The computational modeling and numerical simulation of Friction Stir Welding (FSW) processes is an extremely challenging task due to the highly nonlinear and coupled nature of the physical problem and the complex computational issues that need to be properly tackled in the numerical model

Agelet de Saracibar, Carlos

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metalsEditorialChallenges to Be Tackled in the ComputationalModeling and Numerical Simulation ofFSW ProcessesCarlos Agelet De SaracibarDepartment of Civil and Environmental Engineering, School of Civil Engineering, Technical University ofCatalonia, UPC BarcelonaTech, 08034 Barcelona, Spain; carlos.ageletdesaracibar@gmail.comReceived: 10 May 2019; Accepted: 15 May 2019; Published: 17 May 2019The computational modeling and numerical simulation of Friction Stir Welding (FSW) processes isan extremely challenging task due to the highly nonlinear and coupled nature of the physical problemand the complex computational issues that need to be properly tackled in the numerical model [1–6].1. Physical ModelThe FSW process is a complex problem due to the highly nonlinear and coupled nature of thephysical problem. Different physical phenomena occur during the welding process, involving thethermal and mechanical interactions. The temperature field is a function of many welding parameterssuch as welding speed, welding sequence and environmental conditions. Formation of distortions andresidual stresses in workpieces depend on many interrelated factors such as thermal field, materialproperties, structural boundary conditions and welding conditions. The challenging issues in physicalmodeling of the FSW process are divided into four parts.1.1. Complex Thermal BehaviorHeat transfer mechanisms including convection, radiation and conduction have a significant roleon the process behavior. Convection and radiation fluxes dissipate heat significantly through theworkpieces to the surrounding environment, while conduction heat flux occurs between the workpiecesand the support.1.2. Non-Linear Behavior and Localized NatureThe mechanical behavior during FSW is non-linear due to the high strain rates and visco-plasticmaterial. The strong non-linear region is limited to a small area and the remaining part of the model ismostly linear. However, the exact boundaries of the non-linear zone are not known a priori. Knowledgeof strain rate is important for understanding the subsequent evolution of grain structure, and it servesas a basis for verification of various models as well.1.3. Coupled NatureThe thermal and mechanical problems are strongly coupled. The mechanical effects coupled tothe thermal ones include internal heat generation due to plastic deformations or viscous effects, heattransfer between contacting bodies, heat generation due to friction, etc. The thermal effects are alsocoupled to the mechanical ones; for instance, thermal expansion, temperature-dependent mechanicalproperties, temperature gradients in workpieces, etc. An adequate physical model of the weldingprocess must account for all these phenomena including thermal, mechanical and coupling aspects.Metals 2019, 9, 573; doi:10.3390/met9050573 www.mdpi.com/journal/metalsMetals 2019, 9, 573 2 of 51.4. Thermo-Mechanical Frictional Contact NatureThermo-mechanical frictional contact between the tool and the workpieces plays a crucial role.Interactions between the contacting bodies include impenetrability, frictional stresses, heat generationdue to friction and thermal conduction at the contact interface. An adequate physical model of theFSW process must properly account for all those phenomena.2. Numerical ModelThe numerical simulation of the FSW process by the Finite Element Method (FEM) has manycomplex and challenging aspects that are difficult to deal with. The welding process is described bythe momentum and energy balance equations governing the coupled thermo-mechanical problem.Both governing equations are non-linear and this has important implications upon the complexity ofthe numerical model. Consequently, a robust and efficient numerical strategy is crucial for solvingsuch highly non-linear coupled FE equations. Numerical simulation of the FSW process can be carriedout at a local or global level [1,2]. In local level analysis, the focus of the simulation is the Heat AffectedZone (HAZ). The simulation is intended to compute the heat power generated either by visco-plasticdissipation or by friction at the contact interface. At this level, the relevant process phenomena arethe relationship between welding parameters, the contact mechanisms in terms of applied normalpressure and friction coefficient, the setting geometry, the material flow