COMPUTATIONAL MATERIALS SCIENCE

Computational Materials Science 6 (1996) 92- 102 ELSEVIER

A boundary integral equation technique weld pool shapes in thin

J.-Y. Yeh, L.N. Brush *

for the calculation of plates

Department of Materials Science and Engineering, The Universiry of Washington, Seattle, WA 98195, USA

Received 27 October 1995; accepted 22 January 1996

Abstract

Weld pool shapes are computed via numerical solution of a boundary integral equation used as a model for the autogenous full penetration welding of pure materials. Numerical results show that for a given value of the heat input from the arc, the latent heat of fusion provides a substantial correction for the shape of the weld pool compared to solutions that neglect the latent heat. Moreover, results confirm that the aspect ratio of the weld pool is larger when increasing either the value of the latent heat or the heat input from the arc, and is smaller for higher values of surface convective heat losses. The magnitude of the curvature of the weld pool at the centerline of the trailing edge is increased as the latent heat, the arc velocity or the plate preheat is increased, which may promote weld pool shape transition and/or defect formation.

1. Introduction

The morphology of the solid-liquid interface that develops during arc or laser welding strongly influ- ences the solidification structure in the fusion zone [l-4]. Previous studies have shown that it is possible to group weld pool shapes into a variety of classes [5]. As welding parameters are changed, a weld pool can undergo transition from one shape class to an- other. For example, when the velocity of the heat source is increased during the autogenous, full- penetration welding of thin plates, the weld pool shape changes morphology from an ‘egg’ shape to a ‘teardrop’ shape. This latter weld pool shape is characterized by a nearly V-shaped trailing edge, having a large, possibly discontinuous change in the curvature at the trailing edge of the weld pool near the centerline of the fusion zone. Accompanying

such a change in weld pool morphology can be a transition in grain structure, for example, from the smoothly joining grain structure at the centerline of the fusion zone associated with the resolidification of an ‘egg’ shaped class of weld pool, to a set of grains that wreck together at the centerline of the fusion zone, creating a fault. Such a defect can result in catastrophic failure of the material in application. Other solidification grain structures can result in defects or undesirable structures within the fusion zone, also.

’ Corresponding author. Tel.: + l-206-5437161; fax: + I-206 5433 100.

In order to control the grain structure in the fusion zone it is necessary to understand the reasons for the appearance of different weld pool shapes. Therefore, a model of fusion welding that can couple all of the physical processes that must occur along the trailing edge of the weld pool in order for solidification to take place is required. Previous research has illus- trated the role that some of the physical effects occurring during resolidification play in affecting weld pool shape. For example, it has been shown

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J.-Y. Yeh, L.N. Brush/ Computational Materials Science 6 (1996) 92-102 93

that a model that includes only the effect of heat flow during welding can not be used to predict the appearance of a cusp or a comer along a weld pool interface [a]. This result could imply that the appear- ance of a tear drop shaped weld pool may not be understood unless other physical effects are included in a model of welding. In pure materials, additional physical considerations include for example, the ef- fect of the surface tension, which provides a correc- tion to the equilibrium melting temperature as given by the Gibbs-Thomson equation [7,8]. Additionally, the interfacial attachment kinetics [9] can cause the interface to depart from the condition of local equi- librium. Both of these properties may be anisotropic because they depend on the crystallographic orienta- tion of the solid-liquid interface. It has been argued that crystalline anisotropy is the reason for the com- petitive growth phenomena that gives rise to a pre- ferred orientation in the fusion zone of multi-grained materials [l]. In the case of the welding of Fe-Ni-Cr single crystal alloys, methods for predicting the so- lidification microstructure have recently been devel- oped [lO,ll], which indicate the effect of anisotropy in giving rise to preferred dendritic growth orienta- tions in the fusion zone.

