A study of the effect of laser beam geometries on laser bending of sheet metal by buckling mechanism

Optics & Laser Technology 43 (2011) 183–193

Contents lists available at ScienceDirect

Optics & Laser Technology

0030-39

doi:10.1

n Corr

E-m

(M.S. C

journal homepage: www.elsevier.com/locate/optlastec

A study of the effect of laser beam geometries on laser bending of sheet metalby buckling mechanism

M.S. Che Jamil n, M.A. Sheikh, L. Li

Laser Processing Research Centre (LPRC), School of Mechanical Aerospace and Civil Engineering, University of Manchester, Manchester M60 1QD, UK

a r t i c l e i n f o

Article history:

Received 13 April 2010

Received in revised form

21 June 2010

Accepted 22 June 2010Available online 13 July 2010

Keywords:

Laser forming

Sheet metal

Beam geometries

92/$ - see front matter & 2010 Elsevier Ltd. A

016/j.optlastec.2010.06.011

esponding author. Tel.: +44 1613062365.

ail address: [email protected]

he Jamil).

a b s t r a c t

Laser beam forming has emerged as a new and very promising technique to form sheet metal by

thermal residual stresses. The objective of this work is to investigate numerically the effect of

rectangular beam geometries, with different transverse width to length aspect ratio, on laser bending

process of thin metal sheets, which is dominated by buckling mechanism. In this paper, a

comprehensive thermal and structural finite element (FE) analysis is conducted to investigate the

effect that these laser beam geometries have on the process and on the final product characteristics. To

achieve this, temperature distributions, deformations, plastic strains, stresses, and residual stresses

produced by different beam geometries are compared. The results suggest that beam geometries play

an important role in the resulting temperature distributions on the workpiece. Longer beam dimensions

in the scanning direction (in relation to its lateral dimension) produce higher temperatures due to

longer beam–material interaction time. This affects the bending direction and the magnitude of the

bending angles. Higher temperatures produce more plastic strains and hence higher deformation. This

shows that the temperature-dependent yield stress plays a more dominant role in the deformation of

the plate than the spread of the beam in the transverse direction. Also, longer beams have a tendency

for the scanning line to curve away from its original position to form a concave shape. This is caused by

buckling which develops tensile plastic strains along both ends of the scanning path. The buckling effect

produces the opposite curve profile; convex along the tranverse direction and concave along the

scanning path.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, laser beam forming (LBF) has emerged as anew and very promising technique to form sheet metal bythermal residual stresses. There are many advantages of LBFcompared to conventional sheet bending. Among these are, designflexibility, production of complex shapes (which is not achievableby the conventional methods), forming of thick plates, possibilityof rapid prototyping, etc [1]. However, for industrial applicationsof the process more research work is still needed.

The laser beam forming process is always associated with thelength of time it takes to form a desired bending angle or anintended shape. Recently, much research has been focussedtowards achieving a more efficient LBF process. Studies havebeen carried out on the application of a pre-load on the plate/sheet to produce a higher bending angle [2,3]. Some researchershave also tried to use simultaneous laser scans in order to

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nchester.ac.uk

increase the bending angle [4]. More recently, there has also beena study of the use of unconventional laser beam geometries fortube bending [5]. All these efforts are in a quest to optimise thelaser forming processes.

High power diode laser (HPDL) has been used widely in laserprocessing including laser bending. This laser normally produces arectangular or an elliptical beam shape with various sizes andgeometries. Very limited information is available in the publicdomain about the effect of different beam geometries on laserbending. Casamichele et al. [6] and Chen et al. [7] in theirexperimental studies on the application of diode laser in sheetmetal bending have investigated the effect of elliptical andrectangular beam orientation on the forming of sheet metal. Bothexperienced surface melting when the longer beam axis wasparallel to the scanning direction. This did not allow a directcomparison of beam orientations in terms of bending angle. Also,the beam was kept at its focal point, resulting in small beam sizesranging from 0.6 to 1.8 mm, which favoured the formation of thetemperature gradient mechanism (TGM). It is well known that forthe buckling mechanism (BM) to be dominant, a larger beam sizecompared to sheet thickness is required. Hence the effects ofbeam orientation could be more prominent in the buckling

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εe εt εpεe

αB

vLaser beam

Laser beam

Fig. 1. Buckling mechanism (BM) process.

