A Multi Scale Strategy for Simulation of Microstructural Evolutions in Friction Stir Welding of Duplex Titanium Alloy

Introduction

Friction Stir Welding (FSW) has been invented for more than two decades and quickly applied to automobile, ship, high speed train, aeronautics and astronautics industries [1, 2, 3, 4]. FSW can be suitable for joining aluminum alloys [5, 6, 7, 8, 9], magnesium alloys [10, 11, 12], titanium alloys [13, 14, 15], steels [16, 17, 18]. For the cited industries, the joint of polymer/metal is important as reported [19, 20, 21]. The design on the welding parameters can greatly improve the welding quality of the friction stir welds [22, 23, 24, 25].

The welding temperature in FSW is lower than the melting point (T_m) but higher than the temperature for recrystallization [26, 27]. The heat in FSW is generated by two parts: the frictional and the plastic dissipations [28, 29, 30]. The temperature variations in FSW process have been widely studied in different models. Nandan et al. [31] proposed Computational Fluid Dynamic (CFD) based model to simulate the material flows and the heat generations in FSW of steel. Eulerian mesh is used to avoid possible mesh entanglements in this simulation. Sticking contact or sliding contact with consideration of shear failure criterion can be used for the determination of the material velocities on the contact surfaces [32, 33] in CFD models. Schmidt et al. [34] proposed a fully coupled thermo-mechanical model in which Arbitrary Lagrangian and Eulerian technique is used for the controlling of the excessive mesh distortions. The material flow patterns and the dissipated powers by both friction and plastic deformation can be then determined. Based on the energy dissipations, Zhang et al. [29] found that the FSW for joining of thin plates is much more efficient in energy. Adaptive re-meshing technique can be also used for simulation of FSW process based on solid mechanics. Buffa et al. [35] proposed a finite element model with adaptive re-meshing technique to compare the conical pin with the cylindrical pin in FSW. Zhang et al. [36] proposed a thermo-mechanical model for prediction of the heat generations and the induced macroscopic and microscopic residual stresses evolutions in FSW. The temperatures and the material deformations in FSW are widely investigated by different numerical and analytical models [37, 38, 39, 40, 41].

The determination of heat sources and material deformations in FSW provides the possibility for further investigations on the microstructural evolutions. Microstructural evolution plays the key role for the determination of the friction stir weld quality. The numerical method and the systematic theories for the controlling of microstructural evolutions are essential for the development of the FSW technique. Zener-Hollomon parameter can be used to evaluate the grain sizes in FSW of aluminum alloy [42, 43, 44, 45]. Cellular automata [45, 46] and Monte Carlo models [47, 48] can be used to simulate the recrystallization and the grain growth in FSW. Evolution of the precipitated phases can be further included in Monte Carlo model to investigate the pining effects [49]. Monte Carlo model, as well as cellular automata model, has been verified to be successful for simulation of the formation of equiaxed grains in the stirring zone and the coarsening of the grains in the heat affected zone in FSW of aluminum alloys. But for duplex titanium alloy (α+β), the equiaxed grain can be formed only when the cooling rate is very low [50]. When the cooling rate becomes higher, as revealed in FSW, acicular grain for α phase can be formed. It is necessary to develop new numerical model to accurately describe the mechanism on the microstructural evolutions in FSW of duplex titanium alloy. So, a new Monte Carlo model, which consists of the grain growth model in grain cluster scale and the phase transformation model in one grain scale, is proposed in current work with combination of a successful adaptive re-meshing model to study the microstructural evolutions in FSW of Ti-6Al-4V. The effect of the welding parameters on the grain growths in different welding zones is further investigated.

Finite element model

Adaptive re-meshing model is used to simulate the FSW process, which can provide the necessary strain rates and temperature histories for the further simulation on the microstructural evolutions. The tool shoulder diameter is 10 mm. The pin diameter is 2.5 mm. The rotating speeds are changed from 800 rpm to 1100 rpm. To ensure the success of the model, the translational velocity is selected to be 0.5 mm/s ~ 0.8 mm/s. The welding parameters are summarized in Table 1. According to previous work [51], pre-heating is very important to obtain weld without defects. So, a plunge stage is included for pre-heating of the local region around the welding tool. The plunging speed of the welding tool is 0.4 mm/s. The finite element model of FSW is shown in Figure 1. The initial element sizes are chosen to be 1.03 mm to 2.06 mm. The selection of element size is according to previous work [52]. Adaptive re-meshing technique is used to control the mesh distortions and avoid mesh entanglements. The solutions to the temperatures and the displacements are mapped to the new generated meshes [53].

Figure 1:

Finite element model.

