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CHAPTER 3

THERMAL AND THERMO-MECHANICAL

MODELING OF FSW

3.1 INTRODUCTION

A complex state of thermo-mechanical stresses is developed in a

welded structure as a direct consequence of the non uniform heating cooling

processes taking place in it. Residual stresses are the stresses that remain in a

body after all the external loads have been removed from that body. These

stresses reduce the load carrying capacity of the structures (Canes et al 1995).

The high level of stresses in the neighbourhood of the weld joint can increase

the tendency to brittle fracture and can affect the corrosion and fatigue

behavior (Fratini and Zuccarello 2006). All of this compels the designer to

know about residual stress state quantitatively and qualitatively. This

knowledge would also help them to employ appropriate stress relief

techniques.

Welding is a multi-physics problem and complex in nature. The

complexities include material and thermal properties which vary with

temperature, transient heat transfer, moving heat source, complex residual

stress state and difficulties in making experimental measurements at high

temperatures. The development of mathematical models can greatly

contribute to better understanding of any welding process, particularly FSW.

Evaluation of thermal and residual stresses associated with welding process is

extremely complicated issue. As a first step, the time dependent temperature

39

distribution, associated with the heat supplied during welding has to be

estimated. This temperature distribution will have an associated thermal

stresses which will also be time dependent, leading to residual stresses. The

difficulty in determining these stresses is influenced by the variation of

thermal and mechanical properties of the material with temperature and by the

plastic deformation of weld zone, all of which make analytical approach to

the problem of almost impossible.

Numerical techniques such as FEM can be used to deal this type of

complex problems. A validated FEM model has the potential to produce

reliable information about the thermal cycles, deformation and stress patterns

that are important to understand the mechanism of heating and the bonding of

materials during FSW. The results of the model will also be helpful in

designing FSW tools and thus should be capable of producing welds free of

defects and voids.

Thermal cycle in FSW is due to moving heat source employed in this

welding process. The thermal cycle is characterised by a heating stage, that is,

temperature rise upto a maximum, followed by a cooling process. Heating upto

a maximum temperature can be slow or fast depending upon the welding

process. The high temperature exposure produces varied effects on the

microstructure and properties of the weldments. The actual effect depends on

the temperature duration and the cooling rate after exposure. An exact

knowledge of thermal cycle is important to assess the properties and integrity

of welded joints.

This chapter presents the methodology of developing a thermal and

thermomechanical model to predict the thermal cycles and residual stresses

using finite element code ANSYS. First, a non-linear 3D transient thermal

model to simulate the temperature distribution during FSW of aluminium

40

alloy 2014-T6 is developed. A thermomechanical model to predict the

residual stresses in the welded plate using the predicted temperature field, is

also developed.

3.2 NUMERICAL MODELING

Experimental restrictions on the number of locations at which

temperature and residual stresses could be measured and the high cost of

experimentation necessitated the development of numerical model to predict

the thermal history and residual stresses in FSW. In the present study, an

attempt has been made to predict the residual stresses developed in aluminium

alloy AA2014-T6 during FSW by employing coupled field analysis. The

thermal analysis is performed first to generate temperature history. Thermal

stresses are calculated from the temperature distributions determined by the

thermal model during welding process. The residual stresses in each

temperature increment are then added to the nodal point location to update the

behavior of the nodal point before the next temperature increment. Figure 3.1

presents the analysis procedures.

3.3 THERMAL MODEL

Assumptions made in the thermal analysis are as follows:

The heat generation is due to friction only.

Heat generated during penetration and extraction is negligible.

Heat transfer by radiation is negligible.

The density of the material is not affected by thermal

expansion.

The tool pin is assumed to be cylindrical.

41

Figure 3.1 Flow chart of thermo-mechanical analysis

42

Following process parameters are considered in thermo-mechanical

model (Mahapatra et al 2006):

moving heat source

weld speed

tool rotational speed

axial load and

material properties

The temperature distribution during welding is calculated by

solving the governing equations for heat conduction applying proper

boundary conditions.

3.3.1 Governing Equations

When a volume is bounded by an arbitrary surface S, the balance

relation of the heat flow is expressed by the following equation (Holman

2002):

, , ,, , ,yx zRT x y z t R RC Q x y z t

t x y z

(3.1)

where Rx, Ry and Rz are the rate of heat flow per unit area, T(x, y, z, t) is the

current temperature, Q(x, y ,z, t) is the rate of internal heat generation.

