Big File on Laser Hardening

11 Modeling of Laser SurfaceHardening

� 2008 by Taylor & Fra

Janez Grum

CONTENTS

11.1 Evolution of Laser Materials Processing ......................................................................... 50011.1.1 Laser Beam Mode Structure ............................................................................... 506

11.2 Laser Optics and Beam Characterization......................................................................... 50711.2.1 Focusing with a Single Lens............................................................................... 50711.2.2 Focal Length ....................................................................................................... 50711.2.3 Focal Number...................................................................................................... 50711.2.4 Beam Diameter at Focus..................................................................................... 50711.2.5 Depth of Focus.................................................................................................... 50811.2.6 Laser Beam Characterization .............................................................................. 508

11.3 Laser Light Absorptivity.................................................................................................. 51011.3.1 Temperature Effect.............................................................................................. 510

11.3.1.1 IR Energy Coatings ............................................................................. 51311.3.1.2 Chemical Conversion Coatings ........................................................... 51311.3.1.3 Linearly Polarized Laser Beam ........................................................... 514

11.3.2 Absorption Measuring Technique....................................................................... 51411.3.2.1 Absorptivity ......................................................................................... 51611.3.2.2 Absorption Control.............................................................................. 51711.3.2.3 Absorptivity in Iron and Steel............................................................. 51811.3.2.4 Influence of Roughness ....................................................................... 51911.3.2.5 Influence of Oxidation......................................................................... 52011.3.2.6 Interaction between Laser Beam and Materials

Coated with Absorbers ........................................................................ 52111.3.2.7 Testing of Various Absorbents............................................................ 524

11.4 Laser Surface Hardening.................................................................................................. 52611.4.1 Laser Heating and Cooling ................................................................................. 526

11.4.1.1 Temperature Cycle .............................................................................. 52611.4.2 Metallurgical Aspect of Laser Hardening........................................................... 527

11.4.2.1 Calculation of Thermal Cycle and Hardened Depth........................... 53011.4.3 Austenitization of Steels ..................................................................................... 535

11.4.3.1 Austenitization of Hypoeutectoid Steels ............................................. 53511.4.3.2 Eutectical Temperature Determination................................................ 53811.4.3.3 Phase Transformations at Various Heating and Cooling Rate ........... 54411.4.3.4 Effects of Heating and Cooling Rates on Phase Transformations...... 54511.4.3.5 Determination of Hardness Profiles .................................................... 547

11.4.4 Mathematical Prediction of Hardened Depth ..................................................... 55111.4.4.1 Mathematical Modeling for Microstructural Changes ........................ 555

ncis Group, LLC.

11.4.5 Method for Calculating Temperature Cycle ....................................................... 55811.4.6 Heat Flow Model ................................................................................................ 562

11.4.6.1 Dimensionless Groups......................................................................... 56411.4.7 Thermal Analysis of Laser Heating and Melting Materials ............................... 569

11.4.7.1 Cooling Rate........................................................................................ 57211.4.7.2 Laser Melting ...................................................................................... 57311.4.7.3 Mathematical Description and Solution .............................................. 574

11.5 Residual Stresses After Laser Surface Hardening ........................................................... 57611.5.1 Background ......................................................................................................... 57611.5.2 Determination of Thermal and Transformation Stresses .................................... 578

11.5.2.1 Heat Transfer Analysis ........................................................................ 57811.5.2.2 Thermal and Residual Stress Analyses ............................................... 57911.5.2.3 New 2D Finite Element Model ........................................................... 57911.5.2.4 Simulation Results and Discussion ..................................................... 581

11.5.3 Simple Mathematical Model for Calculating Residual Stresses......................... 58311.5.4 Determination of Stresses by Numerical Simulation.......................................... 58811.5.5 Simple Method for Assessing Residual Stress Profiles ...................................... 59611.5.6 Prediction-Hardened Track and Optimization Process ....................................... 60211.5.7 Application of Modeling..................................................................................... 605

11.5.7.1 Analytical Model ................................................................................. 60511.5.7.2 Case of Cylindrical Workpieces.......................................................... 60611.5.7.3 Prediction of the Heat Treatment Cycle

by Analytical Thermal Model ............................................................. 60811.5.8 Microstructure Analysis after Laser Surface Remelting Process........................ 610

11.5.8.1 Mathematical Modeling of Localized Melting AroundGraphite Nodule .................................................................................. 612

References ..................................................................................................................................... 621

11.1 EVOLUTION OF LASER MATERIALS PROCESSING

Lasers represent one of the most important inventions of the twentieth century. With their devel-opment it was possible to get a highly intensive, monochromatic, coherent, highly polarized lightwave [1,2]. The first laser was created in 1960 in Californian laboratories with the aid of a resonatorfrom an artificial ruby crystal. Dating from this period is also the first industrial application of laser,which was used to make holes in diamond materials extremely difficult to machine. First applica-tions of laser metal machining were not particularly successful mostly due to low capability andinstability of laser sources in different machining conditions. These first applications, no matter howsuccessful they were, have however led to the development of a number of new laser source types.

Laser is becoming a very important engineering tool for cutting, welding, and to a certain extentfor heat treatment. Laser technology provides a light beam of extremely high power densityacting on the workpiece surface. The input of the energy necessary for heating up the surfacelayer is achieved by selecting from a range of traveling speeds of the workpiece and laser beamsource power.

The first laser welds were made around 1963, and involved butt and edge joints in 0.25 mmstainless steel foils, processed with a pulsed ruby laser. Other studies into conduction-limitedwelding in metals report that the technique was applied to joining wires, sheets, and circuit boardsshortly afterward when 0.5 mm was the maximum penetration. The first industrial applicationappeared around 1965, when a pulsed Nd:YAG laser was used to repair broken connectors insideassembled television tubes. Various laser-based conduction joining techniques have been devel-oped, including soldering and brazing, partly in response to the needs of the microelectronicsindustries.

� 2008 by Taylor & Francis Group, LLC.

The first gas-assisted carbon dioxide laser beam was made in 1967. A 300 W laser beam coaxialwith an oxygen jet was used to cut 1 mm thick steel using a potassium chloride lens, an aluminizedbeam turning mirror, and parameters that were very close to those used today.

The first commercial application of continuous wave (CW) CO2 lasers for scribing of ceramicswas demonstrated in 1967. First reports of lasers used for heat treating metals appeared in Germanyand Russia in the early 1960s. Early data in 1966 were mostly a by-product of investigations into theinteraction between materials and the focused beams of pulsed ruby and Nd:glass lasers. Investi-gations of interactions between pulsed ruby laser radiation and a graphite-coated metal surface gavesome indication of the potential for metal hardening. The first mathematical models (1968) of heatflow provided greater insight into the role of the process variables in determining the thermal cyclesinduced and the geometry of the hardened region.

Laser surface melting and the possibilities of surface alloying were investigated in 1963. Earlywork was performed from 1981 to 1985 mainly with pulsed solid-state lasers, in which shallowsurface alloys were produced.

The mechanism of laser-induced vaporization including shock hardening was also studied inearly 1963. At that time, lasers were able to produce pulses only with rise times and lengths on theorder of nanoseconds (10�9 s), and industrial applications were hindered. When ultrafast lasers withpulse lengths on the order of femtoseconds (10�15 s) were developed in the 1980s, interest wasrenewed by the automotive and aerospace industries [3].

The first excimer laser was demonstrated in 1970; liquid xenon was excited with a pulsedelectron beam. Output around a wavelength of 170 nm was demonstrated in high-pressure xenongas shortly afterward.

The 1980s were notable for the development of integrated laser systems, which comprised alaser source, beam handling optics, and workpiece handling equipment. User-friendly interfaceswere developed to provide information and instant control to the operator. In addition, researchersturned to novel methods of using the laser beam for materials processing, rather than a directreplacement for a conventional process.

The early 1980s produced a generation of industrial CO2 lasers that featured higher powers,greater reliability, and more compact designs.

The dominant Nd:YAG lasers available in the early 1980s were pulsed units. Until 1988 themaximum average power available from a commercial unit was 500 W and 1 kW Nd:YAGlaser was produced. This was preceded by the development of fiber-optic cable that couldtransmit a suprakilowatt near infrared (IR) beam, which meant that cumbersome mirror systemsassociated with CO2 laser beam delivery could now be replaced with flexible optics mountedon an industrial robot. With Nd:YAG laser, complex geometry three-dimensional (3D) compon-ents were now able to be treated economically. The industrial Nd:YAG lasers of this time werebased on active media comprising rods of crystals and lamp pumping, which result in lowefficiency of energy conversion (less than 5%) and a poor beam quality in comparison with gaslasers [3].

In the field of laser welding, progress was made with new welded joint designs, novel materialcombinations, and thick section welding; which led to improvements in quality, productivity, andenvironmental friendliness. Progress was made in 1985 by understanding the physics of keyholeformation and stability, which provided greater confidence of welding process. At the same time,reliable high-power industrial lasers were becoming available.

Applying metallic coating to a metallic substrate by the interaction of a directed laser beam anda gas stream containing entrained particles of the coating material was commercialized with theblown powder process for producing hardfaced aeroengine turbine blades. Blown powder claddingis now the most popular laser-based surfacing technique, finding uses in the aerospace, automotive,power generation, and machine tool industries, as well as forming the basis of a rapid manufacturingtechnique. Laser forming was investigated in the early 1980s deliberating distortion through laserheating.


The focus of developments in CO2 laser technology in the early 1990s was on machines ofhigher power, better beam quality, greater reliability, reduced maintenance, improved ease of use,and compactness. Fast development of laser technology was adopted in units up to 20 kW in outputpower and its modular design enabled units up to 60 kW to be produced.

The automotive industry led the way in introducing Nd:YAG lasers on the productionline, where they began to replace CO2 lasers for complex geometry cutting and welding operations.A 10 kW CW Nd:YAG unit was available commercially at the beginning of 2000. Because of theircompact size, and the higher absorptivity of the shorter wavelength diode laser beam by materials,diode lasers were actively investigated in the 1990s as replacements for CO2 and Nd:YAG sourcesin material processing. A major problem was the thermal load, coupled with the requirement tooperate the lasers near ambient temperature, which required efficient cooling [3].

Advantages of laser materials processing are [4–8]

. Savings in energy compared to conventional surface heat treatment welding or cuttingprocedures.

. Hardened surface is achieved due to self-quenching of the overheated surface layer throughheat conduction into the cold material.

. Since heat treatment is done without any agents for quenching, the procedure is a clean onewith no need to clean and wash the workpieces after heat treatment.

. Energy input can be adapted over a wide range with changing laser source power, havingfocusing lenses of different focuses, with different degrees of defocus (the position ofthe lens focus with respect to the workpiece surface), and different traveling speeds of theworkpiece and laser beam.

. Guiding of the beam over the workpiece surface is made with computer support.

. It is possible to heat-treat small parts with complex shapes as well as small holes.

. Optical system can be adapted to the shape or complexity of the product by means ofdifferent shapes of lenses and mirrors.

. Small deformations and dimensional changes of the workpiece after heat treatment.

. Repeatability of the hardening process or constant quality of the hardened surface layer.

. No need for minimal final machining of the parts by grinding.

. Laser heat treatment is convenient for either individual or mass production of parts.

. Suitable for automation of the procedure.

The use of laser for heat treatment can be accompanied by the following difficulties:

. Nonhomogeneous distribution of the energy in the laser beam.

. Narrow temperature field ensuring the required microstructure changes.

. Adjustment of kinematic conditions of the workpiece and laser beam to different productshapes.

. Poor absorption of the laser light in interacting with the metal material surface.

Engineering practice has developed several laser processes used for surface treatment:

. Annealing

. Transformation hardening

. Shock hardening

. Surface hardening by surface layer remelting

. Alloying

. Cladding

. Surface texturing

. Plating by laser chemical vapor deposition or laser physical vapor deposition


Vaporization

Melting

Heating

Log (interaction time) t (s)

1010

108

106

104

103

10010−210−410−610−8

Log

(pow

er d

ensit

y) Q

= P

/D2 (W

/cm

)2

Specific energyES = 102 J/cm2

ES = 100 J/cm2

PulsedHP

lasers

ContinuousHP lasers

FIGURE 11.1 Range of laser heating, melting, and vaporization according to power density (specific energy)and interaction time.

For this purpose, besides CO2 lasers, Nd:YAG and excimer lasers with a relatively low power and awavelength between 0.2 and 1.06 mm have been successfully used. A characteristic of these sourcesis that, besides a considerably lower wavelength, they have a smaller focal spot diameter and muchhigher absorption than CO2 lasers.

Figure 11.1 shows a logarithmic graph defining energy input. For various treatment processes inmechanical engineering, the energy input can be determined by the interdependence of laser-beampower density and the interaction time.

The same treatment process, e.g., transformation hardening, can be performed by varying thepower density and the interaction time.

The lower the energy input required is, the more exacting is the selection of the power densityrequired and adequate interaction time. Thus, a distinction is made among the treatment processeswhere a material is heated below the melting point, the ones where the material is heated betweenthe melting point and the temperature of evaporation, and the ones going on above the temperatureof evaporation. Because of the interdependence of individual parameters of the laser beam, theinteraction time, and the relative travel between the laser beam and the workpiece, the individualprocesses going on in individual states of the workpiece material are separated by straight lines.In the selection of the processing parameters it is usually recommended to select longer interactiontimes with low yet sufficient power densities. Thus, the treatment process is easier to monitor andthe adaptive control of the process easier to carry out. In the opposite case when higher powerdensities are selected, shorter interaction times are required, which ensures higher productivity.Unfortunately, higher productivity makes monitoring and control of the process more difficult.Also, the repeatability of product quality is poorer than with the processes taking longer interactiontimes [8–11].

In heat treatment using laser light interaction, it is necessary to achieve the desired heat input,which is normally determined by the hardened layer depth. Cooling and quenching of the


Drilling Cutting

Transformationhardening

Specific energy Es [J/cm 2]

10010−210−410−610−8103

104

105

106

107

108

109

1010

Log (interaction time) ti (s)

Log

(pow

er d

ensit

y) Q

(W/c

m2 )

Shockhardening

106105104103

102101100

Deeppenetration

welding

Scribing

Cladding

Remelting

Glazing

PulsedHP lasers

FIGURE 11.2 Dependence of power density, specific energy, and interaction time at laser metalworkingprocesses.

overheated surface layer is in most cases achieved by self-cooling since after heating stops, heatconducted into the workpiece material is so intensive that the critical cooling speed is achieved andthus also the wanted hardened microstructure. Figure 11.2 illustrates the dependence of powerdensity and specific energy on laser light interaction time on the workpiece surface in order to carryout various metalworking processes.

Diagonally there are two processes, i.e., scribing and hardening, for which quite the oppositerelationship between power density and interaction time has to be ensured [3,6,7]. In scribingmaterial vaporization at a depth of a few microns has to be achieved, ensuring the prescribed qualityand character resolution. On the other hand, for hardening a considerably lower power density perworkpiece surface unit is required, but the interaction times are the longest among all the mentionedmetalworking processes.

Thus, different power densities and relatively short interaction times, i.e., between 10�1 and10�3 s, are related to remelting of the material. This group thus includes the processes in whicheither the parent metal alone or the parent metal and the filler material are melted. The highest powerdensities are applied in cases where both the parent metal and the filler material are to be melted.Such processes are laser welding and alloying. With cladding, however, the lowest power densitiesare required since the material is to be deposited on the surface of the parent metal and only the fillermaterial is to be melted. The power density required in both welding and alloying of the parentmetal, however, depends on the materials to be joined or alloyed. In hardening of the surface layersby remelting it is necessary, in the selection of the power density, to take into account also the depthof the remelted and modified layer. In laser cutting, a somewhat higher power density is requiredthan in deep welding. In the cutting area, the laser-beam focus should be positioned at the workpiecesurface or just below it. In this way a sufficient power density will be obtained to heat, melt, andevaporate the workpiece material. The formation of a laser cut is closely related to materialevaporation, particularly flowing out of the molten pool and its blowing out due to the oxygenauxiliary gas, respectively [9–16].


TABLE 11.1Laser Processes in the Three Temperature Ranges According to RequiredPower Density and Interaction Time

Power Density Interaction Time Temperature RangeQ (W=cm2) t (s) T Laser Process

105 0.1 T<Tm Transformation hardening

106 (1–10)� 10�3 Surface remelting, surface alloying,soldering, deep welding penetration

107 (1–10)� 10�3 Tm<T< Tv108 (0.1–0.5)� 10�3 T>Tv Shock hardening, scribing, drilling, cutting

In Table 11.1, different laser treatment processes are summarized and classified with referenceto the heating temperature, the power density, and the interaction time required for the materialconcerned. The longest interaction times, ti¼ 0.1 s, are found with transformation hardening.Then follow the interaction times, ti¼ (1.0 – 10)� 10�3 s, found in laser remelting, alloying,cladding, and welding. The shortest interaction times, ti¼ (0.1 – 0.5)� 10�3 s, are found with theprocesses going on above the evaporation temperature of the material, i.e., laser cutting, drilling, andscribing. Shortening of the interaction times with the individual processes, including melting orevaporation of the material, requires a higher power density. A comparison of the processingparameters required in transformation hardening and laser cutting shows that the power density is103 times lower and the interaction times are 103 times longer in transformation hardening than inlaser cutting.

In practical applications, reference is frequently made to the type of laser and its maximumpower. With a given laser power, the main interest is which processes are feasible and whichmaterials can be treated. Figure 11.3 shows a scheme illustrating the given laser power withreference to feasibility of the individual treatment processes with the selected materials.

Marking of glass

Perforating of ceramics Metals weldingCutting/drilling of ceramics

Metals cuttingMarking of plastics

Heat treatment of metals

Cutting of plastic foils

Cutting of paper

Drilling of plastics

Drilling of rubber

Soldering

0 10 100Laser power P (W)

1,000 10,000

FIGURE 11.3 Different laser treatments and corresponding laser.


Consequently, there is a difference between the laser power required in drilling of plastics orrubber and that required in drilling of ceramic materials. The power required for drilling of ceramicscan be 1000 times that required for drilling of plastics.

11.1.1 LASER BEAM MODE STRUCTURE

The transverse power density distribution of the laser beam is very important in the interactionwith the workpiece material. The irradiated workpiece area is a function of the focal distance ofthe convergent lens and the position of the workpiece with reference to the focal distance. Thetransverse power density distribution of the laser beam is also called the transverse electromagneticmode (TEM). Several different transverse power density distributions of the laser beam or TEMscan be shaped [1,3,9].

Each individual type may be attributed a different numeral index. A higher index of the TEMindicates that the latter is composed of several modes, which makes beam focusing on a fine spot atthe workpiece surface very difficult. This means that the higher the index of the TEM is, the moredifficult it is to ensure high power densities, i.e., high energy input. For example in welding, modestructures TEM00, TEM01, TEM10, TEM11, TEM20, and frequently combinations thereof are used.Some of the laser sources generate numerous mode structures, i.e., multimode structures [9–11].

The most frequently used lasers are continuous lasers emitting light with the Gaussian trans-verse power density distribution, TEM00. A laser source operating 100% in TEM00 is ideal forcutting and drilling. A TEM00 beam can be focused with a convergent lens to a very small area thusproviding a very high power density. In practice, TEM01 with energy concentrated at the peripheryof the laser beam with reference to the optical axis is used as well. This mode is applied primarily todrilling and heat treatment of materials because it ensures a more uniform elimination of the materialin drilling and a uniform through-thickness heating of the material in heat treatment. For weldingand heat treatment, very often mixtures of multimode structures, giving an approximately rectangu-lar, i.e., top-hat-shaped, energy distribution in the beam, are used.

Figure 11.4 shows different mode structures of the laser beam, i.e., TEM00 ((A)Gaussian beam) andmultimode beam structures TEM01 (B), TEM10 (C), TEM11 (D), and TEM20 (E), respectively [3,12].

Heat treatment requires an adequate laser energy distribution at the irradiated workpiece surface.This can be ensured only by the correct transverse power density distribution of the laser beam. Thepower density required for heat treatment can be achieved by the multimode structure or a built-inkaleidoscope or segment mirrors, i.e., special optical elements.

Projectionof laser beampower density

Power densityprofile across

diameter of beam

Mode structure (A) TEM00

Radius−r +r

(B) TEM01∗

−r +r

(D) TEM11∗

−r +r

(E) TEM20

QQQQ

−r +r

(C) TEM10

−r +r

Powerdensity Q [W/cm2]

FIGURE 11.4 Basic laser-beam mode structures. (From Dawes, C. Laser Welding, Ablington Publishing andWoodhead Publishing in Association with the Welding Institute, Cambridge, 1992, 1–95.)


Heating of the material for subsequent heat treatment requires an ideal power density distribu-tion in the laser beam providing a uniform temperature at the surface and below the surface to thedepth to which the material properties are to be changed [13–15].

Laser irradiation can usually be applied to an area smaller than the one to be heat-treated;therefore, it is necessary that the laser beam heats the area in several passages and provides auniform temperature. In this case, the effects of the edge heat flow along the preliminary heated andcooling laser trace and, at the opposite side, of that along the cold part of the workpiece should betaken into account.

11.2 LASER OPTICS AND BEAM CHARACTERIZATION

11.2.1 FOCUSING WITH A SINGLE LENS

The beam emitted from a laser is rarely suitable for material processing in its raw form, therefore,seldom in desired size, and the intensity of distribution is not appropriate for the process. It needs to bemanipulated into a suitable optic, which has been incorporated into beam delivery systems [3,16,17].

11.2.2 FOCAL LENGTH

The strength of a lens is measured by its focal length (F). The focal length is the distance from thecenter of the lens to the focal point. The effective focal length is the distance that the designer uses tocalculate the curvature of the lens. It is the distance from the principal plane in which an incomingbeam is bent toward the focal point.

11.2.3 FOCAL NUMBER

The focal number of a focusing optic f characterizes its focusing ability. The f-number defines theconvergence angle of the beam:

f ¼ F

dB(11:1)

whereF is the focal length of the opticdB is the diameter of the beam

11.2.4 BEAM DIAMETER AT FOCUS

The minimum theoretical diameter, df, to which a laser beam of original diameter, dB, and mode,TEM00, can be focused is

df ¼ 4lFpdB

¼ 4lfp

, (11:2)

wherel is the wavelengthdB is the beam diameter at the waist

The effect of the beam mode on the minimum spot diameter can be expressed in terms of thebeam quality factor, K:

df ¼ 4lp

f

K(11:3)


the K factor expresses the beam focusability in terms of a TEM00 beam mode:

K ¼ l

p:4

dBu(11:4)

The diffraction-limited spot size at focus, df, can be calculated from diffraction theory to give

df ¼ 2:44lF

dB(2M þ 1)1=2 (11:5)

wherel is the wavelengthF is the focal length of the opticdB is the diameter of the incident beamM is the number of oscillating modes [3]

11.2.5 DEPTH OF FOCUS

The depth of focus is also known as the depth of field. The depth of focus, zf, is a measure of thechange in the waist of the beam on either side of the focal plane. It is often defined as the distancealong the axis over which the focused spot size increases by 5%, or the distance over which theintensity exceeds half the intensity at focus. For a TEM00 beam, it is defined as

Zf ¼ 8lp

F

dB

� �2

¼ 8l f 2

p¼ 2fdf : (11:6)

For a higher-order beam mode of quality K, the depth of focus defined by separation of the points atwhich the beam waist is

ffiffiffiffiffiffiffi2df

pis given by

Zf ¼ 4lpK

F

dB

� �2

: (11:7)

The depth of focus is proportional to the square of the spot size, i.e., a smaller spot size leads to ashorter depth of focus. A compromise is often sought between these two features in practicalapplications—a small spot size to give a high power density, but a large depth of focus forthrough-thickness processing [3,17].

11.2.6 LASER BEAM CHARACTERIZATION

Knorovsky [18] characterized laser beams used in materials processing. Recent techniques havemade it possible to rapidly and conveniently characterize the size, shape, mode structure, beamquality (M2), and intensity of a laser beam as a function of distance along the beam path. Thisfacilitates obtaining the desired focused spot size and also locating its position.

Recently, an ISO standard [19] for characterizing laser beams has emphasized the use of the M2

parameter to determine beam quality. The equipment required to determine this parameter is complex,but the information gained is also quite comprehensive. While this method is highly recommended,they have found that by using the information gained in determining M2 and replotting it slightlydifferently, increased understanding of the beam–material interaction region can be gained.

The basis for M2 parameter is the beam propagation equation for CW case:

v(z) ¼ v20 þ M2l(z� z0)=pv0

� �2� �0:5, (11:8)


wherev(z) is the radius of the beam as a function of distance along the propagation directionl is the laser light wavelength

The minimum waist and location of the beam are represented by v0(z0), where z’s origin is at thefocusing lens plane. Thus, information on the beam size as a function of beam propagation distancemust be determined, and this parabolic equation fits to the data, with M2, z0, and v0 acting as fitparameters. The size of the beam is determined by the radius of a circle within which 1�(1=e2) ofthe total beam energy or power is contained. The results of such an analysis of data obtained on aPrometec beam scanning profilometer, which uses a rotating wire with small aperture to sample thefocused beam, are shown in Figure 11.5. Curve fitting and plotting were done using software forfour different laser-beam powers.

Four tables are arranged from the highest to lowest power in top-to-bottom order: m1¼minimum beam radius v01, m2¼M2, m3¼ z0, Chisq, and R are statistical parameters measuringthe goodness of fit to the data.

While the lines indicating the beam size versus propagation direction are indicative of theenergy distribution, they are misleading in that the actual value of intensity is a function of z.

To implement this technique, they proceeded assuming that a laser beam with an exponentialshape of beam power versus radius allows relating peak intensity to the total power:

I0(z) ¼ 2P=pv(z)2: (11:9)

y = m1*sqrt (1+(m2*0.00106/(3...Value

m1 0.24056 0.00077598m2 56.11 0.42599m3 −91.294 0.019557Chisq 4.8763e−0.5 NAR 0.99871 NA

Error

y = m1*sqrt (1+(m2*0.00106/(3...Value

m1 0.22228 0.00096307m2 51.656 0.3634m3 −91.679 0.020319Chisq 8.1707e−0.5 NAR 0.99877 NA

Error

y = m1*sqrt (1+(m2*0.00106/(3...Value

m1 0.15003 0.0017144m2 34.879 0.35118m3 −92.543 0.022938Chisq 0.00038214 NAR

−85−90z (mm)

−95

190 Wr350 Wr405 Wr470 Wr

−1000.1

0.15

0.2

0.25

Beam

radi

us (m

m)

0.3

0.35

0.4

0.99846 NA

Error

y = m1*sqrt (1+(m2*0.00106/(3...Value

m1 0.2061 0.0013454m2 48.418 0.38355m3 −91.982 0.023584Chisq 0.00016712 NAR 0.99834 NA

Error

FIGURE 11.5 M2 parameter and beam caustic versus power for 500 W Nd: YAG, CW laser with hard opticbeam delivery system. (From Knorovsky, G.A. and MacCallum, D.O. An alternative form of laser beamcharacterization: E-ICALEO 2000, 92–98.)


The beam’s size v(z) was described by the given beam propagation equation with values of M2 andv0 found experimentally. They also calculated the relation between intensity, radius r, and z asfollows:

I(r, z) ¼ I0(z) exp�2(r=v)2: (11:10)

11.3 LASER LIGHT ABSORPTIVITY

11.3.1 TEMPERATURE EFFECT

With the interaction of the laser light and its movement across the surface, very rapid heating up ofmetal workpieces can be achieved, and subsequent to that rapid cooling down or quenching. Thecooling speed, which in conventional hardening defines quenching, has to ensure martensitic-phasetransformation. In laser hardening, martensitic transformation is achieved by self-cooling, whichmeans that after the laser light interaction the heat has to be very quickly abstracted into theworkpiece interior. While it is quite easy to ensure the martensitic transformation by self-cooling,it is much more difficult to deal with the conditions in heating up. The amount of the disposableenergy of the interacting laser beam is strongly dependent on the absorptivity of the metal. Theabsorptivity of the laser light with a wavelength of 10.6 mm ranges in the order of magnitude from2% to 5% whereas the remainder of the energy is reflected and represents the energy loss. Byheating metal materials up to the melting point, a much higher absorptivity is achieved with anincrease of up to 55% whereas at vaporization temperature the absorptivity is increased even up to90% with respect to the power density of the interacting laser light.

Figure 11.6 illustrates the relationship between laser light absorptivity on the metal materialsurface and temperature or power density [20,21]. It is found that, from the point of view ofabsorptivity, laser-beam cutting does not pose any problems, as the metal takes the liquidor evaporated state, and the absorptivity of the created plasma can be considerably increased.Therefore, it is necessary to heat up the surface, which is to be hardened, onto a certain temperatureat which the absorptivity is considerably higher and enables rapid heating up onto the hardeningtemperature or the temperature that is lower than the solidus line for safety reasons. This wassuccessfully used in heat treatment of camshafts [22].

Poor absorptivity of metal materials subjected to heat treatment can be improved only bydepositing a suitable absorbing agent on their surface. Absorbents have to meet the followingrequirements:

. They have to be cheap and easy to prepare and deposit on the surface.

. They should grant a high degree of absorptivity of the laser light interacting with theworkpiece material in the temperature range of austenitization.

. They should produce no chemical reaction of the base material and should be easilyremoved from the surface if necessary.

The heating up of the workpiece surface material by the laser beam is done very rapidly. Theconditions of heating up can be altered by changing the energy density and relative motion of theworkpiece and the laser beam.

In surface hardening, this can be achieved without any additional cooling and is called self-quenching. The procedure of laser surface hardening is thus simpler than the conventional flame orinduction surface hardening as no additional quenching and washing are required [23].

Bramson [24] defined the dependence between electric resistance and emissivity «l (T) for thelight radiation striking the material surface at a right angle:


Power density Q(W/cm2)

1.55 �104 1.55 �108

CuttingWelding

Meltingtemperature

10

20

30

40

50

60A

bsor

ptio

n A

(%)

70

80

90

100

Temperature T (ºC)

Vaporizationtemperature

Heat treating

Absorptivity ofcoated surface

l = 1.06 μm

l = 1.06 μm

FIGURE 11.6 Effect of temperature on laser light absorptivity. (From Tizian, A., Giordano, L., and Ramous,E. Laser surface treatment by rapid solidification. In: EA Metzbower, Ed., Laser in Materials Processing,American Society for Metals, Conference Proceedings, Metals Park, OH, 1983, 108–115.)

«l(T) ¼ 0:365rr(T)

l

� �1=2

þ 0:0667rr(T)

l

� �þ 0:006

rr(T)

l

� �3=2

, (11:11)

whererr is the electrical resistivity (V cm) at temperature T (8C)«l (T) is the emissivity at T (8C)l is the wavelength of incident radiation (cm)

Reflectivity depends on the incident angle of the laser beam with reference to the polarization planeand the specimen surface. Figure 11.7 shows the reflectivity of CO2 laser light from a steel surface atdifferent incident angles and different temperatures [22,23,33].

The diagram combines experimental data (plotted dots) on reflectivity and absorptivity, andtheoretically calculated values of reflectivity (uninterrupted lines). The variations of absorptivityindicate that absorptivity strongly increases at elevated temperatures due to surface oxidation andat very high temperatures due to surface plasma absorption. Figure 11.8 shows influences exertedon absorptivity of CO2 and Nd:YAG laser light in interaction with specimens made of Ck 45steel [23,33].

The steel specimens were polished, grounded, turned, and sandblasted to Ck 45 heat-treatmentcarbon steel; various methods of surface hardening, and particularly laser hardening, are oftenapplied.

From the column chart, it can be inferred that absorptivity of steel specimens subjected to differentmachining methods is considerably lower with CO2 laser light than with Nd:YAG laser light. Thelowest absorptivity was obtained with the polished specimens. It varied between 3% and 4% with


9060

300 K

SteelCO2 laser lightl = 10 μm

RS ... reflection - normalRP ... reflection - parallel

ExperimentTheory

Rs

Rp3000 K

1000 K

300 K

Angle of incidence f (º)300

0

0.2 0.8

0.6

0.4

0.2

0

1.0

0.4

0.6

Refle

ctiv

ity R

Abs

orpt

ion

A

0.8

1.0

FIGURE 11.7 Variation of reflectivity with angle and plane of polarization. (FromWissenbach, K., Gillner, A.,and Dausinger, F., Laser und Optoelektronic, 3, 291, 1985.)