M.S. Che Jamil et al. / Optics & Laser Technology 43 (2011) 183–193184

mechanism. Many studies have been carried out on laserparameters such as power, feed rate, beam size, etc, but as yetthere is no work reported on the effects of different beamgeometries and their orientation on laser bending by bucklingmechanism. The objective of this work is to investigate numeri-cally the effect of rectangular beam geometries (with differenttransverse width to length aspect ratio) on the laser bending ofthin metal sheets dominated by buckling mechanism. In thispaper, a comprehensive thermal and structural analysis ispresented on the effect of these laser beam geometries on theprocess and on the final product characteristics. Temperaturedistributions, deformations, plastic strains, stresses and residualstresses produced by different beam geometries are compared.

1.1. Laser beam geometries

At present the main laser parameters which control LBF arelaser power, spot size or scanning speed. Variations in theseparameters are often limited by other processing conditions. Forexample, reducing the laser power density might affect theheating depth and changing the scanning speed may affect thecoverage rate. This fact identifies a limit to the control oftemperature distribution during laser processing. One possiblemethod of varying the temperature distribution without changingthe input power or the scanning speed is to modify the geometryof the laser beams [5]. In the area of optics, many studies [8–11]explain the methods and design approaches to produce a varietyof beam shapes such as line, rectangular, star, D-shape, annular,cross, etc. The beam shaping methods range from the most basiclike the use of a mask to the more advanced like optical lens,beam scanners, field mappers and beam integrators.

Several researchers have studied the effects of different laserbeam shape on laser processing. Mucha et al. [12] investigated theeffects of the conventional circular and rectangular beams on thedeformation of plates. Their analytical and experimental workshave found that the bending angle obtained with the use of therectangular laser beam is higher than with a circular beam.Triantafyllidis et al. [13] investigated the effects of different beamgeometries on laser surface treatment of ceramics. They found outthat the cooling rates were modified by changing the geometry ofthe beam. Rhomboid and Pi beam shapes resulted in significantlylower cooling rates. Safdar [14] investigated the effects of non-conventional beam geometry in various laser material processingapplications. Experimental and numerical modelling techniqueswere used for laser surface heating, laser transformation hard-ening, laser tube bending and laser melting of mild steel. It wasconcluded that the laser beam geometry provided a useful tool tomanipulate temperature distribution and hence the optimisationof laser material processing.

1.2. Buckling mechanism in LBF

To date, many forming mechanisms have been suggested[15–21]. The most extensively studied and reported mechanisms

are temperature gradient mechanism (TGM) and buckling mechan-ism (BM). TGM dominates when the temperature gradient acrossthe thickness of the plate is high and hence induces high stressesand strains. The thermal expansion of the layers of material close tothe irradiated surface causes the sheet to bend away from the laserdirection during heating. With the increase in temperature, thecompressive elastic strains at the top surface convert into plasticstrains. During cooling, the sheet bends towards the laser beambecause of thermal contraction of the top surface.

On the other hand, BM dominates when there is small oralmost no temperature gradient across the thickness. When thediameter of the laser beam is significantly larger than thethickness of the sheet (i.e. for thin sheets) and when the ratio ofthermal conductivity to the thickness of the material is large, verysmall temperature gradients are developed in the thicknessdirection. These negligibly small temperature gradients cause auniform thermal strain (et) throughout the thickness (Fig. 1). Thefree expansion of the sheet is restricted by the surroundingmaterial resulting in compressive stresses in the laser heatedregion. These compressive stresses in the sheet result in elasticstrains (ee) in the surrounding material.

In buckling mechanism the buckling stiffness of the sheet isrelatively low whereas the lateral expansion caused by thetemperature field is large. The compressive stress region tendsto buckle when the stresses exceed a certain value. Furtherincrease in temperature results in plastic strains (ep) to develop inthe buckled region. The tendency to buckle is enhanced when thesheet is thin and the temperature dependent flow stress is not toolow.

Once buckling is initiated, it extends along the laser scanningpath due to thermal expansion of the subsequent material. Thetemperature-dependent flow stress in the top irradiated region islow due to higher temperature, thus causing the deformation atthe top irradiated surface to be nearly plastic, whereas thesurrounding region remains nearly purely elastic. When the laserbeam leaves the scanning path, the sheet develops a bendingangle, which is primarily determined by the plastic region in thebend of the sheet.

