The Pennsylvania State University
The Graduate School
Department of Mechanical Engineering
MODELING AND OPTIMIZATION
OF WELDING RESIDUAL STRESS
A Thesis in
Mechanical Engineering
by
Jinseop Song
c© 2004 Jinseop Song
Submitted in Partial Fulfillmentof the Requirements
for the Degree of
Doctor of Philosophy
December 2004
We approve the thesis of Jinseop Song.
Date of Signature
Panagiotis MichalerisAssociate Professor of Mechanical EngineeringThesis AdviserChair of Committee
Richard C. BensonProfessor of Mechanical EngineeringHead of the Department of Mechanical and Nuclear Engineering
Ashok D. BelegunduProfessor of Mechanical Engineering
Tarasankar DebRoyProfessor of Materials Science and Engineering
iii
Abstract
Modeling and optimization of residual stress in fusion and friction stir welding
is investigated using nonlinear finite element analysis and direct sensitivity evaluation
methods.
Fusion welding has been successfully analyzed as a weakly coupled thermal-
mechanical process (Thermal loads evaluated from the heat transfer analysis are applied
to the mechanical analysis) using nonlinear finite element method in Lagrangian frames
where the mechanical process is considered as thermo-elasto-plastic. Sensitivity formu-
lations are developed using direct differentiation method based on the finite element
equations for both thermal and mechanical analysis. These direct sensitivity evaluation
algorithms are verified by comparing with the finite difference sensitivity method. Using
the gradient optimization algorithm, side heaters are successfully optimized for mini-
mum residual stresses in the objective region of the welded structure. Material property
sensitivity to residual stress in a fusion welding is also evaluated using the automatic
differentiation facility, ADIFOR.
An appropriate numerical residual stress prediction algorithm in FSW, which re-
quires a fully-coupled thermal-mechanical analysis because of significant heat generation
from large plastic strain dissipation, is not available. Two Eulerian thermo-elasto-plastic
formulations are developed as candidate algorithms to analyze the stress formation in
FSW: One is based on the rate equilibrium equation, and the other on the standard
equilibrium equation. Each is implemented using a mixed formulation with Streamline
iv
Upwind Petrov-Galerikin (SUPG) stabilization for three-dimensional 8-node brick ele-
ments. Strip drawing examples are simulated to investigate the validity and convergence
of the two algorithms. A combined thermal-viscoplastic and thermo-elasto-plastic anal-
ysis procedure is proposed for steady state analysis and a FSW example is simulated to
show the potential of the Eulerian thermo-elasto-plastic algorithms.
The main contribution of this thesis is as follows: (a) three-dimensional opti-
mization of thermo-elasto-plastic process, (b) evaluation of material property sensitivity
to welding residual stress, (c) Eulerian FE analysis for elastic rate-independent plastic
material with equilibrium equation.
v
Table of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Optimization of Side Heaters in a Fusion Welding Process . . . . . . 2
1.2 Evaluation of Material Property Sensitivity to Welding Residual Stress
using ADIFOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Residual Stress Prediction in Friction Stir Welding . . . . . . . . . . 7
1.4 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2. Lagrangian FE Equations for Weakly-Coupled thermal-mechanical pro-
cesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Transient Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Mechanical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter 3. Sensitivity Formulations of the Thermo-Elasto-Plastic Process . . . . 21
3.1 Thermal Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Mechanical Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Chapter 4. Numerical Implementations for Side Heater Optimization . . . . . . 29
vi
4.1 Welding Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . 39
Chapter 5. Evaluation of Material Property Sensitivity using ADIFOR . . . . . 48
5.1 FE Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Sensitivity Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3.1 Welding Process and Welding Conditions . . . . . . . . . . . 50
5.3.2 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . 52
5.3.3 Response studies . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.4 Sensitivity Studies . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 6. Eulerian fully-Coupled thermal-Mechanical Analysis for FSW Process 62
6.1 Heat Transfer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2 Mechanical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2.1 Flow Equilibrium Equation (FEE) . . . . . . . . . . . . . . . 63
6.2.2 Flow Rate Equilibrium Equation (FRE) . . . . . . . . . . . . 64
6.2.3 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . 64
6.2.4 Boundary Conditions (BC) for FEE and FRE . . . . . . . . . 68
vii
Chapter 7. Numerical Implementations of the Eulerian Thermo-Elasto-Plastic FE
Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1 Voigt Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Mixed Formulation and Smoothing Function . . . . . . . . . . . . . . 70
7.3 Finite Element Equations . . . . . . . . . . . . . . . . . . . . . . . . 71
Chapter 8. Numerical Examples for the Eulerian Thermo-Elasto-Plastic FE For-
mulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.1 Strip Drawing Examples . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.1.1 Example 1: Purely Elastic Example with Frictionless Surface 77
8.1.2 Example 2: Elasto-Plastic Example with Frictionless Surface 77
8.1.3 Example 3: Purely Elastic Example with Velocity Prescribed
BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.2 FSW Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8.2.1 Boundary Conditions for FSW analysis . . . . . . . . . . . . 87
8.2.2 FSW Example . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.3 Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . 92
