120

CHAPTER 5

FINITE ELEMENT MODELING AND SIMULATION

OF FRICTION WELDING PROCESS

5.1 INTRODUCTION

Welding processes involving solid state phenomenon have been

extensively used in steel fabrication since the first patent issued in late

nineteenth century (1885). Although its application is very common, the

physics behind the welding process involving friction is complex because it

involves temperature gradient of thousands of degrees over a distance of less

than a centimeter, occurring on a time scale of a few seconds, involving

multiple phases of solids, liquids, gas and plasma (Eager, 1990). In this

chapter, various physical, metallurgical and numerical aspects such as

equation of continuum for stress, strain and deformation, finite element (FE)

matrix formulation, temperature dependent material properties, were

analyzed. Further, the computation of these entities using finite element

analysis procedure and numerical aspects of simulation etc were addressed for

friction welding process which is a solid state process.

5.2 OVERVIEW OF FINITE ELEMENT ANALYSIS (FEA)

Finite element method (FEM) is one of the most accepted and

widely used tools for the solution of non-linear partial differential equations

which arise in the mathematical modeling of various processes. The FEM was

originally developed for analysis of aircraft structures but now it is applicable

in all fields of engineering and applied sciences like heat transfer, fluid

dynamics, vibrations, magnetism, etc. The procedural steps in the application

121

of typical finite element method are illustrated in Figure 5.1. The process

centerpiece, from which, everything emanates, is the mathematical model.

This is often an ordinary or partial differential equation in space and time. A

discrete finite element model is generated from a variation or weak form of

the mathematical model. The FEM equations are processed by an equation

solver, which delivers a discrete solution (or solutions). The concept of error

arose when the discrete solution was

This replacement is generally called verification. The solution error is the

amount by which the discrete solution fails to satisfy the discrete equations.

This error is relatively unimportant when using computers, and in particular

direct linear equations solvers for the solution step. Discretization error is

more significant, which is the deviation by which, the discrete solution fails to

satisfy the mathematical model. This error was reduced by remodeling the

discretization process so that the FEM model was close to the mathematical

model.

Figure 5.1 Finite element methodologies

Mathematical model

Discrete model Discrete solution

Verification discretization + solution error

FEM

Verification solution error

122

5.3 APPLICATION OF FINITE ELEMENT ANALYSIS TO

FRICTION WELDING PROCESS

The analytical study of any physical phenomenon involves two

major tasks viz., the mathematical formulation of the physical process and the

numerical analysis of the modeled process of system. The mathematical

formulation of a physical process requires good background knowledge in the

related subjects and most often in using mathematical tools. Development of

the mathematical model of a process is achieved through assumptions

concerning how a process works. In a numerical simulation, a numerical

method is used. The finite element method is a powerful numerical technique

devised to evaluate complex physical processes. The method is characterized

by three features (Eager, 1990).

i) The domain of the problem is represented by a collection of

simple sub-domains called finite elements. The collection of

finite elements is called the finite element mesh;

ii) Over each finite element, the physical process is represented

using appropriate functions of desired type, and algebraic

equations relating physical quantities at selective points,

called nodes; and

iii) The element equations are assembled using continuity and /

or balance of physical quantities.

In the process of friction welding, heat flows into the material being

joined and sometimes may cause serious metallurgical changes in the welded

structures, which in turn, may lead to the early failure of the component.

Study of thermal cycles will be the basis for many other analyses like

prediction of distortion and residual stresses, metallurgical analysis, etc.

123

Hence, the study of these entities in the parts being welded in case of friction

welding becomes necessary.

With the advancements in computing field and development of

numerical techniques like finite element methods, it is now possible to

analyze complicated configurations and loadings which vary with respect to

location and temperature or functions of time for any given weldment. The

FEM based analysis is adopted for analyzing the friction welding process

since it is a field-based phenomenon. Computer-based simulations offer the

possibility to examine different aspects of the process without having a

physical prototype of the product. In this work, main parameters of the

friction welding process are considered and the finite element simulation is

performed using ANSYS version 9.0 (Ansys 9.0 User Manual).

5.4 THERMAL ANALYSIS IN FRICTION WELDING PROCESS

A non-linear transient analysis was performed for the first time for

friction welding process to predict the temperature dependent parameters

history of the domain for complete thermal cycle.

5.4.1 Mathematical Model for Heat Transfer Simulation in Welding

The heat transfer in the base material is conceived to be three

dimensional models in nature considering the radial, circumferential and axial

directions.