Shi et al. [22] has derived a critical buckling condition (1) as acriterion to judge whether the forming process is dominated bythe thermal gradient mechanism (TGM) or by the bucklingmechanism (BM).

Pd1=2

h2V1=24

Zp7=2k1=2r1=2c1=2

41:52ð1þmÞAathð1Þ

where P is the laser power, d the laser spot diameter, h thethickness of plate, V the scanning velocity, Z, k, r, c, m, A and ath

denote the correction factor, thermal conductivity, density,specific heat of the material, Poisson’s ratio, the laser absorptioncoefficient and thermal expansion coefficient, respectively. Thiscondition was used in this study in order to set parameters for BMdominant process using a circular beam. The area of the circularbeam was used as reference for other rectangular beamgeometries.

M.S. Che Jamil et al. / Optics & Laser Technology 43 (2011) 183–193 185

2. Numerical modelling

Laser bending experiments were conducted using a LaserlineLDL 160–1500 diode laser with a maximum power of 1.5 kW andoutput wavelength of 808–940 nm. It was coupled with a 1 mmdiameter, 5 m long optical fibre cable and a 1:1 optical imagingsystem with a focal length of 85 mm. The output of the fibrecoupled diode laser was a circular beam with a top-hat intensitydistribution. The delivered power loss using the fibre couplingwas around 28%.

The plates were made of AISI 304 stainless steel withdimension of 100 mm length�50 mm width�0.9 mm thickness.The optical reflectivity of the plate was measured using an SD2000 fibre optic spectrometer manufactured by Ocean Optics Inc.,USA. For the diode laser wavelength (808–940 nm), the reflectiv-ity of the investigated material was found to be 40% at roomtemperature without any surface processing. Therefore, thecoefficient of absorption of the material was assumed to be 0.6.

To investigate the effect of beam geometries on laser bending,five rectangular beam geometries with different dimensions alongthe axial and transverse directions of the scan were chosen, asshown in Fig. 2. A circular beam was also modelled as abenchmark for conventional laser forming. The area of eachbeam was kept constant at 200 mm2 (based on 16 mm circularbeam area) to maintain the power intensity and hence the sameamount of energy transferred into the specimen.

One scan along the x–x0 path (Fig. 3(a)) was simulated for abeam velocity of 10 mm/s. The scanning track started and ended15 mm off the entry and the exit sides so that the total scanninglength was 80 mm, and thus the scanning duration was 8 s. Thepower applied was 800 W. The laser beam intensity had a top-hatdistribution and thus assumed to be constant over the beam area.The laser beam intensity (I) on the workpiece was obtained from:

I¼Z� m� P

Að2Þ

where I is the laser beam intensity, 1.728 W/mm2; Z theabsorption coefficient, 0.6; m the fibre cable efficiency, 0.72;P the power applied, 800 W and A the laser beam area, 200 mm2.

A commercial finite element package (ANSYS) has been used inthis study to simulate the laser bending process using rectangularbeams with different transverse width to length aspect ratio. Thetemperature-dependent material properties are considered in-cluding thermal conductivity, specific heat, thermal expansioncoefficient, Poisson’s ratio, elastic modulus, yield stress anddensity, as shown in Fig. 4.

The temperature field caused by a moving laser beam istransient in nature. If the energy dissipation due to plasticdeformation is neglected, when compared with the energyinvolved in the thermal process, the analysis can be uncoupledinto sequential thermal and stress analyses. The thermal problemis solved first to obtain the temperature field, which is then used

A B

Width = 25mmLength = 8mm


Width=14.14mmLength=14.14mm

CASE

Beamprofile

Aspect ratio 3.125 2.0 1.0

C

Fig. 2. Six beam geometries with similar

as thermal loading for the structural problem. The same mesh isused for both thermal and mechanical calculations.

The finite element equation for transient heat transfer analysiscan be expressed as follows [23]:

½C�½ _T �þ½K�fTg ¼ fQ ðtÞg ð3Þ

where [C] is the specific heat matrix, ½ _T � the time derivative oftemperature, [K] the heat conductivity matrix, {T} the temperatureand {Q(t)} the heat flux column.