The goal of this work is to present a new model of welding that includes a full description of the solidification physics occurring along the trailing edge of the weld pool during autogenous full- penetration welding of pure materials. Physical ef- fects such as the latent heat of fusion and the interfa- cial properties including the anisotropy associated with the underlying registry of the crystal lattice are incorporated into the model, although the isotropic interfacial properties will be shown to have a limited role in affecting weld pool shape in metals. Many other researchers have focused on modeling the weld pool shape in 2D and 3D including the effect of heat flow, convection and other effects [ 12- 191, but the analysis presented here is the first analysis that uses boundary integral techniques to achieve high resolu- tion of weld pool interface shapes computed in a moving reference system. (For a detailed review see also David et al. [20], and the references therein.) Finally, it is noted that in this analysis, grain bound- aries are not explicitly included. Nevertheless, the results presented in this paper should still be relevant to weld pool shapes observed during the welding of

thin polycrystalline plates, at least under the condi- tions in which the individual grains follow trajecto- ries governed primarily by heat flow considerations and therefore lie perpendicular to the maximum tem- perature gradient.

2. The model

For sufficiently thin plates, a two dimensional model of heat flow is derived by defining the aver- aged temperature

qx, y, f) = $yx, y, z, t) dz,

where T’ is the actual temperature and g is the plate thickness, and integrating the fully three dimensional heat equation over the narrow dimension of the thin plate, meanwhile incorporating the appropriate heat transfer conditions at the top and the bottom of the thin plate. The result for steady heat flow in a moving reference system is:

a=u, a=lJ, -i--

v au, at2 ay2 +a, a5 ( 1

- H,U, + F,( 5, Y) = 0

in the solid phase and

a2uL a2uL -+-

v au, at2 ay2 +a, 7 ( 1

-H,_U,_+F,_(t,y)=O ( lb) in the liquid, where 17, (Cl,_) = T, (T,) - To, denotes the solid (liquid) temperature shifted by an amount equal to the ambient temperature To, as (a,) de- notes the solid (liquid) thermal diffisivity, FL (F,) is a net heat flux into the plate (due to the effect of the heat source distribution) and H, (H,) = [ h,(top) + h,(bottom)l/gk, ([ h,(top) + h,(bottom)l/gk,) is an effective solid (liquid) heat transfer coefficient that is the sum of the heat transfer coefficients at the top and the bottom of the plate, respectively, divided by the product of thermal conductivity times the plate thickness g. The variable 6 = x - Vf represents a moving coordinate translating at the constant speed V of the heat source in the positive x direction.

Supplemental interfacial conditions are required

94 J.-Y. Yeh, L.N. Brush/Computational Materials Science 6 (1996) 92-102

in order to determine solutions to Eq. (1). Along the solid-liquid interface, the temperature is continuous

us = lJL (2a)

and is given by the kinetic and capillarity corrected expression:

PI

where q(4) is the anisotropic interface attachment kinetic coefficient, which is a function of (6, angle that the normal vector ii, pointing from solid into the liquid, makes with the cartesian axis, V, is the local normal growth speed of interface, and

the

‘;’ the

(2c)

Here, K is the curvature at a point on the interface, $4) is the anisotropic surface tension, TM is the melting point of the bulk solid, A = TM - To, and L is the latent heat of fusion per unit volume of solid.

The moving interface must also obey a conserva- tion of heat condition:

LV,= -k,_VU,_~fi+ksVUs~R, (24

where V, is the local normal growth speed of the solid-liquid interface, k, ( kL) is the thermal con- ductivity of the solid (liquid) at the interface and the gradient terms are directed along the normal ii to the interface pointing from the solid into the liquid. Finally, the solid temperature Us vanishes far from the interface and the liquid temperature U, is finite at the origin of the coordinate system. The next step will be to reformulate the above description of weld- ing into a boundary integral equation form.

3. A boundary integral equation method

3.1. General formulation

The boundary integral equation format may be derived using standard techniques [21]. The result is a set of boundary integral equations relating the interface temperatures to the normal derivatives of the temperature at the interface:

- uL( ?)vq( x’; zoo) + UL( +y( x’; ?,,)i) ds,

+ hLF,( ?)q( x’; j;o) d”, (3a)

and

~=ps.[us(I: j;,)VU,(,‘) t

- U,( ;)VI.J~( x’; Z,,) + U,( ?)vs( x’; ?“)f] ds,

+I F,(+s(x’; ZO)dV,, VS

Pb)

where the notation x’ and &, denote the points (6, y) and (SO, y,,), respectively, and both Us and us vanish infinitely far from the interface. Here i^ is a unit vector in the positive ‘6 ’ direction, S, is the arclength along the interface contour r,, the normal vector ii, is pointing from the liquid to the solid phase whereas ii, points from solid to liquid,

x K,(G,

and

(5-5,)*+(Y-Yo)*]“*) (44

XK,(G,[(~-~~)*+(Y-Y~)*]“*). (4b) The latter expressions (IQ. (4)) have logarithmic singularity at (&,, y,) [22]. Here, the terms in the angular brackets represent the arguments of the mod- ified Bessel function of the second kind and zero order, which is denoted by the symbol K,,

Generally speaking, the contour integrals are of Cauchy principal value type, since the contour r, over which tbe boundary integration occurs contains both the points ( 5, , yO) and ( 6, y). Due to the source terms, the volume integrals have been re-

J.-Y. Yeh, LA. Brush/Computational Materials Science 6 (1996) 92-102 95

tamed; in this case the subscript ‘0’ appearing in the volume integral terms still refers to a point on the boundary, I,.

3.2. The symmetric model

A dimensionless equation for the interface shape can be derived that retains most of the important features of the model, by assuming that all of the material properties are equal in the solid and liquid phases, i.e., k: = k, = k,, H: = H, = H, and CT: =

oL. Let the heat source be approximated at the z&$ of the coordinate system by F(t, y) = [ q,/(kg)]b( t)S( y), where qt is the Rower input in Watts for example, and 6 is the symbol for the Dirac delta function. Then, addition of the integral Eq. (3) into which the interfacial conditions as given by Eq. (2) have been substituted, scaling lengths by 2cw/V, and introducing the normalized temperatures, Os,, = U&A gives:

= Laj--v( x’; Z”)cos(q5( x’))ds, + 27rQp(& 2’) I

where the relation V, = Vcos(+) has been used and where, although lengths are now measured relative to 2a/V, the notation remains the same as in the dimensional treatment. The variables

V

K= (T,-T,)A’

T,%V

r= 2L(T, - T,)cr ’

2L L, = and Q,=

4t

PC,(T, - T,) 2rk, g(T, - T,)

In addition, the dimensionless variable h = H(20/Vj2 so that:

u a; & ( 1 = $exp( -&)Ko[(l +h)“‘( 6: +Yz>]

in dimensionless form. Also,

Y * = ‘y/x, and q*(4) =clW/A

where ‘yO and A may be for example, the average values of the surface tension and the interface attach- ment kinetic coefficient, respectively. For isotropic materials, Eq. (5) is a five parameter equation relat- ing the shape of the weld pool, a distribution of heat sources located along the interface (due to the effect

of the latent heat) and a heat source located at the origin of the coordinate system in the liquid and reflected in the term Q,u(O, Z$,). Note that Eq. (5) may be solved to determine the unknown shape of the weld pool without having to explicitly calculate the temperature fields anywhere in the two phase region. Thus, the dimensionality of the problem has been reduced from two to one since Eq. (5) is applicable only along the interface. Moreover, for the case in which the interface kinetic processes are infinitely rapid (q(4) = a~>, and neglecting the ef- fects of the latent heat of fusion and the interfacial surface free energy, then the solution to Eq. (5) provides the melting isotherm contour generated by the temperature distribution surrounding a steadily translating heat source and which hereafter will be referred to as the ‘single phase’ solution. This solu- tion is given in Rosenthal [23] and Myers et al. [24].

For metals, < is typically much less than unity, V, is usually significantly less than unity, while L,

and Q, can be of order unity or greater. It is noted that in the absence of capillarity, interface attach- ment kinetic effects, and surface heat losses, this set of dimensionless variables provides an interface shape invariant under a change in the translation rate of the heat source, for constant L, and Q,. More- over, the single phase solutions only depend on Q, (L, = 01, thereby providing a well known and solv- able comparison state to aid in illustrating the effect of L, on pool shape. In the following sections we will establish individually the role that the dimen- sionless parameters L,, Q, and h play in governing the weld pool shape for isotropic systems. Accord- ingly, r is set to 0 in Eq. (5).

