A Modified Analytical Heat Source Model for Numerical Simulation of Temperature Field in Friction Stir Welding

In the conventional analytical model used for heat generation in friction stir welding (FSW), the heat generated at the pin/ workpiece interface is assumed to distribute uniformly in the pin volume, and the heat flux is applied as volume heat. Besides, the tilt angle of the tool is assumed to be zero for simplicity. -ese assumptions bring about simulating deviation to some extent. To better understand the physical nature of heat generation, a modified analytical model, in which the nonuniform volumetric heat flux and the tilt angle of the tool were considered, was developed. Two analytical models are then implemented in the FEM software to analyze the temperature fields in the plunge and traverse stage during FSW of AA6005A-T6 aluminum hollow extrusions. -e temperature distributions including the maximum temperature and heating rate between the two models are different. -e thermal cycles in different zones further revealed that the peak temperature and temperature gradient are very different in the high-temperature region. Comparison shows that the modified analytical model is accurate enough for predicting the thermal cycles and peak temperatures, and the corresponding simulating precision is higher than that of the conventional analytical model.


Introduction
During friction stir welding (FSW), the heat derived from the friction and plastic deformation is the key factor that dictates the final weld quality [1][2][3]. It generally provides two primary functions [4,5]: (i) making the metal material enough soft for the tool to stir to form the solid-state weld and (ii) resulting in the microstructural evolution to influence the joint properties. A precise heat measurement can give the specific peak temperature and thermal cycle to evaluate the weld quality, while large temperature gradient and intense plastic deformation in the stirred zone make it very difficult. Moreover, only limited temperature data can be obtained via experimental measurement [6]. As a contrast, the numerical models can be very efficient and convenient to give global and detailed thermal information [7].
Since the late 1990s, two main kinds of numerical models including computational fluid dynamic (CFD) model and computational solid mechanics (CSM) model have been developed. e CFD models can describe the thermomechanical interaction between the tool and workpiece, and the most important consideration in the frictional boundary condition is the determination of the contact state [8]. In many CFD models, the contact conditions at the tool/workpiece interface are considered as full sticking, which usually leads to an overestimation of peak temperature [8][9][10]. Atharifar et al. [11] assumed a sliding contact state during FSW, in which the velocity of material at the tool/workpiece interface was artificially defined as 60% of the tool velocity to investigate the loads carried by the tool. Chen et al. [12] presented a new frictional boundary condition based on an implementation of the Coulomb friction model, and the model yielded a nonuniform distribution of contact state over the interface. e sticking state was present over a large area at the tool-workpiece interface, while significant interfacial sliding occurs at the shoulder periphery, the lower part of the pin side, and the periphery of the pin bottom. Moreover, the CFD models are based on the Eulerian method irrespective of time, and it is hard to simulate the temperature field and material flow varying with the time [8].