CO2 Nd: YAGl = 1.06 μml =10.6 μm

4%5%−7% 6%−8%

21%−23%

60%−80%

70%−80%

30%

33%−37%36%−43%

46%−51%

60%−80%70%−80%

Material: Ck 45

0

20

40

60

80

100

Abs

orpt

ivity

A (%

)

Polished

Ground

Turned

Sandblasted

Oxidized

Graphitized

FIGURE 11.8 Influence of various steel Ck 45 treatments on absorption with CO2 or Nd:YAG laser light.(From Wissenbach, K., Gillner, A., and Dausinger, F., Laser und Optoelektronic, 3, 291, 1985; Beyer, E.and Wissenbach., K., Oberflächenbehandlung mit Laserstrahlung. Allgemaine Grundlagen, Springer-Verlag,Berlin, 1998, 19–83.)


reference to the laser light wavelength. Absorptivity was slightly stronger with the ground andthen turned surfaces. It turned out in all cases that the absorpitivity of Nd:YAG laser light is seventimes that of CO2. If absorptivity of the two wavelengths is considered, smaller differences may benoticed with the sandblasted surfaces. The oxidized and graphitized surfaces showed the same absorp-tivity of laser light regardless of its wavelength. The latter varied between 60% and 80% [25–27].

11.3.1.1 IR Energy Coatings

To increase laser-beam absorptance at metal surfaces, various methods were used:

. Metal surface painted with absorbing coatings followed by laser processing

. Chemical conversion coatings

. Uncoated metal surfaces processed by a linearly polarized laser beam [34]

IR energy coatings with high absorptance must have the following features for increased efficiencyduring laser heating at heat treatment:

. High thermal stability

. Good adhesion to metal surface

. Chemically passive to material heat conduction from coating to material

. Easily applied and removed

. Lower expenses for coatings is possible

11.3.1.2 Chemical Conversion Coatings

Chemical conversion coatings, such as manganese, zinc, or iron phosphate, absorb IR radiation.Phosphate coatings are obtained by treating iron-base alloys with a solution of phosphoric acidmixed with other chemicals. Through this treatment, the surface of the metal is converted to anintegrally bonded layer of crystalline phosphate. Phosphate coatings may range in thickness from 2to 100 mm of coating surface. Depending on the workpiece geometry, phosphating time can rangefrom 5 to 30 min regarding temperature and concentration of the solution. Phosphate coatings on themetal surface can be prepared with a fine or coarse microstructure.

In terms of chemical passiveness and ease of coating application on metal surfaces, silicatescontaining carbon black are more effective than phosphate coatings. Figure 11.9 schematicallyillustrates the reaction of manganese phosphate with metal surface and subsequent formation of lowmelting compounds, which can penetrate along the grain boundaries over several grains below thesurface of the metal material. This reaction can be prevented by using chemically inert coatings.

CoatingCoating

workpiecereaction

layer

Coating

Impregnationalong grainboundaries

Metal materials grains(b) After processing(a) Before processing

FIGURE 11.9 Potential reaction of IR energy-absorbing coating after laser-treated metal material. (FromGuanamuthu, D.S. and Shankar, V. Laser heat treatment of iron-base alloys. In: CV Draper and P Mazzoldi,Eds., Laser Surface Treatment of Metals, NATO ASI Series–No. 115, Martinus Nijhoff Publishers, Dordrecht,1986, 413–433.)


Unpolarizedlaser beam

Linearlypolarized

laser beam

Plane ofincidence

Direction ofpropagation

y

z

x - Wavelength- Electric vector- Magnetic vector

l

ΦΦ

l

E

Metalmirror

Φ: Angle of incidence

Linearlypolarized

laser beam

HH

E

FIGURE 11.10 Conversion of an unpolarized laser beam to a linearly polarized beam by reflection at aspecific angle. (From Guanamuthu, D.S. and Shankar, V., Laser heat treatment of iron-base alloys. In:CV Draper and P Mazzoldi, Eds., Laser Surface Treatment of Metals, NATO ASI Series–No. 115, MartinusNijhoff Publishers, Dordrecht, 1986, 413–433.)

11.3.1.3 Linearly Polarized Laser Beam

Metals have lower reflectance for linearly polarized electromagnetic radiation. The basis of thisoptical phenomenon has been applied to the CO2 laser heat treatment of uncoated iron-base alloys[30–32]. An unpolarized laser beam can be linearly polarized by using proper reflecting opticalelements. Figure 11.10 shows an unpolarized laser beam with specific incident angle referring to ametal mirror and reflected beam as linearly polarized beam. This angle of incidence is called thepolarizing angle.

When the laser beam is linearly polarized, the dominant vibration direction is perpendicular tothe plane of incidence. The plane of incidence is defined as the plane that contains both the incidentlaser beam and the normal to the reflecting surface.

The electric vector E; of the linearly polarized beam has components parallel Ep and perpen-dicular Es to the plane of incidence. Figure 11.11 illustrates absorptance as a function of the angle ofincidence for iron [34].

At an angle of incidence between 708 and 808, the absorptance is between 50% and 60% for‘‘Ep’’ and 5%–10% for Es. Thus, by directing a linearly polarized laser beam at an angle of incidencegreater than 458, substantial absorptance by iron-base alloys is possible. The possible weakness ofthis method is the important laser-beam power loss during conversion of an unpolarized to a linearlypolarized laser beam.

11.3.2 ABSORPTION MEASURING TECHNIQUE

Rothe et al. [35] determined the absorptivity of metal surfaces on the basis of calorimetricmeasurements. Calorimetric measurements gave an idea of the absorptivity of metallic surfaces.Absorptivity A is defined as

A ¼ m � Cp � DTt � P (11:12)


908070605040Angle of incidence Φ (�)3020100

100

90

80

70

60

50

Abs

orpt

ance

A (%

)

40

30

20

10

0Es

Ep

FIGURE 11.11 Effect of the angle of incidence of a linearly polarized laser beam on absorptance by iron basealloys. (From Guanamuthu, D.S. and Shankar, V. Laser heat treatment of iron-base alloys. In: CV Draper andP Mazzoldi, Eds., Laser Surface Treatment of Metals, NATO ASI Series–No. 115, Martinus Nijhoff Pub-lishers, Dordrecht, 1986, 413–433.)

wherem is the massCp is the specific heatDT is the temperature differencet is the timeP is the power

Testing of various absorption deposits was performed with different steels, i.e., C45 heat-treatment steel, 100Cr6 steel for manufacture of balls of ball bearings, and GGG40 nodular castiron. The absorptivity was determined with a laser-beam power density Q of 2.5� 104 W=cm2 andsquare cross sections of 5� 5, 8� 8, and 12� 12 mm2. The absorptivity attained was stronglydependent on the surface preparation (Table 11.2).

It was found that with a machined, i.e., ground, surface with a roughness Ra of 1 mm absorptivitywas equal to only 8.5%. With an increase in roughness, absorptivity increased as well so that with arelatively strong surface roughness Ra, i.e., 25 mm, it amounted to as much as 18%.

With a sand-blasted surface, absorptivity equaled 35%, which is almost 100% higher than with aroughly ground surface. A surface coated with manganese phosphate showed absorptivity in a largerange from 65% to 85% whereas the one coated with zinc phosphate showed a constant absorptivityof 55%.

The authors stated that in their tests graphite spraying provided a repeatable absorptivity of77%; therefore, this absorbent was used in their further tests of transformation hardening of steels.Experimental results are given in Table 11.2.


TABLE 11.2Absorptivity of Steel Surfaces That Had Been Treatedin Different Ways

Surface Condition Absorptivity A [%]

Grinded Ra¼ 1 mm 8.5Ra¼ 25 mm 18

Sandblasted 35Tungsten powder with glew 37

Mangan phosphate 85–65Zinc phosphate 55Graphite—spraying 77

Source: From Rothe, R., Chatterjee-Fischer, R., and Sepold, G. Hardeningwith Laser Beams. Proceedings of the 3rd International Colloquium

on Welding and Melting by Electrons and Laser Beams, Lyon,France, 1983, pp. 211–218.

11.3.2.1 Absorptivity

Absorptivity dependence on the workpiece traveling speed was more significant. The tests werecarried out at traveling speeds from 1 to 8 m=min. The results have confirmed that at lower travelingspeeds the absorptivity is smaller due to heat transfer to the cold workpiece material and also intothe surrounding area. When the traveling speed is increased from 1 to 8 m=min, an absorptivityincrease from 40% to almost 70% is achieved (Figure 11.12).

Arata et al. [36] also studied the effects of optical conditions on phosphate absorptivity. Thestarting point for this study was the spot size in the direction of the y-axis, which was denoted byDy. The spot size was changed from 1 to 6 mm for the purpose of studying the absorptivitydependence on different workpiece traveling speeds. Thus, at the spot size of Dy¼ 6 mm and

0 2 4 6 8Traveling speed v (m/min)

Abs

orpt

ivity

A (%

)

0

25

50

75

100Ar AIR

Zn3(PO4)2Mn3(PO4)2

Dy = 3 mm

Air

No coating

Ar

FIGURE 11.12 Absorptivity of specimens coated with zinc and manganese phosphates was measured in airand argon atmospheres. (From Arata, Y., Inoue, K., Maruo, H., and Miyamoto, I., Application of laser formaterial processing—Heat flow in laser hardening In: Y Arata, Ed., Plasma, Electron & Laser BeamTechnology, Development and Use in Materials Processing, American Society for Metals, Metals Park, OH,1986, 550–567.)


Dy = 6 mmDy = 4 mmDy = 3 mmDy = 2 mm

201510Traveling speed v (m/min)

500

20

40

60

Abs

orpt

ivity

A (%

) 80

100

Dy = 1 mm

FIGURE 11.13 Spot size was changed from 1 to 6 mm in order to study the absorptivity dependence ondifferent workpiece traveling speeds. (From Arata, Y., Inoue, K., Maruo, H., and Miyamoto, I., Application oflaser for material processing—Heat flow in laser hardening In: Y. Arata, Ed., Plasma, Electron & Laser BeamTechnology, Development and Use in Materials Processing, American Society for Metals, Metals Park, OH,1986, 550–567.)

traveling speed of 1 m=min, the absorptivity was A¼ 65%; at the traveling speed of 8 m=min, it wasA¼ 80% (Figure 11.13).

11.3.2.2 Absorption Control

Pantsar and Kujanpää [37] defined and controlled absorption of the laser in heating metallic materials.They studied the absorption that had been measured by a liquid calorimeter and the surface tempera-ture that had been measured with a dual wavelength pyrometer. The processing parameters usedwere the intensity of the beam, the interaction time, and the angle between the beam and theworkpiece surface. Surface temperatures during hardening varied from the TAc1 temperature to themelting point TM. Tests were done with a 3 kW diode laser with a 10� 5 mm2 hardening optic.

The most important processing parameters in surface hardening are laser power and travelingspeed. The traveling speed was varied so that for each power level tests were made from surfacetemperatures below TAc1 temperature (4908C) to temperatures above melting temperature (16008C).The pyrometer was set to measure the heating and cooling curves of a stationary spot from thecenterline of the hardened track. The angle between the laser’s optical axis and the surface was setfrom 558 to 858 to reduce the amount of power reflected back to the resonator.

When the heated specimen is put to the calorimeter the amount of absorbed energy, EA (J), canbe calculated from the formula

EA ¼ DT1 � C1(m1 þ w)þ DTs � Cs � ms (11:13)

whereDT1 (8C) is the change in temperature of the liquidDTs (8C) is the change in temperature of steel from the start of the test (before laser

processing) to the end of the test (when temperature is even)m1 and ms are the respective weights of liquid and steel (kg)C1, Cs are the respective specific heats of liquid and steel (J=[kg.8C]�1)

The principle of the calorimeter is presented in Figure 11.14.There is some energy loss to the surroundings during measurement. Therefore, the measured

maximum liquid temperature is less than the theoretical maximum. The theoretical maximumtemperature Ttmax can be calculated with an exponential formula:


ttmaxt1

Ta

Tmmax

Ttmax

A2

A1

T

0

FIGURE 11.14 Principle of a liquid calorimeter.

Ttmax ¼ Ta þ (Tmmax � Ta)[ exp (�K=tmax � t1)]�1 (11:14)

whereTtmax (8C) is the theoretical maximum temperatureTmmax (8C) is the measured maximum temperatureTa is the ambient temperatureK is the rate constanttmax is the time to attain Tmmax (s)

Time t1 is set on the t-axis so that TA1 equals TA2.The calorimeter was calibrated and the numeric value for the rate constant K could be composed

with regression analysis: K¼ 2.326� 10�5 for glycol and K¼ 2.028� 10�5 for water.Figure 11.15 shows the absorbed energy plotted against traveling speed and interaction time.

The results of tests in which surface temperature did not exceed the TAc1 temperature are alsopresented.

The angle between the beam and the absorbing surface did not have a significant effect with theangles tested. The absorption varied in these experiments from 53.3% to 56.3%.

The absorption of all workpieces was measured and calculated using the presented method-ology. Surface temperatures were measured for all experiments, except for the workpieces withabsorptive coating. In those cases, the pyrometer failed to measure the temperature of the steel-machined surface, but instead measured the temperature of the absorptive coating.

The measured absorbed energy varied from 27.9% to 68.2% of the laser energy. With equallaser energy per distance, the best absorption was achieved with shorter interaction times andincreased laser power. For example with a laser energy of 114 J=mm, the absorbed energy decreasedby 40.5% when the traveling speed was changed from 1260 to 414 mm=min.

11.3.2.3 Absorptivity in Iron and Steel

Seibold et al. [38] studied absorptivity of Nd:YAG laser radiation on iron and steel.The temperature dependence of absorptivity is little known for temperatures near and above the

melting point, and the data obtained strongly vary up to now. This can be explained among othersby the fact that the interaction of laser radiation with the metal is not only dependent on temperaturebut also considerably influenced by surface conditions like roughness and oxidation.


70

60

50

A (%

)

40

30

200∞

1000 2000 3000 4000 5000 60000.3

V (mm/min)

P (W)

ti (s) 0.15 0.1 0.075 0.06 0.05

436

J/mm

179 J

/mm

143 J

/mm

114 J/mm

92 J/mm

73 J/mm

59 J/mm

47 J/mm

786 983 1229 1536 1920 2400 3000

FIGURE 11.15 Absorption of a diode laser beam with different processing parameters. Laser energy variedfrom 24 to 436 J=mm. All surfaces were machined. (From Pantsar, H. and Kujanpää, V., The absorption of adiode laser beam in laser surface hardening of a low alloy steel.)

A reflectometry device for the measurement of absorptivity at Nd:YAG-wavelength will bepresented. The system is set up in a vacuum chamber, where pressures down to 10�6 mbar can beobtained to enable cleaning of the specimen surface from oxide.

Basic theories of the temperature dependence of absorptivity for ideal, nontechnical surfacesdeal with description of coupling of laser radiation energy with free electrons within the metal. Twomechanisms are relevant, the so called intraband- and the interband-absorption. The first one iscaused by an energy transfer from the electromagnetic wave to the electron that is being acceleratedand damped by collisions. With increasing temperature the time between two collisions decreases.Hence, with increasing temperature this absorption mechanism increases. The second absorptionmechanism is caused by the lift of an electron from the valence band to the conduction band by theenergy of the electromagnetic wave. With increasing temperature it becomes more difficult for theelectron to find an empty place in the conduction band and so this interband mechanism decreaseswith temperature.

In Figure 11.16, the absorptivity of iron at an ambient temperature at perpendicular incidence, ascalculated from data for optical constants given in Ref. [39], is shown. With increasing wavelength anearly steady decrease as a result of the decreasing collision frequency of free electrons is observed,which is interrupted only by a local maximum at ca. 800 nm caused by interband absorption.

11.3.2.4 Influence of Roughness

In Figure 11.17, the measured temperature dependence of absorptivity for pure Fe and steel St37from ambient temperature up to above the melting point is shown. The investigations includedpolished as well as rough surfaces with an average peak to valley height Ra from 0.32 to 0.35 mm.The absorptivity values at ambient temperature were investigated with a calorimetric method. Forhigher temperature, the absorptivity was measured with the reflectometry setup [38].

All curves show a negative temperature dependence up to temperatures of 12008C–13008Cand an increase from that point on. The decrease in absorptivity is caused by intrinsic


1.0

0.8

0.6

Abs

orpt

ivity

A (%

)0.4

0.2

0.00.1 1

Wavelength l (μm)10

CO2

Nd: YAG

Diode laser

FIGURE 11.16 Dependence of absorptivity A on wavelength for Fe. (From Palik, E.D., Handbook of OpticalConstants of Solids I, Academic Press, New York, 1991.)

temperature dependence of absorptivity of the material, which is revealed in the measurement of thepolished samples.

11.3.2.5 Influence of Oxidation

In Figure 11.18, two exemplary measurements with rough iron samples on oxidizing environmentare shown. Absorption was measured at a power level of 560 W. The samples were heated up to themelting point. Absorptivity reaches a maximum of 0.69–0.73 at a temperature of 13758C (1). Afterthis first maximum absorptivity shows a pronounced decrease to a minimum at about 15708C–16508C (2) and a second peak at about 19208C (3) with approximately the same value as the initialpeak. For higher temperatures, absorptivity decreases to values comparable to the unoxidizedsample. The curve shows a deviation of about 4% in the maxima and about 10% in the minimum.

16001400

Fe, polishedFe, Rd = 0.32 μm

St37, Rd = 0.35 μmSt37, polished

12001000800Temperature T (�C)

6004002000

0.30

0.31

0.32

0.33

0.34

Abs

orpt

ivity

A (%

) 0.35

0.36

0.37

FIGURE 11.17 Temperature dependence of absorptivity A for pure Fe (filled symbols) and St37(open symbols) at different roughnesses. (From Seibold, G., Dausinger, F., and Hügel, H., Absorptivity ofNd:YAG-laser radiation on iron and steel depending on temperature and surface conditions: E-ICALEO2000, 125–132.)


280026002400220020001800

3

2

1

1600Temperature T (�C)140012001000800600400

0.35

0.40

0.45

0.50

0.55

Abs

orpt

ivity

A (%

)

0.60

0.65

0.70

0.75

0.80

FIGURE 11.18 Influence of oxidation on absorption. (From Seibold, G., Dausinger, F., and Hügel, H.,Absorptivity of Nd:YAG-laser radiation on iron and steel depending on temperature and surface conditions:E-ICALEO 2000, 125–132.)

11.3.2.6 Interaction between Laser Beam and Materials Coated with Absorbers

J. Grum and T. Kek [40,41] analyzed the voltage signal of IR radiation with different travel speedsand absorbing coating thicknesses in laser hardening. In the study of influence of the travelspeed and the absorbing coating thickness, larger specimens 200� 45� 12 mm3 in size wereused. On the specimens, graphite absorber A was deposited in different thicknesses d¼ 10, 40,and 70 mm. In the treatment, no shielding gas was used. In spite of the high temperatures achieved atthe point of interaction at the moment of the beam passage, the oxidation of the steel specimensurface was negligible because the times of interaction were short [44–46].

The IR electromagnetic radiation from the interaction spot was captured with the photodiode(Figure 11.19). The photodiode permitted monitoring of the electromagnetic radiation in thewavelength range between 0.4 and 1 mm. The wavelength with the highest responsivity ofthe photodiode was 0.85 mm. The active area of the photodiode equaled 1 mm2. The mean valuesof the voltage signal of IR radiation UIR were determined in a time span of 0.3 s.

For an analysis of the data measured in laser surface hardening, the factorial analysis was used.It holds true for the factorial analysis that it is most efficient in the experiments in which theinfluence of two or more factors is studied [42]. The influence of a factor is defined as a change inresponse with a change in the level of a factor.

The factorial analysis is advantageous primarily when there are interactions between the factorsand it permits an evaluation of the influences of the individual factors with different levels of otherfactors. This ensures correct conclusions over the range of laser-hardening conditions chosen.

Figure 11.20 shows the influences of the travel speed and the absorbing coating thickness onthe voltage signal UIR. The significant influences on the voltage signal of IR radiation weredetermined with reference to the factorial analysis of the experimental data. These are linear DL

and square DQ influences of the absorbing coating thickness, linear VL, and square VQ influencesof the travel speed, and the interaction influences, DVLxL and DVLxQ. Taking into account thesignificant influences on the voltage signal, the method of orthogonal polynomials was used to


http://www.crcnetbase.com/action/showImage?doi=10.1201/9781420003581.ch11&iName=master.img-008.jpg&w=273&h=194

Cutting and preparationof samples

Opticalmeasuringmicroscope

- PC- HP 82335 HP-IB- Labview 5

- Voltage signals of IR radiation UIR(t)- Statistical analysis of voltage signals

- Hardened depth and width- Microhardness measurement- Microstructural analysis

Hardened trace

Sample

IR photodiodecentronic-BPX 65

r 45�

OscilloscopeHP 54601A

Amplifier andtransformer

FIGURE 11.19 Experimental setup for capturing and evaluation of voltage signal of IR radiation andfor measuring hardened traces. (From Grum, J. and Kek, T., Thin Solid Films, 453–454(1), 94, 2004; Kek,T.,The influence of different conditions in laser-beam interaction in laser surface hardening of steels: (in Slovene)MSc thesis, University of Ljubljana, 2003, 40–17.)

develop an approximation polynomial of the 3D response surface (Figure 11.21) for the treatmentconditions used:

UIR ¼ �0:122þ 2:51� 10�4 d� 2:51� 10�5 d2 þ 0:001 vþ 6:28� 10�7 v2 þ 2:97

� 10�5 dv� 6:07� 10�8 dv2: (11:15)

The value of R2 among the measured values of the voltage signal of IR radiation and theapproximation polynomial equals 0.98.

500

7040

10

d (μm)

400300Travel speed v (mm/min)

2001000.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Volta

ge si

gnal

UIR

(V)

FIGURE 11.20 Ranges of measured values of voltage signal of IR radiation. (From Grum, J. and Kek, T.,Thin Solid Films, 453–454(1), 94, 2004; Kek, T., The influence of different conditions in laser-beam interactionin laser surface hardening of steels: (in Slovene) MSc thesis, University of Ljubljana, 2003, 40–17.)


10152025303540455055606570

0.16–0.2 0.2–0.24 0.24–0.28 0.28–0.32 0.32–0.36 0.36–0.4 0.4–0.44 0.44–0.48

400340280220

1

10

3050

70

0.120.160.20.240.280.320.360.40.440.48

v (mm/min)d (

μm)

220

280340

400

UIR

(V)

Coat

ing

thic

knes

s d (μ

m)

Travel speed v (mm/min)

Volta

ge si

gnal

(V)

FIGURE 11.21 Voltage signal of IR radiation with regard to travel speed and coating thickness of absorberA. (From Grum, J. and Kek, T., Thin Solid Films, 453–454(1), 94, 2004; Kek, T., The influence of differentconditions in laser-beam interaction in laser surface hardening of steels: (in Slovene) MSc thesis, University ofLjubljana, 2003, pp. 40–17; J Grum, J., J. Achiev. Mater. Manufact. Eng., 24(1), 17, 2007.)

At the interaction between the laser beam and the specimen surface, rapid heating of the graphiteparticles of the absorber occurs. Because of the air surrounding the interaction spot an exothermaloxidation reaction between carbon and oxygen occurs. A product of the oxidation is carbonmonoxide and dioxide. The oxidation reactions represent the burn off of the graphite absorber.This shows in an increased intensity of IR radiation from the interaction spot and in the reduction ofthe absorbing coating thickness. The heated-up microscopic graphite particles, hot gases, and theheated specimen surface at the area of the interaction emit the IR radiation. Based on the measuredmagnitude of the voltage signal of IR radiation, it is estimated that with thicker coatings a largerquantity of the absorber burns off. Similarly, it can be stated that an increase in the travel speed ofthe laser beam results in a larger quantity of absorber burnt off per unit of time, which alsocontributes to higher values of the voltage signal of IR radiation.

Grum and Kek [40,41] also analyzed the depth and width of the hardened trace with differenttravel speeds and absorbing coating thicknesses. In spite of the burn off of the graphite absorber, itspresence produces an increased absorptivity at the interaction spot. It should be considered, however,that the absorptivity of the laser beam is affected also by the coating thickness. This shows in differentdepths and widths of the hardened trace obtained with the same travel speed of the laser beam acrossthe specimen surface, which is with the same energy supplied to the interaction spot.

It follows from the written polynomial of the response surface for the depth of the hardenedtrace that the greatest depths of the hardened trace can be expected with the coating thickness of thegraphite absorber dopt¼ 32 mm. For the given experiment, this thickness may be called the optimumcoating thickness of the graphite absorber A (Figure 11.22).

The measured values of the hardened-trace width were also statistically processed similarly asthe hardened-trace depth. The optimum coating thickness determined by means of the approxima-tion polynomial for the hardened-trace width was very similar, i.e., 35mm.

Heat is conducted through the absorber to the specimen surface. Taking into account thethinning of the absorber due to its burn off and evaporation, the optimum absorber thickness is acombination of the absorber thickness within which the absorption of the laser-beam light occursand the heat conductivity of the absorber. With an absorbing coating thickness smaller than the


10152025303540455055606570

0.54−0.580.5−0.540.46−0.50.42−0.460.38−0.420.34−0.38

Region ofoptimalcoating

thickness

Hardeneddepth d (mm)

Travel speed v (mm/min)220 280 340 400

Coat

ing

thic

knes

s δ (m

)

FIGURE 11.22 Influences of travel speed of laser beam and coating thickness of absorber A on variation ofhardened-trace depth. (From Grum, J. and Kek, T., Thin Solid Films, 453–454(1), 94, 2004, T Kek.The influence of different conditions in laser-beam interaction in laser surface hardening of steels: (in Slovene)MSc thesis, University of Ljubljana, 2003, 40–17.)

optimum one a premature elimination of the absorber in the interaction with the laser beam occurs.This is shown in Figure 11.23. A portion of the laser beam, therefore, falls on the specimen surfacethat has no absorbing coating. Consequently, the quantity of the laser-beam energy absorbed by thespecimen is smaller [43].

With a coating thickness thicker than the optimum one, only a portion of the absorber burns offwhen the laser beam passes over the surface. The remaining portion may present an obstacle to theconduction of heat energy through the absorber to the steel surface. With the optimum absorbingcoating thickness, the laser beam falls during the whole interaction time on the absorber and efficienttransformation of the laser-beam energy into the heat energy accumulated in the specimen isobtained [44–46].

11.3.2.7 Testing of Various Absorbents

The absorptivity of various media was tested with an adapted calorimetric method as used by Borikin Giesen for the determination of the laser light absorption in the lenses used in lasers [48]. Grumand Kek [41] tested various absorption types of media. By measuring the specimen temperature andknowing the laser-beam power, with which the specimen is treated, one can determine the

Sample Sampled < dopt d = dopt

d d Sampled > dopt

d

P, Db, vLaser beam

Absorbingcoating

Absorbingcoating

Absorbingcoating

Burning andevaporationof coating

P, Db, vLaser beam


P, Db, vLaser beam


FIGURE 11.23 Different cases of interaction between laser beam and absorbing coating in terms of burn offand evaporation of absorber and optimum coating thicknesses. (From Grum, J. and Kek, T., Thin Solid Films,453–454(1), 94, 2004; Kek, T. The influence of different conditions in laser-beam interaction in laser surfacehardening of steels: (in Slovene) MSc thesis, University of Ljubljana, 2003, 40–17.)


absorptivity (A) of the absorbent used. The temperature was measured with a thermocouple in thecenter of a C45E steel specimen 11� 15� 44 mm3 in size. Surface roughness Ra of the workpiece atthe area of absorbent deposition amounted to 1.6 mm. Before the absorbent application the surfacewas degreased, the specimen surfaces were coated with different types of purchased or homemadeabsorbents. In order to determine the absorptivity of each absorbent, the absorbent deposits had athickness d of 15 mm, except with a zinc–phosphate deposit where it amounted to 8 mm.

The following types of mainly homemade absorbents were compared [40,41]:

. Mixture of graphite powder with a particle size of 1–2 mm and ethanol in a ratio of 1:4

. Mixture of graphite powder with an average particle size of 1.4 mm and oxide powderFe3O4 with an average particle size of 1.9 mm

. Mixture of graphite powder with an average particle size of 6 mm and ethanol

. Mixture of graphite powder with an average particle size of 6 mm and oxide powder Fe3O4

with an average particle size of 1.9 mm. Silicon-resin black paint with stability up to 6008C. Silicon-resin color paint with an addition of iron oxide Fe3O4 with an average particle size

of 1.9 mm. Industrial spray, product of CRC Industries Europe NV, with a trade name Graphit 33. Zinc–phosphate coating prepared with a thermal phosphate bath of Zn3(PO4)2

Figure 11.24 shows the values for absorptivity of various absorbents calculated with the calorimetricmethod proposed and a comparison of the measured hardened-path depth in the workpieces undergiven transformation-hardening conditions. The results of the determination of absorptivity inaccordance with the calorimetric method were obtained with the use of extrapolation of thetemperature difference DT from the cooling phase of the temperature cycle [4]. The absorptivityof the given laser light at the interaction with the steel surface increased from 3.5% at the ambienttemperature to 28.5%–32% at the transformation-hardening temperature, which with respect to thetype of absorbent represents an increase in absorptivity of 6%–10%.

26A B C D E

Type of absorber

Depth of hardened traceAbsorption A

F G H

27

28

29

Abs

orpt

ion

A (%

)

30

31

32

33

0.2

0.25

0.3

Dep

th o

f har

dene

d tr

ace d

(mm

)

0.35

0.4

0.45

0.5

FIGURE 11.24 Comparison of absorptivity of different absorbents and hardened-path depth achieved at thesame heating conditions for C45E steel. (From Grum, J. and Kek, T., Thin Solid Films, 453–454(1), 94, 2004;Kek, T., The influence of different conditions in laser-beam interaction in laser surface hardening of steels:(in Slovene) MSc thesis, University of Ljubljana, 2003, pp. 40–17; Grum, J., J. Achiev. Mater. Manufact. Eng.24(1), 17, 2007.)


The zinc–phosphate coating shows good absorptivity due to its stability at high temperaturesand good adhesion of the deposit to the workpiece surface. The good adhesion of the absorbentdeposit at the workpiece surface was found also when using high-temperature silicon-resin black-paint deposits. With graphite absorbents, it can be noticed that the absorptivity measured is higherwith larger graphite particles. From voltage-signal measurement of IR radiation in the course ofheating it can be inferred that burn off of the C-type graphite absorbent that has larger graphiteparticles (6 mm) is less intense than the A-type absorbent that has smaller graphite particles(1.4 mm). In all cases when iron oxide was added to the absorbent, the absorptivity of the latterdecreased in spite of its favorable stability at elevated temperatures [40].

The zinc–phosphate coating shows good absorptivity due to its stability at high temperaturesand good adhesion of the deposit to the workpiece surface. The good adhesion of the absorbentdeposit at the workpiece surface was also found when using high-temperature silicon-resin black-paint deposits [44,45]. With the graphite absorbents it can be noticed that the absorptivity measuredis higher with larger graphite particles. From voltage-signal measurement of IR radiation in thecourse of heating, it can be inferred that burn off of the C-type graphite absorbent that has largergraphite particles (6 mm) is less intense than the A-type absorbent that has smaller graphite particles(1.4 mm). In all cases when iron oxide was added to the absorbent, the absorptivity of the latterdecreased in spite of its favorable stability at elevated temperatures [45,46].

11.4 LASER SURFACE HARDENING

11.4.1 LASER HEATING AND COOLING

11.4.1.1 Temperature Cycle

A prerequisite of efficient heat treatment of a material is that the material shows phase transform-ations and is fit for hardening. Transformation hardening is the only heat-treatment methodsuccessfully introduced into practice. Because of intense energy input into the workpiece surface,only surface hardening is feasible. The depth of the surface-hardened layer depends on the laser-beam power density and the capacity of the irradiated material to absorb the radiated light defined byits wavelength. Laser heating of the material being characterized by local heating and fast cooling is,however, usually not suitable to be applied to precipitation hardening, spheroidization, normalizing,and the other heat-treatment methods [4–7,49].

Kawasumi [49] treated laser surface hardening using a CO2 laser and discussed thermalconductivity of a material. He took into account the temperature distribution in a 3D body takenas a homogeneous and isotropic body. On the basis of derived heat conductivity equations, heaccomplished numerous simulations of temperature cycles and determined the maximum temper-atures achieved at the workpiece surface.

Figure 11.25 shows temperature cycles in laser heating and self-cooling. A temperature cyclewas registered by thermocouples mounted at the surface and in the inside at certain workpiecedepths [50]. In this case, the laser beam with its optical axis was traveling directly across the centersof the thermocouples inserted in certain depths. A thermal cycle can be divided into heating andcooling cycles.

