A new smoothed particle hydrodynamics non-Newtonian model for friction stir welding: Process...

International Journal of Plasticity xxx (2013) xxx–xxx

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International Journal of Plasticity

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A new smoothed particle hydrodynamics non-Newtonianmodel for friction stir welding: Process modeling andsimulation of microstructure evolution in a magnesium alloy

0749-6419/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijplas.2013.02.013

⇑ Corresponding author. Tel.: +1 509 375 6686; fax: +1 509 372 4720.E-mail address: [email protected] (W. Pan).

Please cite this article in press as: Pan, W., et al. A new smoothed particle hydrodynamics non-Newtonian model for friction stir wProcess modeling and simulation of microstructure evolution in a magnesium alloy. Int. J. Plasticity (2013), http://dx.doi.org/1j.ijplas.2013.02.013

Wenxiao Pan a,⇑, Dongsheng Li a, Alexandre M. Tartakovsky a, Said Ahzi b,c, Marwan Khraisheh d,Moe Khaleel a

a Pacific Northwest National Laboratory, CSMD, Richland, WA 99352, USAb University of Strasbourg, IMFS-CNRS, 67000 Strasbourg, Francec Georgia Institute of Technology, MSE, Atlanta, GA 30332, USAd Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emirates

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 September 2012Received in final revised form 19 February2013Available online xxxx

Keywords:Friction stir weldingSmoothed particle hydrodynamicsLagrangian particle methodMicrostructure evolution

We present a new smoothed particle hydrodynamics (SPH) model for friction stir welding(FSW). FSW has broad commercial application in the marine, aerospace, rail, and automo-tive industries. However, development of the FSW process for each new application hasremained largely empirical. Few established numerical modeling techniques have beendeveloped that can explain and predict important features of the process physics involvedin FSW. This is particularly true in the areas of material flow and mixing mechanisms. Inthis paper, we present a novel modeling approach to simulate FSW that may have signif-icant advantages over current finite element or finite difference based methods. Unlike tra-ditional grid-based methods, Lagrangian particle methods such as SPH can simulate thedynamics of interfaces, large material deformations, and the material’s strain and temper-ature history without employing complex tracking schemes. Three-dimensional simula-tions of FSW on AZ31 Mg alloy are performed. The temperature history and distribution,grain size, microhardness as well as the texture evolution are presented. Numerical resultsare found to be in good agreement with experimental observations.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Friction stir welding (FSW) is a solid-state joining process in which the joining of similar or dissimilar materials occursbelow the material’s melting temperature. This is achieved by holding the two pieces together and moving a spinning toolalong the joint line. Because of the solid-state nature of the FSW joining process, significant advantages over melt/solidifi-cation joining processes can be realized. FSW has been shown to produce joints that can display lower defect/porosity con-tent, lower residual stresses, greater fatigue life, and better ductility and toughness (Lee et al., 2009). The existing FSWmodels are based on traditional grid-based numerical methods (e.g., finite element method (FEM)) that are able to estimatethe flow field and temperature distribution around the tool but not the interface between materials and the final distributionof the welded materials (Albakri et al., 2011; Aljoaba et al., 2012). For the latter, the grid-based methods require complex andcomputationally expensive front-tracking schemes. In some cases, bends of different materials are created during FSW, andknowledge of the interface position is important in describing degree of mixing as well as the weld quality. Understanding

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2 W. Pan et al. / International Journal of Plasticity xxx (2013) xxx–xxx

material distribution in the weld can be particularly important for joining dissimilar materials. Another important feature ofa successful FSW model is the ability to predict conditions leading to formation of defects, such as wormholes, surface lack offill, root flow defects, etc. Few existing models are able to address these material-flow-related issues, and even fewer cangenerate free surfaces within a solid.

In this paper, we propose a Lagrangian particle model for FSW based on the smoothed particle hydrodynamics (SPH)method (Monaghan, 1992; Monaghan, 2005). SPH was first introduced in the context of astrophysical flow and later appliedto numerous applications in material science, including: high pressure die casting (Cleary and Ha, 2000), injection molding ofliquid-crystalline polymer (Chaubal and Leal, 1988), crack formation and propagation (Benz and Asphaug, 1995), suspensionflows (Tartakovsky et al., 2009), multiphase flow in porous media (Gouet-Kaplan et al., 2009; Tartakovsky et al., 2007) andice sheet dynamics (Pan et al., 2012). Because of its Lagrangian particle nature, SPH has several advantages for modelingFSW: (i) complex interface and free surface flows/deformations, including material coalescence and splitting, can be modeledwithout employing front tracking schemes; (ii) it is easy to track temperature and deformation history of all Lagrangian vol-umes of the welded material, which is important for predicting microstructure evolution that is one of the main focuses ofthe present work; (iii) in a Lagrangian framework, there is no non-linear term in the momentum conservation equation,there is no numerical diffusion in the energy conservation equation, and SPH allows accurate solution of momentum-dom-inated flows; and (iv) coupled, complicated physics such as realistic equations of state, melting/solidification (near-surfaceoverheating), heat generation resulting from viscous dissipation and latent heat generation, strain and strain rate histories,stick–slip conditions at the tool interface, fracturing, void formation, chemistry, and history dependence of material proper-ties can be implemented. Another advantage of the SPH method is it explicitly conserves mass, linear momentum, and en-ergy. In addition, the SPH method is manifestly a Galilean invariant because particle–particle interactions depend on relativeparticle positions and velocity differences. The main disadvantage of the SPH method is that to achieve the same accuracy asin the grid-based methods, the number of discretization points in SPH should be, in general, larger than that in the grid-based methods. This higher computational cost can be partially mitigated by better scalability of the SPH parallel codes.