The governing heat conduction equation in Cartesian coordinate

system can be specified as:

tT

kq

zT

yT

xT 1

2

2

2

2

2

2

(5.1)

124

The heat equation may also be expressed in cylindrical coordinates.

Applying an energy balance to the differential control volume, the following

general form of the heat equation is obtained:

tT

kq

zTT

rrT

rrT 111

2

2

2

2

22

2

(5.2)

tT

kq

zT

zT

rrTr

rr111

2 (5.3)

where pc

k

tTcq

zTk

zTk

rrTkr

rr p2

11 (5.4)

where r , and z refer to radial, circumferential and axial directions; and

Cp respectively refer to thermal conductivity, density and specific heat of the

material; T and t refer to temperature and time variable, respectively and q

depicts internal heat generation per unit time and volume. The associated

initial and boundary conditions can be stated as:

The initial condition is

T (0, , z ,t) = T0 (5.5)

As the process is transient in nature, an essential boundary

condition is defined at time t = 0 as

T (r, , z ,0) = T0 (5.6)

125

More general boundary conditions are used to define energy

exchanges between the base material and its surroundings. The simplest

condition results when the heat flux is specified, i.e.,

sTk qn

(5.7)

Alternatively, a boundary may be insulated from the environment

and this yields the adiabatic no-flux condition

0nTk qn

(5.8)

wheren

denotes differentiation in the direction of the outward normal to the

surface and qn is the heat flux at the surface of the body in the direction of its

outward normal.

A common energy exchange mechanism is convection into / from a

surrounding medium, i.e.

qconv = h(T - T0) (5.9)

where h is the heat transfer coefficient (W/m2K) and T0 is the ambient

temperature of the surrounding environment. Another common exchange

mechanism is radiation

4 40( )radq T T (5.10)

where and are, respectively the emissivity of the material and the Stefan-

Boltzmann constant (= 5.6703 × 10-8W/m2K4).

126

The energy balance equation at the surface of the material is then

qn + qs = qconv + qrad (5.11)

Therefore, the natural boundary condition can be defined by

4 40 0( ) ( ) 0s

Tk q h T T T Tn

(5.12)

The term stands for the imposed heat flux onto the surface. To avoid the

non-linearity due to the term arising out of radiative heat loss, a lumped heat

transfer coefficient is used combining the convective and radiative heat loss

by considering h, as

h (5.13)

In the finite element method (FEM) analysis, these boundary conditions are

applied to the model by specifying the value of heat transfer coefficient and

the surrounding temperatures at the elements and nodes, respectively by

creating a mesh at the boundaries of the domain being studied. In adopting the

Galerkin method of finite element analysis, the governing equation and the

boundary condition are given by equations (5.4) to (5.13) and the system of

equation can be expressed as

[ ] { } [ ] { } { }C T K T Q (5.14)

The above equation (5.14) can be written as

{ } { } [ ]{ } [ ]{ } 0R Q C T K T (5.15)

where {R} is a table containing all the residual nodal values, [K] and [C] are

the conductivity matrix, and the heat capacity matrix, respectively. {T} is the

127

temperature vector, { } /T T t is temperature field, and {Q} is the heat

flow vector and heat exchange on the boundary.

5.4.2 Procedure for Analysis

Procedure for thermal-based parametric analysis is indicated in a

flowchart shown in Figure 5.2. For three-dimensional models, time step is

constant for each lead step during heating and is based on the total heating

time and number of elements in the circumferential direction because heat

source is supposed to stay on each element at least once as recommended by

Lindgren (2001).

The general finite element modeling procedure consists of the

following steps.

Preprocessing

Defining the element type and material properties;

Model building; and

Meshing the model.

Solution

Defining initial condition;

Applying boundary condition;

Applying load; and

Solving for results.

128

Post processing

Examining the solution

The procedure for thermal analysis is depicted in Figure 5.2.

Figure 5.2 Procedure for thermal analysis

5.4.3 Details of Three-Dimensional Model Development

The model for finite element analysis can be built using any of the

two techniques viz.

i) Solid modeling technique; and

ii) Direct generation technique.