At each step, the thermal and mechanical responses are computed,

where C_T is the heat capacity matrix. K_T is the thermal conductivity matrix. T is temperature. P_T is the heat flux vector. K is the tangent stiffness matrix and ΔU is the incremental displacements.

The transient heat conduction equation is solved by the finite difference method,

where β is the parameter controlling the convergence of time.

Tabulated flow stress is used as functions of strain rates and temperatures,

where σ_s is the flow stress. εˉp is the equivalent plastic strain. εˉ∙p is the equivalent plastic strain rate. The curves for flow stress of Ti-6Al-4V are taken from the software database in Deform-3D and shown in Figure 2.

Figure 2:

Flow stress curves at different strain rates in different temperatures.

The total energy in FSW includes the kinematic energy, internal energy and frictional energy in FSW [24],

where σc is the stress without viscous dissipation effect. σv is the viscous stress. ε∙ is the strain rate. v is the velocity vector. τ is the frictional stress. γ∙ is the slipping velocity. t is time. ρ is density. S is the contact surface. V is the volume in the computational region.

Part of the frictional energy can be dissipated into heat,

where δ is the coefficient and is selected to be 0.9.

Part of the internal energy can be dissipated into heat,

where η is the coefficient and is selected to be 0.9 by default.

Deform-3D is used for the simulation of the FSW process. The obtained temperature histories in different welding zones are used for further simulation on the microstructural evolutions in FSW of Ti-6Al-4V.

Microstructural evolution model

Different to the Monte Carlo model for FSW of microstructural changes in aluminum alloys used in previous works [47, 48], two phases including α and β are considered in this new developed Monte Carlo model in grain cluster scale. In order to simulate the grain structure during the FSW process, a two-dimensional matrix with lattice points N×N is used. Each cell in the simulation consists of 1 to q states and q is selected to be 100. m=8 is the number of surrounding cells. The free energy is determined by the present crystallographic orientation and the neighborhood orientations at the randomly selected point. In this grain growth model, the energy of the system in the Monte-Carlo simulation is described as [47],

where J is the lattice energy density. δqiqj is the Kronecker delta function.

The acceptance of the change of the crystallographic orientation is depending on the following Boltzmann probability equation [47],

where ΔE is the change of grain-boundary energy due to reorientation. k_B is the Boltzmann constant. T is the temperature. m₁ and m₂ represent the numbers of different orientations before and after reorientation. If ΔE<0, the random reorientation of this cell is acceptable. Otherwise, the reorientation conforms to Boltzmann probability.

The relationship between the simulated grain size L and the Monte Carlo step can be obtained as [47],

where l is the cell length in the Monte Carlo model. K₁ and n₁ are model constants. The computed relationship between log₁₀MCS and log₁₀L is shown in Figure 3 which is used to determine the constants K₁ and n₁.

Figure 3:

Relation between MCS and L in SZ and HAZ.

The grain growth phenomena in materials can be described by the following equation [54],

where L₀ is the initial grain size which is selected to be 40 mm according to Ref. [55]. t is the heating and cooling time and Q=251 kJ/mol is the activation energy of Ti-6Al-4V [55, 56]. The combination of eq. (10) and eq. (11) can lead to the relationship between the MCS and the time and temperature,

where k=2.02 and n=2.0 are the proportional constants [54, 55]. R=8.3 1J·K·mol⁻¹ is the gas constant. K₁ and n₁ are the intercept and the slope in the Figure 3.

As a solid state joining technique, no melting occurs in the FSW process. But the temperature is still high enough for possible phase transformation from β to α in the FSW of Ti-6Al-4V. When the material is heated to be higher than 980°C, only β phase exists in the Stirring Zone (SZ). The nucleus of α phase is formed preferentially at β grain boundaries and the phase transformation occurs during the cooling progress. The β grain can grow along <110> crystallographic direction. So, a phase transformation model in one grain scale is newly established for the further investigations on the formation of the acicular α phase in the FSW process.

The phase transformation from β to α needs new definition on nucleation rate. The nucleation rate of α phase on β grain boundaries is given by [57],

where h is the Planck constant. N_v is the number of nucleation site per unit volume. ΔG_m is the activation energy for atomic migration across the interface. ΔGC∗ is the activation barrier for nucleation [57],

where γ_αβ is the free interface energy when the connect surface S_αβ is formed. Δg is the free volume energy of nucleation. S(θ) is the shape factor given by [57]

where θ is the wetting angle.

The optimum embryo shape should minimize the total interfacial free energy in two abutted spherical caps with wetting angle θ [57],

where γββ is the free interface energy when the connect surface S_αβ is destroyed. The nucleation rate in cooling period is varying with temperatures according to Ref. [58].