The SPH method has been used before to simulate FSW of similar materials (Tartakovsky et al., 2006). This two-dimen-sional model assumed that effective viscosity depends on temperature only. Despite this simplistic approach, the SPH modelwas able to reproduce temperature profiles, strain profiles, and density distributions that were qualitatively similar to thoseobserved in real systems. In this work, we extend the SPH model to incorporate an effective viscosity model based on theconstitutive relation proposed by Sellars and Tegart (1972), where the flow stress and effective viscosity depend on strainrate and temperature. We use the SPH model to describe the thermomechanical properties of AZ31 Mg alloy. Specifically,we study the temperature distribution and history, average grain size and its distribution, average microhardness in the pro-cessing zone, as well as the texture evolution during FSW. We find a good agreement between results of the SPH simulationsand experimental observations (Darras et al., 2007; Xin et al., 2010). The objective of this work is to validate the model versusexperimental results, so the stirring conditions and material properties are chosen corresponding to the experiment (Darraset al., 2007). To our best knowledge, no other simulation results (e.g., FEM) for the same material and stirring conditions havebeen published yet. Therefore, an inter-model comparison is not included in the present work.

The paper is organized as follows: in Section 2 we describe the governing Eqs. (2.1); SPH discretization (2.2); and themodel setup, including material properties and boundary conditions (2.3). In Section 3, we present our simulation results,including flow mixing pattern; temperature distribution and history; grain size and microhardness of the welding zone; tex-ture evolution; as well as the comparison against experimental results. Section 4 summarizes our findings.

2. Model description

2.1. Governing equations

We model the welded material as a non-Newtonian fluid. Therefore, the deformation of materials during FSW is describedby the Navier–Stokes equations of mass, momentum, and energy conservation:

PleaseProcesj.ijplas

dqdt¼ �q5 �v; ð1Þ

dvdt¼ �5P

qþ5 � r

q; ð2Þ

cpdTdt¼ � 1

q5 ðk5 TÞ þ 1

qðr : 5vÞ � I: ð3Þ

Here, q is the fluid density; v is flow velocity; r ¼ lð5v þ5vTÞ is the deviatoric stress tensor; P and T are the pressure andtemperature, respectively; I is the sink term describing the heat loss to the atmosphere; cp is the specific heat; and k is thethermal conductivity of material.

cite this article in press as: Pan, W., et al. A new smoothed particle hydrodynamics non-Newtonian model for friction stir welding:s modeling and simulation of microstructure evolution in a magnesium alloy. Int. J. Plasticity (2013), http://dx.doi.org/10.1016/.2013.02.013



W. Pan et al. / International Journal of Plasticity xxx (2013) xxx–xxx 3

2.2. SPH discretization

The preceding governing equations (Eq. (1)–(3)) are discretized by the meshless SPH method, in which the welded mate-rial, welding tool, and the backing plate are discretized with separate sets of points or particles. In SPH, a continuous fieldAðrÞ is approximated using values of A at the discretization points, i.e., AðrÞ ¼

Pi

miqi

AiWðr� ri;hÞ. Here, ri is the position ofa discrete point i and Ai ¼ AðriÞ. Each point possesses mass (mi), density (qi), and volume (mi=qi). Therefore, the discretepoints are considered as physical particles. The SPH weighting function, W, is a bell-shaped function with a kernel length,kh. Here the value of k depends on specific functional form of W, and the interpolation scheme assumes summation overall SPH particles within distance kh from r. In the present work, a cubic spline kernel with continuous first and second deriv-atives is chosen. With this form of W, only particles within 2h distance from r contribute to the summation in the SPH inter-polation scheme. Specifically, the kernel is defined as:


Wðr;hÞ ¼ 1

ph3

1� 32jrjh

� �2þ 3

4jrjh

� �30 6 jrjh 6 1

14 2� jrjh� �3

1 < jrjh 6 2

0 jrjh > 2;

8>>>><>>>>:

ð4Þ

where jrj is the magnitude of the vector r. With the SPH interpolation scheme, the gradient of A can be calculated in terms ofthe gradient of W, i.e., rAðrÞ ¼

Pi

miqi

AirWðr� ri;hÞ. The gradient of the cubic spline function, W, has the form:

rWðr; hÞ ¼ rr1

ph4

94jrjh

� �2� 3 jrjh 0 6 jrjh 6 1

� 34 2� jrjh� �2

1 < jrjh 6 2

0 jrjh > 2:

8>>>><>>>>:

ð5Þ

Thus, with the SPH discretization of continuous fields and their spatial derivatives, the mass, momentum, and energy equa-tions (Eq. (1)–(3)) can be written in the form of a system of ordinary differential equations (ODEs) (Monaghan, 2005) as:

dqi

dt¼ qi

Xj

mj

qjvij � riWðri � rj;

