Introduction

As an important solid-state joining technique, friction stir welding (FSW) has been widely used in industries [1, 2]. In FSW, the welding tool is inserted into the welding line. A tight joint can be then formed with the rotation and transverse movement of the welding tool. Both the heat generations and the material flows can affect the microstructural evolutions in the welding zone [3, 4]. According to the different microstructures, the welding zone can be divided into different regions: the stirring zone (SZ), the thermo-mechanically affected zone (TMAZ), and the heat-affected zone (HAZ) [5, 6]. The changes of the microstructures in the welding zone can obviously affect the mechanical properties of friction stir-welded joints, i.e., the hardness [7–9], the tensile strength [10], the fatigue properties [11, 12], the corrosion resistance [13, 14], etc. This is the reason for the development of the numerous numerical models on the microstructural evolutions. Polycrystal plasticity and CFD model were used by Cho et al. [15] for the prediction of the texture evolutions in the welding zone. A smoothed particle hydrodynamics model was established by Pan et al. [16] for the simulation of grain sizes, textures, and microhardness in FSW of AZ31. The cellular automata finite element (CAFE) model was used by Saluja et al. [17] to predict the grain size distribution during FSW and the influence of weld defects on the forming of FSW sheets. Arora et al. [18] simulated the temperature distribution and estimated the grain size during FSW by using finite difference equations. A chain of integrated models were developed by Simar et al. [19] to study various properties related to microstructure evolution during FSW. The hardness profile and the evolution of relevant strengthening precipitates across the welds were studied by Hersent et al. [20] in FSW of 2024-T3 aluminum alloy. Buffa et al. [21] investigated the micro- and macro-mechanical characteristics of friction stir-welded Ti-6Al-4V lap joints numerically and experimentally.

Monte Carlo (MC) method is valid and efficient for the prediction of grain growth and topological features. The application aspects of MC method include the fields of welding [22, 23], abnormal and anisotropic grain growth [24, 25], grapheme growth [26], polycrystalline microstructures [27], etc. Details of the MC technique can be found in the literature [25, 28, 29].

Grain evolution in welding zones is one of the key factors for the determination of the sizes of the HAZ and SZ, which is very important for the controlling of the welding quality and for the investigation on the FSW mechanism. Detailed grain growth process in FSW has seldom been reported. So, MC method with consideration of nucleation rate in each MC step is implemented for the simulation of the grain growth in FSW of AA6082-T6 in the present work. The influences of the rotation speeds and the shoulder diameters on the microstructural changes are studied in detail. The regions of the welding zones are then determined from the predicted grains.

Model description

Finite element model

The dimensions of the welded plates are 80 × 50 × 3 mm. The plates are initially meshed into 6265 nodes and 25152 elements in the commercial FEA code DEFORM-3DTM, as shown in Fig. 1. In accordance with the simulation carried out in the previous work [30], the initial mesh size is taken as 0.75 mm in the regions in and near the welding zone, and the mesh sizes vary from 1.5 to 3 mm in the other regions. The physical and mechanical properties of the welding material AA6082-T6 are considered as the functions of temperatures [31]. Viscoplastic constitutive model is used for the simulation of the material behaviors of the welding material [30],

$$ \dot{\bar{\varepsilon }} = A_{0} \left[ {\sinh \left( {\alpha_{0} \bar{\sigma }} \right)} \right]^{{n_{0} }} e^{{\left( { - Q/RT_{\text{abs}} } \right)}} , $$
(1)

where \( T_{\text{abs}} \) is the absolute temperature; Q is the activation energy; R is the Universal Gas Constant; n 0, α 0, and A 0 are equation coefficients, which are derived from the experimental tests [32]; and n 0, α 0, and A 0 are taken as 6.49, 0.0238, and 8.0 × 109, respectively. \( \dot{\bar{\varepsilon }} \) is the equivalent strain rate. \( \bar{\sigma } \) is the flow stress.

