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Theoretical and experimental studies on laser transformationhardening of steel by customized beam

Michael K.H. Leung a,*, H.C. Man b, J.K. Yu c

a Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, Chinab Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, China

c Environmental Corrosion Center, Institute of Metal Research, Chinese Academy of Sciences, China

Received 27 November 2005; received in revised form 1 March 2007Available online 23 May 2007

Abstract

An exact solution was developed for a quasi-steady-moving-interface heat transfer problem in laser transformation hardening by abeam customized to a flat-top rectangular shape. The absorption and release of the latent heat were taken into account. The solutionprovided the body temperature, heating rate, and subsequent cooling rate. Using the IT and CCT diagrams, the rapid austenitizationand pearlitic/martensitic phase transformation were determined for prediction of the hardening effect. In the experimental study, a dif-fractive optical kinoform (DOK) was used to customize a conventional Gaussian laser spot beam to a flat-top rectangular beam. A rea-sonable agreement between the theoretical predictions and the experimental measurements was obtained. 2007 Elsevier Ltd. All rights reserved.

Keywords: Laser transformation hardening; Flat-top rectangular beam; Moving phase-change interface; Exact solution

1. Introduction

Laser can be used for a wide range of surface heat treat-ment. A high power density laser operating at a slow tra-versing speed can cause the heated surface to melt. Thesurface can thus be treated for glazing, sealing, alloying,cladding, and material homogenization. On the otherhand, a low power density laser operating at a fast travers-ing speed can be used for surface hardening. In this pro-cess, the cooling of the work area can be effected by

rapid internal heat diffusion through the bulk material sur-rounding the heated region. An extrinsic quenching processis thus unnecessary.

In laser transformation hardening of steel, the materialheated by a moving laser beam develops two phase-trans-formation interfaces moving along with the beam. Theleading interface involves material heated above the

austenitization temperature to change phase from cement-ite/ferrite mixture to austenite. The austensite is then rap-idly cooled and transformed into pearlite or martensite atthe trailing interface. As the interfaces absorb and releaselatent heat due to the phase transformations, the problemcan be classified as a moving-interface heat transfer prob-lem. The transient problem can be analyzed for thequasi-steady state that is applicable for modeling manycontinuous industrial production processes.

When the laser heat treatment is needed for a large sur-

face area, the laser equipment can be set up to employ theautomated overlapping method. Conventional overlappingwith a Gaussian laser spot beam often results in adversetempering effect and irregular surface hardness[1]. Alterna-tively, an ordinary spot beam can be customized to a lon-gitudinal shape to alter the heat flux distribution resultingin a uniform surface treatment with less overlapped zone.It was found that a slope intensity profile could hold theheated zone in the austenitization temperature range longerso that the subsequent martensitic transformation wouldbe more uniform within the case depth[2].

0017-9310/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2007.03.022

* Corresponding author. Tel.: +852 2859 2628; fax: +852 2858 5415.E-mail address:mkhleung@hku.hk(M.K.H. Leung).

www.elsevier.com/locate/ijhmt

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substantial overlapping. The intensity profile along the x0

direction can be manipulated in order to control the heat-ing rate and cooling rate for proper phase transformationof the treated material. Since the base of the rectangularbeam has a large length-to-width ratio, the heat transfercan be simulated by a two-dimensional model. This paper

presents the derivation of theoretical solution for a laserbeam customized to a flat-top, rectangular-base beam.The setup shown inFig. 1can be represented by a sta-

tionary slab heated by a moving heat source at a constantvelocity as illustrated in Fig. 2. The fixed coordinates andthe moving coordinates are represented by x00;y00 andx0;y0, respectively. It is assumed that the phase changeis diffusion-driven in a medium of uniform properties indifferent phases. The assumption is justified as the phasetransformation always involves in a solid state, which isentirely different from a solidliquid phase change. Thethermal properties (thermal conductivity, specific heat,etc.) in different phases, i.e. austenite, pearlite, ferrite, and

martensite, do not vary considerably. The ratio of the beamlength to width is large enough to neglect any lateral effectsso that the problem domain becomes two-dimensional. Theenergy equation of this transient heat transfer problem canbe expressed as

r2TXja;b

qLj

k mn;jdn

00 n00j 1

a

oT

ot 1

where the interfaces are situated at n00j and moving at avelocity mn;j perpendicular to the interface. The interfacemovement in the normal direction can be presented inthe x00-direction by using the following substitution as de-

rived by Patel[13]

mn;jdn00 n00j

oS00j

ot dx00 S00j y

00; t; j a;b 2

and

TS00j y00; t;y00; t Tp; j a;b 3

where S00j y00; t denotes the x00-coordinates of the interface

positions and Tpis the phase-change temperature. The ini-tial and the boundary conditions are given by