�hijÞ; ð6Þ

dvi

dt¼ �

Xj

mjPi þ Pj

qiqjþPij

!5iWðri � rj;

�hijÞ þ Fbi ; ð7Þ

cp;idTi

dt¼ �

Xj

mj

qiqj

4kikj

ki þ kjðTi � TjÞ

rij � 5iWðri � rj;�hijÞ

jrijj2� 1

2

Xj

mjPijvij � 5iWðri � rj;�hijÞ � RTidi; ð8Þ

where the viscous term Pij is given by:

Pij ¼ �16bvij � rij

jrijjlilj

ðli þ ljÞhqiqj: ð9Þ

Here, rij ¼ ri � rj;vij ¼ vi � vj;vi is the velocity of particle i; �hij ¼ ðhi þ hjÞ=2; Pi is the fluid pressure at position ri, and li is thefluid viscosity that will be determined through the constitutive relation described herein. The attractive feature of Eq. (7) isthat it has an antisymmetric form with respect to indexes i and j (forces acting between SPH particles satisfy the third New-ton law) and satisfies the linear and angular momentum conservation laws exactly.

To satisfy the no-slip boundary condition, the tool particles are included in the calculation of viscous forces acting on thefluid particles (Eq. (9)). A boundary force Fb

i , acting between the boundary particles (tool/backing plate) and fluid particles(sheet material) near the boundary, is added to Eq. (7) to ensure the normal velocity of fluid particles near the boundary iszero (Monaghan and Kajtar, 2009). The boundary force is calculated as:

Fbi ¼

Xj

V2max

2mj

mi þmjWb jrijj

h

� �rij

jrijj1

jrijj � Dpb; ð10Þ

where, particle j is the neighbor boundary particle of fluid particle i;Wbðjrjh Þ is proportional to a Wendland 1D cubic kernel,and Vmax ¼maxijðjvijjÞ. The boundary particle spacing, Dpb, is the same as the fluid particle spacing, Dp.

Heat is assumed to be lost from the sheet surface, backing plate, and tool surface to the surrounding air. R is the heattransfer coefficient. The constant di in Eq. (8) is set to one for all of the particles at the boundary with air and zero otherwise.

The effective viscosity of non-Newtonian fluid is defined as:

le ¼re

3 _�e: ð11Þ





Thus, li in Eq. (9) is replaced with le;i. Here, re is the effective deviatoric stress, defined as the stress needed to sustain theplastic deformation, and the effective strain rate ( _�e) is defined as:


_�e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi23

_�kl _�lk

r: ð12Þ

Here, _�kl are the components of the strain rate tensor, representing the symmetric part of the velocity gradient tensor:

_�kl ¼ 12

@vk

@xlþ @v

l

@xk

� �k; l ¼ 1;2;3: ð13Þ

The SPH discretization of the strain rate tensor, _�, is:

_�i ¼ �12

Xj

mj

qjvij5iWðri � rj;

�hijÞ� �

þ vij5iWðri � rj;�hijÞ

� �Tn o

: ð14Þ

To avoid unrealistically large values of the effective viscosity due to an occasional small effective strain rate, _�e in Eq. (11)is replaced with _�e þ n, where n is a small number. In this work, we choose n ¼ 0:1h.

The constitutive model proposed by Sellars and Tegart (1972) is used to determine the stress to sustain the plastic defor-mation of AZ31 Mg alloy:

re ¼ rRsinh�1 Zb

� �1=m" #

: ð15Þ

Here, rR ¼ 53:3 MPa; b ¼ 7:78� 108 s�1, and m ¼ 4:36 are material constants that are taken based on a regression anal-ysis of an experimental test of hot extrusion on AZ31 Mg alloy (Tello et al., 2010). Z is the Zener–Hollomon parameter, whichis a function of temperature and strain rates defined as:

Z ¼ _�e expQRT

� �; ð16Þ

where Q ¼ 129� 103 J/mol is the activation energy of lattice diffusion, the other material constant determined from theregression analysis of experimental data on AZ31 Mg alloy (Tello et al., 2010). R ¼ 8:314 J/mol K is the gas constant.

In Eq. (6)–(8),P

j indicates summation over all neighboring particles of particle i. Particles representing the backing plateare fixed in space, and the particles representing the tool are constrained to rotate and translate as a rigid body. They bothenter into the calculation of the density, force, and temperature of fluid particles (sheet material) (Eq. (6)–(8)).