The nonlinear transient dynamic structural equation based onFEM can be written in the matrix form as follows:

½MðTÞ�fu00ðtÞgþ½DðTÞ�fuuðtÞgþ½SðTÞ�fuðtÞg ¼ fFðtÞgþfFthðtÞg ð4Þ

where [M(T)] is the temperature dependent mass matrix, [D(T)] thetemperature dependent damping matrix, [S(T)] the temperaturedependent stiffness matrix, {F(t)} the external load vector, {Fth(t)} thethermal load vector, {u00(t)} the acceleration vector, {u0(t)} thevelocity vector and {u(t)} the nodal displacement vector.

For the case of transient thermal stresses generated due tolaser scanning with no damping or external forces/loadings,Eq. (4) can be reduced to

½SðTÞ�fuðtÞg ¼ fFthðtÞg ð5Þ

Thermal load vector {Fth(t)} can be evaluated by using

Fthn o

¼

Zvol½B�T

EaðTÞDT

1�2vddV ð6Þ

where [B] is the strain displacement matrix, d the [1 1 1 0 0 0]T,E the elastic modulus, v Poisson’s ratio and DT the temperaturedifference.

Total strains can be evaluated from nodal displacements byusing

fBg ¼ ½B�fug ð7Þ

where {u} is the nodal displacement vector.The difference between total strain and thermal strain gives

mechanical strain, which consist of plastic and elastic straincomponents as expressed below:

feg�fethg ¼ feplgþfeelg ð8Þ

where {e} is the total strain vector, {eel} the elastic strain vector,{epl} the plastic strain vector and {eth} the thermal strain vector.

The effect of laser heat flux on the plate is localized around theheating region. Therefore, the problem can be represented by onlyhalf of the plate modelled with symmetric boundary conditions,as shown in Fig. 3(b). This model has also been used by otherresearchers [24–26]. Eight noded 3D brick element (ANSYS:SOLID70) with thermal conduction capability is used for thethermal analysis. For structural analysis ANSYS: SOLID185 typeelement is used which is defined by eight nodes having threedegrees of freedom (ux, uy, uz) at each node. The element hasplasticity, large deflection and large strain capabilities [27].



CIRCULAR

Diameter = 16mm

-0.5 0.32

D E

effective area used in FE simulations.

yz

x

Plane/nodeSymmetric plane:

Node:

Laser irradiation

Line of symmetry

10mm/s z = 0 UZ = 0

Consraint

x = 0, y = 0, z = 0 UY = 0UY = 0x = 50, y = 0, z = 0

x = 25, y = 0, z = 0 UX = 0

Fig. 3. (a) Schematic of the laser bending process and (b) FE mesh, irradiation area and boundary condition for the half plate model.


A dense mesh is used around the laser path with a total of 12 000elements and 16 565 nodes in the mesh. Laser heating is modelledas moving heat flux with ANSYS APDL language used toincorporate the laser beam motion. All exposed surfaces aresubjected to convection, but heat losses due to radiation are notconsidered as they were very small.

The basic assumptions for the model are as follows:

(i)
material is isotropic; (ii) laser intensity distribution is assumed to be uniform;
(iii)
Bauschinger’s effect is neglected; (iv) Von-Misses criterion has been used for plastic yielding; (v) energy dissipation due to plastic deformation is neglected
when compared with the energy involved in the thermalprocess.

2.1. Air cooling vs water cooling

Most of the previous works on laser bending have consideredthe specimen to be cooled by its surrounding air and thus thecoefficient of heat convection, h, is normally assumed to bearound 20 W/(m2

1C). Cooling by surrounding ambient air is avery slow process. Normally after each laser scan it would take

more than one minute to cool the specimen back to the ambienttemperature, which is required to avoid temperature accumula-tion and melting. Therefore, if it is required to have hundred scansto achieve a certain bending angle, then this will be an extremelytime consuming process. Several researchers have conductedstudies to determine the effects of cooling in laser forming. Cheng[28] and Hennige et al. [29] have experimentally and numericallyinvestigated the effects of natural and forced cooling usingpressurised air and water. They concluded that active watercooling reduce the entire processing time and has no detrimentaleffects on the microstructure of the formed parts.

