Mathematical Modelling of Weld Phenomena 13

ANALYSIS OF SOLIDIFICATION CRACKINGCONSIDERING MECHANICAL AND
METALLURGICAL FACTORS
S. MAEDA*, T. KATO*, K. IKUSHIMA*, M. SHIBAHARA**Graduate School of Engineering, Osaka Metropolitan University, Osaka, Japan

DOI 10.3217/978-3-85125-968-1-24
ABSTRACTThe introduction of tack welding by large-heat-input welding instead of using an out-of-plane constraint
has been considered for butt welding automation. However, the occurrence of solidification cracking inwelding is a major issue in large-heat-input welding. Solidification cracking is a welding defect that cansignificantly reduce the strength of weld joints and structures; therefore, it is important to predict its
occurrence for its prevention. Solidification cracking is a phenomenon caused by the interaction of
mechanical and metallurgical factors and it is known to have a tendency to occur where columnar
crystals collide. In the present study, we proposed a numerical analysis method to evaluate theoccurrence of solidification cracking in welding using the thermal elastic-plastic analysis using finite
element method (FEM) while considering mechanical and metallurgical factors. As a mechanical factor,a cracking evaluation index using the increment of plastic strain that occurs in the solidification brittle
temperature range (BTR) during the cooling process is proposed. As a metallurgical factor, thesolidification shrinkage strain and the strength of the solid-liquid coexistence region that changes withthe solid fraction are modelled. In addition, we proposed a simplified method for predicting the directionof columnar crystal growth using temperature gradients in the BTR. The proposed method was applied
to butt welding, and the effect of welding conditions on solidification crack generation was investigated.
It was found that the direction of columnar crystal growth predicted by the simplified method using
temperature gradient in the BTR was in good agreement with the experimental results. In addition, thecrack generation positions obtained with an evaluation index using the plastic strain increment in the
BTR were in good agreement with the cracking positions observed in experiments. The influence of thesolidification shrinkage strain on solidification crack generation was examined in terms of the relation
between the collision angle of columnar crystals and the restraining state around the melting part. Basedon these results, it can be concluded that the effects of mechanical and metallurgical factors on
solidification cracking in welding can be analysed using the proposed method.
Keywords: Solidification cracking in welding, Thermal elastic-plastic FEM, Plastic strain increment in
the BTR, Solid fraction, Solidification shrinkage, Crystal growth direction.

INTRODUCTION
In recent years, the Japanese shipbuilding industry has been attempting to automate
welding due to the difficulty in securing welding workers [1]. This is partly due to theincrease in welding operations as vessels have become larger in order to reducetransportation costs. In automated welding, as shown in Fig. 1, the out-of-plane constraint

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Mathematical Modelling of Weld Phenomena 13
is required to be changed to in-plane tack welding from the viewpoint of operability. In
this case, it is necessary to re-melt the in-plane tack welding from the viewpoint of weld
quality, and welding with a large heat input is required. However, there is concern about
the occurrence of weld solidification cracking during large-heat-input welding [2]. Weld
solidification cracking is a welding defect that occurs when large tensile strain is appliedin the temperature range just below the melting point of the weld metal as it solidifies andcan significantly reduce the integrity of the weld part [3-5]. Therefore, non-destructive
testing is performed after welding on welds where cracking is likely to occur, and if
cracking is confirmed, then repair welding is performed, which is one of the reasons forthe increase in manufacturing costs. Thus, it is important to establish an analytical methodthat can predict the occurrence of weld solidification cracking and study the effects ofvarious factors.
In the present study, a solidification cracking analysis method that can take into
account metallurgical solidification phenomena and mechanical strain behaviour in themolten part is proposed. As a metallurgical factor, the relationship between the brittle
temperature range (BTR) and the solid fraction determined from the metal composition
and the solidification shrinkage strain calculated from the solid fraction are considered in
the evaluation of BTR plastic strain by the thermal elastic-plastic analysis using finite
element method (FEM). A simple method for predicting the direction of columnar crystal
growth, which affects the location of solidification cracking, is also proposed using thetemperature gradient. And proposed method is used to evaluate the collision angle atwhich columnar crystals collide with each other.
Then, the developed method is applied to large-heat-input butt welding in order todemonstrate the validity of the proposed method through comparison with experimental
results. In addition, the effects of welding deformation as a mechanical factor andsolidification shrinkage strain as a metallurgical factor on the solidification cracking arealso discussed.
Through the above studies, we show the utility of the plastic strain increment in the
BTR for the evaluation of solidification cracking initiation and clarify that the crack
initiation location can be predicted by considering both mechanical and metallurgicalfactors.
BeforeFor welding automation

