03 a Mathematical Model for Penetration Laser Welding as a Free Regina

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J. Phys. D: Appl. Phys. 30 (1997) 1300–1313. Printed in the UK PII: S0022-3727(97)77406-0

A mathematical model for penetrationlaser welding as a free-boundaryproblem

P Solana† and J L Oca ˜ na‡

† Departamento de Matematica Aplicada a la Ingenierıa Industrial,ETS Ingenieros Industriales, UPM, C/ Jose Gutierrez Abascal 28006 Madrid, Spain‡ ETSIIMLAS, Departamento de Fısica Aplicada a la Ingenierıa Industrial,ETS Ingenieros Industriales, UPM, C/ Jose Gutierrez Abascal 28006 Madrid, Spain

Received 27 August 1996, in final form 15 January 1997

Abstract. A detailed model is constructed in order to determine the full 3D weldpool and keyhole geometry by setting the appropriate energy and pressurebalances. The energy balance takes into account heat conduction, ablation lossesand evaporation effects at the keyhole open surfaces, as well as the most relevantenergy-absorption mechanisms, namely Fresnel and inverse Bremsstrahlung. Thepressure balance ensures mechanical stability of the keyhole by including ablationpressure against surface tension pressure. The model provides a full description ofthe temperature field, electronic density, degree of ionization and absorptioncoefficient within the plasma, as well as setting the maximum penetration depth fora given set of laser parameters such as power, focusing radius and oscillationtransversal mode. The keyhole boundary is initially taken to be an unknown freeboundary and is obtained as a part of the solution of the problem. For low andmedium welding speeds this boundary is successfully described with a family ofovoids. Good agreement with experimental results is achieved for a wide range oflaser powers and plate thicknesses.

1. Introduction

Various models have been developed in recent years in

order to explain the main physical mechanisms arising in

penetration laser welding. The improvement of industrial

laser techniques requires progressively more accurate

models capable of predicting the laser welding behaviour,

but at the same time it should be easy to compute results

in a reasonable amount of time, avoiding loss of physical

insight with rising complexity so that a complete analysis

of the different processes involved can be carried out in a

straightforward way.

The very first models of laser welding dealt mainly

with the analysis of the weld pool (Swift-Hook and Gick

1973) because the laser power was not high enough at

the time to make keyholing evident. As laser powerincreased, models began to pay attention to vaporizing

effects and keyhole formation (Klemens 1976, Andrews

and Atthey 1976), although considering the vaporized metal

not to be ionized. The complexity arising from the laser–

matter energy coupling at high intensities led researchers to

study different aspects in detail separately: thermocapillary

flow and the influence of alloy elements (Schelhorn 1989),

polarization of the laser light effects (Beyer et al 1988),

plasma absorption within the keyhole (Finke et al 1990)

and shielding gas effects associated with the formation

of a plasma plume outside the workpiece (Dowden et al

1994). Together with these particular studies, some authors

(Kapadia and Dowden 1995, Dowden and Kapadia 1995,

Ducharme et al 1994) developed integrated models in

order to keep a complete vision of the whole problem

while maintaining a useful physical insight throughout the

formulation of the models.

The aim of the mathematical model presented here

is thus to preserve such an insight but to include at the

same time all the interrelated quantities of the problem in a

self-consistent way without avoiding the complexity of the

formulation. The fundamental assumptions of the model

are the following.

(i) The process is considered to occur in a quasi-

stationary way.

(ii) The liquid and solid phases are considered as havingthe same thermophysical properties (heat conductivity

and diffusivity) irrespective of the temperature. The

optical and thermophysical properties of the plasma (heat

conductivity and diffusivity and the absorption coefficient)

are considered temperature-dependent (and thus leading to

a nonlinear heat conduction equation in the plasma). This

assumption affects mainly the weld-pool geometry because

of the constant values for the thermophysical properties

in these phases, but it is likely that the main energy

absorption mechanisms take place within the keyhole, so

0022-3727/97/091300+14$19.50 c 1997 IOP Publishing Ltd



Penetration welding as a free-boundary problem

a more accurate description of this zone is thus necessary

in order to provide a good evaluation of the amount of

energy transferred to the material.

