On the Utility of the Thermal-Pseudo Mechanical Model’s Residual Stress Prediction Capability for the Development of Friction Stir Processing

This paper investigates the thermal-pseudo mechanical (TPM) model’s residual stress prediction capability for its utility in developing friction stir processing (FSP). Specifically, two FSP tests under different processing conditions were conducted, and the corresponding simulations were carried out to verify if the TPM model can predict residual stresses for various tool radii and workpiece materials. The model successfully predicted residual stresses with an error less than 4% for one of the tests but failed to work for the other test. Further simulations under different FSP conditions proved that the TPM model works for cast aluminum alloys and wrought aluminum alloys. In addition, the large FSP tool used was found to be the reason for the model’s failure on one of the tests. This indicates that there is a range of tool radii for which the TPM model is applicable. As a solution, this paper suggests modifications to the TPM model based on calibration to the FSP test temperatures. The resulting residual stress prediction is accurate and differs from the experimentally characterized stress values by only 6.5 MPa. The calibrated TPM model requires FSP to be carried out when using a tool with a different radius. Following that, the effect on residual stresses due to changes in the other process parameters, such as the tool traverse & rotation speeds and the clamping conditions, can be predicted.


Introduction
Reducing vehicle weight is one important strategy to improve energy efficiency. The fact that components are often over-engineered to meet the performance of select regions presents an opportunity. Localized enhancement of properties such as strength and ductility in the select regions could allow weight optimization. Friction stir processing (FSP) is one such suitable process in metal alloys. FSP is a solid-state material processing technique closely related to friction stir welding (FSW) [1]. Like FSW, a non-consumable rotating tool is pushed onto the workpiece and moved along the required surfaces to generate localized heating through friction and material stirring. Unlike in FSW, heat generation and material stirring are not intended to form a joint but to modify the microstructure locally in a single workpiece. Microstructure modifications include grain refinement, homogenization, and primary particle breakdown [2]. Homogenization would also mean the reduction of porosity in cast materials [3]. Through such microstructural modifications, FSP has been shown to improve ductility [4]- [6], fatigue resistance [7], yield stress [8], wearability [9], and corrosion resistance [10]. In some cases, the property enhancement is greater with more passes of FSP [11].
However, FSP and FSW, just like traditional welding techniques, induce undesirable residual stresses. Among these, tensile residual stresses, while still smaller than those from traditional welding techniques [12,13], are detrimental to fracture strength and fatigue life. In addition, the presence of distortion associated with residual stresses promotes buckling failure [14]. Mitigating residual stress would require optimizing the factors influencing the stresses, such as the process parameters, clamping conditions, and tool design, for every workpiece material and geometry. The typical approach to residual stress mitigation is often the expensive trial-and-error method. This involves repeated cycles of adjusting the parameters and conditions of FSP for every workpiece and measuring the residual stresses. As a solution, the use of numerical models as a part of the integrated computational materials engineering (ICME) approach can potentially reduce process development costs by 50% [15].
Various approaches have been developed to model the complex multiphysics processes associated with FSP and FSW, and these efforts were summarized previously in review articles [16]- [19]. Note that in these summaries and from here on in this paper, residual stress modeling approaches for FSW were assessed to be applicable for FSP, and vice versa. For instance, a continuum-based FEM model initially developed for FSW [20] has also been shown to work for residual stress prediction in FSP [21]. With this profusion of models, determining the right tool is not a trivial task. An important criterion often used to assess a model is its representational accuracy, i.e., representing the physical process as well as possible. Using such a metric, models that explicitly consider the material flow to predict the resultant viscous heat generation and frictional heat are superior [22]- [24]. It could be argued that representational accuracy is correlated to the results' accuracy. However, with increasing complexity comes increasing computational costs that can be prohibitive if the workpieces or the process zones are large.
Additional criteria for choosing a simulation model would be subscribing to the adequacy-for-purpose view [25], according to which models should be assessed for their adequacy or fitness for particular purposes. Here is where heat source models prove their mettle. In contrast to the flow models, the heat source models treat the action of the friction stir tool as just a heat flux. Even though it is a poor representation of the physical process, they are adequate for this work's purpose of predicting residual stresses. It is because, essentially, residual stress formation during FSP/FSW is due to heat-induced thermal expansion under mechanical constraints and the resulting plastic deformation [26]. An additional advantage of the heat source models is their low computational expense. However, predicting residual stresses is not a complete definition of the model's purpose. The model needs to predict the effect of various FSP conditions and parameters on the stresses, as summarized in Fig. 1. In this figure, the parameters and conditions were arranged in increasing order of the required modeling complexity from bottom to top. It can be said that the more of these conditions and parameters the model can account for while predicting the residual stresses, the greater the model's utility. Accordingly, the utility of the model determines the cost savings from taking an ICME approach for FSP development.