After

Passable

Impassable
Fig. 1 Transition of constraint states for butt welding automation

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Mathematical Modelling of Weld Phenomena 13
HOT CRACKING ANALYSIS METHOD
The occurrence of weld solidification cracking is affected by phenomena associated withboth the solidification of the molten metal and the welding deformation of the entire
member due to locally large heat input.
In the present study, we propose a solidification cracking analysis method that can take
into account the effects of metallurgical and mechanical factors using thermal elasticplastic analysis using FEM, which is used to analyse weld deformation and residual
stresses. Specifically, the shrinkage associated with the solidification in the BTR and thestrength of the molten metal are considered in thermal elastic-plastic analysis using FEM
while reproducing the progress of solidification based on the relationship between thesolid fraction and temperature in the BTR. An evaluation index for solidification cracking
using the plastic strain in the BTR is also proposed. In addition, we propose a simple
method for predicting the direction of columnar crystal growth, which has a significant
influence on the occurrence of solidification cracking in heat conduction FEM. Since thesolidification cracking analysis requires the detailed elements in the weld part, an efficientsimulation was achieved using a large-scale thermal elastic-plastic analysis methodreferred to as the idealized explicit finite element method (IEFEM) [6].
This section describes the details of the analysis method for solidification cracking
proposed by authors.

MATERIAL CONSTANTS FOR HOT CRACKING ANALYSIS

Material properties (SM490A)

In thermal elastic-plastic analysis using FEM, the temperature and mechanical fields fromthe heating process to the cooling process by welding are sequentially analysed based on
the temperature dependency of the material properties. Fig. 2 shows the temperaturedependent material properties of Japanese Industrial Standards G 3106 SM490A [7]. In
the present study, in addition to setting the material properties used in general welding
mechanics analysis, a method for setting material properties in the BTR was proposed forthe analysis of solidification cracking.
9: Specific heat []: Density []: Thermal conductivity [: Yield stress []: Young’s modulus []: Thermal expansion coefficient [: Poisson’s ratio [
]

6
3
0
0

]

]

300 600 900 1200 1500
Temperature [℃]Fig. 2 Temperature dependent material properties of Japanese Industrial Standards G 3106
SM490A [7]433Mathematical Modelling of Weld Phenomena 13
Relationship between temperature and solid fraction
In solidification cracking analysis, it is necessary to consider the progress of solidificationof the molten metal. In the present study, the progress of solidification was represented bydetermining the solid fraction of each element from the temperature obtained by the heat
conduction FEM. Table 1 shows the chemical composition of JIS G 3106 SM490A [7].
In the present study, the three elements shown in Table 2 were used in a simplified
calculation model of micro-segregation proposed by Clyne et al [8]. The derived
relationship between the temperature and the solid fraction is shown in Fig. 3. The figure
shows that the BTR is 110°C, from 1390°C to 1500°C.
Table 1 Chemical composition of JIS G 3106 SM490A [7]
C

Si

Mn

P

S

0.16

0.24

1.51

0.012

0.005

k

D

α

mi

C0

δ

0.200

6.4e-3

64

90

γ

0.360

6.4e-4

6.4

70

δ

0.130

4.0e-5

0.4

50

γ

0.060

2.5e-6

0.025

50

δ

0.060

1.6e-4

1.6

20

γ

0.015

3.9e-50.39

20Composition(wt%)

Table 2 Thermophysical data [8]Element
C
P
S

0.160
0.012
0.005

Temperature [℃]