4. Numerical solution techniques

The it&grand in the integral Eq. (5) is singular. This poses difficulties when trying to numerically evaluate the integral, whenever evaluation of the integrand at the singular point is required. One method for overcoming this difficulty is to subtract from the integrand the exact strength of the singular- ity, which can be shown to be:

L, cos( &)ln( I s - soI)

in terms of the arclength (s, being the singular point) thereby eliminating the singular point in the

96 J.-Y. Yeh, L.N. Brush/Computational Materials Science 6 (1996) 92-102

integral so that standard quadrature can be applied. This requires addition to the integral equation of the same quantity that has been subtracted. For example,

=2L,cos(&,)[(As)ln(IAsl) -As]

which can be directly evaluated. Two independent methods are used to discretize

the interface. In the first case the solid-liquid inter- face is parameterized by the polar angle 0, so that the interface shape is given by the (unknown) func- tion fle). The integration variable in the integral equation is then transferred to the polar angle theta, the interface is discretized in equal increments of 19 and the integral equation is applied at N discrete points. The integrals are approximated by the trape- zoidal rule. The result is a set of N nonlinear algebraic equations for the interface position at the N discrete evenly spaced values of the polar angle 8. All shape quantities required in the integral equation are obtained from the function fl0 ). Here the accu- racy and convergence of the method is governed by the second order accurate differences that are used to approximate the derivative of the interface shape which is required to compute the normal angle to the interface.

In the second method, the interface is parameter- ized in terms of the relative arclength, a = s/S,, where s is the arclength and S, is the total arclength [25]. The normal angle r#~ is then rewritten:

4=: -7r(2a- 1) + 2 a, sin(27rna). (6a) n= 1

The Fiq. (6a) assumes a plane of symmetry along the 5 axis. Knowledge of the a,‘~, S, and x(O) permits all shape quantities to be calculated, since the Fourier coefficients for

- = -S,sin4= i at aa

c, sin(27rncu) n-0

(6b)

and

ay N

- = -s, cos qb= c f” cos(27rna) aLy n=O

(6c)

can be computed using fast Fourier transforms. ((IX) - c(O) and y<a> may then be determined analyti-

cally. Note that y(O) vanishes. The integrals are discretized using Simpson’s rule. Finally the curva- ture

K= -; + $ c, 27rna,cos(2~na). (6d) T n-

In using either the f(8) or the +(a) representa- tion of the interface, the iterative solver SNSQE ’ [26] is used to compute the unknown solutions of the resultant set of nonlinear algebraic equations. SNSQE uses a modified Powell hybrid method for iterating the equations which minimizes the square of the functions. The Jacobian is calculated internally using finite differences. In the cases shown here the two discretized methods provide agreement within frac- tions of a percent for the values of the Cartesian coordinates, and within two percent for the value of the curvature at the trailing edge of the weld pool along the centerline, when normalized by the differ- ence between the minimum and maximum absolute values of the curvature at all points along the inter- face. Different starting solutions were used, includ- ing the single phase solution as well as circles. When the single phase solution was used to initiate the solution procedure, it was determined by using the iterative solver ZEROIN [27] to find the interface shape as a function of the polar angle theta, while setting V,, f and L, to zero in Eq. (5). Continuation was used for the solution to the nonlinear equations, whereby the output from the solution at a given value of Q,, V,, L, or A was used as the initial guess in determining the next solution at an incre- mentally larger value of the appropriate parameter.

5. Results

Because the dimensional analysis has shown that r and V, are very small for most welding applica- tions, the results shown in the figures of this paper assume that these parameters are zero. However, there is a brief discussion at the end of this section on calculations that were carried out for moderate values of V,.

’ The program SNSQ was written by K. Hiebert and is part of

the SLATEC Common Math Library, National Energy Software

Center, Argonne National Laboratory, Argonne, IL.

J.-Y. Yeh, L.N. Brush/Computational Materials Science 6 (1996) 92-102 97

Effect of Latent Heat

Fig. 1. Plot of a sequence of weld pool shapes for increasing values of the dimensionless parameter L, = 0.0, 0.109, 0.274,

0.438 and 0.602, respectively, with Q, = 0.5 12. Lengths ate mea- sured in units of 2a/V. Increase in L, is according to the direction of the arrow on the plot. Changes in L, can be inter- preted either as differences in the latent heat of fusion at a constant power input and preheat, or as changes in the preheat of the plate while simultaneously changing the power input so as to hold the ratio of the power input to the preheat constant.