Tx00;y00; 0 0 < Tp 4

oTx00; u; t

oy00

hu

k Tx00; u; t fx00 5

oTx00; 0; t

oy00

h0

kTx00; 0; t 0 6

T1;y00; t finite; hu h0 0

0; hu > 0 or h0 > 0

7

T1;y00; t 0 8

Eqs.(1)(8)establish the general differential equation prob-lem for the quasi-steady-moving-interface phenomena of

laser transformation hardening. The derivation of the solu-tion is illustrated for a specimen heated on top by a movingflat-top rectangular beam and insulated on bottom. Theheat source represented by fx00 in Eq.(5)can be writtenas a composite of two step functions

fx00 q

wkUx00 Vt l002 Ux

00 Vt l001

l001 l002

9

whereq is the absorbed laser power; Uz is a step functionwith Uz> 0 1 and Uz6 0 0; l001 and l

002 are the

x00-coordinates of the leading-end and trailing-end of therectangular beam, respectively. The insulation on the bot-

tom surface implies that h0 is zero. The transient problemcan be converted to a dimensionless quasi-steady problemby the following substitutions:

x x00 Vt

u ; y

y00

u ; Sj

S0j

u ; j a;b;

ZwkT

q ; Zp

wkTp

q ; Stj

cTp

Lj;

La L1; Lb L2; Pe Vu

a ; Biu

huu

k 10ah

Accordingly, the governing equation, boundary conditions,and interface conditions can be written in the dimensionless

form:

r2ZXja;b

PeZp

Stjdx Sjy Pe

oZ

ox 11

oZx; 1

oy BiuZx; 1

Ux l1 Ux l2

l 12

oZx; 0

oy 0 13

Z1;y finite; Biu 0

0; Biu > 0

14

Z1;y 0 15

ZSjy;y Zp; j a;b 16

Fig. 2. Problem domain in fixed and moving two-dimensional coordinate

systems.

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With reference to method reported in the previous study[11], Zcan be written as a superposition solution

Zx;y Z1x;y Z2x;y 17

where Z1 and Z2 attribute to the laser beam source in thenonhomogeneous boundary condition (Eq. (12)) and themoving interface in the nonhomogeneous governing equa-tion (Eq. (11)), respectively. Converting the governingequation ofZ1 into a homogeneous KleinGordon equa-tion and solving theoretically yield

Z1x;y e

Pe2x

pl1 l2

Z 1x0

coshny

n2nsinhnBiu coshn

e Pe

2l1 Pe

2 cosxl1 x xsinxl1 x

e Pe

2l2 Pe

2 cosxl2 x xsinxl2 x dx

18

where

n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

Pe2

4

r 19

In solving for the temperature Z2 resulting from theheat absorption and heat release of the moving interfaces,the source-and-sink method is applied to first convert theproblem to a nonhomogeneous KleinGordon equation.With proper expansion in Fourier series and integral com-bination, Z2 can be solved and explicitly written for thefour regions in the problem domain shown in Fig. 3[12]

Region A:

Z2x;y

X1n1

Cn PeZp

St

Z 1yp

eP xSa g eP xSbg

Hpngdg

( )Hpny

20

Region B:

Z2x;y

X1n1

Cn PeZp

St

Z 1yp

ePxSagHpngdg

"(

Z yd x

yp

ePxSb gHpngdgZ 1

yd x

ePxSb gHpngdg#)Hpny21

Region C:

Z2x;y X1n1

Cn PeZp

St

Z ydxyp

ePxSagHpngdg

"(

Z 1ydx

eP xSagHpngdg

Z 1yp

ePxSbgHpngdg

#)Hpny 22

Region D:

Z2x;y X1n1

Cn PeZp

St

Z 1yp

eP xSag(

ePxSbgHpngdg

)Hpny 23

where

Cn 1

2Npnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2n

Pe2

4

q 24Npn

1

2 Bi0 p

2n Bi

20 1

Biu

p2n

Bi2u

25Hpny pncospny 26

and

P

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2n

Pe2

4

r

Pe

2 27

The discrete eigenvaluespn

are obtained by solving the fol-lowing expression:

tanpn Biu

pn28

An iterative process based upon the concept of pertur-bation is employed to find the interface positions Sa andSb. Eqs.(17)(27)can be directly solved by numerical inte-gration. The temperature profile Z Z1 Z2in the movingcoordinates will be used to determine the heating rate andcooling rate for the isothermal-transformation (IT) austen-itization analysis and the continuous cooling transforma-tion (CCT) pearitic/martensitic phase-transformationanalysis, respectively. As a result, the enhanced materialhardness is determined.

3. Experimental procedures

An experimental study on laser transformation harden-

ing by a customized beam was conducted for validation ofFig. 3. Multiple regions in the moving dimensionless coordinates.

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the theoretical model developed. The sample was made ofAISI 1050 steel having a chemical composition ofC(0.470.55), Mn(0.60.9), P(max 0.04), and S(max 0.05).The sample was machined to a 300 mm 25 mm 13 mmrectangular bar. The test surface was ground with 220-gritSiC paper and then sandblasted with 200-grit SiO2. Finally,a graphite dag layer was coated on the test surface. A laser

power meter was used to measure the laser absorptivity ofthe coated surface and it was found to be about 60%.

A high-power CO2 laser was regulated to generate0.8-kW output. The original Gaussian spot beam wascustomized to a flat-top rectangular beam by means of adiffractive optical kinoform (DOK). The DOK was acomputer generated holographic element with speciallydesigned array cell pattern as shown inFig. 4. The custo-mized beam size was 15 mm 2 mm and the beam powerwas specified to be 60% of the incident beam. The lasertransformation hardening experiment was set up as illus-trated inFig. 1. The steel sample bar was insulated on allsides, except the upper test surface. The feed rate was con-trolled at constant 0.83 mm s1 (50 mm min1) for the laserbeam to scan along the entire 300 mm span of the steel sam-ple. The hardening effect was measured by a microindenta-tion hardness tester (model BUEHLER 1600-6301).

4. Results and discussion

In order to avoid any confusion, all the temperaturesmentioned in the remaining discussion are measured abovea common reference of absolute temperature at 273 K. Thematerial properties of AISI 1050 steel and the laser trans-formation hardening test parameters used in both theoret-

ical analysis and experimental test are listed in Table 1.

4.1. Theoretical results

In the theoretical analysis, solving the dimensionlessEqs. (17)(27) followed by a conversion to dimensionalresults yielded the temperature profiles as plotted in

Fig. 5. The case depth of the hardened material could bedetermined from the temperature profiles. Although thematerial at a depth of 650 lm exceeded the phase-transfor-mation temperature Tp (AC3) of 770 C, the materialwould be hardly transformed to the austenitic phase forsubsequent pearlitic/martensitic transformation because

Fig. 4. Diffractive optical kinoform (DOK) used to produce a flat-toprectangular beam.

Table 1Properties of AISI 1050 steel specimen and laser transformation hardeningtest parameters

Material properties

q (kg m3) 7817k(W m1 K1) 30

c (kJ kg1 K1) 0.46

L (kJ kg1

) 83.6Tp (C) 770

Test parameters

u (mm) 13q0 (kW) 0.8DOK efficiency 60%Surface absorption 60%

q (kW) calculated 0.288l(mm) 2

w (mm) 15V(mm s1) 0.83hu (W m

2 K1) 180h0 (W m

2 K1) 0

Dimensionless parameters

Pe 1.29St 4.1Biu 0.078Bi0 0

Zm

1.16l1 0.0769l2 0.0769

Fig. 5. Heating and cooling temperature profiles at various depth.

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the heating process above 770C last for only a shortinstant. With reference to the IT diagram for rapid austen-itization, complete austenitization (100% austenite) wouldbe formed at a depth of 260 lm where the material washeated above 770 C with a mean temperature of 833 Cfor 2 s [14,15]. Therefore, the predicted case depth was

260 lm. The material just below the case depth did notundergo complete austenitization due to insufficient heat-ing time. Thus, the subsequent cooling process, dependingon the cooling rate, could only produce partial martensiteor pearlite that formed a transitional zone with a hardnesslower than that of the upper hardened zone.

