Evaporative material removal process with a continuous wave laser

Cmpurm

Pergamon

0045-7949(95)00043-7

EVAPORATIVE

MATERIAL CONTINUOUS

& .Slrucruws Vol. 57. No. 4. pp. 663-671. 1995 Elsewr Science Ltd Printed in Great Britain 0045.7949195 $9.50 + 0.00

REMOVAL PROCESS WAVE LASER

WITH

A

P. Majumdar, Z. H. Chen and M. J. Kim Department

of Mechanical

Engineering,

Northern

Illinois

University,

Dekalb,

IL 60115, U.S.A.

(Received 15 April 1994) Abstract-The formation of a groove by evaporative removal of material from a moving surface subjected to a high energy Gaussian laser beam is considered. The mathematical model is based on a heat conduction model that includes a moving boundary surface from which material is removed by evaporation. The loss of energy is assumed to be due to conduction and convection. The numerical solution is obtained by using a finite element method. Parameters for this process are identified and the effect of such parameters on the shape and size of the groove are presented. The effect of boundary conditions at the bottom surface of the test piece is also discussed.

NOMENCLATURE

material surface. The performance of this process depends on the type and characteristics of the laser, and optical and thermal properties of the material. Several types of lasers are used as gas (COZ) or solid state (YAG) lasers that operate either in a continuous wave (CW) or pulsed mode. Most of the modelling of laser processes is based on the solution of the heat conduction problem of a stationary heat source acting on a stationary surface. Some analytical works have dealt with the temperature distribution in laser heating of a slab [l-3]. The effects of peak frequency of the laser pulse on the heating of metal sheet were investigated analytically by Rykalin et al. [4]. Dabby and Peak [5] have treated analytically the process of material removal from a transparent surface subjected to a laser beam using one dimensional heat conduction model. Peak and Gagliano [6] presented theoretical analysis of temperature distribution and induced thermal stress in the laser drilling of alumina ceramic substrate subjected to a pulsed laser beam. Effects of beam energy and pulse duration on induced stress were investigated and results were compared with experimental data. Rogerson and Chayt [7] presented the exact solution for the melting time in an ablating slab, using one-dimensional heat conduction model. Von Allmen [8] described the theoretical model of a laser drilling process in which lateral conduction losses were neglected. Results showed agreement with experiment in a certain range of intensities of a rectangular pulsed laser. Nowak and Pryputniewicz [9] investigated experimentally and theoretically pulse laser drilling in a partially transparent media. They used a three-dimensional finite difference model with temperature dependent thermal properties to predict the shape of the drilled hole and demonstrated the importance of the shape and irradiation distribution of the incident laser beam on the quality of the hole.

Biot number laser power density at the center of the beam heat of sublimation convection heat transfer coefficient laser power evaporation parameter conduction parameter laser beam radius laser beam radius at z = 0 groove depth dimensionless groove depth final groove depth temperature evaporation temperature environment temperature dimensionless time laser moving velocity nondimensional laser moving velocity space variables in X. J’ and z directions nondimensional space variables in x, y and z directions length, width and thickness of the specimen in I, 4’ and 2 direction non-dimensional length, width and thickness of the specimen dimensionless temperature absorptivity thermal diffusivity density time variable convergence limit relaxation factor

INTRODUCTION

The use of high energy laser for cutting and machining materials has received considerable interest recently due to high tolerance, rapid, flexible and precise cutting and machining that can be achieved in these processes. Material is removed by melting and/or vaporization as intense highly directional and coherent monochromatic laser light interacts with the 663

