high precision microscale

The use of a pulsed and a CW laser for microscale bending of ceramics, silicon, and stainless steel is demonstrated.. The relation between bending angles and laser operating param-eters,

X Richard Zhang

Xianfan Xu

School of Mechanical Engineering,

Purdue University, West Lafayette, IN 47907-1288

High Precision Microscale Bending by Pulsed and CW Lasers

This paper discusses high precision microscale laser bending and the thermomechanical phenomena involved The use of a pulsed and a CW laser for microscale bending of ceramics, silicon, and stainless steel is demonstrated For each laser, experiments are conducted to find out the relation between bending angles and laser operation param-eters Changes of the ceramics surface composition after laser irradiation are analyzed using an electron probe microanalyzer (EPMA) Results obtained by the pulsed and the

CW laser are compared, and it is found that the CW laser produces more bending than the pulsed laser does However, the pulsed laser causes much less surface composition change and thermomechanical damage to the targets Numerical calculations based on the thermo-elasto-plastic theory are carried out and the results are used to explain the phenomena observed experimentally. 关DOI: 10.1115/1.1580528兴

Introduction

Laser bending or laser forming is a technique of using the

en-ergy from a laser beam to modify the curvature of sheet metals or

hard materials There are several mechanisms for laser bending,

depending on the geometry and the thermophysical properties of

the target material, and the laser processing parameters Common

laser bending mechanisms include the temperature gradient

mechanism and the buckling mechanism关1兴 For the temperature

gradient mechanism, only the surface layer is heated, thus, there is

a temperature gradient with the highest temperature at the surface,

and the residual stress and strain are concentrated in the near

surface region The residual strain is compressive, causing the

target to bend toward the laser beam This type of bending is

preferred when a consistent bending direction is required If laser

heating is uniform throughout the thickness of the target, the

tar-get will bend just like a beam under compression In this case,

bending is caused by the buckling mechanism and the bending

direction depends on the pre-curvature and the initial stress of the

target

Applications of laser bending include ship body construction

关2兴, removing welding distortion and straightening car body parts

关3兴, and rapid prototyping 关4,5兴 Laser bending was also studied

for forming in space stations 关6,7兴 and for bending of

micro-electronic components关8兴 Recently, Chen et al 关9兴 studied high

precision laser bending for manufacturing computer components

They achieved a bending precision of sub-microradian, far

ex-ceeding those obtained by any other method

Numerical simulation using the finite element method is an

ef-fective way to study the influence of laser operating parameters

and the target geometry on bending关10–13兴 While continuous

wave共CW兲 lasers are used in most laser bending operations, a 2D

simulation of pulsed laser bending was also reported关14兴 3-D

models are more appropriate for predicting the actual pulsed laser

bending process, however, the computation of 3-D pulsed laser

bending is inhibited by the computer power This is because in

pulsed laser bending, thousands of laser pulses are irradiated onto

the target Therefore, it is extremely time-consuming to compute

thermal and thermo-mechanical effects caused by all the pulses

Recently, Zhang et al.关15兴 developed an efficient, 3-D finite

ele-ment method to calculate pulsed laser bending In that method,

only a fraction of the total laser pulses instead of all the laser

pulses are calculated, thus the computation time is greatly

reduced

This paper presents high precision bending of ceramics, silicon, and stainless steel specimens using a pulsed laser and a CW laser The relation between bending angles and laser operating param-eters, such as the laser intensity, laser scanning speed, and number

of scanning are studied experimentally The dependence of bend-ing on optical and thermophysical properties of the target material

is illustrated The surface composition of ceramic specimens be-fore and after laser bending is analyzed using an electron probe microanalyzer共EPMA兲 3D numerical simulations of bending of

stainless steel using a pulsed laser and a CW laser are performed Experimental and numerical results are compared to demonstrate the different effects caused by the difference in the laser heating time

