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Scientific Journals

Zeszyty Naukowe

Maritime University of Szczecin

Akademia Morska w Szczecinie

2010, 24(96) pp. 80–87 2010, 24(96) s. 80–87

Mathematical modelling of laser welding in shipbuilding

Modelowanie matematyczne spawania laserowego w budowie

okrętów

Eugeniusz Ranatowski, Krzysztof Ciechacki

University of Technology, Bydgoszcz, Faculty of Mechanical Engineering

Uniwersytet Technologiczno-Przyrodniczy w Bydgoszczy, Wydział Inżynierii Mechanicznej

85-796 Bydgoszcz, ul. Prof. S. Kaliskiego 7, e-mail: ranatow@utp.edu.pl; krzysztof.ciechacki@utp.edu.pl

Key words: laser, welding, modelling, heat source model, differential equation Abstract

A physical model of laser welding process is presented. At the beginning of this paper a short characteristic of correct modelling procedure is shown. In the further part, the form of heat transport in laser welding is described. Also the cylindrical-involution-normal (C-I-N) heat source (H-S) model and the Fourier – Kirchhoff partial differential equation are made and discussed.

Słowa kluczowe: laser, spawanie, modelowanie, model źródła ciepła, równania różniczkowe Abstrakt

W pracy przedstawiono fizyczny model spawania laserowego. W początkowej części scharakteryzowano proces poprawnego modelowania. W dalszej kolejności opisano formę transportu ciepła w procesie spawania laserowego. Również określono cylindryczno-potęgowo-normalne (C-P-N) źródło ciepła (Z-C) i analityczno- -numeryczny proces modelowania i ich zastosowanie w inżynierskiej działalności.

Introduction

The fusion welding is the method of choice for creating most large metal structures. In simple terms, a fusion weld is produced by moving a loca-lised intense heat source along the joint. Further-more such heat sources as laser, electron beam, plasma stream have become an invaluable tool for mankind encompassing such diverse applications as science, engineering, manufacturing and materials processing etc. The joining of large rectangular plates (panels) made by means of butt welding, recently also by laser beam welding, can be found in shipbuilding, in bridge construction and in the construction of large vessels. Such capabilities create new opportunities of changing the configura-tion of typical ship structure and other construcconfigura-tion. Laser-welded panels constitute innovative mo-dular components which already have found appli-cation to ship structures. They are characterised by

considerably greater stiffness at the same mass, as compared with conventional structures, and they are easier in assembling [1]. Temperature fields are required to predict welding and surface treatment processes, microstructures, residual stresses and distortion of workpieces. Experimental determina-tion of the temperature distribudetermina-tion and cooling rates in a weldment during and after welding is extremely difficult. Therefore, in recent years, there has been considerable interest in quantitatively determining the detailed temperature by mathema-tically modelling the physical phenomena that occur.

The recent developments in computational weld mechanics now enable the heat transfer in real welding situations to be analysed or simulated accurately, perhaps more accurately than the data can be measured.

The fundamental rule of any correct modelling procedure is an estimation of physic phenomenon

during welding process, practically results in exa-mining reciprocal relations between extensive an intensive parameters.

The extensive parameters may be often trans-ported and summed up in finish dimension areas and may be the mass – m, entropy – s, volume – V, electric charge – qe etc.

The intensive parameters (temperature – T, pressure p, stress – , chemical potential – ch,

voltage – U) and pseudointensive ones (quotients of two extensive magnitudes – like mass density – m/V) are field magnitudes, creating time – space field where in every space point a real physic magnitude is defined.

Moreover, in calculating process there are material parameters, e.g. specific heat – cp, thermal

conductivity , etc.

The transport process of extensive magnitudes requires observations and estimation of intensive parameters during welding and realised by using such procedures as transient Lagrangian or steady state Eulerian formulations of thermal cycle. A good model for the weld heat source in the analysis of the thermal cycle under laser welding is required [2].