Fig. 1
figure 1

FEM model of case 1

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The welding tool is simplified as a rigid body. Shear model is used for the definition of the contact behaviors on the tool-plate surface,

$$ f = m_{0} \cdot k, $$
(2)

where f is the frictional stress, k is the yield shear stress, and m 0 is the frictional factor and is taken as 0.3 to make sure that the model can match the measured tool forces and temperatures [33]. This value has been validated in the previous work [30, 34]. A limitation on the friction stress by shear failure can be valid especially for the cases in higher rotating speeds [35].

The influences of the different rotation speeds and shoulder diameters on the microstructures are then investigated in the welding zones. The welding parameters of the simulated cases are shown in Table 1. The adaptive remeshing finite element formulations are implemented in the computation process to solve the problems of mesh entanglements and distortions caused by the large deformation and the material flow in FSW. The details of the remeshing technique can be found in Ref. [30].

Table 1 Welding parameters of the simulated cases
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Air cooling condition is used after FSW to obtain the complete temperature histories which are used for the calculations on grain growth in the MC model. The boundaries of the welding plate are set to be at room temperature, and the convective heat transfer coefficient is set to be 30 \( {\text{W/(m}}^{ 2} {\text{K)}} . \)

Monte Carlo model

In the MC model, a two-dimensional matrix with lattice points N × N is used. The values on grid points represent the grain crystallographic orientations (1 ~ q). Adjacent grid points with the same grain crystallographic orientations are regarded as a grain. The free energy is determined by the present crystallographic orientation and neighborhood orientations at the randomly selected point. At each lattice point site, the grain-boundary energy is calculated by the Hamiltonian [22]:

$$ E = - J\sum\limits_{i = 1}^{m} {\left( {\delta_{{q_{i} q_{j} }} - 1} \right)} , $$
(3)

where J is a positive constant which represents the grain-boundary energy, m is the total number of the sites near the lattice point, δ is the Kronecker symbol, and q i is the orientation of the lattice point near the calculated grid site.

The grain growth is activated by the grain-boundary migration kinetics. For the randomly selected grain site, the change of the energy is determined by the change of the present crystallographic orientation to its neighborhood,

$$ p = \left\{ {\begin{array}{*{20}c} {1,} & {\Delta E \le 0} \\ {e^{{ - \frac{\Delta E}{{k_{B} T}}}} = e^{{ - \frac{{(m_{2} - m_{1} )J}}{{k_{B} T}}}} ,} & {\Delta E > 0} \\ \end{array} } \right., $$
(4)

where ΔE is the change of grain-boundary energy due to reorientation, k B is the Boltzmann constant, T is the temperature, and m 1 and m 2 are the number of different orientations before and after the reorientation. If \( \Delta E \le 0 \), the new crystallographic orientation of the randomly selected grain site is accepted; else it is accepted with the Boltzmann probability [23, 36].

In one MC step, the algorithm described above is iterated for N 2 times. Through the MC simulation, the microstructure is represented by the orientation matrix. For the equiaxed grains, the average grain size is measured by the line intercept counting method. For microstructure with unequiaxed grains, the average grain size is computed from the average grain area:

$$ L = \sqrt {\frac{S}{{n_{\text{g}} }}} , $$
(5)

where S is the simulated area and n g is the number of grains in the simulated area. An empirically fitted relation between the simulated grain size and the number of MC steps can be obtained as [23]:

$$ L = K_{1} \lambda (MCS)^{{n_{1} }} , $$
(6)

where λ is the discrete grid spacing of the utilized grid model, and K 1 and n 1 are model constants which can be obtained from the simulated data by regression analysis.