(iii) The temperature gradients along the z axis are

considered constant (namely ∂T/∂z = constant). The value

of this gradient is calculated by averaging the keyhole and

weld-pool temperatures at each depth z. The value of the

constant has to be guessed at the beginning and is iterated

until convergence is obtained. This assumption allows

adjustment of the free boundary at each depth, producinga self-consistent keyhole geometry from the upper surface

of the keyhole to the bottom of the keyhole.

(iv) The laser light is absorbed by the plasma inside the

keyhole by means of inverse Bremsstrahlung. The plasma

is considered as being at 1 atm, as experiments seem to

confirm, and the degree of ionization is calculated using

the Saha equation. Because the plasma state is far from

equilibrium, however, a correction is applied to the degree

of ionization predicted by this formula in order to obtain a

better fit with experimental results. This correction factor

ranges from 0.5 for 1011 W m−2 to unity for 1012 W m−2.

(v) The keyhole wall absorbs the laser light directly

by Fresnel absorption. The formula used assumes thatthe light is circularly polarized and multiple reflections are

neglected.

(vi) The plasma–liquid boundary is adjusted using

energy and pressure balances. The heat flux absorbed

by the plasma and the keyhole wall equals the heat flux

absorbed by the liquid phase plus the heat flux lost by

ablation on the keyhole wall. The melting enthalpy of

the liquid is neglected, because our efforts are focused

on describing the energy-absorption mechanisms inside

the keyhole. Also, the ablation pressure is considered

to be equal to the pressure due to surface tension. The

hydrodynamics of the keyhole are neglected, although they

contribute significantly to the pressure balance at certain

zones of the keyhole (Cuclas et al 1995).

The only inputs of the model are then those related directly

to the laser source and the optical and thermophysical

workpiece properties, quantities which are known totally

independently of the particular process involved. Namely,

the input quantities are the following:

(i) for the laser source, the laser power P L, the intensity

distribution, the focusing radius rF , the beam polarization

and the welding speed U and

(ii) for the workpiece, the liquid heat conductivity κ,

the liquid heat diffusivity χ , the plasma heat conductivity

κg , the plasma specific heat Cpg , the plasma pressure p,

the surface tension at ambient pressure (liquid) σ 0, thevariation in the surface tension with temperature |∂σ/∂T |,the keyhole-wall temperature T s , the iron atomic mass mFe,

the vaporization energy per particle H and the ionization

energy per atom Ei .

The model takes the keyhole boundary to be unknown,

defined by a certain function (x,y,z) = 0 with only

a plane of symmetry; that parallel to the laser beam

containing the welding velocity vector. This free boundary

is to be obtained by posing adequate energy and pressure

balance relations point by point (that is, by imposing

Figure 1. The scheme of the laser-welding procedure.

mechanical stability and conservation of heat flux at every

point of the boundary). The melt-pool geometry is obtained

in a self-consistent way but letting the melting isotherm be

its boundary, without further analysis of the phase change.

As main output quantities, the model gives:

(i) the temperature field in the liquid/solid phase

T(x,y,z),

(ii) the temperature field in the plasma T (x,y,z),

(iii) the electronic density in the plasma ne(x,y,z),

(iv) the absorption coefficient in the plasma α(x,y,z)

(v) the energy absorption in the plasma and keyhole

walls,

(vi) the keyhole geometry (x,y,x) = 0,

(vii) the weld-pool geometry G(x,y,z) = 0,

(viii) the ablation pressure on the keyhole wall pabl ,

(ix) the pressure due to surface tension on the keyhole

wall,

(x) the ablation losses on the keyhole wall qabl ,(xi) the losses due to excess pressure at the open surface

of the keyhole P ev ,

(xii) the variations in temperature and density across

the Knudsen layer and

(xiii) the mean velocity of the gas ejected from the

Knudsen layer u.

2. The model

The welding process is supposed to occur in a quasi-

stationary way, in an x–y–z system of co-ordinates

with its origin at the centre of the laser source at the

workpiece surface, with rising z for increasing depthand relative source–workpiece displacement towards the

positive x axis. A numerical over-relaxed finite-difference

method is applied to solve the three-dimensional steady-

state heat-conduction problem with the free keyhole

boundary. Results are obtained within 15 min using a PC

with Pentium co-processor and 90 MHz. The fundamental

equations and boundary conditions are expressed below.