Characteristic of laser welding

Lastly is provide by Weber [3] a comprehensive, up-to-date compilation of lasers, their properties and original references in a readily accessible form for laser scientists and engineers. Laser action occurs in all states of matter – solids, liquids, gases and plasmas. From practical point of view CO2 lasers are very suitable for some types of use in industry [4]. For example, it is possible to weld metal sheets of substantial thickness in a single pass. It is a characteristic of a laser beam that it can supply power at a very high intensity in a narrow beam.

During laser welding of alloys, the structure and properties of the welds are influenced by coincident occurrence of several important physical processes. In high-power-density beam welding (laser beam, electron beam), which has energy densities as:

 

13 2

10 _{10 Wm}

10  rq   (1)

additional complexities are introduced due to the presence of the “keyhole” that forms during welding in the molten pool. Energy absorption by the work piece from the laser beam can involve a direct interaction with the laser light incident on surface. The absorbed laser light can, however, vaporise material to form a “keyhole” that may contain ionised vapour. The “keyhole” is by far the

most controversial part of the system to model effectively. A partially ionised vapour exist in the “keyhole”. The degree of ionisation depends on the pressure and temperature in the “keyhole” if local thermodynamic equilibrium is assumed. In outline, the laser power is absorbed in the partially ionised vapour in laser generated “keyhole” and transferred to the walls of the “keyhole” by thermal conduction processes and melts the material and forming a weld pool. Finally, heat can transported in liquid molten region by both convection and conduction processes. In the solid region heat is transported relative to material of the work piece by conduction process only [5, 6].

A useful parameter for describing conditions of heat transport in liquid molten region is the Peclet number [5]:  l v P_e (2) 1 1    c_p  (3)

where: v – a constant velocity of translation [ms–1_],

l – characteristic length scale of the process [m], c – the thermal diffusivity [ms–2],  – the thermal conductivity [Wm–1_K–1_{],  – the density of} material [kgm–3_{], c}

p – the specific heat of the

material [Jkg–1_K–1_].

In above expression l is characteristic length scale of the process and it might be the beam radius of the laser, for example, or one of the principal dimensions of the weld pool. Low speeds of welding v or quality of l are naturally linked with small values of Peclet number. It is a dimensionless measure of the relative importance of convected to conducted heat [5]. Dimensionless numbers, for example as Peclet number, can be very useful in deciding the kinds of approximations to be used, and relating problems from different context to one another. Conduction becomes significant and dominate under welding process in weld pool as the Peclet number is much less than unity: Pe < 1.

Because of the laser energy enters to the workpiece from walls of “keyhole”, which to a first approxi-mation can be recognised as a heat source model.

The “keyhole” also can be usually thought of as a line or circular cylinder and melts the material by the process of heat conduction forming a weld pool. Furthermore extent (H-S) model is created by Godlak et. al. [8]. For deep penetration laser and electron beam welds, a conical distribution of power density which has a Gaussian distribution radially and linear distribution axially is esta-blished. Wei and Shian are constituted that under

high-intensity laser beam welding the incident energy rate distribution is assumed to be Gaussian and the cavity (“keyhole”) is idealised by a para-boloid of revolution as H-S model [7].

In this consideration we will be using the cylindrical – involution – normal (C-I-N) heat source model.

Characteristic of the cylindrical – involution – normal heat source model and analytic fundamental of thermal modelling

The mathematical expression of the C-I-N heat source model is [9]:







K z



e







u



z s





Q K k q kx y K z o z z v _{π1exp }    z 1  2 2 (4) where: qv – HS power input in volume [Wcm–3], Q – net power received by weldment [W], u(s – z) –

Heaviside’e function, k – a factor designating the HS concentration [cm–2_{], K}

z – involution factor of

HS [cm–1_{], s – HS penetration [cm].}

The (1 – u (z – s)) factor illustrates that qv = 0,

if z > s.

By changing: s, k and Kz factors, C-I-N model

can represent all several, presently used heat sources.