In order to simulate the grain growth process in FSW, the relation between Monte Carlo steps and real time–temperature cycles of the welded material should be established. In the current work, the changing rate of the average grain size is assumed to be the function of the migration velocity of grain boundaries (v):

$$ \frac{dL}{dt} = \alpha v^{n} , $$
(7)

where α and n are ratio constants, α is taken as 3.0 in HAZ and 1.0 in SZ, and n is taken as 0.1. The equation constants are used to make sure that the grain growth kinetics agree well with the real grain growth. The values used during the calculations are determined by trial and error method. v is defined as [28]:

$$ v = \frac{{AZV_{m}^{2} }}{{N_{\text{a}}^{2} h}}\exp \left( {\frac{{\Delta S_{f} }}{R}} \right)\exp \left( { - \frac{Q}{RT}} \right)\left( {\frac{2\gamma }{L}} \right), $$
(8)

where A, Z, V m , R, N a, and h are physical constants. ΔS f and γ are material parameters of AA6082-T6. The material parameters and the ratio constants of AA6082-T6 used in Eqs. (7)–(8) are summarized in Table 2 [37–39]. From Eqs. (6)–(8), the relation between real time and MC steps can be obtained as the following discrete form:

$$ (MCS)^{{(n + 1)n_{1} }} = \left( {\frac{{L_{0} }}{{K_{1} \lambda }}} \right)^{n + 1} + \frac{{(n + 1)\alpha C_{1}^{n} }}{{(K_{1} \lambda )^{n + 1} }}\sum {\left[ {\exp^{n} \left( { - \frac{Q}{{RT_{i} }}} \right)t_{i} } \right],} $$
(9)

where \( C_{1} = \frac{{2A\gamma ZV_{m}^{2} }}{{N_{\text{a}}^{2} h}}\exp (\frac{{\Delta S_{\text{a}} }}{R}) \), L 0 is initial average grain size, T i is the average temperature in every time interval, and t i represents the value of time interval.

Table 2 Material parameters and ratio constants [37–39]
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In SZ, a 300 × 300 lattices system with randomly initial orientations is used to model the area in SZ of 500 × 500 μm2. Materials in SZ undergo both high temperature and large plastic deformation. So, dynamic recrystallization (DRX) phenomenon should be considered in the simulation of grain growth. In this model, DRX procedure is simulated by implementing a nucleation rate in every MC step. The nucleation rate is considered as a function of both temperature and strain rate according to [40]:

$$ \dot{n} = N_{0} \dot{\bar{\varepsilon }}\exp \;\left( { - \frac{Q}{RT}} \right), $$
(10)

where N 0 is a constant and is taken as 1024 (1/s/m3), \( \dot{\bar{\varepsilon }} \) is the equivalent strain rate in SZ, which is assumed to be uniform in SZ and can be estimated by the following [41],

$$ \mathop {\overline{\varepsilon } }\limits^{ \cdot } = \frac{{w\pi r_{\text{e}} }}{{L_{\text{e}} }}, $$
(11)

where r e and L e are the effective (or average) radius and depth of the dynamically recrystallized zone (SZ), respectively. r e here is represented by 0.78 of the radius of SZ, and L e is assumed to be the length of the pin. Number of nuclei in every MC step is calculated from the nucleation rate, and lattice points at grain boundary are picked to change to random orientations to create new nuclei. From Eq. (10), it can be seen that the nucleation rate is not a constant during the simulation. Higher temperature and equivalent strain rate can lead to higher nucleation rate.

Experimental results reported by Sato [42] are used to validate the simulation method of DRX in SZ. The simulated area is 500 × 500 μm2, and the grid system was 300 × 300 in cases 1–3. The thermal cycles used are the same to the experimental measured values, and equivalent strain rates are calculated according to Eq. (11). Two working conditions with rotation speeds of 1600 rpm and 2450 rpm are modeled. The computed relationship between Log10 MCS and Log10 L is shown in Fig. 2, which are used to determine the constants K 1 and n 1. MC steps calculated from the two thermal cycles [42] are 119 and 216, respectively. Simulated average grain sizes agree well with the measured results, which can validate the simulation method used to model the DRX in SZ, as shown in Table 3. For further validation, friction stir-welded AA6082-T6 plate is modeled with the same welding conditions as the experimental conditions [43]. Average grain size in SZ is calculated by the MC method. The result is compared to the experimental observations [43] as summarized in Table 4. The error between the current numerical model and the experimental value is 6.2 %, suggesting that the present computational procedure can be used to predict grain-microstructure evolution during the FSW process with reasonable accuracy.