For the liquid phase, the heat-conduction equation reads

∂T

∂x= χ

∂2T

∂x2+ ∂

2T

∂y2

. (1)

1301



P Solana and J L Ocana

Table 1. Symbols used in the equations.

A, B , C Clausius–Clapeyron constants q effective heat source (W m−3)c speed of light in vacuum (2.998× 108 m s−1) r m averaged keyhole radius (m)C p liquid specific heat (J kg−1 K−1) T liquid temperature (K)C pg plasma specific heat (J kg−1 K−1) ˜ T plasma temperature (K)e electron charge (1.6022× 10−19 C) T F temperature at the Knudsen layers’s edge (K)E ev vaporization energy per atom (W) T 0 ambient temperature (K)f distribution function for laser intensity T S vaporization temperature (K)F k keyhole upper surface area (m2) u gas speed at Knudsen layer’s edge (m s−1)

H latent heat of vaporization (J kg−1) U welding speed (m s−1)h = h /(2π) = 1.0546× 10−34 J s x , y , z coordinate systemI intensity (W m−2) Z degree of ionizationI 0 laser intensity (W m−2) α absorption coefficient (m−1)k Boltzmann constant (1.3807× 10−23) χ liquid heat diffussivity (m2 s−1)M Mach number ε0 permitivitty of vacuum (8.3542× 10−12)m e electron mass (9.1094× 10−31 kg) γ adiabatic constantm Fe iron atom mass (9.2690× 10−26 kg) keyhole boundaryn gas density at the Knudsen layer’s edge (m−3) ηabs geometric absorption factorn e electronic density (m−3) κg plasma heat conductivity (W m−1 K−1)n i ionic density (m−3) ρ liquid density (kg m−3)n s saturation density (m−3) ρg plasma density (kg m−3)p plasma pressure (Pa) σ surface tension (kg m−1 s2)p abl ablation pressure (Pa) σ 0 ambient surface tension (kg m−1 s2)P ev excess pressure losses (W) ω laser light frequency (s−1)

q abl intensity ablation losses (W m−2) ω0 resonance frequency (s−1)

The term ∂2T/∂z2 vanishes because ∂T/∂z is considered a

constant (C2). For the plasma

ρgCpgU ∂T

∂x= κg

∂2T

∂x2+ ∂

2T

∂y2

+dκg

dT

∂T

∂x

2

+∂T

∂y

2

+ (C1)2

+ q (2)

where C1 is the constant value of ∂T/∂z. The boundary

conditions are

κg grad T − qabl | = κ gradT | (3)

T | = T | = T s (4)

limx,y→∞

T = T 0. (5)

The symbols used are displayed in table 1. See also

figure 1.

The problem consists of solving the system of

equations (1) and (2) with the boundary conditions (3)–(5).

The independent unknown functions of the problem are the

temperature distribution both in the liquid/solid metal and in

the plasma T(x,y,z) and T (x,y,z), as well as the keyhole

boundary (x,y,z)

=0. Ablation losses in the Knudsen

layer near the plasma–liquid boundary qabl (x,y,z) and theeffective heat source term q(x,y,z) have to be evaluated

properly in order to impose the boundary conditions and to

solve the heat-conduction equation in the plasma. These

terms have to be evaluated through the analysis of the

pressure and energy balances. A flow diagram is depicted

in the figure 2 to show how the quantities are obtained in

a self-consistent way. All the quantities in the problem are

related to each other so it is necessary to use an iterative

scheme in most calculations, although each term in the

equations is now to be studied separately.

2.1. The heat source

The heat source per unit volume q is mainly given by

inverse Bremsstrahlung and Fresnel absorption processes

in the plasma within the keyhole. This implies that, for

solving the heat conduction in the plasma, one should know

the spatial distribution of the heat source or, in turn, its

dependence on the temperature.

The collisional energy attenuation in the plasma is given

by an exponential law of the form

I(x,y,z) = I 0(x,y) exp

− z

0

α(x,y,ξ) dξ

(6)

so the effective source term at each point inside the keyhole

can be evaluated. For a proper evaluation of the source

term it is therefore necessary to know the dependence of α

on the spatial co-ordinates or on the temperature inside the

keyhole.

