2005 fem-based process design for laser forming of doubly ...yly1/pdfs/liu fem.pdfjournal of...

Journal of Manufacturing ProcessesVol. 7/No. 2

2005

109

FEM-Based Process Design for Laser Forming ofDoubly Curved Shapes

Chao Liu and Y. Lawrence Yao, Dept. of Mechanical Engineering, Columbia University, New York, New York, USA

AbstractThis study presents a finite element method (FEM) based

three-dimensional laser forming process design methodolo-gy for thin plates to determine the laser scanning paths andheating conditions. The strain fields were first calculated viaFEM. The laser scanning paths were chosen perpendicularto the averaged principal minimum strain direction. The ra-tios of in-plane and bending strain were calculated to helpdetermine the heating condition. Two typical doubly curvedshapes—a pillow shape and a saddle shape—were studied,and the overall methodology was validated by experiments.

Keywords: Laser Forming, Process Design, Sheet Metal,Doubly Curved Shapes, Finite Element Method

1. IntroductionSignificant progress has been made in analyzing

and predicting laser forming processes of sheet metal.To advance the process further for realistic formingapplications in an industrial setting, it is necessaryto consider the issue of process design, which is con-cerned with determination of laser scanning pathsand heating conditions given a desired shape to form.

A number of approaches have been attempted inthe past few years in laser forming process design.Genetic algorithm based design (Shimizu 1997; Chengand Yao 2001) and response surface methodologybased robust design (Liu and Yao 2002) have beenreported. They are either for 3-D shapes with signifi-cantly simplified design tasks or for quasi 3-D shapessuch as cylindrical shapes and axisymmetrical shapes.However, these methods are not appropriate for gen-eral 3-D process design of doubly curved shapes dueto their geometry complexity.

In a series of reports, Ueda et al. (1994a,b,c) ad-dressed issues in the development of a computer-aided process planning system for plate bending byline heating. In the first report, the concept of inher-ent strain was emphasized, but its significance wasnot adequately articulated or demonstrated. Scan-

ning paths were determined based on FEM-deter-mined in-plane strain, and heating condition deter-mination was not addressed. This approach is notdirectly applicable to laser forming process designbecause no consideration for the characteristics ofthe laser forming process was given. The secondreport examined forming procedures for three simplecurved shapes often encountered in shipyards. Theprediction was found to be in agreement with thereal practice of skilled workers. However, the ap-proach too heavily relied on prior experience. More-over, when bending strain is relatively large, amechanical means such as a rolling machine wasused to generate the bending strain, which is notsuitable for an automatic laser forming process.

Jang and Moon (1998) developed an algorithmto determine the heating lines based on the lines ofcurvature of a design surface and the extrema of prin-cipal curvatures along them. This method, however,may only work for simple surfaces. Heating condi-tion, such as power and speed of the heat source,also has not been addressed. Edwardson et al. (2001)and Watkins et al. (2001) aimed to establish rulesfor positioning and sequencing the scanning pathsrequired for the 3-D laser forming of a saddle shapefrom rectangular sheet material. Various scanningpatterns were first postulated and then modeled viaFEM. However, the work too heavily relied on priorexperience in coming up with the patterns in the firstplace and may become even less effective when theshapes to be formed become more complex. Fur-thermore, a constant scanning speed was assumedthroughout an entire scanning pattern of a shape,and this severely limits the realization of the full pro-cess capability of laser forming.

Cheng and Yao (2004) presented a methodologyto design laser scanning paths and heating condi-tions of laser forming for a general class of 3-Dshapes from thin sheet metal. The strain field wascalculated by FEM. In determining laser scanningpaths and heating conditions, however, this method

This paper is an expanded version of a paper published in theTransactions of NAMRI/SME, Vol. 31, 2003.


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only considered the in-planecomponent of the total strainand, therefore, may incur largererrors for thicker plates. Liu,Yao, and Srinivasan (2004) pro-posed strategies for solving es-sentially the same processdesign problem as above. Thestrain field was determinedbased on differential geometryand an optimization approach.This method, however, con-cerned the middle surface only,again neglecting the effect ofbending strains across thethickness of a plate, and themethod therefore only workedwell for very thin plates.

This study presents an FEM-based 3-D laser forming processdesign method, which extends the method’s appli-cability to relatively thicker plates by consideringbending strain in addition to in-plane strain. Thestrain field was calculated by way of an elastic, large-deformation FEM model; the rationale of using theelastic model is given later in the paper. The laserscanning paths were chosen taking into account theprincipal minimum strain direction of both the topsurface and the middle surface of a plate. The ratiosof in-plane and bending strain were calculated tohelp determine the heating condition. Two typicaldoubly curved shapes were studied, and the overallmethodology was validated by experiments.