1520
1500
1480
1460
1440

BTR

1420
1400
1380
0.0

0.2

0.4

0.6

0.8

Solid fraction
Fig. 3 Relationship between temperature and solid fraction in BTR

434

1.0

Mathematical Modelling of Weld Phenomena 13
Solidification shrinkage strain
During solidification of a molten metal, the liquid phase is pulled toward the solid
phase due to the solidification shrinkage strain εsh caused by the density difference
between the liquid and solid metals. Specifically, in carbon steel, it is known that there isa density difference of approximately 3.0% [9,10] and its influence cannot be ignoredcompared to the critical strain for crack initiation [11]. This causes a solidification
shrinkage strain of approximately 1.0 % in one direction.
The increment of the solidification shrinkage strain Δεsh when the solid fraction fschanges in the BTR can be expressed as follows: sh = f s  s − 1
 l


(1)

where ρs is the density of solid, ρl is the density of liquid, and (ρs /ρl -1) on the right-handside is the solidification shrinkage coefficient [12,13].
In the present study, (ρs/ρl -1) is defined as 0.03, and, based on the relationship between
temperature and solid fraction shown in Fig. 3, solidification shrinkage is considered asthermal strain by introducing the increment of solidification shrinkage strain into thethermal expansion coefficient, as shown in Fig. 2.

Strength of molten metal
When considering shrinkage around molten metal, it is important to reproduce thedifference in strength between solid and liquid metals. For example, rather than the solid
phase which has higher strength, the liquid phase which has lower strength, is tensile due
to solidification shrinkage.
It has been confirmed through high-temperature tensile tests that molten metals canretain a stress of approximately 1.0 MPa near the solidus temperature due to partial
crystal bonding caused by solidification [14]. Therefore, the yield stress at 1390°C, whichis the solidus temperature, was set to 1.0 MPa in the present study. In the temperature
range above the liquidus temperature, the strength of the molten metal is considered to beapproximately zero. It is thought that the strength of the molten metal can be reproducedby making the yield stress very small above the liquidus temperature in the thermal
elastic-plastic analysis using FEM. However, if the yield stress is set to be too small, theanalysis becomes unstable. In the present study, therefore, a value of 0.1 MPa, which is
sufficiently smaller than 1.0 MPa, was used. In the BTR, the yield stress is assumed to be
proportional to the solid phase ratio, as shown in Fig. 4.

435

Mathematical Modelling of Weld Phenomena 13

Yield stress [MPa]

1.0
0.8
0.6
0.4
0.2
0.1
0.0
0.0

0.2

0.4
0.6
Solid fraction

0.8

1.0

Fig. 4 Relationship between Yield stress in BTR and solid fraction

HOT CRACKING EVALUATION INDEX
Solidification cracking is evaluated by the displacement and strain acting on the weld
metal in the BTR, and cracking occurs when these values exceed the critical ductility
curve, as shown in Fig. 5 [15]. In other words, the amount of strain generated in the BTR
is important to evaluate the occurrence of solidification cracking.

BTR

StrainLiquidSolid(Brittle Temperature Range)
Ductility curve

Temperature()

Fig. 5 Schematic illustration of initiation of solidification cracking

Strain in the thermal elastic-plastic analysis using FEM is defined as the strain used inordinary elastic-plastic analysis and the thermal strain. In addition to these strains, it isnecessary to consider the solidification shrinkage strain associated with the solidificationof the molten metal in solidification cracking analysis, as described above. In other words,
the total strain increment is the sum of the elastic strain increment ∆εe, the plastic strain
increment ∆εp, the thermal strain increment ∆εT and the solidification shrinkage strain
increment ∆εsh, as shown in the following equation:

436

Mathematical Modelling of Weld Phenomena 13

 =  e +  p +  T +  sh(2)

On the other hand, solidification shrinkage is a phenomenon in which liquid metal
shrinks during solidification. Therefore, the solidification shrinkage strain increment ∆εshshown in Equation (2) is a negative strain that represents shrinkage in the BTR andcontributes to solidification cracking. Accordingly, it is necessary to use an index that canevaluate the effect of the solidification shrinkage strain increment on solidification
cracking.
p
In the present study, the plastic strain increment in the BTR  BTR shown in Fig. 6 isconsidered as an evaluation index of solidification cracking. As shown in the figure, thisrepresents the difference in plastic strain when the liquidus and solidus temperatures arereached during the cooling process. Note that the weld part can be evaluated as tensile dueto the negative strain increment caused by solidification shrinkage strain. In the present
study, the plastic strain increment in the BTR was calculated by the thermal elastic-plastic
analysis using FEM as follows:
nt 1

p
(I )
 BTR
 iip ( i = x, y, z ),i = 

(3)