Let’s use equation (4) to analyse the shape of the surface that the volume of effective C-I-N affect. Surfaces of constant values can be obtained by comparing:



2 2



_

₁

_

_

_

_const max     _{q e}    _{u z s} q kx y K z v v z (5) where:







K s



Q K k q z z vmax _{π1exp } (6) presuming z < s we receive:





_A q ekx y Kzz   max const 2 2



x y



K z

 

A B k   _z    2 2 ln





z K y x k B z  2 2 (7) The solution (7) gives a family of paraboloids of revolution. Taking into consideration (1 – u (z – s)) factor, it is obvious that some of the paraboloids may be “cut”. A few examples of the paraboloids are shown in figure 1. All of them have the same height h = s.

Fig. 1 Examples of the paraboloids of constant values of qv

Rys. 1. Przykłady kształtu źródła ciepła w postaci paraboloid obrotowych o tej samej wartości qv

The paraboloid that limits the volume of effec-tive input affect can be obtained using Rykalin’s suggestion that power effectivity comes to an end when:





max max 0.05 2 2 v z K y x k v e q q     z 



_x2_y2



_{K z}3 k _z



2 2



3 y x K k K z z z    (8)

The equation (8) is a expression that describes effective input affect volume. In dependence on

h = 3/Kz value, mentioned paraboloids can be more

or less slim (see figure 2a). Furthermore source power in dependence on distance from centre and depth is presented – figure 2b. At extreme con-ditions we can obtain several, well known heat sources like:

a) Kz  0, k  0 – cylinder with height = s –

discuss heat source,

b) Kz  , s  0, k  0 – surface heat source.

The possibility of changing Kz, s, k makes C-I-N

source a universal one.

The transport process of extensive magnitudes requires observations and estimation of intensive parameters during welding and is realised by using such procedures as transient Lagrangian and steady state Eulerian formulations. We define an Eulerian (moving) frame with origin at the centre of the source and coordinates (x, y, z) – figure 3. For a Cartesian coordinate system (xo, yo, zo) which

remains stationary for all time t and loading history, we define as a Lagrangian coordinate reference.

Suppose a heat source is moving at a constant speed

v in the positive xo direction. The transformation

from (xo, yo, zo) to (x, y, z) is given by x = xo – vt, y = yo, z = zo.

The energy flux transport mainly by thermal conduction is described in a Lagrangian reference frame by Fourier – Kirchoff (F-K) parabolic differential equation:





q



x y z t



t T c T _{p o}, _o, _o, grad div        (9) where: t – time [s].

Finally putting equation (4) to (9) we obtain:

 

























t T z K y x k s z u s K t Q K k z T y T x T o z o o o z z o o o                          1 exp 1 exp 1 π 2 2 2 2 2 2 2 2 (10)

where: (t) – Dirac’s distribution [s–1_].

The solution of general form of F-K equation (10) in stationary and moving coordinates system with appropriate boundary conditions enable to establish for C-I-N heat source temperature fields.

An analytic-numerical evaluation of the thermal cycle

Using Rykalin’s addivity method [10, 11] we may obtain the summary temperature field gene-rated by moving heat source.

The manner of solution of equation ) ( d ) ( 0



  o t t T t

T which was described previously – enable to establish for C-I-N heat source temperature fields [9]:

– stationary coordinates system (xo, yo, zo):





_

_

_

_

 



























αr t t



D C B t t k α y t v x k t t k α t t u s K c K k q ,t ,z ,y x T i i i i i t z z                                           





    exp 1 4 exp 4 1 d exp 1 π 1 2 2 0 0 0 0 0 (11) where: ) sin( ) cos( 0 ₀ 0 rz r z r B _i i i i _    (12)                                    ₀ 2 2 2 1 1 2 2 2 0 2 ) ( 2 i i i i r g r r C (13)          ) ( ) ( ) sin( ) cos( ) ( ) sin( ) cos( ) exp( 2 2 0 2 2 0 0 2 2 2 i z z i i z i z i i i i z i i i z z i r K K r r K s r K s r r r r K s r r r rs K s K D                     (14) where: r1, r2, r3 ... (ri) – roots of equation:

 

_

_

1 0 1 0 2 2 ctg          i i i _r r g r (15)

– moving coordinates system (x = xo – vt, y = yo, z = zo):

a)

b)

