequenched end agrees better with the experiments at the beginning of quench, but is slightly lower than theexperiments as the quenching time approaches 60 sec.

Fig. 8 shows the computed volume fractions of the phases at various positions on the circumferential

surface of the specimen as functions of the distancesfrom the quenched end after 60 minutes of endquenching. The austenite phase near the quenched endwas rapidly transformed into martensite phase due to thewater spraying at the bottom surface. Towards the topsurface of the specimen, austenite phase grows and themartensite phase ceases to exist at the normalizeddistance of 0.19. The volume fraction of the bainite phase increases sharply at the normalized distance of 0.15, reaches its peak at the normalized distance of 0.23,then decreases. The lower cooling rate in thisneighborhood than that near the quenched end results inthe growth of the ferrite, pearlite and bainite phase.The experimental volume fraction of the microstructural phases will be determined by micrographs taken fromthe specimen 6.35 mm deep and parallel to the specimencylindrical axis and will be compared to the computedresults based on the developed finite element program.

Fig. 9 shows the computed residual stresses atvarious positions on the circumferential surface of thespecimen as functions of the distances from thequenched end after 60 minutes of end quenching. Dueto steep decrease in temperature at the bottom surface,tensile stress was generated near the quenched end,while compressive stress was generated at the middleregion of the specimen. Absolute values of the stresseswere decreased gradually towards the top surface of thespecimen due to smaller change in the temperaturecompared to that in the region near the quenched end.

5. ConclusionsA finite element program was developed to

determine the temperature, volume fraction of microstructural phases and stress of the Jominy testspecimens. The computed results based on the program have a good agreement with the experimentalresults of the Jominy test. The finite element programcoupled with the phase transformation model and the

elastoplastic formulation can be applied for prediction

of residual stress and distortion of quenched steel parts.

6. AcknowledgementsThis work is funded by China Steel Corporation

under grant RE101019.

7. References1. S.-H. Kang and Y.-T. Im , Three-dimensional

thermo-elastic-plastic finite element modeling of quenching process of plain-carbon steel in couplewith phase transformation , International Journal of Mechanical Sciences, Vol. 49, pp. 423-439, 2007.

2. C. imir and C. H. Gr, 3D FEM simulation of steel quenching and investigation of the effect of asymmetric geometry on residual stressdistribution, Journal of Materials ProcessingTechnology, Vol. 207, pp. 211-221, 2008.

3. L. S. Darken and R. W. Gury, Physical chemistry of metals, McGraw-Hill, New York, 1953.

4. T. Ericsson, S. Sjostrom and M. Knuuttila,Predicting residual stresses in cases. In: DiesburgDE, editor. Case hardened steels, MetallurgicalSociety of AIME, 1983.

8. Figures and Tables

Fig. 1 A schematic of a TTT diagram.


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Fig. 2 Flowchart for prediction of the temperature andvolume fraction of each phase.

Fig. 3 Flowchart for the elasto-plastic analysis.

Fig. 4 Schematic of the Jominy test considered in thisinvestigation.

Fig. 5 A mesh of the axisymmetric model.


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Paper ID: 1396

Fig. 6 Temperature history calculated for variousdistance from the quenched end based on the freeconvection heat transfer coefficients.

Fig. 7 Temperature history calculated for variousdistance from the quenched end based on the forcedconvection heat transfer coefficients.

Fig. 8 Volume fractions of the phases at various positions on the circumferential surface of the specimenas functions of the distances from the quenched end.

Fig. 9. Residual stresses at various positions on thecircumferential surface of the specimen as functions of the distances from the quenched end.

Table 1 Free convection heat transfer coefficients.T [oC] h c [J/m

2s oC] of freeconvection for water

hc [J/m2s oC] of free

convection for air 0 4350 5200 8207 25300 - 40400 11962 50430 13492 -500 12500 -560 10206 -600 7793 75

700 2507 90800 437 110900 135 -1000 - 175

1* 2 2 2

1

2

: [email protected]

Scheil Johnson-Mehl-Avrami-Kolmogorov

S50C

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compModeOfJominyTestPaperID1396