e CSM models can overcome the above-mentioned shortcomings, and the Lagrangian and the Arbitrary Lagrangian-Eulerian (ALE) formulations are mainly adopted. In most models using the ALE formulations, the grids are remeshed in the simulating process, and the tool/workpiece interaction including friction and plastic deformation can be defined to reflect the welding process to some extent. But excessive remeshing operations in the simulating process will employ extremely large amounts of computational time [13,14]. Feulvarch et al. [15] and Dialami et al. [16] used the new moving mesh strategies to avoid remeshing. In the strategies, the mesh is partitioned into 3 parts: a first one which is fixed around the stirring zone, a second one which includes the base material in contact with the tool, and a third one (the tool). e coupling between each part can be performed using a node-to-node link approach, and only the connection between the moving and the fixed parts changes at each time step. erefore, there are no mesh distortions and the formalism leads to satisfying computing time. e models adopting the Lagrangian methodology do not need the remeshing operations and demonstrate enormous advantages in computational costs and efficiency, especially for the cases in which only the thermal information is concerned [17,18]. In all thermal models, one of the main tasks is to reveal the physical nature of FSW and give the appropriate analytical model for heat generation.
In the early stage, Chao and Qi [19] thought that the heat source generated by the friction at the shoulder/ workpiece interface was the major one and assumed that the downward pressure from the tool distributed uniformly at the interface. e authors built the relationship between the surface heat flux and the pressure based on the classical Coulomb's friction law. Song and Kovacevic [20] assumed that the heat at the tool/workpiece interface was only caused by friction, and the friction coefficient was constant. e heat from the tool pin was taken into account and divided into three parts: (1) heat generated by shearing of the material; (2) heat generated by the friction on the threaded surface of the pin; and (3) heat generated by the friction on the vertical surface of the pin. Zhang et al. [17] proposed an analytical model for heat generation during the plunging stage, and the heat flux generated by the friction at the pin/workpiece interface is applied as volume heat. Khandkar et al. [21] assumed a uniform shear stress at the tool/workpiece interface and built a correlation between the moving heat flux and the actual machine power input. Schmidt et al. [22] established an analytical model for heat generation based on the modified Coulomb's friction law.
ree contact conditions at the tool/workpiece interface including sliding, sticking, and partial sliding/sticking are defined to describe the respective mechanisms of heat generation. e experimental results demonstrated that the sticking or close to sticking condition was present at the tool/ workpiece interface.
However, some common problems remain unsolved in the above-mentioned analytical model. First, the volumetric heat flux was considered to distribute uniformly in the pin volume, and it was discrepant with the real FSW process. Second, only a few authors revealed the temperature field in the plunge stage and always set the tilt angle of the FSW tool to be zero for simplicity, which brings about some simulating deviation. e analytical model, in which the volumetric heat flux distributes uniformly in the pin volume and the tilt angle is neglected, is called the conventional analytical model (CAM). A modified analytical model (MAM) in the current research was developed to consider the nonuniform volumetric heat flux and the tilt angle. Two analytical models are adopted in this paper to describe the specific heat generation, and the temperature fields including the temperature distribution and thermal cycle were compared to identify the simulating accuracy.
is work can help the investigation on the physical nature of heat generation and can contribute to the numerical studies on FSW. In order to justify the analytical models, the AA6005A-T6 aluminum hollow extrusions were friction stir welded, and the comparisons between measured and numerical results in the CAM and MAM were analyzed in this work.