In this work, the specimen was assumed to have active coolingby water immediately after the heating period. The convectioncoefficient used for heat exchange between the specimen andwater was 2000 W/(m2

1C). Simulations were performed to findthe difference between water cooling and normal convection tothe surrounding air for a circular laser beam. It is shown from thesimulation that more than 200 s of cooling time is required to cooldown to the ambient temperature when it is exposed to thesurrounding air. Water cooling, on the other hand only requires5 s of cooling time. The transient temperature measured at point(25, 0.9, 0) and displacement measured at point (0, 0.9, 50) for thetwo cooling methods are shown in Fig. 5(a) and (b), respectively.It can be seen from Fig. 5(b) that the final displacement for both

Specific heat

450

500

550

600

650

700

750

0Temperature (C)

Spe

cific

Hea

t (J/

kg. °

C)

Thermal conductivity

141618202224262830

0

Temperature (C)

Ther

mal

Con

duct

ivity

(W/m

. °C

)

Thermal expansion

161718192021222324

Temperature (C)

Ther

mal

Exp

ansi

on (x

10e-

6)

Poisson ratio

0.260.270.280.290.3

0.310.320.33

Temperature (C)

Poi

sson

Rat

ioDensity

7200

7400

7600

7800

8000

0Temperature (C)

Den

sity

(kg/

m3 )

Young's modulus

0

40

80

120

160

200

Temperature (C)

You

ng's

Mod

ulus

(MP

a)

Yield stress

0

50

100

150

200

250

Temperature (C)

Yie

ld S

tress

(Mpa

)

200 400 600 800 1000

0 200 400 600 800 1000 0 200 400 600 800 1000

250 500 750 1000 1250 1500

500 1000 1500 0 500 1000 1500

0 500 1000 1500

Fig. 4. Temperature dependent thermal and mechanical properties of AISI 304 stainless steel [33].

0100200300400500600700

1

Tem

pera

ture

(°C

) Water coolingAir cooling

Time (s) Time (s)

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Dis

plac

emen

t, U

Y (m

m)

Water coolingAir cooling

10 100 1000 1 10 100 1000

Fig. 5. (a) Temperature vs time measured at point (25, 0.9, 0) and (b) displacement vs time measured at point (0, 0.9, 50).



methods are similar, which is around �2.6 mm. However, byusing water the total processing for one scan is reduced from200 s to only 10 s. In terms of FE modelling, this technique hasalso significantly reduced the calculation effort. All thesimulations in this study consider water cooling.

2.2. Experimental validation

To validate the numerical model, the experimental worksusing the circular beam from the fibre coupling were used tocompare with the circular beam used in the numerical works.Laser power of 400, 600, 800 and 1000 W was applied. A circularbeam of 16 mm diameter was obtained when the workpiece was140 mm below the lens. Other parameters were identical toparameters discussed in the numerical section, with laser beamspeed, v, of 10 mm/s. Each test was repeated five times.

The plate material was AISI 304 stainless steel which is anaustenitic low carbon steel. The 0.9 mm thick plate was cut into110 mm length�50 mm width. The plate was clamped over10 mm at one end, as shown in Fig. 3(a). The clamp was mountedon a CNC table, controlled by a CNC Automation Controller C116-4isel. The values of the plate bending angles were measured using aprotractor with a resolution of 1/121 (50), along the centre line ofthe plates.

Fig. 6 shows the magnitude of the plate bending angle from theexperiments and the numerical study. The bending angle shown isthe average of five tests, and the error bar represents the standarddeviation. It shows that the numerical results are generally ingood agreement with the experimental results. This validates thenumerical model developed here for the laser bending problem.From the experiments, the bending directions obtained aremixture of concave and convex bending. This variation is causedby the experimental uncertainty due to the internal residualstresses and the plate geometrical imperfections that couldtrigger bending in either direction for the buckling mechanism.