When the acicular α phase is formed, the migration of the interface separating the two phases (α/β) is depending on the flux of vanadium atoms between α and β phases. The corresponding flux across the α/β interface is [58],

where M is the interface mobility and V_m is the molar volume of the α phase. Δμv is given as [58],

If a steady state exists at the interface, the fluxes must be balanced [58],

where JVβ is the flux of vanadium in the interior of β phase,

where Cβeq is the equilibrium vanadium concentration in the β phase.

Fortran is used to compile the program for the calculations on the microstructural changes in FSW of Ti-6Al-4V. Matlab is used for the further data processing and image plotting.

Results and discussions

In order to validate the microstructural evolution models on the grain growth and the phase transformation, the welding temperature curve [59] shown in Figure 4 is used in the new established models. The grain growth model is used for calculations of the volume fractions of α and β phases in the welding process. The phase transformation model is further used for the simulation on the formations of the acicular α phase. The comparisons with experimental data [59] are shown in Figure 5. The comparisons of the two models and the experimental data can show the validities of the proposed grain growth and phase transformation models. The errors between the phase transformation model and the experimental data range from 1% ~ 5%, which can validate the proposed models. In comparison with the grain growth model, the errors of the phase transformation model are smaller. It can be apparently seen that the decrease of the temperature results in higher volume fraction of α phase and longer acicular α grains in the cooling process.

Figure 4:

Temperature used in experiment [59].

Figure 5:

Variation of volume fraction of β in SZ with time.

After the validation of the proposed grain growth and phase transformation models, the temperature curves calculated from the adaptive re-meshing model of FSW are used.

Adaptive re-meshing model is used to obtain the temperature histories in different welding zones, as shown in Figure 6. The accuracy for the prediction of temperatures in FSW has been validated in previous work [52]. Figures 6 (a) and (b) illustrate the temperature histories in SZ and HAZ respectively in the FSW processes and the subsequent cooling processes in different cases. The welding parameters for all cases are listed in Table 1. The temperature curve in Figure 6 (a) is selected in the middle of the welding line and the location of the point in Figure 6 (b) is 8 mm far away from the welding line. The welding temperature in SZ is much higher than HAZ, which is higher enough for recrystallization, which has been validated and observed in experiments [60, 61]. The welding temperature in HAZ can lead to coarsening of the grains. The increase of rotating speeds and the decrease of the welding speeds can lead to the increase of the maximum welding temperatures in SZ and HAZ, which has been widely observed in both experimental and numerical results [62, 63].

Figure 6:

Temperature histories.

The temperature distributions perpendicular to the welding line in all 3 cases are shown in Figure 7. It shows that the maximum temperature occurs under the tool shoulder. When the maximum temperature reaches over 980°C (e.g. v=0.5 mm/s, ω=1100 rpm), all the phases can be transformed into β phase. When the temperature drops to 600°C, the phase transformation from β to α stops [63]. With combination of the obtained temperature histories in both SZ and HAZ and the new proposed microstructural evolution model, the grain growth and the formation of acicular α phase can be further simulated.

Figure 7:

Temperature distribution during the weld in different cases.

The grain growth in 1100 rpm and 0.5 mm/s in SZ are shown in Figure 8. In current case, the maximum welding temperature is higher than 980°C (as shown in Figures 6 and 7), α and β phases can be transformed into β phase before cooling. The initial grain size in HAZ is considered to be 40 μm [55]. Due to the recrystallization, the grains in SZ are broken into finer and smaller grains which are about 4 μm [47, 49] by the rotation and the translation of the welding tool with mixing effects for the initial state for the grain growth. The embryos for α grains are generated on the boundaries of β grains and becomes bigger in the following air cooling process. Both α and part of β grains become coarsened but the grain growth rate of α grains is faster. The other part of β grains can be replaced by α grains.

Figure 8:

Grain growth of α/β phases in SZ in 1100 rpm and 0.5 mm/s.

Figure 9 shows the grain growth in HAZ in 1100 rpm and 0.5 mm/s. Due to the lower temperatures in HAZ, no phase transformation takes place and only grain coarsening occurs. In comparison with the smaller grains in SZ, the average grain size in HAZ is 138 μm.

Figure 9:

Grain growth of α/β phases in HAZ in 1100 rpm and 0.5 mm/s.

