A Computational Model for the Numerical Simulation

A Computational Model for the Numerical Simulation of

FSW Processes C. Agelet de Saracibara, M. Chiumentia, D. Santiagob, M. Cerveraa, N. Dialamia and G. Lomberab a

International Center for Numerical Methods in Engineering (CIMNE)

Building C1, Campus Norte, UPC, Gran Capitán s/n, 08034 Barcelona, Spain

b Grupo de Ingeniería Asistido por Computadora, Universidad Nacional de Mar del Plata,

J.B. Justo 4302, 7600 Mar del Plata, Argentina

Abstract. In this paper a computational model for the numerical simulation of Friction Stir Welding (FSW) processes is presented. FSW is a new method of welding in solid state in which a shouldered tool with a profile probe is rotated and slowly plunged into the joint line between two pieces of sheet or plate material which are butted together. Once the probe has been completely inserted, it is moved with a small tilt angle in the welding direction. Here a quasi-static, thermal transient, mixed multiscale stabilized Eulerian formulation is used. Norton-Hoff and Sheppard-Wright rigid thermo­ viscoplastic material models have been considered. A staggered solution algorithm is defined such that for any time step, the mechanical problem is solved at constant temperature and then the thermal problem is solved keeping constant the mechanical variables. A pressure multiscale stabilized mixed linear velocity/linear pressure finite element interpolation formulation is used to solve the mechanical problem and a convection multiscale stabilized linear temperature interpolation formulation is used to solve the thermal problem. The model has been implemented into the in-house developed FE code COMET. Results obtained in the simulation of FSW process are compared to other numerical results or experimental results, when available. Keywords: Friction Stir Welding, Thermomechanical Modeling, Stabilization Methods.

PACS: 02.60 Cb, 02.70 Dh, 44.05.+e, 46.15.-x, 47.11.Fg

INTRODUCTION Friction Stir Welding (FSW) is a new method of welding in solid state, created and patented by “The Welding Institute” (TWI) in 1991 [1]. In FSW a cylindrical, shouldered tool with a profiled probe is rotated and slowly plunged into the joint line between two pieces of sheet or plate material, which are butted together. The parts have to be clamped onto a backing bar in a manner that prevents the abutting joint faces from being forced apart. Once the probe has been completely inserted, it is moved with a small tilt angle in the welding direction. The shoulder applies a pressure on the material to constrain the plasticized material around the probe tool. Due to the advancing and rotating effect of the probe and shoulder of the tool along the seam, an advancing side and a retreating side are formed and the softened and heated material flows around the probe to its backside where the material is consolidated to create a high-quality solid-state weld. The maximum temperature reached is of the order of 80% of the melting temperature. Despite the simplicity of the procedure, the mechanisms behind the process and the material flow around the probe tool are very complex and the computational modeling of FSW processes is one of the most interesting research topics over the last years [2-7]. The material is extruded around the rotating tool and a vortex flow field near the probe due to the downward flow is induced by the probe thread. The process can be regarded as a solid phase keyhole welding technique since a hole to accommodate the probe is generated, then filled during the welding sequence. The material flow depends on welding process parameters, such as welding and rotation speed, pressure, etc., and on the characteristics of the tools, such as materials, design, etc. The first applications of FSW have been in aluminum fabrications. The weld quality is excellent, with none of the porosity that can arise in fusion welding, and the mechanical areOF at EACH least as good as the best achievable CREDIT LINE (BELOW) TO BE INSERTED ON THEproperties FIRST PAGE PAPER EXCEPT THE PAPER ON PP. 837 TO 840

CP1252, NUMIFORM 2010, Proceedings of the10th International Conference edited by F. Barlat, Y. H. Moon, and M.G. Lee © 2010 American Institute of Physics 978-0-7354-0800-5/10/$30.00

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by fusion welding. The process is environmentally friendly, because no fumes or spatter are generated, and there is no arc glare or reflected laser beams with which to contend. Another major advantage is that, by avoiding the creation of a molten pool which shrinks significantly on re-solidification, the distortion after welding and the residual stresses are low. With regard to joint fit up, the process can accommodate a gap of up to 10% of the material thickness without impairing the quality of the resulting weld. As far as the rate of processing is concerned, for materials of 2 mm thickness, welding speeds of up to 2 m min-1 can be achieved, and for 5 mm thickness up to 0.75 m.min-1. Recent tool developments are confidently expected to improve on these figures.