Fig. 2. Characteristic of heat source model: a) the paraboloids limiting C-I-N effective input power volume; b) source power in dependence on distance from centre and depth

Rys. 2. Charakterystyka modelu źródła ciepła: a) obszar efektywnego działania źródła ciepła ograniczony poprzez; b) wpływ odległości od osi źródła na rozkład wydajności w funkcji zmiennej

Fig. 3. Scheme of the coordinate system under welding Rys. 3. Schemat procesu spawania





_

_

_

_

 

































                                    































1 2 2 2 0 exp 4 1 d exp 1 π , , ,

1

4

exp

i i i i i t z z t t r α D C B t t k α t t u s K c K k q t z y x T

t

t

k

α

y

t

t

v

x

k

(16) where:







r z



r z r B _i i o i i cos  

_

_ sin 



₍₁₇₎

Ci, Di, ri – values are the same like in stationary

system.

The above solutions assume that physic para-meters describing the process model are constant whereas they are unlinear for real welding system as a result of dependence on temperature.

In order to fulfil similarity criterions it is neces-sary to execute computer calculations of equations (11), (16) with temperature dependent physical parameters: (T), (T), cp(T), (T) and the above

algebraic expressions must be transformed. Expressions (11), (16) must be also discretised in order to make computer calculations possible.

Therefore the following assumptions were done:  the heat source energy is being input to the metal during time t, not impulsively t  0. HS inputs are being summed up in points in distance x = v t. Considering this t’ = (j – 1) t (j = 1, 2, 3 … n),

 the integrals were replaced by finished sums assuring sufficient exactness.

In first sequence we obtain the following com-puting expressions for linear heat flow solutions:

– stationary co-ordinates system:





















































( ( 1) )





exp ) sin( ) (cos( 1 4 1 1 exp 1 4 1 exp 1 π , 0 , 1 if , , , 2 1 0 0 0 2 0 2 0 1 0 0 0 t j t r C z r r z r D t j t k y t v j x k t j t k c s K t qkK t j t t z y x T i i m i i i _i i z z n j      _                                                    (18)

– moving co-ordinates system:













































 

 











r

t

j

t





C

z

r

r

z

r

D

t

j

t

k

y

t

v

j

vt

x

k

t

j

t

k

c

s

K

t

qkK

t

j

t

t

z

y

x

T

i i i i i m i i z z n j



































































              

1

exp

sin

cos

1

4

1

1

exp

1

4

1

exp

1

π

,

0

,

1

{

if

,

,

,

2 0 1 2 2 1











(19) The algorithms base on the numerical method of calculation of the constitutive equations (18), (19) with use mathematical program MathCAD were made [12].

In order to execute computer with temperature dependent physical parameters: (T), (T), cp(T),

(T) the above algebraic expressions transformed. Therefore the additional assumptions were done:  the integrals are changed to an algorithm which executes proper summing with physical para-meters upon temperature change control,

 as (T), (T), cp(T), (T) values in defined

increments are know, the matrices containing T and corresponding (T), (T), cp(T), (T) values

are defined.

With use of linear interpolation procedure, the continuous functions (T), (T), cp(T), (T) were

created and built-in inside calculation sheet in MathCAD.

Examples of thermal modelling of laser welding

For analysing the performance of a welded structure, information about the weld shape, the microstructure of the weld and the heat affected zone (HAZ) metal is required. Generally laser beam welding is characterised by a low heat input and high cooling rate. In welded steel joints, the austenite decomposes at relatively low temperature. Then the aim of this consideration is to analyse the temperature fields and microstructured state in high power laser beam welding of butt – joints. The laser beam power Q and welding speed v were as follows: 5 kW and v = 0.5 cm s–1_{. The material is} low alloy 09G2 steel and the thick of plate is

g = 0.4 cm. Finally, the thermal characteristic

values of low alloy 09G2 steel such as (T), (T),

matrices containing T and the corresponding values of (T), (T), cp(T), (T) were presented previously

in [12].