The variations of the temperature cycles in the individual depths indicate that

. Maximum temperatures have been obtained at the surface and in the individual depths

. Maximum temperature obtained reduces through depth

. Heating time is achieved at the maximum temperature obtained or just after

. Greater the depth at which the maximum temperature is obtained, the longer is the heatingtime required


250

0 0.2 0.4 0.6Interaction time t (s)

z = 1.0 mm

z = 0.5 mm

Power density: Q = 4000 W/cm2

Traveling speed: v = 1200 mm/minTrace area: A = 5 mm � 5 mm = 25 mm2

z = 0.0 mm

Coolingcycle

Heatingcycle

Thermocouplez

z = 0, 0x

Surface z = 0.5 z = 1.0

0.8 1.0

500

750

1000Su

rface

and

inte

rior t

empe

ratu

re T

(�C)

FIGURE 11.25 Temperature cycle on the workpiece surface and its interior versus interaction time. (FromKawasumi, H., Metal surface hardening CO2 laser. In: EA Metzbower, Ed., Source Book on Applications of theLaser in Metalworking, American Society for Metals, Metals Park, OH, 185–194.)

. In the individual depths, the temperature differences occurring are greater in heating than incooling.

. Consequently, in the individual depths the cooling times are considerably longer than theheating times to obtain, for example, maximum temperature.

Figure 11.26 shows two temperature cycles in laser surface heating [50]. In each case the maximumtemperature obtained at the surface is higher than the melting point of the material; therefore,remelting will occur. The remelting process includes heating and melting of the material, fastcooling, and material solidification. As the maximum temperature at the surface is higher than themelting point, a molten pool will form in the material surrounding the laser beam.

Because of the relative travel of the laser beam with reference to the workpiece, the molten pooltravels across the workpiece as well whereas behind it the metal solidifies quickly. The depth of theremolten material is defined by the depth at which the melting point and the solidificationtemperature of the material have been attained.

The depth of the remelted layer can be determined experimentally by means of opticalmicroscopy or by measuring through-depth hardness in the transverse cross section.

11.4.2 METALLURGICAL ASPECT OF LASER HARDENING

Prior to transformation hardening, an operator should calculate the processing parameters atthe laser system. The procedure is as follows. Some of the processing parameters shall be chosen,some calculated. The choice is usually left to the operator and his experience. He shall select anadequate converging lens with a focusing distance f and a defocus zf taking into account the size ofthe workpiece and that of the surface to be hardened, respectively. Optimization is then based onlyon the selection of power and traveling speed of the laser beam. The correctly set parameters oftransformation hardening ensure the right heating rate, then heating to the right austenitizingtemperature TA3, and a sufficient austenitizing time tA. Consequently, with regard to the specified



Traveling speed: v = 2400 mm/minTrace area: A = 5 mm � 5 mm = 25 mm2

1500

1000

500

Surfa

ce an

d in

terio

r tem

pera

ture

T (º

C)

Tm

TA4

TA3


Traveling speed: v = 600 mm/min

titi

0 0.2 0.4 0.6Interaction time t (s)

0.8 1.0

FIGURE 11.26 Effect of laser light interaction time on the temperature cycle during heating and coolingat various power densities and traveling speeds. (From Kawasumi, H., Metal surface hardening CO2 laser.In: EA Metzbower, Ed., Source Book on Applications of the Laser in Metalworking, American Society forMetals, Metals Park, OH, 185–194.)

depth of the hardened layer, in this depth a temperature little higher than the transition temperatureTA3 should be ensured. Because of the high heating rate the equilibrium diagram of, for example,steel is not suitable; therefore, it is necessary to correct the existing quenching temperature withreference to the heating rate. Thus with higher heating rates, a higher austenite transformationtemperature should be ensured in accordance with a time–temperature-austenitizing (TTA) diagram.

The left diagram in Figure 11.27 is such a TTA diagram for 1053 steel in the quenched-and-tempered state (A) whereas the right diagram is for the same steel in the normalized state (B) [51].As the steel concerned shows pearlitic–ferritic microstructure, a sufficiently long time should beensured to permit austenitizing. In fast heating, austenitizing can be accomplished only by heatingthe surface and subsurface to an elevated temperature. For example with a heating time t of 1 s, fortotal homogenizing a maximum surface temperature Ts of 8808C should be ensured in the first caseand a much higher surface temperature Ts, i.e., 10508C, in the second case. This indicates thataround 1708C higher surface temperature DTs should be ensured in the second case (normalizedstate) than in the first case (quenched-and-tempered state).

Figure 11.28 shows a space TTA diagram including numerous carbon steels with differentcarbon contents. The TTA diagram gives particular emphasis to the characteristic steels, i.e., 1015,1035, 1045, and 1070 steels and their variations of the transition temperature TA3 with reference tothe given heating rate and the corresponding heating time [51].

When the laser beam has stopped heating the surface and the surface layer, the austeniticmicrostructure should be obtained. Then the cooling process for the austenitic layer begins. Toaccomplish martensite transformation, it is necessary to ensure a critical cooling rate, which dependson the material composition. Figure 11.29 shows a continuous cooling transformation (CCT)diagram for EN19B steel including the cooling curves.


600

700

800

900

1000

1100

Tem

pera

ture

T (�

C)

Heating rate (�C/s)

Quenchedand

tempered

Homogeneousaustenite


Normalized

TA3

TA4

TA1

TA3

1200

1300

1400103 102 101 100

Heating rate (�C/s)

103 102 101 100

104103102101

Heating time t (s)

Quenched and tempered Normalized

10010−1 104103102101

Heating time t (s)10010−1

(A) (B)

FIGURE 11.27 TTA diagram of 1053 steel for various states. (From Amende, W., Chapter 3: Transformationhardening of steel and cast iron with high-power lasers. In H Koebner, Ed., Industrial Applications of Lasers,John Wiley & Sons, Chichester, 1984, 79–99.)

0.1 1 1.1011.102

Heating time t (s)1.103

1.1040.7

0.60.5

0.4

Carbon co

ntent C

(%)0.3

0.2

700

800

Tem

pera

ture

T (º

C)

TA1

TA3900

9.103 9.1029.101 9

Heating rate (�C/s)ΔTΔt

0.9

AISI 1015

AISI 1035AISI 1045

AISI 1070

9.10−3

0.1

FIGURE 11.28 Influence of heating rate and carbon content on austenitic transformation temperature. (FromAmende, W., Chapter 3: Transformation hardening of steel and cast iron with high-power lasers. In H Koebner,Ed., Industrial Applications of Lasers, John Wiley & Sons, Chichester, 1984, 79–99.)


As carbon steels have different carbon contents, their microstructures show different contents ofpearlite and ferrite. An increased carbon content in steel decreases the temperature of the beginningof martensite transformation TMS as well as of its finish TMF. Figure 11.30 shows the dependencebetween the carbon content and the two martensite transformations.

Consequently, the increase in carbon content in steel results in the selection of a lower criticalcooling rate required. In general, the microstructures formed in the surface layer after transformationhardening can be divided into three zones:

. Zone with completely martensitic microstructure

. Semi martensitic zone or transition microstructure

. Quenched-and-tempered or annealed zone with reference to the initial state of steel

Sometimes, particularly in the martensitic zone, retained austenite occurs too due to extremely highcooling rates and the influence of the alloying elements present in steel.

11.4.2.1 Calculation of Thermal Cycle and Hardened Depth

Com-Nougue and Kerrand [52] presented in the theoretical part an extension of the numericalsolution of the steady-state heat conduction equation for constant thermal properties.

01 10 102 103 104

HVHV

HV

105 106

100001000100

10010h1

10minLog time t (s)

HRc 58 58

85 7575

38

8 5 10

718 70

Austenite to

Austenite to

Austenite tobainite

Austenite to martensite

Ferrite

30

Austenitizing temperature

TA3 = 850 ºC

TAC3

pearlite

TAC1

tA = 10 min

40 4060 60

34 28 21 230 328

300

53

1

100

200

300

400

500

600

A

TMSTem

pera

ture

T (º

C)

700

800

900

1000

FIGURE 11.29 CCT diagram of steel EN19B. (From Amende, W., Chapter 3: Transformation hardening ofsteel and cast iron with high-power lasers. In H Koebner, Ed., Industrial Applications of Lasers, John Wiley &Sons, Chichester, 1984, 79–99.)


Austenite

TMS

TMF

Martensitictransformation

910

800

600

400

Tem

pera

ture

T (�

C)

200

−200

0

0 1.21.00.80.6Carbon content C (%)

0.40.2

FIGURE 11.30 Influence of carbon content in steels according to start and finish temperature of martensitictransformation.

The objective of their investigation was to provide a numerical analysis for the 3D heatconduction equation for moving a beam with a random energy distribution. In order to accountfor the random energy profile, the beam is considered as composed of N1�N2 zones in which thepower density of each zone is supposed to be constant and independent in the surrounding zones.

If Xi and Yi are the center point coordinates of an elementary zone and Bi, Lj, respectively, itshalf-length and width, the temperature increase at (x, y) point due to all surface elements is

T ¼ T0 þ 1lffiffiffiffip

pXi,j

Qi,j

ð10

exp� z2

16«2

� erf

y� yj þ Lj4«

� �� erf

y� yj � Lj4«

� ��

� erfx� xi þ Bi

4«þ V«

a

� �� erf

x� xi � Bi

4«þ V«

a

� �� d« (11:16)

wherei, j subscripts denoting two principal directionsQi,j heat actually absorbed in (i, j) area: Pi,j=4 BiLj« is emissivitya thermal diffusivityl thermal conductivityr densityCp specific heat a ¼ k

r �CpV workpiece traveling speedT0 is ambient temperature prior to heating and (i, j), respectively, vary from 1 to N1 and

1 to N2


The different terms of the equation were evaluated by numerical integration to determine thetemperature distribution.

The random energy distribution was determined by an accurate molding of a burn pattern onplexiglas obtained by irradiation with a CO2 laser beam, and in each zone the power density wasevaluated from the incident power using the relative depths of the pattern.

The computer model predicts the workpiece maximum temperature reached during the treat-ment and its correlation with the case depth. In order to allow a comparison with the experimentaldata, the temperature increase was calculated for the defocused beam of the transverse laser withB¼ 8.25 mm and L¼ 6 mm at selected processing power and speed values. The chosen grid meshincludes nine zones (beam with two energy peaks in the travel direction) and the theoretical analysiswas performed for a chromium steel (S2) with the following thermal properties (300 K):

Thermal conductivity: l¼ 29 W=m=8CThermal diffusivity: a¼ 0.06 cm2=s

Figure 11.31 gives the relation between experimental and theoretical case depths for the defocusedbeam. A very good agreement was obtained between the experimental results and the mathematicmodel, in spite of an approximation in the determination of the workpiece initial temperature beforeeach trial and of the beam profile. Figure 11.32 shows the temperature increases as a function oftime at the top and the bottom of the HAZ. The experiment was conducted with 3 kW and 10 mm=son a surface coated with a black paint (56% absorptivity). The temperature curve exhibits two peakscorresponding to the energy peaks in the beam spot. The influence of such energy distribution showsthat it can be used to increase the time duration above the TA3 temperature for formation of uniformaustenite with an appropriate shaping of the beam.

Further, the calculation developed with the computer model indicates that the maximumtemperature reached on the workpiece surface never exceeds the melting point of the materialwhen the hardening treatment is performed at 3 kW and 10 mm=s on a surface coated with graphiteor black paint. Consequently, the slight melting actually observed for these conditions on graphite-coated surfaces probably results from a reaction of the graphite spray itself under the laser beam.

A numerical solution of the 3D heat-conduction equation has been determined for a laser beamwith a random intensity distribution. A very good agreement was obtained between the experimentalresults and theoretical values.

00

0.2

0.4

0.6

Theo

retic

al d

epth

z (m

m)

0.8

1.0

1.00.8Experimental case depth z (mm)

0.60.40.2

FIGURE 11.31 Relation between experimental and theoretical hardened depths. (From Nougue, J.C. andKerrand, E. Laser surface treatment for electromechanical applications: NATO ASI Series. In: C.W. Draper,P. Mazzoldi, and M. Nijhoff, Eds., Laser Surface Treatment of Materials, Publishers, Dordrecht, 1986, 497–511.)


1500

500

00 1

Time t (s)

z = 0 mm

z = 1.03 mm

Tem

pera

ture

T (�

C)

2

1000 TAC3

TMs

FIGURE 11.32 Calculated thermal cycles at the top and the bottom of the HAZ (defocused beam, P¼ 3 kW,v¼ 10 mm=s, T0¼ 208C, black paint coating, steel S2. (From Nougue, J.C. and Kerrand, E. Laser surfacetreatment for electromechanical applications: NATO ASI Series. In: C.W. Draper, P. Mazzoldi, and M. Nijhoff,Eds., Laser Surface Treatment of Materials, Publishers, Dordrecht, 1986, 497–511.)

Figure 11.33 shows a complete diagram including possible processing parameters and thedepths of the hardened layers obtained in transformation hardening [53].

The data are valid only for the given mode structure of the laser beam (TEM), the given area ofthe laser spot (A), and the selected absorption deposit. In this case, the processing parameters are

1008060Traveling speed v (mm/s)

4020

0.25

0.5

0.75

1.0

Har

dene

d de

pth

z (m

m)

1.5

1.75

2.0Material: SAE 4140Laser spot:A = 15.24 mm � 15.24 mm

Surface meltingarea

Laserpower

8 kW

7 kW

6 kW

5 kW4 kW3 kW

Transformationhardening area

FIGURE 11.33 Influence of laser power and traveling speed on depth of hardened layer at given laser spot.(From Belforte, D. and Levitt, M., Eds., The Industrial Laser Handbook. Section 1, 1992–1993 ed. Springer-Verlag, New York, 1992, 13–32.)


selected from the laser-beam power P (W) and the traveling speed of the laser beam v (mm=s). Theupper limit is a power P of 8 kW. This is the limiting energy input that permits steel melting.Although the data collected are valid only for a very limited range of the processing parameters,conditions of transformation hardening with the absorptivity changed due to the change in thedeposit thickness of the same absorbent, or other type of absorbent, can efficiently be specified.Greater difficulties may occur in the selection of the laser trace that can be obtained in different waysand can be varied too. In case the size of the laser spot changes due to optical conditions, it isrecommended to elaborate a new diagram of the processing conditions of transformation hardening.

Figure 11.34 shows a shift of the transformation temperature, which ensures the formation ofinhomogeneous and homogeneous austenite within the selected interaction times [54].

A shorter interaction time will result in a slightly higher transformation temperature TA1 andalso a higher transformation temperature TA3. To ensure the formation of homogeneous austenitewith shorter interaction times, considerably higher temperatures are required. Figure 11.34A showsa temperature=time diagram of austenitizing of Ck 45 steel.

The isohardnesses obtained at different interaction times in heating to the maximum temperatureensures partial or complete homogenizing of austenite; this is plotted in the figure. Figure 11.34Bshows the same temperature=time diagram of austenitizing of 100Cr6 hypereutectoid alloyed steel.

The diagram indicates that with short interaction times, which in laser hardening vary between0.1 and 1.0 s, homogeneous austenite cannot be obtained; therefore, the microstructure consists ofaustenite and undissolved carbides of alloying elements, which produce a relatively high hardness,i.e., even up to 920 HV0.2. After common quenching of this alloyed steel at a temperature of

1200

1100

1000

Tem

pera

ture

T (�

C)

Tem

pera

ture

T (�

C)

900

800

7001 10 102

Time t (s)(A) Material Ck 45 (B) Material 100Cr6

103 104 1 10 102

Time t (s)103 104

Ferrite + perlite Ferrite + carbide

Inhomogeneousaustenite



HV

HV

840

900

850

800

750

800700600820

800780

TAC1

TAC1b

TAC1a

TAC

TAC3

FerriteFerrite

+ perlite

+ carbide

Carbide+

austenite

Inhomogeneousaustenite

Austenite + melting

+ austenite

+ austenite

FIGURE 11.34 Temperature–time-austenite diagrams with lines of resulting hardness for various steels.(From Meijer, J., Kuilboer, R.B., Kirner, P.K., and Rund, M., Laser beam hardening: Transferability ofmachining parameters. Proceedings of the 26th International CIRP Seminar on Manufacturing Systems –

LANE’94. In: M Geiger and F Vollertsen, Eds., Laser Assisted Net Shape Engineering, Meisenbach-Verlag,Erlangen, Bamberg, 1994, 243–252.)


homogeneous austenite, a considerably lower hardness, i.e., only 750 HV0.2, but a relatively highcontent of retained austenite were obtained. Retained austenite is unwanted since it will produceunfavorable residual stresses and reduce wear resistance of such a material.

11.4.3 AUSTENITIZATION OF STEELS

11.4.3.1 Austenitization of Hypoeutectoid Steels

The phenomenology of austenitization of hypoeutectoid steels with ferrite and pearlite microstruc-ture is more complex than that of the same transformation in eutectic or hypereutectic alloys.Pearlite in two-dimensional (2D) section has the appearance of alternate lamellae of ferrite andcementite. Ferrite has a very low solubility of carbon and transforms into austenite at hightemperature. However, cementite decomposes and yields its carbon to the transformation zone atlower temperature from ferrite to austenite. Initiation of austenitization in a hypoeutectoid steel is inpearlite, where the diffusion distances for carbon are small.

When the austenite starts to grow into ferrite, carbon has to partition to the austenite=ferriteinterface for the reaction to proceed, so the diffusion rate of carbon in austenite becomes one of thelimiting factors. The diffusion process range is much larger and the rate of transformation willdepend on the morphology, distribution, and volume fractions of the phases present. Thermo-dynamic equilibrium of the transformation process runs at long times, while nucleation of austenitein pearlite colonies and diffusive processes are expected to control the rates of transformation.

Gaude-Fugarolas and Bhadeshia [55] describe a model including nucleation of new austenitegrains at the edges of pearlite colonies. These grains will be assumed to grow until the pearlite hascompletely transformed into austenite, after which ferrite transforms.

The mechanisms of the structural changes during the hardening process involve the followingprocesses [56]:

. Transformation of pearlite to austenite.

. Homogenization of carbon in austenite during the heating cycle.

. Decomposition of austenite to ferrite and pearlite.

. Transformation of austenite to martensite during the cooling cycle. The former threechanges are diffusion controlled, but the last change is displacive and does not dependon diffusion.

The transformation of pearlite to austenite takes place when the workpiece is heated above theconventional temperature TAc1. Cementite lamellae in the pearlite colonies first dissolve and carbondiffuses outward into the surrounding ferrite. If the distance between the cementite lamellae within thecolony is d and carbon diffuses laterally, it might be thought that the diffusion of carbon over adistance d would be sufficient to convert the colony to austenite. To achieve such an extent of carbondiffusion, at a high heating rate, superheating is required. Additionally, part or all of the ferriteoriginally surrounding the pearlite colonies may transform to austenite, depending on the value of thetemperature relative to the TAcl, where considerable superheating of the ferrite might also occur. Thepearlite becomes austenite containing 0.8% carbon, and the ferrite becomes austenite with negligiblecarbon content. Thereafter, carbon diffuses from a high concentration to a low concentration region,homogenizing the carbon distribution. On subsequent cooling, all austenite with carbon contentabove a critical value displacively transforms to martensite, and the rest reverts to ferrite.

However, due to the finite rate of carbon diffusion, at a high heating rate a superheating isrequired to enhance the diffusion rate in order to achieve austenite transformation.

Assuming that the distribution of phases is more accentuated in steels with chemical segregationand banded microstructure after rolling [55], the microstructure of the steel can be defined by fourindependent parameters, as shown in Figures 11.35 and 11.36.


2lp

PPP

αα

2lh

2lα

FIGURE 11.35 Definition of microstructure parameters denoting pearlite ferrite regions. (From Fugarolas,D.G. and Bradeshia, H.K.D.H., J. Mater. Sci., 38, 1195, 2003.)

The parameters la and lp define the thickness of the ferrite and pearlite layers and the sum ofboth gives lh. Assuming that the carbon content of ferrite is zero, and that pearlite contains eutectoidcarbon composition (0.77 wt%), lp can be defined as,

lp ¼ (la þ lp)wc

0:77¼ lhwc

0:77(11:17)

Where wc is the wt% of carbon.The dimension of the pearlite colonies lcol may be defined as a typical colony. The distance

is the periodicity distance between neighboring pearlite cementite in lamellae form, measured bythe linear interception method. Finally there is 2le between neighboring pearlite cementite inlamella form.

lcol

2lc

FIGURE 11.36 Definition of pearlite microstructure parameters. (From Fugarolas, D.G. and Bradeshia,H.K.D.H., J. Mater. Sci., 38, 2003, 1195–1201.)


Austenite nucleates at the pearlite surface colonies according to classic nucleation theorynucleation rate can be calculated at the given temperature:

I ¼ C0 � N06icol

k:T

hexp �G* þ Q

RT

!(11:18)

whereI is the nucleation rate per unit time in a single colonyN0 is the number of nucleation sites per unit area of colony interfaceT is the absolute temperatureG* is the activation free energy for nucleationC0 is a fitting parameter

The active nuclei are all assumed to be located at the surface of the pearlite colonies; hence theratio between colony surface to volume, which gives the factor 6=lcol, k is the Boltzmann constant, Rthe gas constant, h the Planck constant, and Q is an activation energy representing the barrier for theiron atoms to cross the interface.

Once the new grains of austenite have nucleated, their growth rate up to the equilibrium state isdetermined by the decomposing cementite and the diffusion of carbon in austenite. The velocity ofthat interface can be determined from mass balance and the relevant diffusion equation.

The atoms involved in the boundary are determined by (cga – cay) dr, where cga and cay are thecomposition of austenite and ferrite and rint the position of the interface.

Finally, the velocity of the interface can be calculated as follows:

vint � D

r

cgu � cga

cga � cag

� �, (11:19)

where the diffusion distance of carbon in austenite is important.The nucleation rate of austenite in a pearlite colony is the equation for I. Each active nucleus

develops in one of the layers of ferrite surrounded by cementite. The newly nucleated grains growand reach a size of the order of te and a steady growth rate, and start growing into the colony.

The average velocity of pearlite colony growth rate can be determined as

�vint ¼ 1rf � r0

( ln rf � ln r0) � D � Cgu � Cga

Cga � Cag

� �, (11:20)

where�vint is the averaged velocity of the interfacerf and r0 are in this case the distance to the center of the cementite layer

In many cases many austenite nucleus will start to grow by Avrami and Cahn vg¼ 1 – exp(�ve)where vg is the real volume fraction and ve is the extended one.

In order to compare the predictions of the model with the transformation behavior of steel, astandard set of experiments have been designed by Grande-Fugarolos and Bhadeshia [54]. This setof experiments can then be used to compare the capability of the model to predict the effects of otherparameters like composition and microstructure. A series of six experiments was conducted onsteels A and B with the chemical composition given in Table 11.3. They where heated at 508C=s tovarious maximum temperatures and various heating time above the TAc1 temperature and variousheating times. All the temperatures equated are in the intercritical range so only transformation isexpected, although the extent of austenite varied from very little to almost complete transformation.


TABLE 11.3Composition of Steels Used

Steel

Chemical Composition in wt%

C Si Mn Cr Ni Mo V

Steel A 0.55 0.22 0.77 0.20 0.15 0.05 0.001Steel B 0.54 0.20 0.74 0.20 0.17 0.05 0.001

Source: From Fugarolas, D.G. and Bradeshia, H.K.D.H., J. Mater. Sci., 38, 1195, 2003.

TABLE 11.4Microstructure of Steels Used

Steel

Microstructure Parameters

2la�s (m) 2lp (m) le�s (m) lcol�s (m)

Steel A [2.55� 1.36]� 10�6 6.38� 10�6 [0.51� 0.05]� 10�6 [19.73� 0.95]� 10�6

Steel B [1.85� 0 97]� 10�6 4.34� 10�6 [0.25� 0.05]� 10�6 [18.46� 0.95]� 10�6

Source: From Fugarolas, D.G. and Bradeshia, H.K.D.H., J. Mater. Sci., 38, 1195, 2003.

Experimental data had been collected using a dilatometer, using hollow steel samples followingthe thermal history described for the same thermal conditions. Microstructural features are given inTable 11.4 at the same thermal conditions for various steel compositions.

11.4.3.2 Eutectical Temperature Determination

Chen et al. [57] studied and developed a new method for determining eutectoid temperature TAc1 ofcarbon steel during laser surface hardening. They used 3D heat flow model with temperature-dependent physical properties and solved temperature distribution by employing the finite elementmethod (FEM).

There are two significant features during laser surface hardening:

1. High heating and cooling rate reaching 104 8C=s2. Period of steel heating above the critical temperature is extremely short, often about 0.1 s

or less, leading to a nonequilibrium transformation during austenitization

The eutectoid temperature in such nonequilibrium transformation is determined by matching theprofiles of the experimental-hardened and melted zones with isotherms depths numerically calculatedby FEM. The melted zone profile was used as a calibrator, and the unknown surface absorptivity wasadjusted by parameter estimation until the calculated isotherm melt depth. The eutectoid temperatureis designated as the temperature where isotherm depth best fits the experimental profile of thehardened zone (HZ).

Additionally, the numerical model was used for predicting hardened depth at various lasertransformation hardening conditions.

Figure 11.37 shows a sketch of the heat transfer model for a workpiece subjected to a Gaussianlaser beam (TEM00). The laser beam is stationary, while the workpiece moves at constant travelingspeed toward negative x-direction. The origin of the x–y–z coordinate system is fixed at the center ofthe laser beam. When the laser beam impinges on the surface of the workpiece, part of the laser lightis absorbed and the remainder is reflected. The following approximations were applied:


X

WorkpieceZ

Y

Traveling speedvx (cm/s)

Laser beam

Rb

FIGURE 11.37 Sketch of heat transfer model. (From Chen, C.C., Tao, C.J., and Shyu, L.T., J. Mater. Res.,11(2), 458, 1996.)

. Energy loss from the surface due to convection and radiation to the environment isnegligible compared to the energy conducted into the interior of the workpiece.

. Quasisteady state is established during laser scanning.

. Workpiece mass is large enough to that all surfaces except the surface subjected to the laserbeam remain at room temperature.

. Energy transfer within the melt zone is dominated by conduction and the convectivetransfer due to fluid motion induced by buoyant and thermocapillary forces is neglected,the energy equation describing the heat transfer in the system is simplified and given as

@

@xl@T

@x

� �þ @

@yl@T

@y

� �þ @

@zl@T

@z

� �¼ �rCpvx

@T

@x(11:21)

� 2008 b

wherel is thermal conductivityr is the densityCp is specific capacityvx traveling speed

The boundary conditions are

l@T

@x¼ A(x, y)P

pR2b

exp � x2 þ y2

R2b

� �, at z ¼ 0 (11:22)

whereA(x, y) is the absorptivity of the laser beam at the workpiece surfaceP is laser powerRb is beam radius

Because of the high traveling speed and short interaction time does not penetrate deep into theworkpiece, the input heat flux does not penetrate deep into the workpiece nor does it spreadsignificantly in the lateral direction. Thus, in the numerical method, the temperature at all locationsfar away from the beam center were regarded as remaining at initial temperature T0 and in the case

y Taylor & Francis Group, LLC.

of analytical solution, the heat transfer in the workpiece is regarded as a transfer to a semi-infinitesystem, even though it is physically neither very thick nor very wide.

11.4.3.2.1 Analytical SolutionCline and Anthony [58] have obtained an expression of the temperature distribution for a semi-infinite system with constant physical properties and absorptivity while scanned workpiece surfacewith a Gaussian beam, i.e.,

T ¼ T0 þ T1

ð10

exp� x=Rb þ T2m2=4ð Þ2þ(y=Rb)2=1þ m2

� �þ (z=Rb)

2=m2� �h i

p3=2(1þ m2)dm (11:23)

with

m2 ¼ 4lkt

rCpR2b

, T1 ¼ aP

lRb

, T2 ¼ rCpRbvxl

(11:24)

11.4.3.2.2 Numerical SolutionThe FEM was employed to solve the temperature distribution in the workpiece from given equation.The thermal physical properties of AISI 1042 steel were used in the computation. The latent heat ofmelting=solidification was incorporated by the specific heat method, i.e., incorporationed as anapparent increase in the specific heat over the melting=solidification range and expressed as

Cp(T) ¼ Cps(Tm)þ Hf =DT (11:25)

for T¼ Tm to TmþDTThe choice of DT affects the resulting temperature distribution, but negligible effects were

observed as DT was reduced from 608C to 308C.After the computer model was developed, its accuracy was verified with the analytical solution

for the cases of constant physical properties and absorptivity. A very good agreement between thenumerical and the analytical solutions was obtained, as shown in Figure 11.38, thus supportingthe validity of the computer model.

The determination of TAc1 is equivalent to determining the temperature where isotherm depth bestfits the lightly etched HZ. Figure 11.39 shows profiles comparison of the hardened and remelted zoneswith macroetched-hardened trace on the workpieces with three depth isotherms. Isotherm depth fortemperature 7678C is in good agreement with the profile of the HZ in the central part, while isothermdepth for temperature 7238C shows a discrepancy, leading to the conclusion that the start of austenitictransformation occurs at a temperature of 7678C instead of the conventional TAcl is equal to 7238C.

Numerical simulation in Figure 11.40 shows that the maximum depth and width of the hardenedisotherm depth increase monotonically with absorbed power. However, the maximum depthschange negligibly (<1%) if the position variation of a is less than �15%. Thus, when theabsorptivity is not strongly position dependent, the use of the remelted zone profile as a calibratorto determine TAc1 can be simplified to using the maximum depth of the remelted zone. For a betterapproximation, the central part of the remelted zone profile can be used, and this significantlyreduced the effort required to find a proper a(x, y) by parameter estimation.

As the beam size increases, more workpiece surface is irradiated by laser light, but at lowerintensity. As a result, shallower depth is expected, and this is confirmed by simulation given results.The results show a different mode of variation for the width. Figure 11.41 shows more conductedsimulations and results. The maximum width increases with increasing beam radius, reaches amaximum, then decreases sharply at larger beam radius, while the maximum depth decreases almost


0.15

0.10Y/Rb = 1.5Y/Rb = 1.0Y/Rb = 0.5Y/Rb = 0.0T2 = 5.0

0.05

T/T 1

0.00−6 −4 −2 0

X/Rb

2 4

FIGURE 11.38 Comparison of the numerical solution (- - - -) and the analytical solution (-) where physicalproperties and absorptivity were assumed constant. (From Chen, C.C., Tao, C.J., and Shyu, L.T., J. Mater. Res.,11(2), 458, 1996.)

200100

767�C

723�C

1491�C

Conditions:P = 49.3 WRb = 118 μmvx = 2.87 cm/s

0Y-axis (μm)

−100−200100

80

60

40

Z-ax

is (μ

m)

20

0

FIGURE 11.39 Comparison of the calculated depth isotherms, corresponding to the temperature of 14918C,7678C, and 7238C (. . . . ), respectively, with the profiles of the hardened and melted zones (-) of the macro-etched hardened trace in the workpiece. (From Chen, C.C., Tao, C.J., and Shyu, L.T., J. Mater. Res., 11(2),458, 1996.)


100

150

200D

epth

z (μ

m)

250

300Rb = 100 μmRb = 165 μm

vx = 0.5 cm/svx = 2.0 cm/s

5030 40

Absorbed power PA (W)50

FIGURE 11.40 Variation of the predicted depth and width of the hardened depth isotherm as a function ofpower absorbed for two beam radii, obtained assuming a eutectoid temperature of 7708C and using thetemperature-dependent physical properties and constant a, where the depth is represented by solid lines andthe width by dashed lines. (From Chen, C.C., Tao, C.J., and Shyu, L.T., J. Mater. Res., 11(2), 458, 1996.)

0.050.040.03Beam radius (μm)0.020.010.00

0

50

CalculatedDepthWidth

PA = 50 W

vx = 0.5 cm/svx = 0.5 cm/svx = 1.5 cm/svx = 2.0 cm/s

723�C770�C770�C770�C

100

150

200

Dep

th/w

idth

Z/Y

(μm

)

250

300

FIGURE 11.41 Variations of the calculated depth and width of the hardened depth isotherms as functions ofbeam radius for various traveling speeds of 50 W using the temperature-dependent physical properties andconstant a. (From Chen, C.C., Tao, C.J., and Shyu, L.T., J. Mater. Res., 11(2), 458, 1996.)


864Traveling speed vx (cm/s)

2

DepthCalculated

Width

00

100

150

200

Dep

th/w

idth

Z/Y

(μm

)

723ºCPA = 30 W PA = 40 W

770ºC830ºC

FIGURE 11.42 Variations of the calculated depth and width of the hardened depth isotherms as functionsof traveling speed for two absorbed powers at a beam radius 0.1 mm, assuming various eutectoid temperatureand using the temperature-dependent physical properties and constant a. (From Chen, C.C., Tao, C.J., andShyu, L.T., J. Mater. Res., 11(2), 458, 1996.)

monotonically. At small beam radius, the maximum depth and width corresponding to the TAc1 ofthe temperature 7708C is very close to that corresponding to the conventional temperature TAclof 7238C. The figure also shows that there is an optimum combination of process variables thatproduce a larger width of HZ without surface remelting.