For the experiments, a 2 W Nd:VA nanosecond pulsed laser and

a 9 W CW fiber laser are used The operation parameters of these two lasers are summarized in Table 1 For pulsed laser, the laser power level is controlled by using a polarizing beam splitter so that the pulse format can be kept the same The laser beam diam-eter on targets is measured with a knife edge technique The mean value of the beam diameter is given in Table 1 For the CW laser, beam diameters of 80␮m and 40 ␮m have been used in

experi-ments However, the numerical calculations are conducted for an

80␮m diameter beam only

Figure 1 illustrates the experimental setup for performing laser bending as well as for measuring the bending angle The laser beam is expanded by a beam expander and then focused onto the target using a focusing lens One end of the target is clamped, and the distance between the scanning path and the clamped edge is about 5 mm The focused laser beam scans the target in the

y-direction, causing bending in the z-direction A He-Ne laser

beam is focused at the free end of the specimen to measure the bending angle The reflected He-Ne laser beam is received by a position sensitive detector 共PSD兲 with 1 ␮m sensitivity in the

position measurement Bending of the specimen causes the re-flected He-Ne laser beam to move across the PSD The position change of He-Ne laser beam measured at the PSD surface is then converted to the bending angle of the specimen using geometrical calculations The distance between the specimen and the PSD is set to 750 mm, resulting in an accuracy of the bending angle measurement of about⫾1.5␮rad The whole apparatus is set on a

vibration-isolation table

Ceramics, silicon, and stainless steel共AISI 301兲 sheets are used

as specimens The dimensions and some key material properties are given in Table 2 Before laser treatment, all the samples are polished and cleaned with acetone The Al2O3/TiC ceramics is

Contributed by the Manufacturing Engineering Division for publication in the

J OURNAL OF M ANUFACTURING S CIENCE AND E NGINEERING Manuscript received

July 2001; Revised December 2002 Associate Editor: L Yao.

used in computer hard disks as the material for the read/write

head Compositions of ceramic specimens are recorded with

EPMA before and after laser treatment High magnification photos

of the surface topography and quantitative composition analyses

of the surface area irradiated by the lasers are also obtained with

EPMA

2 Experimental Results and Discussion. Bending angles

of the ceramic specimens are obtained at various laser processing

conditions Figures 2 and 3 compare results of pulsed and CW

laser bending as a function of laser intensity and laser scanning

speed, respectively As expected, the bending angle increases

when the input laser intensity increases, and decreases with an

increase of the scanning speed For the pulsed laser, the heat

dif-fusion length in ceramics is about 1␮m, and for the CW laser, the

heat diffusion length is about 50␮m at a scanning velocity of 130

mm/s With such a small heat diffusion length in pulsed laser

bending, bending is always controlled by the temperature gradient

mechanism In the experiments, it is found that the specimens

always bend toward the laser beam However, for CW laser

bend-ing, the bending direction could be controlled by different

mecha-nisms depending on the scanning velocity It is found that the

specimen bends away from the laser beam due to the buckling

mechanism at a laser intensity of 3.98⫻105

W/cm2 and a scan-ning speed lower than 26 mm/s

Figure 4 indicates that for both pulsed and CW laser, scanning over the specimen surface repetitively at the same location would increase the total bending angle However, the amount of addi-tional bending decreases with the number of scanning lines For the pulsed laser, the bending angle obtained by the second laser scan drops to about 25% of the angle obtained by the first scan, and no additional bending occurs after four scans For the CW laser, the bending angle of the second scan is about 70% of the first scan, and no additional bending occurs after six scans This is because bending depends on the initial stress/strain status in the specimen—in this case, the residual stress/strain from a previous scan The laser scan generates tensile residual stress关9兴, thus, it

requires higher temperature to reach the compressive yield stress

On the other hand, the residual strain is compressive关9兴, and the

area scanned by the laser beam becomes hardened which is known

as the effect of strain hardening Consequently, less additional compressive strains or bending can be obtained in the subsequent scans

Table 1 Parameters of pulsed laser and CW laser

Pulsed laser CW laser Laser wavelength 1.06␮m 1.10␮m

Laser pulse full width 120 ns —

Laser pulse repetition 22 kHz —

Laser maximum power 2.0 W 9.0 W

Laser beam diameter 55␮m 40,80␮m

Fig 1 Experimental setup 1 Laser, 2 shutter, 3 polarizing

beam splitter, 4 mirror, 5 beam expander, 6. x - y scanner, 7.

specimen, 8 beam splitter, 9 position sensitive detector, 10.

lens, 11 He-Ne laser

Table 2 Specimen parameters and material properties at 300 K

Specimen material Ceramics

Al 2 O 3 /TiC

Silicon Stainless steel

301 Specimen length 共mm兲 10.0 8.0 10.0

Specimen width 共mm兲 1.25 1.50 1.00

Specimen thickness 共mm兲 0.35 0.20 0.10

Thermal diffusivity 共m 2 /s 兲 6.8 ⫻10 ⫺6 9.9⫻10 ⫺5 4.0⫻10 ⫺6

Optical absorption depth at

⫺4 2⫻10 ⫺8 Thermal expansion coefficient

⫺6 2.7⫻10 ⫺6 14⫻10 ⫺6

Fig 2 Bending angle of ceramics as a function of laser inten-sity For pulsed laser, v Ä 3.25 mm Õ s, d Ä 55␮m; for CW laser,

v Ä 130 mm Õ s, d Ä 40␮m.