within the HAZ, its size andthe corresponding consequences on the microstructure evolution, etc. A simulation carried out atglobal level studies the entire component to be welded. In this case, a moving heat power source isapplied to a control volume representing the actual HAZ at each time-step of the analysis. The effectsinduced by the welding process on the structural behavior, such as distortions, residual stresses orweaknesses along the welding line, are the target of this kind of study. The challenging issues innumerical modeling of the FSW process are divided into the following seven parts.2.1. Mechanical ProblemThe mechanical problem is governed by the momentum balance equation. A quasi-static mechanicalanalysis can be assumed as the inertia effects in welding processes are negligible due to the high viscositycharacterization. At local level, the volumetric changes are found to be negligible, and incompressibilitycan be assumed. To deal with the incompressible behavior, a very convenient and common choice is todescribe the formulation splitting the stress tensor into its deviatoric and volumetric parts. Dealing withthe incompressible limit requires the use of mixed velocity-pressure interpolations. The problem suffersfrom instability if the standard Galerkin FE formulation is used, unless compatible spaces for the pressureand the velocity fields are selected (LBB stability condition). Due to this, pressure instabilities appear ifequal velocity-pressure interpolations are used. Thus, the challenging issue of pressure stabilization risesup [1–3,6]. The welding process is characterized by very high strain rates as well as a wide temperaturerange going from the environmental temperature to the melting point. Hence, the constitutive lawsadopted should depend on both variables. At typical welding temperatures, the large strain deformationis mainly visco-plastic. Depending on the scope of the analysis, rigid-visco-plastic or elasto-visco-plasticconstitutive models can be used. Not only the prediction of the temperature evolution, but the accurateresidual stress evaluation field generated during the process is the objective of the FSW simulation.The selected constitutive model must appropriately define the material behavior and has to be calibratedby the temperature evolution. The challenge arises from the extremely non-linear behavior of theseconstitutive models and, therefore, from the numerical point of view, a special treatment is obligatory.Moreover, the localized large strain rates usually involved in FSW processes make the problem evenmore complex.Metals 2019, 9, 573 3 of 52.2. Thermal ProblemThe thermal problem is defined by the energy balance equation. In FSW simulation, the plasticdissipation term appearing in the energy equation has a critical role on the process behavior and itis the main source of internal heat generation. The definition of the heat source is one of the keypoints when studying the welding process. In global level simulations, the mesh density used todiscretize the geometry is not usually fine enough to define the welding pool shape or a non-uniformheat source. This is only done if the simulation of the welding pool is the objective itself (local levelanalysis). If the global structure is considered (global level analysis), the size of the heat source is of thesame dimension than the element size generally used for a thermo-mechanical analysis. Therefore,in a global level analysis the resulting mesh density is usually too coarse to represent the actualshape of the heat source. Depending on the kinematic framework used to describe the formulation ofthe coupled thermo-mechanical problem, a convective term might appear in the thermal governingequations. Therefore, convection instabilities of the temperature appear for convection dominatedproblems [3,6]. It is well known that in diffusion dominated problems, the solution is stable. However,in convection dominated problems, the stabilizing effect of the diffusion term becomes insufficient andoscillations appear in the temperature field. The threshold between stable and unstable solutions isusually expressed in terms of the Peclet number.2.3. Kinematic FrameworkEstablishing an appropriate kinematic framework for the simulation of FSW processes is a keyissue. If the welding process is studied at global level, the use of a Lagrangian framework is anappropriate choice for the description of the problem. The Lagrangian reference frame allows easytracking of free surfaces and interfaces between different materials. In a local simulation, the main focusof the simulation is the HAZ where the use of a Lagrangian framework is not always advantageous.In the HAZ, the large