On the other hand, increasing traveling speed shortens the interaction time of a workpiece withlaser light. Figure 11.42 shows that if less power is absorbed, it results in narrower width andshallower depth. The figure shows that the maximum depth and width decrease with increasedtraveling speed.

When superheating is required for the transformation, higher temperatures need to be reached.The simulation shows that the maximum depth and width of the hardened depth isotherm decreaseapproximately linearly with superheating. Figure 11.42 also shows variations of the maximum depthand width as functions of traveling speed for three TAc1’s; temperatures 8308C and 7708C aresmaller than those corresponding to the conventional temperature TAc1 of 7238C. For the TAc1temperature of 7708C, the difference is less than 10 mm. Thus, the use of the conventionaltemperature TAc1¼ 7238C in the prediction of the hardened depth leads to an error less than 10%when the hardened depth is more than 100 mm.

11.4.3.2.3 Martensite Volume FractionAn expression relating the volume fraction of martensite to the laser temperature cycle in termsof the carbon content of the steel being hardened has been given previously by Ashby and


Easterling [56]. In their treatment, it is assumed that all austenite resulting from the temperaturecycle with a carbon content greater than Cc will transform to martensite on subsequent cooling. It isalso assumed that the maximum carbon content does not exceed 0.8%. In the Ashby–Easterlingmodel the volume fraction of austenite is calculated by considering the effect of the temperaturecycle on the transformation of pearlitic–ferritic steel and the subsequent redistribution of carbon inthe growing austenite. This volume fraction will obviously vary as a function of the temperaturecycle experienced by the steel, being most severe at the surface. By knowing the severity of thetemperature cycle at each point below the surface according to heat flow theory, the volume fractionof austenite, which contains average value of carbon, and the subsequent volume fraction ofmartensite can then be calculated. On this basis, the full expression for the total volume fractionof martensite f is

f ¼ fm � ( fm � f i) exp� 12 f 2=3i =p1=2g� �

� ln=Ce=2Cc) D0at exp (�Q=RTp) �1=2n o

(11:26)

wherefm is the maximum permitted austenite volume fraction, as given by the Fe–C phase diagramfi is the initial volume fraction of martensitefi is assumed in hypoeutectoid steels to be equal to the initial volume fraction of pearlite in the

steel, the pearlitic regions transforming to austenite first (at the eutectoid temperature).

Thus,

fi ¼ (C � Cf )=(0:8� Cf ) � C=0:8 (11:27)

whereC is the carbon content of the steel being treatedCf is the carbon content of the ferrite

The first term of the main exponential in equation, containing fi and g (the mean distance betweenpearlite islands), is concerned with the growth of the austenitized regions with an initial eutectoidcarbon content of Ce and a lower critical carbon content (Cc¼ 0.05%), below which the austenite isassumed to transform simply to equiaxed ferrite. The second exponential term refers to the diffusionof carbon in austenite (temperature-independent coefficient D0), with an activation energy of Q. R isthe gas constant, Tp refers to the peak temperature of the temperature cycle, and a and t are kineticconstants of the temperature cycle [56,59].

11.4.3.3 Phase Transformations at Various Heating and Cooling Rate

Miokovi�c et al. [60] implemented in the finite-element-program ABAQUS as user-definedmaterial laws, allowing a coupled calculation of temperature and phase development during heatingand cooling. The influence of heating and cooling rate on the time-dependent temperature fields andphase transformations within the affected zone was investigated. A good correspondence betweenthe results of simulation and experimental data was obtained in respect of the resulting hardnessprofiles and the degree of homogeneity of the microstructure.

The mathematical description of transformation hardening is of great significance for lasersurface hardening due to locally different austenizing and quenching conditions in the HAZ. Themodeling of the laser hardening process allows the determination of time-dependent temperaturefields and phase transformations due to the heat impact and supports the understanding of thehardening process. The initial state of austenite is of great importance for development of the finalmicrostructure and its mechanical properties. There are many studies on the simulation of surface


hardening of steels considering transformation hardening with homogeneous austenite formation[59,61]; the modeling of the effect of the state of austenite involving inhomogeneous austeniteformation on the kinetics of phase transformation during cooling has been taken by Rödel et al. [62].

Systematic experimental investigations on phase transformation processes during heating andcooling with highest rates, including the effect of inhomogeneous austenite formation on theformation of martensite of AISI 4140 (EN 42CrMo4), in a state quenched and tempered at4508C, will be used to deduce models for the description of these effects. Additionally, thesemodels were implemented in the commercial finite-element-system ABAQUS and applied tosurface hardening using a 3 kW high-power diode laser, which allows to control surface temper-atures every millisecond. By modeling the development of the martensite start and martensite finishtemperatures for different surface hardening conditions, the effect of heating and cooling rate on themicrostructure in the surface layer and austenite homogenization was analyzed.

11.4.3.4 Effects of Heating and Cooling Rates on Phase Transformations

In order to study the short-time austenizing and quenching behavior of AISI 4140 dilatometricexperiments with varying heating rates nheat, cooling rates ncool, and maximum temperatures TA,max

were performed. The strains measured during heating are shown in Figure 11.43A and allowed todetermine the thermal expansion coefficients of the ferritic–carbidic initial microstructure and ofaustenite, the transformation strain for the transformation of ferrite into austenite, and the kinetics ofthis transformation. Figure 11.43A also shows that the transformation is shifted to higher temper-atures with increasing heating rate and that the strain during the transformation is reduced due to thehigher thermal expansion coefficient of austenite than ferrite. Using an Avrami approach fortransformation and parameters mentioned earlier, the data measured is described in Figure 11.43B.

Dilatometric experiments showed that austenization continues during the first part of cooling ifit was not completed at maximum temperature. This allowed the determination of the effects ofheating and cooling rates and maximum austenite content, as shown in Figure 11.44. The analysis ofthe dilatometric strains allowed the determination of the thermal expansion coefficient of martensiteand considering the austenite formation during cooling, the determination of martensite contentversus temperature as depicted in Figure 11.45 for different heating and cooling rates, as well asmaximum temperatures. These values are given in Figure 11.46 and show that the martensite startand finish temperatures increase with increasing heating and cooling rate due to the inhomogeneityof the austenite formed and the increasing amount of dissolved carbides, leading to growing meancarbon contents soluted in the austenite. Therefore, the martensite start and finish temperatures


800

Model

1,000 K/svheat AISI4140

TA = 1150�CAISI4140TA, max = 1150�C

3,000 K/s6,000 K/s10,000 K/s

Model10,000 K/s6,000 K/s3,000 K/s1,000 K/s

vheal

7507001.0

1.1

1.2

1.3

Ther

mal

stra

in (%

)

1.4

950900Temperature T (�C)(B)(A)

8508007500

20

40

60

80

Aus

teni

te co

nten

t (vo

l %)

100

FIGURE 11.43 Thermal expansion «(T) versus temperature (A) and volume fraction of austenite fA(B) for different heating rates. (From Miokovi�c, T., Schulze, V., Vohringer, O., and Lohe, D., Mater. Sci.Eng. A, 435–436, 547, 2006.)


Max

. aus

teni

te co

nten

t af (

vol%

)

Heating rate vheat (K/s)10,0008,000

vcoal = 1000 K/svcoal = 3000 K/sAustenite contentduring heating fA

H

6,0004,000

AISI 4140TA,max = 850�C

2,00000

20

40

60

80

100

FIGURE 11.44 Maximum volume fractions of austenite fHA nheatð Þ and fA max nheatð Þ for heating process andduring heating and subsequent cooling with cooling rates. (From Miokovi�c, T., Schulze, V., Vohringer, O., andLohe, D., Mater. Sci. Eng. A, 435–436, 547, 2006.)

350

AISI 4140vcool = 1000 K/s

AISI 4140vcool = 3,000 K/s

TA,max = 850�CTA,max = 1150�C

TA,max = 1150�C

TA,max = 1150�CModel

TA,max = 1150�C

TA,max = 1150�C

TA,max = 1150�CTA,max = 850�C

TA,max = 850�CTA,max = 1150�CTA,max = 850�C

TA,max = 850�C

ModelTA,max = 850�C

TA,max = 850�C

1,000 K/s

6,000 K/s

10,000 K/s

1,000 K/s3,000 K/s

10,000 K/s6,000 K/s

vreal

vreal


200150 350 400 450 500300250Temperature T (�C)(B)(A)

200150

0

20

40

60

Mar

tens

ite co

nten

t mf (

vol%

)

80

100

0

20

40

60

Mar

tens

ite co

nten

t mf (

vol%

)

80

100

FIGURE 11.45 Experimentally determined and calculated volume fraction of martensite for TA,max¼ 8508Cand 11508C and different heating and cooling rates (A) 1000 K=s and (B) 3000 K=s. (From Miokovi�c, T.,Schulze, V., Vohringer, O., and Lohe, D., Mater. Sci. Eng. A, 435–436, 547, 2006.)

360

355

350

345

340

850(A)

900 950Temperature TA,max (�C)

1000 1050 1100 1150

T Ms (�

C)

160AISI 4140

Model

vcool = 1000 K/s

1,000 K/s6,000 K/s10,000 K/s

vheat

AISI 4140vcool = 1000 K/svheat

155

150

145

140

850(B)

900 950Temperature TA,max (�C)

1000 1050 1100 1150

T Mf (�

C)

Model

1,000 K/s6,000 K/s10,000 K/s

FIGURE 11.46 Experimentally determined and calculated development of (A) T Ms(TA,max) and (B) TMf(TA,max) for different heating rates at a cooling rate of 1000 K=s. (From Miokovi�c, T., Schulze, V.,Vohringer, O., and Lohe, D., Mater. Sci. Eng. A, 435–436, 547, 2006.)


depend on the austenizing and quenching conditions like heating rate, cooling rate, and themaximum austenizing temperature and have to be described mathematically. This is done byapplying

�TMs=f TA,max ,vheat,vcoolð Þ ¼ �TMs=f�Cstð Þ exp (11:28)

x exp exp� TA,max � TA1, homð Þk

a0,s=f þ a1,s=fvcool� �

ln vheat=n0 þ 1ð Þ

!" #(11:29)

whereTA1,hom¼ 7308C is the equilibrium temperature of initiation of austenite formationTMs (�Cst)¼ 3408C and TMf (�Cst)¼ 1408C are the martensite start and martensite finish tem-

peratures for martensite formation after homogeneous austenization, corresponding to thefull carbon content �Cst¼ 0.42 wt% of the steel

k is the Boltzmann constantn0 ¼ 1 K=s is a standard heating rate used for normalizinga0 and a1 are fit parameters

This model was used to describe the transformation of austenite into martensite combined with theKoistinen–Marburger approach. The determination of martensite start and finish temperaturesaccording to the equation needs the availability of mean heating and cooling rates at the positionof interest. The values �nheat=cool,iþ 1 in the time increment iþ 1 were determined incrementally bythe temperature–time-course applying

�nheat=cool,iþ1 ¼�ni(T � dT)þ dT

dt dT

T

�� (11:30)

Where�nheat=cool,i+1 is the mean rate in time increment iT is the temperaturedT and dt are the temperature and time increment, respectively

The temperature gradient dT=dt> 0 determines the onset for the calculation of the mean heating ratenheat and dT=dt< 0 the onset for calculating ncool.

11.4.3.5 Determination of Hardness Profiles

Due to locally different austenizing and quenching conditions during the laser impact, differentzones within the surface area occur after the hardening process. At the top, a completely martensiticmicrostructure comes up, which is referred to as the HZ. The transition zone (TZ) consists of partlyaustenized and eventually hardened microstructure and the remaining base material that does nottransform during the laser impact. Considering mainly the martensite transformation during coolingand assuming that carbon diffusion is dominant due to rapid heating and cooling, the carbon contentin the martensitic matrix formed after the quenching process allows the determination of the finalhardness of the microstructure in the laser affected zone. By associating the hardness calculationwith the phase transformation calculation, the final hardness H is obtained by accumulating thecontributions of the different constituents formed along cooling and can be estimated using a simplerule-of-mixture according to

H ¼ fMHM þ faHa þ H(Fe,M)-carbides þ Hundissol:carbides½ � (11:31)


The terms fM and fa are the volume fraction of martensite and ferrite and HM and HF the hardness ofthe martensitic and ferritic matrix, respectively. The hardness HM of the locally forming martensiticmatrix is related to its mean carbon content �CM, which can be determined from the local martensitestart temperature, the inverse of the relationship:

T �Ms(�CM )¼ TMs(�Cst)

þ 350�C=%(�Cst � �CM) (11:32)

Therefore, the hardness of the martensite is given by the polynomial:

HM(HV) ¼ 150þ 1667(�CM%)� 926(�CM%)2 �

(11:33)

The carbon content in the small amount of undissolved special carbides is regarded as small and wasnot taken into account. The decrease in H(Fe,M)-carbides due to the proceeding carbide dissolvingprocesses during austenizing is determined as a logarithmic function according to

H(Fe,M)-carbides(HV) ¼ [�183:6 ln [1� {[�Cst%]� [�CM%]}] ] (11:34)

Figures 11.47 and 11.48 give an example of a simulation of the laser hardening process with heatingand subsequent quenching of the specimen surface applying heating and cooling rates ofnheat¼ ncool¼ 1000 K=s. The resulting temperature–time-courses T(t) at different distances to thesurface x are illustrated in Figure 11.47A. As can be seen, the peak of the temperature cycle isshifted toward smaller values and larger times with increasing distance to the surface. The coolingprocess is similar for all depths except for the initial region down to� 6008C. Figure 11.47B showsthe maximum temperature TA,max versus distance to the surface. As expected TA,max decreases withincreasing depths due to the lower heat impact below the surface than the top of the material. Thecourse of TA,max(x) determines the volume fraction of austenite and degree of austenite homogen-ization. Deep below the surface the base material remains unaffected by the laser impact.

The developing amount of austenite fA(t) at different depths during heating and the resultinglevels of bainite fB(t), martensite fM(t), and ferrite=pearlite fF(t) during the cooling process areillustrated in Figure 11.48 for the surface and some depths below the surface in the region of theTZ. With increasing distance to the surface the diffusion processes occurring are limited due todecreasing maximum temperature, and only a part of the base material transforms into austeniteand subsequently into martensite. Considering x¼ 0.72 mm the material consists of fM¼ 40%martensite and of fF¼ 50% untransformed base material.

1200 T(t)-course at the surface T(t)-course at the surface

TA,max = 1150�Cvheat = 1000 K/svcool = 1000 K/s

TA,max = 1150�Cvheat = 1000 K/svcool = 1000 K/s

x = 0 mmx = 0.36 mmx = 0.69 mmx = 0.72 mmx = 0.81 mm

1000

800

600

400

200

0

1200

1100

1000

900

800

700

6000 0.2 0.4 0.6

Depth below the surface z (mm)0.8 1.0 1.2 1.40

(A) (B)1 2 3

AISI 4140Spot hardening


Time t (s)

Tem

pera

ture

T (�

C)

Max

. tem

pera

ture

Tm

ax (�

C)

4 5 6

FIGURE 11.47 Temperature cycles at various depth below the surface (A) and maximum temperature depthbelow the surface (B) during laser surface hardening with a heating and cooling rate of 1000 K=s. (FromMiokovi�c, T., Schulze, V., Vohringer, O., and Lohe, D., Mater. Sci. Eng. A, 435–436, 547, 2006.)


100

80

T(t)-course at the surface


T(t)-course at the surfaceTA,max = 1150�Cvheat = 1000 K�svcool = 1000 K�s

TA,max = 1150�Cvheat = 1000 K�svcool = 1000 K�s

TA,max = 1150�Cvheat = 1000 K�svcool = 1000 K�s

60

40

Aus

teni

te a

f (vo

l%)

Bain

ite b

f (vo

l%)

20

00

(A) (B)

(C) (D)

2 4Time t (s)

6 8




x = 0 mmx = 0.69 mmx = 0.72 mmx = 0.81 mm

x = 0 mmx = 0.69 mmx = 0.72 mmx = 0.81 mm

x = 0 mmx = 0.69 mmx = 0.72 mmx = 0.81 mm

10

100

80

60

40

Mar

tenz

ite m

f (vo

l%)

20

00 2 4

Time t (s)6 8 10

T(t)-course at the surfaceTA,max = 1150�C

vheat = 1000 K�svcool = 1000 K�s


x = 0 mmx = 0.69 mmx = 0.72 mmx = 0.81 mm

100

80

60

40

Ferr

ite +

carb

ides

fcf (

vol%

)

20

00 2 4

Time t (s)6 8 10

0 2

0.5

0.4

0.3

0.2

0.1

04Time t (s)

6 8 10

FIGURE 11.48 Volume fraction of (A) austenite, (B) bainite (C) martensite, and (D) ferrite versus timeduring laser surface hardening with a heating and cooling rate 100 K=s. (From Miokovi�c, T., Schulze, V.,Vohringer, O., and Lohe, D., Mater. Sci. Eng. A, 435–436, 547, 2006.)

In order to investigate the influence of the process parameters on time-dependent temperaturefields and phase transformations within the affected zone, modeling of the laser hardening processwas performed applying different heating and cooling rates at the surface of the model. Figure 11.49illustrates the simulated temperature field TA,max versus distance to the surface for laser surfacehardening with heating rates between nheat¼ 1000 and 10,000 K=s at cooling rates of ncool¼ 1000and 3000 K=s. Owing to the shorter effective heating time due to rapid heating and cooling, the drop

1200

1100

1000

900

800

700

600

5000

(A) (B)0.2



0.4

T(t)-course at the surface T(t)-course at the surface

0.6Depth below the surface z (mm)

Max

. tem

pera

ture

Tm

ax (�

C)

1200

1100

1000

900

800

700

600

500Max

. tem

pera

ture

Tm

ax (�

C)

0.8 1.0 1.2 0 0.2 0.4 0.6Depth below the surface z (mm)

0.8 1.0 1.2

TA,max = 1150�Cvcool = 1000 K�svheat = 1,000 K/svheat = 6,000 K/svheat = 10,000 K/s

TA,max = 1150�Cvcool = 3000 K�s

vheat = 3,000 K/svheat = 6,000 K/svheat = 10,000 K/s

FIGURE 11.49 Maximum temperature versus depth below during laser surface hardening with variousheating and cooling rates (A) vcool¼ 1000 K=s and (B) vcool¼ 3000 K=s. (From Miokovi�c, T., Schulze, V.,Vohringer, O., and Lohe, D., Mater. Sci. Eng. A, 435–436, 547, 2006.)


0.5

0.4

0.3

0.2

0.1

00 2,000

Dep

th z

(mm

)

4,000



vcool = 1,000 K/svcool = 3,000 K/s

6,000Heating rate vheat (K/s)

8,000

TA,max = 1,150�C

10,000

FIGURE 11.50 Depth z in which homogeneous martensite is expected versus in the various applied heatingand cooling rate of vcool¼ 1000 K=s and 3000 K=s. (From Miokovi�c, T., Schulze, V., Vohringer, O., and Lohe,D., Mater. Sci. Eng. A, 435–436, 547, 2006.)

in the maximum temperature with increasing distance to the surface is higher, the higher the heatingand cooling rate are.

Owing to undissolved carbides during austenizing, the austenite formed shows a lower carbonconcentration. The lower carbon content of the austenitic matrix leads to an increase in themartensite transformation temperatures compared to equilibrium conditions. From the course ofTMs depths Xhom below the surface are determined as an estimation of the size of the surface area inwhich the mean carbon content in the austenite becomes CM¼ 0.42%, neglecting the small amountof carbon in the remaining special carbides. Therefore, the austenite formed is homogeneous duringheating. In this region, after subsequent quenching a homogeneous martensitic microstructure isexpected with special carbide. In Figure 11.50, these depths Xhom are plotted versus the appliedheating rate at the surface for laser surface hardening with heating and cooling rates to a depth ofXhom¼ 0.42 mm. With an increase in heating rate of the transformation temperatures for thebeginning and end of austenite formation to higher temperatures, the area with homogeneousaustenite and martensite is shifted toward lower depths, the higher the heating and cooling rate.For laser surface hardening with extremely high rates of heating and cooling of nheat¼ 10,000 K=s atncool¼ 3000 K=s, homogeneous martensite formation appears only down to Xhom¼ 0.05 mm.The evaluation of the depths Xhom (nheat) from the course of TMf leads to comparable results.

The depths Xhom evaluated from the results of simulation show a fine agreement with themicrostructural investigations presented and validated by the model.

Consider mainly the martensitic transformation during cooling assuming that carbon diffusion isdominant in Figure 11.51, which illustrates the calculated final hardness H and its differentconstituents using the given equations. The microhardness profiles are fairly similar for each caseof heating and cooling rate. As expected the course of the hardness of the martensitic matrix HM

has the highest influence on the final hardness. HM increases with its distance below the surface andis greatest when the carbon content in martensite corresponds to the carbon content of the steelof �Cst¼ 0.42 wt%. On the other hand, the hardness of the dissolving ferrous and mixed carbidesH(Fe,M)-carbides decreases toward the surface due to proceeding carbide dissolving during theaustenizing process.

As seen in Figure 11.51 the comparison of microhardness profiles determined for experimentand simulation shows a fine agreement for all heating and cooling rates.


800T(t)-course at the surfaceAISI 4140spot

hardeningspot

hardening

TA,max = 1150�C

fM HM

HExperiment

fα Hα

H(Fe,M)-carbidesHundissol.carbides

fM HM

HExperiment

fα Hα

H(Fe,M) - carbidesHundissol.carbides

vheat = 1000 K/svcool = 1000 K/s

TA,max = 1150�Cvheat = 3000 K/svcool = 3000 K/s600

400

Mic

roha

rdne

ss H

V

200

00

(A) (B)0.5 1.0Depth below the surface z (mm)

AISI 4140T(t)-course at the surface

1.5 2.0 2.5

800

600

400

Mic

roha

rdne

ss H

V

200

00 0.5 1.0

Depth below the surface z (mm)1.5 2.0

FIGURE 11.51 Microhardness and the contributions the different constituents versus distance below thesurface after laser surface hardening and cooling (A) vheat¼ vcool¼ 1000 K=s and (B) vheat¼ vcool¼ 3000 K=s.(From Miokovi�c, T., Schulze, V., Vohringer, O., and Lohe, D., Mater. Sci. Eng. A, 435–436, 547, 2006.)

11.4.4 MATHEMATICAL PREDICTION OF HARDENED DEPTH

The classical approach to modeling the heat flow induced by a distributed heat source moving overthe surface of a semi-infinite solid starts with the solution for a point source, with integrations overthe beam area [63,69]. This method requires numerical procedures for its evaluation. Thesesolutions are rigorous, so their computations are complex and the results difficult to be applied.Bass [64] gives an alternative approach by presenting temperature field equations for variousstructural beam modes. Analytical results show good response to various materials. Ashby et al.[56] and Li et al. [61] developed a further analytical approach and developed an approximatesolution for the entire temperature field. Comparison of the analytical results with numericalcalculations shows adequate description of laser transformation hardening. The variability of lasertransformation hardening parameters and changing material properties with temperature result insome scatter. With the use of dimensionless parameters to simplify computation, results applicableto all materials are obtained. There exist many such examples in the analysis of welding [65] and inlaser surface treatment [66–68].

In mathematical modeling the following assumptions were considered:

. Surface absorptivity A is constant.

. Latent heat of the a to g transformation is negligible.

. Thermal conductivity l and thermal diffusivity a of steel are constants.

. Eutectioid temperature TA1 is as given by the phase diagram.

. Radius of Gaussian beam rB is the distance from the beam center to the position where theintensity is 1=e times the peak value.

The origin of the coordinate system is the beam center. The laser of total power P moves in thex-direction with traveling speed v, with the y-axis across the track and z-axis in the distance belowthe surface.

The temperature field equation from Ashby and Easterling [56] is valid for the Gaussianline source:

T � T0 ¼ Aq

2plv[t(t þ t0)]1=2� exp� 1

4a(zþ z0)

2

tþ y2

(t þ t0)

, (11:35)


whereq[w] is the beam powern[ms�1] is the traveling speedt[s] is the timet0[s] is the heat flow time constantl[Jm�1 s�1 K�1] is the thermal conductivity of the steela[m2 s�1] is the thermal diffusivity

The equation contains two reference parameters, defined by t0 ¼ r2B=4a and z0, which is acharacteristic length, as a function to limit the surface temperature.

Shercliff et al. [67] defined the following dimensionless parameters:

T*¼ (T – T0)=(TA1 – T0) is the dimensionless temperature rise.q*¼Aq=rBl (TA1 – T0) is the dimensionless beam power.v*¼ vrB=a is the dimensionless traveling speed.t*¼ t=t0 is the dimensionless time.(x*, y*, z*)¼ (x=rB, y=rB, z=rB) are dimensionless x, y, z coordinates.

whereT* is the normalized temperatureTA1 is the eutectoid temperature of the steelT is the temperaturerB is the radius of a Gaussian beamA is the surface absorptivityx*, y*, z* are normalized x, y, z coordinates

The distance z0 is normalized as follows:

z*0 ¼ z0=rB and the dimensionless temperature parameter is then

T* ¼ (2=p)(q*=v*)

[t*(t*þ 1)]1=2exp� (z*þ z0*)2

t*þ y*2

(t*þ 1)

: (11:36)

The time to peak temperature tp* at a (x*, y*, z*) position is found by differentiating with respectto time:

tp* ¼ 14

2(z*þ z0*)2 � 1þ ½4(z*þ z0*)

4 þ 12(z*þ z0*)2 þ 1�1=2

h i(11:37)

At a stationary laser beam of uniform intensity Q produces a peak surface temperature given byBass [28]:

Tp � T0 ¼ 2A Q

p1=2l(at)1=2: (11:38)

wheret is the interaction timeQ is the average intensity of the laser beam


The average intensity of a Gaussian beam is Q ¼ q=pr2B, so the previous equation can be rewritten:

Tp � T0 ¼ 2Aq

p3=2 r2Bl(at)1=2 (11:39)

or in dimensionless form t is 2rB=v and

(Tp* )z* ¼ 0 ¼ (2=p)3=2 q*=(v*)1=2: (11:40)

The conditions for the first hardening Tp¼ TA1 or the onset of melt Tp¼ Tm are thus defined by aconstant value of the process variables as follows:

q*=Tp*(v*)1=2 ¼ (p=2)3=2 ¼ constant (11:41)

with Tp* taking the value appropriate to the peak temperature of interest TA1 or Tm.Bass [64] gives a more general solution for the peak surface temperature at a stationary

Gaussian beam acting in dimensionless form:

(Tp*)z* ¼ 0 ¼ (1=p)3=2q* tan�1 (8=v*)1=2: (11:42)

A constant value of a single dimensionless parameter defines the first hardening and the onset ofmelt as follows for all v*:

(q*=Tp*) tan�1 (8=v*)1=2 ¼ p3=2 ¼ constant: (11:43)

Four dimensionless parameters define laser hardening with a Gaussian beam. The aim is to producea diagram from which process variables may be readily selected. A convenient plot is the dependentvariables zc*, v*, q*, and Tp* ¼ 1, which give Tp¼ TA1 as shown in Figure 11.52. As surface meltingis generally undesirable, the contours are dashed if the surface melts

Tp* ¼ (Tm � T0)=(TA1 � T0) at z* ¼ 0: (11:44)

10

10−1

10−2

10−2 10−1 1 10

Material: carbon steel 0.4%

Dimensionless traveling speed v∗

Tp∗ = 1,0 Tp∗ = 2.12

z∗ = 0Tp = TmTp = TA1

q∗ = 4 5 8 15 30

Onset ofsurfacemelting

Dim

ensio

nles

s dep

th Z

∗ c

102 10310−3

1

FIGURE 11.52 Dimensionless hardened depth ZC* against laser beam traveling speed v* with curvesof constant laser-beam power q* for Gaussian power density cross section. (From HR Shercliff and MFAsby. The prediction of case depth in laser transformation hardening. Metallurgical Transactions A, 22A,1991, 2459–2466.)


Figure 11.52 shows the depth at which surface melt commences for a 0.4 pct carbon steel atTp* � 2:12.

A non-Gaussian source may be simulated by superposing a number of Gaussian sources.The total power is shared between the Gaussian sources, to give the best fit to true energy profiles.For a particular location and given time the temperature rises due to each source contributing toheating. This is valid if the thermal properties of the material are independent of the temperature,meaning that the differential heat flow according to the equation is linear. This method will bepresented by the same authors for laser-beam heating with rectangular sources defined as laser-beamspot ratio.

A laser-beam ratio R is defined by length l in travel direction x and the width w, of track in crossdirection y (R¼ l=w). In practice, fixed width is normally used and the length is varied, so it issensible to normalize the process variables using the beam width as follows:

qR* ¼ Aq=wl(TA1 � T0): (11:45)

vR* ¼ vw=a, (11:46)

zR* ¼ z=w, (11:47)

where the subscript R refers to a rectangular laser beam.Figure 11.53 shows a dimensionless diagram for medium carbon steel according to various ratio

power to track width q=w (W=mm) and spot ratio R¼ l=w (dimensionless).The diagram shows the nondimensional ratio of the hardened-layer depth to the laser-spot width

for a rectangular laser beam as a function of a product of the traveling speed and the laser-tracelength. Three characteristic power densities per unit of laser-spot width, i.e., q=w¼ 50, 200, and800 W=mm, were chosen, which equals a ratio of 1 to 4 to 16.

1

10−1

10−2

Power

2.0

0.2

1.0 2.0

2.0

0.2

I/w

1.0

Onset of surface melting

0.2

1.0

I/w

I/w

(W/mm)

Track widthq/w = 50 200

First hardening

80010−3

10−4

Case

dep

th/tr

ack

wid

th Z

c/w

1 10 102

Speed � spot length vl (mm2/s)103 104

FIGURE 11.53 Dimensionless hardened depth ZC=w against laser beam traveling speed parameter vl withcurves of q=w¼ constant for rectangular power density and spot ratio l=w¼ constant. (From HR Shercliff andMF Asby. The prediction of case depth in laser transformation hardening. Metallurgical Transactions A, 22A,1991, 2459–2466.)


In the diagram there are curves plotted for individual power densities q=w valid with certainratios of the laser-spot dimensions of the rectangular beam, i.e., l=w¼ 0.2, 1.0, and 2.0. In individualcases laser-hardening conditions and boundary conditions of laser hardening, i.e., laser remelting,were known too. The individual curves in the diagram indicate that by increasing the laser-beampower density per unit of laser-spot width the boundary conditions will be achieved with smallerdepths of the hardened layer. The diagram shown is of general validity and permits the determin-ation, i.e., prediction, of the hardened-layer depth based on the variation of the power density perunit of laser-spot width and the traveling speed.

It can be summarized that the approximate heat flow model of Ashby and Easterling [56] forlaser transformation hardening has been presented, describing Gaussian and non-Gaussian sourcesover a wide range of process variables.

The following advantages can be noted:

. Simplification by choosing dimensionless parameters

. Surface temperature calibration, extending the approximate Gaussian solution to all laser-beam traveling speeds

. High traveling speed solution was found to be acceptable for rectangular, uniform sources

. General Gaussian solution and enabling extension of the model to non-Gaussian sources

. Identification of constant dimensionless parameters containing all of the process variablesfor both sources that determine the position of the first hardening or the onset of melting

. Process diagrams for rectangular sources allow the choice of the process variables such asthe beam power, track width, traveling speed, and spot dimensions

11.4.4.1 Mathematical Modeling for Microstructural Changes

Ashby et al. [56] presented their results as laser-processing graphs that show the microstructures andhardnesses for process variables. In their experiments, they treated two steels, i.e., a Nb micro-alloyed and a medium-carbon steel. They varied the laser power P, the beam spot radius rB, and itstraveling speed v.

They carried out laser surface heat treatments using a 0.5 and a 2.5 kW, CW CO2 laser usingGaussian and top hat energy profiles.

The microhardness of individual martensitic and ferritic regions in the low-carbon steel could beestimated fairly well from a simple rule-of-mixtures

HV(mean) ¼ fmHm þ (1� fm)Hf , (11:48)

whereHm is the mean microhardness of the martensiteHf is the mean microhardness of the ferrite at that depth

Figure 11.54 shows the hardness profiles for the two steels with different energy densities q=vrBand interaction time rB=v. It can be noted that worse hardening is obtained when the carbon contentof the steel is low.

For the high-carbon steel, complete carbon redistribution occurs within the austenitizationprocess, which gives a uniform high hardness. They developed simple models for pearlite dissol-ution, austenite homogenization, and martensite formation. The heat cycle T(t) at the depth causesmicrostructural changes if high enough temperature is achieved.

Some of the microstructure changes are diffusion controlled, such as the transformation ofpearlite to austenite and the homogenization of carbon in austenite.