An expression for the absorption coefficient in a plasma

due to inverse Bremsstrahlung is (Ocana et al 1992)

α(T ) = Z2e6neni

6πε3

0m2

echω31

− ω0

ω21/2

me

2πkT

1/2

×

1− exp

− hωkT

1

2ln(1+2) (7)

with

= 12√

2πε0me

e3ω0

kT

me

1/21

Z

Z

1+ Z

1/2

. (8)

The plasma is supposed to be described by means of a single

temperature, with equal density for electrons and ions. So,

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Figure 2. A simplified flow diagram for the resolution of the free-boundary problem.

the equation for the pressure reads

p = nekT Z + 1

Z

(9)

which, together with the corrected Saha equation, provided

that the plasma is at atmospheric pressure, gives the value

of the electronic density as a function of the temperature,

which is necessary in turn to obtain an expression for the

absorption coefficient dependent only on the temperature.Fresnel absorption is calculated from a prescribed

initial keyhole boundary, supposing that the laser light is

circularly polarized. One way of obtaining such an initial

keyhole boundary is taking at first only the absorption of

laser light in the plasma bulk into account, including the

Fresnel absorption on the keyhole walls through an iterative

procedure until convergence is obtained.

It is regarded that not all the intensity hitting the upper

surface of the workpiece will be absorbed by the material:

the part hitting the liquid phase is taken to be fully reflected;

on the other hand, the open keyhole surface leads to a

certain amount of evaporated material being dissipated by

its excess pressure. The first of these two effects is taken

into account by calculating, at each step of the iteration

procedure, the ratio of the laser intensity actually entering

the keyhole and the total amount of flux (the geometrical

absorption factor):

ηabs =1

I 0

sf(r,θ)r dr dθ (10)

where the integral of the spatial distribution of intensity

is computed over the whole of the upper surface S of the

keyhole.

As a first approach, losses due to excess pressure at

the open keyhole surface are calculated by means of a

Clausius–Clapeyron equation of state, depending on the

temperature at the keyhole walls (Delacroix 1960):

P ev = 133.310BT Cs 10A/T s

(2πkT smFe)1/2EevF k. (11)

1303




Figure 3. The Mach number versus laser power for different welding speeds with a Gaussian beam, using a focusing radiusof 100 µm on steel 304.

Once the whole calculation has been carried out, the losses

due to excess pressure are found to be completely negligible

(in the range 5–12 W) in the overall energy balance.

2.2. The ablation process

Close to the liquid phase, inside the keyhole on its walls,

there is a sheath within which vaporization of material takes

place. This so-called Knudsen layer is governed basically

by the transition from a non-equilibrium state at the keyhole

wall to an equilibrium state a few mean free paths distant

from the liquid boundary. In fact, the velocity distribution

of particles at the keyhole walls is to be regarded as semi-

Maxwellian (Finke et al 1990), a Maxwellian distribution

been attained at the edge of the Knudsen layer with mean

velocity u.

The existence of such a layer provokes a rapid change

in the main quantities describing the vapour state (whichis considered as a thin sheath which is not ionized, due

its relatively low temperature); namely the density and

temperature. The variations in such quantities throughout

the Knudsen layer are to be regarded as associated with

rapid surface vaporization (Knight 1979) and are given by

T F

T s=

1+ πγ − 1

γ + 1

m

2

21/2

−√ π γ − 1

γ + 2

m

2

2

(12)

n

ns=T s

T F

1/2 m2 + 1

2

em

2

erfc(m)− m√ π

+ T s

2T F [1−√ πm em

2

erfc(m)] (13)

with m = n/(2RT f )1/2, erfc the complementary error

function and γ the relationship between specific heats of

the gas.The number density of the vapour at the edge of the

Knudsen layer n is to be obtained by iteration of the

former equations if a certain value of the Mach number

is assumed. The possible values for this number lie

between 0.05 (below this value the conduction effects are of

dominating importance) and unity (for higher values shock

waves are expected to arise). Although in many papers

the gas flow has been taken to be sonic as an a priori

condition (Anisimov 1968), in this model the Mach number

is calculated in order to fulfil the pressure balance, thus

ensuring the mechanical stability of the keyhole.