The model may then be completed by introducing the Fourier heat

flow as

xTkR xx (3.2)

43

yTkR yy (3.3)

zTkR zz (3.4)

where T is the temperature and t is the time. kx, ky and kz are the thermal

conductivity along three directions. The heat transfer equation for the

workpiece is modified as:

, , ,x y z

T x y z t T T TC k k kt x x y y z z

(3.5)

The final system of finite element equations of (3.5) is as follows:

.

[ ]{ } [ ]{ } { }K T C T F (3.6)

where

1

[ ] [ ]M

e

eK K

(3.7)

1

[ ] [ ]M

e

eC C

(3.8)

M

e

eFF1

}{}{ (3.9)

and

1

[ ] [ ] [ ][ ] [ ] [ ]e T T

v S

K B D B dv h N N dS (3.10)

[ ] [ ] [ ]e TpC C N N dV (3.11)

2 3

{ } [ ] [ ] [ ]e T T T

V S S

F Q N dV q N dS hT N dS (3.12)

The equation (3.6) is the required basic matrix differential equation.

44

3.3.2 Initial and Boundary Conditions

The solution of the heat conduction equation involves a number of

arbitrary constants to be determined by specified initial and boundary

conditions. These conditions are necessary to translate the real physical

conditions into mathematical expressions (Hsu 1986). The conduction and

convection coefficients on various surfaces play a key role in determination of

the thermal history of the workpiece in FSW. Figure 3.2 shows the various

thermal boundary conditions applied in the model.

Figure 3.2 Thermal boundary conditions

Initial boundary conditions are required only when dealing with

transient heat transfer problems in which the temperature field in the material

changes with time:

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(i) The common initial boundary condition in a material for the

calculation can be expressed mathematically as

0, , ,T x y z t T (3.13)

The initial temperature of the workpiece is assumed to be

atmospheric temperature 303 K.

Specified boundary conditions are required in the analysis to

all transient or steady state problems. The energy balance at

the work surface leads to a few other boundary conditions. A

specified heat flow qs and qp are supplied from the shoulder

and pin over the instantaneous surface of the work. The

other surfaces except bottom are exposed to atmosphere,

where heat loss qcon takes place owing to convection.

(ii) The heat flux boundary condition at the tool and workpiece,

tool pin and workpiece interface are expressed as:

sTk qn

and p

Tk qn

(3.14)

(iii) The convective boundary condition for all the workpiece

surfaces exposed to the air is expresses as:

( )oTk h T Tn

(3.15)

where n is the normal direction vector of the boundary. To

account for convection, all of the surfaces exposed to the

atmosphere were allocated a uniform convection coefficient

of 15 Wm-2K-1 (Peel et al 2006a). The surface of the

workpiece in contact with the back plate is approximated to

the convection condition with an effective coefficient of

convection 300 Wm-2K-1. The mode of heat transfer between

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workpiece and back plate under tool is modeled (Vijay et al

2005) with contact conductance of 3000 – 4000 Wm-2 K-1.

(iv) The two plates to be welded are assumed to be identical. At

the centerline of the workpiece, the temperature gradient in

the transverse direction (∂T/∂y) equals zero due to the

symmetrical requirement.

3.3.3 Modeling Heat Input

The heat generated between the tool shoulder and the workpiece

and heat generated between the tool pin and the workpiece are both

considered as heat inputs in this finite element model. The heat source in

FSW is considered to be the friction between the tool shoulder and the

workpiece, the tool pin and the workpiece surfaces. The local heat generated

(dQ) over the interface of the tool shoulder and workpiece surface is

calculated using the standard equation (3.16) (Dong et al 2001, Fonda et al

2002).

22 ad Q F r d r (3.16)

Even though the coefficient of friction µ varies with temperature, in

this model an effective coefficient of friction of 0.3- 0.5 is considered. High

and low values of co-efficient are used for low and high values of tool

rotation respectively (Peel et al 2006a).

Colegrove has proposed an expression (Song and Kovecevic 2003)

on calculating the heat generated between the tool pin and workpiece. The

heat generated by the tool pin (Q) given in equation (3.17) consists of three

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parts. They are (1) heat generated by the shearing of the material; (2) heat

generated by the friction on the threaded surface of the pin; and (3) heat

generated by friction on the vertical surface of the pin.

CosVFVhrYkVYkhrQ mpprtpmtp

4

13

2

32

2

(3.17)

where

190 tanm (3.18)

sin

sin 180m pV v

(3.19)

sin

sin 180r p pV v

(3.20)

pp rv (3.21)

Since the traverse force Ft is not measured during welding, the third

component in the equation is neglected while calculating the heat input due to

tool pin.