1,000

800

600

400

200

00 0.2 0.4

Depth z (mm)

Mic

roha

rdne

ss H

Vm

0.6

0.6 wt % C steel

0.1 wt % C steel

0.8

qv rB

= 16MJ/m2

qv rB

=19MJ/m2

qv rB

= 14MJ/m2

qv rB

= 17MJ/m2

FIGURE 11.54 Hardness profiles for the two steels with different energy densities, E¼ q=vrB and interactiontime, ti¼ rB=v. (From Asby, M.F. and Easterling, K.E., Acta Metall., 32, 1935, 1984.)

The microstructure changes depend on the total number of diffusive jumps that occur during thetemperature cycle. It is measured by the kinetic strength I of the temperature cycle, defined by

I ¼ð10

exp� QA

RT(t)dt, (11:49)

whereQA is the activation energy for the typical microstructural transformationR is the gas constant

This can be conveniently represented as

I ¼ at exp� QA

RTp, (11:50)

whereTp is the maximum temperaturet is the thermal constant

The constant a is well approximated by

a ¼ffiffiffiffiffiffiffiffiRTpQA

3

r: (11:51)

At rapid heating, the pearlite first transforms to austenite, which is followed by carbon diffusionoutward, and increases the volume fraction of high-carbon austenite.

If the cementite and ferrite plate spacing within a colony distance is l, it might be thought thatlateral diffusion of carbon to austenite occurs. In an isothermal heat treatment, this would require atime t given by l2¼ 2Dt

l2 ¼ 2D0at exp� QA

RTp, (11:52)


where D is the diffusion coefficient for carbon. In a temperature cycle T(t) the quantity Dt is wherethe maximum temperature required to cause the transformation is Tp.

The modeling of the carbon redistribution in austenite is most important to the understanding oflaser transformation hardening.

When a hypoeutectoid, plain-carbon steel with carbon content C is heated above the TA1temperature, the pearlite transforms instantaneously to austenite. The pearlite transforms to austenitecontaining Ce¼ 0.8% carbon and the ferrite becomes austenite with negligible carbon content Cf.Thereafter, the carbon diffuses from the high to the low concentration regions, to an extent thatdepends on temperature and time. On subsequent cooling of the steel with a carbon content greaterthan a critical value Cc 0.05 wt%, it transforms to martensite and the rest reverts to ferrite.

The volume fraction occupied by the pearlite colonies is equal

fi ¼ C � Cf

0:8� Cf

� C

0:8, (11:53)

where cf is the carbon content of the ferrite.The volume fraction of the martensite is as follows [29]:

f ¼ fm � ( fm � fi) exp� 12f 2=3iffiffiffiffiffiffipg

p lnCe

2Cc

� � ffiffiffiffiffiDt

p" #, (11:54)

where g is the mean grain size (m).The hardness of the transformed surface layer depends on the volume fraction of martensite and

its carbon content. The authors [56] calculated the hardness of the martensite and ferrite mixture byusing a rule of mixtures.

H ¼ fHm þ (1� f )Hf (11:55)

and suggested the following formula for calculating hardness:

H ¼ 1667C � 926C2=f þ 150: (11:56)

Figure 11.55 shows three measured hardness profiles for different energy parameters (solid lines)and the calculated profiles (dashed line).

9000.6 wt% C steelq = 0.45 kW, rB = 2 mm

800

700

600Mic

roha

rdne

ss H

Vm

5000 0.1

q51

MJ/m2 MJ/m266vrB

0.2Distance from surface z (mm)

Measured

Legend

Calculated

Input energyparameter

0.3 0.4

qvrB

= 37 MJ/m2

FIGURE 11.55 Measured and calculated hardness profiles for the 0.6 wt% carbon steel compared with thosepredicted. (From Asby, M.F. and Easterling, K.E., Acta Metall., 32, 1935, 1984.)


10−3

10−2

10−1

1

1010 20

A = 0.7g = 50 μm

Surface melting

Dep

th z

(mm

)

.77.82 .87 .92 .98 = fm

.715 744 770 790 810 = HV

Hardenedzone

TA1

TA3

30

0.6 % C steel

50 100

Energy density q/vrB (MJ/m2)

200 300 500 1000Beam radius r B (m

m)

2

4

8

q = 1 kW

FIGURE 11.56 Laser processing diagram for 0.6 carbon steel. (From Asby, M.F. and Easterling, K.E., ActaMetall., 32, 1935, 1984.)

Figure 11.56 shows a laser-processing diagram for a 0.6 wt% plain carbon steel. The horizontalaxes present energy density q=vrB and beam spot radius rB. These variables determine the tempera-ture cycle in the transformed layer. The vertical axis is the depth below the surface.

Within the shaded region, melting occurs outside the transformation hardening process.The diagram also shows contours of martensite volume fraction.

The volume fraction and carbon content of the martensite are used to calculate the hardness HVafter laser surface transformation hardening.

The author [56] showed that

. Steels with a carbon content below about 0.1 wt% do not respond to transformation hardening

. Optimum combination of process variables gives maximum surface hardness withoutsurface melting

The method could be used for laser glazing and laser surface alloying.

11.4.5 METHOD FOR CALCULATING TEMPERATURE CYCLE

Several methods exist to solve the heat conduction equations for various conditions. Interestingdescriptions are given by Carslaw and Jaeger [63,69]. Most of the computing methods to calculatetemperature cycles are based on one of the many cases that are modified to suit the particular case [70].

Gregson [71] discussed a one-dimensional (1D) model using a semi-infinite flat-plate solutionfor idealized uniform heat source, which is constant in time. Expressions used for the temperatureprofile separated for heating and cooling are as follows.

Heating temperature–time profile

T(z, t) ¼ «zQAV

l

ffiffiffiffiffiat

p � ierfc z

2ffiffiffiffixt

p� �

, (11:57)


Q(t) ¼ Q for t > 00 for t < 0

� �: (11:58)

Cooling temperature–time profile

T(z, t) ¼ 2QAV

ffiffiffia

pl

ffiffit

p � ierfc z

2ffiffiffiffiat

p � ffiffiffiffiffiffiffiffiffiffiffit � tL

p � ierfc z

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia(t � tL)

p� ��

, (11:59)

f (t) ¼ Q for 0 < t < tL0 for 0 > t > tL

� �, (11:60)

whereT is the temperature (8C)z is the depth below the surface (cm)t is the time (s)« ffi 1 is the emissivityQAV is the average power density (W=cm2)l is the thermal conductivity (W=cm 8C)a is the thermal diffusivity (cm2=s)to is the time start for power on (s)tL is the time for power off (s)ierfc is the integral of the complementary error function

These equations for description of laser heating and cooling process are valid if the thickness ofthe base material is greater than t ffiffiffiffiffiffiffi

4atp

and they could be approximately described for thehardened layer.

These 1D analyses may be applied to a laser transformation hardening process with idealizeduniform heat sources, which are produced by using optical systems such as laser-beam integrator orhigh-power multimode laser beam with a top-hat power density profile. These equations present 1Dsolutions and provide only an approximate temperature time profile. For better description ofthermal conditions, a 2D or 3D analysis considering actual input power density distribution andvariable thermophysical properties treated material are required.

Sandven [72] presented the model that predicts the temperature time profile near a moving ring-shaped laser spot around the periphery of the outer or inner surface of a cylinder. This solution canbe applied to the transformation hardening processes using toric mirrors.

Sandven [72] developed his model based on a flat-plate solution and assumed that the tempera-ture–time profile T (t) for cylindrical bodies can be approximated by

T ¼ uI, (11:61)

whereu depends on workpiece geometryI is the analytical solution for a flat plate

The final expression for a cylindrical workpiece that is derived from this analysis is

T � 1� 0:43ffiffiffiffif

p� � 2Qoa

p�K �vðxþB

x�B

eu � Ko(z2 þ u2)1=2du, (11:62)


whereþ sign means the heat flow into a cylinder� sign means the heat flow out of a hollow cylinderQo is the power densityv is the laser-beam traveling speed in the x-directionKo is the modified Bessel function of the second kind, and 0 orderu is the integration variable2b is the width of the heat source in the direction of motionz is the depth in radial direction

B ¼ vb2a

, Z ¼ vz2a

, X ¼ vx2a

,

f ¼ at=R2,(11:63)

where R is the radius of the cylinder.Sandven [72] provided graphical solutions for Z¼ 0 for various values of B. To estimate an

approximate depth of hardness, the maximum temperature profile across the surface layer is the onlyitem to be interested.

Cline and Anthony [58] presented a most realistic thermal analysis for laser heating. They used aGaussian heat distribution and determined the 3D temperature distribution by solving the equation:

@T=@t � ar2T ¼ Q=Cp, (11:64)

whereQ is the power absorbed per unit volumeCp is the specific heat per unit volume

They used a coordinate system fixed at the workpiece surface and superimposed the knownGreen function solution for the heat distribution. The following temperature distribution is

T(x, y, z) ¼ P(CparB)�1f (x, y, z, v), (11:65)

where f is the distribution function.

f ¼ð10

exp (�H)

(2p3)1=2(1þ m2)dm and (11:66)

H ¼X þ tm2

2

� �2þ Y2

2(1þ m2)þ Z2

2m2,

m2 ¼ 2at0=rB; t ¼ vrB=a

(11:67)

wherem2¼ 2at0=rB, t¼ vrB=aX¼ x=rB, Y¼ y=rB, Z¼ z=rBP is the total powerrB is the laser-beam radiust0 is the earlier time when laser was at (x0, y0)v is the traveling speed

The cooling rate can be calculated as follows:

@T=@t ¼ �v x=g2 þ v=2a(1þ x=g) �

T , (11:68)

where g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

p.

The given cooling rate is calculated only when point heat source is used.


This 3D model is a great improvement over 1D models because it includes temperature-dependent thermophysical properties of the material used for numerical solutions.

Grum et al. [73] obtained a relatively simple mathematical model describing the temperatureevolution T (z, t) in the material depending on time and position, where the heating cycle and thecooling cycle are distinguished between.

For reasons of simplifying the numerical calculations, it is necessary tomake certain assumptions:

. Latent heat of material melting is neglected.

. Material is homogeneous with constant physical properties in the solid and liquid phase.Therefore, it is assumed that material density, thermal conductivity, and specific heat areindependent of temperature.

. Thermal energy is transferred only through transfer into the material; thermal radiation andtransfer into the environment are disregarded.

. Laser light absorption coefficient to workpiece material is constant.

. Limiting temperatures or transformation temperatures are assumed from phase diagrams.

. Remelted surface remains flat and ensures a uniform heat input.

Thus, a relatively simple mathematical model can be obtained describing the temperature evolutionT(z, t) in the material depending on time and position, where the heating cycle and the cooling cycleare distinguished between.

1. The heating cycle conditions in the material can be described by the equation:

T(z, t)¼ T0þ A �P2 �p �l � vB �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � (tiþ t0)

p � e�(zþz0 )

2

4�a�t� �

þ e�(z�z0 )

2

4�a�t� �

� erfc zþ z0ffiffiffiffiffiffiffiffiffiffiffiffiffi4 �a � tp

� �(11:69)

� 2008 b

for 0< t< ti.

2. The cooling cycle conditions in the material can be expressed by the equation:

T(z, t)¼ T0þ A �P2 �p �l � vB �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � (tiþ t0)

p � e�(zþz0 )

2

4�a�t� �

þe�(z�z0 )

2

4�a�t� �

�e� (z�z0 )

2

4�a�(t-ti )� �" #

� erfc zþ z0ffiffiffiffiffiffiffiffiffiffiffiffi4 �a � tp

� �(11:70)

for t> ti.

In upper equations, the variable t0 represents the time necessary for heat to diffuse over adistance equal to the laser-beam radius on the workpiece surface and the variable z0 measures thedistance over which heat can diffuse during the laser-beam interaction time [56]. C is a constant, inour case defined as C¼ 0.5.

Figure 11.57A presents the time evolution of temperatures calculated according to equationsat a specific depth of the material in the nodular iron 400-12 at a laser-beam traveling speedvB¼ 12 mm=s. Figure 11.57B illustrates the variation of heating and cooling rates during theprocess of laser remelting in the remelted layer and in deeper layers of the material.

The temperature gradient is at the beginning of laser-beam interaction with the workpiecematerial, i.e., on heating up very high, on the surface achieving values as high as 48,0008C=s.The results show that the highest cooling rate is achieved after the beam has passed by half the valueof its radius rB across the measured point.

Knowing the melting and austenitization temperatures, the depth of the remelted and modifiedlayer can be successfully predicted (Figure 11.58). Considering the fact that on the basis of limitingtemperatures it is possible to define the depth of particular layers and that these can be confirmed bymicrostructure analysis, the success of the proposed mathematical model for the prediction ofremelting conditions can be verified.

y Taylor & Francis Group, LLC.

1600 60,000

40,000

20,000

0

−20,000

−40,000

140012001000

800

Tem

pera

ture

T (�

C)

600400200

00 0.1 0.2

2.0

10.50.3

0.15

1.0

0.50.3

0.15z = 0 mm

z = 0 mmNodular iron 400–12 Nodular iron 400–12TMelting = 1190�C

TAustenitization = 810�C

0.3Time t (s) (B)(A) Time t (s)

0.4 0 0.1 0.2 0.3 0.4

Cool

ing

rate

(�C

/s)

ΔT Δt

FIGURE 11.57 Temperature cycles and cooling rate versus time at various depths. (From Grum, J. andŠturm, R., Calculation of temperature cycles heating and quenching rates during laser melt - Hardening of castiron. In: LAJL Sarton and HB Zeedijk, Eds. Proceedings of the 5th European Conference on AdvancedMaterials and Processes and Applications, Materials, Functionality & Design, Vol. 3. Surface Engineering andFunctional Materials. Maastricht, The Netherlands 1997, 3=155–3=159.)

1600Nodular iron 400-12

Tmelting = 1190�C

Taustenitization = 810�C

V = 2mm/s

12mm/s24mm/s36mm/s

Modifieddepth

Remelteddepth

1400

1200

1000

Tem

pera

ture

T (�

C)

800

600

400

200

00 0.2 0.4 0.6 0.8 1

Depth z (mm)1.2 1.4 1.6 1.8 2

FIGURE 11.58 Maximum temperature drop as a function of depth in nodular iron 400–12. (From Grum, J.and Šturm, R., Calculation of temperature cycles heating and quenching rates during laser melt-Hardening ofcast iron. In: LAJL Sarton and HB Zeedijk, Eds., Proceedings of the 5th European Conference on AdvancedMaterials and Processes and Applications, Materials, Functionality & Design, Vol. 3. Surface Engineering andFunctional Materials. Maastricht, The Netherlands 1997, 3=155–3=159.)

Thus, a comparison is made in Figure 11.59 between the experimentally obtained results for thedepth of particular zones of the modified layer and the results calculated according to the math-ematical model.

As can be seen, the calculated depths of the remelted and HZs correlate well with the experi-mentally measured values. Too big deviations in the depth of the modified layers are found only ongray iron at very low workpiece traveling speeds and they are probably due to the occurrenceof furrows on the workpiece surface.

11.4.6 HEAT FLOW MODEL

Kou et al presented [74] a theoretical and experimental study of heat flow and solid-state phasetransformations during the laser surface hardening of 1018 steel. In the theoretical part of the study,


1.61.41.2

10.8

Dep

th z

(mm

)

0.60.40.2

00 5 10 15 20 25

Traveling speed vB (mm/s)

Modified measuredModified calculated

Remelted measuredRemelted calculated

30 35 40 45

1.4Gray iron grade 200 Nodular iron 400–121.2

10.8

Dep

th z

(mm

)

0.60.40.2

00 5 10 15 20 25

Traveling speed vB (mm/s)

Modified measuredModified calculated

Remelted measuredRemelted calculated

30 35 40 45

FIGURE 11.59 Comparison of experimentally measured remelted and HZ depths with those calculated withthe mathematical model, for gray iron grade 200 and nodular iron 400-12. (From Grum, J. and R Šturm, R.,Calculation of temperature cycles heating and quenching rates during laser melt-Hardening of cast iron.In: LAJL Sarton and HB Zeedijk, Eds. Proceedings of the 5th European Conference on Advanced Materialsand Processes and Applications, Materials, Functionality & Design, Vol. 3. Surface Engineering andFunctional Materials. Maastricht, The Netherlands 1997, 3=155–3=159.)

a 3D heat flow model was developed using the finite difference method. The surface heat loss, thetemperature dependence of the surface absorptivity, and the temperature dependence of thermalproperties were considered. This heat flow model was verified with the analytical solution of Jaeger[63] and was used to provide general heat flow information, based on the assumptions of no surfaceheat loss, constant surface absorptivity, and constant thermal properties. The validity of each ofthese three assumptions was evaluated with the help of this heat flow model.

The energy balance equation, the boundary conditions, and the finite difference equation used inthe heat flow simulation are introduced in the following formulation of finite difference equation.The laser beam is stationary while the workpiece moves at a constant velocity v. The energy balanceequation for a stationary coordinate system (x–y–z) is given below:

@(rH)

@t¼ r � (lrT)� v

@(rH)

@x, (11:71)

wheret is the timeT is the temperatureH is the enthalpyl is the thermal conductivityr is the density of the workpiece material

The equation is generally applicable to any heat-conduction type problem in which the materialbeing considered is moving at a constant velocity v in x-direction. The left-hand side of the equationrepresents the rate of enthalpy change per unit volume. The first term on the right-hand side of thesame equation corresponds to heat transfer due to conduction, while the last term corresponds toheat transfer due to the motion of the material.

Integrating the equation over the volume element dxdydz and applying the divergence theoremto the first term on the right-hand side, the following integral equation is obtained:ð ð ð

@(rH)

@tdx dy dz ¼

þ ðlrT � d~S�

ð ð ðv@(rH)

@xdx dy dz, (11:72)

where~S is the total surface area of the volume element.


11.4.6.1 Dimensionless Groups

If all thermal properties are considered constant, both the energy balance equation and the boundaryconditions can be rewritten in dimensionless forms, and the results of heat flow calculations becomegenerally applicable to different materials as well as different operating conditions. The equationcan be written in the following dimensionless form:

ð ð ð@T*@F0

dx* dy* dz* ¼þ þ

r*T* � d~S*�ð ð ð

@T*@x*

dx* dy* dz* (11:73)

with

T* ¼ la(T � T0)=Qv as the dimensionless temperature

F0 ¼ tv2=a as the dimensionless time (11:74)

x* ¼ xv=a

y* ¼ yv=a

z* ¼ zv=a

9>=>; as the dimensionless distances (11:75)

S* ¼ Sv2=a2 as the dimensionless area (11:76)

r* ¼ (a=v)r as the dimensionless gradient (11:77)

wherea is the thermal diffusivityQ is the power actually absorbed by the workpiece

Note that in the derivation of equation [75] the definition of dH¼Cp dT has been used, whereCp is the specific heat of the material.

With the help of the definition of the dimensionless variables shown above, the boundaryconditions can be rewritten in the following:

1: @T*=@y* ¼ 0 at y* ¼ 0 (11:78)

2: T* ¼ 0 ! lo(x*2 þ y*2 þ z*2)1=2 ! 1 (11:79)

3: � @T*=@z* ¼ h*=4a*b*,

if z* ¼ 0 and jx*j � a* and jy*j � b* (11:80)

4: � @T*=@z* ¼ Bi(T*� Ta*)

if z* ¼ 0 and jx*j > a* or jy*j > b* (11:81)

where

a* ¼ av=a is the dimensionless length of the laser beam

b* ¼ bv=a is the dimensionless width of the laser beam

h* ¼ h=h0 is the absorptivity ratio (11:82)

Bi ¼ heffa=kv is the Biot number

Ta* ¼ l�a(Ta � To)=Q � v

In the above equation h0, i.e., the overall absorptivity, is defined as Q¼h0Q.


1.00.5z∗/a∗ = z/a

00

5

10

T∗ �

103

15

a∗ = 10

15

20

25

30

50

FIGURE 11.60 Peak temperature, at y¼ 0, as a function of depth and the beam size. (From Kou, S., Sun,D.K., and Le, Y.P., Metall. Trans., 14A, 643, 1983.)

Figure 11.60 shows the distribution of the dimensionless peak temperature as a function ofdepth along the central plane of the workpiece. The heat source is a square laser beam of uniformenergy distribution. The relationship between dimensionless maximum workpiece temperatureTmax* (i.e., at y*¼ z*¼ 0) and the dimensionless size of the square laser beam a* can be determined.Figure 11.61A shows the plot of Tmax* versus a*�1.5; the maximum workpiece temperaturepredicted by 1D heat flow equation approaches that of 3D equation when a* is greaterthan about 20. As a* becomes smaller, Tmax* deviates further away from the value predicted byGreenwald’s equation.

As can be seen in Figure 11.61B when Tmax* is replotted against a*�1.4, the following simplerelation can be obtained:

Tmax* ¼ 0:293a*�1:4 (11:83)

The above equation works best when a* is between 5 and 50. For very high values of a*, however,Greenwald’s equation is preferred. Since melting is undesirable during transformation hardening,Tmax* can be used as a guide to make sure that the maximum workpiece temperature stays below itsmelting point Tm. In other words,

Tm* ¼ la(Tm � T0)

Qv¼ 0:293

av

a

� ��1:4(11:84)

On rearranging, the following is obtained:

Q

a1:4v0:4¼ cm, (11:85)


0 25 50 75 100(A)

0

10

20

30

T∗ m

ax �

103

T ∗max = ( )

*1.5−12πa

1D heat flow 3D heat flow

Limit of 1Dapproximation

a∗−1.5 � 103 (B)

0

10

20

30

T∗ m

ax �

103

T ∗max= 0.293a*−1.4

3D heat flow

0 25 50 75 100a∗−1.4 � 103

Simplified equation forpredicting maximum

temperature

(C)

Melting

Onset of meltin

g

No melting

Slope =k(Tm−T0)

0.293a∗0.4

0

Q

a1.4U0.4

FIGURE 11.61 Maximum workpiece temperature and the onset of surface melting. (A) Comparison betweenthe 1 D heat flow calculation and the 3D heat flow calculation (B) simplified equation for predicting T*max

(C) the onset of surface melting. (From Kou, S., Sun, D.K., and Le, Y.P., Metall. Trans., 14A, 643, 1983.)

1.00.5Max. depth of heat-affected zone (mm)

Transformation hardening

00

5

Onset of melting

ExperimentCalculation

1018 SteelTravel speed: 38 mm/sLaser beam size: 12 � 12 mm2

Melt

ing

10

Beam

pow

er Q

0 (KW

)

FIGURE 11.62 Experimental results of the laser transformation hardening of 1018 steel. The beam size andthe travel speed were kept constant. The slope of the melting curve is much higher than that of thetransformation hardening curve due to the heat of melting. The dashed line represents the calculated resultsbased on high temperature thermal properties and an absorptivity of 84 pct. (From Kou, S., Sun, D.K., and Le,Y.P., Metall. Trans., 14A, 643, 1983.)


where

cm ¼ k(Tm � T0)

0:293a0:4: (11:86)

Figure 11.61C shows a plot of Q versus a1.4 v0.4. For a given material, the coefficient cm is constant,and Figure 11.61C can be used as a simple guide to select proper operating conditions so that theproblem of surface melting can be avoided.

A plot of the beam power versus the maximum depth of the heat affected zone (HAZ) is shown inFigure 11.62 for a group of specimens scanned by the laser beam at a travel speed of 38 mm=s. Theonset of melting in one of the specimens was confirmed by the presence of numerous tinymelted spotson its surface. The slope of the melting curve is higher than that of the transformation hardening curvedue to the additional heat that is needed as the latent heat of melting [76–78].

Kou et al. [74] presented a theoretical and experimental study of heat flow and solid-state phasetransformation during laser surface hardening of steel. Similar calculations were made for thespecimen in which the onset of surface melting was observed. An absorptivity of 88.6 pct wasfound to best fit both the depth of the HAZ observed and the maximum temperature at the workpiecesurface, i.e., the melting point. The results are shown in Figure 11.63. The average heating rate at thebottom of the heat affected zone, i.e., 22008C=s, is much greater than 408C=s, again suggesting that7808C is the effective A1 temperature on heating. As shown in this figure, the bottom of the HAZ cooleddown from 7808C to 5008C in 0.41 s, again suggesting the formation of martensite. The microhardnessprofile of the HAZ is shown in Figure 11.64. As can be seen, the microhardness continues to increase as

1500

1000

Tem

pera

ture

T (�

C)

500

00 0.5

Time t (s)

0.34 s

0.41 s

0.44 s

at y = 0

1 mm(A)

(B)

780870

1.0

FIGURE 11.63 Calculated results for beam power 5.7 kW and traveling speed 38.1 mm=s. (A) Calculatedand the measured (hatched area) sizes of the HAZ and (B) calculated thermal cycles at the top and the bottomof the HAZ. The austenitization temperature of the corresponding CCT diagram is 8708C. (From Kou, S., Sun,D.K., and Le, Y.P., Metall. Trans., 14A, 643, 1983.)


1.51.0Depth z (mm)

0.500

200

400

Knoo

p ha

rdne

ss (5

00 g

) [kg

/mm

2 ]

FIGURE 11.64 Microhardness of the HAZ. (From Kou, S., Sun, D.K., and Le, Y.P., Metall. Trans., 14A,643, 1983.)

the top surface; however, the microhardness drops, perhaps due to the onset of surface melting. Themaximum microhardness is about 435 daN=mm2 (0.5 daN Knoop), which is slightly lower than thehardness of a 0.18 pct Cmartensite. Themicrostructure near the top surface of the specimen, as shown inFigure 11.65, is essentially martensite. A small amount of ferrite is also present, because the timeallowed for carbon atoms to diffuse in austenite was still not quite sufficient. However, such amicrostructure is essentially consistent with the calculated thermal cycle shown in Figure 11.63B,since the critical cooling time for the formation of martensite in 1018 steel is greater than 0.44 s.

FIGURE 11.65 Microstructure near the top of the HAZ. The Knoop hardness is 435 kg=mm2 (magnification370�). (From Kou, S., Sun, D.K., and Le, Y.P., Metall. Trans., 14A, 643, 1983.)


http://www.crcnetbase.com/action/showImage?doi=10.1201/9781420003581.ch11&iName=master.img-035.jpg&w=276&h=202

Experiment10

5Be

am p

ower

Q0 (

kW)

00 10 20

a1.4 V 0.4 � 105 (m1.8 s−0.4)

Calculation

Melting

Onset of melting

No melting

FIGURE 11.66 Onset of surface melting. The solid line represents experimentally observed onset of surfacemelting while the dashed line represents the calculated result based on high-temperature thermal properties andan absorptivity of 84 pct. (From Kou, S., Sun, D.K., and Le, Y.P., Metall. Trans., 14A, 643, 1983.)

Finally, the calculated results based on high-temperature thermal properties were compared withthe experimental results, as shown in Figures 11.62 and 11.66. The absorptivity used in thecalculation was 84 pct. As can be seen in these figures, the agreement between the experimentalresults and the calculated results based on constant absorptivity and high-temperature thermalproperties seems surprisingly good.

11.4.7 THERMAL ANALYSIS OF LASER HEATING AND MELTING MATERIALS

HE Cline and TR Antony [58] presented a thermal analysis for laser heating and remelting materialsfor a Gaussian source moving at the constant traveling speed. Heat flow for a rapidly moving high-powered laser is dominated by conduction in the solid, which is related to the thermal diffusivity Dand specific heat per unit volume Cp. A semi-infinite geometry is a reasonably good approximationif the laser beam is small compared to the object heat treated. In practice, the surface is made highlyabsorbing to the laser radiation by sand blasting and coating with colloidal graphite. The analysiscan be formulated in terms of the power absorbed at the surface P (which depends on the absorptionof the surface), which is smaller than the laser output power.

@T

@t� Dr2T ¼ Q

Cp(11:87)

wherel is the absorption depthh(z)¼ 1 for 0< z< l and h(z)¼ 0 for z> l

A moving Gaussian beam normalized to give a total power P for a spot radius R is of the form

Q ¼ Pexp � (x� vt)2 þ y2

�(2R2)�1

�2pR2

h(z)

l, (11:88)

The temperature distribution along the x-axis is calculated from the equation for differenttraveling speeds (Figure 11.67). As the traveling speed increases, the maximum temperature


0.18

0.16

0.14

0.12

0.10f

0.08

0.06

0.04

0.02

0−6 −4 −2 0

X/R

V = 4RD

2 4 6

V = 1RD

V = 0RD

FIGURE 11.67 Calculated temperature distribution function f along the x-axis, which is the path of thelaser beam for different scanning traveling speed. (From Cline, H.E. and Anthony, T.R., J. Appl. Phys., 48,3895, 1977.)

decreases and shifts behind the center of the moving laser, and at various depths below thesurface the temperature decreases (Figure 11.68). The temperature under the laser beam(Figure 11.69) decreases with increasing traveling speed because less time is available to heatthe material.

0.16

0.14

0.12

0.10

0.08f

0.06

0.04

0.02

0−3 −2 −1 0

X/R1 2 3

= 1ZR

= 0ZR

= 1RVD

= 2ZR

FIGURE 11.68 Temperature distribution f at different depths below the surface at a constant traveling speed.(From Cline, H.E. and Anthony, T.R., J. Appl. Phy., 48, 3895, 1977.)


0.20

0.18

0.16

0.14

0.12

0.10f

0.08

0.06

0.04

0.02

00 1 2 3 4 5 6 7

= 0ZR

ZR

ZR

ZR = 1

RVD

= 0.2

= 0.5

FIGURE 11.69 Effect of travel speed on the temperature distribution f at different depths below the surface.(From Cline, H.E. and Anthony, T.R., J. Appl. Phy., 48, 3895, 1977.)

By curve fitting, the calculated penetration curves (Figure 11.70) are approximated by a relation:

T ¼ T0 exp (�z=z0), (11:89)

whereT0 is the value of the temperature at the surface beneath the beamz0 is the depth parameter, approximately 10 R (0, 0, 0, v)

0.18

0.16

0.14

0.12

0.10

f

0.08

0.06

0.04

0.02

00 1 2

3.02.0

1.51.0

0.5

RVD = 0

105.0

Z/R

FIGURE 11.70 Penetration of the distribution function f into the material at different velocities. (From Cline,H.E. and Anthony, T.R., J. Appl. Phy., 48, 3895, 1977.)


11.4.7.1 Cooling Rate

The cooling rate is derived from the temperature distribution that moves in steady state with thelaser. The cooling rate is related to the thermal gradient in the direction of motion given by

@T

@t¼ �v

@T

@x, (11:90)

which is the term that relates a fixed coordinate system to a moving coordinate system. The gradientis found by differentiation of the equation with respect to x to yield an integral expression that mayonly be evaluated numerically.

However, far from the center of the beam compared to the spot size, the gradient approaches thatof a point source, and the cooling rate is found to be

@T

@t¼ � vP

CpD2p@

@x

exp [�v=2D(r þ x)]

r

� �: (11:91)

After differentiation with respect to x, the cooling rate becomes

@T

@t¼ �v

x

r2þ v

2D1þ x

r

� �h iT : (11:92)

A plot of the temperature distribution and cooling rate for 304-stainless steel with a 100 W pointsource moving at 0.5 cm=s shows that the cooling-rate distribution is quite similar to the temperaturedistribution (Figure 11.71). The cooling rate is zero at the temperature maximum:

@T=@x ¼ 0: (11:93)

1400

1200

1000

800

δT/δ

f (�C

/s)

600

400

200

0−1.0 −0.8 −0.6 −0.4

X (cm)(A)

0.20

0.15

0.10

0.05

Z (cm) = 0

304-Stainless steelv = 0.5 cm/sCp = 3.5 W/cm2 �C

D = 4 � 10−2 cm2/sP = 100 W

−0.2 0

304-Stainless steelv = 0.5 cm/sCp = 3.5 W/cm2 �C

D = 4 � 10−2 cm2/sP = 100 W

1400

1200

1000

800

Tem

pera

ture

T (�

C)

600

400

200

0−1.0 −0.8 −0.6 −0.4

X (cm)(B)−0.2 0

0.30

0.250.20

0.15

0.10

0.05Z (cm) = 0

FIGURE 11.71 Cooling rate and the temperature along the x-axis at different depths for 304-stainless steel.(From Cline, H.E. and Anthony, T.R., J. Appl. Phy., 48, 3895, 1977.)