I = nt0

where t0 is the solidification start time, t1 is the solidification completion time, nt0 is thesolidification start time step, nt1 is the solidification completion time step, and(I ) iip is

the plastic strain increment in direction i at time step I.

Plastic strain

Temperature

Solid

BTR

LiquidFig. 6 Schematic illustration of plastic strain increment in BTR during cooling

Solidification cracking is classified into cracking at the position at which the dendrite
tips collide and cracking that occurs between neighbouring dendrites. Both types ofcracking are considered to be caused by strain perpendicular to the liquid film. In thepresent study, the plastic strain increment in the BTR during the cooling process was used
to evaluate the solidification cracking at the position at which the dendrite tips collide.

437

Mathematical Modelling of Weld Phenomena 13
SIMPLIFIED PREDICTION OF COLUMNAR CRYSTAL GROWTH DIRECTION
The solidification morphology of molten metal during welding is determined by thetemperature gradient and cooling rate and is generally columnar solidification consistingof cells or dendrites [16].
In the research field of metallurgy, the influence of the direction of crystal growth ofthe molten metal on solidification cracking has been studied. The relationship between thecollision angle of columnar crystals and crack initiation has been evaluated based on grainboundary energy [17,18], and it has been reported that solidification cracking is more
likely to occur when the collision angle is large [19,20]. In the present study, the directionof solidification growth of columnar crystals and the collision angle between columnar
crystals are calculated simply as shown in Fig. 7 using a simplified prediction method forthe direction of columnar crystal growth based on the heat conduction analysis using
FEM. The direction of growth of columnar crystals in the molten pool is assumed to be
the direction of the maximum temperature gradient when each element reaches the BTR
in the cooling process, which is defined as the temperature gradient vector in the BTR
GBTR:m N 
N 
N   
GBTR =  
i+
j+
k T
y
z 
 =1  x(4)

where Nα is the shape function in the FEM, m is the number of nodes per element, i, j andk are basis vectors in the directions, and Tα is the temperature of each node when eachelement reaches the BTR in the cooling process.

1500 1350 1200 1050 900 750 600 450 300 150

0［℃］

Temperature gradient in BTR≒ Solidification growth direction

Fig. 7 Temperature gradient vector in BTR for columnar crystal growth direction

By using the temperature gradient vector in the BTR GBTR, the direction of columnar
crystal growth can be easily predicted. Furthermore, the position of cracking generation isevaluated using the collision angle of the columnar crystals. It has been reported that thecollision angle of columnar crystals has a significant effect on crack initiation [19,20].
Moreover, it is known that solidification cracking is likely to occur where columnar
crystals collide and a liquid film which remains over a wide area.

438

Mathematical Modelling of Weld Phenomena 13
In the present study, the solidification form that is prone to cracking is investigated in
the relationship between the temperature gradient vector in the BTR GBTR and the plasticpstrain increment in the BTR  BTR generated in the weld zone.

IDEALIZED EXPLICIT FEM
Analysis of solidification cracking requires detailed temperature and strain distribution
histories in the weld zone. As such, it is necessary to place fine elements around themelting zone. The IEFEM is an analytical method based on the dynamic explicit FEM
that can perform large-scale thermal elastic-plastic analysis at high speed and with little
memory [6]. This section provides an overview of the analysis method.
In the IEFEM, the following steps are followed, as shown in Fig. 8.
(1) In the welding transient state, load and temperature increments are applied andthe state is maintained.
(2) Calculate the displacement until static equilibrium is reached based on Equation(5) of the dynamic explicit FEM.
(3) When a static equilibrium state is obtained, return to step (1) to calculate the next
load step.
Ne
T
1
 1

C   U  t+t =  F t −  V e  B    dV
 2 M  +
2 t
 t

e =1
2
1
 1
+ 2  M U t −  2  M  −C   U t −t
t
2 t
 t


(5)

2. Displacement is calculated.
3. Load increment is given
after convergence is achieved

Displacement③
1. Load (temperature)
increment is given.