Fig. 2
figure 2

Growth kinetics in SZ

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Table 3 Validation of DRX simulation in SZ
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Table 4 Validation of predicted grain size
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The initial grain size in HAZ is selected to be 80 μm according to experimental observation [43], and a 150 × 150 lattices system is used to model the area in HAZ of 1000 × 1000 μm2. Randomly distributed Voronoi elements are used to model the initial distribution of the grains in HAZ. Material particles in TMAZ are affected by both mechanical elongation and temperature rises. For the determination of the stretch in TMAZ, two series of material particles on top and bottom surface of the plate at the cross section are traced. The initial distance between the two particles is 0.5 mm, as shown in Fig. 3. The flow behaviors of the traced material particles are used to distinguish the welding zones and to calculate the elongation of grains in TMAZ. A randomly generated configuration of grains with the same average elongation ratio calculated from traced particles is used to simulate the initial grain microstructure in TMAZ, and the lattice system used in TMAZ is the same of HAZ.

Fig. 3
figure 3

Traced material particles

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Results and discussion

Temperature distributions and materials flow

The calculated temperature fields of three cases are shown in Fig. 4. In the simulated cases, the temperature fields are symmetrical to the welding seam. The peak temperatures appear at the contacting surfaces of welding tools and the welding plates. In case 1, the peak temperature is 676 K. Both the increases in rotation speeds and shoulder diameters can lead to increased heat generation. When the rotation speed increased from 1000 to 1500 rpm in case 2, the peak temperature reaches 778 K. When the shoulder diameter increases from 10 to 13 mm, the peak temperature reaches 788 K in case 3. Peak temperature histories of the traced material particles in different cases are shown in Fig. 5.

Fig. 4
figure 4

Temperature distributions on welding plates. a Case 1, b case 2, c case 3

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Fig. 5
figure 5

Temperatures of traced particles along welding seam

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In order to validate the calculated temperature fields, numerical results in case 1 and the experimental observations reported by Buffa [44] are compared. The experimental conditions are the same with case 1. The comparison of the peak temperature profile on the cross section is shown in Fig. 6. As seen in this figure, the calculated temperature profile matches well with both the numerical and experimental results in Ref. [44].

Fig. 6
figure 6

Validation of peak temperatures in case 1

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Ten material particles in the middle layer of the work piece with the distance intercept of 0.5 mm are traced in the three cases to study the detail of material flows. Figure 7 shows the movements of particles when the welding tool reaches the particles. In case 1, as shown in Fig. 7a, P1, P2, and P3 at the advancing side are stirred and finally moved back to the advancing side. This flow phenomenon indicates a sufficient material flow, and the defects are not likely to happen at the advancing side by this condition [45]. P4 and P5 do not rotate with the tool pin, but experience considerable amounts of deformation. So, P4 and P5 are most likely to belong to the TMAZ. At the retreating side, particles have not been stirred to rotate around the pin. P6 moves to nugget zone, although the rest of particles still remain at the retreating side and belong to TMAZ. It indicates that the welding joint is formed by the materials from the retreating side. These results agree well with the studies of material flows reported by Marzbanrad et al. [46] and Shojaeefard et al. [47].

Fig. 7
figure 7figure 7

Material flows in different cases. a Case 1, b case 2, c case 3

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In case 2 and case 3, as shown in Fig. 7b and c, the ranges of materials particles affected by welding tool are obviously increased with the increase of rotation speed and shoulder diameter. When the rotation speed increases to 1500 r/min in case 2, material particles P1, P3, and P4 are adhered to the stir pin and transverse with the pin for some distance. It indicates that at higher rotation speed, the materials flow more severely and experience larger deformations.