11.4.7.2 Laser Melting

In heat treating a solid, the maximum temperature T0 was below the melting point Tm. In this sectionthe case where the maximum temperature exceeds the melting point but does not exceed the boilingpoint is considered. The scanning laser produces a weld puddle that moves with the beam to meltand subsequently resolidify material near the surface. Latent heat absorbed at the melting interface isliberated at the solidifying interface, which alters the temperature distribution somewhat but doesnot significantly affect the penetration depth. The effect of latent heat or the effect of differences inthermal conductivities between the liquid and solid is not considered. The solid–liquid isotherm isgiven by

Tm ¼ P CpDR� ��1

f (s, y, z, v), (11:94)

and the liquid penetrates to a depth

Zm ¼ Z0 ln P=Pmð Þ (11:95)

wherePm is the power that is absorbed just before melting occursz0¼ 10 f(0, 0, 0, v)R

These equations are used to estimate the traveling speed versus absorbed-power curves for differentpenetration depths in 304-stainless steel (Figure 11.72). The curves are similar in shape to the

20

18

16 0.02 c

m

0.015

0.01

0.00

5

14

12

Velo

city

(cm

/s)

10

8

6

4

2

00 40 80 120 160

Absorbed power (W)200 240 280

Z m =

0

304-Stainless steelR = 0.01 cmCp = 3.5 W/cm2 ºC

D = 4 � 10−2 cm2/s

FIGURE 11.72 Calculated relationship between velocity and power absorbed for different depths of pene-tration of the liquid zone using constants for 304-stainless steel. (From Cline, H.E. and Anthony, T.R., J. Appl.Phy., 48, 3895, 1977.)


experimental curves for welding 304-stainless steel. An accurate comparison with the experiment isdifficult because of uncertainties in reflectivity and spot size. These curves are in reasonableagreement with the experiment.

Festa et al. [79] presented simplified thermal models in laser and electron beam surfacehardening. The following requirements must be fulfilled in laser hardening:

. Limited power of the heat source

. Assurance of hardened depth

. Hardening process should be in solid state of the steel

To this aim, various simplified models are developed for industrial applications, allowing theprediction of maximum surface temperature hardening depth and reliable correlations with regardto physically meaningful dimensionless parameters.

Thermal models for laser surface hardening of steels in solid state are simplified considering thefollowing assumptions:

. No surface remelting occurs.

. Austenite transformation occurs at any depth at various temperature cycles.

. Cooling rate should assure hardening of austenitized layer, using (2Dy, and (1D)t,models. Hardening depth is determined by (2D)v v and (1D)t model as a functionof austenitization temperature and Peclet number. Maximum temperatures attained atvarious depths and hardening depths are nondimensionally correlated with processparameters according to (1D)t model. For the (2D)v model a correlation is derived,which predicts that hardening depths as well as the maximum surface temperaturesfor given process parameters are simply correlated with the Peclet number according tothe model.

11.4.7.3 Mathematical Description and Solution

The exact solutions of linear heat conduction problems in a semi-infinite isotropic homogeneousbody at uniform initial temperature heated by a surface heat source for a stationary uniform constantheat flux are presented in ref. [79] over the whole surface during a finite time, (1D)t, and in ref. [63]for a moving uniform constant strip 2b wide, (2D)v.

With reference to uniform heat flux for a finite amount of moving heat and source 2b wide,respectively, thermal properties being assumed independent of the position, the solution of the (1D)tproblem is

T1�D,t(z, t) ¼ 2q0a1=2

lt1=2 � ierfc z

(4at)1=2

� d(t)(t � t)1=2ierfc

z

4a(t � t)1=2� �" #( )

(11:96)

where

d(t) ¼ 0 for t � t1 for t � t

�(11:97)

t being the dwell time; that is, the amount of time the spot on the surface is exposed to the uniformand constant heat flux (in the [2D]v problem it can be expressed as t¼ 2b=v). The solution of the(2D)v problem is


T2�D,v(x,z,t) ¼ q0a1=2

2l � p1=2

ð10

exp � z2

4am

� �� erf

v

(2a)1=2� (xþ b)=v� t þ m

(2m)1=2

�

� erfv

(2a)1=2� (x� b)=v� t þ m

(2m)1=2

�dm

m1=2(11:98)

The above equations can be written in dimensionless form. Let

X ¼ x

(4at)1=2, Z ¼ z

(4at)1=2, z ¼ m

t, j ¼ t

t(11:99)

Pe ¼ vb

2a¼ 2b

(4at)1=2

2(11:100)

Tþ ¼ T

2bq0=l(11:101)

t being the dwell time, which is the amount of time the spot on the surface is exposed to the uniformand constant heat flux in the (2D)v problem, it can be expressed as t¼ 2b=v. The solution of the(2D)v problem is the equation; furthermore, it makes the maximum temperature independent ofthe Peclet number.

The hardening depth can be obtained by inverting equations:

T1,m* (Z) ¼ p1=2F(Z) (11:102)

T2,m* ¼ (Z,Pe) ¼ G(Z, Pe): (11:103)

This yields for the (1D)t

Zh ¼ F�1(T1,m* ) (11:104)

when T1,m* is equal to Tc* and, for the (2D)v

Zh ¼ G�1z (T2,m* ,Pe) (11:105)

when T2,m* is equal to Tc*

Since the inversion of equations cannot be performed analytically, it was carried out byempirical expressions. Zh values were calculated iteratively by equations as a function of Pe andT1,m* and T2,m* for (1D)t and (2D)v, respectively.

Results are presented in dimensionless temperature diagram form according to the equation,which allows a direct comparison between the predictions of (1D)t and (2D)y models. The hard-ening depth as a function of Peclet number at various austenitization temperatures is presented inFigure 11.73 for (2D)v model and in Figure 11.74 for (1D)t model. For both models it stands that ata given hardening depth (Zh) a higher maximum temperature Tc is achieved at lower Peclet number(Pe). This means that in the (2D)v model maximum temperature, the lower the heat flux, the lower isthe traveling speed for the given thermal diffusivity of the material. Figures also show that the lowerthe Tþ

c , the higher the slope of the curves are. As expected the value at the hardened depth predictedby the two models is in good agreement with high Peclet numbers (>10), while at low value (<1)the (1D)t over predicts the hardened depth.


4.0

3.0

0.075T

∗C = 0.05

0.15

0.1

0.2

0.30.40.50.6

1.0

0.70.8

0.252.0

1.0Har

deni

ng d

epth

zh (

mm

)

1.0−2 1.0−1 1.00

Peclet number Pe

(1D)t

1.01 1020

FIGURE 11.73 Hardening depth versus Peclet number at various austenitization temperatures in (2D)ymodel. (From Festa, R., Manca, O., and Naso, V., Int. J. Heat Mass Transfer, 33(11), 2511, 1990.)

4.0

3.0

2.0

1.00.80.70.6

0.5

0.4

0.30.25

0.20.15

0.10.075

T∗C

1.0

010−2 10−1 101 102100

Peclet number Pe

(2D)v

Har

deni

ng d

epth

zh

(mm

)

=0.05

FIGURE 11.74 Hardening depth versus Peclet number at various austenitization temperatures in (1D)tmodel. (From Festa, R., Manca, O., and Naso, V., Int. J. Heat Mass Transfer, 33(11), 2511, 1990.)

11.5 RESIDUAL STRESSES AFTER LASER SURFACE HARDENING

11.5.1 BACKGROUND

From the technology viewpoint, various surface hardening processes are very much alike since theyall have to ensure adequate energy input and the case depth required. In the same manner, regardlessof the hardening process applied, in the same steel, the same microstructure changes, very similar


microhardness variations, and similar variations of residual stresses within the hardened surfacelayer may be achieved.

Normally the surface transformation hardening process also introduces compressive stressesinto the surface layers, leading to an improvement in fatigue properties. Hardened parts—alwaysground because of their high hardness—require a minimum level of final grinding. This can only beachieved by a minimum oversize of a machine part after hardening, thus shortening the finalgrinding time and reducing cost of the final grinding to a minimum. With an automated manufac-turing cell, one should be very careful when selecting individual machining processes as well asmachining conditions related to them.

Ensuring the required internal stresses in a workpiece during individual machining processesshould be a basic criterion of such a selection. In those cases when the internal stresses in theworkpiece during the machining process exceed the yield stress, the operation results in workpiecedistortion and residual stresses. The workpiece distortion, in turn, results in more aggressiveremoval of the material by grinding as well as a longer grinding time, higher machining costs,and a less-controlled residual-stress condition. The workpiece distortion may be reduced bysubsequent straightening, that is, by material plasticizing, which, however, requires an additionaltechnological operation, including appropriate machines.

This solution is thus suited only to exceptional cases when a particular machining processproduces the workpiece distortion regardless of the machining conditions. In such cases the solesolution seems to be a change in shape and product dimensions so that material plasticizing in themachining process may be prevented.

It is characteristic of laser surface hardening that machine parts show comparatively highcompressive surface residual stresses due to a lower density of the martensitic surface layer. Thecompressive stresses in the surface layer act as a prestress, which increases the load capacity ofthe machine part and prevents crack formation or propagation at the surface. The machine partstreated in this way are suited for the most exacting thermo mechanical loads since their susceptibilityto material fatigue is considerably lower. Consequently, much longer operation life of the parts canbe expected.

Residual stresses are the stresses present in a material or machine part when there is no externalforce=or external moment acting upon it. The residual stresses in metallic machine parts haveattracted the attention of technicians and engineers only after manufacturing processes improvedto the level at which the accuracy of manufacture exceeded the size of strain, that is, distortion of amachine part.

Thus, it was in the 1850s that the effect of internal stresses on distortion of machine parts aftermechanical machining was known. It was then that experts introduced the measurement of indi-vidual dimensions of machine parts. For a given type of machining process, they connected theinfluence of the selected machining conditions with the size of deviations of dimensions. This wasalso the beginning of an expert approach to the selection of the most suitable machining conditionsbased on the criterion of minimum deviations of dimensions, that is, minimum machine partdistortion. However, at the time of publication, measurement of individual machine part dimensionsis a very practical, uncomplicated, and reliable method of machine part quality assessment.

The surface and sub surface layer conditions of the most exacting machine parts, however, aremonitored increasingly by means of the so-called surface integrity [80].

This is a scientific discipline providing an integral assessment of the surface and surfacelayers defined at the beginning of the 1960s. For high-quality machine parts and products subjectedto heavy thermomechanical loads, different levels of description of the surface integrity were given.A basic level of the surface-integrity description includes the measurement of roughness and theanalysis of microstructure and microhardness in the thin surface layer resulting from the machiningprocess under the given machining conditions. The second level of the surface-integrity descriptionincludes studies of residual stresses in the surface layer and of mechanical properties of the givenmaterial. The third level of the surface-integrity description includes test behavior of the given part


during machining. More detailed information on the levels of surface-integrity description may beobtained [80,81].

11.5.2 DETERMINATION OF THERMAL AND TRANSFORMATION STRESSES

Yang et al. [92] proposed the model for determination of thermal and transformation residualstresses after laser surface transformation hardening of a medium-carbon steel. Thermal stresseswere induced due to thermal gradients and martensitic phase transformation causing prevailingresidual stresses. The dimensions of the specimen for simulation were 6 mm in length, 2 mm inthickness and the width of the sliced domain was 0.2 mm. To carry out heat treatment, laser powerP¼ 1.0 kW, traveling speed v¼ 25 mm=s, absorptivity A¼ 0.15, and characteristic radius of heatflux r¼ 1.0 mm were chosen. At rapid heating austenitic transformation temperature started at8308C and ended at 9508C.

The cooling rate was also high enough in the heated material and austenite transformed tomartensite in the temperature range from 3608C to 1408C. In the simulation the volume changesat phase transformation base microstructure to austenite at heating (aFþP!A¼ 2.8� 103) and alsovolume changes in phase transformation of austenite to martensite at cooling (aA!M¼ 8.5� 10�3)were considered. Figure 11.75 shows numerous temperature cycles at the surface and at differentdepths below the surface through the center of the laser-hardened track [82].

11.5.2.1 Heat Transfer Analysis

The calculation of the transient temperature distribution was based on the attainment of quasi-stationary conditions occurring when the heat source moves at constant speed on a regular path andend effects resulting from either initiation or termination of the heat source are neglected.

The energy transfer from the laser beam to the workpiece was simulated by the heat flux. InFigure 11.75 there are two typical heat flux distributions that are commonly used, namely, theGaussian beam mode and the square beam mode. In the case of the Gaussian beam mode, the heatflux q(r) on the top surface of the workpiece can be expressed by the following equation:

q(r) ¼ 3Qpr�2

exp �3r

�r

� �2� �(11:106)

r

rr

d1

d2

(A) (B)

p r3Qq =

2

−3d1d2

Qq =

r

( ([ [

FIGURE 11.75 Schematic diagram of two different laser-beam shapes. (A) Gaussian beam mode and(B) square beam mode. (From YS Yang and SJ Na. A study on residual stresses in laser surface hardeningof a medium carbon steel. Surface and Coatings Technology, 38, 1989, 311–324.)


wherer is the radial distance from the laser-beam centre�r is the characteristic radius, defined as the radius at which the intensity of the laser beam falls

to 5% of the maximum intensityQ is the power transferred into the substrate where A is the absorptivity and P is the laserpower

The heat flux of the square beam mode can be expressed simply as follows:

q(r) ¼ Q

d1d2(11:107)

where d1 and d2 are the beam width and length, respectively.

11.5.2.2 Thermal and Residual Stress Analyses

Material subjected to the thermal cycle of the laser surface hardening is postulated to behavemechanically as an initially isotropic, elastoplastic, strain-hardening continuum, such that a com-ponent of total strain «ij is given by the following formula:

«ij ¼ «eij þ «pij þ «thij (11:108)

where «eij, «pij, «

thij are the components of elastic, plastic, and thermal strain, respectively [90].

The equation indicates that the effect of temperature is considered by an additional term in thewell-known relationship, the Prandtl–Reuss equation, between strain rate and deviatoric stress.For the elastic part, the following formula was adopted:

«eij ¼1þ n

Esij � n

Eskkdij, (11:109)

wheredij is the Kronecker symboln is the Poisson ratioE is Young’s modulus

The thermal strain can be expressed as follows:

«thij ¼ a T � Trð Þ, (11:110)

wherea is the coefficient of linear thermal expansionTr is the reference temperature

11.5.2.3 New 2D Finite Element Model

For calculating the thermal and residual stresses of an unconstrained workpiece effectively, a new2D modeling scheme was introduced in this study [90].

In this model, the longitudinal total strain distribution «x(y, z) in the solution domain was used asthe boundary condition. The «x(y, z) values, which make the resultant force along the longitudinal


direction, were considered as the exact set of the boundary values. To determine «x(y, z), thecompatibility equations that satisfy all the assumptions of this model were considered as follows:

@2«x@y2

¼ 0, (11:111)

@2«x@z2

¼ 0, (11:112)

@2«x@y@z

¼ 0, (11:113)

XMz ¼ 0: (11:114)

As the longitudinal total strain distribution «x(y, z) must be symmetric about the z-axis, it couldalways be satisfied and «x(y, z) must be an even function of y. From the above relationships, thefollowing equation can be obtained:

«x(z) ¼ aþ bz (11:115)

By considering the equations and Hooke’s law, the constants a and b can be determined as follows:

a ¼ 8AD� 12BWD2

and (11:116)

b ¼ 24B� 12ADWD3

, (11:117)

where

A ¼ðD0

ðW=2

0

«px þ «thx � n

1� n«ey þ «ez

� �dy dz (11:118)

B ¼ðD0

ðW=2

0

«px þ «thx � n

1� n«ey þ «ez

� �z dy dz (11:119)

andW is the width of the workpieceD is the thickness of the workpiece

As thermal stress is caused mainly by the uneven temperature distribution, the longitudinal totalstrain «x(y, z) is related to the temperature distribution in the workpiece. Hence, the gradientof longitudinal total strain «x(y, z) along the y-direction was assumed to be the same as that alongthe z-direction. As the longitudinal total strain distribution «x(y, z) must be symmetric about thez-axis, «x(y, z) could be formulated as follows:

«x(y, z) ¼ aþ bjyj þ bz: (11:120)

From the above relationship, the exact values of the boundary condition «x(y, z), which satisfy theequilibrium condition along the longitudinal direction, can then be calculated.


11.5.2.4 Simulation Results and Discussion

In the laser surface heat treatment, the austenitic transformation temperature is changed because ofits very high heating rate. The beginning of the austenitic transformation temperature range shouldbe approximately 8308C and the end of austenitic transformation temperature range should be9508C. In addition, the cooling rate is high enough for all the materials that undergo transformationinto austenite to be transformed into martensite. The martensitic transformation, which causes theexpansion of the volume, was assumed to occur in the temperature range from 3608C to 1408C. Inthe simulation, the volume changes in the austenitic and the martensitic transformations wereconsidered by the thermal dilatation for which 2.8� 10�3 were adopted, respectively.

The incremental analysis of elastoplastic stress was carried out using the finite element mesh.The dimensions of the workpiece for the simulation are 16� 5 mm2, and the thickness of the sliceddomain was 0.2 mm. In this study, the basic processing parameters chosen for the analysis were asfollows: laser power, P¼ 1110 W; absorptivity, A¼ 72.7%; traveling speed, v¼ 30 mm=s; andbeam size, 4.5� 4.5 mm2 (square beam mode). Then the beam type (Gaussian mode, square mode),the beam shape of the square mode, the traveling speed, and the laser power were changed toobserve the residual stress distribution for variable processing parameters.

Figure 11.76 shows the temperature changes with time at various points on the workpiecesurface for the Gaussian and square beam modes. In the Gaussian beam mode, as shown inFigure 11.76A, the temperature history at any selected point can be described by a continuoussmooth curve because the heat flux on the sliced solution domain increases and decreases continu-ously. In contrast, the temperature history in the square beam mode appears as a discontinuous curvebecause of the suddenly decreasing characteristics of the heat flux as shown in Figure 11.76B.

The longitudinal thermal stress distributions on the top surface during heating and cooling areshown in Figure 11.77 and 11.78. The simulation conditions of Figure 11.77A and B were selectedto be the same. Upon the initial heat-up, the localization of severe temperature gradients in theimmediate vicinity of the beam line produces the compressive yielding in this region. However, thestate of thermal stress changes to the tensile stress as the distance from the centerline increases. Asmore energy is being supplied by the laser beam, the temperature increases and the yield strengthdecreases quickly, so that the compressive thermal stress becomes lower and lower in the vicinity ofthe beam line. Upon cooling, the yield strength increases and the unloading from the yield surfaceproceeds elastically until the material yields in tension. Hence, as the cooling proceeds, the portionof the workpiece that is longitudinal in tension grows steadily. The residual stress in laser surface

1500

1200

900

600

Tem

pera

ture

T (�

C)

300

00.0 0.1 0.2

Time t (s)0.3

y = 0.00 mm

1.80 mm

2.20 mm

2.84 mm4.89 mm

0.4 0.5

Tem

pera

ture

T (�

C) y = 0.00 mm

1500

1200

900

600

300

00.0 0.1 0.2

Time t (s)0.3 0.4 0.5

1.80 mm

2.20 mm

2.84 mm4.89 mm

(A) (B)

FIGURE 11.76 Temperature histories of (A) Gaussian beam mode (characteristic radius of heat flux�r¼ 3 mm) and (B) square beam mode (beam size 4.5� 4.5 mm2) at various distances from the centerlineon the top surface. (From YS Yang and SJ Na. A study on residual stresses in laser surface hardening of amedium carbon steel. Surface and Coatings Technology, 38, 1989, 311–324.)


600

400 27.69 � 10−2 s

15.09 � 10−2 s

0.35 � 10−2 s

200

−200

−400

−6000 2

Residual stress

4Distance from center line y (mm)

Long

itudi

nal s

tres

s sL (

N/m

m2 )

6 8

0

FIGURE 11.77 Longitudinal stress distributions on top surface during heating and cooling.(From YS Yangand SJ Na. A study on residual stresses in laser surface hardening of a medium carbon steel. Surface andCoatings Technology, 38, 1989, 311–324.)

hardening is induced from the temperature gradient and phase transformation. The residual stress inthe HZ is always tensile as long as only the temperature gradient is considered, while the residualstress arising from the austenite–martensite phase transformation causes the surface compression.Accordingly, in laser surface hardening the resulting residual stress in the HZ is a function of thesetwo opposing factors. As shown in Figure 11.77, the compressive residual stress was generated inthe HZ within about 1.8 mm from the centerline. Therefore, it can be suggested that the martensitictransformation is the major influence on the residual stress in the laser surface heat treatment.

Although the compressive residual stress at the surface of the laser-hardened workpiece has aprofitable effect on the mechanical characteristics like wear resistance, fatigue strength etc., it should

1500

1200y = 0 mm

0.45 mm

0.90 mm1.52 mm

2.73 mm

900

600

300

00.00 0.05 0.10 0.15

Tem

pera

ture

T (�

C)

Time t (s)0.20 0.25 0.30

FIGURE 11.78 Temperature cycles of laser surface hardening at various distances from the surface. (FromYS Yang and SJ Na. A study on residual stresses in laser surface hardening of a medium carbon steel. Surfaceand Coatings Technology, 38, 1989, 311–324.)


4.63 � 10−2 s

2.85 � 10−2 s1.35 � 10−2 s

60

40

20

0

−20

−40

−600.0 0.5

Long

itudi

nal s

tres

s (N

/mm

2 )

1.0 1.5Distance from centerline y (mm)

2.0 2.5 3.0

FIGURE 11.79 Longitudinal stress distributions at the surface laser heating. (From YS Yang and SJ Na. Astudy on residual stresses in laser surface hardening of a medium carbon steel. Surface and CoatingsTechnology, 38, 1989, 311–324.)

be noted that the tensile residual stress in the interior of the workpiece possesses a potential togenerate internal microcracks.

In comparison with the Gaussian distribution of the beam power, the square beam mode resultsin a wider, but shallower, HZ. The calculation results also showed that the high-power beam withthe high traveling speed is more suitable for laser surface hardening than the low power beamwith the low traveling speed if the heat input per unit length of the workpiece is maintained constant.

Figure 11.79 shows longitudinal stress profiles in the direction of the laser-beam travel at varioustime moments in heating, namely at t1¼ 1.35�10�2 s, t2¼ 2.85�10�2 s, and t3¼ 4.65�102 s [92].

From the graph it can be seen that the compressive stresses are in the immediate vicinity of thelaser beam. Nevertheless, the thermal stresses at heating change to tensile stresses according to thedistance from the laser beam.

As the laser provides more energy according to the heating time, the temperature increases, sothat the compressive thermal stresses become progressively lower in the vicinity of the laser-beamcenter.

Figure 11.79 shows longitudinal stress profiles in the direction of the laser-beam travel atvarious cooling times during laser surface hardening [92].

The residual stress in laser surface hardening is induced from temperature gradients and phasetransformations. During the cooling process longitudinal stresses are always tensile and change tocompressive during phase transformation. This means that longitudinal residual stress is always ofcompressive nature. As shown in Figure 11.80 the compressive residual stress was generated in thehardened depth within 0.5 mm from the centerline of the exposed laser beam.

11.5.3 SIMPLE MATHEMATICAL MODEL FOR CALCULATING RESIDUAL STRESSES

Li and Easterling [83] developed a simple analytical model for calculating the residual stresses atlaser transformation hardening. Figure 11.81 shows a single track across the surface of the flatspecimen. Following the temperature cycle of the laser beam traveling, a certain volume of the heat-affected material expands as a result of the martensitic transformation. The created martensitemicrostructure has higher specific volume than matrix microstructure, which causes residual stressesin the surface layer.


−600.0 0.5 1.0 1.5

Distance from centerline y (mm)

6.54 � 10−2 s

9.70 � 10−2 s

Residual stress

2.0 2.5 3.0

−40

−20

0Lo

ngitu

dina

l str

ess (

N/m

m2 )

20

40

60

FIGURE 11.80 Longitudinal stress distributions and residual stress distributions at the surface during coolingin laser surface hardening. (From YS Yang and SJ Na. A study on residual stresses in laser surface hardening ofa medium carbon steel. Surface and Coatings Technology, 38, 1989, 311–324.)

The magnitude of this residual stress is calculated as a function of the laser input energy and thecarbon content of the steel.

To simplify the calculations, the following assumptions can be made:

. Stresses induced by thermal expansion near the surface can be neglected.

. Plastic strains caused by the martensitic transformation can be considered negligible.

. Dilatation in the HAZ is completely restrained along the x and y directions during the lasertransformation hardening.

Consider an elementary volume DV¼DxDyDz in the HAZ (Figure 11.81). After laser surfacehardening the elementary volume single track expands due to martensitic transformation.

G

HAZH

b

B

A C

E

F

x

z

D

y

c

a

FIGURE 11.81 HAZ of single pass at laser heating. (From Li, W.B. and KE Easterling, K.E., Surf. Eng.2, 43, 1986.)


From the author’s equations, different factors and parameters influence residual stress calcula-tions, in spite of the simplifying assumptions made. These factors include

. Dimensions of the specimen

. Properties of the material and composition

. Laser processing variables

The residual stress sxx should equal

sxx ¼ � b f E

1� nþ 1bc

ðc0

dz

ðþb=2

�b=2

b f E

1� ndyþ 12(0:5c� z)

bc3

ðc0

dz

ðþb=2

�b=2

b f E

1� n(0:5c� z)dy: (11:121)

This equation is a complete description of the residual stress in the x-direction, where the first term isnegative and decreases with depth below the surface, and the second and third terms are bothpositive. It shows that these terms resulted in a compressive residual stress at the surface andchanged to a tensile residual stress at some depth below the surface.

The y-component of the residual stress syy can be expressed as

syy ¼ � b f E

1� n: (11:122)

As several laser runs overlap in the y-direction, there is some stress relief in the overlapping zone.The maximum syy, given by the equation, is still valid.

Assuming that expansion in the z-direction proceeds freely, the z-component of residual stressszz is zero:

szz ¼ 0: (11:123)

It is impossible to change the volume because of phase transformation; therefore, shearing strains donot occur, and the shear components of residual stresses are zero:

sxy ¼ syz ¼ szx ¼ 0: (11:124)

The given equations present simplified ways of calculating residual stresses at laser transformationhardening.

Figure 11.82 shows the effect of carbon content on sxx residual stress profile from the centerlineof the single laser track [83].

It can be seen that the value of the compressive residual stress is very dependent upon carboncontent, increasing by about 200 N=mm2 for each 0.1% C. The crossover from compressive totensile residual stress in the x-direction occurs approximately at the depth where the steel has beenheated to the TA1 temperature. Figure 11.84 also shows the profile for syy in the steel with 0.44% C.As it can be seen syy lie at slightly higher compressive stresses than the corresponding profile for thex-direction.

The effect of changing the input energy density of the laser process on residual stress profile sxx

is shown in Figure 11.83. It can be seen that for the 0.44% C steel changes in laser input energy donot substantially affect the value of the compressive residual stress but do affect the depth do of thecompressive layer.

Figure 11.84 shows changes in compressive residual stress sxx and depth of compression layerd0 as a function of laser input energy density.

In practical laser transformation hardening, a surface may be hardened not by a single laser trackbut by a number of overlapping tracks. This obviously results in local stress relaxation in the


100

−100

−300

−400

s zz = 0

sxx

(N/m

m2 )

−5000 0.5 1.0

s yy (0.

44% c)

C = 0.44

%

C = 0.33%

C = 0.23%

1.5Distance from surface z (mm)

Input energy50 J/mm2

2.0 2.5 3.0

−200

0

FIGURE 11.82 Predicted residual stress distribution sxx as a function of depth below surface and steel carboncontent C. (From Li, W.B. and Easterling, K.E., Surf. Eng. 2, 43, 1986.)

y-directions in the overlapping zones, although the residual stresses calculated at the center of eachlaser track will not be affected and remain as presented above. If the overlapping tracks cover theentire surface, end effects, as discussed in conjunction with sxx, should be included in thecalculation of syy for completeness.

Com Nogue et al. [52] analyzed the through-depth variations of residual stresses and micro-hardness of a laser-hardened layer on S2 chromium steel. Heating was carried out by a laser beamwith a power P of 2 kW and a traveling speed v of 7 mm=s. A semi elliptical power densitydistribution in the laser beam with defocusing was chosen so that a constant depth of the hardenedlayer d, i.e., 0.57 mm, was obtained with a width W of 10.8 mm. Figure 11.85 shows thethrough-depth variations of the measured hardness and experimentally determined residual stressin the hardened layer.

100

−100

−200

−300

−400

−5000 0.5

33.340

5066.7

q/vR

b [J/m

m2 ]

100

1.0 1.5Distance from surface z (mm)

s xx (

N/m

m2 )

2.0 2.5 3.0

0

FIGURE 11.83 Predicted residual stress sxx as function of depth below surface and laser input energy density(q=vRb) for 0.44% C steel. (From Li, W.B. and Easterling, K.E., Surf. Eng. 2, 43, 1986.)


490

440

390

1.5

1.0

d 0 (m

m)

0.530 40 50

d0

q/vRb (J/mm2)

s xx

(N/m

m2 )

60 70

sxx

FIGURE 11.84 Predictions of surface compressive residual stress sxx and depth d0 as a function of laser inputenergy density for 0.44% C steel. (From Li, W.B. and Easterling, K.E., Surf. Eng. 2, 43, 1986.)

The variations of both hardness and residual stresses in the surface-hardened layer depend onthe microstructure of the latter. Microstructural changes can be affected by the heating conditions,which in laser heating, can be determined by the selection of the laser power and the area of thelaser-beam spot diameter dB or AB with a given mode structure. The traveling speed of the laserbeam was then suitably adapted. The measured average surface hardness was 725 HV. It slowlydecreased to around 300 HV, i.e., hardness of steel in a soft state, in a depth d of 0.57 mm.The variation of microhardness permitted an estimation that heating was correctly chosen to achievethe maximum surface hardness and a very distinct width of area with the transition microstructure(a mixture of hardened and base microstructure).

The through-depth variation of residual stresses of the hardened layer depended on the residualstresses occurring in the material prior to heating. Then followed relaxation of the existing stressesdue to laser heating for further hardening. Finally the variation of the residual stresses after laserheating was also affected by the initial microstructure and the cooling rate of steel. A higheraustenitizing temperature and an ensured homogeneity of carbon in austenite and as long as possibleheating of the material for the given depth of the hardened layer ensured efficient tempering of the

400Semi elliptic beamSteel S2Power: P = 2.0 kWSpeed: v = 7mm/s

800

700

600

500

400

300

200

Resid

ual s

tres

ses s

(MPa

)

Mic

roha

rdne

ss H

Vm

−200

−400

Depth below the surface z (mm)

01 2 3

HVms

FIGURE 11.85 Residual stress and microhardness profiles through the laser surface-hardened layer. (FromNougue, J.C., and Kerrand, E., Laser surface treatment for electromechanical applications: NATO ASI Series.In: C W Draper, P Mazzoldi, and M Nijhoff, Eds., Laser Surface Treatment of Materials, Publishers,Dordrecht, 1986, 497–511.)


preceding residual stresses and introduced new low-tensile stresses. Thus after laser surface hard-ening, comparatively high compressive residual stresses could be ensured. They could be confirmedby computations as well.

The through-depth variation of residual stresses of the hardened layer was, as expected,coordinated with the variation of hardness. It was very important that the maximum residual stressoccurred just below the surface and amounted to �530 N=mm2. Then it changed the sign at a depthof 0.57 mm and became the maximum tensile stress at a depth of 0.9 mm. In the range between themaximum compressive and compressive tensile residual stresses, i.e., between the depths of 0.57and 0.9 mm, a linear variation of the residual stresses occurred.

11.5.4 DETERMINATION OF STRESSES BY NUMERICAL SIMULATION

Denis et al. [84] presented an analysis of the development of residual stresses for laser surfacehardening of steel by numerical simulation.

A 2D calculation has been performed for laser-hardened plates made of a hypoeutectoid plaincarbon steel with 0.42% carbon. The finite element code SYSWELD in which their metallurgicalmodel has been included has been used. The necessary input data have been determined: on onehand the parameters of the laser treatment as total power and energy distribution of the laser beamand traveling speed, on the other hand the surface absorptivity and thermophysical, metallurgical,and mechanical properties of the steel were chosen from Refs. [85–88].

Figure 11.86 shows the calculated temperature evolutions in the middle of the laser trace atdifferent depths for a treatment with 960 W total power and 4.5 mm=s traveling speed. Figure 11.87shows that the distribution of the microstructures at the end of cooling is characterized by a fullymartensitic domain until 0.7 mm depth with homogeneous martensite near the surface and hetero-geneous martensite below. Figure 11.88 shows calculated hardness profiles below the surface in themiddle of the trace [84].

Two laser treatments with two different traveling speeds (3.5 and 5.5 mm=s) have beensimulated. All the other parameters are the same as those of the case of reference. The increase inthe traveling speed leads essentially to a decrease in the maximum temperature reached on heatingand consequently to a decrease in the hardened depth. Figure 11.89 shows the residual stressprofiles and that the stress level at the surface decreases when the traveling speed decreases [84].