2.3. The pressure balance

The main forces acting on the keyhole walls are supposed

to be the ablation pressure opposed by the surface tension

effects. In an approximate analysis fluid motion is ignored,

at least for medium welding speeds not above 10 m min−1.

As has been stated already, the plasma hydrodynamics

inside the keyhole can play a major role in the overall

pressure balance (Clucas et al 1995), although the complete

formulation of the Navier–Stokes equations for the plasma

would add a great amount of complexity, increasing the

computing time dramatically. Furthermore, the plasma

1304




(a )

(b )

Figure 4. A comparison with experimental data with a Gaussian beam, using a focusing radius of 100 µm on steel 304:(a ) 4 kW and (b ) 10 kW.

1305




(a )

Figure 5. Keyhole profiles for a Gaussian beam of 6 kW, focusing radius of 100 µm on steel 304: (a ) welding speed2 m min−1, penetration 8.2 mm and maximum plasma temperature 17 200 K and (b ) welding speed 6 m min−1, penetration4.0 mm and maximum plasma temperature 16 950 K.

motion has been evaluated for open keyholes of simplegeometry, whereas the present model always provides blind

keyholes and predicts the maximum penetration depth for

a semi-infinite plate.

Thus, the pressure balance reads

pabl =σ 0 −

dσ dT

T srm

(14)

and the ablation pressure is given in terms of the density at

the edge of the Knudsen layer and the square of the ejected

gas mean velocity:

pabl = mFenu2. (15)

This equation lead us to the determination, in a self-

consistent way, of the conditions at the edge of the sheath,

as well as providing the amount of energy lost by the

ablation process qabl , which in turn forms part of the

boundary conditions of the problem posed:

qabl = mFenuH. (16)

1306




(b )

Figure 5. (Continued)

2.4. The free boundary

During each step of the iterative procedure, the free

boundary limiting the keyhole must be adjusted in order

to fulfil the heat-flux boundary condition coupled with the

pressure balance. The points on the mesh have then to

be re-distributed along the nodes; thus a fitting criterion is

needed in order to give a continuous analytical description

of the boundary.

Although a classical cubic spline method has been

implemented, convergence was achieved at a level of 90%

only after a great amount of computing time. In order to

avoid problems of this kind, it is to be noted that, for the

vast majority of cases (for velocity ranges up to 9 m min−1

and laser power values of 10 kW), the keyhole geometry

can be described by curves of practically ovoid shape (at

each depth). A slight deviation of the centre of the ovoid

from the laser source arises for low translational speeds,

which effect increases with speed. Thus, an ovoid familyis used to fit the keyhole geometry at each depth in the

form

M(z) e−N(z)xK0[N(z)(x2 + y2)1/2] = 1 (17)

with M(z) and N(z) parameters to be determined

depending on the location of the boundary points on the

mesh. This kind of ovoid fitting has been found to be

satisfactory for most cases, although it fails to represent

the keyhole geometry properly for high welding speeds.

The class of functions chosen in this way will influence the

intensity absorbed by the plasma inside the keyhole and at

the keyhole wall. This would be specially true if multiple

reflections were taken into account, but it is also so at high

translational speeds; in these cases, the keyhole is longer

and thinner in cross section and the chosen family of ovoids

fails to represent that shape.The stability of such a boundary is also limited by the

presence of shock waves generated in the Knudsen layer:

as a result of the increasing welding speeds, the energy

coupling between laser source and plate deteriorates and

so the keyhole becomes narrower, leading to large surface

tension forces arising at the keyhole boundary; insofar as

this value must be compensated by the ablation pressure,

the mean velocity of the ejected gas at the edge of the

Knudsen layer must increase its value, thus causing shock

waves to appear (see figure 3).

3. Results and discussion

The model is intended to be valid for the description of

the quasi-stationary behaviour of the laser welding, for a

great variety of laser power values and plate thicknesses.

In order to test the validity of the model for a wide range,

the results were compared with experimental data (Kaplan

1994) obtained for laser powers of 4 and 10 kW, relating

the welding speed and the plate thickness. Figures 4(a) and

(b) show the good agreement achieved for both cases. For

a fully penetrated workpiece at a certain laser power, the

experiment gives a relationship between the welding speed

1307




(a )

(b )

Figure 6. Averaged inverse Bremsstrahlung and Fresnel absorption mechanisms versus depth for a 4 kW, Gaussian beam,using a focusing radius of 100 µm on steel 304: (a ) welding speed 6 m min−1 and (b ) welding speed 2 m min−1.

and the plate thickness. This is assumed to be equal to

the relation between the welding speed and the maximum

penetration depth in the model. Figure 5 shows the keyhole

profile for welding speeds of 2 and 6 m min−1.