Figure 8 shows the traced material particles on top and bottom surfaces in the three cases after welding. The region of SZ can be determined by the material flows. The width of SZ in each case is summarized in Table 5. Both the increases of rotation speed and the shoulder diameter can lead to the increase of the width of SZ. The grains in HAZ are only affected by heat generation. The movements of the traced material particles are not affected by the stirring effect from the welding tool. This is the criterion for the determination of inner border of HAZ. The outer border of the HAZ is determined by the grain growth process. Between the SZ and the HAZ, the TMAZ exists as a transient region. Certainly, in some cases, TMAZ is ambiguous and hard to determine [2].

Fig. 8
figure 8figure 8

Traced material particles. a Case 1, b case 2, c case 3

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Table 5 Width of SZ (mm)
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TMAZ is quite narrow and can be only distinguished by several particles which are close to the inner border of HAZ. In case 1 shown in Fig. 8a, for example, TMAZ at RS is represented by point P and point Q. The elongated ratio is computed from the distance elongated by the welding tool. The elongated ratios of the two points are 276 and 335 %, respectively. The width of SZ at the bottom surface is narrower than that on top surface. The larger welding tool can obviously increase the width of SZ, as shown in Fig. 8b, c and Table 5.

Simulation of grain growth

Figure 9 shows the simulated grain growth kinetics of the 2D grid systems in different welding zones of the three cases. The values of grain growth exponents in HAZ and TMAZ agree well with 0.41, reported by Anderson et al. [48] and Grest et al. [49] in similar models. The grain growth exponents in SZ are smaller than that in HAZ and TMAZ, and this phenomenon is due to the refinement effects of DRX nuclei. Different thermal cycles and equivalent strain rates lead to different nucleation rates in every MC step, and this causes variations in the growth exponents.

Fig. 9
figure 9

Growth kinetics of the three cases

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The thermal cycles of the traced material particles together with the annealing times are shown in Fig. 10. At t = 0 s, the temperature starts to raise up and reaches the peak temperature at about t = 50 s for case 1. Material particles with higher peak temperatures need more time to cool down to room temperature. Based on the thermal cycles, MC steps of the positions at simulated cross sections can be computed according to Eq. (9).

Fig. 10
figure 10

Temperature histories of traced material particles

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As described before in the model description part, DRX phenomenon in SZ is affected by the temperature and equivalent strain rate. Figure 11 illustrates the DRX microstructure evolutions in the three cases, 1 mm from welding seam on RS. The nucleation process happen mainly during the high temperature periods, and at the iteration steps in cooling periods with low temperatures, there are nearly no new nuclei to form. In case 1, temperature and equivalent strain rate keep on a low level and the nucleation rate is the smallest, the highest nucleation rate is only 6133 1/mcs at mcs = 28. With higher rotation speed and shoulder diameter, the nucleation rates are much higher, reaching 24051 and 19085 1/mcs in case 2 and case 3, respectively. Figure 11 also shows the microstructure in SZ at the highest nucleation rate points. From the grain-microstructure graphs, the DRX phenomenon can be easily recognized by the small nuclei on the grain boundaries. The density of nucleated cores indicates the nucleation rate in different cases. The nuclei formed from DRX phenomenon also grow during the grain growth simulation, and this is the main effect of grain refinement caused by DRX.

Fig. 11
figure 11

DRX microstructure evolution in SZ. a Case 1, b case 2, c case 3

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Microstructures of grain growth in 3 cases on top surface are shown as examples in Figs. 12, 13, and 14. Compared to TMAZ, SZ and HAZ are wider and more important in determining the quality of welding joints, thus only SZ and HAZ are investigated in case 2 and case 3 to study the influence of shoulder diameters and rotation speeds.