It assume that the radius rw of laser beam is

change in between 0.010.1 cm and take such values as: rw1 = 0.01 cm, rw2 = 0.015 cm, rw3 =

0.020 cm, rw4 = 0.03 cm, rw5 = 0.06 cm, rw6 =

0.1 cm. In far sequence, taking into consideration the radius of laser beam incident on surface, which equals [11]: wn n r w k r2 _ 3 ₍₂₀₎

that allows to define the concentration factor wn r k of HS as: 2 3 n wn w r _r k  (21)

In accordance with eq. (21) values of wn

r k are

presented in table 1.

Table 1. Values of krwn = f(3 / r2wn)

Tabela 1. Wartości dla krwn = f(3 / r2wn)

rwn

cm 0.01 0.015 0.02 0.03 0.06 0.1

krwn

cm–2 _3104 1.(33)104 7.5103 3.(33)103 8.(33)102 3.0102

Than in agreement with use equation (4) we can assess the energy density qv (at z = 0, Kz = 1) for

laser beam incident on a surface. At z = 0, Kz = 1

equation (4), take form:



2 2



π y x k v e Q k q     (22a) or



2 2



3 π y x k w v e r Q q n     (22b)

and to be Gaussian in nature. Figure 4 shows the relationship between qv(r, z, k), Wcm–2, and

dimen-sion of laser beam rw. It should be noticed that the

energy density qv it is highly dependent on laser

beam radius rw at the same power Q.

The modelling procedure and heat transport in workpiece having a Peclet number Pe < 1 is provided. If the radius of rw = 0.10.01 cm

(0.0010.0001 m) at a welding speed of v = 0.5 cms–1 _{(0.005 ms}–1_{) and thermal diffusivity for} liquid  = 0.55110–5 _m2_s-1_{, the value of Peclet} number is Pe = 0.09070.90 respectively. Further-more is assume that s = g and Kz is equal [10]: Kz = 3/s, cm–1, then Kz = 7.5 cm–1.

Fig. 4. Power density qv (r, z, k) on surface plate (z = 0)

Rys. 4. Rozkład gęstości mocy qv (r, z, k) na powierzchni

spawanej laserem (z = 0)

The results of non-linear analytic-numerical evaluation of temperature with application equ-ations (13), (14) and algorithms [5] in figures 5 a, b and 6 are presented.

Fig. 5. Stabilised temperature fields in sections y – z: a) rw =

0.1 cm; b) rw = 0.01 cm

Rys. 5. Rozkład izoterm w płaszczyźnie y – z: a) rw = 0,1 cm;

b) rw = 0,01 cm

Fig. 6. Course of change of temperature in stationary coordi-nates system (xo , yo, zo) for rw = 0.01 cm and following points:

A. (xo = 2.0 cm, yo = 0.1 cm, zo = 0 cm); B. (xo = 2.0 cm, yo =

0.2 cm, zo = 0 cm); C. (xo = 2.0 cm, yo = 0.5 cm, zo = 0 cm);

D. (xo = 2.0 cm, yo = 0.9 cm, zo = 0 cm)

Rys. 6. Przebieg zmian temperatur T w układzie nieruchomym (x0, y0, z0) dla rw = 0,01 cm, w punktach: A. (x0 = 2,0 cm, y0 =

0,1 cm, z0 = 0 cm), B. (x0 = 2,0 cm, y0 = 0,2 cm, z0 = 0 cm),

C. (x0 = 2,0 cm, y0 = 0,5 cm, z0 = 0 cm), D. (x0 = 2,0 cm, y0 =

Isothermal lines in y – z section and in moving coordinates for radius of laser beam a. rw = 0.1 cm;

b. rw = 0.01 cm are shown in figures 5 a, b.

In figures 6 we have the temperature change T(t) in stationary coordinates in time in one section and different points.

Distribution of isothermal lines in two cases (Figs 5a, b and 6) indicate that energy density

qv (Q, K, rw) decide on the condition under welding

process in weld and HAZ. Analysis the temperature values T(t) in the given point we can approximately obtain the instantaneous cooling rate wT (Ks–1)

using the following expression:













t T t z y x T T t z y x T t z y x w o o o o o o o o o T                     2 , , , 2 , , , , , , (23) In figure 7 instantaneous cooling rate wT (Ks–1)

is shown.