Fig 3 Bending angle of ceramics as a function of scanning speed For pulsed laser, P Ä 1.75 Ã 10 7 W Õ cm 2 , d Ä 55␮m; for

CW laser, P Ä 3.98 Ã 10 5 W Õ cm 2 , d Ä 40␮m.

There are many ways to compare bending angles induced by

the pulsed laser and the CW laser Here bending angles are

com-pared when two adjacent scans do not influence each other in

terms of the resulting bending angle In practice, this information

is useful when one intends to obtain a large bending angle with a

minimum number of scans The bending angle as a function of the

separation distance between two scans is measured As shown in

Fig 5, for the pulsed laser bending at a scanning speed of 3.25

mm/s and a laser intensity of 1.75⫻107W/cm2, a minimum

sepa-ration distance of 100␮m is needed for not decreasing the

bend-ing angle For the same separation distance, the scannbend-ing speed of

the CW laser is 130 mm/s and its intensity is 3.98⫻105W/cm2

However, the resulted bending angles from the two lasers are

different The bending angle obtained from the CW laser is about

twice of that obtained from the pulsed laser

The backscattered electron 共BSE兲 images of laser irradiated

Al2O3/TiC ceramic specimens are shown in Fig 6 TiC grains are

white irregular ‘‘islands’’ and alumina grains are the background

‘‘sea.’’ The bending angle obtained is 9␮rad for the pulsed laser

and 30␮rad for the CW laser After laser irradiation, the surface

becomes gray A few microcracks can be seen in Fig 6共a兲 in the

pulsed laser-irradiated area After the CW laser irradiation, an ex-tensive gray color region appears as shown in Fig 6共b兲 A 30␮m

wide band of homogeneous material replaces the original ceramic composite material, and a curved, 1␮m wide microcrack is

lo-cated at the center of the scanning line, connected with several transverse microcracks Formation of the gray substance is possi-bly due to diffusion of TiC into Al2O3, and/or oxidation of TiC to form TiO2or TiAl2O5 Quantitative analyses of the surface com-position shown below will provide further explanations Appar-ently, more material damages are produced by the CW laser than the pulsed laser, although a larger bending angle is obtained by the

CW laser

Weight percentage changes of Al and Ti are also obtained using EPMA Two sets of experiments are carried out For the first set, the pulsed laser is used with an intensity of 1.75⫻107W/cm2and

a scanning speed of 13 mm/s Figure 7共a兲 shows the element

weight percentage change versus the number of scanning lines at the same location It can be seen that the weight percent of Al decreases slightly with an increase in the number of scanning

Fig 4 Additional bending angle of ceramics as a function of

the number of laser scanning lines For pulsed laser, P Ä 1.75

à 10 7 W Õ cm 2 , v Ä 3.25 mm Õ s, d Ä 55␮m; for CW laser, P Ä 3.98

à 10 5 W Õ cm 2 , v Ä 130 mm Õ s, d Ä 40␮m.

Fig 5 Bending angle of ceramics as a function of distance

between adjacent scanning lines For pulsed laser, P Ä 1.75

à 10 7 W Õ cm 2 , v Ä 3.25 mm Õ s, d Ä 55␮m; for CW laser, P Ä 3.98

à 10 5 W Õ cm 2 , v Ä 130 mm Õ s, d Ä 40␮m.