Static equilibrium state

②

①

Δt
ΔT

Virtual time step

Fig. 8 Schematic illustration of Idealized Explicit FEM (IEFEM)

where [M], [C], [B], and {σ} are the mass matrix, damping matrix, displacement-strain
relation matrix and stress vector, respectively. {U}t+Δt, {U}t, {U}t-Δt and {F}t are thedisplacement vectors at times t+Δt, t and t-Δt and the load vector at time t. Moreover, Ne
is the number of elements in the analysis model, and Ve is the volume of the elements. By
making the mass matrix [M] and the damping matrix [C] nodal-intensive diagonal matrix,

439

Mathematical Modelling of Weld Phenomena 13
the matrices operations in Equation (5) are no longer simultaneous equations and can beanalysed with less memory.
Since the effects of the inertia and damping terms become negligible once the static
equilibrium state is obtained by the above calculation procedure, in the process ofobtaining the static equilibrium state in the calculation procedure of steps (1) and (2), themass matrix [M] and the damping matrix [C] can be adjusted to reduce the time steps
required to reach static equilibrium.
By using the above calculation procedure, the IEFEM can be analysed based on the
dynamic explicit FEM while considering convergence to a static equilibrium state. The
accuracy of the analysis is equivalent to that of the static implicit FEM, while achieving
high speed and reduced memory requirements.

APPLICATION OF PROPOSED METHOD TO LARGE-HEAT-INPUT BUTTWELDINGANALYSIS MODEL AND CONDITIONSIn order to investigate the validity of the solidification cracking analysis method presented
in the previous section, the proposed method was applied to large-heat-input butt welding.
A JIS G 3106 SM490A steel plate was used as the test specimen. Downward single-sided
welding was performed using GMAW. A JIS Z 3313 T49J 0 T5-1 C A-U flux-core wire
was used, and a groove filler was sprayed in the groove to stabilize the molten pool. As
shown in Fig. 9, the process of joining two steel plates of 600 mm in length, 150 mm inwidth and 25 mm in thickness by butt welding was studied.

Nodes: 1,421,065
Elements: 1,336,036zyx
200 mmTotal step of analysis: 10,520 steps

Plate thickness: 25 mm

Fig. 9 Shape and size of test plates of butt welding

Tab plates of 150 mm in length, 150 mm in half-width and 25 mm in thickness wereattached to the start and end of the base plates. Tack welds of 50 mm in length and 5 mm

440

Mathematical Modelling of Weld Phenomena 13
in thickness were installed at x = 125 mm, 425 mm and 725 mm, and constraint plates
were attached on the opposite side of the welding at x = 200 mm, 440 mm and 680 mm,
as shown in Fig. 9 (b). The constraint plate has a U-shape with a width of 300 mm, aheight of 200 mm and a thickness of 20 mm. The weld start position is x = 50 mm and theweld end position is x = 850 mm, i.e., the weld length is 800 mm. The 800 mm welding
line is divided into 800 elements in the direction of the weld line and unequal elements inthe direction of the plate width. The weld part is a V-groove with a 35° groove angle anda root gap of 5 mm. The model shape of the weld metal is based on the experimental
cross-sectional macro photograph. The number of nodes and elements are 1,122,416 and1,059,664, respectively. The current, voltage, welding speed and thermal efficiency were
set to 520 A, 43 V, 250 mm/min, and 0.6, respectively, meaning that the heat input is3,219.8 J/mm. To show the validity of the proposed method, a comparison was made withexperimental results under the same conditions. At the start of the heat conduction
analysis using FEM, the element corresponding to the weld metal was disabled, and theelement was activated when the torch reached the weld during welding. At the beginningof the heat conduction analysis using FEM, the element corresponding to the weld metal
was disabled, and the element was activated when the torch reached the weld during
welding. The room temperature was set at 20°C.