The equation of state, Pi ¼ P0qi=q0, is used to close the system of Eq. (6)–(8). In the equations of state, q0 is the equilib-rium density, P0 is the corresponding equilibrium pressure, and their relation to the speed of sound is given by c2 ¼ P0=q0.Furthermore, the smoothing length, h, determines the resolution and number of neighbors that contribute to the propertiesat a point. Therefore, the efficiency and accuracy would be greater if h is chosen to depend on the local particle number den-sity (Monaghan, 1992). Mass of SPH particle i is constant (does not change with time) and proportional to qih

3i . Then, hi can

be calculated from (Kajtar and Monaghan, 2008):

dhi

dt¼ � hi

3qi

� �dqi

dt: ð17Þ

Finally, the time step is chosen as the minimum of the Courant–Friedrichs–Lewy (CFL) condition,

Dt 6 minij

2hci þ cj

; ð18Þ

the viscosity condition (Cleary and Monaghan, 1999),

Dt 6 0:1minij

h2ðqi þ qjÞðli þ ljÞ4lilj

ð19Þ

and the condition imposed by the boundary forces (Monaghan and Kajtar, 2009),

Dt 6 minij

jrij � DpbjVmax

: ð20Þ

2.3. Model setup

The model includes a sheet, a rotating and translating tool, and a backing plate as shown in Fig. 1. Two interfaces are de-fined: the first is between the tool and the sheet, and the second is between the sheet and the backing plate. The dimensionsof the AZ31 Mg sheet are 4 � 2� 0.2 in, and the H13 steel tool has a 0.5 in diameter shoulder. The pin diameter is 0.25 in. Theselected materials and dimensions are chosen to be identical to those in the experimental work (Darras et al., 2007). The




Table 1Summary of material parameters used in the present work. The data for the H13 steel and AISI 4142 steel are obtained from www.Matweb.com.

Materials Density (q) Thermal conductivity (k) Specific heat (cp)

Sheet material (AZ31 Mg alloy) (Hu et al., 2009) 1777 kg/m3 400 w/m K 1164 J/kg KTool material (H13 steel) 7800 kg/m3 24.3 w/m K 460 J/kg KBacking plate (AISI 4142 steel) 7800 kg/m3 42.6 w/m K 473 J/kg K

Fig. 1. Three-dimensional setup of SPH model for FSW. Here, ‘‘AS’’ represents the advancing side, and ‘‘RS’’ represents the retreating side.


backing plate is assigned the properties of AISI 4142 steel. The thermal conductivity and specific heat of AZ31 Mg are deter-mined according to the measurements of Hu et al. (2009) and Yang et al. (2010). Specific parameter values of material prop-erties are listed in Table 1.

A partial slip boundary condition is imposed between the tool and the welding sheets. Energy conservation equation (Eq.(3)) assumed that the heat generated in FSW is due to plastic deformations. Imposing a no-slip boundary condition leads toover-estimation of the heat generation in FSW numerical models. The partial slip is introduced by adding a constant D < 1into Eq. (15) as:

1 For


re ¼ DrRsinh�1 Zb

� �1=m" #

: ð21Þ

Here, we set D to 0.1. The free slip boundary condition is set between the sheet material and the backing plate (fixed frame),i.e., the viscous force acting on fluid particles from the backing plate is zero.

3. Results

In FSW, the tool moves along the adjacent edges of welding materials with prescribed rotational and translational velocityto form the joint. Metal in the welded sheets heats up and softens near the advancing edge and gets mixed in the ‘‘wake’’ ofthe tool (see Fig. 2). To illustrate the mixing flow during FSW, a different color (yellow/green)1 represents each side of thewelding material. The tool is represented by the dark gray particles.

3.1. Temperature variation

To validate our SPH model, we compare the temperature variation obtained from our simulations against that observed inthe experiment by Darras et al. (2007). In the simulations, we also use the same process parameters as those in theexperiment.

Fig. 3 shows the temperature variation obtained from the simulations (top panel) at a fixed location (marked as a smallwindow by blue solid lines in Fig. 2) on the top surface of the sheet material, for different translational speeds at a fixed rota-tional speed of 1200 rpm. For comparison, results from a similar measurement in the experiment (Darras et al., 2007) arealso provided (bottom panel).

Temperature at the marked location increases as the tool approaches it and decreases as the tool moves away from thislocation. Both the experimental and numerical results show higher translational speed leads to lower temperatures in thestirring zone. Similarly, the simulated and experimental results of temperature variation at different rotational speeds for

interpretation of color in Fig. 2, the reader is referred to the web version of this article.




0 1 2 3 4 5 6 7 8

300

400

500

600

700

800

900

Processing Time (sec)

Tem

pera

ture

(K)

SPH Simulation

1200 rpm−22ipm1200 rpm−30ipm

0 1 2 3 4 5 6 7 8

300

400

500

600

700

800

900


Tem

pera

ture

(K)

Experiment


Fig. 3. Temperature versus time for two translational velocities at the fixed rotational speed of 1200 rpm. The temperatures are obtained from the SPHmodel (top) and experiments (Darras et al., 2007) (bottom).

Fig. 2. Mixing flow during FSW.


a constant translational speed of 22 inches per minute (ipm) are shown in Fig. 4. Notably, increasing the rotational speedresults in increased heat generation and higher temperatures due to the increased strain rate and plastic dissipation in

Please cite this article in press as: Pan, W., et al. A new smoothed particle hydrodynamics non-Newtonian model for friction stir welding:Process modeling and simulation of microstructure evolution in a magnesium alloy. Int. J. Plasticity (2013), http://dx.doi.org/10.1016/j.ijplas.2013.02.013



0 1 2 3 4 5 6 7 8

300

400

500

600

700

800

900


Tem

pera

ture

(K)

SPH Simulation


0 1 2 3 4 5 6 7 8

300

400

500

600

700

800

900


Tem

pera

ture

(K)

Experiment1200 rpm−22ipm1600 rpm−22ipm

Fig. 4. Temperature versus time for two rotational speeds at the fixed translational speed of 22 ipm. The temperatures are obtained from the SPH model(top) and experiments (Darras et al., 2007) (bottom).


the stirring zone. The maximum temperatures are observed in the range of 430–560 �C, which is approximately 0.7–0.9 ofthe melting point (618 �C) of AZ31 Mg alloy.