Typically, models that explicitly treat the flow can consider all the parameters and conditions in Fig. 1, but the same cannot be said about the heat source models. The simplest heat source models need prior knowledge of the heat generation from the same FSP experiment. For these models, the heat source is calibrated to either the temperatures measured on the workpiece [27] or the torque on the tool [28]. As noted by Schmidt and Hattel [29], such a requirement would "in some situations conflict with the very objective for thermal modelling of FSW." The situations where the simple heat source models do serve their objectives would be when residual stresses need to be predicted for different clamping conditions, weld paths, or workpiece geometries, i.e., the bottom row in Fig. 1. Nevertheless, they cannot predict the effect of process parameters, tool radii, and workpiece materials. Since these factors in rows 2 and 3 of Fig. 1 are known to be highly influential on the residual stresses, the utility of such simple heat source models can be deemed to be low and insufficient.
In light of such issues with residual stress modeling approaches, the potential of the thermal-pseudo mechanical (TPM) model [29] is promising. The TPM model represents an improvement to the simple heat source models. In (4) Fig. 1 The chart, from bottom to top, shows an increasing number of FSP parameters and conditions a model can consider while predicting residual stresses this model, the heat generated depends on the local thermal and mechanical conditions, i.e., there is a pseudo-coupling between thermal and mechanical aspects. With this feature, it can recreate the observed self-stabilizing effect in the maximum FSP temperatures [30] without resorting to physically unjustified means [31] [28]. Consequently, the TPM model can accurately predict the heat generated for various tool rotation speeds quantified as rotations per minute (RPM) and traverse speeds, i.e., row 2 in Fig. 1. The TPM model was also used to predict the residual stresses for different tool radii and workpiece materials [32,33], i.e., row 3 in Fig. 1. However, the range of tool radii was narrow, and all the workpiece materials were aluminum-wrought alloys.
In addition, the physical basis for the applicability of the approximations made in the TPM model's heat source formulation remains to be investigated. Therefore, it is unclear if the approximations hold for all tool radii and materials.
In this work, we aim to verify if the TPM model can predict the effect of the tool radius and workpiece material on the residual stresses and determine the utility of the model for FSP development. With such an objective, two FSP tests were carried out, after which residual stresses were characterized. One FSP test is on aluminum alloy A7075, and the other will use aluminum alloy A380, a cast alloy and different from the ones previously used with the TPM model. The second test will also use a much larger tool radius than previous. The study finds the limitations of the TPM model, proposes the modifications needed to make it work within these limitations, and then discusses the resulting consequences on its utility.

Friction stir processing
This study performs two FSP tests. The first is on a workpiece made of a wrought aluminum alloy; the second is on a cast aluminum alloy workpiece using a different FSP tool and process parameters. Both the tests were performed in a precision friction stir welding machine at the Pacific Northwest National Laboratory (PNNL) under force-controlled conditions. Table 1 summarizes the process parameters for both these tests. FSP test #2 on A380 alloy is run using a much slower moving and larger tool than FSP test #1 on A7075 alloy. The z-force (in the thickness direction) for FSP tests #1 and #2 was 10kN and 17kN, respectively. The tool tilt for FSP tests #1 and #2 was 2° and 1.33°, respectively.
The workpieces in both tests were plates, and FSP was carried out on a line in the center. Figure 2 shows the workpieces and their dimensions. The A7075 plate for test #1, as Fig. 2(a) shows, is 127 mm in the weld direction, 100 mm wide, and 2.5 mm thick. Weld is made such that there is a gap of 12 mm from the ends of the processed zone to the edge of the plate. The plate for test #2, shown in Fig. 2(b), was made of aluminum alloy A380 cast using high-pressure diecasting. The plate is 250 mm long in the weld direction, 60 mm wide, and 3.5 mm thick. The processed zone is such that there is a gap of 25 mm from the end of the processed zone to the edge of the plate and 30 mm from the plunge location to the edge of the plate. The material properties for these two different aluminum alloys are stated in the appendix.