Fig. 1 is a plot of a sequence of weld pool shapes calculated by varying L, while maintaining fixed Q, = 0.512. (The choice of dimensionless Q has been motivated by welding conditions set forth in the studies of Ganaha et al. [4] and Kou and Le [5], for example, for the welding of Al alloys.) Fig. 2 shows the variation in the weld pool shape for identical values of L, as in Fig. 1 except that the value of Q, is fixed at 0.855. In both Figs. 1 and 2 the values of (dimensionless) L, are 0.0, 0.109, 0.274, 0.438 and 0.602, in order of increasing aspect ratio. As the value of the parameter L, is increased, the weld pool elongates, in other words its aspect ratio (the ratio of the maximum length measured parallel to the direc- tion of heat source motion relative to the maximum width as measured perpendicular to the direction of heat source motion) increases. The arrow in each figure indicates the sense of increasing L, for the sequence of contours. The pools with the lowest aspect ratios in Figs. 1 and 2 (identified as the innermost contour at the centerline of the trailing

edge) correspond to the single phase solution for which L, is identically zero. Changes in L, can be most easily interpreted as a change in the latent heat of fusion, all other variables held constant. There- fore, materials with higher latent heats will have higher aspect ratios, provided that A remains the same, and all other variables are fixed. However, the results can also be interpreted as a change in the preheat (T, - T,,) of the plate, so that higher values of (T, - T,) correspond to lower values of L,. How- ever, in changing the preheat, one would also have to change the dimensional value of the power input in such a way as to keep the dimensionless parameter Q, constant.

Fig. 3 is a plot of the dimensionless curvature of the weld pool at the centerline of the trailing edge of the weld pool as a function of L, for the two sets of weld pool interface shapes corresponding to condi- tions identical to those given in Figs. 1 and 2. The curvature corresponding to the single phase solutions shows that in dimensionless units, the magnitude of

Effect of Latent Heat

_._ Q=.tCE

Ob-

g o.o-

-Od-

-1.0 I I

4.0 -1.0 & 0.0 1

Fig. 2. Plot of a sequence of weld pool shapes for increasing values of the dimensionless parameter L, = 0.0, 0.109, 0.274, 0.438 and 0.602, respectively, with Q, = 0.855. Lengths are mea- sured in units of 2o/ V. Increase in L, is according to the direction of the arrow on the plot. Changes in L, can be. inter- preted either as differences in the latent heat of fusion at a constant power input and preheat, or as changes in the preheat of the plate while simultaneously changing the power input so as to hold the ratio of the power input to the preheat constant.

98 J.-Y. Yeh, L.N. Brush/Compututional Materials Science 6 (1996) 92-102

L,

Fig. 3. Plot of the dimensionless curvature of the weld pool at the

centerline as a function of the dimensionless parameter L, (in- creasing in increments of 0.0547) including the cases shown in

Figs. 1 and 2 with Q = 0.5 12 and 0.855, respectively. Note that

the curvature corresponding to the higher power input decreases

more rapidly than the curvature for the case of the lower power

input; and in fact the curvature becomes more sharp after the point

of crossing. The higher the magnitude of the curvature the more

susceptible the material will be to tearing or to the development of

a centerline grain structure.

the curvature at the trailing edge of the weld pool is smaller for the higher heat input at this limit. In both the cases Q, = 0.5 12 and 0.855, as the value of L, increases (in increments of 0.0547) the curvature becomes more negative. In other words the trailing edge becomes more sharp at the centerline as the value of L, increases, for a fixed value of Q,. However, as the dimensionless parameter L, in- creases the magnitude of the curvature at the trailing edge increases more rapidly for the case in which Q, = 0.855 than the sequence of weld pools corre- sponding to the lower value of Q, = 0.512. At an intermediate value of L,, the magnitude of the curva- ture becomes larger for the higher heat input. The conclusion is that the dimensionless parameter L, causes more dramatic changes in the centerline cur- vature for higher values of Q,.