Fig 6 BSE images of ceramic specimen surface after „a…

pulsed laser bending, PÄ 1.75 Ã 10 7 W Õ cm 2 , vÄ 13 mm Õ s, d

Ä 55␮m, and „b… CW laser bending, P Ä 3.98 Ã 10 5 W Õ cm 2 , v

Ä 130 mm Õ s, dÄ 40␮m.

lines For the second set of data shown in Fig 7共b兲 the CW laser

is used with a scanning speed of 130 mm/s It can be seen that

there is almost no change in the weight percent of Al when the

laser intensity is increased from 7.96⫻104

W/cm2 to 3.98

⫻105W/cm2 The weight percent of Ti increases slightly from

23.8% to 24.8% when the CW laser intensity is increased from

7.96⫻104W/cm2 to 2.39⫻105W/cm2, and then decreases

sig-nificantly to 17.0% at 3.98⫻105W/cm2

The behavior of the Al2O3/TiC ceramics under laser irradiation

can be qualitatively understood by analyzing its optical and

ther-mal properties In the ceramic specimen, the alumina grains are

transparent to the laser irradiation, while the TiC grains absorb the

laser energy Heating of alumina is through heat conduction from

the TiC grains to the alumina grains On the other hand, alumina

has a lower melting temperature共2345 K兲 and vaporization

tem-perature共3803 K兲 compared to those of TiC 共3413 K and 5093 K兲

关16兴 It is estimated using a finite element calculation that during

pulsed laser irradiation, the boiling temperature of TiC is reached

at an intensity of about 1.5⫻107W/cm2 Therefore, at the laser

intensity of 1.75⫻107W/cm2 used in this experiment, a certain

amount of alumina and TiC grains are evaporated TiO2and other

oxides such as TiAl2O5can be produced due to oxidation TiC and

its oxides form a gray pattern on the surface as shown in Fig 6共a兲.

Overall, the evaporation and oxidation are weak because of the short heating time, which cause a small change in the element weight percentages For the CW laser irradiation, at low laser intensity (⬍2.39⫻105W/cm2), the temperature reached could be lower than the vaporization temperature of TiC 共5093 K兲 but

higher than the vaporization temperature of alumina 共3803 K兲

Therefore, more alumina is evaporated and TiC and its oxides are left on the surface Evaporation should be rather weak since the element percentage changes are small On the other hand, at higher laser powers (⬎2.39⫻105

W/cm2), a higher peak tem-perature is reached on the surface of the TiC grains, causing stron-ger evaporation and oxidation However, it is noticed from Fig

7共b兲 that the slight increase in Al does not offset the large decrease

in Ti, indicating other compounds are formed such as oxides Forming TiO2 from TiC would reduce the weight percentage of the Ti element Thus, it is concluded that more oxides are formed when the laser power is high, resulting in reduced C and Ti ele-ment weight percentages

Figure 8 shows the bending angle of silicon as a function of power of the pulsed laser and the CW laser, respectively Com-paring Fig 8 with Fig 2, it can be seen that the bending angles of silicon are comparable or less than those of the ceramics specimen

at most power levels, even the thickness of silicon specimen is only 57% of that of ceramics According to material properties given in Table 2, the optical absorption depth of silicon at the laser wavelengths共1064 nm and 1100 nm兲 is about 0.25 mm and that of

ceramics is less than 100 nm Also, the heat diffusion length of silicon is about 4 times longer than that of ceramics for the same laser parameters The longer optical absorption depth and heat diffusion length of silicon result in a much lower peak tempera-ture and smaller temperatempera-ture gradient Therefore, smaller bending angle of silicon is obtained when using identical laser parameters Figure 9 shows the bending angle of stainless steel as a function

of the laser intensity The scanning speed of the pulsed laser is 195 mm/s Surface melting occurs when the laser intensity is higher than 2.3⫻106W/cm2at this scanning speed Compared with the pulsed laser bending of ceramics and silicon, it is found that com-parable bending angles can be obtained for steel at a scanning speed about 60 times higher but only half of the laser intensity This is because that the steel is much more ductile than ceramics and silicon

Fig 7 Al and Ti weight percent changes versus„a…number of

pulsed laser scanning lines, P Ä 1.75 Ã 10 7 W Õ cm 2 , v Ä 13 mm Õ s,

d Ä 55␮m, and „b… CW laser intensity, v Ä 130 mm Õ s, d

Ä 40␮m.

Fig 8 Bending angle of silicon as a function of laser intensity

at v Ä 3.25 mm Õ s For pulsed laser, d Ä 55␮m; For CW laser, d

Ä 40␮m.

