PDF
Welding of a Titanium Plate
Introduction
During welding of metals, the material will undergo phase transformations due to the thermal transients around the moving melt pool. In this example, welding of a titanium plate is considered. The titanium material is a heat treatable alpha-beta alloy, and the welding process is modeled using a double-ellipsoidal Goldak heat source (Ref. 1). The model shows how to define the temperature-dependent metallurgical phase transformations that are involved in this process, and how to compute the heterogeneous phase composition in the plate subsequent to a single weld pass.
Model Definition
The plate is 5 cm in length, 3.5 cm in width, and with a thickness of 5 mm. In this example, the plate is to be welded to a second, identical plate, along their lengths. Because of symmetry, only one plate is modeled. The plate is made from an alpha-beta titanium alloy. This is a class of materials that can be found in for example automotive and aerospace applications. The material modeled here represents a fictitious alloy, however the material properties and the phase transformation kinetics are representative of commercial alpha- beta titanium alloys.
Material Properties
The material properties of the alpha-beta alloy are temperature-dependent, and also phase composition dependent. The Alpha-Beta Phase Transformation physics interface automatically averages these properties into effective properties that define a compound material. The compound material is used in the thermal analysis.
Phase Transformation Analysis
Thermal transients involving heating and cooling can give rise to different phase transformations in an alpha-beta titanium alloy. The Alpha-Beta Phase Transformation physics interface automatically sets up a beta phase and two alpha phases — Widmanstätten alpha, and Martensitic alpha, and then defines phase transformations relevant to cooling and to heating. The physics interface also generates several model parameters, such as transformation temperatures. As this model concerns a fictitious alpha-beta titanium alloy, these default values are used without modification. The phase transformations that need to be supplemented with phase transformation data are:
•
Beta to Widmanstätten alpha
•
Alpha (Widmanstätten alpha and Martensitic alpha) to beta
•
Martensitic alpha to beta
•
Martensitic alpha to Widmanstätten alpha
•
Beta re-formation
In addition, the temperature dependent equilibrium fraction of alpha phase is required. The phase transformation data and the data for the equilibrium fraction of alpha phase are defined in text files that can be loaded as interpolation functions.
Beta to Widmanstätten Alpha
The transformation of beta phase into Widmanstätten alpha phase during cooling below the beta transus temperature is modeled using the JMAK phase transformation model. It is characterized using TTT data for the 1% and 50% transformation levels, here given as interpolation functions. The equilibrium phase fraction of the alpha phase is also given as an interpolation function. The TTT data and the equilibrium phase fraction data are taken from Salsi and others (Ref. 2); see Figure 1 and Figure 2
Figure 1: TTT data for the Beta to Widmanstätten alpha phase transformation.
Figure 2: Equilibrium phase fraction for the alpha phase.
Alpha to Beta
The dissolution of alpha phase is modeled using the Hyperbolic rate phase transformation model. The temperature dependent data for the alpha dissolution is taken from Kelly (Ref. 3), where a power-law expression was used to describe a so-called parabolic thickening:
with the implied unit of 1/s1/2, and the temperature in Kelvin. This analytical form for the data is converted into a numerically less challenging interpolation function fDiss(T), and when it is used to model the phase transformation using the Hyperbolic rate phase transformation model, the rate term becomes:
This phase transformation represents the simultaneous dissolution of Widmanstätten alpha and Martensitic alpha. This is accomplished using the Additional source phase subnode.
Martensitic Alpha to Beta & Martensitic Alpha to Widmanstätten Alpha
Above the martensite dissolution temperature, the Martensitic alpha phase dissolves into a combination of Widmanstätten alpha and beta phase. This transformation is modeled using two separate Phase transformation nodes. The JMAK phase transformation model is used for each phase transformation, and TTT data is used to characterize the rate of dissolution. The amount of available data in the literature is scarce. Guided by the experimental work in Qazi and others (Ref. 4), the transformation is approximated with TTT data for 1% and 99% transformation levels, see Figure 3.
Figure 3: TTT data for the dissolution of Martensitic alpha.