Fig. 7. Instantaneous cooling rate wT for T(t) for the same

points A, B, C, D as shown in figure 6: rw = 0.01 cm

Rys. 7. Przebieg zmian chwilowej prędkości chłodzenia dla przebiegu temperatur jak na rysunku 6 (w tych samych punk-tach A, B, C, D) dla rw = 0,01 cm

Fig. 8. Temperature versus heating and cooling rate diagram of low-alloy 09G2 steel

Rys. 8. Wykres temperatury względem wskaźnika nagrzewania i chłodzenia dla stali niskostopowej 09G2

Finally, welding time – temperature transfor-mation diagrams and instantaneous cooling rate wT

value enable to give an information about the microstructure of weld and HAZ.

We can directly to assess the change of micro-structure in welded joints. Figure 7 shows the tem-perature versus heating and cooling rate diagram of low-alloy steel 09G2 [13]. From our accounts of cooling rate wT and figure 8 results that martensite

or mixed martensite – bainite microstructure will be present in welded joint.

Conclusions

The following conclusions are drawn:

1. Realistic descriptions of physical phenomena of laser welding is made.

2. Characteristic of correct modelling procedure and heat transport in workpiece having a Peclet number less than unity is provided.

3. The C-I-N heat source model was constituted and analysed. This heat source model is favourable for imitation of the welding process with high concentrated energy such as laser or electron beam welding.

4. An appropriate form of parabolic differential equation to assess the temperature fields is established.

References

1. METSCHKOW B.: Sandwich panels in shipbuilding. Polish Maritime Research S1, 2006. Special issue: S1, 2006, 5–8. 2. RANATOWSKI E.: Thermal modelling of laser welding. Part

I: The physical basis of laser welding. Part II: An assess-ment of thermal cycle. Advances in material science. 2003, 3, 1(3), 34–50.

3. WEBER M.J.: Handbook of laser. The CRS Press. New York 2001.

4. DULEY W.W.: CO2 Lasers: Effects and applications.

Aca-demie Press. London 1986.

5. DOWDEN J.M.: The mathematics of thermal modelling. An introduction to the theory of laser material processing. Chapman & Hall. London 2001.

6. PAUL A.,DEBROY T.: Free surface flow and heat transfer in conduction mode laser welding. Metallurgical Transactions 1988, 19B, 851–858.

7. WEI P.S.,SHIAN M.D.: Three-dimensional analytical tem-perature field around the welding cavity produced by a moving distributed high-intensity beam. Journal of Heat Transfer 1993, 115, 848–856.

8. GOLDAK J.,GU M.: Computational weld mechanics of the steady state. Mathematical Modelling of Weld Phenomena. 2. Edited by Cerjak. Institute of Materials. Cambridge. Book 594, UK 1995.

9. RANATOWSKI E., POĆWIARDOWSKI A.: An analytic-nume-rical evaluation of the thermal cycle in the HAZ during welding. Mathematical Modelling of Weld Phenomena IV. Edited by H. Cejrak. Cambridge. Book 695, UK 1998, 379–395.

10. RYKALIN N.N.: The solutions of thermal processes during welding. Moscow 1951. wT [C/s] 800 600 400 200 0 200 400 0 2 4 6 8 10 _{t [s]}

11. RYKALIN N.N.: Energy sources for welding. Welding in the World, 1974, 9/10, 12, 1–23.

12. RANATOWSKI E., POĆWIARDOWSKI A.: An analytic-nume-rical estimation of the thermal cycle during welding with various heat source models application. Mathematical Modelling of Weld Phenomena 5. Cambridge. Book 738. UK 2001, 725–742.

13. KARKHIN V.A. ET. AL.: Effect of low-temperature phase transformations on residual stress distributions in laser welded joints. Mathematical Modelling of Weld Pheno-mena 5. Cambridge. Book 738. UK 2001, 597–614.

Recenzent: dr hab. inż. Zbigniew Matuszak, prof. AM Akademia Morska w Szczecinie