The FSP tools used for the two tests are different. Figure 3 shows the photos and drawings necessary to represent the two tools' essential features. Figure 3(a) shows a photo of the tool's surface from test #1; the tool's shoulder diameter is 12.7 mm, and the diameter of the pin tool at its base is 5.09 mm. The tool pin has a height of 2.3 mm and a taper of 15°. The tooltip is threaded like a screw, and the surface of the tool's shoulder is scrolled. Also, note the presence of a hole on the surface of the shoulder at 2 mm from the edge of the shoulder. This hole is used to insert a K-type  thermocouple to measure temperature while FSP is being carried out. This is often an indicator of the max temperatures and therefore assists in monitoring the process. Figure 3(b) is a three-dimensional drawing of the FSP tool used for test #2. The shoulder diameter is 30 mm, and the diameter of the pin tool at its base is 10 mm. The tool pin has a height of 3.3 mm and a taper of 15°. The tool pin is threaded, and the surface of the tool's shoulder is scrolled, as shown. The thermocouple is inserted in the tool shoulder through a hole at 10 mm from the center to measure temperature. Figure 4 and 5 describe the clamping conditions in detail since they are known to significantly affect residual stress distribution [34]. Figure 4. shows a photo of the clamping setup from the top. Clamping is intended to securely hold the FSP plate or workpiece while FSP is happening. The workpiece is held on the machine bed using clamps and additional steel plates. Figure 5 shows the clamping conditions using a schematic of the cross-section normal to the weld direction. Two clamping plates on both sides of the weld and the top surface of the workpiece are pushed down using clamps. These clamping plates on the top are in contact with the workpiece all along its length, spanning a width of approximately 12.7 mm on each side. In addition, all six surfaces at the edges of the FSP plate are constrained using dummy plates, as shown in Fig. 4 and 5. The machine bed also acts to constrain the movement of the FSP plate toward the bottom. Note that these clamping conditions also determine how the heat is diffused from the workpiece.
Temperature measured during FSP often serves to validate the heat source in the simulation. Figure 6 shows the temperatures recorded by the probes in the tool during FSP, plotted with respect to the distance traversed by the tool. For FSP test #1 on alloy A7075, temperature increases rapidly at first and then saturates to the value of around 490℃. For FSP test #2 on alloy A380, the temperature saturates to a value of around 475℃.

Residual stress characterization
Residual stresses arising from FSP were characterized using an Xstress DR45 X-ray diffractometer. In this nondestructive technique, the interplanar distance and the lattice distortion are first determined by measuring the change in the diffraction peak position using the modified χ mode [35]. The Cr K α radiation source was powered using an electrical current of 9 mA at a 30 kV voltage and emitted a 1 mm-sized beam. The exposure time was 1 s, the number of tilts was 5/5, and the tilting range was −45 • to 45 • . Stresses were then determined from lattice distortion (strain) by using Aluminum's elastic properties. The uncertainty in residual stress measurement was estimated by considering beam/detector misalignment and errors in measuring the peak location and lattice spacing. All the settings were specified in the Xstress Studio v1.1 software, which also automates the analysis. Stresses on the surface of the workpieces were characterized at specified locations. As Fig. 2 shows, stresses were measured on a line normal to the weld direction. Note that the line location is important because the magnitude of residual stresses varies in the weld direction. The spacing of measurements on this line is 1 mm, and this value was set by the beam spot size. Typically, residual stresses are largest near the center of the processed zone length and much lower near the end of the processed zone [1]. The characterization locations in this study were designed to capture these stress distributions.

Simulation using TPM model
The residual stresses from FSP were simulated using the TPM model [29] within a coupled thermo-mechanical finite element framework in ABAQUS/ Standard software. In TPM, the tool's action is reduced to a moving surface heat flux. This heat flux q is defined as a function of the radial position with respect to the tool's center r , the tool's angular velocity , and the temperature-dependent yield stress In Eq. (1), R shoulder is the radius of the tool's shoulder. Note that the inclusion of the temperature-dependent yield stress in the heat source model is what makes TPM a pseudo-mechanical model. The materials' properties, including the yield stresses, are in the appendix (Tables 4 and 5). This heat source model is incorporated in ABAQUS FEA as a user-defined subroutine DFLUX.