1308




Figure 7. Fresnel absorption versus the keyhole depth z and azimuthal angle for a 4 kW, Gaussian beam, using a focusingradius of 100 µm on steel 304, with a welding speed 5 m min−1.

3.1. Absorption mechanisms

Inverse Bremsstrahlung and Fresnel absorption are found

to be mechanisms operating with effects of the same order

of magnitude, although Fresnel absorption becomes more

important for higher welding speeds (see figure 6). This

can be explained in terms of the fact that, for a fixed

laser power, the increasing welding speed leads to keyhole

geometries of minor depth and with a smaller averaged wall

slope, that is, a greater area of the keyhole walls, at thetop of the keyhole, to absorb laser energy by direct impact.

Nevertheless, the total amount of energy absorbed by direct

impact is clearly related both to the impact angle and to

the laser polarization. Figure 7 shows the behaviour of

the energy absorbed along the keyhole wall (with different

impact angles at each point) for a circularly polarized laser

beam.

Inverse Bremsstrahlung is closely related to the plasma

state (ion and electron densities, degree of ionization and

temperature) and it is observed, at the top of the keyhole,

as a rapidly increasing absorption of light from the walls

to the centre of the keyhole (due to the effect of plasma

ionization), the absorption coefficient reaching a maximum

value relatively close to the keyhole wall; the increasingtemperature towards the centre of the keyhole tends to

decrease the value of the absorption coefficient, because

the electron density decreases there. With increasing depth

in the keyhole the temperature gradient decreases and thus

the leading effect on the absorption coefficient as one moves

towards the centre of the keyhole is not the electron density

but rather the effect of increasing ionization. Figure 8

shows the absorption coefficient at the upper surface of

the keyhole, whereas figure 9 shows the variation in the

averaged absorption coefficient with depth. It can be

observed that the absorption coefficient goes to a maximum

of 100 m−1 for 4 kW and up to 185 m−1 for 10 kW.

This effect is closely related to the correction taking the

degree of ionization into account and is in good agreement

with other authors’ results for low laser powers (Fabbro

and Poueyo (1995) gave an absorption coefficient for the

plasma of 1 cm−1) and high laser powers (Kaplan 1994).

In order to pursue a simpler way of carrying out

the overall calculations, the averaged Fresnel absorption

was fitted to an exponential curve, so that both Fresneland Bremsstrahlung mechanisms are to be described,

approximately, by means of a simple and unique term.

It was found that the integrated absorption coefficient,

once it had been averaged inside the keyhole, was in the

range 250–375 m−1, depending on the process parameters.

The implementation of a constant absorption coefficient of

325 m−1 dramatically reduces the computation time while

keeping an acceptable agreement with experimental data.

3.2. Energy losses

The most important loss mechanism is directly related to

the amount of laser intensity hitting the upper keyhole

surface. The relationship between the total intensityactually absorbed by the keyhole and that hitting the

liquid/solid phase is given by the geometrical factor ηabs .

Its value decreases rapidly with welding speed as the shift

between the beam axis and the centre of the keyhole

becomes larger.

The geometrical factor for a Gaussian beam is depicted

in figure 10 versus the welding speed for various laser

power sources. It can be observed that, for the most

optimistic case (very low welding speeds), the energy losses

are up to 40%.

1309




(a )

(b )

Figure 8. The absorption coefficient in the plasma at the upper keyhole surface for a Gaussian beam, using a focusingradius of 100 µm on steel 304, at welding speed 4 m min−1 (a ) 4 kW and (b ) 10 kW.

In the overall energy balance, losses due to excess

pressure at the open keyhole surface are totally negligible.

Ablation losses, although they are very much higher than

the former ones, do not play a major role in the energy

balance, the ablation process being dominant in the pressure

balance.

1310




Figure 9. The averaged absorption coefficient in the plasma for different laser sources (4, 6 and 10 kW) versus depth, for aGaussian beam, using a focusing radius of 100 µm on steel 304.