COMPUTATIONAL MODEL The flow of the material around a FSW tool is characterized by a Reynolds number which is much smaller than 1, typically around 1.0E-04 due to the small length scale, the low velocities and the very high viscosity of the material. For these values of the Reynolds number, the inertial forces of the linear momentum balance equation can be neglected and a quasi-static analysis can be performed [7]. The deformation of the material taking place around a FSW tool is extremely high. The computational modelling of the material flow around a FSW tool using a Lagrangian formulation requires continuous remeshing to avoid extremely distorted mesh elements. Therefore, the use of alternative formulations, such as ALE [4,6] or Eulerian formulations [2-3, 7], is a better choice. In this work we will use an Eulerian formulation. On the other hand, the Peclet number for a FSW process typically ranges from 1.0E+01 to 1.0E+03. Within this range, the convective term of the spatial energy balance equation cannot be neglected and transient conditions will be considered [7]. Coupled thermo-mechanical rigid-viscoplastic material constitutive models, such as the Norton-Hoff [2-3,7] or the SheppardWright [6,7] models, will be considered. The resulting coupled thermo-mechanical problem will be solved using a product formula algorithm, leading to a staggered solution algorithm. A mechanical problem, involving mechanical variables as unknowns, is defined at constant temperature and a thermal problem, involving the temperature as unknown, is defined at constant configuration [7]. A multiscale pressure stabilized mixed linear velocity/linear pressure finite element interpolation formulation will be used to solve the mechanical problem [8-16] and a convection stabilized linear temperature interpolation formulation will be used to solve the thermal problem [7].

Mixed Strong Form of the Quasi-Static Transient Coupled Thermomechanical Problem in Eulerian Form



Let us consider a spatial velocity vector field v ( x,t ) , a spatial pressure field p ( x,t ) and a spatial temperature field θ ( x,t ) . Within the framework of a coupled thermo-mechanical mixed velocity-pressure formulation, the Cauchy stress tensor field is given by an appropriate constitutive equation as a function of the velocity, pressure and temperature fields, such that,

σ ( v, p, θ ) = p 1 + s ( v, θ )

(1)

where s ( v, θ ) := dev σ ( v, p, θ ) is the deviatoric part of the Cauchy stress tensor. Appropriate boundary conditions are considered such as a prescribed velocity field v ( x,t ) on ∂ v Ω , prescribed traction field t ( x,t ) on ∂σ Ω , a prescribed temperature field field per unit of surface q ( x,t ) on

θ ( x,t )

on ∂θ Ω and prescribed normal heat flux

∂qΩ .

The strong form of the quasi-static transient coupled thermo-mechanical problem in mixed form and using a spatial Eulerian formulation is defined by the momentum balance equation, the incompressibility equation and the energy balance equation (neglecting elastoplastic heating terms) and can be stated as [7]: Find a velocity vector field v ( x,t ) ∈V , a pressure field p ( x, t ) ∈Q and a temperature field

θ ( x,t ) ∈ T

, such that the following equations hold in

Ω:

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∇p + ∇ ⋅ s ( v, θ ) + f = 0 ∇⋅v −

c0 where

(2)

∂eθ (θ ) − v ⋅ ∇eθ (θ ) = 0 ∂t

(3)

∂θ + c0 v ⋅ ∇θ = −∇ ⋅q (θ ) + R + D ∂t

(4)

f is the body forces vector per unit of volume, eθ (θ ) is the volumetric thermal strain (taking into account

the thermal expansion in solid state and the thermal shrinkage during the liquid-solid phase change), c0 is the heat capacity, q is the heat flux vector per unit of surface, R is an internal heat source rate per unit of volume and D is the internal dissipation rate per unit of volume. Additionally, appropriate constitutive equations for the deviatoric part of the Cauchy stress, as a function of the velocity (or the velocity and the temperature), and for the heat flux per unit of surface, as a function of the temperature, have to be supplied.