Fig. 12
figure 12

Microstructure of grain growth in case 1. a SZ on top surface, 1 mm from welding seam, RS; b TMAZ on top surface, 4.5 mm from welding seam, RS; c HAZ on top surface, 5 mm from welding seam, RS

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Fig. 13
figure 13

Microstructure of grain growth in case 2. a SZ on top surface, 1 mm from welding seam, RS; b HAZ on top surface, 6 mm from welding seam, RS

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Fig. 14
figure 14

Microstructure of grain growth in case 3. a SZ on top surface, 1 mm from welding seam, RS; b HAZ on top surface, 7 mm from welding seam, RS

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In SZ, the materials undergo severe plastic flow and broke into fine equiaxed small grains early during the stirring process. The average grain size at the initial time (MCS = 0) was 5.4 μm in the three cases. Microstructure evolved with the progress of the MC simulation, the average grain size increased visibly as shown in Figs. 12a, 13a, and 14a. Higher rotation speed and larger shoulder led to the increase in temperatures, and also influenced the average grain size in different cases. DRX phenomenon happened during the early periods when the temperature was high, and the DRX phenomenon led to a microstructure consisting of a mixture of large grains and small nuclei, as shown in Fig. 11. But in the cooling periods, the nucleation level decreased quickly. DRX-formed nuclei and normal grains were both simulated to grow according to the same algorithms. The final average grain sizes in SZ of case 1, case 2, and case 3 were 17.6, 18.9, and 20.8 μm, respectively. The average grain sizes can be increased due to the increase of the shoulder diameters and the rotation speeds. This is in accordance with the work reported by Shojaeefard et al. [50]. Similar to the grains in SZ, grains in HAZ were equiaxed grains and their size increased during the post-weld cooling period.

The original grain microstructure of the TMAZ, a narrow zone between the SZ and HAZ, is represented by randomly elongated grains and is shown in Fig. 12b. The evolution of the TMAZ grain microstructure was completed in 90 MC steps, and the average grain size increased from 60.1 to 77.5 μm. The final TMAZ microstructure contained unequiaxed grains, a finding which is consistent with the experimental results of Liu et al. [51].

Figure 15 illustrates the average grain sizes at the simulated cross sections after welding. The FSW-joint is clearly composed of distinct weld zones according to the average grain sizes, as summarized in Table 6. Relationship between the rotation speed and the HAZ width in current work is similar to the optical microscopic investigations reported by Shojaeefard et al. [52]. In the SZ, the average grain sizes are almost uniform in the three cases. The grains on top surface are slightly larger than those on the bottom surface; similar results were reported by Chang et al. [41]. Due to the friction between the shoulder and the workpieces, the SZ is wider on the top surface than the bottom surface. The largest grains appear at the inner border of the HAZ. With increasing distance from the welding seam, the grains in the HAZ become smaller and finally transit to grains in base metal at the outer border of the HAZ. Compared to case 1, higher rotation speed and larger shoulder can clearly increase the width of welding zones.

Fig. 15
figure 15

Distribution of average grain size and welding zones. a Case 1; b case 2; c case 3

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Table 6 Width of welding zone (mm)
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Conclusions

  1. 1.

    Monte Carlo method is employed to simulate the grain growth process in FSW of AA6082-T6.

  2. 2.

    The predicted grain size distributions are symmetrical about the welding seam, and the grain sizes on the top surface of the weld are larger than that those on the bottom surface.

  3. 3.

    Increasing the rotation speed and shoulder diameter can lead to larger grains. Higher rotation speed and larger shoulder can also increase the width of welding zones.

  4. 4.

    It has been demonstrated that the width of the SZ can be estimated by monitoring the motion of traced material particles. The SZ is wider on the top surface of the weld than the bottom surface.

  5. 5.

    The inner border of the HAZ can be determined by material behavior and the outer border can be distinguished by the predicted grain size distribution after welding.