2. Problem DescriptionThe flexural properties of a plate depend greatly

on its thickness in comparison with other dimensions.A thin plate normally has a ratio of 10 < a/h < 100,where h is a plate thickness and a is a typical dimen-sion of a plate (Ventsel and Krauthammer 2001). Inthis study, the ratios a/h of the desired shape rangefrom 16 to 89 (thickness ranges from 0.89 mm to 5mm; width and length are both 80 mm), as opposedto about 123 in Cheng and Yao (2004) and about 90in Liu, Yao, and Srinivasan (2004). Although thin-plate theories are still applicable in the present study,additional considerations, especially for bendingstrains, need to be given.

The overall design strategy is outlined in Figure1. First, a strain field is obtained via a large-defor-mation, elastic FEM by flattening the desired shape.A large-deformation model is used because the de-sired shape has a large deflection relative to its thick-ness. The elastic FEM is utilized because strain-fielddevelopment from the desired shape to planar shapeis primarily geometrical and it simplifies the compu-tation without altering the problem. This point willbe elaborated on in section 3.1. After obtaining thestrain field, the principal minimum strains and di-rections in both the middle and top surfaces are cal-culated by solving the eigenvalues and eigenvectorsof the strain tensor. Because in the laser forming pro-cess the highest compressive strains occur in the di-rection perpendicular to a scanning path, it is naturalto place a laser path perpendicular to the principalminimum strain direction. In this study, scanningpaths are chosen to be perpendicular to the aver-aged minimal principal strain directions between topand middle surfaces.

Different from forming a singly curved shape,forming a doubly curved shape requires not onlybending strains (angular distortion) but also in-planestrains (stretching/contraction of the middle surfaceof a plate). Therefore, at each material point in theplate, the principal minimum strain can be decom-posed into in-plane strain and bending strain com-ponents. To match the required strain field in a laser

Figure 1Flow Chart of Laser Forming Process Design

Desired shape

Solve large deformation FEM

Scanning path direction

Heating condition

Laser forming FEM

Calculate and average the principal strain

direction at the top and middle surface

Decompose into principal in-plane and

bending strain

Decompose into principal in-plane and

bending strain

Choose scanning path normal to the averaged principal minimum strain direction

Calculate the ratio of bending and in-plane strain

Match Database of ratios of bending and in-

plane strain

Heating condition determination


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cubic-spline cross section sweeping along a path ofa cubic-spline curve. The sweep surface is definedby: S(x,y) = A(x) + B(y), x,y � [0,80] in mm. For thepillow shape, A(x) is defined by (0,0,0), (40,0,1.7),and (80,0,0), and B(y) is defined by (0,0,0),(0,40,2.9), and (0,80,0). For the saddle shape, A(x)is defined by (0,0,0), (40,0,1.95), and (80,0,0), andB(y) is defined by (0,0,0), (0,40,–1.95) and (0,80,0).The adjacent sides of the pillow shape are not iden-tical, with the one along the x direction curvingslightly higher. The saddle shape bends up at a pairof opposite sides and down at the other pair with thesame magnitude. The desired shapes are shown inFigures 2a and 2b, respectively.

3. Strain Field AttainmentAs indicated in the previous section, the first step

in the laser forming process design is to determinethe strain field required to form into the desired shape.The strain field of the planar development is solvedby a large elastic deformation FEM model under dis-placement constraints. The desired shape is placed

forming process, not only did the principal minimumstrain in the middle surface need to be satisfied, butalso the strains across the thickness, namely, the dis-tribution of bending strain. Therefore, the introduc-tion of ratio of in-plane strain with bending straincharacterizes the nature of strains not only in themiddle surface but also all across the thickness ofthe plate. For a specific principal minimum strain,there exist multiple heating conditions (different com-binations of laser power, velocities, and so on). How-ever, among them, only one satisfies both of in-planeand bending strain across the thickness. It is impor-tant to determine heating conditions (that is, laserpower P and scanning speed V) of a laser formingprocess such that not only the total strain at eachpoint but also the ratio between the in-plane andbending strains are realized. This is possible becausea strain field generated by the laser forming processhas its own characteristics; such characteristics mayvary even under the same line-energy input level(that is, the same P/V ratio) and, therefore, providethe possibility of matching the desired strains andratio. To get more understanding on the strain fieldgenerated by the laser forming, once againthe FEM is utilized. A database of laserforming FEM results based on the simpleststraight scan, but under a variety of condi-tions and validated experimentally, is estab-lished to aid the heat conditiondetermination. To match the required strainfield, not only did the principal minimumstrain in the middle surface need to be sat-isfied, but also the strains across the thick-ness. Therefore, the introduction of the ratioof in-plane strain to bending strain charac-terizes the nature of strains not only in themiddle surface, but also all across the thick-ness of the plate.