3 Numerical Calculations

CW and pulsed laser bending induced by the temperature

gra-dient mechanism is calculated using 3-D finite element models A

thermal analysis and a stress analysis are conducted The two

analyses are treated as uncoupled since the heat dissipation due to

deformation is negligible compared with the heat provided by the

lasers In an uncoupled thermo-mechanical model, a transient

tem-perature field is obtained first in the thermal analysis, and is then

used as a thermal loading in the subsequent stress analysis to

obtain transient stress, strain, and displacement distributions

For the CW laser bending simulation, the laser beam is

consid-ered as a constant heat source moving at a constant speed The

time step in the calculation must be small enough so that the

continuously moving laser flux can be calculated accurately For

pulsed laser bending, direct simulation using a 3-D model is

im-practical in terms of the computer power, since there are too many

pulses along a single laser scanning line An efficient method for

simulating pulsed laser bending was recently developed by Zhang

et al.关15兴, which is briefly described as follows In a pulsed laser

bending process, although a single laser pulse generates

non-uniform stress and strain distributions, in practice, laser pulses

with the same pulse energy, separated by a very small distance

compared with the laser beam diameter are used Thus, the

laser-induced stress and strain vary little along the scanning direction

Also, the stress and strain fields induced by a laser pulse are

contained within a short distance from the laser-irradiated area

Therefore, it is only necessary to calculate several laser pulses

until the stress and strain fields in an x-z cross-section area are not

changed by a new laser pulse 共see Fig 1 and Fig 10 for the

coordinate system兲 Then, the residual strain field in this

cross-section can be imposed onto the whole domain to calculate the

deformation distribution In other words, a strain field, which can

be used to calculate displacements of the target after pulsed laser

scanning, is generated by calculating only a fraction of the total

pulses

The thermal analysis is based on solving the 3D heat

conduc-tion equaconduc-tion The initial condiconduc-tion is that the whole specimen is at

the room temperature 共300 K兲 The laser flux is handled as a

volumetric heat source decreasing exponentially from the target

surface with an optical absorption depth of 20 nm Using the

transient temperature data obtained from the thermal analysis as

thermal loading, the transient stress, strain, and displacement

dis-tributions are obtained by solving the quasi-static force

equilib-rium equations The material is assumed to be linearly

elastic-perfectly plastic The Von Mises yield criterion is used to model

the onset of plasticity The left edge of the specimen is completely constrained, and all other boundaries are force free Material prop-erties including thermal conductivity, thermal expansion coeffi-cient, density, yield stress, and Young’s modulus are considered as temperature dependent关17兴

For both CW and pulsed laser calculations, a dense mesh is used around the laser path and a coarse mesh is used outside the primary processing region Transition elements are created to con-nect the dense mesh and the coarse mesh Eight-node linear brick elements are used The mesh used for the CW laser bending simu-lation is shown in Fig 10 The Cartesian coordinate system is attached to the computational domain and the center of laser beam

moves along the y-direction at x⫽0 The computational domain

has the same dimensions of the steel sheet used in the experi-ments, 10 mm⫻1 mm⫻0.1 mm It is constrained at the left edge

(x⫽5 mm) The total element number is 20,790 The mesh for the

pulsed laser bending simulation is similar except smaller elements are used and the total element number is 99,440 The step size between two adjacent laser pulses is 9␮m, which is sufficiently

small to provide uniform residual strain and stress along the

y-direction For each case, the same mesh is used for both thermal

and stress analyses

The non-linear finite element solver, ABAQUS is employed for both CW and pulsed laser bending simulations Only bending of stainless steel is simulated since more property data of stainless steel are available compared with the other two materials The maximum temperatures obtained in all simulations are lower than the melting point of steel共1650 K兲 Details of calculations and

experimental validations were provided elsewhere 关12,15兴 Here

the focus is on comparing pulse vs CW laser heating, and ex-plaining the difference between the two cases observed experimentally

Figure 11 shows the temperature distributions along the

z-direction induced by the pulsed laser with the laser intensity of

1.54⫻106

W/cm2 Only the range from 0 to 10␮m in the z

di-rection is presented so that temperature variations can be seen clearly The maximum temperature is obtained at the upper

sur-face and reaches 890 K at t⫽87.7 ns The heat propagation depth

is around 4 ␮m at 2.2 ␮s and the temperature gradient during

heating period is as high as 290 K/␮m This sharp temperature

gradient results in the temperature gradient mechanism of bend-ing Figure 12共a兲 shows the off-plane displacement w and the

residual stress␴xxon the top surface of the specimen at the same

laser intensity The center of the laser beam is located at x⫽0 A

‘‘V’’ shape profile is obtained after laser scanning, with the valley

located at about 10␮m from the center of the scanning line The

positive off-plane displacement at the center of the scanning line

Fig 9 Bending angle of stainless steel as a function of laser

intensity For pulsed laser, v Ä 195 mm Õ s, d Ä 55␮m; for CW

la-ser, v Ä 8 mm Õ s, d Ä 40␮m.