3. Numerical results

3.1. Thermal analysis

Fig. 7(a) shows the temperature vs time for all the six beamgeometries measured at the centre of scanning path. For each

0

1

2

3

4

5

6

7

8

9

10

300Power (W)

Ben

ding

Ang

le (°

)

FEAExperiments

Plate thickness = 0.9mm Beam diameter = 16mm Speed = 10mm/s

500 700 900 1100

Fig. 6. Bending angle vs power for the circular laser beam with a diameter of

16 mm.

beam, the temperature starts rising approximately when thebeam enters the reference point defined by the coordinate (25,0.9, 0). The temperature increases rapidly until the beam leavesthe reference point and the temperature starts to drop. Thetemperature profile for laser bending using the conventionalcircular beam was found to be almost identical to the squarebeam (beam C—14.14�14.14 mm2). This can be explained by thefact that the lateral and longitudinal dimensions of both beamsare almost similar, and hence the distribution of energy is thesame along the scanning width.

Fig. 7(a) shows that the beams having a longer dimension inthe scanning direction produces higher temperatures despite thefact that the power intensity and beam effective area are the samefor all beams. This is caused by the longer beam–materialinteraction for beams with longer dimension in the scanningdirection. It is also interesting to note that the narrower the beamtransverse width, the higher the cooling rate after the beam leavesthe reference point.

Beam E which has the longest axial length produces thehighest temperature of 857 1C. However, its temperature spreadalong the z-direction is narrower compared to the other beams.Fig. 7(b) shows the isotherm for each beam. The beam–materialinteraction duration, ts, is the ratio of the axial length, dax, to thescanning speed, v, (ts¼dax/v). The interaction time and themaximum temperatures obtained for each beam at the referencepoint are given in Table 1. Fig. 8 shows the relationship betweenthe interaction time and the maximum temperature. It shows thateven though the beam intensity and the effective area for allbeams are similar, the beam interaction time produces a largedifference in the maximum temperature attained.

3.2. Structural analysis

Final surface deformation for each beam case is shown inFig. 9(a)–(f). In terms of the bending direction, only beam Aproduces concave bending (towards the laser direction) while allthe other beams produce convex bending. As expected, thedeformation produced by the circular beam is almost identicalto the square beam (beam C—14.14�14.14 mm2). This is due tothe similarity in the temperature profiles for the two beams, asdiscussed in the previous section. From here onwards, the circularbeam results will not be discussed further due to its similarthermal and deformation characteristics to the square beam.Vertical displacements for each beam along paths z–z0 and pathx–x0 are shown in Fig. 10(a) and (b) for clarity. The pattern showsan increasing vertical displacement from beam A to E. This showsthat the maximum temperature significantly influences themagnitude of deformation.

Variation of bending angle along the bending edge and thecurvature of the bending edge or the scanning line are twoinaccuracies which have been termed as ‘edge effect’ by previousresearchers [30–32]. It is interesting to note that by usingrectangular laser beams with different aspect ratios give thevariation of edge effect characteristic as shown in Fig. 9. From thefigure it can be seen that the bending angle at the beam entrancepoint (x¼0) is higher compared to the beam exit point (x¼50) forbeam A (Fig. 9(a)). This result is against the general finding wherebending angle at the beam exit point is higher due to the highertemperature at the exit point than that at the entrance [30]. As thebeam aspect ratio reduces as for the case of beam E, this trendreverses and the bending angle at the exit is higher than at theentrance (Fig. 9(e)).

Another edge effect characteristic is the curvature of thebending edge. Fig. 10(b) shows the vertical displacement alongthe path x–x0 (plane z¼0). The results for each beam are presented

0

200

400

600

800

1000

0Time (s)

Tem

pera

ture

(°C

)

A

B

C

D

E

Circular

Circular beam - 16mm diameter

Beam B – 20 x 10mm2

Beam D – 10 x 20mm2

XY

Z

XY

Z

A = 100 B = 200 C = 300 D = 400 E = 500 F = 600 G = 700 H = 800 x

zy

XY

Z

XY

Z

XY

ZX

Y

Z

2 4 6 8 10 12 14

Beam E – 8 x 25mm2

Beam C – 14.14 x 14.14mm2

Beam A – 25 x 8mm2

Fig. 7. (a) Temperature vs time at ref point (25, 0.9, 0) and (b) isotherms of temperature for each beam at t¼5 s.

Table 1Beam interaction time (ts) and maximum temperature at the reference point

(25, 0.9, 0).

