3D numerical models using a fluid or a solid formulation of FSW processes with a noncylindrical pin
 Philippe Bussetta^{1},
 Narges Dialami^{2},
 Michele Chiumenti^{2},
 Carlos Agelet de Saracibar^{2},
 Miguel Cervera^{2},
 Romain Boman^{1} and
 JeanPhilippe Ponthot^{1}Email author
DOI: 10.1186/s4032301500482
© Bussetta et al. 2015
Received: 30 May 2015
Accepted: 15 October 2015
Published: 18 November 2015
Abstract
Friction stir welding process is a relatively recent welding process (patented in 1991). FSW is a solidstate joining process during which materials to be joined are not melted. During the FSW process, the behaviour of the material is at the interface between solid mechanics and fluid mechanics. In this paper, a 3D numerical model of the FSW process with a noncylindrical tool based on a solid formulation is compared to another one based on a fluid formulation. Both models use advanced numerical techniques such as the Arbitrary Lagrangian Eulerian formulation, remeshing or the Orthogonal SubGrid Scale method. It is shown that these two formulations essentially deliver the same results.
Keywords
Friction stir welding (FSW) Finite element method Remeshing Arbitrary Lagrangian Eulerian (ALE) formalismBackground
This paper deals with the extension to 3D of the works exposed in [11]. This article presents and compares two different 3D numerical approaches of the FSW process. The first model is based on a solid approach written in terms of nodal positions and nodal temperatures. The second model is based on a fluid approach written in terms of the velocity, the pressure and the temperature fields. Both models use advanced numerical techniques such as remeshing and the ALE formulation. 2D models are useful to test and easily compare both numerical formulations. Nevertheless, the FSW process is a fully 3D thermomechanical process. The effect of the shoulder and the thermal boundary conditions have a great influence on the FSW process, but they cannot be considered in the 2D models.
This article is split into three parts. First, both 3D numerical approaches are presented. Secondly, the 3D models are compared and the differences with the 2D models are exposed. Finally, some conclusions and explanations about the differences between the results of both models are made.
3D numerical modelling of FSW process
The tool is described by a classical Lagrangian mesh. Due to high deformations in the neighbourhood of the tool, the use of a Lagrangian formalism would lead very quickly to mesh entanglement. Then, the plates are modelled using the ALE formulation. On top of this, the ALE formulation allows the model to take into account tools with no rotational symmetry. In relation with the distance from the rotation axis of the tool, the plates are split in three zones. In the closest zone around the tool (red region in Fig. 3), the mesh has the same rotational speed as the tool. In the model, this region is limited by the value of the distance from the rotation axis of the tool equal to the value of the radius of the shoulder. In the furthest zone from the tool, the grey zone in Fig. 3, the mesh is fixed and the material is flowing through the mesh. The last zone is a transition zone, white region numbered 2 in Fig. 3. This zone connects the meshes of both other zones. Therefore, the quality of the mesh in this zone decreases with the simulation time. The numerical techniques used to overcome this problem are explained in the section named “The transition zone”.
Thermomechanical formulation
The numerical models presented here are based on the finite element method. In this paper, two numerical formulations are compared (for more information about these formulations see [11]). The first one is based on a solid mechanics approach. It is written in terms of nodal positions and temperatures. The second one is based on a fluid mechanics approach. The equilibrium is written as a function of nodal velocities, pressures and temperatures.
Solid approach
The position and temperature fields are computed at each node of the elements. The mesh is composed of 21,980 linear hexahedral elements. The stresses and the internal variables are computed at each quadrature point of the element (8 Gauss points). To overcome the locking phenomenon, the pressure is considered constant over the element and computed only at a central quadrature point. The thermomechanical equations are split into a mechanical part and a thermal part. At each time step, the mechanical equations are first solved using a constant temperature field. This temperature field is the one obtained at the previous increment. Then, the thermal equations are solved on the frozen resulting geometrical configuration that has just been obtained.
Fluid approach
The fluid approach is based on a stabilized mixed linear temperaturevelocitypressure finite element formulation. This formulation is stabilized adopting the Orthogonal SubGrid Scale method (OSS) [19–21] to solve both the pressure instability induced by the incompressibility constraint and the instabilities coming from the convective term. A mesh of 74,127 linear tetrahedral elements is used for the domain discretisation. The velocity, the pressure and the temperature fields are computed at each node of the elements. The deviatoric stresses and the other internal variables are computed at each quadrature point of the element. Finally, the coupled thermomechanical problem is solved by means of a staggered timemarching scheme where the thermal and mechanical subproblems are solved sequentially, within the framework of the classical fractional step methods [22, 23].
The transition zone
Solid approach
In the solid approach, the transition zone is a ring with a finite thickness (region 2 in Fig. 3). In this region, the evolution of the rotational speed of the mesh, which differs from the material velocity, is linearly interpolated between the ALE region and the Eulerian zone. As the mesh distortion grows with time, a remeshing operation is periodically required. The remeshing operation can be divided into two steps. First, a bettersuited mesh, called the new mesh, is created. In this case, the simple geometry of this region allows an easy generation of the new hexahedral mesh. Then, to carry on the computation over this new mesh, the data are transferred from the old mesh to the new one (for more informations about the data transfer see [24]).
Fluid approach
Thermomechanical constitutive model
Solid approach
In the solid model, the value of the variation of the pressure (dp) is computed thanks to the variation of the volume (dV) and the bulk modulus (K): \(dp = K dV\). In addition, with the solid approach, it is possible to replace the NortonHoff constitutive model with a thermoelastoviscoplastic one, see e.g. [25]. With this kind of constitutive model, it is possible to compute the residual stresses.
Fluid approach
In the fluid model, the material is assumed to be incompressible and this constraint is incorporated into the equations to be solved.
Thermomechanical contact
A perfect sticking thermomechanical contact is considered between the tool and the workpiece. It means that the temperature field and the displacement field are continuous through the interface between the tool and the workpiece. Like some authors [26–28], we assume that the heat produced by the friction between the tool and the workpiece is negligible versus the heat generated by plastic deformations.
Comparison of numerical results
In this paper, the numerical results of the solid approach are compared with the already validated model based on the fluid approach (see [2, 6, 10, 15]). In this example, the section of the pin is an equilateral triangle (Figs. 4, 5). The dimensions of the tool are presented in Fig. 4. The width of the two plates is 50 mm, the thickness is 4.7 mm and the simulated length is 100 mm. The rotation axis of the tool is located at the centre of the simulated region (see Fig. 5).