3.4 THERMO-MECHANICAL MODEL

Residual stresses in welding are produced by (Murthy et al 1996):

(1) Plastic strains due to heating and cooling owing to large

thermal gradients and low yield strength corresponding to that

temperature.

(2) Material dilatation during solid phase transformation.

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These are dependent on the rate of cooling. Calculation of the

residual stress due to thermal gradient induced strains, strains caused by

microstructural changes and strains due to mechanical loading requires

thermo elasto-plastic formulations.

The following assumptions apply to the formulations:

1. The material is treated as a continuous medium or a

continuum.

2. The material is isotropic, with its properties independent of

direction.

3. The material has no memory such that the effects on the

material in previous event do not impact the current event.

In the stress analysis, the temperature history obtained from the

thermal analysis is taken as input for determining thermal strains and stresses.

Thermal strains and stresses are calculated in addition to the mechanical

strains and stresses due to the axial load of the tool at each time increment and

the cumulative stresses are found at each load step. Before the next

temperature increment, these stresses are then added to those at nodal points

to update the behavior of the model.

3.4.1 Governing Equations

Some of the plasticity constitutive laws for metals are discussed in

this section. Thermal effects are accounted for in the stress analysis by

including thermal strains and temperature dependent material properties. Also

the thermal state affects the physical yield criterion. The constitutive laws

49

relating stresses, strains, and temperature are nonlinear in the theory of plastic

deformation of solids.

Two basic sets of equations relating to the elasto-plastic model, the

equilibrium equations and the constitutive equations are considered as follows

(Teng et al 1998, Sadhu Singh 2005).

(1) Equations of equilibrium (in Cartesian co-ordinates):

Consideration of the variation of the state of stress from point to point leads to

the equilibrium equations given by

0

xxzxyx Bzyx

(3.22)

0

y

yzyxy Bzyx

(3.23)

0

zzyzxz Bzyx

(3.24)

where Bx, By, and Bz are the components of the body force (in N/m3), such as

gravitational, centrifugal or other inertia forces.

(2) Constitutive equations for an elasto-plastic material: For an

isotropic material, we have only two independent elastic constants K and G,

known as Lame’s constants and the generalized Hook’s law gives the

following stress –strain relations.

1x x y zE

(3.25)

1y y x zE

(3.26)

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1z z x yE (3.27)

xy

xy G

(3.28)

yz

yz G

(3.29)

xz

xz G

(3.30)

The governing equations in elastic and plastic range are obtained

based on non-isothermal formulations as given below (Murthy et al 1996):

(a) For elastic range: The stress strain relations for thermo-elastic

analysis

eD (3.31)

0D T T (3.32)

The incremental form of the equation (3.32), considering

small deformation in the elastic range is given by

0ed D d dT T T d dD

(3.33)

where [dD] is the change in material matrix corresponding to a

change in temperature ΔT in time increment Δt.

(b) For plastic range: The stress – strain relation for thermo-

elasto-plastic analysis is given by

0p pD D T T (3.34)

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where the additional term p is the plastic strain.

The incremental form of equation (3.34) considering small

deformations in the plastic range, is given by

0p ed D d dT T T d d dD (3.35)

The thermal elasto-plastic model, based on von Mises yield

criterion with the associated flow rule and the multi-linear kinematic

hardening is considered in determining the stresses.

3.4.2 Boundary Conditions Figure 3.3 shows the mechanical boundary conditions employed

while developing the finite element model.

Figure 3.3 Mechanical boundary conditions

The degrees of freedom of the work piece arrested in x, y and z

directions are shown in Figure 3.3 by triangles. The axial load acting on the

FSW tool is transferred to the weld specimen in addition to the thermal load.

52

3.5 FINITE ELEMENT MODEL

This investigation develops a non-linear three dimensional thermo-

mechanical model to estimate the thermal history and residual stresses of the

butt-welded joint using the finite element code ANSYS. This model employs

three dimensional eight noded brick elements (SOLID70 and SOLID45) for

both thermal and stress analysis (ANSYS user manual). Symmetry along the

weld line is assumed in the numerical modeling, so one half of the welded

plate is taken for analysis to expedite the model run time.

