Arc physics based heat source modelling for numerical simulation of weld residual stress and distortion

Simulation model

The process using this model is stationary tungsten inert gas (TIG) welding using a tungsten cathode 3·2 mm in diameter with a 60° conical tip. The calculation domain, described in two-dimensional cylindrical coordinates with rotational symmetry around the arc axis, is shown in Fig. 1. The anode is 490 MPa class high tensile strength structural steel, JIS SM490YB. Shielding gas is supplied from outside the cathode on the upper boundary at a certain flowrate. Two-dimensional calculations are conducted. In a two-dimensional axisymmetrical model of stationary TIG welding, the effect of welding speed is virtually considered by converting moving of welding torch into moving of welded plate, and thermal energy stored in anode material is deprived with the increase in moving speed of the welded plate. Thermal energy stored in anode material is deprived with the increase in moving speed of the welding torch. It is based on the fact that when the moving speed of the welded plate becomes higher, the temperature rise in the anode material becomes lower. The governing equations used in the model are as follows.

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Figure 1

Calculation domain of TIG arc plasma simulation

The mass continuity equation is (1) The radial momentum conservation equation is (2) The axial momentum conservation equation is (3) The energy conservation equation is (4) The current continuity equation is (5) and (6) where r is the radial distance, t is the time, H is the enthalpy, p is the pressure, v_r and v_z are the radial and axial velocities, j_r and j_z are the radial and axial components of the current density, g is the acceleration due to gravity, c is the specific heat, λ is the thermal conductivity, ρ is the density, τ is the viscosity, ψ is the radiative emission coefficient and κ is the electrical conductivity. E_r and E_z are the radial and axial components of the electric field defined by (7) where U is the electric potential.

The azimuthal magnetic field B_θ induced by the arc current is evaluated by Maxwell's equation (8) where μ₀ is the permeability of free space.

It is necessary to consider the effects of energy transfer at the surfaces of the electrodes. The additional energy fluxes at the cathode and anode are described as follows: (9) (10) where γ is the surface emissivity, α is the Stefan–Boltzmann constant, φ_K is the work function of the tungsten cathode, V_i is the ionisation potential of the arc plasma gas, j_e is the electron current density, j_i is the ion current density, φ_A is the work function of the anode and T is the temperature. For the cathode surface, F_K needs to be included in equation (9) to account for thermionic cooling by the emission of electrons, radiative cooling and ion heating. Similarly, for the anode surface, F_A is required in equation (10) to account for radiative cooling and thermionic heating. Furthermore, for the cathode surface, the electron and ion currents are considered separately and defined based on the Richardson–Dushman equation of thermionic emission as follows (11) where k_B is Boltzmann's constant and A is Richardson's constant, which are dependent on the cathode material. The ion current density j_i is then assumed to be j_i = j−j_R, where the total current density j is j = j_i+j_e. The governing and auxiliary equations are solved iteratively by the numerical procedure,9, 10 and the other approximation and boundary conditions are given in previous papers.14^– 16

Effect of welding conditions on TIG arc heat source

The welding conditions in the numerical simulation of arc plasma process are shown in Table 1. The welding method is TIG welding. The shielding gas is 100%Ar, and the flowrate is 1·67×10⁵ mm³ s⁻¹. To investigate the effect of welding current and welding speed individually, five welding conditions are examined. The base conditions are a 200 A welding current, a welding speed of 1·67 mm s⁻¹ and a 3 mm arc length. Two approaches for the reduction of weld heat input Q ( = IV/v)(J mm⁻¹) are implemented. One is reducing the welding current (I-changed), and the other is increasing the welding speed (v-changed). These welding conditions are applied in both 6 and 12 mm plate thicknesses. As a result, the influence of plate thickness is negligibly small.