In solid modeling, the models are built using geometric primitives,

which are fully developed lines, areas and volumes. The solid model is

meshed to get the finite element model. Direct generation method is

Deformation

Parametric geometry

Solid model generation with predetermined weld bead

geometry

Meshing

FE model generation

Temperature dependent material properties

Boundary conditions

Stress Strain

129

convenient for small and simple models, and also provides complete control

over the geometry and numbering of every node and every element. In this

work, direct generation technique was adopted. The rod was modeled using

solid 70. In addition, a quarter section of the joint was used for the calculation

because of the axial symmetry of the joint. A free mesh was adopted for the

calculation, which includes a total of 85,885 nodes and 25,668 elements for

the calculation domain. The calculation domain is increased and selected

through a series of calculations for this size showing a uniform distribution of

heat flux.

The analysis for deformation, stress and strain was carried for two

different cases of interlayers. The advantages of interlayers during friction

welding of rods were already explained in the previous chapters. Case 1 uses

silver as the interlayer while case 2 has copper being used as an interlayer.

5.5 FEM FOR FRICTION WELDING OF RODS WITH SILVER

INTERLAYER

The material properties of the aluminum alloy, stainless steel and

silver are used for the FEM analysis.

5.5.1 Boundary Condition and Heat Input

For this study, an assumption is made that the friction welding

process is carried out at room temperature with air as the medium in the set-

up. The necessary mathematical equations for the FEA are adopted here.

5.5.2 Results and Discussion

With the above-mentioned numerical model, deformation, stress

and strain distribution during friction welding was predicted and illustrated.

The details were discussed in three subsections.

130

5.5.2.1 Deformation analysis

A unique pattern and characteristic prevalent in the Figures

5.3,5.4,5.5 and 5.6 are described as follows: At a very far distance from the

weld centerline, the von Mises stress is found to be similar to the von Mises

stress of material at room temperature, since the effects of temperature and

deformation on the material are negligible. At a closer distance to the weld

centerline, in the heat-affected zone, the temperature increases and the yield

stress at temperature decreases which in turn, reduces the von Mises stress. At

a closer distance to the weld centerline, the increase of temperature produces

more plasticity while the rest of the material resists against the deformation

which in turn, increases the von Mises stress. At the weld centerline, where

the two surfaces are in contact with each other, the von Mises stress reduces

and the peak plastic strain increases due to temperature softening produced at

the weld centerline. At distances away from the weld zones due to resistance

of the material against the deformation, the von Mises stress increases.

The von Mises stress obtained for the friction welding of aluminum

alloy and stainless steel was found to be higher than that of the individual

materials. This matter is attributed to friction welding of two dissimilar

materials, which in turn produces more resistance to deformation due to the

s variations at temperatures. The deformation increase

while moving towards aluminum alloy and contrarily decreases towards the

stainless steel rod.

131

Figure 5.3 Deformation along X axis

132

Figure 5.4 Deformation along Y axis

133

Figure 5.5 Deformation along Z axis

134

Figure 5.6 Total deformation

135

5.5.2.2 Stress analysis

In the friction welding process, continuous rubbing of contact

surfaces generates the heat at the weld interface. The temperature of the

material increases with heat and therefore subsequent softening of the

material occurs. The material at the interface starts to flow plastically and

forms an upset collar.

Figure 5.7 Equivalent von-Mises stress

136

Figure 5.8 Maximum principal stress

137

Figure 5.9 Vector principal stress

138

Therefore, the type of equation that can be used to describe the material

behavior during the process must include the effect of temperature as well as

strain and strain rate. Therefore, the Johnson-Cook equation is used in which

the von Mises yield stress is defined as a function of strain, strain rate, and

temperature, i.e.,

1 .ln 1n o ompA B c T (5.16)

op is the plastic strain rate, T° is the

homologous temperature and T is the absolute temperature.

room

melt room

T TT T

(5.17)

A, B, n, c, and m are five constants. The expression in the first set of brackets

gives the stress as a function of strain for op =1 and T°=0. The expressions in

the second and third sets represent the effects of strain rate and temperature,

respectively. The Johnson-Cook parameters for the materials are used in this

study.

Simulation results showed that the final pressure did not

significantly affect the stress obtained at the fusion zone and on the fixed bar.

However, the stress increases slightly in the rotating bar with the final

pressure. In other words, the HAZ stress increases in the rotating bar with the

final pressure. Figures reveal that initial pressure is a more important

parameter than final pressure in affecting the stress. While on other portions,

the stress appears to decrease. However, in the aluminum alloy rod the stress

is reasonably higher than in the steel rod.

139

5.5.2.3 Strain analysis

The strain rate is an important physical parameter in plasticity.