The temperature profiles shown in Fig. 5could also beused to predict the hardness. The temperature profiles ofthe material within the case depth, labeled by solid squares,solid triangles, and solid diamonds, were very close duringthe cooling process belowTpof 770 C. The cooling rate at770 C was 130 C s1 and gradually decreased with time.The average cooling rate between 770 C and 600 C was

67.4 C s1. With reference to the CCT diagram [16], allthe austenite would be transformed to pearlite withoutany formation of martensite. The predicted hardnesswithin the case depth of 260 lm was 370 HV.

4.2. Experimental results

In the experimental study, detailed hardness measure-ments were conducted for the material at mid-length ofthe steel sample. The surface hardness profile across thesample width, i.e. thez00 direction, was measured. The mea-surements presented in Fig. 6 showed a uniform hardness

distribution within the irradiated area, 7.5 mm 6 z00 67.5 mm, with a mean value of 355 HV. The results vali-dated the simplification of the problem to a two-dimen-sional model as the majority hardening effect wasindependent of z00. The hardness values at both ends,z00 < 7.5 mm andz00 > 7.5 mm, were relatively low because

the heat dissipation in thez00 direction diminished the heat-ing effect for austenitization. In comparison with the theo-retical prediction, the predicted hardness of 370 HVdeviated from the mean measurement of 355 HV by 4.2%.

The sample was cut in the middle for more hardnessmeasurements along the depth of the centerline. The mea-

surements are presented inFig. 7. At a depth greater than240 lm, the hardness was around 280 HV showing nonoticeable hardening effect. From 240 to 180 lm depth,an obvious increase in hardness occurred in this transi-tional zone. Additionally, as illustrated in Fig. 8 showingthe micrograph of the cross-sectional view, the case depthcould vary between 190 lm and 320 lm. Such variationcould be caused by variation in laser absorption due tononuniform properties of the surface prepared by grinding,sandblasting, and coating of laser absorbing layer. As aresult, the theoretically predicted case depth of 260lmwas consistent with the experimental measurements.Fig. 9 clearly shows the interface between the austenitic

and pearlitic microstructures.

Fig. 6. Surface hardness after laser transformation hardening.

Fig. 7. Hardness versus depth.

Fig. 8. Micrograph showing the case depth of hardened material.

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The hardness measurements shown in Fig. 7were alsocompared with the theoretical predictions. The maximum,minimum, and mean of the microhardness values measuredbetween 0 and 180 lm depth were 370 HV, 334 HV, and349 HV, respectively. The predicted hardness of 370 HVjust fell within the measurement range and deviated fromthe mean measurement of 349 HV by 6.0%. A reasonableagreement between the theoretical and experimental resultswas obtained for the laser transformation hardening effectsin terms of case depth and hardness. Although the modelvalidation was done for pearlitic transformation, it impliedthat the theoretical model could be used for martensitictransformation because the two phase-transformation

mechanisms are basically the same.For the process parameters used in the above illustra-

tion, only pearlite was formed without any martensite inthe hardened zone because of the low cooling rate. Thus,the hardness was not increased by a significant amount.The hardening process could be improved by increasingthe feed rate as well as the laser power. Using the exactsolution developed in this study, it was found that a1.5-kW laser source with a feed rate of 3.33 mm s1

(200 mm min1) could yield surface material cooling from770 C to 250 C at a mean rate of 130 C s1. The conse-quent hardening effect would be 100% martensitic transfor-

mation for the material within a case depth of 130 lm witha surface hardness of 820 HV.

5. Conclusion

This paper presents a derivation of an exact solution formodeling the moving-interface heat transfer in a laser trans-formation hardening process. The solution is illustrated fora customized laser beam with a flat-top rectangular config-uration. The techniques used to solve a KleinGordonequation with expansion in Fourier series and integral com-bination were employed in the derivation. The theoreticalpredictions in terms of case depth and increased material

hardness agreed well with the experimental results. Besideslaser transformation hardening, the exact solution can beapplied to other laser heat treatment processes involvingmoving phase-change interfaces, such as glazing, sealing,alloying, and cladding.

Acknowledgments

The work described in this paper was fully supportedby a grant from the Research Grants Council of the HongKong Special Administrative Region, China (Project No.HKU 1008/01E) and a grant from the University Re-search Grants (No. 10206014) of the University of HongKong.

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Fig. 9. High-resolution micrograph of the phase-change interface.

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