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Table 1

Welding conditions in TIG arc plasma simulation

	Base	I-changed		v-changed
Welding current I/A	200	160	100	200	200
Welding speed v/mm s−1	1·67	1·67	1·67	2·33	3·33

The temperature distributions in TIG arc plasma are shown in Figs. 2a and 3a for each of the three cases of varying welding current and speed for the 6 mm plate thickness. According to these figures, the temperature distribution in TIG arc plasma is significantly affected by the welding current. When the welding current is 100 A, the peak temperature is not more than 14 000 K, and the spread of the area of increased temperature in the radial direction is almost within 12 mm. In contrast, when the welding current is 200 A, the peak temperature is more than 14 000 K, and the spread of the increased temperature area in the radial direction is much greater than 14 mm. The difference in welding speed does not affect either the peak temperature or the spread of the increased temperature area in the radial direction in TIG arc plasma. The heat transfer from arc plasma to a welded plate should be considered based on the differences of temperature distributions in TIG arc plasma due to certain welding conditions. The effects of welding current and speed on the distribution of heat transfer from arc plasma to a welded plate are quantitatively evaluated, as shown in Figs. 2b and 3b. The effects of welding conditions on the heat transfer from arc plasma to a welded plate are summarised as follows: the welding current affects the distribution of heat transfer from arc plasma to a welded plate, and the welding speed does not affect the distribution of heat transfer from arc plasma to a welded plate.

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Figure 2

Effect of welding current on characteristics of TIG arc heat source

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Figure 3

Effect of welding speed on characteristics of TIG arc heat source

Simulation model

The universal expression of the three-dimensional thermal conduction is as follows (12) where T is the temperature of the base metal and w is the internal heat generation.

In the elastic–plastic stress analysis, the governing equations are as follows. The equation of the strain–displacement relation is (13) where {ϵ} and {u} are the tensor of strain and displacement respectively and [B] is the matrix that includes the differential operator.

The constitutive equation is (14) where {σ} is the tensor of stress, and {ϵ^e} and {ϵ^t} are the tensors of elastic strain and thermal strain respectively. [D^e] is the elastic stress–strain matrix.

The virtual work principle, shown in the following, can substitute for the equation of equilibrium (15) where is the tensor of body force per unit volume and is the tensor of surface force per unit area. V_σ is the volume of the body and S_σ is the area for which the boundary conditions are given. δ{ } is the amount of virtual change and { }^T is the transposed tensor. Equations (13)–(15) are essential in the elastic–plastic stress analysis. Combined work hardening of the material and temperature dependence of the mechanical properties, which include yield stress, Young's modulus and the thermal expansion coefficient, are considered in the mechanical analysis. No restraint is used during the welding process. Details of the mechanical computation have been described in the previous papers.17^– 23

One of the benefits associated with the use of numerical simulation is the diversification possible in generating the heat source model for thermal conduction in welding.24^– 34 At present, to establish the heat source model from the welding conditions, an experiment to obtain information about weld penetration and the temperature profile near the melted zone is imperative. However, a reasonable heat source modelling from the welding conditions should, in nature, be established based on the consideration of weld physical phenomena. In this study, the heat source modelling for welding is established based on heat transfer from arc plasma to a welded plate, which was already calculated by the numerical simulation of TIG arc plasma. The distribution of heat transfer from arc plasma to a welded plate is approximately expressed by the Gaussian distribution (16) where q is the weld heat input per time (J s⁻¹), R is the radius of the Gaussian distribution and t is the time. The weld heat input q and the radius of Gaussian distribution R can be determined based on the result of the TIG arc plasma simulation, shown in Table 2. The distribution of heat input in weld heat source used in the numerical simulation of weld residual stress and distortion is also shown in Figs. 2b and 3b. The Gaussian distribution of the heat input is denoted by a solid line, compared with the distribution of heat transfer obtained by arc plasma simulation. Both distributions demonstrate good agreement. Therefore, these distributions are used in the numerical simulation as the heat flux input on the surface. The simulation model used in the numerical simulation is a bead-on-plate model of a 490 MPa class high tensile strength structural steel, JIS SM490YB. The plate has a length of 200 mm, a width of 500 mm and a thickness of 6 or 12 mm. The weld length is 150 mm, which leaves unwelded sections of 25 mm at both ends of the plate. The finite element model used is a symmetrical model to the weld line due to the symmetry of weld phenomena.