Softening due to material microstructure evolution during FSP was also considered in these simulations because yield stress evolution significantly affects residual stress prediction. While various microstructure evolution mechanisms are active, precipitate dissolution is known to be the most influential on the hardness in age-hardenable alloys such as A7075 alloy [1,36]. Therefore, softening is modeled using only the Myhr & Grong model for precipitate dissolution [37]. The model considers the softening induced by the dissolution of precipitates in A7075 alloy. This model is restricted to predicting the hardness change and does not explicitly track the evolution of the precipitates' size, shape, and distributions. Here, the fraction of the remaining precipitate is given as where f is the remaining precipitate volume fraction, f 0 is the initial precipitate fraction, t is the time, and t * is the time needed for complete precipitate dissolution at the same temperature. t * in turn, according to classical kinetic theory is given as where Q is the effective activation energy for precipitate dissolution, t r is the time for total dissolution at the reference temperature T r , and R is the gas constant. The yield stress is then a linear interpolation betweesssn the original state and the fully dissolved state, given as where max and min are the yield stresses of material in the T6 and solution-treated conditions, respectively. For the sake of numerical implementation, Eq. 2 is used in the following form where Δt i represents the timestep i and t * i is the time needed for complete precipitate dissolution at the temperature in this time step. The parameters for A7075 alloy, adopted from previous studies [45,46], are stated in the appendix (Table 3).
Hardening through natural aging that happens over a larger period, in the order of days, is ignored because it does not affect the distribution of residual stresses [38]. This softening mechanism is incorporated as a subroutine USD-FLD. As far as A380 cast alloy is concerned, there is usually an increase in the hardness/yield stress resulting from , microstructure evolution in the nugget region during FSP. However, the degree of hardness change is usually not substantial. In one case, the degree of hardening is only around 23% [4]. In comparison, the degree of softening in A7075 alloy could be as high as 50% in 7xxx alloys [39]. In addition, TPM lacks the capability necessary to determine the strain and strain rates in the nugget zone. Only such strain data would have enabled modeling the grain size evolution, causing the nugget zone's hardening [39]. Therefore, modeling the microstructure evolution in A380 alloy is neglected in this work because of the expected smaller degree of influence and the lack of a suitable model. Figure 7 illustrates the critical details of the threedimensional FEA model used for FSP simulation. Since symmetry is assumed in the TPM model, only half the plate is modeled, and symmetry boundary conditions were assumed across the weld line. The initial temperature of the workpiece and ambient air temperature were set to be 20℃. The simulations consist of two steps. The first step is where FSP is simulated while the plate is clamped. In this step, the FSP tool is modeled as a moving surface heat flux, illustrated in Fig. 7. In this model, both mechanical and thermal boundary conditions were set to recreate conditions in the experiment. Thermal conductivity boundary conditions were assigned a value of 1000 W∕m 2 K on all surfaces in contact with external metal surfaces. This value will recreate the heat transfer to a much larger metal part, and such a strategy is commonly used [40]. The surfaces in contact with external metal bodies are the bottom surface of the plate, the three side edges, and a part of the top surface in contact with the clamping. In addition, all the surfaces exposed to ambient air were set with convection and radiative heat transfer conditions. Displacement boundary conditions were set to mimic the effect of clamping. On the bottom surface of the plate, displacement in the direction normal to the plate is set to be zero. On the three surfaces on the sides in contact with the dummy plates, the displacements in the directions normal to these surfaces are set to zero. On the section of the top surface under the clamping plate, the displacements in all directions were set to zero. These boundary conditions are similar to those set in literature [32]. After completing the first step, the second step is used to mimic the unclamping process. In this step, all the displacement and thermal boundary conditions were removed. The stresses in the workpiece after unclamping would be the residual stresses studied. Figure 8 shows the meshes used for FSP simulations. The dimensions match the samples shown in Fig. 2. Note that the dimension in the transverse direction is half because symmetry is assumed. Eight node hexahedral elements were used to mesh. Figure 8(a) shows the model for FSP test #1 on A7075 alloy, where the thickness dimension is uniformly divided by six elements. The element size in the weld direction is 0.75 mm. The element size in the transverse direction in the areas directly under the tool shoulder is also 0.75 mm. Also shown in Fig. 8(a), the mesh size in the transverse direction is progressively increased further away from the weld line to a size of 6 mm at the edge of the plate. In total, there are 11,830 elements. Figure 8(b) shows the model for FSP test #2 on A380 alloy, where the thickness dimension is uniformly divided by eight elements. The element dimensions in the in-plane directions are 2 mm everywhere. Also shown in Fig. 8(b), the element length in the transverse direction is gradually refined to 1 mm close to the edge of the top clamp. In total, there are 15,210 elements.