The variations of the average grain sizes in both SZ and HAZ in all the used 3 cases are compared in Figure 10. Figure 10 shows that the grain sizes in SZ and HAZ can be increased with the increase of the rotating speed but decreased with the increase of the transverse speed. When the rotating speed is increased from 800 rpm to 1100 rpm, the grain size in SZ can be increased from 14.5 μm to 17.2 μm and the grain size in HAZ is increased from 129 μm to 138 μm. When the transverse speed is increased from 0.5 mm/s to 0.8 mm/s, the grain size in SZ is decreased from 17.2 μm to 16 μm and the grain size is HAZ is also decreased from 138 μm to 132 μm. With consideration of the temperature curves shown in Figure 6, it can be seen that higher welding temperature can lead to the increase of grain sizes in both SZ and HAZ. This phenomenon has been clearly observed in experimental tests [64], which can further validate the current proposed model. From the temperature curves, it can be also obtained that higher welding temperature in FSW means longer cooling time in air, which can lead to larger grains in both SZ and HAZ.

Figure 10:

Variations of grain sizes with time in different welding conditions.

Figure 11 illustrates the distributions of the average grain size on the cross sections after welding in different welding conditions. Due to the recrystallization in SZ, the grain size in SZ is much smaller than HAZ, which is caused by the stirring effect on the initial grains and the new nuclei generated in SZ. There is no obvious thermal-mechanical affected zone observed in this figure, which can be also confirmed by experiments [65]. The grains near the outer border of the HAZ gradually become similar to the grains in base metal.

Figure 11:

Distribution of average grain size in welding zones.

The morphologies of α and β phases during the cooling progresses in different welding conditions are shown in Figure 12. The α phase (white zone) and β phase in the process of the air cooling are taken at 850°C, 750°C and 600°C. It is obvious that α grains occur on the boundaries of β grains and both of them can be increased in the cooling process in FSW. Due to the higher cooling rate above 600°C, the transformation from β phase to α phase is incomplete in all the studied cases.

Figure 12:

α and β phases during the cooling progresses in different welding conditions.

The volume fractions of α phase in different welding conditions are shown in Figure 13. It is shown that the volume fraction of α phase can reach 81.3% in 1100 rpm and 0.5 mm/s and 71.1% in 1100 rpm and 0.8 mm/s when the temperature is cooled down to 600°C. However, the α phase fraction in 800 rpm and 0.5 mm/s is only 54.0%. This is due to the fact that the maximum welding temperature in this case is the lowest and the cooling rate is the highest above 600°C. The volume fraction of α phase is significantly determined by both welding temperature and cooling rate. With the increase of the rotational speed, the volume friction of α phase gets increased in SZ. With the decrease of the translational speed, the volume fraction of α phase also gets increased. This phenomenon observed in this numerical model is fitted well with the experimental observations [66].

Figure 13:

Predicted volume fractions of α in SZ.

In SZ, a 300×150 lattice system (20 μm×10 μm) is used to model the formation of acicular α phase inside one grain in different temperatures (750°C and 600°C), as shown in Figures 14 and 15. The variations of the average length of acicular α phase with welding temperatures in different welding conditions are further summarized in Figure 16. The length of acicular α grain is strongly dependent on the maximum welding temperature and the corresponding cooling rate during the FSW process. Actually, higher cooling rate can limit the time for growth of α grain. This is the reason that the length of acicular α grain can be decreased with the increase of the transverse speed. For the case in 800 rpm and 0.5 mm/s, the welding temperature is the lowest and the cooling rate is the highest. This is the reason for the formation of the smallest α grain in this case. With the decrease of welding temperature in the FSW process, the average length of acicular α grain is gradually increased.

Figure 14:

Formation of acicular α phase morphology in 750°C in different welding conditions.

Figure 15:

Formation of acicular α phase morphology in 600°C in different welding conditions.

Figure 16:

The average length of acicular α grains in different welding conditions.

Conclusions

A Monte Carlo model for grain growth in FSW of duplex Ti alloy is newly proposed. A phase transformation model is further established to simulate the formation of the acicular α grain. Comparisons with the experimental data and phenomena show the validity of the proposed models. The main observations and results are summarized as follows,

1. The volume fraction of α phase can be predicted by both the proposed grain growth model in grain cluster scale and the phase transformation model in one grain scale. Comparison with experimental data shows that the results from the phase transformation model in one grain scale is more accurate.

2. The grain growth and the length of the acicular α grain are dependent on both the maximum welding temperature and the corresponding cooling rate. Lower rotational speed or higher transverse speed can lead to lower welding temperature, which can lead to smaller grain sizes in SZ.

3. Due to the recrystallization in SZ, the grain size in SZ is much smaller than HAZ, which is caused by the stirring effect on the initial grains and the new nuclei generated in SZ.

4. Due to the higher cooling rate above 600°C, the transformation from β phase to α phase is incomplete in FSW of Ti-6Al-4V.