Fig. 4 is a plot of the aspect ratio of the weld pools as a function of L, for the two sets of weld pool interface shapes given in Figs. 1 and 2. In both cases the aspect ratio increases with increasing L,

and the sequence of weld pool shapes generated at the higher heat inputs also increases more rapidly than those generated at the lower value of the heat input. Moreover, at any fixed value of L, the weld pool having the higher value of the dimensionless power input has a higher aspect ratio than the pool with the lower value of the dimensionless heat input.

Figs. 5 and 6 are plots of a series of dimension- less weld pool shapes found by varying the plate preheating A = TM - T,, holding all other values constant. Unlike the weld pool shapes shown in the previous figures, variation in the dimensional preheat causes variation in both the values of L, and Q,. In Fig. 5, the values of Q, are 0.512, 0.542, 0.591 and 0.650, respectively, and the corresponding values of L, are 0.580, 0.614, 0.670 and 0.737, respectively, in order of increasing weld pool area. In Fig. 6 the values of Q, are 0.855 and 0.905, respectively, and the corresponding values of t, are 0.580 and 0.614, respectively, in order of increasing weld pool area. Comparison can be made of Figs. 5 and 6 with Figs. 1 and 2. If Al were being welded, then the dimen-

Aspect Ratio

22

20

1.5

2

1: Q = .512

*s c 2Q=.65!5

1 lb-

2 1.4 -

1

Fig. 4. Plot of the aspect ratio of the weld pools as a function of

the dimensionless parameter L, (increasing in increments of

0.0547) including the cases shown in Figs. 1 and 2 with Q = 0.5 I2

and 0.855, respectively. Note that the aspect ratio corresponding

to the higher power input increases more rapidly than the aspect

ratio for the case of the lower power input.

J.-Y. Yeh, LN. Brush/Computational Materials Science 6 (1996) 92-102 99

Effect of Preheat

Fig. 5. Plot of a sequence of weld pool shapes for increasing values of the preheat temperature. T,. or decreasing values of the parameter A = Tm - T,. Ihe change in the preheat affects both the value of the dimensionless parameters Q, and L,. lk. values of Q, are 0.512, 0.542, 0.591 and 0.650, respectively, and the corresponding values of L, am. 0.580, 0.614, 0.670 and 0.737, respectively, in order of increasing weld pool area. ‘Ihe innermost contour could correspond to a weld pool shape in aluminum with no preheat, and the successive weld pools correspond to values of A = 600°C. 550°C and 5OO”C, respectively. Lengths are measured in units of 24/V.

sionless shapes in Fig. 5 correspond to values of A = 635”C, 6OO”C, 550°C and 5OO”C, in order of increasingly large weld pool area, respectively, whereas those in Fig. 6 correspond to the two lowest values of plate preheat temperature used to calculate the shapes shown in Fig. 5. The result is that rela- tively minor amounts of preheat cause dramatic changes in the amount of molten metal, resulting in a larger fusion zone.

Fig. 7 plots the curvature of the trailing edge of the weld pool shapes shown in Figs. 5 and 6 for the values of the preheat indicated above. In both cases, the curvature of the trailing edge of the weld pool becomes increasingly sharp as the ambient tempera- ture is increased. The pool with the higher heat input has a centerline curvature that decreases more rapidly than the lower power input counterpart. The plot in

Fig. 6 and the curvature in Fig. 7 indicate that the weld pool becomes very sharp near the centerline, which is more characteristic of a tear drop shaped weld pool than an egg shape weld pool.

Fig. 8 is a plot of a series of weld pool shapes calculated for values of the surface heat loss term h = 0.0, 0.5 and 1.0, respectively, in order of de- creasing weld pool area. Each of the weld pools shown in Fig. 8 is for the same value of L, = 0.58 and Q, = 0.855. For given values of Q, and L,, as the surface heat losses are increased, the aspect ratio decreases, as the weld pool tends to a more circular shape. However, the area of the molten weld pool becomes smaller with increasing surface heat loss, also.