Apart from the two FSP tests, two additional simulations were performed with conditions specified in Table 2. Simulation #3 is FSP test #1, but the tool is from FSP test #2. Simulation #4 is FSP test #2, but the tool is from FSP test #1.  Fig. 9(b) shows the peak tensile stresses to be near the edge of the tool's shoulder and the stresses away from the weld line to be compressive. This is the M-shaped distribution of residual stresses often mentioned in the literature [1]. An important observation is the variation of the stresses along the weld line. Near the ends of the processed areas, the residual stresses were much smaller than near the middle of the weld line. This means, based on where stresses were characterized, the value of residual stresses can be very different. Figure 10 is used to examine the distribution of residual stresses further and validate these simulation results. Here, residual stresses were extracted from a line on the surface normal to the weld direction from both simulations and experimental characterization. The location of this line is shown in Fig. 2. The uncertainty in the characterized stress values are also shown. Figure 10 shows both transverse and longitudinal residual stresses plotted with respect to the distance to the weld line. The predicted and characterized stress distributions can be observed to be similar. An aspect of residual stresses that is often the most consequential is the location and value of the peak tensile stresses in the longitudinal direction. The experimentally observed value is 140.3 MPa, and its location is 5 mm from the weld line. The simulated peak value is 135.2 MPa, and it is 6 mm away from the weld line. The prediction error for the value is less than 4%, and the predicted location is around the spatial resolution of the stress characterization technique. In both simulations and experiments, the residual stresses gradually decrease in value as the distance from the processed zone increase increases. Towards the edge of the workpiece,  Comparing the simulated residual stresses with the experimentally characterized results for FSP test #1 on the A7075 alloy workpiece shows that TPM predictions are correct. As noted in previous studies, the few inaccuracies observed can be overcome through improvements in the material evolution model for the nugget zone and a higher fidelity model for heat generation [18].
The main objective of this study is not the accuracy of the predictions but to establish the utility of the TPM model. Considering this objective, it should be noted that for the A7075 workpiece, the simulation required no calibration to the FSP test carried out. All the parameters for the model were known before the FSP test. This same predictive capability was demonstrated previously for other wrought alloys and tools of different radii at different process parameters. Therefore, this supports the assertion that the TPM model can predict the effect of different tool radii.

Simulation #2: FSP test #2 on A380 alloy
The simulation carried out using the TPM model for FSP test #2 on A380 alloy fails to complete. The displacement fields in the simulation fail to converge less than 2 s into the process. Temperature and stress fields were investigated to identify the problem. Figure 11 shows maximum temperatures from the simulation with respect to time. The temperature rapidly rises to more than 550℃ before the simulation fails. The maximum temperature observed in the experiment as per Fig. 6 is only 475℃. Figure 12 shows the temperature and von mises stress fields in the last step before the simulation fails. Figure 12(a) shows that temperatures exceed 500℃ in an area almost as large as the tool. As shown in Fig. 12(b), the stresses are severe near the clamped region and the tool's edge.
As per Fig. 11 and Fig. 12, a large plate region is under temperatures higher than the melting of the A380 alloy, and at the same time some of these regions are also under severe stress. Therefore, the material deforms uncontrollably due to a combination of low yield stresses at elevated temperatures and large stresses, thus explaining the displacement field convergence issues.
The convergence issue encountered in the simulation indicates problems with the heat generation model. FSP test #2 must have presented conditions at which the approximations made in the TPM model breakdown. Compared to FSP test #1 and other previous demonstrations of TPM use, FSP #2 is unique in a few ways. All the previous uses of TPM were for wrought alloy. Another reason could be the large tool size. At a diameter of 30 mm, the tool is more than twice as large as the tool used in FSP test #1 and the tool sizes encountered in the literature. As of now, there is no evidence to point out which of these factors are out of the predictive range of TPM. The subsequent simulations will identify the factor responsible. In addition, TPM will be modified in Sect. 4 to make the prediction of residual stresses possible in cases where TPM fails, such as FSP test #2. First, the heat source model will be modified to leverage the temperatures recorded during FSP tests and simulate realistic process temperatures. In addition, clamping conditions will be relaxed because the severe stressing near the clamps might be unrealistic.