Beta Re-Formation
When the temperature exceeds the beta transus temperature, the alpha phases dissolve to beta phase. Here, this phase transformation is simplified, as the objective is simply to “reset” the material above the beta transus. The JMAK phase transformation model is used, and a fixed time constant and Avrami constant are used. The Additional Source Phase subnode is used to include both alpha phases as source phases. If a more detailed description of the alpha dissolution is required, the phase transformation data would need a more careful definition.
Thermal Analysis
The heat transport in the plate is described by the heat equation:
where T is the temperature, k represents the thermal conductivity, ρ denotes the density, Cp denotes the specific heat capacity, and Q is a heat source. The thermal conductivity, the density, and the specific heat capacity are in general temperature dependent, but in the presence of metallurgical phase transformations, they also depend on the current phase composition. In the present thermal analysis, phase transformation latent heat is neglected. Thus, the only heat source that contributes is that of the externally applied weld arc. The densities, specific heat capacities and heat conductivities of the individual phases are defined below. Note that the equations are solved on the material frame, meaning that the density of each phase is constant, corresponding to the density at the volume reference temperature. With regard to thermal conductivities, they are taken as linear functions of temperature. The specific heat is assumed to be equal across all three phases. Table 1 lists the thermal properties for the three phases. The temperature T is assumed to be in kelvin.
Table 1: Temperature-Dependent Thermal Material Properties.
Phase
 ρ
  Cp
 k
Beta
4700kg/m3
480J/(kg·K) + 0.24m/W · T
0.3W/(m·K) + 0.021m/W · T
Widmanstätten
Alpha
4400kg/m3
480J/(kg·K) + 0.24m/W · T
0.1W/(m·K) + 0.016m/W · T
Martensitic
Alpha
4400kg/m3
480J/(kg·K) + 0.24m/W · T
0.1W/(m·K) + 0.016m/W · T
During welding, the plate is subjected to a moving heat source that represents the weld arc. As the plate heats up, it also releases heat to the exterior through natural convection and thermal radiation.
The Goldak Double-Ellipsoid Heat Source
The center point of the weld arc moves along the X axis, at a velocity vel = 1 mm/s. Its current position is thus given by X0 = vel·t. The heat source by Goldak is defined by two regions that join at X0, and whose shapes are ellipsoidal. The widths a and depths b of these regions are equal, but the front and rear lengths, cf and cr, may differ, see Figure 4. The heat source is given by:
where q0 is the power density of the weld, given by
where the weld power is Qw = 800 W, a = 4 mm, b = 4 mm, cr = 8 mm, cf = 4 mm, and the condition that fr + ff = 2 gives fr = 4/3 and ff = 2/3.
Figure 4: The Goldak double-ellipsoid.
Natural Convection
The convection of heat to the exterior is characterized by an assumed constant heat transfer coefficient of 10 W/(m2·K).
Thermal Radiation
Thermal radiation is modeled using a surface emissivity of 0.4.
Results and Discussion
As the weld arc moves along the plate, it heats up the material. In reality, a melt pool forms, but this is not modeled here. Instead, we concern ourselves with solid state phase transformations that result from heating the material, and subsequently when the material cools. Figure 5 shows the phase composition evolution over time, halfway along the weld path. The initial composition is 89% Widmanstätten alpha phase, and 11% beta phase. As the temperature increases past the beta transus, the alpha phase dissolves into beta phase. As the weld arc passes, the material cools below the beta transus, and the beta phase transforms into Widmanstätten alpha. On further cooling below the Martensite start temperature, Martensitic alpha phase forms. This example involves a single weld pass. In a situation of multiple weld passes, the phase composition will change with each weld pass. By performing simulations of so-called multipass welding, it is possible to gain insight into how the material develops in the heat affected zone (HAZ).
Figure 5: Phase composition evolution over time.
References
1. J. Goldak, A. Chakravarti, and M. Bibby, “A New Finite Element Model for Welding Heat Sources,” Metall. Trans. B, vol. 15, pp. 299–305, 1984.
2. E. Salsi, M. Chiumenti, and M. Cervera, “Modeling of Microstructure Evolution of Ti6Al4V for Additive Manufacturing,” Metals, vol. 8, pp. 633–657, 2018.