Figs. 3 and 4 show that the general behavior of the temperature and peak temperatures found in the SPH simulationsagree with the experimental measurements of Darras et al. (2007). However, some differences exist between the simulatedand experimental results at the pre- and post-stirring stages. These discrepancies can be explained by the fact that the posi-tion in the experiment (Darras et al., 2007) where the temperature history was recorded using an infrared camera might bedifferent from the fixed location specified in our simulation (marked as a small window by blue solid lines in Fig. 2). Theexact coordinates of the former were not reported in the experiment (Darras et al., 2007). Therefore, the comparison is onlylimited to the trends and peak values in temperature distribution along processing time.

A typical temperature distribution on the top surface and cross section of the welded metal sheet are shown in Fig. 5 andFig. 6, respectively. Fig. 5 shows the temperature distribution on the surface in the zone marked by black dash lines in Fig. 2.Fig. 6 depicts the temperature distribution on the cross section of the processing zone marked by the green dash-dot line inFig. 2. Our results show that higher temperature occurs in the zone closer to the tool pin, and maximum temperature isreached under the tool shoulder on the advancing side of the sheet. In essence, compared to the retreating side, the relativemotion between the tool and the sheet material on the advancing side yields higher strain rates, thus yielding higher plasticdissipation and heat generation.




Fig. 5. Typical temperature distribution on the top surface of the welding sheet, illustrated in a selected zone (as marked by black dash lines in Fig. 2), andcalculated at a translational speed of 11 ipm and rotational speed of 1200 rpm.

Fig. 6. Typical temperature distribution on the cross section of the processing zone (as marked by the green dash-dot line in Fig. 2) calculated at atranslational speed of 11 ipm and rotational speed of 1200 rpm.


3.2. Grain size and microhardness

In this section, we study the metallurgical and microstructural aspects of FSW. We use the Zener–Hollomon parameterand Hall–Petch relationship to predict variations in grain size and microhardness resulting from different rotational andtranslational speeds. By averaging both the working temperature and strain rate values, the Zener–Hollomon parametercan be evaluated through Eq. (16). For AZ31 Mg alloy, Chang et al. (2004) proposed an empirical relation between the Ze-ner–Hollomon parameter and the average recrystallized grain size:


lnðdÞ ¼ 9� 0:27 lnðZÞ; ð22Þ

where d is the average grain size in lm. They used experimental measurements of the average grain size and temperature ofthe processed AZ31 Mg alloy for the same stirring conditions. This equation is only valid in the affected zone, which wascharacterized by the effective strain rate ( _�e) higher than 0.5 s�1 and a temperature higher than 500 K. The recrystallizationtemperature of AZ31 Mg alloy ranges from 523–753 K, according to Fatemi-Varzaneh et al. (2007). We assume the unaf-fected base material has a grain size of 50 lm.

Figs. 7 and 8 show the distribution of the calculated Zener–Hollomon value and grain size, respectively, on the cross sec-tion passing through the green dash-dot line as marked in Fig. 2, for the simulation with translation velocity of 11 ipm androtational velocity of 1200 rpm. The effective strain rate ( _�e) distribution on the cross section of the processing zone is alsoillustrated in Fig. 9, which shows dynamic strain rates that exceed 103 s-1. The finest grains (see Fig. 8), i.e., the highest Ze-ner–Hollomon values (see Fig. 7), are observed in the heat-affected zone (HAZ), located between the thermomechanicallyaffected zone (TMAZ) and unaffected zone. The retreating side (RS) shows finer grain size than the advancing side (AS). Thisresults from the fact that in the affected zone, the lower temperature and higher strain rate are more favorable to grainrefinement. For more details of this physical phenomenon, see Nandan et al. (2008).

The Vickers microhardness (Khan et al., 2012), Hv , in the processing zone was evaluated using the Hall–Petch relationship,which for AZ31 Mg alloy is given by Chang et al. (2004) as:




Fig. 7. The Zener–Hollomon value distribution around the pin on the cross section of the processing zone (as marked by the green dash-dot line in Fig. 2)calculated at a translational speed of 11 ipm and rotational speed of 1200 rpm. Here, zero is assumed to be the Zener–Hollomon value for the unaffectedzone.

Fig. 8. Grain size distribution around the pin on the cross section of the processing zone (as marked by the green dash-dot line in Fig. 2) calculated at atranslational speed of 11 ipm and rotational speed of 1200 rpm. Here, 50 lm is assumed to be the grain size of the unaffected zone.

Fig. 9. Effective strain rate ( _�e) distribution around the pin on the cross section of the processing zone (as marked by the green dash-dot line in Fig. 2)calculated at a translational speed of 11 ipm and rotational speed of 1200 rpm.