Simulations #3 and #4
These simulations were carried out to verify the TPM heat source model, and no corresponding experiments were performed. Therefore, only the simulated temperatures are of interest. The maximum temperatures on the workpieces were extracted and plotted in Fig. 13. In simulation #3,, the temperatures rapidly increase beyond unrealistic temperatures of 800℃. At this point, the simulation fails to converge and stops. The temperature saturates to around 450℃ in simulation #4 and it completes successfully. These two simulations identify which factors, A380 cast alloy material or the large 30 mm diameter tool, caused the TPM's heat formulation to fail. For simulation #4, the use of the smaller tool of 12.7 mm diameter on the A380 alloy workpiece poses no problems. The saturation temperature is within the range of typical temperatures expected during FSP/FSW of aluminum alloys [30]. However, simulation #3, where the larger 30 mm tool is used on the A7075 alloy workpiece, fails to converge. As the unrealistically large temperatures indicate, the same problem encountered during the simulation of FSP test #2 causes the simulation to fail. From this observation, it can be inferred that TPM applies to only a range of tool radii, and a tool with a 30 mm diameter does not lie within this range. Furthermore, it can also be inferred that there is no reason why TPM cannot be applicable for A380 alloy material in addition to all the wrought aluminum alloys on which TPM has been previously used.

Changes to the TPM model
The TPM heat source model relies on multiple assumptions, among which the contact condition at the tool and workpiece interface is the most significant [41]. It assumes that at the interface, the contact shear stress, contact is where friction is the shear friction stress and yeild is the shear yield stress. In general, the heat source equation should have been where is the degree of sticking. Determining and the friction needs detailed monitoring of the flow and extensive experimentation. Therefore, it is no trivial task and is not within the scope of this study. In addition, the yeild values used are for quasistatic conditions, and strain rate dependency is ignored. Another assumption is that most heat generated is transferred to the workpiece. As a solution, many models have added an efficiency factor to consider the heat lost to the tool [42].
We propose to use an approach that has been suggested before but not expounded [43]. As a modification to the TPM model, a calibration parameter is added to the heat source. Heat flux is now is adjusted to make the simulated maximum temperatures match the observed tool temperatures. Such an approach is similar to some models where the unknown parameters in the thermal model were determined to fit the observed temperatures [38,44].
The clamping conditions were also modified to correct the severe stresses simulated near the top surface clamp. Under severe stresses, clamping might not be able to effectively constrain the movement of the workpiece and result in slipping under the clamps. Therefore, displacement boundary conditions were modified on the area on the top surface of the workpiece corresponding to the top clamp. A total of three different simulations with three different constraints at the top clamp were carried out. In addition to the fully constrained condition representing perfect clamping, two other constraint conditions were used. In one condition, displacement in the Y direction is unconstrained to specify slipping in the transverse direction. In the other condition, both X and Y directions were unconstrained to specify slipping in all the in-plane directions.  Figure 14 shows the maximum simulated temperatures for various calibration factors. The calibration parameters of 0.2 to 0.25 succeed in simulating a saturating maximum temperature of less than 500℃. A calibration parameter of 0.25 results in the experimentally observed temperature of 475℃. Simulations using the determined calibration parameters were carried out until completion to find the residual stresses. Figure 15 shows the simulated longitudinal stresses on a line on the surface in the exact location as the experimental characterization. Three simulations' results with different clamping conditions on the top surface were plotted. When no slipping happens, the longitudinal stresses are negligible in most of the plate and present only near the clamped region, the most severe being near the edge of the clamping location. This could mean the greatest plastic deformation in the workpiece happened in the clamped regions. In such conditions, an assumption that clamping that could effectively prevent slipping might be unrealistic. In the second case, slipping is assumed to happen in the weld direction. The stresses now are tensile in most of the sheet and compressive in the clamped regions. The tensile stresses in the region traversed by the tool (< 15 mm) are approximately constant with a value of 32 MPa. In the third case, slipping also happens in the transverse direction, and the clamp only prevents the deformation of the plate in the thickness direction. In this case, the transverse residual stresses are significant and as large as 80 MPa. Figure 16 shows the experimentally characterized residual stresses, both transverse and longitudinal, compared to the simulated stresses. The uncertainty in the characterized stress values are also shown. The stresses within the regions traversed by the tool (< 15 mm) are tensile and are approximately the same. The average of the observed longitudinal stresses within this region is 38.5MPa . These observed stresses are only 6.5 MPa larger than those predicted from a simulation where the workpiece under clamping slips in the weld direction. The predicted stresses from other clamping conditions differ from the observed stresses by a larger degree. Therefore, it can be inferred that the workpiece clamping to prevent slipping in the welding direction was inadequate, and the simulation results shown in Fig. 16 are from such clamping conditions.