3. S.M. Kelly, Thermal and Microstructure Modeling of Metal Deposition Processes with Application to Ti-6Al-4V, PhD Thesis, Virginia Tech, 2004.
4. J.I. Qazi, O.N. Senkov, J. Rahim, and F.H. (Sam) Froes, “Kinetics of Martensite Decomposition in Ti-6Al-4V-xH alloys,” Mater. Sci. Eng., vol. A359, pp. 137–149, 2003.
Application Library path: Metal_Processing_Module/Titanium_Phase_Transformations/welding_of_a_titanium_plate
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Heat Transfer > Heat Transfer in Solids (ht) and Heat Transfer > Metal Processing > Alpha–Beta Phase Transformation (abp).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
Lx
5[cm]
0.05 m
Plate length
Ly
3.5[cm]
0.035 m
Plate width
Lz
5[mm]
0.005 m
Plate thickness
Qw
800[W]
800 W
Weld power
vel
1[mm/s]
0.001 m/s
Welding speed
a
4[mm]
0.004 m
Goldak ellipsoid width
b
4[mm]
0.004 m
Goldak ellipsoid depth
cr
8[mm]
0.008 m
Goldak ellipsoid length, rear
cf
4[mm]
0.004 m
Goldak ellipsoid, front
fr
2/(cf/cr+1)
1.3333
Goldak parameter
ff
2-fr
0.66667
Goldak parameter
q0
6*sqrt(3)*fr*Qw/a/b/cr/pi/sqrt(pi)
1.5553E10 W/m³
Goldak power density
Definitions
Variables 1
1
In the Model Builder window, expand the Component 1 (comp1) > Definitions node.
2
Right-click Definitions and choose Variables.
3
In the Settings window for Variables, locate the Variables section.
4
In the table, enter the following settings:
Name
Expression
Unit
Description
X0
vel*t
m
Current weld arc position
Analytic 1 (an1)
1
In the Definitions toolbar, click  Analytic.
2
In the Settings window for Analytic, locate the Definition section.
3
In the Expression text field, type q0*exp(-3*( dX^2/C^2 + dY^2/A^2 + dZ^2/B^2 )).
4
In the Arguments text field, type dX, dY, dZ, q0, A, B, C.
5
Locate the Units section. In the Function text field, type W/m^3.
6
In the table, enter the following settings:
Argument
Unit
dX
m
dY
m
dZ
m
q0
W/m^3
A
m
B
m
C
m
Interpolation 1 (int1)
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
In the Function name text field, type xieqAlpha.
4
Click  Load from File.
5
Browse to the model’s Application Libraries folder and double-click the file welding_of_a_titanium_plate_xieqalpha.txt.
6
Locate the Units section. In the Function table, enter the following settings:
Function
Unit
xieqAlpha
1
7
In the Argument table, enter the following settings:
Argument
Unit
t
degC
Interpolation 2 (int2)
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
In the Function name text field, type ttt1.
4
Click  Load from File.
5
Browse to the model’s Application Libraries folder and double-click the file welding_of_a_titanium_plate_ttt1.txt.
6
Locate the Interpolation and Extrapolation section. From the Extrapolation list, choose Linear.
7
Locate the Units section. In the Function table, enter the following settings:
Function
Unit
ttt1
1
8
In the Argument table, enter the following settings:
Argument
Unit
t
degC
Interpolation 3 (int3)
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
In the Function name text field, type ttt50.
4
Click  Load from File.
5
Browse to the model’s Application Libraries folder and double-click the file welding_of_a_titanium_plate_ttt50.txt.
6
Locate the Interpolation and Extrapolation section. From the Extrapolation list, choose Linear.
7
Locate the Units section. In the Function table, enter the following settings:
Function
Unit
ttt50
1
8
In the Argument table, enter the following settings:
Argument
Unit
t
degC
Interpolation 4 (int4)
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
In the Function name text field, type fDiss.
4
Click  Load from File.
5
Browse to the model’s Application Libraries folder and double-click the file welding_of_a_titanium_plate_fdiss.txt.