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Table 2

Information for heat source modelling used in finite element analysis

	Base	I-changed		v-changed
Welding current I/A	200	160	100	200	200
Welding speed v/mm s−1	1.67	1.67	1.67	2.33	3.33
Weld heat input q/J s−1	1500	1150	685	1505	1505
Radius R/mm	2.8	2.7	2.5	2.8	2.8
Weld heat input Qnet/J mm−1	900	690	410	645	450

Effect of welding conditions on weld residual stress and distortion

Weld residual stress and distortion are evaluated at the centre of the plate in the welding direction. The relationship between welding conditions and weld residual stress is quantified by the weld heat input Q_net, as shown in Fig. 4. The spread of the area of tensile stress and the local minimum point of compressive stresses are increased with the increase in weld heat input Q_net. The relationship between welding conditions and weld distortion is quantified by the heat input parameter Q_net/h², where h is the plate thickness, as shown in Fig. 5. The heat input parameter is the influential dominant factor of angular distortion theoretically derived by the similarity rule of the temperature distribution on plate thickness section.35 Weld distortion is not represented by this heat input parameter. Thus, the difference in the effect of welding current and welding speed on weld distortion becomes clear. When the heat input parameter is small, the weld distortion increases with the increase in welding speed. In contrast, when the heat input parameter is large, the weld distortion decreases with the increase in welding speed. This means that the increase in welding speed is equal to the increase in the apparent heat input parameter. The developed numerical simulation using arc physics based heat source modelling thus generates a clearer understanding of the relationship between welding conditions and weld distortion.

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Figure 4

Simulation result on effect of welding conditions on weld residual stress

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Figure 5

Simulation result on effect of welding conditions on weld distortion

Experimental procedure

Verification of the developed numerical simulation of weld residual stress and distortion using arc physics based heat source modelling is performed by comparing the experimental result of weld distortion with those in the numerical simulation. This is because weld distortion is more sensitive to temperature distribution than weld residual stress. The steel used in the experiment is a 490 MPa class high tensile strength structural steel, JIS SM490YB. The chemical composition of the steel is shown in Table 3. The configuration of the weld joint used in the experiment is the same as that used in the numerical simulation. The experimental plate is placed on a sill plate because of the heat transfer from the bottom of the welded plate to the air. Weld distortion in the experiment is measured using contact type displacement gauge in ambient temperature. The welding conditions for the verification experiments are the same as those of the numerical simulation, as shown in Table 1.

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Table 3

Chemical compositions of JIS SM490YB, mass-%

Plate thickness/mm	C	Si	Mn	P	S
6	0·15	0·25	1·42	0·020	0·004
12	0·16	0·28	1·45	0·015	0·003

Comparison of weld penetration and distortion

The measured weld distortions in all the experimental conditions, including welding current, welding speed and plate thickness, are compared to those simulated, as shown in Fig. 6. In the figure, the configuration of weld penetration is also shown in order to compare the experimental results with the simulation results. The penetration shape generated by numerical simulation does not strictly conform to that of the experiment. This is because convective heat transfer in the weld pool is not considered yet in the present numerical simulation. It is known that the penetration shape in TIG welding is influenced by convective heat transfer in the weld pool.36^– 40 The simulation results, however, correspond approximately with the experimental results in both weld distortion and penetration. Especially, the weld distortion simulated conforms very closely to that in the experiment. Thus, evaluating the characteristics of weld heat source highly increases the accuracy of predicting the weld distortion. Furthermore, a more detailed understanding of the effects of the welding conditions on weld distortion is possible. It can be concluded that the combination of arc plasma simulation and mechanical computation is highly effective in the numerical simulation of weld distortion.

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Figure 6

Comparison between simulated and measured weld penetration and distortion