Welding of a Titanium Plate

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:

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Beta to Widmanstätten alpha

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Alpha (Widmanstätten alpha and Martensitic alpha) to beta

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Martensitic alpha to beta

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Martensitic alpha to Widmanstätten alpha

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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 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 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 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 multi-pass 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,” Mat. Sci. Engng., vol. A359, pp. 137–149, 2003.

Metal_Processing_Module/Titanium_Phase_Transformations/welding_of_a_titanium_plate

Modeling Instructions

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Global Definitions

Parameters 1

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In the window for , locate the section.

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	6sqrt(3)fr*Qw/a/b/cr/pi/sqrt(pi)	1.5553E10 W/m³	Goldak power density