P. Majumdar et ul

664

A three-dimensional numerical model for predicting the temperature distribution and melt depth in a welding process using a laser beam was developed by Mazumdar and Steen [IO]. The problem of a twophase moving interface and surface tension driven convection in the molten pool was considered by Chan et al. [I I] in their two-dimensional model. The two-dimensional steady-state laser melting problem was considered by Basu and Srinivasan [l2]. They investigated the surface tension driven flow pattern in the molten pool and its effects on the heat transfer and pool shape were analyzed for varying laser power. Yoshizawa ef al. [ 131 studied experimentally the process of cutting concrete and reinforced concrete with both continuous and pulsed beam using I kW COz laser. The effects of laser output power and lens parameters were investigated. Masumoto et ul. [I41 showed experimentally the relation between process parameters and cut quality in laser cutting of aluminum alloys. Considerable research on laser cutting processes has been conducted by Modest and Abakians. They [ 151 developed a numerical model for evaporative removal of material from a semi-infinite solid subjected to a moving Gaussian CW laser. In this model thermal losses due to conduction and convection are assumed to be minor and treated in an approximate manner, and results are presented for different laser and solid parameters. Biyikli and Modcst [I61 used the same model to investigate the effect of focusing parameters of the laser optical system on the cutting process. Abakians and Modest [I71 considered partial evaporation of moving semi-infinite and semi-transparent solid based on the assumption made by Boersch-supan [18] that material removal is a volume rather than a surface phenomenon. Their results show the laser deposition of energy may be treated as surface phenomena unless the medium is very transparent. Bang and Modest [I91 included in their study the effects of multiple reflections and beam

channeling using a one-dimensional model. Their results show increased material removal rates for considering multiple reflections and increased effectiveness of using beam channeling for high reflectivity materials and/or deep grooves cases. In this paper the material removal process using a high energy Gaussian CW laser beam is analyzed using a two-dimensional finite element model. The effect of boundary conditions and parameters on the shape and size of the groove will be investigated. MATHEMATICAL

FORMULATION

A typical laser cutting installation is shown in Fig. I. The physical processes involved in the laser cutting process are basically thermal in nature. When a laser beam strikes on a material surface, several effects take place: reflection and absorption of the beam; conduction of heat in the material and loss of heat by convection and/or radiation from the material surfaces. The amount of energy absorbed and utilized in removing the material depends on the optical and thermo-physical properties of the material. The mathematical mode1 describing the process of material removal from the surface subjected to high intensity laser beam is presented here. The process of material removal by high intensity laser can be described by heat transfer mode1 similar to the one presented by Modest and Abakians [I 51. It is assumed that there are three different regions on the surface of the material: Region I is assumed to be too frlr away to have reached evaporation temperature; Region II is assumed to be the area in which evaporation takes place; Region III is assumed to be the region in which evaporation has already been taken place. The following assumptions are also made in deriving the model: (I) the laser is assumed to be the Gaussian continuous beam type: (2) the material moves at constant relative velocity: (3) the

Gaussian laser Beam

Fig. 1. Laser cutting

installation.

Evaporative material removal process material is isotropic and opaque with constant thermal and optical properties; (4) material removal is assumed to be a surface phenomena and phase change from solid to vapor occurs in one step; evaporated material and inert assisted gas are assumed to be transparent and do not interfere with the incident laser beam; (5) heat losses by convection and radiation from the surfaces to the environment can be approximated by using single constant coefficients. Based on these assumptions, the mathematical statement of the problem is as follows:

Boundary condition 2. At the upper surface, outside laser beam incident area, convection condition gives ao --B,O=O SZ

F,

-Y,,,ax< Y< Y,,,.

-=O

or

az

(1)

O=O

atZ=Z,.

-x,
(6)

Boundary condition 3. Boundary conditions at the bottom surface could be either insulated or at constant temperature with infinite surface heat transfer coefficient, i.e

ao

ao a20 a20 a% g+ux=~+~+rlz 2'

atZ=O,

X,,,(Y)
Gooerning equations *

665

-YE<

Y<

Y,.

(7)

conditions

Boundary condition 1: at the upper surface. Region I-no evaporation at the surface. Balance of energy absorbed at the surface with conduction and convection losses gives

Boundary condition 4. Constant temperature boundary conditions are assumed at the positive and negative x planes O=O

atX=*X,.

(8)

Boundary condition 5. Convection or constant perature boundary conditions can be assumed planes as

-x,

-=cx -z X,,"(Y).- Y, < Y < Y,

(2)

a0 z+B,O=O

Region II-evaporation takes place at the surface. Balance of energy absorbed at the surface with energy utilized in evaporating materials and energy losses due to conduction and convection gives

or

O=O

atY=+Y,,

-x,
temat y

o
(9)

Ltariables.

x=x

@y-L)

(T,-TX)’

s=&).

R(O)’

Y=& a,7

z=&),

(10)

f=R?(o).