Beam dax (mm) ts (s) Max temp (1C)

A 8 0.8 407.4

B 10 1.0 500.0

C 14.1 1.4 648.7

D 20 2.0 771.4

E 25 2.5 857.2

E

D

C

B

A350

450

550

650

750

850

950

0.5Interaction time (s)

Max

imum

tem

pera

ture

(°C

)

1 1.5 2 2.5 3

Fig. 8. Maximum temperature vs interaction time at reference point (25, 0.9, 0).


separately for clarity. The figure shows an increasing trend for thescanning line to curve away from its initial position to form aconcave shape. This is contributed by the buckling effect thatdevelops plastic strain in the x-direction along the scan path.During the heating process, thermal expansion initiates bucklingwhich produces compressive strain in the middle and tensilestrain at both end of the scanning line, as shown in Fig. 10(c).During cooling, the thermal strains reduce, leaving the residualplastic strains which push both ends upwards to produce a

concave shape. This trend becomes more apparent as the laserbeam length along the scanning axis (x-direction) increases, dueto the fact that a higher temperature develops higher plastic

Fig. 9. (a)–(f) Final surface deformation of the plate for all the beams considered. (a) Beam A-25�8 mm2; (b) beam B-20�10 mm2; (c) beam C-14.14�14.14 mm2; (d)

beam D-10�20 mm2; (e) Beam E-8�25 mm2 and (f) circular beam Ø16 mm diameter.


strains. The buckling effect produces opposite curvature; convexalong the z-axis and concave along the x-axis. This result might beuseful in many applications of 3D laser bending such as formationof a saddle shape and could be investigated further.

Fig. 11 shows the plastic strain along the path z–z0. As thisplastic is strain developed only around the heated region, thefigure shows plastic strain within 25 mm of the laser scanning

path. The values are taken at the end of the cooling period(t¼13 s). The results for each beam are presented separately forclarity. The plastic strain profiles show the local effect of laserbeam transverse width on the spread of the plastic region alongz–z0 direction. Generally, the top surface produces tensile strainwhiles the bottom surface produces compressive plastic strain,except for beam A which produces compressive strain on both the

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0Distance (mm)

Dis

plac

emen

t (m

m)

ABCDE

0.30.20.1

0-0.1-0.2-0.30.30.20.1

0-0.1-0.2-0.30.20.1

0-0.1-0.2-0.3-0.4

0-0.1-0.2-0.3-0.4-0.5-0.6

0-0.1-0.2-0.3-0.4-0.5-0.6

0Distance along scanning path (mm)

Dis

plac

emen

t, U

Y (m

m)

Beam A

Beam B

Beam C

Beam D

Beam E

Heating

Laserbeam

Bucking

CoolingY

X

10 20 30 40 50

10 20 30 40 50

Fig. 10. (a) Vertical displacement (Uy) along the path z–z0(transverse to the scanning direction); (b) vertical displacement (Uy) along the path x–x0(scanning direction) and

(c) buckling effect on the plate.

0

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

Distance (mm)

Pla

stic

stra

in, ε

z

A

Top surfaceBottom surface

25 25 25 25 250 00 0

B C D E

Fig. 11. Plastic strain along the path z–z0 (transverse to the scanning direction).


top and the bottom surface. The compressive strain on the bottomsurface increases from beam A to E. Beam A does not show anysignificant plastic strain on the top surface. This is believed to becaused by the longer buckling length along z-axis (contributed bya wider beam transverse width), which creates larger bucklingradii and thus lower tension. This is also contributed by lowertemperature distribution and higher flow stress. The magnitude ofplastic strains and hence the deformations are mainly influencedby the maximum temperature attained by specific beams. Thisshows that the temperature-dependent yield stress plays adominant role in the deformation of the plate, more than thespread of the beam in transverse direction.

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0Time (s)

Pla

stic

stra

in, ε

z

A topA bottomB topB bottomC top

C bottomD topD bottomE topE bottom

2 4 6 8 10 12 14

Fig. 12. Transverse plastic strain, ez, vs time at the reference point [top (25, 0.9, 0),

bottom (25, 0, 0)].

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0

Time (s)

Dis

plac

emen

t, U

Y (m

m)

A

B

C

D

E

2 4 6 8 10 12 14

Fig. 13. UY displacement vs time at point (0, 50, 0.9).