Hv ¼ 40þ 72d�1=2: ð23Þ

The predicted average grain size and hardness values at different translational and rotational speeds are plotted in Figs. 10and 11, along with the available experimentally measured values (Darras et al., 2007). In the simulations, the average grainsize and hardness were estimated by calculating their local values in each SPH particle and then evaluating the average onlyfor the particles in the processing zone, which was characterized by the effective strain rate ( _�e) higher than 0.5 s-1 and atemperature higher than 500 K. Fine grain sizes in the range of 4 lm to 11 lm, predicted by the SPH model, are in goodagreement with the experimental data of Darras et al. (2007). The calculated average hardness also agrees with the exper-imentally observed values in Darras et al. (2007). The dependence of the calculated average hardness on the spatial resolu-tion is further examined. Fig. 12 shows that hardness values converge with decreasing particle spacing for all consideredrotational speeds. The decrease in initial particle spacing Dp from 0.025 to 0.0125 in results in an L2 relative difference ofabout 4%, and the decrease in Dp from 0.0125 to 0.00625 in produces an L2 difference of less than 1%. Based on this conver-gence study, we conclude that the resolution with Dp ¼ 0:0125 (inch) is sufficient to produce an accurate solution, and wechose this resolution in all the simulations reported here. This resolution requires 820,000 SPH particles to discretize thesimulated domain.

Our results suggest an increase in the translational speed yields finer grains and increased hardness. It has been observedexperimentally that excessive heat generated during FSW promotes abnormal grain growth, leading to heterogeneity ofgrain size distribution (Mansoor et al., 2009). Therefore, based on the results of our model, we may conclude that fastertranslation speed should cause a reduction in temperature, minimal grain growth, and better homogeneous grain refine-ment. Figs. 10 and 11 show the calculated grain size and hardness, as well as the experimental values for variable rotationalspeeds. It can be seen that an increase in the rotational speed results in increasing grain size and decreasing hardness, whichis due to the higher temperature generated by faster rotation in the welding zone. In summary, in the range of rotational and




1000 1200 1400 1600 1800 2000 22002

4

6

8

10

12

14

Rotational Speed (rpm)

Gra

in S

ize

(μm

)

translational speed = 30 ipm

SPH Simulation

10 15 20 25 30 352

4

6

8

10

12

14

Translational Speed (ipm)

Gra

in S

ize

(μm

) rotational speed = 1200 rpm

Experiment (Darras et al., 2007)SPH Simulation

Fig. 10. Calculated average grain size in the processing zone, compared against the experimental results (Darras et al., 2007).


translational speeds studied in this work, higher translational speeds and lower rotational speeds produce finer grains, pro-vided enough heat is generated to promote dynamic recovery and recrystallization.

3.3. Simulation of texture evolution

In this section, we examine the texture evolution during FSW, particularly in the TMAZ. Here, we have made several sim-plifying assumptions. We consider each SPH particle to represent a polycrystalline representative volume element (RVE), andwe assume a weak coupling between the SPH simulations and crystal plasticity modeling. Particle velocities obtained fromthe SPH solution at each time step are used to calculate the plastic velocity gradient for the polycrystalline deformationbehavior. Texture evolution is then computed using crystal plasticity modeling (Hama and Takuda, 2011; Khan et al.,2011, 2012).

The considered polycrystalline plasticity model is the viscoplastic Taylor model (Kocks, 1970; Taylor, 1938), where themechanisms of deformation are mainly crystallographic on the basal, prismatic, and pyramidal planes. It is well establishedthat tensile twinning can be activated during deformation of magnesium alloys such as AZ31 (Jain and Agnew, 2007). How-ever, the FSW process induces high temperature level in the TMAZ (as observed in Fig. 6) for which twinning activation canbe neglected (Janecek and Chmelik, 2011). At these temperatures, recrystallization occurs, but it is neglected in our crystal-lographic texture analysis.

To model crystallographic slip, we assume that the plastic velocity gradient Lp is expressed as the sum of a number ofcrystallographic slip rates:





1000 1200 1400 1600 1800 2000 220050

55

60

65

70

75

80

85

90


Har

dnes

s (H

V)



10 15 20 25 30 3550

55

60

65

70

75

80

85

90

Translational Speed (ipm)

Har

dnes

s (H

V)

rotational speed = 1200 rpm


Fig. 11. Calculated average microhardness in the processing zone compared against the experimental results (Darras et al., 2007).

1000 1200 1400 1600 1800 2000 220050

55

60

65

70

75

80

85

90


Har

dnes

s (H

V)


Δp = 0.025 inΔp = 0.0125 inΔp = 0.00625 in

Fig. 12. Average microhardness in the processing zone versus rotational speed as a function of particle spacing, Dp.