distortions would require continuous re-meshing. The alternative is to useEulerian or Arbitrary Lagrangian Eulerian (ALE) methods. The Eulerian formulation facilitates thetreatment of large distortions in the fluid motion. Its handicap is the difficulty to follow free surfacesand interfaces between different materials or different media. An Arbitrary Lagrangian Eulerian(ALE) formulation is particularly useful in flow problems involving large distortions in the presenceof mobile and deforming boundaries. In the simulation of FSW, it is adroit to introduce an aproposkinematic framework for the description of different parts of the computational domain [4,6]. Despitethe efficiency of the idea, the mesh moving strategy and the treatment of the domains’ interactionare challenging.2.4. Thermo-Mechanical Frictional Contact ProblemThe computational modeling of the thermo-mechanical frictional contact between the tool andworkpieces is a key issue in the numerical simulation of the FSW process [1,2,6]. The computationalmodel must accurately deal with contact impenetrability, frictional behavior, heat generated by frictionand heat transfer due to thermal contact at the contact interface. Penalty-based methods, such as thepenalty method or the Uzawa’s version of the augmented Lagrangian method, Lagrange multipliersor direct elimination methods, can be used to model the mechanical frictional contact interaction.Within the framework of a fluid mechanics approach, a Norton thermo-frictional contact model can beused to compute the tangential component of the traction vector at the contact interface in terms ofthe variation of the relative slip velocity. The heat flux generated by friction at the contact interfacebetween the tool and the workpieces can be split into two parts, that is, a part absorbed by the tool anda part absorbed by the workpieces, where the amount of heat absorbed by the tool and the workpiecesdepends on the thermal diffusivity of the two materials in contact. Alternatively, as a limit case, fullstick thermo-mechanical contact conditions between the tool and the workpieces can be also considered.Metals 2019, 9, 573 4 of 5In this case, the temperature and velocity fields are continuous through the contact interface betweenthe tool and the workpieces.2.5. Coupled ProblemThe numerical solution of the coupled thermo-mechanical problem involves the transformation ofan infinite dimensional transient system into a sequence of discrete non-linear algebraic problems [6].This is achieved by means of the FE spatial discretization procedure, a time-marching scheme for theadvancement of the primary nodal variables and a time integration algorithm to update the internalvariables of the constitutive equations. Regarding the time-stepping schemes, two types of strategiescan be applied to the solution of the coupled thermo-mechanical problems. The first possibility is to usea monolithic (simultaneous) time-stepping algorithm which solves both the mechanical and the thermalproblems together. It advances all the primary nodal variables of the problem simultaneously. The mainadvantage of this method is that it enables stability and convergence of the whole coupled problem.However, in simultaneous solution procedures, the time-step as well as time-stepping algorithm has tobe equal for all subproblems, which may be inefficient if different time scales are involved in the thermaland the mechanical problem. Another important disadvantage is the considerably high computationaleffort required to solve the monolithic algebraic system and the necessity to develop software andsolution methods specifically for each coupled problem. A second possibility is a staggered algorithm(block-iterative or fractional-step), where the two sub-problems are solved sequentially. Usually,a staggered solution, arising from an operator split and a product formula algorithm (PFA), yieldssuperior computational efficiency. Staggered solutions are based on an operator split, applied tothe coupled system of non-linear ordinary differential equations, and a product formula algorithm,which, within the framework of classical fractional step methods, leads to a splitting of the originalmonolithic problem into two smaller and better conditioned sub-problems. This leads to the partitionof the original problem into smaller and typically symmetric (physical) subproblems. After this, theuse of different standard time-stepping algorithms developed for the uncoupled sub-problems isstraightforward, and it is possible to take advantage of the different time scales involved. The majordrawback of these methods is the possible loss of accuracy and stability. However, it is possible toobtain unconditionally