Constitutive Equations In this work rigid plastic Norton-Hoff [2-3,7] and Sheppard-Wright [6-7] constitutive material models are considered. Furthermore, Fourier law is used as thermal constitutive equation. The rigid-plastic Norton-Hoff constitutive model is given by [7],

s ( v,θ ) = 2μ (θ ) where

μ (θ )

(

3ε& ( v )

)

m (θ ) −1

ε& ( v )

(5)

is a temperature dependent viscosity parameter, m (θ ) is a temperature dependent rate sensitivity

s parameter, ε& ( v ) = ∇ v is the deformation rate, defined as the symmetric part of the spatial velocity gradient, and

ε&

is the equivalent strain rate given by,

ε& = ( ε& : ε& )

ε& = 2 3 ε&

12

(6)

The Sheppard-Wright constitutive material model is given by [6-7],

s ( v, θ ) = 2 μ ( ε& ( v ) , θ ) ε& ( v ) where the strain rate and temperature-dependent viscosity parameter

μ ( ε& ( v ) ,θ ) =

(7)

μ ( ε& ( v ) , θ ) is given by,

σ e ( ε& ( v ) , θ ) 3ε& ( v )

(8)

where the strain rate and temperature-dependent yield stress is given by, (9)

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1n 2n ⎡⎛ ⎛ Z ( ε& ( v ) ,θ ) ⎞ ⎤⎥ Z ( ε& ( v ) ,θ ) ⎞ ⎢ ⎟ + 1+ ⎜ ⎟ σ e ( ε& ( v ) , θ ) = log ⎢⎜ ⎜ ⎟ ⎜ ⎟ ⎥ A A α ⎠ ⎝ ⎠ ⎥⎦ ⎢⎣⎝

1

(

)

where the Zener-Hollomon parameter Z ε& ( v ) , θ , representing the temperature compensated effective strain rate, is given by,

⎛ Q ⎞ Z ( ε& ( v ) , θ ) = ε& ( v ) exp ⎜ ⎟ ⎝ Rθ ⎠

(10)

where α , A and n are material parameters, R is the universal gas constant and Q is the activation energy. The thermal constitutive equation for the heat flux per unit of surface is defined by the Fourier law given by,

q (θ ) = −k (θ ) ∇θ

(11)

where k (θ ) is the temperature-dependent thermal conductivity.

Time-Discrete Multiscale Stabilized Mixed Variational Form of the Quasi-Static Transient Coupled Thermomechanical Problem in Eulerian Form In the time-discrete setting, the multiscale stabilized discrete mixed variational equations can be written as [7,12­ 16],

(p (∇ ⋅ v

h , n +1

h , n +1

, ∇ ⋅ δ v h ) + ( s h , n+1 ,∇ sδ v h ) = lmh ,n+1 (δ v h ) ∀δ v h ∈V0,h

nelem ⎛ 1 ⎞ , δ ph ) − ⎜ ( ehθ,n +1 − ehθ, n ) + v h ,n +1 ⋅∇ehθ, n +1 , δ ph ⎟ − ∑ e=1 τ e(1), n +1 Ph⊥ ( ∇ph , n +1 ) ,∇δ ph ⎝ Δt ⎠

(

(

− ∑ e=1 τ e(1),n +1 Ph⊥ ( ∇ph ,n +1 ) ⋅∇eθ (θ h ,n +1 ) , δ ph nelem

)

e

= 0 ∀δ ph ∈Q h

(12)

)

− e

(13)

⎛ c0 ⎞ ⎜ Δt (θ h ,n +1 − θ h ,n ) + c0 v h ,n+1 ⋅ ∇θ h ,n+1 , δθ h ⎟ − ( q h ,n +1 , ∇δθ h ) − ( D h ,n +1 , δθ h ) + ⎝ ⎠ + ∑ e=1 τ e(2) , n +1 ( c0 v h , n +1 ⋅∇θ h , n +1 , c0 ∇ ⋅ v h , n +1δθ h ) + nelem

e

+ ∑ e=1 τ nelem

(2) e , n +1

(c v 0

h , n +1

⋅∇θ h ,n +1 , c0 v h ,n +1 ⋅∇δθ h ) +

(

+ ∑ e=1 τ e(1),n +1 c0 Ph⊥ ( ∇ph , n +1 ) ⋅∇θ h , n +1 , δθ h nelem

)

(14)

e

e

= lth ,n+1 (δθ h ) ∀δθ h ∈ T0, h

where, within the framework of the sub-grid scale (SGS) stabilization method [8], a zero sub-grid scale pressure field has been assumed and the sub-grid scales for the velocity and temperature fields for linear velocity, linear pressure and linear temperature elements, at the element level, have been approximated as [7], (15)

84

v% n+1 e = τ e(1),n +1 Ph⊥ ( ∇ph ,n +1 ) , θ%n+1 = −τ e(2) , n +1 c0 v h , n +1 ⋅∇θ h , n +1 e

⊥

where Ph

(•)

⊥

is the orthogonal projection operator defined as Ph

( • ) = ( • ) − Ph ( • ) ,

Ph ( • ) is the L2 ( • )

projection operator onto Vh , and τ e , n+1 and τ e , n+1 are mesh-dependent stabilization parameters defined as [7], (1)

τ where

(1) e , n +1

:= c1h

(2)

2

∇ s v h , n +1 s h , n +1

,

τ e(2) , n +1 := c2

he c0 v h , n +1

(16)

c1 and c2 are constants and h and he denote mesh element sizes.