Parameters. and assuming temperature temperature of evaporation 0 = I

at the surface gives

equal

to

at Z = S(X. Y),

B

x,,, ( Y) < x < XX,, ( Y). - y,,, < y < y,,, Region IIIevaporation place and similar balance

u

has already been of energy gives

(4) taken

=

)

=

uR(O) -a, hR(O)

-,

k

x, F

&=R(O). Nk=

y

_ b

k(T, - T, ) R (OW

l’aF R(O) ’

.

ph,u

NC=-.

G

(11)

Here B, is the Biot number representing the ratio of convective to conduction losses. U represents the ratio of relative speed of the work specimen to the thermal diffusivity of the material. N, gives the ratio of energy utilized in evaporating the material and the absorbed laser energy. NL approximates the ratio of conduction losses to the absorbed laser energy. Finite element formulation

atZ=S,(Y),

x,,,(Y)
- Y,,, < Y < Y,,, . CAS Vi&H

(5)

The governing equation and boundary conditions are solved using a finite element method for the two-dimensional case with negligible heat loss and

P. Majumdar

666

c/ crl

The unknown

function

0(X,

0 is approximated

Y) =

c

as

0,Y)”

(13)

,=I

where YF is the interpolation function for a finite element. The global finite element model of the problem is expressed as [A]{@) +[B]{Oj={P}

Fig. 2. Two-dimensional

finite element

mesh.

large dimension in the I’ direction. A detailed description of the finite element formulation for the discretization of the governing differential equation and boundary conditions is given by Kim et ul. [lo]. The discretized equations are summarized as follows. The problem domain is initially divided into a mesh of rectangular elements as shown in Fig. 2. The variational formulation of the equations for a finite element in matrix form is expressed as:

forO
Gt,,

(14)

where [A]. [B], [P) are known matrices and {0} is the column vector of the unknowns. The set of first-order differential equations is transformed into a set of algebraic equations by approximating the first-order time derivative. The first-order time derivatives can be approximated by using a scheme given as

for O
< 1.

(15)

Using this approximating scheme, eqn (14) is transformed into a set of algebraic equations given in matrix form as:

where Ml”

= ,I

s

La:@L?+,= @II’@;.,,+ {p}..,!+I

Y, Y, dX d 1

(16)

Qlcl

where

[kl=[Al+cAt,,+,[Bl xdXdY-

[e] = [A] - (I - c)At,,+ , [B].

B, Y, ‘4, ds J l,Cl

FCC’= _

Y’JdX j
dy - j

The solution at time t = t,,+ , is obtained in terms of the solution known at time t,, At t = 0 the solution is known from the initial conditions of the problem,

qY, ds

,-,c,

U=O, Ned,

Bi=O.CNXlI.Nk =0.5

Position along the length x Fig. 3. The groove

shape with different

number

of meshes.

Evaporative

material

removal

process

661

Transient Groove Shapes U=O),Ne=O, Bi=0.0001. Nk=O.5 2.50

I

I

-

-2.40

-4.00

-0.80

I

I

0.80

2.40

4.0

Position along the leag~ X

Fig. 4. The groove and therefore eqn (16) can be used to obtain solutions with marching in time.

Computational

shape using transient

the

methods

Since the center region will be subjected to higher temperature gradients as it is located right below the laser beam, smaller elements are used in this section. However, as the location of the surface at Z = S moves due to the evaporation of the materials, we need to relocate this surface based on the criterion that evaporation takes place when temperature at any node reaches unity, and redistribute the meshes in the new domain. In order to establish the domain at every time step an iterative procedure is utilized. At any section a new position of interface location Z,,,,, = S is identified and calculated based on linear interpolation of two other points, between which the value of 0 crosses over unity, representing nondimensional evaporation temperature. This position is expressed as below: Z polnl

=

-z,

z,+ 2 +’ _ *I (0, -,+I o

- I).

(17)

analysis.

OnceZ,,,,, is identified, the length between Z = Sand Z, is divided into equal lengths among the elements and then this new mesh distribution is used to obtain a new temperature distribution. This iteration procedure is continued until the percent relative error for temperature at each nodal point and also the percent relative error for Z,,,,, fall below the prespecified tolerance oft, = 0.05%. Also, after each new value of temperature is calculated, that temperature value is modified by an underrelaxation factor 1 as a weighted average of the previous and the present iterated values to dampen out any oscillation and to enhance convergence rate, as o,=Aoi+(l

Mesh sizes were also refined until prediction of the groove shape did not differ significantly. Figure 3 shows groove shapes obtained for the case with U = 0 using different number of mesh distribution. Curves 1 and 2 show considerable changes in groove shapes using 10 x 5 and 18 x 5 size distribution.