Surfaces of many engineering structuresare commonly fabricated as doubly curvedshapes to fulfill hydrodynamic, aesthetic,or structural functional requirements. In thisstudy, two distinctive doubly curved sur-faces are chosen as desired shapes in thisstudy. They are a pillow shape, which haspositive Gaussian curvature over the entiresurface, and a saddle shape, which has nega-tive Gaussian curvature over the entire sur-face. The two shapes are specified by a

Figure 2Doubly Curved Shapes To Be Laser Formed (80 × 80 × 1.4 mm)

(b) Saddle shape

(a) Pillow shape


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between two flat rigid bodies and compressed intoa planar shape to generate the strain field. The toprigid body is given a step-by-step displacement to-ward the bottom one along the thickness directionsuntil the gap between them is equal to the thick-ness of the desired shape. It is assumed that there isno friction between the rigid bodies and the de-sired shape.

The development of a strain field from one shapeto another is primarily a geometrical problem and isindependent of material properties. In fact, it has beenshown that the strain field determination is indepen-dent of Young’s modulus (Cheng and Yao 2004).Elastic FEM is used instead of elastic/plastic FEMfor this geometrical step because less material prop-erties need to be specified in elastic FEM. The large-deformation model is employed for the followingreasons. In the case of small deflection, the normaldisplacement component of the midplane (w0) issmall compared with the plate thickness (h), and thusthe in-plane strain can be neglected. However, if themagnitude of deflection increases beyond a certainlevel (w0 > 0.3h), these deflections are accompaniedby stretching or contraction of the midplane andtherefore cannot be neglected (Ventsel andKrauthammer 2001). In this study, the ratio w0/hreaches 3.3 and, therefore, large-deformation FEMis necessary. In addition, as discussed in section 2,the sheet thickness dealt with in the paper falls withinthe usual definition of thin plates although the bend-ing strain cannot be neglected.

The governing relations for elastic large defor-mation of a thin plate are briefly summarized. It isassumed that the material of the plate is elastic, ho-mogenous, and isotropic; the straight lines, initiallynormal to the middle surface before bending, remainstraight and normal to the middle surface during thedeformation, and the length of such element is notaltered. Deflection w0 in the z (thickness) direction isassumed large relative to the thickness h of the plate,and therefore, membrane forces (Nx, Ny, and Nxy) be-come more pronounced. In the middle surface, thestrain-displacement equations can be expressed as(Ventsel and Krauthammer 2001):

22

0 00 0 0 0

0 0 0 0 0

1 1, ,

2 2xx yy

xy

u w v w

x x x y

u v w w

y x x y

⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞ε = + ε = + ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂ ∂ ∂

γ = + +∂ ∂ ∂ ∂

(1)

where u0, v0, and w0 are the displacement compo-nents, and 0

xxε , 0yyε , and 0

xyγ are the strains at themiddle surface, respectively. The total strains in thelayer of the plate parallel and a distance z from themiddle surface can be written as

2 20 00 0

2 2

20 0

, ,

2

xx xx yy yy

xy xy

w wz z

x y

wz

x y

∂ ∂ε = ε − ε = ε −∂ ∂

∂γ = γ −∂ ∂

(2)

The equilibrium equation for the large-deflectionplate analysis follows Hookes law and can be ex-pressed as

( ) ( ) ( )22 2

2 2

22 2 20 0 0

2 2

12 1 xy

x y y x

NN N N N

Eh y x x y

w w w

x y x y

⎡ ⎤∂∂ ∂− ν + − ν − + ν⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦

⎛ ⎞∂ ∂ ∂= −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(3)

where ( )0 1xx x yN N

Ehε = − ν , ( )0 1

yy y xN NEh

ε = − ν , and

0 xyxy

N

Ghγ = ; E and G are Young’s modulus and shear

modulus, respectively, and � is Poisson’s ratio.

The strains, membrane forces, and displacement,w0, can be solved by the above set of equations, yield-ing two governing differential equations:

22 2 24 4 40 0 0

4 2 2 4 2 22

w w wh

x x y y x y x y

⎡ ⎤⎛ ⎞∂ ∂ ∂∂ ϕ ∂ ϕ ∂ ϕ ⎢ ⎥+ + = −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦(4)

4 4 40 0 0

4 2 2 4

2 2 22 2 20 0 0

2 2 2 2

2

12

w w w

x x y y

w w wP

D y x x y x y x y

∂ ∂ ∂+ +

∂ ∂ ∂ ∂

⎡ ⎤∂ ∂ ∂∂ ϕ ∂ ϕ ∂ ϕ′= + + −⎢ ⎥′ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦

(5)

where 3

212(1 )

hD′ =

− ν,

2

2xNy

∂ φ=∂

, 2

2yNx

∂ φ=∂

,2

xyNx y

∂ φ= −∂ ∂

, Eϕ = φ , and ′( )P x y, is the lateral

load in the z direction P(x,y) divided by E.