INVESTIGATION OF PENETRATION SHAPE AND COLLISION ANGLE OF COLUMNAR CRYSTALSIn this section, we investigate the validity of a simple method for predicting the directionof columnar crystal growth. This method is a simplified prediction that assumes the
direction of macroscopic epitaxial growth and does not assume the morphology of thebase metal, anisotropy in the direction of preferential growth or crystal size. Fig. 10 (a)shows a cross-sectional macro photograph and Fig. 10 (b) shows the penetration shapeand the distribution of the temperature gradient vector in the BTR obtained from the heat
conduction analysis using FEM. Based on these figures, it can be confirmed that thepenetration shape obtained by proposed analysis is in good agreement with theexperimental results.

(a)

［℃］
1500
1350
1200
1050
900
750
600
450
300
150
0

(b)

Fig. 10 Penetration shape and direction of columnar crystal growth: (a) Experimental result,(b) Analysis result441

Mathematical Modelling of Weld Phenomena 13

Distance from the bottom [mm]The collision angles were calculated at each of the 14 points at which the vertical line
passing through the centre of the welding line. The collision angles obtained fromexperiments and from the temperature gradient vector in the BTR are compared, as shownin Fig. 11.

11
10
9
8
7
6
5
4
3
2
1
0

A’

A’

A

Analysis
Experiment 1
Experiment 2
Experiment 3

A

0

60

120 180 240 300 360
Collision angle[°]Fig. 11 Collision angle as determined by analysis and experimental results along A-A'

The experimental results were derived from cross-sectional macro photographs
obtained from three different cross sections of the same test. The figure shows that theexperimental results are highly repeatable with little variation. It can also be seen that thecollision angles are close to 180 degrees in the entire plate thickness direction, especiallyat the upper part, where columnar crystals collide. It can also be found that the
distribution trends of the collision angles in the experimental result and analysis results
are in good agreement. These results suggest that the temperature gradient vector in the
BTR can be used to predict the growth direction and collision angle of columnar crystals
with a certain degree of accuracy.

EVALUATION OF SOLIDIFICATION CRACKING INITIATIONIn this section, the analysis of the proposed method considering solidification shrinkage
strain is performed and compared with the experimental results for the cracking position.
Fig. 12 (a) show the distribution of the BTR plastic strain increment in the transverse
section at x = 425 mm. Fig. 12 (b) shows the experimental cross-sectional macro
photographs at x = 425 mm.

442

Mathematical Modelling of Weld Phenomena 13
Solidification crack

［%］
3.6
2.8
2.0
1.2
0.4
-0.4
-1.2
-2.0
-2.8
-3.6

(a)

(b)

Fig. 12 Distribution of plastic strain increment in BTR and cracking position on transverse
cross section at x = 425 mm: (a) Analysis result, (b) Experimental result

In this figure, large tensile plastic strain increment in the BTR of approximately 3.0%
is observed at the upper part. As shown in Fig. 12 (b), the experimental results show thatcracking occurs at the upper part, and the cracking position is in good agreement with theposition of the large tensile plastic strain increment in the BTR obtained by using the
proposed method.
Fig. 13 (a) shows the distribution of the plastic strain increment in the BTR in thelongitudinal section along the welding line, and Fig. 13 (b) shows the cracking position in
the experiment. In Fig. 13 (a), a large plastic strain increment in the BTR can be observed
over a wide area behind the tack weld position at the middle and end of the welding line.
In addition, from Figs. 13 (a) and 13 (b), it is shown that cracking occurs at the same
locations as those with a large plastic strain increment in the BTR. From these figures, it
is found that the proposed method can evaluate solidification cracking initiation in thelongitudinal direction as well.
［%］
2.7
2.1
1.5
0.9
0.3
-0.3
-0.9
-1.5
-2.1
-2.7

Large tensile
Welding direction

(a)
(b)
z
x

: Tack welds: Cracked areayFig. 13 Distribution of plastic strain increment in BTR and experimentally obtained cracking
position: (a) Analysis result, (b) Experimental result

443

Mathematical Modelling of Weld Phenomena 13
In the above results, it was shown that the proposed method can predict the crack
position in both the thickness direction and the longitudinal direction in the solidification
cracking evaluation.