Finally, the effect of the kinetics of attachment have been studied, which could be important in metals if the plate preheat is very high, or if the velocity of the heat source is very high, or if the material is a non-metallic material for which the

Effect, of preheat

.a

-0.75

i

-LOO-l 1 I I

-1.7 -te iiga

-02

Fig. 6. Plot of weld pool shapes for two values of the preheat temperature T,. or the parameter A = T, - T,. ‘Ihe change in the preheat affects both the value of the dimensionless parameters Q, and L,. ‘Ihe values of Q, are 0.855 and 0.905, respectively, and the corresponding values of L, are 0.580 and 0.614, respectively, in order of increasing weld pool area. The innermost contour could correspond to a weld pool shape in aluminum with no preheat, and the larger weld pool corresponds to a value of A = 600°C. Lengths ate measured in units of 2 n/ V.

100 J.-Y. Yeh, L.N. Brush/Computational Materials Science 6 (1996) 92-102

interface attachment kinetic processes are often rate controlling under normal solidification conditions. By solving

it can be shown that an increase in the kinetic effect through an increase in V,, causes the weld pool to elongate because the interface temperature lags in- creasingly further behind the bulk equilibrium melt- ing point along the trailing edge of the weld pool, the more sluggish the attachment kinetics become.

6. Discussion

The numerical results have shown that the latent heat of fusion is an important factor that must be included when considering the shape of the weld pool. For pure materials, the surface tension and the kinetic effects can be neglected under most welding situations therefore, the interface temperature under these conditions is still given by the bulk equilibrium

Preheat

1: Q. = 512

2: Q, =.8!%

-O.lO

-o.lS- 1

E__

-Us-

_- 2

o.00 0.w u”wham 0m Oas Fig. 7. Plot of the curvature of the trailing edge of the weld pool

at the weld centerline, for the cases shown in Figs. 5 and 6. The

horizontal axis is a ratio of the difference between the preheat

temperature and 25°C divided by the difference between the

melting point and 25°C.

Surface Losses

-0.50

-0.75 1 -‘.oo-#

-1.00 -120 -1.00 -0.75 -060 -025 0.00 020 I

xv/ro: io

Fig. 8. Plot of a sequence of weld pool shapes for increasing

values of the dimensionless heat transfer coefficient h = 0, 0.5 and 1 .O. The value of Q, = 0.855 and the value of L, = 0.58, in

all cases. The innermost contour could correspond to a weld pool

shape in aluminum with no surface heat losses. The increase in the

heat transfer coefficient causes the weld pool aspect ratio to

decrease for all other variables held constant. Lengths are mea-

sured in units of 2 (Y / V.

melting temperature TM. Exceptions might be very low melting point, non-metallic materials. The latent heat causes the melting isotherm to distort signifi- cantly away from the solutions as given by Rosen- thal for example, or by other models of welding in pure materials or in alloys.

The implication of an increasing magnitude of the curvature at the centerline of the trailing edge of the weld pool is to promote the appearance of a center- line resolidification grain structure in the fusion zone. The reason is that as the magnitude of the curvature of the weld pool increases at the centerline, the weld pool takes on more closely the appearance of a teardrop shaped weld pool. Thus grains following the trailing edge must undergo large changes in orienta- tion over an increasingly smaller region of the trail- ing edge of the weld pool interface in order to maintain growth parallel to the direction of the maxi- mum temperature gradient. Eventually, if the magni- tude of the weld pool curvature becomes sufficiently

J.-Y. Yeh, L.N. Brush/Computational Materials Science 6 (1996) 92-102 101

large near the centerline of the fusion zone the grains apparently cannot accommodate one another as they meet the grains from the opposite side of the fusion zone or the weld pool, therefore they wreck together to form a centerline structure. One candidate for a criterion denoting the point at which grains will begin to wreck together at the trailing edge forming a centerline structure would be the condition that the radius of curvature of the weld pool at the centerline becomes similar in magnitude to the average width of the grains following the trailing edge of the weld pool. At this point the weld pool interface at the centerline could undergo a sharpening transition al- though the validity of such a criterion has not been addressed in this work.