Experimental Details.
e AA6005A-T6 aluminum hollow extrusions with a butting depth of 4 mm were friction stir welded. e tool was composed of a shoulder with the scrolled feature and a right-hand conical screwed pin, as shown in Figure 1. e tool shoulder radius R 0 was 10 mm, and an Archimedes scrolled groove was made on the flat shoulder. e root radius R 1 and length H of the pin are 4 mm and 3.8 mm, respectively, and the right-hand threads and three flutes were machined on the pin side to promote the material flow. e welding experiments were conducted on the ESAB SuperStir ™ FSW machine, as shown in Figure 2. e process parameters in different stages were listed in Table 1. e transient temperatures at 5 locations were recorded using K-type thermocouples embedded into the predrilled holes, and the locations of thermocouples (or holes) were shown in Figure 3. e depth and diameter of each hole were 2.0 mm and 1.0 mm, respectively.

Numerical Details.
It should be noted that the analytical model only expounds the mechanism of heat generation, and it must be implemented in numerical software in order to analyze the temperature field. In this paper, the commercial FEA software MSC. MARC, which is a Lagrangian implicit code, is used to model the FSW. Two hollow extrusions are meshed due to the structural asymmetry, and the detailed mesh generation is shown in Figure 4. In order to balance the computational accuracy and efficiency, the dimensions of meshes in A zone including the weld zone are 1 mm × 1 mm × 1 mm, and the meshes in B (B1 and B2) and C (C1 and C2) zones will gradually become coarser along the transverse direction (TD) and normal direction (ND) except the welding direction (WD). e finite thermal model consists of 212, 700 elements and 266, 686 nodes.
As a three-dimension quasisteady heat transfer process, the governing equation for thermal conduction in Cartesian coordinates in FSW can be written as follows: where k(T) is the thermal conductivity, q(x, y, z, t) is the inner heat generation rate per unit volume, ρ(T) is the mass density, and c(T) is the specific heat. e convection boundary condition for all the surfaces of the workpiece is as follows [23]: where n is the normal direction vector of the boundary Γ, h is the heat convection coefficient, and T 0 is the ambient temperature. e heat convection from the workpiece to the backing plate or clamps is complex and uncertain due to the contact gap conductance [21]. erefore, the steel backing plate and clamps are removed in this model, and the corresponding contact surfaces are assumed to be cooled by convection from a cool fluid owning a heat convection coefficient of 500 W·m − 2 ·°C − 1 [24,25]. On the other surfaces, a heat convection coefficient of 30 W·m − 2 ·°C − 1 is used to define the natural convection between the workpiece and air [26,27]. e initial boundary condition describing the detailed value at a specific time can be expressed as follows: e value of thermal-physical parameters at different temperatures are taken from [28,29], and the relationship  Advances in Materials Science and Engineering 3 between each parameter and temperature can be deduced as follows: where σ(T) is the yield strength. Plots of the thermophysical parameters given by equations (4)∼(7) are shown in Figure 5.

Heat Generation
e heat generation from the friction and plastic deformation is governed by the contact condition between the tool and the workpiece, and a contact state variable δ is used to describe the different conditions: where ω ′ and ω are the angular velocity of the matrix and the tool. δ � 0 (full sliding) or δ � 1 (full sticking) is not likely to occur at the tool/workpiece interface, and the actual value of δ is in the range of 0-1 (partial sliding/sticking) [30]. e analytical expression used for heat generation is established in the form of heat flux, and the total heat flux can be expressed as follows [31]: where r is the distance from the heat source center, and τ yield (T) is the shear yield stress, which is equal to σ(T)/ � 3 √ according to the von Mises yield criterion.

Plunge Stage in the CAM.
e plunge stage is subdivided into two stages: it is called the pin-plunging stage before the shoulder contacts with the workpiece, and the rest is called the shoulder-plunging stage. In the CAM, the volumetric heat flux (yellow zone) is assumed to distribute uniformly in the pin volume [32,33], and the concrete distribution characteristics in the pin-plunging stage at different steps are illustrated in Figure 6. e maximum sectional radius r s and the effective pin volume V ep can be calculated as follows:

Advances in Materials Science and Engineering
where R 5 and α are the tip radius and semicone angle of the pin, respectively, and h is the transient plunge depth. Figure 7 shows how to get an infinitesimal segment area on the pin surface, and the heat from the pin side Q ps and pin tip Q pt can be derived as follows: where σ is the nodal yield strength. en the volumetric heat flux q p can be expressed as follows: It is well known that the heat generated from the shoulder plays a major role in the total heat generation, and the interaction between the shoulder and workpiece can significantly influence the temperature field in the plunge stage. In the CAM, the tilt angle is always considered to be zero for simplicity [34,35], and the shoulder immediately contacts with the workpiece entirely once the pin reaches the target depth, which is inconsistent with the real FSW process. e contact conditions between the shoulder and workpiece in the shoulderplunging stage at different steps are exhibited in Figure 8. e surface heat flux from the shoulder q s can be described by equation (9) where R 1 is the bottom radius of the pin, and R 0 is the radius of the shoulder. It should be noted that part of the heat at the shoulder/workpiece interface will "flow" into the tool, and the ratio of the heat "flow" into the workpiece can be calculated as follows [36]: In FSW of the 6XXX aluminum alloys, the temperature at the shoulder/workpiece interface is approximately in the range of 400-500°C [37,38]. e thermal-physical parameters of the workpiece and the tool at 450°C are substituted to equation (14) and a value of 0.62 can be obtained. e surface heat flux from the shoulder can be expressed as follows:

Traverse Stage in the CAM.
As the tool moves forward, a micropore will occur and then is filled with the plasticized material in a short time, and the friction stir welding can be regarded as micropores generation and disappearing [39]. erefore, only about half of the pin works to generate the heat [40], and the volumetric heat flux q p can be given by equation (12): where H is the pin length. e surface heat flux is calculated by using equation (15).