(

Let us introduce now a new vector field Π h , n +1 := Ph ∇ph , n+1

) defined as the projection of the pressure

ϒ = H1 and ϒ h ⊂ ϒ as the space of pressure gradient projection and its finite element associated space, respectively. Taking Π h , n +1 ∈ ϒ h as an independent

gradient onto the finite element space [7,9-16]. Let us also introduce

continuous variable, the orthogonal projection of the discrete pressure gradient can be written as [7, 9-16],

Ph⊥ ( ∇ph , n +1 ) = ∇ph , n +1 − Π h , n +1

(17)

Operator Split of the Time-Discrete Stabilized Mixed Variational Form of the Quasi-Static Transient Coupled Thermomechanical Problem in Eulerian Form The time discrete stabilized variational coupled thermo-mechanical problem defined by Equations 12-14 can be solved using a staggered algorithm arising from an operator split and a product formula algorithm (PFA) [7]. Within this context, a mechanical problem and a thermal problem are defined. The mechanical problem, getting the discrete velocity, pressure and continuous pressure gradient projection as mechanical variables, is defined holding constant the discrete temperature field, while the thermal problem, getting the discrete temperature as thermal variable, is defined holding constant the mechanical variables [7]. A staggered algorithm is defined such that for any time step, the mechanical problem is solved first at constant temperature and then the thermal problem is solved keeping constant the mechanical variables, velocity, pressure and pressure gradient projection [7].

COMPUTATIONAL SIMULATIONS This example shows the 2D transient coupled thermo-mechanical computational simulation of a FSW process [7]. Two rectangular aluminium plates of 10 cm x 5 cm and a circular tool of 0.6 cm diameter are considered. An advancing velocity of 40 cm/min and different rotational velocities of 0, 20, 40 and 80 rpm have been considered. An initial and environmental temperature of 20 ºC has been assumed. A Norton-Hoff material model has been used for the aluminium alloy plates. The two parameters of the Norton-Hoff material model, assumed to be constants, are given in Table 1. The thermal properties of the aluminium alloy are given in Table 2. Figure 1 shows the finite element discretization of the plates and tool using P1/P1 GLS pressure stabilized elements, including a detail of the finite element discretization around the tool area. Three different finite element mesh discretizations, with 4000, 5800 and 8100 elements, representing element sizes at the limit layer of 0.05, 0.01 and 0.005 cm, respectively, have been considered in the simulations. Influence of the dilatation parameter has been analysed and simulations assuming a zero-dilatation coefficient have been also carried out. Table 3 shows the maximum temperature obtained in the simulations for the 5800 element mesh and the four different rotational velocities considered, using the in-house developed FE software COMET [17]. Those results are

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FIGURE 2. From left to right: Temperature maps for rotational velocities of 20, 40 and 80 rpm, using COMET (Top) and FW (Bottom).

FIGURE 3. Temperature map distributions. Tool rotational velocity of 80 rpm using COMET; (a) α=0 ºC-1; (b) α=2.40e-5 ºC-1.

( ) ( ) FIGURE 4. Pressure map and streamlines. Tool rotational velocity of 80 rpm using COMET; (a) α=0 ºC-1; (b) α=2.40e-5 ºC-1.

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ACKNOWLEDGMENTS

This research work has been carried out within the framework of the STREP project Detailed Multi-Physics Modelling of Frictional Stir Welding (DEEPWELD) of the 6th Framework Programme of the EC, the PROFIT project “Nuevas Herramientas para Optimizar el Proceso de Soldadura por Fricción” (FSWNET) of the “Ministerio de Educación y Ciencia” of Spain and the project “Simulación Numérica del Proceso de Soldadura Mediante Batido por Fricción” (FSW) of the “Plan Nacional de I+D+I (2004-2007)” of the “Ministerio de Educación y Ciencia” of Spain. Those financial supports are gratefully acknowledged.

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