Once a desired shape is given, that is, deflection

w0 and curvatures ∂∂

20

2

w

x,

∂∂

20

2

w

y, and

∂∂ ∂

20w

x y are known,


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� and P� and, in turn, the in-plane

strains, εxx0 , εyy

0 , and γ xy0 can be calcu-

lated under appropriate boundary con-ditions, and the calculation isindependent of Young’s modulus. How-ever, Eq. (3) indicates that εxx

0 , εyy0 , and

γ xy0 depend on Poisson’s ratio �, which

is a geometric parameter. The total strain�xx, �yy, and �xy can also be calculated[Eq. (2)]. The FEM model was imple-mented in ABAQUS.

3.1 Decomposition of Strains

The above formulation assumes thinplates and therefore involves only �xx,�yy, and �xy, while the FEM implementa-tion gives 3-D strains in terms of ten-sor. To obtain principal strains andbending strains directly from the tensorE, the steps are briefly outlined below.E can be expressed in terms of �1n1n1

T

+ �2n2n2T + �3n3n3

T, where principalstrains, �1, �2, and �3 ε ε ε1 2 3≤ ≤( ) , andthe orientation of the principal strain,n1, n2, and n3, correspond to the eigen-values and eigenvectors of the straintensor E at that material point. There-fore, �1, �2, and �3, and n1, n2, and n3

can be obtained by solving the eigen-value problem

En = �n (6)

where � and n are principal strain andcorresponding principal strain direction,respectively. Figure 3 shows the contourplots of the minimal principal strain ε1

0( )distribution in the middle surface of thedesired shape obtained from the FEMmodel. In a thin plate, the variation ofstrain across the thickness is assumed tobe linearly distributed [Eq. (2)]; there-fore, the strain in the middle of the platecan be taken as the in-plane strain. As seen, the strainfield of the pillow shape shown in Figure 3a is onlysymmetrical about the x and y axes, while the strainfield of the saddle shape is symmetric about the ori-gin. This is obviously because the pillow shape speci-fied in section 2 is not symmetric about the origin.

Figures 4a through 4d show the magnitude andorientation of minimum principal strain at the middlesurface and top surface of both pillow and saddleshapes. The length of a bar in the plots representsthe magnitude of minimum principal strain at thatlocation. These strains represent the required strain

Figure 3Minimum Principal Strain in Middle Surface (80 × 80 × 1.4 mm)

(a) Pillow shape

(b) Saddle shape


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field to be realized by the laser forming process atthese planes. Clearly, the minimum principal strain(�1) and direction (n1) at a physical location repre-sents the magnitude and direction, respectively, ofmaximum shrinkage at that point.

The bending strain is caused by the nonuniformdistribution of strain in the thickness direction. Givenan (x, y) location, the bending strain at that locationvaries along the thickness direction and is generallydefined as the difference between strain at that z value

and the strain at the mid-plane of the sheet. For athin plate, the bending strain linearly increases withthickness and reaches the maximal value at sheetsurfaces [Eq. (2)]. The minimal principal bendingstrain at the top surface (z = h/2) is calculated basedon the minimal principal strains found above.

��b = �1h/2n1

h/2 – �10n1

0 (7)

where �1h/2 and n1

h/2 are magnitude and direction of

Figure 4Vector Plots of Minimum Principal Strain at (a) Middle Surface and (b) Top Surface of Pillow Shape, and (c) Middle Surface and

(d) Top Surface of Saddle Shape. (Length represents strain magnitude; orientation represents strain direction.)

Pillow Shape Saddle Shape

(a) Middle surface (c) Middle surface

(b) Top surface (d) Top surface


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the principal minimum strain at z, and �10 and n1

0 aremagnitude and direction of the principal minimumstrain at the mid plane. Figures 5a and 5b show themagnitude and orientation of the minimal principalbending strain at the top surface of the pillow shapeand the saddle shape, respectively. The length of abar in the plots represents the magnitude of bendingstrain at that location. The regions within the dashedlines (also in red) represent positive bending strain,while the rest of areas represent negative bendingstrains. The physical meaning of negative bendingstrain is that the top surface is subject to more com-pressive strain than the middle. In the laser formingprocess operating under a temperature-gradientmechanism, the plate bends toward the laser beam.

This indicates that the laser should be placed on thetop surface of the plate for this case. If the positivebending strains are much smaller than the negativeones and occur in much smaller regions, they there-fore may be neglected as in the case of the pillowshape, but not in the saddle case.