Lp ¼XN

a¼1

_cana �ma; ð24Þ

where na and ma are the vectors representing slip direction and slip plane normal of slip system a, and N is the number of slipsystems. A viscoplastic power law was utilized to relate the slip rate of a given slip system a; _ca, to the corresponding re-solved shear stress, sa, as:

_ca ¼ _c0sa

sa0

1=q

signðsaÞ; ð25Þ

where _c0 is the reference strain rate and defined as _c0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi23 Lp

klLpkl

qwith k; l ¼ 1;2;3. sa

0 is the slip resistance of the slip system

a; sa is the resolved shear stress along slip system a, and q is the rate sensitivity exponent. This leads to a non-linear rela-tionship between the plastic strain rate tensor (elasticity is neglected) and the deviatoric Cauchy stress tensor (Ahzi andM’Guil, 2008; M’Guil et al., 2011). In this single crystal constitutive law, two important parameters are called for: the inverserate sensitivity coefficient, assumed to be the same for all slip systems, and the critical resolved shear stress of each slip sys-tem a, i.e., sa

0. The evolution of this threshold stress can be described by the microscopic Voce hardening law:

sa0ðCÞ ¼ s0 þ ðs1 þ h1CÞ 1� exp �C

h0

s1

� � �; ð26Þ

where C is the total accumulated plastic strain. The parameters used in hardening law and power law are obtained fromexperimental results using curve fitting. The inverse rate sensitivity coefficient is taken with a value of 19. The values ofparameters s0; s1; h0, and h1 for each slip system are chosen and listed in Table 2. The details of parameter fitting accordingto the experimental stress–strain curves can be found in previous studies (Wang et al., 2010; M’Guil et al., 2006; Tome et al.,1987). The change rate of crystallographic axes is further determined by the lattice spin S�, expressed in terms of total crys-tallographic spin S and the slip rate ( _ca) of slip system a as:

S� ¼ S� Sp ¼ S�XN

a¼1

12

_caðna �ma �ma � naÞ; ð27Þ

where Sp is the plastic spin with no contribution to the rigid body rotation.Orientation of crystal, g, represented by three Euler angles (/1;/, and /2), is updated using the lattice spin S�. For poly-

crystalline materials, texture is defined as a distribution function of crystal orientation (g), i.e., f ðgÞ ¼ f ð/1;/;/2Þ, and thus isupdated from the lattice spin of each crystal. Texture is usually represented by pole figures as 2D projections of the crystalaxes on the reference plane. It will evolve during mechanical deformation due to the lattice spin from the velocity gradient.Using the Taylor model to predict texture evolution, we considered an RVE composed of 140 grains with an initial rollingtexture corresponding to the initial texture of the welded plates (Park et al., 2003). The initial texture is represented bythe pole figures shown in Fig. 14 (a). Particularly, the (0001) pole figure shows that the c-axis of most crystals is aligned alongnormal direction, while the other two pole figures show that the other crystallographic axes normal to the c-axis are distrib-uted randomly.

We selected the RVE to be a part of the TMAZ in the retreating side (RS), which is represented by the particle A in Fig. 13.This figure shows the displacement of particle A, among other particles, during FSW. There are two stages in this particledisplacement. The first stage is very short and is characterized by a large displacement and deformation accompanied bysignificant changes in the microstructure. In the second stage, the displacements and deformations are relatively smallbut still cause material property and microstructure evolution. It should be noted that the texture is determined by thedeformation history and the initial position of the studied particle. The particles in the same zone at the beginning may fallinto different zones after processing.

Texture evolution is modeled using an implicit time-integration model for hyperelastic–viscoplasticity. This model wasoriginally implemented in the finite element program (Kaudindi and Anand, 1992; Webej and Anand, 1990). In this originalapproach, the history of the local deformation gradient in the neighborhood of a material point of interest was first extractedfrom FEM and used as an input in a Taylor-type polycrystal model. The model was then run independently as a post-process-ing operation to predict the evolution of crystallographic texture at a material point of interest. Convergence of this algo-rithm has been shown by Ling et al. (2005). Similar to this weakly coupled Taylor-FEM model, our weakly coupledTaylor-SPH model uses one way coupling, i.e., the texture is affected by material deformation, but the materials flow is

Table 2Model parameters describing critical resolved shear stress and hardening responses of the three slip systems used insimulation.

Slip system s0 (MPa) s1 (MPa) h0 (MPa) h1 (MPa)

basal 16 19 50 30primastic 64 8 8 30pyramidal 112 6 130 2




0 0.01 0.02 0.03 0.04 0.05 0.06−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

x (m)

y (m

)RS

AS

A

→

0 0.01 0.02 0.03 0.04 0.05 0.06−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

x (m)

y (m

)

RS

AS

A

→

0 0.01 0.02 0.03 0.04 0.05 0.06−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

x (m)

y (m

)

RS

AS

A

→

0 0.01 0.02 0.03 0.04 0.05 0.06−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

x (m)

y (m

)

RS

AS

A

→

0 0.01 0.02 0.03 0.04 0.05 0.06−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

x (m)

y (m

)

RS

AS

A

→

0 0.01 0.02 0.03 0.04 0.05 0.06−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

x (m)

y (m

)

RS

AS

A

→

Fig. 13. Materials flow on the surface in the retreating side (RS). (a) t = 0 s (b) t = 0.9 s (c) t = 1.7 s, (d) t = 2.5 s, (e) t = 3.2 s, and (f) t = 5.3 s.


independent of the texture. The convergence of the SPH model with Dp for hardness values has been demonstrated in Fig. 12.Given that the SPH transport model converges, the Taylor-SPH model is expected to converge.