6
Locate the Units section. In the Function table, enter the following settings:
Function
Unit
fDiss
1
7
In the Argument table, enter the following settings:
Argument
Unit
t
K
Interpolation 5 (int5)
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
In the Function name text field, type ttt1m.
4
Click  Load from File.
5
Browse to the model’s Application Libraries folder and double-click the file welding_of_a_titanium_plate_ttt1m.txt.
6
Locate the Interpolation and Extrapolation section. From the Extrapolation list, choose Linear.
7
Locate the Units section. In the Function table, enter the following settings:
Function
Unit
ttt1m
1
8
In the Argument table, enter the following settings:
Argument
Unit
t
degC
Interpolation 6 (int6)
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
In the Function name text field, type ttt99m.
4
Click  Load from File.
5
Browse to the model’s Application Libraries folder and double-click the file welding_of_a_titanium_plate_ttt99m.txt.
6
Locate the Interpolation and Extrapolation section. From the Extrapolation list, choose Linear.
7
Locate the Units section. In the Function table, enter the following settings:
Function
Unit
ttt99m
1
8
In the Argument table, enter the following settings:
Argument
Unit
t
degC
The imported TTT data uses a logarithmic time axis, so we need to convert the data to use it in the phase transformation modeling.
Analytic 2 (an2)
1
In the Definitions toolbar, click  Analytic.
2
In the Settings window for Analytic, locate the Definition section.
3
In the Expression text field, type 10^ttt1(x).
4
Locate the Units section. In the Function text field, type s.
5
In the table, enter the following settings:
Argument
Unit
x
K
Analytic 3 (an3)
1
In the Definitions toolbar, click  Analytic.
2
In the Settings window for Analytic, locate the Definition section.
3
In the Expression text field, type 10^ttt50(x).
4
Locate the Units section. In the Function text field, type s.
5
In the table, enter the following settings:
Argument
Unit
x
K
Analytic 4 (an4)
1
In the Definitions toolbar, click  Analytic.
2
In the Settings window for Analytic, locate the Definition section.
3
In the Expression text field, type 10^ttt1m(x).
4
Locate the Units section. In the Function text field, type s.
5
In the table, enter the following settings:
Argument
Unit
x
K
Analytic 5 (an5)
1
In the Definitions toolbar, click  Analytic.
2
In the Settings window for Analytic, locate the Definition section.
3
In the Expression text field, type 10^ttt99m(x).
4
Locate the Units section. In the Function text field, type s.
5
In the table, enter the following settings:
Argument
Unit
x
K
Geometry 1
Block 1 (blk1)
1
In the Model Builder window, expand the Component 1 (comp1) > Geometry 1 node.
2
Right-click Geometry 1 and choose Block.
3
In the Settings window for Block, locate the Size and Shape section.
4
In the Width text field, type Lx.
5
In the Depth text field, type Ly.
6
In the Height text field, type Lz.
7
Locate the Position section. In the z text field, type -Lz.
8
Click to expand the Layers section. Find the Layer position subsection. Select the Front checkbox.
9
Clear the Bottom checkbox.
10
In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
0.2*Ly
11
Click  Build Selected.
Heat Transfer in Solids (ht)
Heat Source 1
1
In the Physics toolbar, click  Domains and choose Heat Source.
2
In the Settings window for Heat Source, locate the Domain Selection section.
3
From the Selection list, choose All domains.
4
Locate the Heat Source section. In the Q0 text field, type (X>=X0)*an1(X-X0,Y,Z,q0,a,b,cf) + (X<X0)*an1(X-X0,Y,Z,q0,a,b,cr).
Heat Flux 1
1
In the Physics toolbar, click  Boundaries and choose Heat Flux.
2
Select Boundaries 1, 3–5, and 7–11 only.
3
In the Settings window for Heat Flux, locate the Material Type section.
4
From the Material type list, choose Solid.
5
Locate the Heat Flux section. From the Flux type list, choose Convective heat flux.
6
In the h text field, type 10.
Surface-to-Ambient Radiation 1
1
In the Physics toolbar, click  Boundaries and choose Surface-to-Ambient Radiation.
2
Select Boundaries 1, 3–5, and 7–11 only.
3
In the Settings window for Surface-to-Ambient Radiation, locate the Surface-to-Ambient Radiation section.