Plunge Stage in the MAM.
It can be seen from equation (12) that the volumetric heat flux in the CAM depends on only the yield strength and transient plunge depth, and the values of all nodes are calculated using the same equation. However, the heat flux of each node varies with the distance from the heat source center (see equation (9)), and the hypothesis of uniform volumetric heat flux will bring an error about the prediction of the nodal temperature. So, a modified analytical model is developed in the current research to reduce the error. Specifically, the volumetric heat flux (color zone) is segmented into several parts in the pinplunging stage at different steps (i � 1 − 4), as depicted in Figure 9. e bottom radius of the j layer r j can be given as follows:    Advances in Materials Science and Engineering from the pin side at the i step Q psj can be calculated via equation (9): where σ ij is the yield strength from the j layer at the i step. e tip radius of the pin is 3.4 mm, and it is unreasonable to assign a constant yield strength for the whole area. e pin tip is divided into one circle and three concentric rings, as shown in Figure 10(b). In each circle or ring, the heat flux is assumed to be uniform. e heat generation in the k circle or ring Q ptk can be expressed as follows: where σ 5,k and R 5,k are the yield strength and radius of the k circle or ring, respectively.
Meanwhile, the pin volume of the j layer at the i step V ij is as follows: πhj r 2 j + r j r j+1 + r 2 j+1 , (j < i), where h j is the height of the j layer. So, the volumetric heat flux of the j layer at the i step q ij can be written as follows: In fact, the pin volume V ij is occupied by the pin and not the base material during FSW, and it is theoretically more accurate to use a surface heat flux on the pin surface while it is unable to apply a surface heat flux on the specific elements in the pin volume, and the assumption that the heat distributes uniformly in each layer inhibits the heat flux from transferring through the pin/workpiece interface. In other words, the base material in pin volume does not have a significant impact on the temperature distribution, and adopting the volumetric heat flux does not reduce the simulating accuracy obviously. It is observed from equation (21) that the volumetric heat flux of each node is directly affected by the yield strength, and how to calculate the value of different layers is a key point. is equation only gives the mathematical form of the volumetric heat flux, and how to calculate its value and apply the value to the specific elements in the MARC software will be elaborated in the following.
In view of computational cost and efficiency, the meshes cannot be refined infinitely. e butting depth is 4 mm, and the weld zone can be divided into 4 layers of elements combined with 5 layers of nodes, as shown in Figure 11(a). When calculating the q ij (j < i), the average temperature of 8 nodes in the m layer (m � j) is selected to deduce the yield strength according to equation (6), as shown in Figure 11(b). As for calculating the q ij (j � i) of the last layer, the yield strength from the side and tip of the pin needs to be considered. e calculation of the yield strength from the pin side is the same as other layers. e yield strength in each circle or ring from the pin tip is calculated using the average temperature of 4 nodes in the m layer and 4 nodes in the m + 1 layer, as shown in Figure 11(c). For the convenience of programming, a set of numbering rules is set up to describe the node number. L and k are the distance of any node from the heat source center around the pin side and pin tip, respectively. Four directions (d � 1 − 4) are assigned for telling the difference among the nodes with the same L. For example, n jdL represents the node L mm away from the heat source center at the d direction in the j layer, and n mdk represents the node k mm away from the heat source center at the d direction in the m layer. en the average temperature of the j layer from the pin side at the i step T ij can be given as follows: T ij � 1 8 n j,1,4 + n j,2,4 + n j,3,4 + n j,4,4 + n j,1,5 + n j,2,5 + n j,3,5 + n j,4,5 .
(23) Figure 12 depicts the contact conditions between the shoulder and workpiece in the shoulder-plunging stage at different steps. When the trailing edge of the shoulder just contacts with the workpiece, the heat only originates from the pin/workpiece interface. en the heat starts to generate at the shoulder/workpiece interface when they are in contact with each other, and the proportion becomes more and more as the contact area (blue thick line and red zone) increases gradually until the shoulder contacts fully with the workpiece. e specific surface heat flux generated at the shoulder/workpiece interface can be directly described by equation (15). e specific area for heat production S s can be given as follows: where (x, y, z) is the coordinate of the welding center, and IW is the plunging speed.