Fig 10 Mesh for the 3D simulation of CW laser bending„laser irradiates on the z Ä 0 surface…

is produced by thermal expansion along the positive z-direction.

The stress␴xxis tensile and its value is around 1.04 GPa in the

region within 15␮m from the center This agrees with the

theo-retical prediction that the tensile residual stress will be obtained

near the center of the laser-irradiated area due to the thermal

shrinkage during cooling关9兴 The total high stress region is about

30 ␮m and is comparable to the radius of the laser beam 共27.5

␮m兲 The residual stress and strain distribution along the thickness

direction are shown in Fig 12b The high tensile stress is induced

by the laser near the surface and it becomes compressive at depths greater than 0.6␮m The residual compressive strain ␧xxreaches a minimum value of⫺0.00031 within 1␮m from the surface, and

then increases gradually with the depth The compressive strain indicates that the bending is toward the laser beam

Figure 13共a兲 shows the residual stress distribution along the x-direction induced by the CW laser In order to compare with

pulsed laser bending shown in Fig 12共a兲 only a region of 200␮m

instead of 5 mm along the x-direction is plotted As shown in Fig.

13共a兲 the largest residual stress is located at the center line (x

⫽0) and the value is about 700 MPa, which is smaller than that

induced by the pulsed laser 共1.04 GPa兲 However, the total

stressed zone in the x-direction is more than 100␮m, much larger

than the radius of the laser beam共40␮m兲 and the stressed zone

induced by a pulsed laser This is because the heat diffusion length

of CW laser is much longer Figure 13共b兲 shows the peak tensile

residual stress and strain distribution along the z-direction It can

be seen by comparing Fig 13共b兲 with Fig 12共b兲 that the residual

stress and strain depth of CW laser bending are about 5 times larger than that of pulsed laser bending The wider and deeper stressed and strained region explain the more visible mechanical damages observed in the CW laser experiments

This work demonstrated using pulsed and CW lasers for mi-croscale bending of ceramic, silicon, and stainless steel samples Experimental studies were conducted to find out relations between bending angles and laser operation parameters Results obtained

Fig 11 Transient temperature distributions along the

z -direction induced by a laser pulse. P Ä 1.54 Ã 10 6 W Õ cm 2 , d

Ä 55␮m.

Fig 12 Simulation results of pulsed laser bending. P Ä 1.54

à 10 6 W Õ cm 2 , v Ä 195 mm Õ s, d Ä 55␮m.„a…Residual stress and

off-plane displacement distributions along the x -direction;„b…

residual stress and strain distributions along the z -direction.

Fig 13 Simulation results of CW laser bending simulation. P

Ä 1.59 Ã 10 5 W Õ cm 2 , v Ä 8 mm Õ s, d Ä 80␮m. „a…Residual stress distributions along the x -direction; „b… residual stress and strain distributions along the z -direction.

by a pulsed and a CW laser were compared For the ceramic

specimen, when two adjacent laser scans do not influence each

other, the CW laser produced more bending than the pulsed laser

did However, the pulsed laser caused much less surface

compo-sition change and thermomechanical damage Numerical

calcula-tions using the thermo-elasto-plastic theory were conducted and

the results are used to explain the phenomena observed

experi-mentally

Acknowledgments

Support of this work by the National Science Foundation is

acknowledged The authors are most grateful to Dr Andrew C

Tam of IBM Almaden Research Center for his contribution to this

work The authors also thank SDL, Inc for providing the CW

fiber laser system, and Mr Carl Hager of the Department of Earth

and Atmospheric Sciences at Purdue University for his help on the

EPMA measurements

Nomenclature

P ⫽ laser intensity

d ⫽ laser beam diameter on targets

v ⫽ laser scanning speed

w ⫽ off-plane displacement

x, y, z ⫽ Cartesian coordinates

␧xx ⫽ residual stress along the x-direction

␴xx ⫽ residual stress along the x-direction

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