4
From the ε list, choose User defined. In the associated text field, type 0.4.
Alpha–Beta Phase Transformation (abp)
1
In the Model Builder window, under Component 1 (comp1) click Alpha–Beta Phase Transformation (abp).
2
In the Settings window for Alpha–Beta Phase Transformation, locate the Material Properties section.
3
Click Create Compound Material in the upper-right corner of the section.
Beta
1
In the Model Builder window, under Component 1 (comp1) > Alpha–Beta Phase Transformation (abp) click Beta.
2
In the Settings window for Metallurgical Phase, locate the Phase Material section.
3
Click Create Phase Material in the upper-right corner of the section.
Widmanstätten Alpha
1
In the Model Builder window, click Widmanstätten Alpha.
2
In the Settings window for Metallurgical Phase, locate the Phase Material section.
3
Click Create Phase Material in the upper-right corner of the section.
Martensitic Alpha
1
In the Model Builder window, click Martensitic Alpha.
2
In the Settings window for Metallurgical Phase, locate the Phase Material section.
3
Click Create Phase Material in the upper-right corner of the section.
Beta to Widmanstätten Alpha
1
In the Model Builder window, click Beta to Widmanstätten Alpha.
2
In the Settings window for Phase Transformation, locate the Phase Transformation section.
3
From the Formulation list, choose TTT diagram data.
4
In the ξeqd text field, type xieqAlpha(abp.T).
5
Find the TTT curve 1 subsection. In the X1 text field, type 0.01.
6
In the t1 text field, type an2(abp.T).
7
Find the TTT curve 2 subsection. In the X2 text field, type 0.5.
8
In the t2 text field, type an3(abp.T).
Alpha to Beta
1
In the Model Builder window, click Alpha to Beta.
2
In the Settings window for Phase Transformation, locate the Phase Transformation section.
3
In the Ps −> d text field, type 0.5*fDiss(abp.T)^2 * 1[1/s].
Martensitic Alpha to Beta
1
In the Model Builder window, click Martensitic Alpha to Beta.
2
In the Settings window for Phase Transformation, locate the Phase Transformation section.
3
From the Formulation list, choose TTT diagram data.
4
Find the TTT curve 1 subsection. In the X1 text field, type 0.01.
5
In the t1 text field, type an4(abp.T).
6
Find the TTT curve 2 subsection. In the X2 text field, type 0.99.
7
In the t2 text field, type an5(abp.T).
Martensitic Alpha to Widmanstätten Alpha
1
In the Model Builder window, click Martensitic Alpha to Widmanstätten Alpha.
2
In the Settings window for Phase Transformation, locate the Phase Transformation section.
3
From the Formulation list, choose TTT diagram data.
4
Find the TTT curve 1 subsection. In the X1 text field, type 0.01.
5
In the t1 text field, type an4(abp.T).
6
Find the TTT curve 2 subsection. In the X2 text field, type 0.99.
7
In the t2 text field, type an5(abp.T).
Beta Re-Formation
1
In the Model Builder window, click Beta Re-Formation.
2
In the Settings window for Phase Transformation, locate the Phase Transformation section.
3
From the Phase transformation model list, choose Johnson–Mehl–Avrami–Kolmogorov (JMAK).
4
In the τs −> d text field, type 5.
5
In the ns −> d text field, type 2.
Global Definitions
Beta (mat2)
In the Model Builder window, expand the Global Definitions > Materials node.
Analytic 1 (an1)
1
In the Model Builder window, expand the Beta (mat2) node.
2
Right-click Global Definitions > Materials > Beta (mat2) > Basic (def) and choose Functions > Analytic.
3
In the Settings window for Analytic, locate the Definition section.
4
In the Arguments text field, type T.
5
In the Expression text field, type 0.3+0.021*T.
6
Locate the Units section. In the Function text field, type W/m/K.
7
In the table, enter the following settings:
Argument
Unit
T
K
8
In the Function name text field, type k.
Beta (mat2)
1
In the Model Builder window, under Global Definitions > Materials > Beta (mat2) click Basic (def).
2
In the Settings window for Basic, locate the Output Properties section.