1200

1000

800

600

400

Tem

pera

ture

T (�

C)

200

00 1 2 3 4

Time t (s)

Depth (mm)0.000.5651.0651.542

5 6 7 8

FIGURE 11.86 Temperature evolutions at different depths in the middle of the track. (From Denis, S.,Boufoussi, M., Chevrier, J.Ch., and Simon, A., Analysis of the development of residual stresses for=surfacehardening of steel by numerical simulation. Proceedings of the International Conference on Residual Stresses(ICRS4), Society of Experimental Mechanics, Baltimore, MD, 1994, 513–519.)


Homogeneousmartensite

1.0

0.8

0.6

0.4

0.2

0.00 0.5

Depth z (mm)

Abs

orpt

ivity

A (%

)

1 1.5

High carbonmartensite

Low carbonmartensite

Retainedaustenite

Pearlite

Ferrite

FIGURE 11.87 Final distribution of microstructures in the middle of the track. (FromDenis, S., Boufoussi, M.,Chevrier, J.Ch., and Simon, A., Analysis of the development of residual stresses for=surface hardening ofsteel by numerical simulation. Proceedings of the International Conference on Residual Stresses (ICRS4),Society of Experimental Mechanics, Baltimore, MD, 1994, 513–519.)

Calculation

Har

dnes

s HV

Depth z (mm)

1000

800

600

400

200

00 0.5 1 1.5

Experiment HV 0.3

FIGURE 11.88 Hardness profiles versus depth in the middle of the track. (From Denis, S., Boufoussi, M.,Chevrier, J.Ch., and Simon, A., Analysis of the development of residual stresses for=surface hardening ofsteel by numerical simulation. Proceedings of the International Conference on Residual Stresses (ICRS4),Society of Experimental Mechanics, Baltimore, MD, 1994, 513–519.)


Power P = 960 W

Depth z (mm)

400

300

200

100Tr

ansv

erse

stre

ss s

(MPa

)

−100

−200

−300

−4000 0.5 1 1.5 2 2.5 3

0

Travel speed (mm/s)3.54.55.5

FIGURE 11.89 Residual stress distributions for different travel speeds. (From Denis, S., Boufoussi, M.,Chevrier, J.Ch., and Simon, A., Analysis of the development of residual stresses for=surface hardening ofsteel by numerical simulation. Proceedings of the International Conference on Residual Stresses (ICRS4),Society of Experimental Mechanics, Baltimore, MD, 1994, 513–519.)

Moreover, for the lowest traveling speed, the maximum compressive stress is found beneath thesurface. Figure 11.90 shows summarized results about transversal residual stress at various travelingspeeds. It can be seen that in the domain of laser hardening (for which melting must be avoided and

1.45

400

200

Mel

ting

No

phas

e tra

nsfo

rmat

ion

−200

−400

−6002 3 4 5

Travel speed (mm/s)

Surface

Transitionhardened zone-base metal

PowerP = 960 W

Tran

sver

se st

ress

s (M

Pa)

6 7 8 9

0

1.10 0.80 Hardened depth (mm)

FIGURE 11.90 Transverse stress versus travel speed. (From Denis, S., Boufoussi, M., Chevrier, J.Ch., andSimon, A., Analysis of the development of residual stresses for=surface hardening of steel by numericalsimulation. Proceedings of the International Conference on Residual Stresses (ICRS4), Society of Experimen-tal Mechanics, Baltimore, MD, 1994, 513–519.)


600

400

300

200

100

Tran

sver

se st

ress

s (

MPa

)

−100

−200

−300

−400

0

700 800 900

0.40 1.10Hardened depth (mm)

1.60

1000

Transitionhardened zone-base metal

Surface

Mel

ting

v = 4.5 mm/s

No

phas

e tra

nsfo

rmat

ion

Total power P (W)1100 1200 1300

FIGURE 11.91 Transverse stress versus total power of the laser-beam travel speed of 4.5 mm=s. (FromDenis, S., Boufoussi, M., Chevrier, J.Ch., and Simon, A., Analysis of the development of residual stressesfor=surface hardening of steel by numerical simulation. Proceedings of the International Conference onResidual Stresses (ICRS4), Society of Experimental Mechanics, Baltimore, MD, 1994, 513–519.)

a desired hardened depth must be kept) an increase in the traveling speed leads only to a relativelysmall increase in the compressive residual stresses in the HZ [84].

A similar diagram in Figure 11.91 shows simulating transversal residual stress at various totallaser power. An increase in the total power leads to a larger hardened depth when the maximumtemperature reached on heating increases and to a decrease in the surface residual compressivestress. Again, it follows that the maximum level of the stress that can be reached by decreasing thetotal power remains a relatively small compressive residual stress in the amount of �400 MPa [84].

From the study of the results, particularly from the analysis of the plastic strains and transform-ation plasticity strains generated during the treatment, it has been concluded that the relatively smallsensitiveness of the level of residual stress to the process parameters is essentially related to thecooling phase of the treatment, which occurs by heat conduction in the inside of the workpiece.This type of cooling sets limits to the thermal gradients that develop during cooling and conse-quently to the plastic strains that are generated and to the level of the residual stresses. If the level ofthe compressive residual stresses in the HZ is to be increased, it is necessary to have higher coolingrates through additional cooling of the surface [85–88].

In order to confirm this analysis, laser heating followed by quenching has been simulatednumerically and a high constant heat transfer coefficient (700 W=m2) behind the laser beam has beenimposed. Figure 11.92 gives the comparison between the temperature evolutions at the surface andat 0.5 mm beneath it for both cases (reference and simulation). Much higher cooling rates areobtained with quenching. It can be noted that the temperature evolutions are very similar toinduction surface hardening treatments. Figure 11.93 shows that the calculated residual stressprofiles at laser surface hardening with quenching giving higher compressive stresses in themartensitic zone have doubled for the same hardened depth. This residual stress has been relatedmainly to the increase in the plastic strains generated in austenite during cooling. Figure 11.94


1200

1000

Classical laserhardening

Laser heating+ quenching

800

600

400Tem

pera

ture

T (�

C)

200

00 1

Depth 0.5 mm

Surface

2 3Time t (s)

4 5

FIGURE 11.92 Temperature evolutions at the surface and at 0.5 mm depth in the middle of the track. (FromDenis, S., Boufoussi, M., Chevrier, J.Ch., and Simon, A., Analysis of the development of residual stressesfor=surface hardening of steel by numerical simulation. Proceedings of the International Conference onResidual Stresses (ICRS4), Society of Experimental Mechanics, Baltimore, MD, 1994, 513–519.)

shows the permanent traverse strain profiles below the surface, which gives a good correlation toresidual stress profiles [84].

Fattorini et al. [89] assessed the possibility of predicting the residual stresses distribution inducedby laser surface-hardening treatment. A finite element thermomechanical model was used to simulate

400

200

−200

−400

−6000 0.5 1 1.5

Depth z (mm)

Classical laserhardeningLaser heating+ quenchingTr

ansv

erse

stre

ss s

(MPa

)

2 2.5 3

0

FIGURE 11.93 Residual stress distributions for a classical laser surface hardening with laser surfaceheating with quenching. (From Denis, S., Boufoussi, M., Chevrier, J.Ch., and Simon, A., Analysis of thedevelopment of residual stresses for=surface hardening of steel by numerical simulation. Proceedings ofthe International Conference on Residual Stresses (ICRS4), Society of Experimental Mechanics, Baltimore,MD, 1994, 513–519.)


0.004

0.003

0.002

Perm

anen

t tra

nsve

rse s

trai

n e (

%)

0.001

−0.001

0

0 0.5 1.5Depth z (mm)

Classical laserhardeningLaser heating+ quenching

2 32.51

FIGURE 11.94 Transverse permanent strain distributions at the end of cooling. (From Denis, S., Boufoussi,M., Chevrier, J.Ch., and Simon, A., Analysis of the development of residual stresses for=surface hardening ofsteel by numerical simulation. Proceedings of the International Conference on Residual Stresses (ICRS4),Society of Experimental Mechanics, Baltimore, MD, 1994, 513–519.)

surface hardening treatments of flat faces of cylindrical samples of 3.5 NiCrMoV steel. Temperatureand stress trends were followed throughout the heating and cooling cycles. The simulated heattreatments were also performed experimentally on samples coated with colloidal graphite, in orderto improve absorption of laser radiation to the metal surface. The samples were characterized from themicrostructure and hardness profile aspects. X-ray diffraction analyses were also performed toascertain what microstructural phases are present in the hardened layer, to measure residual stresses,and to evaluate any eventual interstitial C enrichment on the sample. Comparison stresses reveal verygood agreement between the predicted values and those measured experimentally in the case ofsamples treated at Tsur¼ 13008C, while there is marked disagreement where samples treated atTsur¼ 10008C are concerned. The lack of agreement has been attributed to the diffusion of coatingcarbon into the base material in g phase and to the possible partial inhibition of the g¼>atransformation owing to the occurrence of excessive compressive stresses during cooling.

The mathematical model provides for two distinct operating phases: temperature calculation toascertain the spatial and temporal distribution of temperatures; mechanical calculation that uses theresults of the first phase as input data and examines the evolution of stresses and strains in the elasto-plastic range during the thermal transient. Both phases are based on the FEM and, particularly in theactual case, axial-symmetric triangular elements have been simulated on cylinders 10 mm indiameter and 10 mm tall, irradiated on a flat surface. Radiation conditions involved specific heatinput of 2 kW=cm2 and flux–material interaction times permitting maximum surface temperature of10008C and 13008C. The samples were then coated with graphite to increase the amount of laserradiation impinging on the metal. The surface treatments were performed with a continuous CO2

laser, utilizing a beam focused with an integrator to ensure more even distribution of power on theradiated surface. The laser-beam=material interaction times were taken to be 0.36 and 0.6 s, neededto attain maximum surface temperatures of 10008C and 13008C according to the mathematicalsimulations. The test pieces were characterized from the aspects of microstructure and hardnessprofiles in the hardened layer down to the base material. Samples were also subjected to x-raydiffraction analysis. The latter has been developed on the basis of the following analytical andstructural characterizations: in depth phase recognition, residual stress plotting determination of the


a (g) lattice parameter, assignment of the average interstitial carbon t content, and residual stresseslevel determined by means of the sin2c technique.

A finite element mathematical model can be a useful tool for predicting hardened thicknesses ofsamples subjected to laser radiation. Anyway, it is of prime importance to be able to evaluate withreasonable accuracy the absorption capacity of the coating, which depends on the operationalmethods adopted for the heat treatment.

Mathematical models permit the prediction of internal stress distribution in the componenttreated, provided that the microstructures obtained within the hardened layer are homogeneousand predictable on the basis of the chemical composition of the steel treated, so that the hotthermomechanical properties can be calculated with reasonable accuracy.

When graphite is used as coating for CO2 laser radiation, the C diffusion into the g phaseinduces structural modifications due to local surface melting and massive concentrations of retainedaustenite. In this case, preprediction of the residual stresses is still possible only if the layer ofthe steel affected by these inhomogeneities is very small compared with the total hardened layer.Figure 11.95 shows the hardness profiles obtained in the samples [89].

As predicted by the mathematical simulation, it ensures that the hardened thicknesses measure0.4–0.5 and 0.8–0.9 mm, with surface temperatures of 10008C and 13008C.

Microstructure analysis revealed very heterogeneous structures, particularly in the layers imme-diately beneath the surface. In the samples treated at 13008C, islands of residual austenite (400 HV)surrounded by well-developed, coarse needle like martensitic structures are observed.

Figure 11.96 shows the sample treated at Tsur¼ 10008C. The amount of retained austenite onthe surface is about 40%, reducing gradually with depth until it disappears altogether at about0.20 mm. At the same time, the Fe3C phase reaches about 12% on the surface and disappearsafter about 0.03 mm. In the samples treated at Tsur¼ 13008C, retained austenite (about 0.40%)is limited to a skin layer measuring about 10 mm. The measurements of the lattice parameter a (g)in austenitic phase point to the presence of very large amounts of interstitial C (1.4þ 1.5%) inthe skin layers, thus providing further confirmation of the C diffusion from the graphite coating.

600

400

600

400

0.2 0.4 0.6Depth z (mm)

Har

dnes

s HV

Specimen treated at Tsur = 1300�C


0.8 1.0

0.2 0.4 0.6 0.8 1.0

FIGURE 11.95 Hardness profiles. (From Fattorini, F., Marchi Ricci, F.M., and Senin, A., Internal stressdistribution induced by laser surface treatment. In: BL Mordike, DGM, Ed., Proceedings of the EuropeanConference on Laser Treatment of Materials (ECLAT), 1992, 235–242.)


Austenite

Fe3C

0.1

40

20Fe3C

-aus

teni

te (%

)

0.2Depth z (mm)

FIGURE 11.96 Phase analysis Tsur¼ 10008C. (From Fattorini, F., Marchi Ricci, F.M., and Senin, A., Internalstress distribution for the specimen treated at laser surface treatment. In: BL Mordike, DGM, Ed., Proceedingsof the European Conference on Laser Treatment of Materials (ECLAT), 1992, 235–242.)

Figure 11.97 reports the results of the residual stress measurements. The values obtained bymathematical model simulations are also included for comparison. There is very good agreementbetween the predicted and measured values in the case of samples treated at Tsur¼ 13008C, whilethere is marked lack of accord in the case of Tsur¼ 10008C.

Ericsson et al. [90] studied residual stresses, the content of retained austenite, and themicrostructure supported by microhardness measurements. The experiment consisted in making alaser-hardened trace on a cylinder with a diameter of 40 mm and a length of 100 mm. In theexperiment AISI 4142 and AISI 52,100 steels in quenched and tempered (320 HV) and fullyannealed conditions (190 HV) were used. Laser surface hardening was carried out with a CO2 laserwith a power of 3 kW in CW mode. Several laser surface hardening parameters were chosen.The laser power, the laser-beam spot diameter, and the traveling speed of the workpiece, however,were varied. In cooling of the specimens, two methods were chosen, i.e., self-cooling and waterquenching. In the course of hardening, i.e., the heating and cooling cycle, the temperature wasmeasured with thermocouples located in two depths, i.e., z1 ffi 0.82 mm and z2 ffi 4.98 mm.

Figure 11.98 shows the results of calculations in laser surface hardening with reference tothrough-depth distribution of austenite in the heated layer after 24 s and then the through-depthvariations of martensite and residual stresses in the hardened layer after quenching.

In the calculations of the austenite and martensite contents and the variation of residualstresses in surface hardened AISI 4142 steel with 55.2 mm in diameter, a power density Q of6.6 MW=m2, a traveling speed v of 0.152 m=min, and width of hardened trace W of 8.175 mmwere taken into account. After hardening, up to 35% of martensite was found in the surface layerto a depth of 0.5 mm. Then the martensite content decreased in a linear manner to a depth of 1.0mm. was The through-depth variation of residual stresses of the compressive character wasvery similar with a maximum value of around �150 MPa in the axial and tangential directions.This is followed by a transition to tensile residual stresses in the unhardened layer amounting toþ350 MPa. Next, there was a transition to the central part of the specimen with a constant stressof around �200 MPa.

The radial residual stresses were almost all the time constant, i.e., sr¼ 0, from the surface to adepth z of 20 mm, and then they increased to around �200 MPa in the center of the cylinder.An efficient indicator of the variation of residual stresses was the martensite content. The martensite




0.2 0.4 0.6 0.8

Depth z (mm)

+400

+200

+200

−200

−400

−600

−800

−200

−400

−600

−1000

0

0

0

Depth z (mm)

RS measurements

Resid

ual s

tres

ses s

(MPa

)

Simulations

1.0 1.2

0.2 0.4 0.6 0.8 1.0 1.2

FIGURE 11.97 Residual stress profiles for the specimen treated at 10008C and 13008C. (From Fattorini, F.,Marchi Ricci, F.M., and Senin, A., Internal stress distribution induced by laser surface treatment. In: BLMordike, DGM, Ed., Proceedings of the European Conference on Laser Treatment of Materials (ECLAT).1992, 235–242.)

content showed the variation of the residual stresses occurring in the axial and tangential directions,which means that the martensite transformation had a decisive role in determination of the variationof residual stresses.

11.5.5 SIMPLE METHOD FOR ASSESSING RESIDUAL STRESS PROFILES

Grevey et al. [91] proposed a simple method for assessment of the degree of residual stresses after laserhardening of the surface layer. The method proposed was based on the knowledge of the parameters ofinteraction between the laser beam and the workpiece material, taking into account laser power and thetraveling speed of the laser beam across the workpiece and the thermal conductivity of the materialconcerned. The authors of the method maintained that in the estimation of residual stresses with low-alloy and medium-alloy steels, the expected deviation of the actual variation from the calculatedvariation of residual stresses in the thin surface layer did not exceed 20%.

On the basis of the known and expected variation, the authors divided the residual-stress profilesthrough the workpiece depth into three areas as shown in Figure 11.99 [91]:


100

80

60

40

Aus

teni

te (%

)

20

0

(A) (B)0 1 2 3 4

Mar

tens

ite (%

)

100

80

60

40

20

00 1 2 3 4

Depth z (mm) Depth z (mm)

(C)

Resid

ual s

tres

s sRS

(MPa

) 500

250

−250

−500

−1000

−750

0

sz

s rsϕ

0 8 15Depth z (mm)

22 30

FIGURE 11.98 Austenite distribution after 24 s (A), martensite distribution at the end of cooling (B) and sz,sf, and sr residual stress profiles (C). (From Ericsson, T., Chang, Y.S., and Melander, M., Residual stressesand microstructures in laser hardened medium and high carbon steels. Proceedings of the 4th InternationalCongress on Heat Treatment of Materials, Vol. 2, Berlin, 1985, 702–733.)

Tension

Depth below the surface z

Compressive

e0

Resid

ual s

tres

s sRS

(MPa

)

1200�C

TAC3

h

III

smax. css

smax

III

Tre-tempered

FIGURE 11.99 Residual stress profile, classified in typical areas. (From Grevey, D., Maiffredy, L., andVannes, A.B., J. Mech. Working Technol., 16, 65, 1988.)

. Areas I and II make up a zone from the surface to the limiting depth with compressivestresses defined in accordance with a TTA diagram (Orlisch diagram) for the given steel.

. Area III is the zone extending from the limiting depth, thus being an adjoining zonewhere the specific volume of the microstructure is smaller than that at the surface and,consequently, acts in the sense of relative contraction, which produces the occurrence of


tensile stresses in this zone. The zone showing tensile residual stresses is stronglyexpressed in the specimens, which were quenched and tempered at 4008C or 6008C andthen laser surface hardened. Because of a thermomechanical effect occurring duringsurface heating and fast cooling, certain residual stresses persist in the material aftercooling as well. This thermomechanical effect was related to the state of the materialprior to laser surface hardening.

Area III is known as the re-tempered zone of the material at the transition between the laser-hardened surface layer and the base metal, which is in a quenched-and-tempered state in the givencase. The thermal effects in laser heating occurring in the re-tempered zone were calculated by theauthors from the variation of the temperature in the individual depths of the specimen on a half-infinite solid plate:

T(z) ¼ Ts 1� erfz

2ffiffiffiti

p� �

(11:125)

whereTs is the surface temperature (8C)T(z) is the temperature at depth zTs¼ rPoa

1=2ti1=2=S l

h is the total energy efficiency (%)Po is the average power required (W)S is the spot area of the beam (cm2)a is the thermal diffusivity (cm2=s)ti is the interaction time (s)v is the traveling speed (cm=s)

An error function was approximated with (1� exp (� ffiffiffiffiffiffipu

p), which with 0.2< u< 2.0 provided

a favorable agreement between the theoretical and experimental results. The depth at which thetransition from compressive residual stresses to tensile residual stresses occurred was calculatedusing the following equation:

z ¼ � 4p

ffiffiffiffiffiffiffiffiffia � ti

plnT(z) � n1=2 � p � r3=2o � K

ra1=2 � Po

: (11:126)

The mathematical description of the variation of thermal cycle permitted them to predict the depth atwhich the austenite transformation TA3 occurred and the temperature range in which re-temperingoccurred.

The difficulty of the theoretical method consists in the requirement for knowledge of threeparameters: thermal conductivity l(W�1

cmC�1), thermal diffusivity a(m2=s), and total energy effi-

ciency h (%).Concerning the first two parameters, they depend on the type of material used, on its initial

microstructural state, and also on its temperature. They are linked by the relationship

a ¼ l=(r) � Cp (11:127)

wherer¼ 7.8 g=cm3 is the material densityCp is the specific heat, which depends on the temperature (J g�18C)l is thermal conductivity (W�

cmC�1)


Material: 35 NCD 16Power P = 1.75 kW Material: 35 NCD 16

Traveling speed: v = 5.5 mm/s

0.6(A) (B)

0.7

19 19

1817

16

1.4 1.6 1.7P0 (kW)

r (%

)

r (%

)

1817

16

0.8 v (cm/s)

FIGURE 11.100 Variation of efficiency recording (A) traveling speed and (B) laser-beam power. (FromGrevey, D., Maiffredy, L., and Vannes, A.B., J. Mech. Working Technol., 16, 65, 1988.)

In the calculations only the effective energy was taken into account; therefore, the efficiency ofmaterial heating and the global coefficient r (%), including optical losses along the hardened traceand the reflection of the laser beam from the specimen, were taken into account. Thus, the displaypower P0 was treated and calculated, by means of the global coefficient of efficiency r, the so-calledeffective power, Pe¼ r�P0. The authors experimentally verified the effective power and confirmedthe linear relationship as shown in Figure 11.100A and B. Figure 11.101 shows the results ofcalculations and measurements of longitudinal residual stresses.

The deviations could be defined with reference to the depth of the transition of the compressivezone into the tensile zone (Dz¼ zEXP – zEST) and the deviation of the size of the maximum tensilestress in the subsurface (DsmaxT¼smaxEXP�smaxEST< 4%).

Estimated

Long

itudi

nal r

esid

ual s

tres

s sL R

S (M

Pa)

Experiment

Depth z (mm)

sm

Material: NF 35 NCD 16(4% Ni alloyed steel)

quenched and temperedat 600�C

Laser surfacehardening condition

P0 = 1.75 kWv = 5.5 mm/s

e 0.5 1.0p h 1.5

400

200

−200

−400

FIGURE 11.101 Comparison of the experimental and estimated longitudinal residual stress profiles. (FromGrevey, D., Maiffredy, L., and Vannes, A.B., J. Mech. Working Technol., 16, 65, 1988.)


In the calculations the deviation between the experimental and estimated powers (DP¼PEXP�PEST< 3%) was taken into account.

A very simple and practical method of determination of the through-depth variation of longi-tudinal residual stresses in the laser-hardened trace was proposed. The procedure was based onmeasurement of longitudinal residual stress at the surface ss and measurement of different charac-teristic depths by means of metallographic photos or from the through-depth hardness profiles,which are defined by e, p, and h (Figure 11.99), i.e.,

sm ¼ � pþ e

h� pss: (11:128)

An interesting study was made by Yang et al. [92] in his paper titled ‘‘A study on residual stresses inlaser surface hardening of a medium carbon steel’’ by using a 2-D finite element model. By using theproposed model, the thermal and residual stresses at laser surface hardening were successivelycalculated. The phase transformation had a greater influence on the residual stress than thetemperature gradient. The simulation results showed that a compressive residual stress regionoccurred near the hardened surface of the specimen and a tensile residual stress region occurredin the interior of the specimen. The maximum tensile residual stress occurred along the center of thelaser track in the interior region.

The compressive residual stress at the surface of the laser-hardened specimen has a significanteffect on the mechanical properties such as wear resistance and fatigue strength.

The size of the compressive and tensile regions of the longitudinal residual stress for variousspot ratios of the square beam mode is shown in Figure 11.102. It should be observed that withincreasing beam width the compressive region becomes wide but shallow.

Transverse direction y (mm)0

0

1

CompressionTension

Compression

Beam shape

Compression

Dep

th d

irect

ion

z (m

m)

Compression

TensionCompression

Beam shape

Compression

Tension

Beam shape

2 d3

d(4.5mm)

1.5 d2

30

1

2

30

1

2

3

1 2 3 4

FIGURE 11.102 Compressive and tensile regions of the longitudinal residual stress for various square beamstructural modes at constant laser power. (From Yang, Y.S. and Na, S.J., Surf. Coatings Technol., 38, 311,1989.)


00

1

2

30

1

2

30

1

2

3

1Compression

Compression

Dep

th d

irect

ion

z (m

m)

Transverse direction y (mm)

Tension

Compression

Tension

Tension

Compression

Compression

Compression

P = 1110 Wv = 30 mm/s

P = 740 Wv = 20 mm/s

P = 1480 Wv = 40 mm/s

2 3 4

FIGURE 11.103 Compressive and tensile regions of the longitudinal residual stress for various laser powersand traveling speeds at given input energy. (From Yang, Y.S. and Na, S.J., Surf. Coatings Technol., 38, 311,1989.)

From the comparison of the results, it is recommended that wide laser-beam spots are used forobtaining the desirable heat-treated region.

Figure 11.103 shows the sizes of the compressive and tensile regions of the longitudinal residualstress for various laser-beam powers and traveling speeds at the given input energy. Although theinput energy is constant, the compressive residual stress region increases according to increasedlaser power and traveling speed. This means that it is desirable to use the high-power beam and hightraveling speed at laser surface hardening.

Estimation and optimization of processing parameters in laser surface hardening was explainedby Lepski et al. [93]. Optimum results were obtained if the processing was based on temperaturecycle calculations, taking into account the material properties and input energy distribution. User-friendly software is required in industry application of laser surface hardening. The software shouldfulfill the following criteria:

. Ability to check any given hardening problem

. Ability to estimate without experiments which laser power beam shaping or beam scanningsystem is selected

. Ability to predict the laser hardening results at given application

. Ability to calculate the processing parameters with minimum cost at desired hardening andannealing zone size

. Graphic presentation of relationship between the processing parameters and the hardeningzone characteristics


3.00.1

z

z

z

z

1 1060

50

40

30

20

w

w

w

w

10

0

2.5

2.0

1.5

1.0

0.5

0.00.1

Spot axis ratio Y : X

Har

deni

ng d

epth

z (m

m)

Lase

r tra

ck w

idth

W (m

m)

1 10

v = 1,000 mm/minv = 2,000 mm/minv = 5,000 mm/minv = 10,000 mm/minCooling rate too smallPower P = 10 kW80% Absorbtionsteel C45

FIGURE 11.104 Influence of laser spot axis ratio and various traveling speeds on obtained depth and widthof single hardened track. (From Lepski, D. and Reitzenstein, W., Estimation and optimization of processingparameters in laser surface hardening. Proceedings of the 10th Meeting on Modeling of Laser MaterialProcessing, Igls=Innsbruck, 1995, 18 pp.)

The integration of laser hardening in complex manufacturing systems requires hardening with hightraveling speeds. In order to get a sufficient hardening depth both surface maximum temperature andlaser interaction time must not fall below certain limits even for high traveling speeds.

This may be achieved to a certain degree by laser spot stretching along the traverse direction. InFigure 11.104 the track depth as well as the track width are represented for the steel C45 asfunctions of the spot axis ratio SAR (SAR¼ y=x¼ 1 . . . 10) for various values of the travelingspeeds (v¼ 1–10 m=min) and a laser power of 10 kW. Values less than unity of the ratio SARcorrespond to a spot stretched along the traverse direction.

11.5.6 PREDICTION-HARDENED TRACK AND OPTIMIZATION PROCESS

Marya et al. [94] reported the prediction of hardened depth and width and the optimization processof laser transformation hardening. They used a dimensionless approach for Gaussian and rectangu-lar sources to find laser heating parameters at given dimensions of the hardened layer.

Laser transformation hardening has been performed on a 0.45% carbon steel, coated with acarbon to maintain the surface absorptivity to about 70%. Figure 11.105 calculated results fromMarya et al. [94] according to their model of predicted dimensions of the hardened layer and processoptimization.

The diagram shows the influence of dimensionless power (q*) and traveling speed (v*) ondimensionless hardened depth (Zh*) for Gaussian beam. Thus, any useful combination of laserprocessing parameters must maximize heat diffusion to required depth of hardened layer. It isnecessary that surface melting temperature is reached.

Figures 11.105 and 11.106 show that a low dimensionless traveling speed (v*) is necessary toallow heat conduction in depth and achieves high dimensionless depth Zh*¼ Zh=R and dimension-less width Wh*¼Wh=R. The dimensionless power parameter (q*) is determined with respect to


q∗ = 08.6 (Li)q∗ = 10.5 (Present work)q∗ = 15.0 (Present work)q* = 17.9 (Li)q∗ = 29.8 (Steen)q∗ = 46.2 (Steen)q∗ = 58.0 (Present work)q∗ = 69.3 (Steen)

Data from Li and Steen

Dimensionless speed v∗

1 10 100 1000

1

Gaussianbeam

Decreasing accuracy

Onset ofsurfacemelting

q∗ = 10

0.01

0.1

q * = 8

q∗ = 15 q∗ = 30 q∗ = 50Dim

ensio

nles

s har

dene

d de

pth

Zh∗

FIGURE 11.105 Influence of dimensionless power (q*) and traveling speed (v*) on dimensionless hardeneddepth (Zh*) for Gaussian beam. (Marya, M. and Marya, S.K., Prediction & optimization of laser transformationhardening. In: M. Geiger and F. Vollersten, Eds. Proceedings of the 2nd Conference ‘‘LANE’97’’: LaserAssisted Net Shape Engineering 2. Erlangen, 1997, Meisenbach-Verlag GmbH., Bamberg, 693–698.)

dimensionless traveling speed (v*) to reach the melting onset. If the laser beam moves faster, greatervalues (q*) must be selected to reach surface melting.

Similar calculations were realized for square laser-beam power density. Hardened widths shouldcorrespond rather well to beam spot diameter because the step energy gradient of the beam edgeshould produce an evenly steep temperature gradient.

Present workq∗ = 10.5

q* = 10

q∗ = 5 q∗ = 8 q∗ = 15 q∗ = 30q∗ = 50

q∗ = 15 q∗ = 58.0

Gaussianbeam

Onset of surface melting

0.1 1 10 100 1000Dimensionless speed v ∗

1

2

0.3

3

4

Wh∗

Dim

ensio

nles

s wid

th

FIGURE 11.106 Influence of dimensionless power (q*) and traveling speed (v*) on hardened width (w*) forGaussian beam. (Marya, M. and Marya, S.K., Prediction & optimization of laser transformation hardening. In:M. Geiger and F. Vollersten, Eds. Proceedings of the 2nd Conference ‘‘LANE’97’’: Laser Assisted Net ShapeEngineering 2. Erlangen, 1997, Meisenbach-Verlag GmbH., Bamberg, 693–698.)


3.5

2.5

1.5

0.5

0

3

2

1

Transition remelting/hardening

Decreasingcooling

rates

Remelting

3000 W

ΔHV = 140

ΔHV = 420

ΔHV = 540

2000 W

1 2 3 4 5Spot radius R (mm)

Har

dene

d de

pth

z (m

m)

6 7 8 9 10

1000 W

FIGURE 11.107 Influence of spot radius and laser-beam power on hardened depth. (Marya, M. and Marya,S.K., Prediction & optimization of laser transformation hardening. In: M. Geiger and F. Vollersten, Eds.Proceedings of the 2nd Conference ‘‘LANE’97’’: Laser Assisted Net Shape Engineering 2. Erlangen, 1997,Meisenbach-Verlag GmbH., Bamberg, 693–698.)

Results in stationary laser beam hardening show that melting cannot be achieved at dimension-less power below 7.6. Similar conclusions have already been drawn for Gaussian beams. Indeed, asthe beam speed approaches zero, the beam profile contribution decreases since heat tends todissipate more uniformly.

Figure 11.107 shows optimized beam spot dimensions at the surface, which are very wellpredicted by the theoretical analysis. Moreover, the results show that hardened depth increases asheat input energy. Authors experimentally verified that melting conditions are proportional to thespot dimension.

Although increasing power and spot beam diameter produced a wider hardened layer, thecooling rates decreased significantly.

Figure 11.107 shows the variation in surface hardness according to the hardness of base material(HV¼ 205). In an optimization process, a compromise between a high-hardened depth and asignificant hardness increase must therefore be found.

In common with Gaussian, spot radius is defined as the distance from the beam spot center tothe position at which the intensity has fallen to (I=e) times the peak value. This intensity is definedas follows:

q(x, y) ¼ A � n � qp � R2

� exp �nx2 þ y2

R2

(11:129)

For rectangular beams, the spot size depends on two variables, and therefore other dimensionaltransformations must be done. For convenience, the spot radius R used for a Gaussian beam issimply replaced by (Ly=2) in (q*), (Lx=2) in (v*), (x*), and (z*).