Position along the length x Fig. 5. The groove

shape

-RI)@,_,

using steady-state

analysis.

P. Majumdar ef a/

668

Position along the length x Fig. 6. The groove

depth

Curve 3 shows identical groove shapes using 20 x 5 and 20 x 10 mesh size distributions and Curve 4 shows smoother and identical groove shapes using 40 x 5. 40 x 10 and 40 x 20 mesh size distribution.

RESULTS AND DISCUSSION The finite element model is used to study the process of material removal from a solid surface subjected to a Gaussian laser beam of constant power. The groove shapes and sizes are studied for various operating parameters and boundary conditions. Figures 4 and 5 show the groove shapes predicted by transient and steady-state analyses for the case of U = 1, respectively. For the transient case. Fig. 4 shows that the rate of increment of the groove size decreases with increase in time and reaches a steadystate value. Figure 5 shows that the groove shape and size obtained by steady-state analysis are almost identical to those obtained by transient analysis with t =8.

with different

values of Nk.

Figures 6-9 show results for a case with insulated boundary condition at the bottom surface and at moving nondimensional velocity of U = 1.O. Figure 6 shows the effects of conductive loss parameter Nk on the groove shape size. It can be seen that at higher Nk the groove becomes shallower due to higher conductive losses for a given surface irradiation. Figure 7 shows the effect of surface convection resistance on the groove shape and size by varying Biot number. Results show an increase in groove size with decrease in Biot number due to lower convection heat losses. A major increase in groove shape and depth can be noticed when Biot number is decreased from 0.1 to 0.01. The effect of nondimensional velocity on the material removal process is shown in Fig. 8. A higher nondimensional velocity U is obtained by either increasing the velocity of the moving laser or using material with lower thermal diffusivity. For a given material, higher laser velocity will cause lower residence time for heating the surface, resulting in reduced groove depth and size. Effect of laser power and latent heat of evaporation of the material on this

=

Ne 0.01

Position along the length x Fig. 7. The groove

depth

with different

values of II,.

Evaporative material removal process 0.0

-

0.5

-

1.0

-

669

N

2

4 Q x

5@

1.5 -

1. Nk = 0.80 2. Nk = 0.75

u= 1.0

3. N!S= 0.70

Bi = 0.0001

4. Nk = 0.65

Ne = 0.01

2.0 -

I

2.5 -4.0

I

-2.0

I 2.0

0.0

4

0

Position along the length x Fig. 8. The groove

depth

process is shown in Fig. 9 by varying nondimensional parameter N,. As expected a decreasing N, will increase groove size due to the higher effective energy available for evaporation of material with low heat of sublimation. It can be noticed that groove shape becomes narrower and deeper as N, is decreased. Figure 10 shows the shape of groove for two different boundary conditions at Z = Z,, representing two limiting cases. Curve 1 is for boundary condition:

with different

values of (1.

be increased or evaporation rate can be increased considerably by insulating the bottom of the work piece, and consequently, lower laser power will be required to reach the same groove size. Figure 11 shows the groove shape during the laser cutting process where the work piece is moving with a nondimensional moving speed of U = 1. Results show clearly the cutting Region II, where the material is being removed by evaporation and also the Region III where a constant cutting depth S, is reached for the operating parameters chosen.

de3 az

-=OatZ=Z,, CONCLUSIONS

which means that the bottom surface of the work piece is assumed to be adiabatic or perfectly insulated. Curve 2 is for boundary condition 0 = 0 at the bottom surface, representing perfect heat transfer with environment, which means that the temperature of the work piece is kept the same as the temperature of the surrounding at Z = Z,, X = &XF and Y = + Yr. It can be seen that the groove depth will

Various laser and material parameters and operating conditions affect the shape, size and time required to form a groove by the process of evaporating materials subjected to high intensity laser beam. A two-dimensional finite element model is developed to anlayze this material removal process using a CW Gaussian laser beam. The effect of various nondimensional parameters and type of boundary conditions

1. Bi = 0.1 2. Bi = 0.01 3. Bi=O.OOl 4. Bi = 0.0001

3, 4 Ne = 0.01 N!=z= 0.7

2.0

0.0

-2.0

Position along the length x

Fig. 9. The groove

depth

with different

values of N,.