3
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Thermal conductivity
k_iso ; kii = k_iso, kij = 0
k(T)
W/(m·K)
3x3
4
Locate the Model Inputs section. Click  Select Quantity.
5
In the Physical Quantity dialog, select General > Temperature (K) in the tree.
6
Click OK.
7
In the Settings window for Basic, locate the Output Properties section.
8
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Density
rho
4700
kg/m³
1x1
Analytic 2 (an2)
1
In the Home toolbar, click  Functions and choose Global > Analytic.
2
In the Settings window for Analytic, type C in the Function name text field.
3
Locate the Definition section. In the Arguments text field, type T.
4
In the Expression text field, type 480+0.24*T.
5
Locate the Units section. In the table, enter the following settings:
Argument
Unit
T
K
6
In the Function text field, type J/kg/K.
Beta (mat2)
1
In the Model Builder window, under Global Definitions > Materials > Beta (mat2) click Basic (def).
2
In the Settings window for Basic, locate the Output Properties section.
3
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Heat capacity at constant pressure
Cp
C(T)
J/(kg·K)
1x1
Analytic 1 (an1)
1
In the Model Builder window, expand the Widmanstätten Alpha (mat3) node.
2
Right-click Global Definitions > Materials > Widmanstätten Alpha (mat3) > Basic (def) and choose Functions > Analytic.
3
In the Settings window for Analytic, locate the Definition section.
4
In the Arguments text field, type T.
5
In the Expression text field, type 0.1+0.016*T.
6
Locate the Units section. In the Function text field, type W/m/K.
7
In the table, enter the following settings:
Argument
Unit
T
K
8
In the Function name text field, type k.
Widmanstätten Alpha (mat3)
1
In the Model Builder window, under Global Definitions > Materials > Widmanstätten Alpha (mat3) click Basic (def).
2
In the Settings window for Basic, locate the Output Properties section.
3
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Thermal conductivity
k_iso ; kii = k_iso, kij = 0
k(T)
W/(m·K)
3x3
4
Locate the Model Inputs section. Click  Select Quantity.
5
In the Physical Quantity dialog, select General > Temperature (K) in the tree.
6
Click OK.
7
In the Settings window for Basic, locate the Output Properties section.
8
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Density
rho
4400
kg/m³
1x1
Analytic 2 (an2)
1
In the Home toolbar, click  Functions and choose Global > Analytic.
2
In the Settings window for Analytic, type C in the Function name text field.
3
Locate the Definition section. In the Arguments text field, type T.
4
In the Expression text field, type 480+0.24*T.
5
Locate the Units section. In the table, enter the following settings:
Argument
Unit
T
K
6
In the Function text field, type J/kg/K.
Widmanstätten Alpha (mat3)
1
In the Model Builder window, under Global Definitions > Materials > Widmanstätten Alpha (mat3) click Basic (def).
2
In the Settings window for Basic, locate the Output Properties section.
3
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Heat capacity at constant pressure
Cp
C(T)
J/(kg·K)
1x1
Analytic 1 (an1)
1
In the Model Builder window, expand the Martensitic Alpha (mat4) node.
2
Right-click Global Definitions > Materials > Martensitic Alpha (mat4) > Basic (def) and choose Functions > Analytic.
3
In the Settings window for Analytic, locate the Definition section.
4
In the Arguments text field, type T.
5
In the Expression text field, type 0.1+0.016*T.
6
Locate the Units section. In the Function text field, type W/m/K.
7
In the table, enter the following settings:
Argument
Unit
T
K
8
In the Function name text field, type k.
Martensitic Alpha (mat4)
1
In the Model Builder window, under Global Definitions > Materials > Martensitic Alpha (mat4) click Basic (def).
2
In the Settings window for Basic, locate the Output Properties section.
3
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Thermal conductivity
k_iso ; kii = k_iso, kij = 0
k(T)
W/(m·K)
3x3
4
Locate the Model Inputs section. Click  Select Quantity.
5
In the Physical Quantity dialog, select General > Temperature (K) in the tree.
6
Click OK.
7
In the Settings window for Basic, locate the Output Properties section.