The comparison between the experimental and simulated stresses in the rest of the regions is favorable. Close to the weld, longitudinal and transverse stresses from both simulation and experiment are around 30 − 40MPa . They all gradually decrease further away from the weld line and steeply decrease to be compressive stresses beyond the area traversed by the tool. A more precise comparison is not appropriate because the uncertainty in the measurements, at around 10-25%, is not negligible. The most discrepancy in the stress prediction is near the tool's edge. This could be because the workpiece surface is the roughest here, and    16 Characterized and simulated residual stresses for FSP test #2 on the A380 sample. Plot also shows the uncertainties in the measurements X-ray diffractometer characterization might not accurately measure residual stresses at this location. This also explains why measurements are noisier at this location.
It can be inferred from the results in this section that the addition of a parameter calibrated to the temperatures from the FSP experiment corrected the issue with TPM predicting unrealistic temperatures for FSP test #2 on A380 alloy. Because of the modified heat source model, the simulation can correctly predict the observed residual stresses. As mentioned for the first FSP test case on A7075 alloy, the minor discrepancies in residual stress predictions could be overcome with improved heat generation and material evolution models.
The more interesting aspect of this study is the implication of these modifications on the utility of the TPM model. Adding a factor that requires calibration to the temperatures recorded during the FSP experiment is a decrease in terms of utility. It means the test must be first performed when using FSP tools whose radii are not within a range of radii before residual stresses can be predicted. The precise range of tool radii for which calibration is not needed requires extensive investigation and is left for future work. Nonetheless, the existence of one case proves that the TPM model cannot predict the effect of tool radii on the residual stresses. Some researchers have suspected there would be parameters and conditions that the TPM model cannot predict, and they have used a similar calibration approach [32,43]. It should be noted that the calibrated TPM model can still predict the effect of process parameters (row 2 in Fig. 1) on the residual stresses as a virtue of its ability to simulate the selfstabilizing effect in FSP temperatures [29].
The clamping issue could also potentially decrease the TPM model's utility. In our paper, we could predict the residual stresses only after inferring the clamping conditions from the characterized residual stresses on an already processed workpiece. This limits the use cases of the TPM model. However, this issue might be unique to this specific test case. The clamping issue only arose because the clamp was too close to the FSP tool. That is probably the reason we see severe stresses and the clamp slipping. Most other simulations in the literature assumed a perfect constraint due to the clamp, and the residual stress predictions were accurate.

Conclusion
This study investigated the TPM model's capability to predict the effect of tool radii and workpiece materials on the residual stresses arising from FSP. This investigation involved two different FSP tests and a series of simulations under different parameters and conditions. We find that the TPM model is applicable to cast aluminum alloys in addition to wrought aluminum alloys. The peak longitudinal residual stress value was predicted with an error of less than 4% for FSP on the wrought alloy. However, TPM is applicable for only a range of FSP tool radii. The TPM model was modified using a calibration factor to overcome this limitation. The calibration approach works; the predicted peak stress values were within 6.5 MPa of the characterized values. However, calibrating the model to measured temperatures implies that an FSP test needs to be carried out before residual stresses can be predicted.
The progress made in this work in establishing the utility of the TPM model would increase confidence in this model's predictive capabilities and help stimulate its better integration into the ICME approach for FSP development. Future work would involve improving the fidelity of the heat generation model and integrating advanced models of material microstructure evolution. Table 4 Temperature-dependent yield stresses of aluminum alloy A7075 [47,48] and A380 [49] The yield stresses of the softened A7075 alloy material in the Myhr & Grong model were assumed to be the same as the yield stresses of the A7075-O alloy.