Traverse Stage in the MAM.
In the traverse stage, the pin can be divided into 4 parts, as shown in Figure 13(a). e volumetric heat fluxes of 1-3 layers are calculated via dividing the heat from the pin side by the volume of each layer; the heats from the side and tip of the pin are used to calculate the volumetric heat flux of 4 layers. As only about half of the pin contacts with the workpiece, six nodes in three directions (1, 3, and 4) are selected to deduce the average temperature for calculating the volumetric heat flux from the pin side of each layer, as shown in Figure 13(b). e bottom radius of the j layer R j is as follows: e volumetric heat flux of the j layer q j can be summarized as follows: e average temperature on the side of the j layer T j can be described as follows: T j � 1 6 n j,1,4 + n j,3,4 + n j,4,4 + n j,1,5 + n j,3,5 + n j, 4,5 . (29) e average temperature in the k circle (or ring) of 4 layer T k is given as follows: T k � 1 6 n 4,1,k + n 4,3,k + n 4,4,k + n 5,1,k + n 5,3,k + n 5,4,k . (30) e surface heat flux can be written as equation (15). e specific area for heat production S s can be given by equation (26).

Model Verification.
In order to verify the simulating precision of two thermal models, friction stir welding was carried out and the transient temperatures were recorded. Figure 14 shows the transient temperatures from the experimental measurement and numerical simulation in different stages. e numbers 1∼5 in Figure 14 represent  the locations 1∼5 in Figure 3, and the centers of location 1, location 2, and location 3∼5 are located in the plunge stage, traverse stage (unsteady stage), and traverse stage (steady stage), respectively. Figure 14(a) shows that the heating rate in welding experiments is higher than that in the CAM and MAM during the plunge stage (0∼11.55 s). is could be attributed to that the plunging pin gives rise to the "escaping" of the plasticized material from its original position in the actual welding process, and part of the material contacts with the shoulder to generate heat, which can increase the heating rate to some degree. Figure 14(b) shows the predicted and measured thermal cycles of location 2 in the traverse stage (unsteady stage). e maximum temperature in welding experiments is 248.1°C, and the maximum temperatures in the CAM and MAM are 230.0°C and 243.6°C, respectively. e predicted and measured thermal cycles of locations 3∼5 in the traverse stage (steady stage) are shown in Figure 14(c). e simulated thermal cycles in the MAM show better agreement with the magnitude and shape of the experimental ones than that in the CAM. e peak temperatures and the corresponding relative tolerances of locations 1∼5 are revealed in Figure 15. e maximum temperature differences of the CAM and MAM are 25.2°C and 4.5°C, respectively. e relative tolerance between the predicted and measured value of the MAM are lower than that of the CAM, and the maximum value in the two models is 1.8% and 7.3%, respectively. ese results demonstrate that the MAM owns a relatively higher simulating accuracy than that of the CAM. Figure 16 plots the calculated temperature distributions on the transverse sections (yoz, x � 0) and longitudinal sections (xoz, y � 0) in the pin-plunging stage. e temperature contour bands in the two models are different, and the maximum temperature in the MAM is 27.7°C higher than that in the CAM. e temperature difference between the CAM and MAM is mainly caused by the different calculating methods of the volumetric heat flux. During the pin-plunging stage, FSW results in intense plastic deformation around the rotating pin and friction between the pin and workpiece. at is to say, the heat is generated by plastic deformation and friction occurred in the region around the pin/workpiece interface.