Figure 10. The geometrical absorption factor versus the welding speed for various laser sources (4, 6 and 10 kW) for aGaussian beam, using a focusing radius of 100 µm on steel 304.

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Figure 11. The domain of validity for the model, for a Gaussian beam, using a focusing radius of 100 µm on steel 304.

3.3. The validity of the model

The light striking the liquid/solid surface at the top of

the workpiece is considered to be entirely reflected and,

for low as well as for medium welding speeds, the main

energy source per unit volume is the heat absorbed by the

keyhole. Nevertheless, as the welding speed increases,

the shift between the laser beam and the centre of the

keyhole also increases (thus decreasing the value of ηabs )

and thus a greater amount of energy strikes the liquid/solid

surface. It reaches a level such that energy absorption

by the liquid/solid phase can no longer be neglected. It

is considered that, for plasma absorption less than 30%

of the total intensity, the model does not represent the

process with accuracy. Thus, an upper velocity limit has

to be established, depending on the laser source and metal

properties.

On the other hand, the gas ejected from the boundary

inside the keyhole can achieve the local speed of sound.

The model presented here does not deal with suchdiscontinuities, but the effects of the laser power, welding

speed and shock-wave formation should be taken into

account.

As the laser power decreases, the keyhole tends

to close, increasing the surface tension in the wall

dramatically. In order to maintain mechanical stability, a

great amount of energy is employed in the ablation process,

permitting the gas from the Knudsen layer to acquire great

velocity, generating instability associated with shock-wave

formation.

A net intensity loss is experienced by the plasma as the

welding speed increases (for a constant laser power, ηabsdecreases with the welding speed). This loss in the effective

source term leads to a diminishing averaged keyhole radius,

thus increasing the pressure due to surface tension and

resulting again in large velocities for the gas ejected fromthe wall of the keyhole.

Figure 11 shows the validity domain of the model for

laser power values in the range 4–10 kW. The validity

boundary is fitted with the quadratic expression:

U = − 1

24D2 + 11

12D + 9 (18)

with U the welding speed in m min−1 andD the penetration

depth in millimetres. The theoretical limits to the model

reflect the actual departure of the results from experimental

data observed for high welding speeds, suggesting that

a different approach to the problem is required when

analysing welding processes involving high translationalspeeds.

4. Conclusions

The three-dimensional geometry of the keyhole and melt

pool is obtained by solving the energy and pressure balance

equations in a self-consistent way, taking the plasma–liquid

interface to be a free boundary. The boundary is obtained

as a result of the solution of the problem, without any shape

or location being prescribed for it in advance. It is found

1312




that, for medium welding speeds, the keyhole boundary can

be successfully fitted with a family of ovoids at each depth,

reducing the computation time substantially. A further

useful simplification can be made in order to take into

account absorption phenomena inside the keyhole, namely

fixing a global absorption coefficient of about 325 m−1

for most cases, although this averaged value is actually

dependent on the laser source and the material.

The establishment of a detailed pressure balance at the

keyhole wall takes into account that Knudsen-layer effectsassociated with non-equilibrium states provide an instability

regime for the keyhole, with shock-wave formation at the

wall due to intense vaporization. These instabilities provide

an upper limiting value for the welding speed for a fixed

laser source, thereby restricting the model predictions.

Good agreement with experimental results has been

achieved for a wide range of laser sources and plate

thicknesses. The results obtained for plasma temperatures

and densities were very similar to those from other,

more detailed, theoretical approaches (Tix 1993), showing

that heat conduction is probably the main energy-

transport mechanism inside the keyhole, although radiative

phenomena must be included. Plasma temperatures werehigher than the experimentally determined values. This

could have been due partially to the fact that the plasma was

described by means of a one-fluid model (although it was

not considered as fully ionized). However, it should also be

noticed that the plasma temperature gradients at the keyhole

mouth were very large, so that a direct measurement of

the temperature near the keyhole upper surface would not

necessarily have indicated what was actually happening

inside the keyhole.

Acknowledgments

We would like to thank Professor P Kapadia and Dr J M

Dowden for their critical reading of the manuscript, as

well as for their scientific support and discussions about

the subject.

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