4.0

P. Majumdar et al.

670

1.5

-

1. $0

Bi = O.ooOl

otz=z~

Nk = 0.5

2. e=o afz=z, 2.0

Ne = 0.01

Position along the length x Fig. 10. The groove depth with different bdundary conditions

8,

1.5

P 8

6

2.0

U = 1. Ne = 0.01, Bi = O.OCOl, Nk = 0.5

Position along the length x Fig. I I. The groove shape with increase

maintained at the bottom surface on the effectiveness of this process is discussed. Future improvement of the computational model would be to extend to a three-dimensional model with variable optical properties of the materials, effects of parameters of the lens focusing system, and laser beam characteristics.

I. 8. 9.

REFERENCES IO. 1. K. Brugger. Exact solutions for the temperature rise in a laser-heated slab. J. uppI. P&s. 43(2) (1972). 2. R. E. Warren and M. Sparks, Laser heating of a slab having temperature-dependent surface absorption. J. appl. Phys. 50(12) (1979). A. A. Uglov and I. Yu. Smurov. 3. N. N. Rykalin, Nonlinearities of laser heatig of metals. Sop. Phys. Dokl. 27(11) (1982). 4. N. N. Rykalin, A. A. Uglov and N. I. Makarov. Effect of peak frequency in a laser pulse on the heating of metal sheets. Sou. Phys. 12(6) (1967). 5. F. W. Dabby and U.-C. Pack, High-intensity laserinduced vaporization and explosion of solid material. IEEE J. Quantum Electron. QE-8(2) (1972). 6. U.-C. Paek and F. P. Gagliano, Thermal analysis of

Il.

12.

13.

14.

in time from

t = I to t = 8.

laser drilling processes. IEEE J. Quantum Electron. QE-S(2) (1972). J. E. Rogerson and G. A. Chayt. Total melting time in the ablating slab problem. J. appl. Phys. 42(7). (1971). M. Von Allmen. Laser drilling velocity in metals. J. appl. Phys. 47 (1976). T. Nowak and R. J. Pryputniewicz. Theoretical and experimental investigation of laser drilling in a partially transparent medium. J. Electronic Packq. 114/71 (1992). J. Mazumdar and W. M. Steen. Heat transfer model for CW laser material processing. J. appl. Phys. 51(I) (1980). C. Chan, J. Mazumdar and M. M. Chen, A two dimensional transient model for convection in laser melted pool. Metal. Truns. 15A (1984). B. Basu and J. Srinivasan. Numerical study of steadystate laser melting problem. Int. J. Heat Mass Trunsfer 31( I I) (1988). H. Yoshizawa. S. Wignardjah and H. Saito, Study on laser cutting of concrete. Trans. Jup. Weld. Sot. 20(l) (1989). I. Masumoto. M. Kutsuna and K. Ichikawa. Relation between process parameters and cut quality in laser cutting of aluminum alloys. Jup. Weld. Sot. 23(2) (1992).

Evaporative

material

15. M. F. Modest and H. Abakians, Evaporative cutting of a semi-infinite body with a moving CW laser. Trans. ASME 108 (1986). 16. S. Biyikli and M. F. Modest, Effect of beam expansion and focusing on evaporative cutting with a moving CW laser. J. Heal Transfer 110 (1988). 17. H. Abakian and M. F. Modest, Evaporative cutting of a semitransparent body with a moving CW laser. ASME J. Hear Transfer 110 (1988).

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18. W. Boersch-Supan, L. W. Hunter and J. R. Kuttler, Endothermic gasification of a solid by thermal radiation absorbed in depth. Int. J. Hear Mass Transfer 27 (1984). 19. S. Y. Bang and M. F. Modest, Multiple reflection effects on evaporative cutting with a moving CW laser. ASME J. Heat Transfer 113 (1991). 20. M. J. Kim, Z. H. Chen and P. Majumdar, Finite element modelling of the laser cutting process. Compur. Struct. 49(2) (I 993).