Temperature Distribution.
e region belongs to the low-temperature region in the stirred zone, and the corresponding yield strength is higher than in other regions. While in the CAM, the volumetric heat flux of each node which is mainly located in the high-temperature region is calculated using its own yield strength (temperature), and thus leads to the decrease of the volumetric heat flux. In the MAM, the average temperature at the pin/workpiece interface is used to calculate the volumetric heat flux of each node via equations (21)- (23), which can reflect a more accurate temperature distribution. Additionally,  thermal characteristics can be found on the trailing side (TS) and the leading side (LS) because of the gradual heat accumulation behind the tool. Figure 17 depicts the temperature distributions on the tip surfaces in the shoulder-plunging stage at different times. When the pin plunges into the workpiece for 8.9 s in the MAM, the trailing edge of the shoulder contacts with the workpiece (white box in Figure 17(a)), and the maximum temperature is 361.6°C. At the 9.5 s, half of the shoulder contacts with the workpiece, and the maximum temperature increases to 385.9°C. When the time is 10.1 s, the shoulder contacts fully with the workpiece, and the resulting maximum temperature is 415.9°C. In contrast, the maximum temperature and heating rate in the CAM is significantly lower than that in the MAM before the shoulder contacts with the workpiece (10.1 s). In addition, the temperature distributions in the CAM and MAM are asymmetric with respect to the tool center, and the hightemperature zone (yellow zone) in the MAM is less uniform than that in the CAM. e phenomena result from two reasons. First, the yoz plane (x � − 25) is the end surface exposed to air whose thermal conductivity is extremely low compared with the aluminum alloys, and the different cooling conditions on the two sides of the tool center cause the nonuniform temperature distribution in the two models. Second, the gradually plunging shoulder in the MAM leads to the nonuniform heat input on the two sides, which promotes the occurrence of the nonuniform temperature field, especially the high-temperature zone. Figure 18 displays the temperature distributions on the tip surfaces and transverse sections across the tool center in the traverse stage. e temperature gradients in the front of the tool are much larger than that behind the tool, and the preheating width (WP) is only about 5 mm. e high-temperature zone is mainly located at the back of the tool due to the gradual accumulation of heat input. e maximum temperatures in the CAM and MAM are 457.8°C and 478.1°C, respectively. e peak temperatures on the transverse sections in the two models are lower than the maximum temperature, and the values are 424.3°C and 444.1°C, respectively. Figure 19 shows the thermal cycles of different zones in the traverse stage, including nugget zone (NZ), thermomechanically affected zone (TMAZ), and heat affected zone (HAZ). It can be seen that the heating and cooling rates in the CAM and MAM are very similar, except the high-temperature regions that are marked by the purple ellipses. e heating rate is rather large, and the transient temperatures of different zones reach the peak value in 3 seconds; the cooling rate is much lower than the heating one, and the transient temperatures show a slowly decreasing trend. Additionally, the peak temperatures of three zones in the CAM are all lower than that in the MAM, especially that of NZ.

Conclusions
Two analytical models, including the conventional and modified models, are compared to describe the heat generation and predict the temperature field. e main conclusions are drawn as follows.
(1) In the modified analytical model, the nonuniform volumetric heat flux and the tilt angle of the tool are considered. e side and tip of the pin are divided into several parts, and the volumetric heat flux of each part is calculated separately, which can reflect a more accurate heat generation than that of the conventional one. (2) Comparison of the predicted and measured thermal cycles and peak temperatures in two models demonstrates that the MAM owns a higher simulating accuracy than the CAM. (3) In the pin-plunging stage, the temperature distributions in the two models are different, and the maximum temperature in the CAM is 27.7°C lower than that in the MAM. Meanwhile, some thermal characteristics are similar: the temperature distributions are asymmetric with respect to the weld center, and the temperatures on the RS and LS are lower than that on the AS and TS. (4) In the shoulder-plunging stage, the maximum temperature and heating rate in the MAM are much higher than that in the CAM, but the distribution of the high-temperature region is less uniform. (5) In the traverse stage, the high-temperature zone is mainly located at the back of the tool, and the peak temperature does not appear in the transverse section. e thermal cycles show that the peak temperature of different zones in the MAM are all higher than that in the CAM.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.