3.5.1 Defining Material Properties

The base metal is aluminium alloy AA2014-T6. The material

properties that have to be included when temperature simulation is to be

performed are specific heat, heat conductivity and density. It is reported that

the transient temperature in a welded plate is significantly affected by the

value chosen for the thermal conductivity (Little and Kamtekar 1998). The

conductivity is usually temperature dependent. The specific heat for most

welding materials is strongly temperature dependent. Since the material

density is little affected by temperature field (Zhu et al 2002), a constant value

of material density is enough for simulation process. The temperature

dependency of the yield stress must be considered in a welding process

simulation, to improve the accuracy of results.

Young’s modulus and the thermal expansion co-efficient have little

effects on the residual stress and distortion respectively in welding

deformation simulation. It is found that the numerical results obtained by

using the room temperature value of Young’s modulus are much better than

those using average value over temperature history (Zhu et al 2002). The

Poisson’s ratio has been assumed to be constant, since its influence is usually

53

not significant (Tekriwal and Mazumdar 1988). Temperature dependent

thermal and mechanical properties of the material included in the model are

illustrated in Figures 3. 4 and 3.5.

0

200

400

600

800

1000

1200

0 100 200 300 400 500 600 700

Temperature, K

Thermal conductivity (W/m K)

Specific Heat ( J/kg K)

Figure 3.4 Thermal properties of aluminium alloy AA2014-T6

0

100

200

300

400

500

0 0.002 0.004 0.006 0.008

Strain

Stre

ss, M

Pa

303 K

366 K

422 K

477 K

533 K

588 K

Figure 3.5 Stress-strain curves of aluminium alloy AA2014-T6

54

3.5.2 Meshing Scheme and Time Step

Once the material properties are defined, the next step in an

analysis is generating a finite element model that adequately describes the

model geometry. There are two methods to create the finite element model:

solid modeling and direct generation. With solid modeling, once the

geometric shape of the model is described, then the FEA code automatically

meshes the geometry with nodes and elements. Size and shape of the elements

that the program creates can be controlled by the user.

Before meshing the model, and even before building the model, it is

important to consider about whether a free mesh or a mapped mesh is

appropriate for the analysis. A free mesh has no restrictions in terms of

element shapes, and has no specified pattern applied to it. Compared to a free

mesh, a mapped mesh is restricted in terms of the element shape it contains

and the pattern of the mesh. A mapped area mesh contains either only

quadrilateral or only triangular elements, while a mapped volume mesh

contains only hexahedron elements. In addition, a mapped mesh typically has

a regular pattern, with obvious rows of elements. To use mapped meshing, the

geometry has to be built as a series of fairly regular volumes that can accept a

mapped mesh. In the current FE model mapped volume mesh is used. Fine

mesh along the weld line and coarse mesh away from the weld line was used

in the simulation model to reduce computational time (Figure 3.6).

The element selected for the 3D transient thermal analysis is

SOLID70 a eight noded tetrahedral thermal solid. Solid70 which is a three

dimensional thermal solid, is used as the element type for analysis. Solid70

has a three-dimensional thermal conduction capability. The element has eight

nodes with single degree of freedom, temperature, at each node.

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Figure 3.6 Discretisation of the weld specimen

The element is applicable to a three-dimensional, steady-state or

transient thermal analysis. The element also can compensate for mass

transport heat flow from a constant velocity filed. If the model containing the

conducting solid element is also be analysed structurally, the element should

be replaced by an equivalent structural element such as SOLID45. The

element is defined by eight nodes and the orthotropic material properties.

Orthotropic material directions correspond to the element coordinate

directions. Specific heat and density are ignored for steady state solutions.

Convection or heat flux (but not both) and radiation may be input as surface

loads at the element faces as shown by the circled numbers in Figure 3.7.

(ANSYS user manual) Heat generation rates may be input as element body

loads at the nodes. Convection heat flux is positive out of element; applied

heat flux is positive into the system.

Consider the heat conduction equation (3.1). For isotropic material

with time independent thermal conductivity, no internal heat generation, and

heat transfer in one dimension the equation reduces to

xT x RCt t

(3.36)

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Figure 3.7 Details of eight noded tetrahedral solid element

For the same change in temperature, equation (3.36) can be used to

estimate the relationship between the spatial and time increments as

2Xt

(3.37)

where diffusivity is given by

kC

Utilising the data for aluminium alloy AA2014 at 400 K,

k =155 W/m K, ρ = 2800 kg/m3, C = 880 J/kg K, the thermal diffusivity

κ = 6.2910-5 m2/s. Taking a characteristic mesh size of 2 mm, the estimate of

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the time step is 0.07 s. Therefore, for a characteristic mesh size of 2 mm, a

time step of about 0.07 s will be sufficient to properly capture the temporal

thermal variations in the model. Convergence studies utilizing the 2 mm mesh

also confirmed a time step of 0.1 s yields numerically acceptable results. But

analysis using a time step of 0.1 s proved too time consuming, a time step of

0.5 s was used.