The curvature along the trailing edge of the weld pool at the centerline is affected differently by the different physical effects. The dimensionless parame- ters Q, and h can act to increase and decrease respectively, the curvature of the weld pool upon their increase at constant velocity. Higher weld speeds will cause an increase in the magnitude of the dimensional curvature at the weld pool trailing edge, although the dimensionless curvature does not vary, at constant L, and Q, for vanishingly small r, V, and h as the heat source velocity is increased. How- ever, if one considers an average ‘dimensionless’ grain size of the material (the average dimensional size scaled by 24V1, this becomes larger as V

increases; therefore, a criterion for tear-drop pool shape formation as given in the previous paragraph can be developed even though the pool shape re- mains invariant in terms of the dimensionless vari- ables introduced in this work. The results show that forced convection at the top and the bottom of the plate could be useful in controlling the aspect ratio and/or the curvature at the centerline of the trailing edge of the weld pool. The results of this work also show that the higher the value of Q,, the more quickly the effect of the latent heat appears to in- crease the curvature at the trailing edge.

It is also interesting to note that the weld pools with the largest aspect ratios shown in Fig. 2 and to a lesser degree in Fig. 6, (corresponding to the largest L, and to the largest preheating) have in addition to the most pointed trailing edge at the centerline, a trailing edge that is more linear in appearance than other weld pool shapes shown. Linear trailing edges

have been most often associated with competitive growth phenomena, wherein grains at the trailing edge at the centerline having favorable orientations win out over grains with less favorable orientations. The results of this paper show that an increasingly straight isothermal trailing edge appears without con- sideration of the effect of competitive grain growth, instead only the effect of the latent heat, the power input and the preheat temperature have been in- cluded.

Examination of the (isotropic) interface attach- ment kinetic effect given in Eq. (7) reveals there is no solution for V, > 1. The length of the pool is related to the magnitude of the departure of the interface from local equilibrium. Thus, the lack of solution means that the interface attachment kinetic effects are too sluggish and the ambient plate tem- perature CT,,) is too high to ever allow for sufficient interfacial undercooling that would enable the growth of the interface as dictated by the attachment kinetic processes to acquire the velocity of the heat source, no matter how elongated the pool becomes.

Finally, it is noted that the calculations performed in this work involved between 50 to 600 nodes to calculate a given interface shape, depending upon the method used, and the values of the physical parame- ters involved. In order to compute weld pool shapes using standard methods would require the following iterative method. Given a guess of the interface shape, the interface temperature could be calculated by the Gibbs-Thomson Eq. (2~). Then for a pre- scribed outer boundary temperature in the solid at some finite cutoff distance from the solid-liquid interface, the solution to the steady state heat equa- tions in the liquid and the solid would need numeri- cal calculation using for example, finite difference methods. If the required resolution of the interface was one hundred nodes then this could involve cal- culation on the order of 1002 unknowns in both the liquid and the solid. Then the derivatives of the temperature would be calculated at the interface and substituted into the heat conservation condition. Gen- erally, the heat conservation condition would not be obeyed and a new guess for the interface shape would be required and the iterative process repeated until a self-consistent solution is found. Thus, at each iterative step large systems of equations would re- quire solution, and all of this only to calculate one

102 J.-Y. Yeh, L.N. Brush/Computational Materials Science 6 (1996) 92-102

interface shape. The boundary integral method over- comes these problems and allows accurate detailed computation of individual weld pool shapes usually in minutes, but at most an hour or so on a worksta- tion, depending on the desired accuracy. Clearly an advantage in computational efficiency has been achieved by the development of the boundary inte- gral method.

In summary, the weld pool shapes presented in this paper were calculated by the numerical solution to a boundary integral equation assuming steady state conditions in a moving reference system. The results indicate that the magnitude of the heat source, the latent heat of fusion and the amount of plate preheating are significant factors governing the ap- pearance of the weld pool shape. In pure materials, the interfacial properties play little role, except in the case of very high preheat or for a heat source traveling exceedingly rapidly. The results have shown how variations in a variety of these parameters can decrease the magnitude of the curvature at the cen- terline along the trailing edge of the weld pool which could help to control the defect formation in the fusion zone.

Acknowledgements

Support from NSF Grant No. DDM-9110923 is acknowledged. Helpful discussions and consultation with G.B. McFadden and R.F. Sekerka are gratefully acknowledged.

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