3.2 Laser Forming FEMAfter analyzing the strain field on the mechanical

pressing side, the strain field in the laser formingprocess needs to be explored as well. In an effort toestablish a database about the characteristics of la-ser forming induced strain distribution under vari-ous heating conditions, a simple straight line scanningalong a centerline of 80 × 80 × 1.4 mm plates isassumed in the FEM simulation of the laser formingprocess. Due to symmetry about the centerline, onlyhalf of the plate is simulated. The workpiece mate-rial is assumed isotropic. Material properties such asYoung’s modulus, yield stress, heat transfer proper-ties, thermal conductivity, and specific heat are tem-perature dependent. The material is 1010 steel. Theheat flux of the laser beam is assumed to follow aGaussian distribution. No melting is involved andno external forces are applied in the forming pro-cess. The symmetric plane is assumed to be adia-batic. Commercial FEM software, ABAQUS, is usedto solve the thermal mechanical problem. Liu andYao (2002) give more details.

The simulation results have been validated byexperiments, which will be described in more detailin a subsequent section. As shown in Figure 6, thebending angles calculated from FEM under various

20 30 40 50 600.4

0.8

1.2

1.6

2.0

2.4

Beam size=6mm

FEM result Experimental resultP=1400W

P=1200W

P=1000W

Ben

ding

Ang

le (

Deg

ree)

Velocity (mm/s)

Figure 6Experimental Validation of Numerically Determined Bending

Angle. (Error bars represent standard deviation of two samples.)(a) Pillow shape

(b) Saddle shape

Figure 5Vector Plots of Bending Strain


116

conditions are within the experimental errors. A da-tabase of in-plane and bending principal minimumstrains under these conditions is established as shownin Figures 7a and 7b, respectively. Figure 7c showsthe ratios between the in-plane and bending strainsunder these conditions. Obviously, many combina-tions of laser power, P, and scanning speed, V, willgive an identical ratio. This is indicative of the feasi-bility to choose a combination that matches not onlythe ratio but also the in-plane and bending strainvalues required by forming a desired shape.

4. Scanning Path DeterminationIt has been discussed in the previous sections that

the scanning path should be placed perpendicular tothe minimum principal strain direction because it iswell known that the maximal compression occurs inthe direction perpendicular to a laser scanning pathin laser forming. For a very thin plate where bend-ing strains are small, the in-plane minimal principalstrain direction, n1

0, should be used to determine thescanning paths (Cheng and Yao 2004). For the platesused in this study, there are sizable bending strains

10001050

11001150

12001250

13001350

1400 20

2530

3540

4550

5560

-0.016

-0.014

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

Inpl

ane

Prin

cipa

l Min

imum

Str

ain

Velocit

y (m

m/s)

Power (W)

10001050

11001150

12001250

13001350

140020

2530

3540

4550

5560

-0.007

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

Prin

cipa

l Min

imal

Ben

ding

Str

ain

Velocit

y (m

m/s)

Power (W)

10001050

11001150

12001250

13001350

1400 2025

3035

4045

5055

600.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Prin

min

ben

ding

/inpl

ane

stra

ins

Velocity (m

m/s)Power (W)

(a) (b)

(c)

Figure 7(a) Principal Minimum In-Plane Strain, (b) Principal Minimum Bending Strain, and (c) FEM Determined Ratio of

Principal Minimum In-Plane Strain and Bending Strain (laser beam diameter is 6 mm)


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in some regions of the plate (Figure 5), and it seemsmore reasonable to vector average the in-plane straindirection, n1

0, and the total strain direction, n1h/2, at

the top surface, where the highest bending strain isfound, as the basis for the laser path planning. Fig-ures 8a and 8b show a quarter (due to symmetry) ofthe plate with averaged �1

0n10 and �h/2nh/2, which is

used for determining scanning path direction.In determining the spacing of adjacent scanning

paths, a number of guidelines are followed. In gen-eral, the smaller the spacing, the more precise thedesired shape can be formed. Practically, however,the adjacent paths cannot be too close and also be-cause being too close will violate the underlying as-sumption when using the laser forming FEM. Theassumption is that adjacent paths should be inde-pendent with each other. The regions on a shape thathave larger strains need to be scanned more and moreexactly and therefore require denser paths. Roughlyspeaking, spacing between two adjacent paths, Dpaths,should be equal to average strain generated by laserforming, �laser, multiplied by laser beam spot size,dlaser, and divided by the average principal minimalstrain over the spacing. Figures 8a and 8b showsscanning paths determined following these guide-lines. As seen, they are perpendicular to the minimalprincipal strain directions everywhere and denseraround the center of the quarter due to larger strainsthere. Heating conditions are also indicated alongthe paths; their determination will be explained inthe next section.