The history of SPH computed velocity gradient of the particle A is used to compute the lattice spin in the viscoplastic Tay-lor model. Fig. 14 shows the resulting texture evolution of particle A at six different times, represented in terms of (0001),(10�10), and (11�20) pole figures. Fig. 13 depicts positions of the particle A and several surrounding particles at different times.It can be seen that the texture changes rapidly in the beginning and slowly at the end of the simulation. Correspondingly,displacements of the particle A are larger at the early time than at the later time. In the first stage (Fig. 14 (b)), the textureof particle A evolves the most dramatically. Texture keeps evolving after 0.9 s, as shown in Fig. 14 (c) and (d). After 2.5 s,there is not much change in the texture any more, as seen in Fig. 14 (e) and (f). Fig. 15 shows comparison of the predictedfinal (0001) and (10�10) pole figures with the experimental observation in TMAZ by Xin et al. (2010). In this figure, we rotatethe coordinate system to fit with the experimental results. We can see an agreement in the general trend of texture evolutionwith the experiment. The (0001) pole figure shows the texture component of the c-axis moved away from the normal direc-tion in both experiment and simulation. However, the component shown from the simulation is more tied to the processing




Fig. 14. Evolution of texture. (0001), (10�10), (11�20) pole figures, of particle A on the surface in the retreating side (RS). (a) t = 0 s, (b) t = 0.9 s, (c) t = 1.7 s, (d)t = 2.5 s, (e) t = 3.2 s, and (f) t = 5.3 s.


direction, i.e., rolling direction here, compared with the experimental result. The intensity in pole figures of (10�10) is veryweak from both experiment and simulation, showing that the (10�10) crystallographic axes are randomly distributed. Severalreasons may be attributed to the discrepancies observed. First, the position of RVE relative to the weld may not be exactly thesame. Second, some processes, including recrystallization, void formation, and grain refinement, are neglected in this model.The influence of these simplifications should be explicitly evaluated in the future. Third, the anisotropic crystal plasticity andSPH transport model may require a closer coupling.

4. Conclusion

We present a new SPH-based model for FSW. In this model, the elastic deformations of the welding material are disre-garded, and the material is modeled as a non-Newtonian fluid, satisfying the constitution relation proposed by Sellarsand Tegart (1972). We have shown the essential dynamics of FSW can be accurately captured by the SPH model. The SPHmodel was validated using data from a laboratory experiment by Darras et al. (2007) for AZ31 Mg alloy.

The SPH model was used to study the effects of tool rotational and translational speeds on temperature distribution andevolution, grain size distribution, material hardness, and texture in the welding zone. Grain size and hardness were related to




Fig. 15. Simulated final (a) (0001) and (b) (10�10) pole figures of particle A in the thermomechanically affected zone (TMAZ) on the surface at the retreatingside (RS) after FSW, compared with the experimental observations (Xin et al., 2010) (with courtesy permission from Trans Tech Publications).


the strain rate and temperature using the Zener–Hollomon parameter and Hall–Petch relationship. With the data obtainedfrom the experiment by Darras et al. (2007), we found good agreement between results of the SPH model for temperature,grain size, and microhardness.

Our results show that a combination of high-translational and low-rotational tool speeds are ideal to cause effective grainrefinement and higher hardness of the material. This occurs because lower rotational and higher translational speeds gen-erate less heat and prevent abnormal grain growth.

Texture evolution was modeled using an implicit time-integration model for hyperelastic–viscoplasticity that was weaklycoupled with the SPH model. Due to computational limitation, the texture evolution was modeled only in one Lagrangianvolume, which was arbitrary selected in the welding zone. A strong coupled model for crystal plasticity and SPH is underdevelopment to substitute the isotropic strain rate tensor in Eq. (14) with the viscoplastic power law for anisotropic poly-crystalline materials. The initial texture of the rolled magnesium sheets has the c-axis strongly aligned along the normaldirection. After processing, it becomes more isotropic. Texture component intensity decreased dramatically, and the c-axisof crystals moved away from the normal direction. The simulation results agree with experimental observation qualitatively.To improve the texture prediction, a closer coupling of the crystal plasticity and SPH transport models may be required. Inaddition, analysis of texture evolution at different locations in the processing zone, should be performed, and results shouldbe compared to experiments.

Additional future developments of the SPH modeling framework for FSW should include a realistic treatment of elasticdeformations. Furthermore, metallurgical aspects should be considered in more detail (Lee et al., 2009; Fernandez et al.,2011), and the effect of initial grain size, grain growth and recovery, and recrystallization should be included by constitutivemodeling to fully describe the phenomena occurring during FSW.

Acknowledgment

This research was supported by the Advanced Scientific Computing Research Program and the Scientific Discoverythrough Advanced Computing Program of the Office of Science, U.S. Department of Energy at the Pacific Northwest NationalLaboratory. The work of texture evolution is funded by Vehicle Technologies Program, DOE Office of Energy Efficiency andRenewable Energy at the Pacific Northwest National Laboratory. The Pacific Northwest National Laboratory is operatedfor the U.S. Department of Energy by Battelle under Contract DE-AC05-76RL01830.

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