8
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Density
rho
4400
kg/m³
1x1
Analytic 2 (an2)
1
In the Home toolbar, click  Functions and choose Global > Analytic.
2
In the Settings window for Analytic, type C in the Function name text field.
3
Locate the Definition section. In the Arguments text field, type T.
4
In the Expression text field, type 480+0.24*T.
5
Locate the Units section. In the table, enter the following settings:
Argument
Unit
T
K
6
In the Function text field, type J/kg/K.
Martensitic Alpha (mat4)
1
In the Model Builder window, under Global Definitions > Materials > Martensitic Alpha (mat4) click Basic (def).
2
In the Settings window for Basic, locate the Output Properties section.
3
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Heat capacity at constant pressure
Cp
C(T)
J/(kg·K)
1x1
Mesh 1
Mapped 1
1
In the Mesh toolbar, click  More Generators and choose Mapped.
2
Select Boundary 1 only.
Size 1
1
Right-click Mapped 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Extremely fine.
Free Quad 1
1
In the Mesh toolbar, click  More Generators and choose Free Quad.
2
Select Boundary 5 only.
Size 1
1
Right-click Free Quad 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Extra fine.
Swept 1
In the Mesh toolbar, click  Swept.
Size 1
1
Right-click Swept 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Extra fine.
4
Click  Build All.
Study 1
Step 1: Time Dependent
1
In the Model Builder window, under Study 1 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,2.5,500).
4
In the Study toolbar, click  Compute.
Result Templates
1
In the Results toolbar, click  Result Templates to open the Result Templates window.
2
Go to the Result Templates window.
3
In the tree, select Study 1/Solution 1 (sol1) > Heat Transfer in Solids > Isothermal Contours (ht).
4
Click the Add Result Template button in the window toolbar.
5
In the Results toolbar, click  Result Templates to close the Result Templates window.
Results
Cut Point 3D 1
1
In the Results toolbar, click  Cut Point 3D.
2
In the Settings window for Cut Point 3D, locate the Point Data section.
3
In the X text field, type Lx/2.
4
In the Y text field, type 0.
5
In the Z text field, type 0.
6
From the Snapping list, choose Snap to closest edge.
7
Click  Plot.
1D Plot Group 6
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, locate the Data section.
3
From the Dataset list, choose Cut Point 3D 1.
4
Click to expand the Title section. From the Title type list, choose None.
5
Locate the Plot Settings section. Select the Two y-axes checkbox.
6
Locate the Legend section. From the Layout list, choose Outside graph axis area.
Point Graph 1
1
Right-click 1D Plot Group 6 and choose Point Graph.
2
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Alpha–Beta Phase Transformation > Phase composition > abp.phase1.xiGp - Phase fraction - 1.
3
Click to expand the Legends section. Select the Show legends checkbox.
4
From the Legends list, choose Manual.
5
In the table, enter the following settings:
Legends
Beta
Point Graph 2
1
In the Model Builder window, right-click 1D Plot Group 6 and choose Point Graph.
2
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Alpha–Beta Phase Transformation > Phase composition > abp.phase2.xiGp - Phase fraction - 1.
3
Locate the Legends section. Select the Show legends checkbox.
4
From the Legends list, choose Manual.
5
In the table, enter the following settings:
Legends
Widmanstätten Alpha
Point Graph 3
1
Right-click 1D Plot Group 6 and choose Point Graph.
2
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Alpha–Beta Phase Transformation > Phase composition > abp.phase3.xiGp - Phase fraction - 1.
3
Locate the Legends section. Select the Show legends checkbox.
4
From the Legends list, choose Manual.
5
In the table, enter the following settings:
Legends
Martensitic Alpha
Point Graph 4
1
Right-click 1D Plot Group 6 and choose Point Graph.
2
In the Settings window for Point Graph, locate the y-Axis section.
3
Select the Plot on secondary y-axis checkbox.
4
Locate the y-Axis Data section. In the Unit field, type degC.
5
Locate the Legends section. Select the Show legends checkbox.
6
From the Legends list, choose Manual.
7
In the table, enter the following settings:
Legends
Temperature
8
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
9
In the 1D Plot Group 6 toolbar, click  Plot.