Another consideration is which side of the plateshould be scanned. The physical meaning of nega-tive bending strain is that the top surface is subjectto more compressive strain than the middle, whichcorresponds to a concave curvature in a given shape.In the laser forming process, the plate always bendstoward the laser beam; therefore, the laser should beplaced on the top surface of the plate where the bend-ing strain is negative and be placed on the other sideof the plate if the bending strains are positive. Asshown from Figure 5a, the positive bending strainsof the pillow shape are much smaller than the nega-tive ones; because they are negligible, the laser shouldbe scanned at top surface of the plate. On the otherhand, the bending strains in the saddle shape re-veal a different scenario. The positive bendingstrains and negatives strains are almost equally dis-tributed, both in amount and magnitude. Therefore,

it is necessary to place the laser path on both sidesof the plate according to the distribution of the signof the bending strain.

5. Heating Condition DeterminationIf the laser spot size is given, the heating condi-

tions to be determined include laser power, P, andscanning velocity, V. While it is possible to continu-ously vary them to generate the strain field required

(a)

(b)

Figure 8Laser Paths and Heating Conditions Determined for (a) Pillow

Shape and (b) Saddle Shape. Only a quarter of the plate isshown due to symmetry. The strain field shown represents

minimal principal in-plane strain averaged between top andmiddle surfaces (numbers next to segments on each scanning

path represent scanning speed in mm/s).


118

to form the desired shape, this study adopts the strat-egy of constant power and piecewise constant speedfor a given path in favor of implementation simplic-ity. The procedure is summarized below. A path isbroken down to a few segments such that within eachsegment the range of strain variation (highest minuslowest strain) is about the same as in other segments.

The segment having the largest strain, which hasthe strongest influence on the final shape, is firstchosen. In determining the strain, strains betweenadjacent scanning paths are lumped together becauseall of these strains are to be imparted by the paths.The highest and lowest in-plane minimal principalstrains in the segment is averaged and compared withthose from the database established by the laser form-ing FEM, resulting in a group of (P,V) combinations.The process is repeated for the bending strains, re-sulting another group of (P,V) combinations. Theintersection of these two groups gives a unique (P,V)combination for the segment. The determined P valueis also adopted for the entire path. Ratios of princi-pal minimum in-plane strains and bending strainsversus laser power and velocity are calculated andplotted in Figure 7c. The magnitude of the ratio var-ies from 0.3 to 1.8. As indicated by the chart, heat-ing conditions with lower velocities tend to havelower ratio numbers. This can be explained by thefact that, at lower speed level, more heat will be al-lowed to be dissipated and thus heat energy is moreuniformly distributed across the thickness, and there-fore, in-plane shrinkage will be dominant. On theother hand, the faster the speed, the less time allowedfor heat to dissipate, the temperature gradient mecha-nism will be more pronounced, and bending strainwill be more dominant.

To determine scanning speed, V, for another seg-ment on the path, a level of compromise is neededbecause, with P chosen, to find a V value that satis-fies the required ratio of in-plane and bending mini-mal principal strains (using Figure 7) as well as thestrains themselves is generally unattainable. Assum-ing V0 and V1 are the scanning speeds determined bythe in-plane strain, 0

1ε , and bending strain, / 21hε , from

the database, the scanning speed for the segment is

0 10 11 1

0 1 0 11 1 1 1

V V Vε ε= +

ε + ε ε + ε(8)

The procedure is repeated for other segments ofthe path as well as for all other paths. Figures 8a and

8b shows the heating conditions determined follow-ing the above strategy. Due to symmetry, only onequarter of the plate is shown. As seen, a constant laserpower is chosen for a path and a constant speed (unit:mm/s) for a segment. In the case of the saddle shape,both sides of the plate need to be scanned. Due to thegeometric symmetry of the given shape, the heatingconditions are kept the same on both sides of the plate.The region below in the dashed line (in red) needs tobe scanned on the top side, while the region abovethe line is scanned on the other side, based on theexplanations associated with Figure 5.

6. Experimental ValidationExperiments were conducted on 1010 steel cou-

pons with dimensions 80 × 80 × 1.4 mm. The scan-ning paths and heating conditions in the experimentswere determined as described above and indicatedin Figure 8. The laser system used is a PRC-1500CO2 laser, which is capable of delivering 1,500 Wlaser power, and the laser beam diameter on the topsurface of workpiece is 6 mm. The motion ofworkpieces was controlled by a Unidex MMI500motion control system, which allows easy specifica-tions of variable velocities along a path with smoothtransitions from segment to segment.