density: 2700 kg m\(^{3}\)

bulk modulus: 69 GPa (used only with the solid approach)

thermomechanical NortonHoff law (presented in the page 4) with \(\mu = 100\) MPa, \(m = 0.12\),

heat conductivity: 120 W m\(^{1}\) K\(^{1}\)

thermal expansion coefficient: \(1 \times 10^{6}\) K\(^{1}\)

heat capacity: 875 J kg\(^{1}\) K\(^{1}\)

density: 7800 kg m\(^{3}\);

heat conductivity: 43 W m\(^{1}\) K\(^{1}\);

heat capacity: 460 J kg\(^{1}\) K\(^{1}\).

Conduction on the lower side of both plates (approximation of the thermal behaviour of the backing plate), exchange coefficient: 4500 W m\(^{2}\) K\(^{1}\);

Convection and radiation on the free upper side of both plates (except the part in contact with the tool), convection coefficient: 10 W m\(^{2}\) K\(^{1}\), emissivity coefficient: 0.2.
Figures 6 and 7 expose respectively the mesh of the solid model and the one of the fluid model. Figures 8, 9, 10 and 11 show the evolution of the pressure and the evolution of the temperature computed by the two models with the rotation speed of 40 RPM at the control points and along the control line defined in Fig. 5. Figures 12, 13, 14 and 15 present the same comparison with the rotation speed of 100 RPM. Points 1 and 2 have the same rotational speed as the tool (these points move according to the mesh). Point 3 is fixed in space.
After a transient phase which depends on the numerical strategy adopted for each approach the results of both models are very similar for the two values of the rotation speed of the tool (see Figs. 8, 9, 10, 12, 13, 14). The difference of frequency between the pressure at point 3 and the pressure and the temperature at points 1 and 2 is explained by the fact that point 3 is fixed in space while points 1 and 2 have the same rotational velocity as the tool. On the one hand, the pressure at point 3 is affected by the three corners of the pin. On the other hand, the frequency of the pressure and the temperature at points 1 and 2 are controlled by the rotation speed of the tool. Consequently, the pressure frequency at point 3 is three times higher than the frequency of the pressure or the temperature at points 1 or 2.
CPU time in hours versus the model type
Rotation speed  Solid model  Fluid model 

40 RPM  59  37 
100 RPM  131  79 
Conclusion
The phenomena happening during the friction stir welding (FSW) process are at the interface between solid mechanics and fluid mechanics. In this paper, two different formulations are presented to simulate the FSW process numerically. One 3D model is based on a solid approach which computes the position and the temperature fields and another one is based on a fluid approach written in terms of velocity, pressure and temperature fields. Both models use advanced numerical techniques such as the Arbitrary Lagrangian Eulerian formalism or remeshing operations or an advanced stabilization algorithm. These advanced numerical techniques allow the simulation of the FSW process with a tool with no rotational symmetry. The aim of the paper is to compare two computational models based respectively on a solid and a fluid approach for the solution of FSW process. Based on the authors’ point of view, being able to simulate a process using a solid model and at the same time a fluid model, is numerically very interesting and represents a further verification of the implementation in both approaches. The presented example (with a triangular pin) shows that the two formulations essentially deliver the same results. Nevertheless, each model has its specificities. The computation of the next time step with the fluid model only requires the nodal values. The history of the internal variables is not necessary. This specificity allows the fluid model to use less CPUintensive numerical techniques. Thus, the fluid model is more efficient from a computational point of view. The downside is that this model is limited to a thermoviscoplastic constitutive model. On the other hand, the model based on the solid approach has the advantage that it can be used with any thermoelastoviscoplastic constitutive model. Therefore, the solid model can be used to predict the FSW process and to also compute the residual stresses after the end of the process.
Declarations
Author's contributions
The Belgian authors worked on the model based on the solid approach and the Spanish authors worked on the one based on the fluid approach. All authors have read and approved the final manuscript
Acknowledgements
The Belgian authors wish to acknowledge the Walloon Region for its financial support to the STIRHETAL project (WINNOMAT program, convention number 0716690) and to the FSWPME project (Programme de recherche collective 2013 – convention number 1217826) in the context of which this work was performed.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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