Therefore, the analysis was carried out to consider the strain rate distributions

in the friction welding process. The most deformation occurred in the

interface as a result of high transient temperature and subsequent material

flowing. Figures 5.10, 5.11and 5.12 showed the strain rate distributions for

friction welding of aluminum alloys and stainless steel materials.

Figure 5.10 Maximum principal elastic strain

140

Figure 5.11 Minimum principal elastic strain

141

Figure 5.12 Equivalent elastic strain

142

However, the Figures 5.10, 5.11and 5.12 showed that the strain rate

was zero at the weld interface due to temperature softening and might be

positive and negative in the heat-affected zone, decreasing from zero to a

minimum value and reaching again at x = 0; to rise to a maximum positive at

the heat-affected zone. This variation of the strain rate was seen in modeling

of all samples. The strain rate approached zero at a far distance, since no

deformation occurred at distances away from the weld and HAZ zones. More

fluctuations of the strain rate were -hand side

than in the right-hand side. The reason was attributed to fixing of the right-

hand-side bar and rotation of the left hand-side bar. The strain rate

fluctuations were increased with the bar rotation velocity. The simulation

results showed that the effective strain distributions are not affected by the

final pressure. Figures 5.10, 5.11 and 5.12 revealed that initial pressure was

more effective than final pressure on changing the effective strain profiles.

The strain was higher in the steel rod and decreased while moving towards

aluminum alloy rod.

5.6 FEM FOR FRICTION WELDING OF RODS WITH COPPER

INTERLAYER

The material properties of the aluminum alloy, stainless steel and

copper were used for the FEM analysis.

5.6.1 Boundary Condition and Heat Input

For this study, an assumption was made that the friction welding

process was carried out at room temperature with air as the medium in the set-

up. The necessary mathematical equations for the FEA were adopted here.

143

5.6.2 Results and Discussion

With the above mentioned numerical model, deformation, stress

and strain distribution during friction welding was predicted and illustrated.

The details are discussed in three subsections.

5.6.2.1 Deformation analysis

Figure 5.13 Deformation along X axis

144

Figure 5.14 Deformation along Y axis

145

Figure 5.15 Deformation along Z axis

146

Figure 5.16 Total deformation

The deformation increased while moving towards aluminum alloy and

contrarily decreased towards the stainless steel rod. The copper interlayer

yielded a deformation value similar to that of silver interlayer. This

emphasizes the fact that copper interlayer has insignificant influence on the

deformation.

147

5.6.2.2 Stress analysis

Figure 5.17 Equivalent von-Mises stress

148

Figure 5.18 Maximum principal stress

149

Figure 5.19 Vector principal stress

The stress induced at the weld line in the presence of copper

interlayer was considerably higher than the silver interlayer. While on other

portions, the stress appeared to decrease. However, in the aluminum alloy rod,

the stress was reasonably higher than in the stainless steel rod.

150

5.6.2.3 Strain analysis

Figure 5.20 Equivalent elastic strain

151

Figure 5.21 Maximum principal elastic strain

152

Figure 5.22 Minimum principal elastic strain

The strain was higher in the stainless steel rod and decreased while

moving towards aluminum alloy rod. In copper interlayer, strain was higher

than silver interlayer.

5.7 SUMMARIZING THE CHAPTER FINITE ELEMENT

MODELING AND SIMULATION

Considering the mathematical model of continuous effects of the

mechanics and heat transfer, the distributions of temperature, deformation,

von Mises stress, strain, and strain rate during the continuous friction welding

process were numerically analyzed. Using finite element method (FEM) in

153

this study, distribution of welding temperature, flow stress, and plastic strain

and strain rate could be obtained. Thus, this model was found to be an

industrial tool to predict evolution of temperature, stress, strain, and final

geometry of the welded parts. The conclusions drawn from this study may be

summarized as follows:

Simulation results predicted a high value of plastic strain

produced at the weld centerline due to increasing the

temperature;

The peak von Mises stress was produced at distances away

from the weld centerlines;

The von Mises stress reduced in the fusion zone due to

temperature softening;

The von Mises stress obtained for friction welding of steel to

aluminum alloys was found to be high. This was attributed

to friction welding of two dissimilar materials, which in turn

produced more resistance to deformation, and was more due

to the materials yield stress variations at temperatures; and

Silver interlayer appeared to induce less stress and strain as

compared to copper interlayer.