Figure 9 shows the formed pillow and saddleshapes under these conditions. A coordinate mea-suring machine is used to measure the geometry ofthe formed shapes. Figure 10 compares the geom-etry of the formed shape under the determined con-ditions and the desired shape. Only the top surfaceof the plate is measured and compared. A generalagreement can be seen from the figure, and themiddle of the plate shows some discrepancy. Pos-

Figure 9Laser-Formed AISI1010 Steel Plate Using Scanning Paths and

Heating Condition Indicated in Figure 8


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sible sources contributing to the discrepancy includethe lumped method used to sum strains between ad-jacent paths; finite number of paths to approximatea continuous strain field; and constant power for eachpath and constant velocity for each segment. In form-ing the saddle shape, each side of the sheet metal isscanned alternatively, that is, scanning a path on aside, then scanning a path on the other side, and soon, to achieve the thermal symmetry to the maximalpossible extent (Hennige 2000).

7. Further Discussion on Bending Strainand Thickness Effect

In the approach to obtain the bending strain pre-sented in section 3.1, if the in-plane minimum prin-cipal strain has a similar direction as that at the topsurface, the resultant bending strain has a similardirection as the in-plane strain such that there is noambiguity to determine the direction of the scanningpath. To guarantee that the bending strain and in-plane strain lie in the same direction, the bending

strain is alternatively calculated as the appropriatestrain in the bending strain tensor, projected alongthe minimum principal in-plane strain direction.

In a general 3-D space, the bending strain tensoris obtained by

Eb = E1 – E0 (9)

where E1 is the 3×3 strain tensor at the top surfaceand E0 is the 3×3 strain tensor in the middle surface.Once the minimum in-plane strain, �1

0n10, is obtained,

the magnitude of the bending strain can be calcu-lated as the following:

0 01 1

T

b bn E nε = ⋅ ⋅ (10)

In this expression, �b is a scalar, with either posi-tive or negative value, representing the magnitudeof the bending strain whose direction is the same asn1

0. The expression of the bending strain can be sim-plified for the case of thin plates. Under the assump-tion of thin plate theory, the strain in the middlesurface can be written as in Eq. (1), and the totalstrains in the layer of the plate parallel and a dis-

tance z from the middle surface can be written as in

Eq. (2). As seen from Eq. (2), 0w

x

∂∂

and 0w

y

∂∂

repre-

sent the slope of the deformed shape in the x and y

direction, and 2

02

w

x

∂∂

and 2

02

w

y

∂∂

are the local curva-

tures in the x and y direction. Therefore, the bending

strain can be expressed by subtracting strain in themiddle surface from the total strain, namely,

2 20 0

2

2 20 0

2

b

w wz z

x x y

w wz z

x y y

⎛ ⎞∂ ∂− −⎜ ⎟∂ ∂ ∂⎜ ⎟=⎜ ⎟∂ ∂− −⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠

E (11)

Figures 11a and 11b show the magnitude and ori-entation of the bending strain obtained from Eq. (10),projected onto the x-y plane, for the pillow and saddleshapes, respectively. The length of a bar in the plotsrepresents the magnitude of bending strain at thatlocation. As seen in the plot, the bending strains arein the same direction as in-plane strain (Figures 4aand 4b). In these plots, the red region representspositive bending strain, while the black region rep-

Figure 10Comparison of Formed Shape (solid lines) and Desired Shape

(dashed lines). Formed shape was measured by CMM.

(a)

(b)


120

resents the negative bending strain. The positivebending strain means that the strain is more com-pressive in the middle surface than the top surface,vice versa for the negative bending strain. Also wor-thy of mentioning is that the magnitude of the bend-ing strain obtained based on Eq. (11) is almostidentical to that obtained based on Eq. (10) andshown in Figure 11. Therefore, for a thin plate, it isadequate to use the plane strain formula [Eq. (11)]to obtain the bending strain.

As seen in Eq. (11), the magnitude of bendingstrain is proportional to the thickness of the plate.Therefore, whether a desired shape can be precisely

formed or not depends on not only the given shapebut also its thickness. The characteristics of in-planestrain and bending strain with various thicknesseswere studied in this paper using two more thicknesslevels, 0.89 mm and 5 mm, in addition to the thick-ness 1.4 mm discussed so far for the saddle shape.The plate size remains the same as 80 × 80 mm. Asexpected, the magnitude and distribution of in-planestrains obtained for the 0.89 mm and 5 mm thickplates are essentially the same as that for the 1.4 mmplate shown in Figure 4c. However, this is not truefor the bending strains. As shown in Figures 12aand 12b, the magnitude of bending strain changed

Figure 11Vector Plot of Bending Strain of (a) Pillow Shape and

(b) Saddle Shape Obtained Based on Eq. (10).(Plate thickness = 1.4 mm)

(a)

(b)

Figure 12Vector Plot of Bending Strain of Saddle Shape with (a) 0.89 mmThickness and (b) 5 mm Thickness. [Note: (b) scale is half of (a).]