stable schemes using this approach, providing that the operator split preservesthe underlying dissipative structure of the original problem.2.6. Particle TracingOne of the main issues in the study of FSW at local level is the heat generation. The generatedheat must be enough to allow for the material to flow and to obtain a deep HAZ. Insufficient heatforms voids as the material is not softened enough to flow properly. The visualization of the materialflow is a very useful tool to understand its behavior during the weld. It can be used to investigatethe appropriate process parameters to create a qualified joint. However, following the position of thematerial during the welding process is not an easy task, neither experimentally (needs metallographictools) or numerically. This is why establishing a numerical method for the visualization of the materialtrajectory in order to gain insight to the HAZ and the material penetration within the thickness ofthe workpieces is one of the key issues of the numerical simulation. Particle tracing is a methodused to simulate the motion of material points, following their positions at each time-step of theanalysis [5]. In the Lagrangian framework the trajectories are given by the displacement field. Whenusing Eulerian and ALE framework the solution does not give directly information about the materialposition. However, the velocity field obtained can be integrated to get an insight of the extent ofmaterial mixing during the weld. Integration of the velocity field is proposed at post-process levelto follow the material motion. An appropriate time integration method for the solution of the ODEin order to track the particles is needed. Moreover, a search algorithm must be executed to find theposition of the material points if Eulerian or ALE meshes are used.Metals 2019, 9, 573 5 of 52.7. Residual StressesGenerally, FSW yields fine microstructures, absence of cracking, low residual distortion, and noloss of alloying elements. Nevertheless, as in the traditional fusion welds, a softened HAZ and a tensileresidual stress field appear. Although the residual stresses and distortion are smaller in comparisonwith those of traditional fusion welding, they cannot be ignored, especially when welding thin platesof large size. In the local level analysis, the focus of the study is the HAZ and a visco-plastic model isused to characterize the material behavior. Elastic stresses are neglected, and thus, the calculation ofresidual stresses is not possible. However, at global level, the residual stresses are one of the mainoutcomes of the process simulation using an elasto-visco-plastic constitutive model. The use of alocal-global coupling strategy has been proposed as a method to obtain the residual stress field, as thisa challenging issue [2].Conflicts of Interest: The authors declare no conflicts of interest.References1. Dialami, N.; Chiumenti, M.; Cervera, M.; Agelet de Saracibar, C. Challenges in thermo-mechanical analysisof Friction Stir Welding processes. Arch. Comput. Methods Eng. 2017, 24, 189–225. [CrossRef]2. Dialami, N.; Cervera, M.; Chiumenti, M.; Agelet de Saracibar, C. Local-global strategy for the prediction ofresidual stresses in FSW processes. Int. J. Adv. Manuf. Technol. 2017, 88, 3099–3111. [CrossRef]3. Agelet de Saracibar, C.; Chiumenti, M.; Cervera, M.; Dialami, N.; Seret, A. Computational modeling andsub-grid scale stabilization of incompressibility and convection in the numerical simulation of friction stirwelding processes. Arch. Comput. Methods Eng. 2014, 21, 3–37. [CrossRef]4. Dialami, N.; Chiumenti, M.; Cervera, M.; Agelet de Saracibar, C. An apropos kinematic framework for thenumerical modeling of friction stir welding. Comput. Struct. 2013, 117, 48–57. [CrossRef]5. Dialami, N.; Chiumenti, M.; Cervera, M.; Agelet de Saracibar, C.; Ponthot, J.P.; Bussetta, P. NumericalSimulation and Visualization of Material Flow in Friction Stir Welding via Particle Tracing. In NumericalSimulations of Coupled Problems in Engineering; Idelsohn, S., Ed.; Springer: Cham, Switzerland, 2014; Volume33, pp. 157–169. ISBN1 978-3-319-06135-1. (Print); ISBN2 978-3-319-06136-8. (Online). [CrossRef]6. Chiumenti, M.; Cervera, M.; Agelet de Saracibar, C.; Dialami, N. Numerical modeling of friction stir weldingprocesses. Comput. Methods Appl. Mech. Eng. 2013, 254, 353–369. [CrossRef]© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Challenges to be tackled in the computational modeling and numerical simulation of FSW processes

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Challenges to be tackled in the computational modeling and numerical simulation of FSW processes

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