3.5.3 Applying Loads

The word loads as used in ANSYS includes boundary conditions

(constraints, supports or boundary field specifications) as well as other

externally and internally applied loads. Loads in the ANSYS program are

divided into six categories: DOF constraints, forces, surface loads, body

loads, inertia loads and coupled-field loads

Most loads of these can be applied either on the solid model (on

keypoints, lines and areas) or on the finite element model (on nodes and

elements). For example, forces can be applied at a keypoint or a node.

Similarly, convections (and other surface loads) can be specified on lines and

areas or on nodes and element faces. No matter how the loads are specified,

the solver expects all loads to be in terms of the finite element model.

Therefore, if loads are specified on the solid model, the program

automatically transfers them to the nodes and elements at the beginning of

solution.

3.5.4 Stepped Load versus Ramped Load Step

When more than one substep in a load step is specified, the

question of whether the loads should be stepped or ramped arises. If a load is

stepped, then its full value is applied at the first substep and stays constant for

58

the rest of the load step, as shown in Figure 3.8(a). If the load is ramped, then

its value increases gradually at each substep, with the full value occurring at

the end of the load step as shown in the Figure 3.8(b). In this study, stepped

load step option is used while applying thermal and mechanical loads to

closely simulate the loading.

(a) (b)

Figure 3.8 Loading a) Stepped b) Ramped

3.6 METHODOLOGY

In this present work non-linear equations are solved, including

temperature-dependent thermal properties with all complexity related to their

solutions. This assumption is made observing that the temperature changes

(gradients) encountered in the heat-affected-zone is so large that the change of

thermal properties could not be neglected. To determine temperatures and

other thermal quantities that vary over time there is the need to perform a

transient thermal analysis. Implicit method of time discretisation is employed

which allows for larger time steps.

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Using the finite element analysis the thermal and stress analysis are

uncoupled while in reality thermal effect and mechanical deformation occur at

the same time. The de-coupling of the analyses becomes acceptable if one

assumes that dimensional changes (mechanical deformation) during welding

process are negligible and the dimensional changes due to thermal energy are

predominant over mechanical work done during welding. Therefore to

evaluate the stresses, thermal analysis is performed first in order to find nodal

temperatures as a function of time. Once temperature history is predicted for

each node, nodal temperature loads are applied to the structural model. The

plate absorbs a major portion of the heat generated. There are losses from the

surfaces in the form of convection. The radiation effect is neglected because

of the low melting point of aluminium. The load applied is considered as

stepped.

A moving cylindrical coordinate system is used to move the heat

source. Instead of moving source continuously, it is moved in steps, that is,

heat is applied at each successive point one after the other chronologically. By

making the distance between successive points very small, a close

approximation to the continuous movement can be achieved. In the current

model an incremental distance of 0.5, 1, 2 mm is assumed. The distance

between the centers of the overlapping circles is 2 mm. the coordinate system

is moved after each load step. At every load step a set of elements in the

shape of the tool are selected and the calculated heat flux is applied on the

surface of the elements. A time step of 0.5 s is used. APDL (ANSYS

parametric design language) is used to write a subroutine for a looping

transient moving heat source model. The ADPL programs for thermal and

stress analysis are given in Appendix 1 and 2.

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3.7 SUMMARY

A detailed non linear thermo-mechanical finite element model has

been developed to study the thermal history and stress distribution in FSW of

aluminium alloy 2014-T6. Nine welding cases with three tool rotations of

355, 710 and 1120 rpm and three weld speeds of 30, 60 and 120 mm/min

have been considered for simulation. The following points are summerised

from the analysis.

1. Thermal model is useful to predict the temperature under the

tool shoulder in FSW which could not be predicted

experimentally.

2. For the nine welding cases, the simulated maximum

temperature at weld zone is less than the melting point of the

material aluminium alloy AA2014-T6. The difference of the

maximum temperatures at the same location (i.e., weld center)

is less than 170 K for the entire set of parameters considered.

3. Using 3D thermo-mechanical model, the stress fields in the

welded plates are simulated to find the nature of distribution.

The effect of simulation fixture release after welding is

included in the model. The longitudinal stress perpendicular to

the weld direction is predicted.

Results of this model will provide prior knowledge about residual

stress contours along with thermal history in order to design stress relief

techniques, while designing FSW based aluminium alloy structures.