As a result of these variable changes, the conditions for hardening (T¼ TA1) and onset ofmelting (T¼ Tm) are respectively defined by the dimensionless temperatures (T*¼ 1) and (T*¼ 2.1)for mild carbon steels. For thick plates in adiabatic conditions with constant thermal properties,the temperature field developed in quasi stationary state from the superposition of punctual heatsources over a Gaussian region should adequately simulate the process. Accordingly, the tempera-ture field equation, after being transformed in the dimensionless form T* (x*, y*, z*), can beformulated [94].


T* ¼ q*2p

ffiffiffiffip

pð10

1

t*þ 1=nð Þ ffiffiffiffit*

p � exp � x*þ v* � t*=4ð Þ2þ y*ð Þ2t*þ 1=n

� z*ð Þt*

" #dt (11:130)

To maintain the necessary (q*) of surface melting, increasing beam powers associated with reducedspeeds are required.

q

Rl Tm � T1ð Þ ¼ exp 1,4826 � vR

a

� �0,1709" #

(11:131)

Under these particular conditions of surface melting, the largest transformed depths and widths aresimultaneously encountered. However, the mechanical properties of the transformed zone, morespecifically its hardness, are governed by cooling times. As the beam moves faster, cooling isenhanced, and the ensuing quench is capable of yielding harder transformed zones [94].

11.5.7 APPLICATION OF MODELING

To achieve a precise and controlled hardening laser process, a thorough analysis of the thermalbehavior of the material is necessary. Yánez et al. [95] presented a numerical simulation of the laserhardening process using both analytical solutions and the finite element code ANSYS to solve the heattransfer equation inside the treatedmaterial. The knowledge of the thermal cycles has enabled suitableprocessing parameters to be ascertained, thus improving surface properties when metallic alloysirradiated. A simpler analytical method is also used to determine processing parameters more quickly[96–100].

11.5.7.1 Analytical Model

An analytical model provides a description of the time-dependent temperature field induced by theincident laser beam on the surface and inner parts of the workpiece. Given that the heat conductionequation for a certain source f(r, t) is

rC@T

@tþr(�lDT) ¼ f (r, t) (11:132)

wherer is the densityC is the specific heatT is the temperaturet is the timel is the thermal conductivity

If r, C, and l are temperature and position independent, the equation is simplified to

1k

@T

@t�r2T ¼ f (r, t), (11:133)

where a¼ l=rC is the thermal diffusivity.If a flat and semi-infinite workpiece, initially at room temperature T0, is treated with a time-

dependent laser heat source on plane z¼ 0, the temperature field takes the following shape:

T(r, t) ¼ 2al

ð10

ð1�1

ð1�1

f (x0, y0, t0): (11:134)

XG(=r ¼ r0=, t � t0) dx0 dy0 dt (11:135)


where G is the Green function with a shape derived from the Fourier transformation:

G(=r � r0=, t � t0) ¼ e�=r�r0==4a(t�t0)(4p0a(t � t0))�3=2, (11:136)

the solution factorizes

Gx(=x� x0=, t � t0 ¼ e�=x�x0=4a(t�t0)(4pa(t � t0))�1=2 (11:137)

f(x0, y0, t0) is the beam intensity distribution, which in the TEM01* case has the shape

f (x0, y0, t0) ¼ g(t0)4[(x0 � vt0)2 þ y

02]

pv4e�2[(x0�vt0)2þy

02]=v2(11:138)

If a workpiece of finite depth (h) is treated, a new boundary condition shows up, which can beattained through application of the image method. An infinite number of images is needed but as thedistance from the surface increases a corresponding decrease in their importance occurs and a smallnumber of them is enough to ensure a good approach to the solution. An equivalent procedure isapplied when limits in length and width are taken into account.

Authors Yánez et al. [95] extended the method to the cylindrical case taking into account thefollowing changes: A (h) thickness, (R) outer radius (h� R) ring considered as a thin slab of (2pR)length and non adiabatic borders located at x¼ 0 and (2pR), and periodic temperature field with(2pR) period

T(x, y, z, t) ¼ T(xþ 2pR, y, z, t) (11:139)

The non adiabatic edges are connected in such a way that the outgoing heat flow at x¼ 0 becomesincoming heat flow at x¼ 2pR. The effect is achieved by adding sources in x-direction withperiod (2pR).

11.5.7.2 Case of Cylindrical Workpieces

When processing a ring, the way in which the beam moves over the surface must guarantee auniform treatment. Once the helix has been chosen as the appropriate method and the beam size andintensity have been fixed along with the relative velocity of the beam workpiece, the only parameterto be determined is the displacement in the y-direction when a round is completed. This parameter isconnected with the overlap between two consecutive passes and is essential to arrive at the desireduniformity.

The traveling of the beam over the cylindrical surface causes it to harden as well as increases theoverall temperature.

A combination of analytical and numerical techniques appears an efficient way to ascertain theprocess parameters needed to modify and improve the surface properties of stainless steels. Gettingthe different beam passes to overlap correctly turns out to be a difficult task from the experimentalpoint of view; the same is true for the determination of the time-dependent input power, which isnecessary to compensate for the variation in heat resulting from previous scans. Maintaining themaximum surface temperature at a constant value guarantees a good degree of homogeneity [96–100].

All these results can also be obtained using the finite element code ANSYS. The analyticalmodel is faster when calculating the evolution of a small number of points but slows down whenthe number of points increases because the computer time is proportional to this number.A treatment close to that simulated by the analytical model was performed using a constant inputpower of 1900 W, with a mesh of 32,000 nodes and a calculation of 4000 time steps. Under theseconditions, the absolute error in the temperature is less than 50 K. In Figure 11.108, the temperature


1600

1275

950

625

Tem

pera

ture

T (K

)

30050 50.5 51

z = 1 mm

2 mm3 mm

4 mm5 mm

6 mm

Time t (s)51.5 52 52.5

FIGURE 11.108 Temperature evolution at various depths below the surface. (From Yanez, A., Alvarez, J.C.,Lopez, A.J., Nicolas, G., Perez, J.A., Ramil, A., and Saavedra, E., Appl. Surf. Sci., 186, 611, 2002.)

evolution of points with the same x- and y-values, but with different z is shown; it is clear thatbelow 1 mm depth below the surface the maximum temperature goes beyond both TAc1 and TAc3values, and hardening is achieved. Two different points of the treated surface were chosen to testhomogeneity:

1. In the trajectory described by the center of the intensity distribution2. On the edge of that distribution. Figure 11.109 shows results of the hardness measurement

(HRC), which proves that heat treatment is effective with homogeneous microstructure

80

60

40

20

00

(A)1 20.5 2.51.5

Depth z (mm)

Rock

wel

l har

dnes

s HRC

80

60

40

20

00

(B)1 20.5 2.51.5

Depth z (mm)

Rock

wel

l har

dnes

s HRC

FIGURE 11.109 Rockwell HRC hardness as a function of depth in two selected points A and B. (FromYanez, A., Alvarez, J.C., Lopez, A.J., Nicolas, G., Perez, J.A., Ramil, A., and Saavedra, E., Appl. Surf. Sci.,186, 611, 2002.)


1000

900

800

700

600

Tem

pera

ture

T (�

C)

500

400

300

20010−2 10−1 100

TypicalAnnealingTempering

101

Time t (s)

Ac3

Ac1

Ms

Mf

a b

cFerrite

Perlite

Bainite

d

e

102 103

FIGURE 11.110 Typical heat treatment used to develop DP steel and two potential heat treatments toimprove DP steel formability. (From Capello, E. and Previtali, B. Enhancing dual phase steel formability bydiode laser heat treatment. Paper 510, Laser Materials Processing Conference, ICALEO, Congress Proceedings,2007; Li, M.V., Niebuhr, D.V., Meekisho, L.L., and Atteridge, D.G., Metall. Mater. Trans. B, 29(3), 661,1998; Gould, J.E., Khurana, S.P., and Li, T., Welding J., 85(5), 111s, 2006.)

Capello and Previtali [101] studied various surface treatment effects by diode laser on the localformability of dual phase (DP) steel. An analytical thermal model allowed the temperature andcooling rate curves to be predicted and used to select the process parameter conditions. Microhard-ness measurements, microstructure observations, mechanical tensile test, and Erichsen cup testallowed the positive effects of laser heat treatment on dual phase steel formability to be quantified.

This research was aimed at studying new local heat treatment of dual-phase steel sheet bydiode laser. Advantages of diode laser being mainly almost uniform power distribution, the highabsorption can be usefully applied to develop different heat treatment processes as well laserhardening [102–106].

The increase in formability should be obtained by a local change of the microstructureproperties. Dual phase steel microstructure is mainly constituted by ferrite and martensite phases.

This particular microstructure is obtained by controlled cooling from the intercritical region(a–b segment in the continuous line of Figure 11.110, to then transform some austenite to ferrite(b–c segment), before rapid cooling to transform the remaining austenite to martensite (d–esegment) [105,106].

The treatment profiles in Figure 11.110 are only indicative because different paths can comeout, in terms of maximum temperature and cooling rate for both the annealing and temperingtreatments.

Therefore, a thermal model of the diode laser heat treatment (DLHT) is needed, in order to findtwo feasible thermal cycles with features similar to those ideally required by annealing or temperingtreatment.

11.5.7.3 Prediction of the Heat Treatment Cycle by Analytical Thermal Model

Although numerical analysis is a powerful tool to reproduce realistic temperature fields, it requiressubstantial calculation time. Analytical approach to thermal modeling offers many advantages due


02

4

6

8sTm

ax-M

s

10

12

1416

1820

P = 400 W v = 1 mm/s

P = 500 W v = 1 mm/s

P = 1400 W v = 12 mm/s

P = 2200 W v = 12 mm/s

Them A Them B Model A, B

FIGURE 11.111 Comparison of modeled and measured cooling times (power P[W], traveling speed[mm=s]). (From Capello, E. and Previtali, B. Enhancing dual phase steel formability by diode laser heattreatment. Paper 510, Laser Materials Processing Conference, ICALEO, Congress Proceedings, 2007.)

to its simplicity and short time requirements. Rosenthal [102] was the first to develop a simplifiedthermal model applied to metal treatments based on the simple theory of a moving heat source.Since then a number of modified thermal models have appeared, incorporating more realisticassumptions.

Woo and Cho [104] derived quite an accurate 3D transient temperature model for laser heattreatment processes, which seems to be the appropriate tool for predicting the temperature field inDLHT process.

ANOVA analysis allowed the graphical results of Figure 11.111 to be confirmed, because nosignificant difference in dTmax – TMs between the modeled and the measured temperature profileswas detected [101]. The figure shows that the analytical thermal model also predicts the coolingtime in all the process parameter conditions with good accuracy.

If a generic thermal profile, experienced by a point of the workpiece during the DLHT process,is plotted on the CCT curves in Figure 11.112, two main considerations can be pointed out. Theheating and cooling thermal cycles produced by the DLHT process have very different shapes fromthe ideal profiles. In particular, the heating phase is very fast; the soaking period, where thetemperature is kept constant is absent, and the cooling phase is also very rapid. Moreover,the maximum temperature reached and the rapidity of the cooling time vary in accordance withall the inputs in the solution in equation. In particular, Tmax and dTmaxTMsvary with the pointposition and with the process parameters P and n [101].

As a result, different temperature profiles can be drawn on the CCT curves, which can be validas an annealing treatment. Similarly, different process conditions allow different tempering treat-ment profiles.

Figure 11.112 depicts two thermal profiles that can be representative of the annealing (A) andtempering treatment (T) for the DLHT process, obtained according to process parameters given inTable 11.5. In Figure 11.110, the continuous line shows the temperature profile experienced by thepoints belonging to the traveling direction of the beam center on the top surface. Similarly, thedashed line is the temperature profile of those points on the centerline of the bottom surface.

Therefore, the DLHT process is a valid technique, when the local ductility of the DP 800 steelhas to be increased, in view of future forming or drawing processes.


1000

Ac3

Ac1

DLHT-T

DLHT-A

FerritePerlite

BainiteMS

Mf

900

800

700

600

Tem

pera

ture

T (�

C)

500

400

300

20010−2 10−1 100 101

Time t (s)102 103

FIGURE 11.112 Temperature profile of the annealing DLHT-A and tempering DLHT-T process. (FromCapello, E. and Previtali, B. Enhancing dual phase steel formability by diode laser heat treatment. Paper 510,Laser Materials Processing Conference, ICALEO, Congress Proceedings, 2007.)

TABLE 11.5DLHT Experimented Conditions and Thermal Attributes(Top and Bottom Surfaces)

Type P (W) v (mm=s) E (kJ) Tmax (8C) dTmax-Ms

DLHT-A 400 1 21.2 996–904 21–16

DLHT-T 1400 12 5.3 623–573 2.1–1.4

Source: From Capello, E. and Previtali, B. Enhancing dual phase steel formability by

diode laser heat treatment. Paper 510, Laser Materials Processing Conference,

ICALEO, Congress Proceedings, 2007.

Capello et al. [101] presented a new analytical model, able to reproduce the DLHT processquite accurately. It was validated and used to define the process conditions, which produce twodifferent treatments of the material. In view of future development, the thermal model can bevery helpful in selecting and optimizing other process conditions that allow the tempering orannealing.

11.5.8 MICROSTRUCTURE ANALYSIS AFTER LASER SURFACE REMELTING PROCESS

Ductile iron is commonly used in a wide range of industrial applications due to its good castability,mechanical properties, and low price. By varying the chemical and microstructure composition ofcast irons, it is possible to change their mechanical properties as well as their suitability formachining. Ductile irons are also distinguished by good wear resistance, which can be raisedeven higher by additional surface heat treatment. With the use of induction or flame surfacehardening, it is possible to ensure a homogeneous microstructure in the thin surface layer; however,


200 μm

FIGURE 11.113 Cross section of a single laser-modified trace; remelting conditions: P¼ 1.0 kW, zs¼22 mm, and vb¼ 21 mm=s. (From Grum, J. and Šturm, R., Mater. Charact., 37, 81, 1996.)

this is possible only if cast irons have a pearlite matrix. If they have a ferrite–pearlite or pearlite–ferrite matrix, a homogeneous microstructure in the surface hardened layer can be achieved only bylaser surface remelting [108–112].

After the laser beam had crossed the flat specimen, a microstructurally modified track wasobtained, which was shaped like a part of a sphere (Figure 11.113).

To achieve a uniform thickness of the remelted layer over the entire area of the flat specimen(Figure 11.114), the kinematics of the laser beam were adapted by 30% overlapping of theneighboring remelted traces [107].

The microstructure changes in the remelting layer of the ductile iron are dependent on tem-perature conditions during heating and cooling processes. In all of the cases of the laser surfaceremelting process two characteristic microstructure layers were obtained, i.e., the remelted layer andhardened layer. Figure 11.114 shows the microstructure in the remelted surface layer, which is finegrained and consists of austenite dendrites, with very fine dispersed cementite, together with a smallportion of coarse martensite [113,114].

X-ray phase analysis of the remelted layer showed the average volume percentages of theparticular phases as follows: 24.0% austenite, 32.0% cementite, 39.0% martensite, and 5.0%graphite. Figure 11.115 shows the microstructure of the hardened layer consisting of martensitewith the presence of residual austenite, ferrite, and graphite nodules. Graphite nodules are sur-rounded by ledeburite or martensite shells.

150 μm

FIGURE 11.114 Laser surface modified layer at 30% overlap of the width of the remelted traces;remelting conditions: P¼ 1.0 kW, zs¼ 22 mm, and vb¼ 21 mm=s. (From Grum, J. and Šturm, R., Mater.Charact., 37, 81, 1996.)


30 μm

FIGURE 11.115 Microstructure of the hardened layer. (From Grum, J. and Šturm, R., Mater. Charact., 37,81, 1996.)

11.5.8.1 Mathematical Modeling of Localized Melting around Graphite Nodule

Roy et al. [115] described in their paper mathematical modeling of localized melting aroundgraphite nodules during laser surface hardening of austempered ductile iron. Similar findingswere presented by Grum et al. [116] for laser surface remelting in the TZ, while heating at lowpower beam P¼ 700 W and traveling speed v¼ 60 mm=s, where dissolution of the graphite nodulesat depth z¼ 100 mm below the surface occurred. At the heating process in the region with dissolutedgraphite in austenite, lower melting temperature was reached, which resulted in local remelting.

Roy and Manna, [115] presented mathematical modeling of localized melting around graphitenodules during laser surface hardening of austempered ductile iron. In order to correlate themicrostructural features of partial=complete local melting around graphite nodules with laser surfacehardening LSH parameters, an attempt has been made to analytically solve the concerned heatbalance equation and predict the thermal profile generated within the laser-irradiated zone. Themodel and solution are based on an earlier approach reported by Ashby and Easterling [56].Accordingly, the heat balance equation for heating=cooling a metallic sample following laserirradiation with a CW CO2 laser with a Gaussian energy deposition profile is given by

r2T � 1a

@T

@tþ rr

l¼ 0, (11:140)

whereT is the temperaturea is the thermal diffusivityrr is the amount of heat energy injected in the samples per unit volume per unit timel is the thermal conductivity of the samplet is the time

Figure 11.116 schematically shows the laser-beam profile and location of a spherical graphitenodule (G) at a given vertical depth (z) from the surface. Here, (ym) is the maximum width of theannular melt zone around the nodule. During laser surface hardening, the sample stage moves alongthe x-direction with a linear speed (n) and allows an average laser-material radiation time (t¼ 2r=n,


I0

I = I0 exp − —

(+) y

dzG

ymx

y

z(+) z

(+) y(−) y

(−) y

y2

r2

FIGURE 11.116 Schematic diagram showing the energy deposition profile (Gaussian) of the CW CO2 laserbeam and configuration (semicircular) of the laser-irradiated zone on the yz plane, respectively. G denotes apartially melted graphite nodule with an annular melted zone ofmaximumwidth ym and located directly under thecenter of the laser beam. dz is the case depth along depth z. (From Roy, A. and Manna, I., Optics and Lasers inEngineering, 34, 369–383, 2000.)

where r is the beam radius) over a circular region on the surface. The other necessary assumptions tosolve the equation are

. Radiative heat loss from the surface is negligible.

. Thermal and optical properties of the material are not functions of temperature.

. Heat flow takes place under a quasi-stationary, state implying that the heated zone of aconstant volume will move together with the heat source at the same velocity.

. Melting initiates once the temperature exceeds the solidus temperature by 508C and latentheat is supplied.

It may be pointed out that preliminary studies have earlier shown that a minimum superheating of508C difference, between the peak temperature generated by the laser radiation and concernedsolidus temperature, is necessary to initiate melting during the present set of laser surface hardeningexperiments. Ideally, a precise estimation of the thermal profile including that during the laser-induced melting is possible through a numerical solution of the heat balance equation.

The equation may be analytically solved under the following boundary conditions:

@T=@z ¼ 0 for z ¼ 0, T ¼ 0 at z ¼ 1, and T ¼ 0 at t ¼ 0: (11:141)

The analytical solution equation under these assumptions and boundary conditions yields

T � T0 ¼ Aq=n

2pr t t þ t0ð Þ½ �1=2exp � 1

4az2

tþ g2

t þ t0ð Þ� �

, (11:142)


whereT0 is the initial temperature of the substrateA is the absorptivity at the sample surface materialq is the incident laser powera¼ l=rX is the thermal diffusivityr is the densityc is specific heatr is the radius of the beamt0 t¼ r2=4a is the time taken for heat to diffuse over half the beam width

Temperature profiles as a function of time have been determined for different locations (x, y),with incident power (P) varying from 500 to 1000 W, and traveling speed (v) varying from 20 to60 mm=s.

To calculate the carbon concentration profile around the graphite nodules the following equationwas used:

C(y, t) ¼ Cf � Ce

21� erf

y

2ffiffiffiffiffiDt

p

þ Ce, (11:143)

where Cf is defined as the carbon content of austenite at the graphite–matrix interface and isassumed equal to 2.2 wt% carbon as per the maximum solubility of carbon in austenite, and Ce isthe matrix carbon concentration assumed to be 0.72 wt%. Since laser surface hardening=lasersurface remelting involves transient or dynamic heating=cooling, the thermal effect for agiven interaction time is approximated by replacing the quantity Dt in the equation by D0ai(edp–[Q=RTp]) [10], where a is the kinetic strength (it is the characteristic time constant of thetemperature pulse generated by the laser beam), Tp is the maximum (peak) temperature, and D0

(taken as 10�5 m2=s) is the preexponential factor for carbon diffusion in austenite.Figure 11.117A and B show the concentric regions around the graphite nodule as follows:

dendritic retains austenite interfacial region (RAþmartensite) and hardened region (martensite). Itcan be noted that the higher volume fraction of the retained austenite causes detrimental effects onwear properties of the surface.

Figure 11.118 shows temperature cycles during laser surface hardening at P¼ 700 W andv¼ 60 mm=s at different depths below the surface.

5 μm(A) (B)

H

I

G HMG. . GraphiteM. . Remelted zoneI. . . Interfacial zoneH. . Hardened matrix

ym

FIGURE 11.117 SEM (A) showing the partial incipient melting of graphite nodule due to laser remeltingwith P900N and v¼ 60 mm=s and a schematic presentation of microstructural regions (B). (From Roy, A. andManna, I., Optics and Lasers in Engineering, 34, 369–383, 2000.)


0 0.050

400

800

1200

0.10Time t (s)

Tem

pera

ture

T (�

C)

RemeltingconditionsP = 700 Wv = 60 mm/s

z = 0 μm50100150200300400

0.15 0.20

FIGURE 11.118 Temperature cycles at the surface and at different depths for given remelting conditions.(From Roy, A. and Manna, I., Optics and Lasers in Engineering, 34, 369–383, 2000.)

Figure 11.119 gives the variation of carbon concentration as a function of the graphite noduledistance from the surface. From the diagram it can be concluded that the higher carbon concentra-tion in the matrix is on the surface and lower with increasing the depth below the surface.The dashed line shows 1.6 wt% carbon, which determines the minimum level of carbon enrichmentand the maximum width ym of the localized remelted zone around the graphite nodule.

Figure 11.120 gives the matrix carbon concentration and maximum temperature as a function ofgraphite interface distance dy. The dashed horizontal line denotes the effective temperature andmaximum carbon solubility 1.6 wt% for austenite, and the vertical one represents the given remeltwidth (ym) from the graphite surface.

Figure 11.121 shows the changing of remelted width (ym) as a function of the depth below thesurface (z) at given laser surface hardening conditions. Solid symbols in the diagram showtheoretically predicted maximum melt width ym around the graphite nodule as a function of thedepth below the surface. The open symbols note the experimental data at given laser hardening

0 20 40 60Distance from graphite–matrix interface

dy (μm)

Mat

rix ca

rbon

conc

entr

atio

n (w

t%)

1.4

1.6

1.8

2.0

2.2

z = 0 μm50 μm100 μm

150 μm200 μm250 μm300 μm350 μm

80 100

FIGURE 11.119 Variation of matrix carbon concentration as a function of distance from graphite matrixinterface. (From Roy, A. and Manna, I., Optics and Lasers in Engineering, 34, 369–383, 2000.)


01120

1130

1140

1150

1160

1170

1180

20 40 60

1.6 wt% carbon

1147�C

Mel

t with

(ym

)(E

xper

imen

tal d

ata)

Distance from graphite−matrix interface dy (μm)

Tem

pera

ture

T (�

C)

80 100

Peak temperatureMatrix carbon concentration

1200.5

1.0

1.5

2.0

Mat

rix ca

rbon

conc

entr

atio

n(w

t%)

2.5

FIGURE 11.120 Variation of the peak temperature and matrix carbon concentration according to the distancefrom graphite matrix interface. (From Roy, A. and Manna, I., Optics and Lasers in Engineering, 34, 369–383,2000.)

conditions and depth below the surface as well. Figure 11.122 shows the microhardness profilesbelow the surface at given laser surface hardening conditions.

Based on all given data for laser surface hardening of austempered ductile iron (ADI), it can beconcluded that

. Depth of hardened layer is directly proportional to laser-beam power and interaction time

. Laser surface hardening primarily gives martensitic microstructure up to the lower limitlevel of the dissoluted carbon

. Higher dissolution of the carbon in the austenite around the graphite nodules at heatinggives the retained austenite after cooling

. Mathematically determined maximum width around the graphite nodules compares wellaccording to the experimental results

10080

100

80

60

40

20

120 140Depth bellow the surface z (μm)

P = 700 Wv = 60 mm/s

P = 1000 Wv = 20 mm/s

Rem

elte

d w

idth

ym

(μm

)

FIGURE 11.121 Variation of remelted width ym according to remelting depth z theoretical predicted-solidsymbols and experimental data-open symbols. (From Roy, A. and Manna, I., Optics and Lasers in Engineering,34, 369–383, 2000.)


1500200

400

600

800

1000

1200

1400

300 450Depth z (μm)

Mic

roha

rdne

ss H

V

P = 650 W

P = 500 W

FIGURE 11.122 Microhardness profiles according to remelting conditions. (From Roy, A. and Manna, I.,Optics and Lasers in Engineering, 34, 369–383, 2000.)

Transition between the remelted and hardened layers. Grum et al. [116] studied rapidsolidification of a microstructure in the remelting layer and microstructure changes in thehardened layer.

The application of laser surface remelting to nodular iron 400-12 causes the material to undergomicrostructural changes. A newly created austenite–ledeburite microstructure with the presence ofgraphite nodules in the remelted layer and a martensite–ferrite microstructure with graphite nodulesin the hardened layer have been observed. Microscopy of the hardened layer was used to analyze theoccurrence of ledeburite shells and martensite shells around the graphite nodules in the ferritematrix. The thickness of the ledeburite and martensite shells was supported by diffusion calcula-tions.

The qualitative effects of the changed microstructures were additionally verified by microhard-ness profiles in the modified layer and microhardness measurements around the graphite nodules inthe hardened layer.

The tests involved the use of an industrial CO2 laser with a power of 500 W and Gaussiandistribution of energy in the laser beam. The optical and kinematic conditions were chosen so thatthe laser remelted the surface layer of the specimen material. The specimens were made fromnodular iron 400-12 with a ferrite–pearlite matrix that contained graphite nodules.

A characteristic of the transition area between the remelted and the hardened layer is that localmelting occurs around the graphite nodules. Grum et al. [116] showed in Figure 11.123 a schematicrepresentation of the sequence of processes occurring in the phases of heating and cooling:

. Matrix transforms into a nonhomogeneous austenite

. Follows diffusion of carbon from graphite nodules into austenite

. Increased concentration of carbon in the austenite around the graphite nodule lowered themelting-point temperature and local melting of part of the austenitic shell around thegraphite nodule occurred

. After rapid cooling, a ledeburite microstructure is formed locally; this is then furthersurrounded by a martensite shell

The transition area is very narrow and very interesting from the microstructural point of view.However, it is not significant in determining the final properties of the laser-modified surface layer.

Figure 11.124 shows the measured and calculated thickness of martensite shells around graphitenodules with respect to their distance from the remelted layer.


Energy input

Aus

teni

tic-p

hase

rich

in ca

rbon

Matrixtransforms

into austenite

Heating Hardening layer Hardening layer

Pearlite Martensite

Martensite Martensite

Martensite

Martensiteshell Ledeburite

shell Martensiteshell

Microstructurenodular iron

400–12

FerriteFerrite Ferrite

Austenite Austenite

Pearlite

Graphite

Aus

teni

tizat

ion

tem

pera

ture

Room

tem

pera

ture

Local temperature isless than melting

point temperature

Local temperature riseonto material melting

point temperatureCarbon diffusesfrom graphite

Carbon diffusesfrom graphite

(A) (E) (F)

(B) (C) (D)

FIGURE 11.123 Schematic presentation of the microstructural changes in the transition area between theremelted and the HZs. (From Grum, J. and Šturm, R., Mater. Charact., 37, 81, 1996; Grum, J. and Šturm, R.,J. Mater. Eng. Perform., 10, 270, 2001.)

00

5

10

15

20Nodular iron 400-12

Surface

Martensite shell

Ledeburite shellGraphite

d

MeasurementCalculation

100 200Nodule graphite distance from remelted layer z (μm)

Dist

ance

from

gra

phite

d (μ

m)

300 400

z

FIGURE 11.124 Comparison of measured and calculated thicknesses of martensite shells aroundgraphite nodules in the HZ. (From Grum, J. and Šturm, R., Mater. Charact., 37, 81, 1996; Grum, J. andŠturm, R., J. Mater. Eng. Perform., 10, 270, 2001.)


For an assessment of the martensite shells around the graphite nodules, the shell sizes were alsomeasured with respect to the distance of the graphite from the melted zone. Gradually, withincreasing depth, the thickness of the martensite shell decreased to zero, which also represents theboundary between the HZ and the base material. In laser heat treatment, the time–temperaturevariation at particular depths with a simple temperature function can be defined, and by consider-ation of the melting-point temperature or the austenitization temperature, determine the processesoccurring at specific points of observation. On this basis, the thickness of the martensite shell in theHZ can be calculated using diffusion equations and the results compared with the measured values.In the calculation, for different temperature ranges, different activation energy values (QA) anddiffusion constants (D0) were chosen according to the literature [117].

The carbon diffusion distance x in time t is calculated according to the diffusion equations:

x ¼ffiffiffiffiffiffiffiffiffiffiffiffi(2Dtt)

p(11:144)

Dt ¼ D0e�QA=RT (11:145)

whereDt is the diffusion coefficient (m2=s)t is the time (s)T is the temperature (K)R is the universal gas constant (8314 J kmol�1 K�1)

Our estimation is that the differences between the measured and the predicted martensiteshell thickness are within the expected limits since the data on heat conductivity and diffusivitywere chosen from the literature. The calculations confirmed the validity of the mathematical modelfor the determination of temperature T and time t of remelting in a given temperature range, whichenabled us to define the diffusion path of the carbon or, in other words, the thickness of themartensite shell.

Circumstances for rapid solidification process of cast iron. Three features are significant for therapid solidification process after laser remelting [109]:

. Cooling rate _« ¼ dT=dt

. Solidification rate R¼ dx=dt, which characterizes crystal grains growth per time unit in theliquid=solid interface

. Temperature gradient G¼ dT=dx across the liquid=solid interface to given location

These parameters are connected by the equation

_« ¼ RG: (11:146)

Depending on the values of the above variables, different microstructures can be produced afterthe solidification process. It is important that different microstructures can be obtained in the samematerial at various solidification conditions. In Figure 11.125, one set of parallel representslines equal to G=R ratios and the other, rectangular to the former that represents equal RG products(equal _«) [109].

TheG=R parallel lines give similar solidification conditions before solidification increases with _«.Therefore, an increasing nucleation frequency is obtained resulting in a finer microstructure of thesame morphology. For the set of lines where _« is constant, the same grain size occurs, with changedsolidification.


Solidification rate R (mm/s)

Tem

pera

ture

gra

dien

t G (K

/mm

)

10−310−1

101

103

Planar solidification, single face

Dendritic solidification

(g)

Cellular solidificationPlanar solidification, two faces

10−1 101

GIR = consta

nt

e • = constant

FIGURE 11.125 Variation of microstructure of a cast iron with solidification conditions. (From Bergmann,H.W., Surf. Eng., 1, 137, 1985.)

At laser remelting, it is difficult to verify experimentally the two limiting cases of solidification.Therefore, special conditions must occur at rapid solidification:

. Substantial superheating of the melt, which influences the heterogeneous nucleation

. Extreme temperature gradients, which assure rapid, directional solidification

. Epitaxial growth on substrate crystals

In Figure 11.126, the cooling rate, remelted depth, and dendrite arm spacing are correlated [109].Laser processing models relate to beam characteristics, material properties, and processing param-

eters. Effective models include combination of various physical and chemical processes enabling

10−2 10−1 100

Dendrite spacing l (μm)101 102

100

102

104

106

108 0.001

0.0050.01

0.050.1

0.5

E(λ)

e• (λ)

Dep

th z

(mm

)

Cool

ing

rate

e• (K/

s)

FIGURE 11.126 Variation of dendrite arm spacing with solidification parameters. (From Bergmann, H.W.,Surf. Eng., 1, 137, 1985.)


insight into various laser treatments of the material. Models provide a better understanding of theprocess and interactions between the process variables. The process can be simulated meaningthe reduction of expensive testing during the optimization and certification of procedure. Modeling isa valuable tool in the design process, particularly in the selection of material and processing parametersand the scheduling of production sequences.

The solution to a model can be obtained by using analytical or numerical methods. Analyticalmodels using justifiable assumptions allow a relationship that enables the effects of variations inprocess variables on the product characteristics to be visualized.

Numerical techniques require fewer assumptions, can produce more exact results if input dataare known accurately, but require a more sophisticated method of solution. The details needed in themodel, and the method by which the formulation is solved, should correspond to the complexity ofthe problem, the reliability of input data, and the accuracy required in the results.

The approach adopted is predominantly analytical. The methods can be solved by using apersonal computer, and results are obtained quickly. The expected accuracy is �5%, which reflectsthe typical accuracy of data available for modeling.

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