EFFECT OF SOLIDIFICATION MORPHOLOGY ON CRACK INITIATION

Distance from the bottom [mm]In this section, the effect of solidification shrinkage strain on the distribution of plasticstrain increment in the BTR is examined. As described above, in the transverse section at
x = 425 mm, the crystals collided in the upper part, which is a cracking-prone
solidification condition.
Analyses were conducted using the conventional method without consideringsolidification shrinkage strain and the proposed method with solidification shrinkage
strain. Fig. 14 shows the distribution of the plastic strain increment in the BTR along line
A-A’ of the cross section at x = 425 mm for the comparison of the influence without orwith solidification shrinkage strain.

15
12
9

Without solidification shrinkage.
With solidification shrinkage.

A’
A’

6
3
0
-1.0

A
0.0

A

1.0

2.0

3.0

4.0

Plastic strain increment in BTR [%]Fig. 14 Influence of solidification shrinkage on plastic strain increment in BTR along A-A’

Fig. 14 shows that the effect of solidification shrinkage strain is not uniform
throughout the weld metal, but rather varies depending on the cooling conditions at eachposition. With the conventional method that does not consider solidification shrinkage
strain, it can be confirmed that a tensile plastic strain increment in the BTR of less than
0.2% is distributed throughout the entire plate thickness direction. With the proposed
method, which takes into account the solidification shrinkage strain, a large tensile plasticstrain increment in the BTR of approximately 3.0% can be confirmed at the upper part. In
other words, a large plastic strain increment in the BTR was generated due to the large
influence of solidification shrinkage strain at upper part.
Fig. 15 shows the temperature distributions on lines X-X' and Y-Y' in the figure at thestart of solidification. Line X-X' is located at the upper part where the crystals collide, and444

Mathematical Modelling of Weld Phenomena 13
line Y-Y' is located at the backside where a small plastic strain increment in the BTR isgenerated.

Temperature [ ℃]1550

: X-X’: Y-Y’

Y

1500

X
1450

BTR
X

1400
1350
0.0

X’
Y

Y’

X’

Y’

1.5
3.0
4.5
Coordinate in y-direction [mm]Fig. 15 Temperature distribution in transverse cross section at the start of solidification at x=425 mmFig. 15 shows that the temperature gradient in the width direction is large on line X-X'
at the upper part, i.e., the distance from the region of the solid phase where stiffness isrecovered to the BTR is small. On line Y-Y' of the backside of the weld metal where thecollision angle is smaller, the temperature gradient in the width direction is small, i.e., thedistance from the region of the solid phase to the BTR is large. In other words, whensolidification shrinkage occurs at the upper part of the weld metal, there is greater
restraint from the region of solid phase. This is considered to cause cracking
susceptibility, due to the greater effect of solidification shrinkage strain.
The above suggests that solidification cracking is likely to occur at the position atwhich columnar crystals collide, because the effect of solidification shrinkage strain
becomes large due to the large surrounding restraints.

CONCLUSIONSIn the present study, a new evaluation method for solidification cracking initiation basedon the thermal elastic-plastic analysis using FEM considering mechanical andmetallurgical factors is proposed. As metallurgical factors, the direction of solidification
growth was simply predicted using the temperature gradient vector in the BTR, and thecracking position was evaluated based on the collision angle of columnar crystals.
Analysis was also conducted considering solidification shrinkage strain based on therelationship between temperature and solid fraction in the BTR. From a mechanical pointof view, the strain generated in the weld zone was evaluated using the plastic strain
increment in the BTR. Furthermore, the proposed method was applied to large-heat-input
butt welding.
The results showed the validity of a simple evaluation method for the direction ofcolumnar crystal growth using the temperature gradient vector in the BTR and a445

Mathematical Modelling of Weld Phenomena 13
solidification cracking evaluation method using the plastic strain increment in the BTR.
This indicates that the proposed method can be used to predict the cracking position.
In addition, it was found that the position at which the collision angle is close to 180°
and the crystals collide provides greater restraint against solidification shrinkage,suggesting that the tensile strain at the weld is large, i.e., such solidification morphology
is prone to solidification cracking.
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