(a)

(b)


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significantly as thickness changed. Compared withFigure 11b for the thickness of 1.4 mm, Figure 12afor the thickness of 0.89 mm shows smaller bendingstrains, while Figure 12b for the thickness of 5 mmshows much larger bending strain (note the scalehere is halved for viewing clarity). The magnitudevariation of the maximum value of bending strainand in-plane strain is plotted in Figure 13, whichshows almost constant for the in-plane strain andalmost linear change for the bending strain with thick-ness. This also means that when thickness changesone may need to change to a different beam spotsize, or another laser forming mechanism such as abuckling mechanism, to obtain the required in-planeand bending strains for that thickness. For processdesign, this means that there might be a need to es-tablish different databases for different plate thick-ness values.

8. ConclusionsThe FEM-based 3-D laser forming process de-

sign methodology for thin plates considering thebending strain effect is experimentally shown ef-fective. A strain field required to form a desireddoubly curved shape is obtained through FEM first,decomposed to in-plane and bending strains, andboth used to determine the scanning paths. An al-ternative way to obtain the bending strain is alsopresented. In determining heating conditions, theconcept of in-plane and bending strain ratio is use-ful in approximately matching a strain distributionrequired to form a desired shape with that producedby a laser forming process.

ReferencesCheng, J. and Yao, Y.L. (2001). “Process synthesis of laser forming

by genetic algorithms.” Proc. of ICALEO 2001, Section D 604.Cheng, J. and Yao, Y. (2004). “Process design of laser forming for

three-dimensional thin plates.” ASME Trans., Journal of Mfg. Sci-ence and Engg. (v126, n2), pp217-225.

Edwardson, S.P.; Watkins, K.G.; Dearden, G.; and Magee, J. (2001).“3D laser forming of saddle shape.” Proc. of LANE 2001.

Hennige, T. (2000). “Development of irradiation strategies for 3D-laser forming.” Journal of Materials Processing Technology (v103),pp102-108.

Jang, C.D. and Moon, S.C. (1998). “An algorithm to determine heat-ing lines for plate forming by line heating method.” Journal ofShip Production (v14, n4), pp238-245.

Liu, C. and Yao, Y.L. (2002). “Optimal and robust design of laserforming process.” Journal of Manufacturing Processes (v4, n1),pp000-000.

Liu, C.; Yao, Y.L.; and Srinivasan,V. (2004). “Optimal process plan-ning for laser forming of doubly curved shapes.” Trans. of ASME,Journal of Mfg. Science and Engg. (v126, n1), pp1-9.

Shimizu, H. (1997). “A heating process algorithm for metal formingby a moving heat source.” Master’s thesis. Cambridge, MA: Mas-sachusetts Institute of Technology.

Ueda, K.; Murakawa, H.; Rashwan, A.M.; Okumoto, Y.; and Kamichika,R. (1994a). “Development of computer-aided process planningsystem for plate bending by line heating (report 1) – relationbetween final form of plate and inherent strain.” Journal of ShipProduction (v10, n1), pp59-67.

Ueda, K.; Murakawa, H.; Rashwan, A.M.; Okumoto, Y.; and Kamichika,R. (1994b). “Development of computer-aided process planningsystem for plate bending by line heating (report 2) – practice forplate bending in shipyard viewed from aspect of inherent strain.”Journal of Ship Production (v10, n4), pp239-247.

Ueda, K.; Murakawa, H.; Rashwan, A.M.; Okumoto, Y.; and Kamichika,R. (1994c). “Development of computer-aided process planningsystem for plate bending by line heating (report 3) – relationbetween heating condition and deformation.” Journal of ShipProduction (v10, n4), pp248-257.

Ventsel, E. and Krauthammer, T. (2001). Thin Plates and Shells:Theory, Analysis and Applications. New York: Marcel Dekker,Inc.

Watkins, K.G.; Edwardson, S.P.; Magee, J.; Dearden, G.; and French, P.(2001). “Laser forming of aerospace alloys.” AeroSpace Mfg.Technology Conf.

Authors’ BiographiesChao Liu was a PhD candidate with the Dept. of Mechanical

Engineering at Columbia University. His area of research interestincludes optimal and robust process design methodologies as appliedto laser forming processes.

Y. Lawrence Yao is a professor of mechanical engineering at Co-lumbia University, where he directs the Manufacturing Research Lab-oratory. His area of research interests includes laser forming, laserpeen forming, microscale laser shock peening, and lasermicromachining. He has a PhD in mechanical engineering from theUniversity of Wisconsin-Madison.

1 2 3 4 50.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Maximum value of inplane strain Maximum value of bending Strain

Str

ain

Thickness (mm)

Figure 13Variation of Maximum Value of Bending Strain and In-PlaneStrain with Different Plate Thickness (from FEM simulation)

2005 fem-based process design for laser forming of doubly ...yly1/pdfs/liu fem.pdfjournal of...

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