-400

-300

-200

-100

0

100

200

300

0Time (s)

Stre

ss, σ

z (M

Pa)

A topA bottomB topB bottomC top

C bottomD topD bottomE topE bottom

2 4 6 8 10 12

Fig. 14. Transverse stress, sz, vs time at the top (25, 0.9, 0) and bottom (25, 0, 0)

surfaces.


Fig. 12 shows the transverse plastic strain, ez, vs time at the top(25, 0.9, 0) and bottom (25, 0, 0) reference points at the centre ofthe scanning path. Generally a slight tensile strain is produced atboth top and bottom reference points [(25, 0.9, 0), (25, 0, 0)]before the beam reach the points due to the material expansionahead of the beam. After being heated, the top reference pointgenerally produces a tensile strain while the bottom referencepoint produces a compressive strain. Only beam A producescompressive strain at both top and bottom surfaces.

Fig. 13 shows the transient vertical displacement of the freeend of the plate measured at point (0, 0.9, 50). The verticaldisplacement at the beginning of the scan until about a third ofthe scanning time (total scanning time is 8 s) is very small. Afterabout 3 s the displacement develops rapidly. This is caused by thebuckling triggered by out-of-plane instability of the expandingheated material. This causes the plate to bend upwards for thecase of beam A, and downwards for all the other beams. Thedirection of bending is determined by a slight out-of-planeexpansions in the material contribute by the temperaturedistributions. In real laser bending process, internal residual

stresses and the plate geometrical imperfections are other factorsthat could affect the bending direction.

The reference point in Fig. 13 reaches its peak position at about5.2 s. Then for beams A, B and C the displacement reduces slightlybefore it increases again during the cooling periods. For beams Dand E the displacement continues to rise until the end of thecooling period. During the cooling period, as the temperaturedrops to the ambient temperature, the thermal strains decrease tozero and leave the residual elastic and plastic strains. The finaldeformation depends on the amount of plastic strain developedduring the heating cycle.

Fig. 14 shows the transverse stress, sz, vs time measured at thetop (25, 0.9, 0) and bottom (25, 0, 0) reference points at the centreof the scanning path. At the beginning of the scan the stress at themeasured points at both the top and the bottom surfaces istensile. This is due to the expansion of the heated region behindthe measured point. The stress increases rapidly until the beamhits the reference point. During heating the stress rapidly changesinto compression at both the top and the bottom surfaces due toconstrained thermal expansion. When buckling occurs, the topsurface is in tension whilst the bottom surface experiences furthercompression. After the beam leaves the reference point, thethermal expansion of the successive region will cause the stress tobe tensile. At the end of the cooling period, the top surface is incompression whilst the bottom surface is still in tension.

4. Conclusions

The proposed FE model serves as an efficient tool for analysisof laser bending process and process optimisation. Comparison ofthe numerical results with the experimental data clearly showsthe effectiveness of the model. The results of simulations areanalysed to determine the effect of different beam geometries onlaser bending of thin metal sheets. The results show that beamgeometries play an important role in the resulting temperaturedistributions on the workpiece. Longer beam dimensions in thescanning direction (in relation to its lateral dimension) producehigher temperatures, despite the fact that the power intensity and


beam effective area is the same for all beams. This is due to higherbeam–material interaction time for beams with longer dimensionin the scanning direction.

The temperature distributions affect the bending direction andthe magnitude of the bending angles. Higher temperaturesproduce more plastic strains and hence higher deformations. Thisshows that the temperature-dependent yield stress plays adominant role in the deformation of the plate, more than thespread of the beam in transverse direction. All beams in this studyproduced convex bending except beam A (25�8 mm2). Themagnitude of deformation increases from beam A to E.

Beam geometries also influence the ‘edge effect’ character-istics. The longer the beam axial length, the higher is the tendencyfor the scanning line to curve away from its original position toform a concave shape. This is caused by buckling which developstensile plastic strain along x-axis (i.e. along the scanning path) atboth ends of the scanning path. The buckling effect producesopposite curve profile; convex along the z-axis (transverse) andconcave along the x-axis (scanning path). This result could beuseful in the applications of 3D laser bending such as theformation of saddle shape and should be investigated further.

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