Chapter 9. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Appendix A. Basic Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A.1 Isotropic hardening plasticity in deviatoric space . . . . . . . . . . . 95
A.2 Summary of the radial return algorithm . . . . . . . . . . . . . . . . 96
Appendix B. Detailed Derivation of Plastic Sensitivity Equations . . . . . . . 98
viii
B.1Dσ
hDφ
iand
∂σh
∂U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
B.2 DmDφ
iand ∂m
∂U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
B.3Dσ
YDφ
iand
∂σY
∂U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Appendix C. Voigt Transformation . . . . . . . . . . . . . . . . . . . . . . . . 101
Appendix D. FE Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 104
D.1 Mapping to The Master Element . . . . . . . . . . . . . . . . . . . . 104
D.2 Field Variable Interpolators . . . . . . . . . . . . . . . . . . . . . . . 105
D.3 Gradient Interpolators . . . . . . . . . . . . . . . . . . . . . . . . . . 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
ix
List of Tables
4.1 Boundary conditions for mechanical analysis ( see Figure 4.1 for P1, P2,
and P3 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Design variables (see Figure 4.1) . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Normalized longitudinal residual stress sensitivities of initial design by
direct(Sd) and finite difference(S
f) methods in the objective region . . . 45
4.4 Computation times for the initial design (Real time∗ reflects the efficiency
of the parallelization) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.1 Specifications of the strip drawing examples . . . . . . . . . . . . . . . . 77
x
List of Figures
1.1 Welding setup with side heaters . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Welded structure without side heaters (12” X 12” X 1/8”) . . . . . . . . 4
1.3 Welded structure with side heaters (12” X 12” X 1/8”) . . . . . . . . . . 4
1.4 A schematic of friction stir welding operation . . . . . . . . . . . . . . . 7
4.1 Configuration of welding and side heating setup. . . . . . . . . . . . . . 31
4.2 Conductivity (k), specific heat (Cp), and air convection (h) for A36. . . 32
4.3 Elastic modulus (E), Poission’s ratio (ν), thermal expansion coefficient
(α), and yield strength (σY
) for A36. . . . . . . . . . . . . . . . . . . . . 32
4.4 Side heater shape parameters Mx
and Mz
( see Equation (4.3) and Equa-
tion (4.4)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.5 3D Lagrangian analysis model : 12” × 12” × 1/8” . . . . . . . . . . . . . 34
4.6 Temperature profile of the initial design [ ◦C] . . . . . . . . . . . . . . . 40
4.7 Normalized sensitivity of temperature with respect to side heat source
(φ1) of the initial design [ ◦C] . . . . . . . . . . . . . . . . . . . . . . . . 40
4.8 Normalized sensitivity of temperature with respect to transverse position
(φ2) of the initial design [ ◦C] . . . . . . . . . . . . . . . . . . . . . . . . 41
4.9 Normalized sensitivity of temperature with respect to longitudinal dis-
tance (φ3) of the initial design [ ◦C] . . . . . . . . . . . . . . . . . . . . 41
4.10 Longitudinal residual stress of the initial design [MPa] . . . . . . . . . . 42
xi
4.11 Normalized sensitivity of longitudinal residual stress with respect to side
heat source (φ1) of the initial design [MPa] . . . . . . . . . . . . . . . . 42
4.12 Normalized sensitivity of longitudinal residual stress with respect to
transverse position (φ2) of the initial design [MPa] . . . . . . . . . . . . 43
4.13 Normalized sensitivity of longitudinal residual stress with respect to lon-
gitudinal distance (φ3) of the initial design [MPa] . . . . . . . . . . . . 43
4.14 Error of sensitivity in longitudinal stress w.r.t. φ1 . . . . . . . . . . . . 44
4.15 Error of sensitivity in longitudinal stress w.r.t. φ2 . . . . . . . . . . . . 44
4.16 Error of sensitivity in longitudinal stress w.r.t. φ3 . . . . . . . . . . . . 45
4.17 Longitudinal residual stress of the optimum design [MPa] . . . . . . . . 46
4.18 Longitudinal residual stress comparison along the ”Transverse Center
Line” (see Figure 4.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.19 Variation of the normalized objective function F/F0 during optimization 47
5.1 Welding conditions for double fillet welding . . . . . . . . . . . . . . . . 50
5.2 Temperature-dependent thermal and mechanical properties for AL-6XN;
(a) Conductivity k, specific heat Cp, and convection coefficient h (b)
Elastic modulus E, yield strength σy, Poisson’s ratio ν, and thermal
expansion coefficient α . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 2D Lagrangian analysis model . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4 Time histories of the temperature and von-Mises stress for the welded
joint shown in Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
xii
5.5 Snapshots of the normalized temperature and von-Mises stress for the
welded joint shown in Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . 54
5.6 Time histories of the first-order sensitivity coefficients of the temperature
with respect to thermal properties ki, c
pi, and h
ifor the welded joint
shown in Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.7 Time histories of the first-order sensitivity coefficients of von-Mises stress
with respect to mechanical properties Ei, σ
y0i, and α
ifor the welded
joint shown in Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.8 Snapshots of the first-order sensitivity coefficients of the temperature
with respect to thermal properties k4 and cp4 for the welded joint shown
in Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.9 Snapshots of the first-order sensitivity coefficients of von-Mises stress
with respect to mechanical properties E5 and σy05 for the welded joint
shown in Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.10 Time histories of the second-order sensitivity coefficients of the temper-
ature with respect to k3k3, cp2c
p2 and cp2k3 for the welded joint shown
in Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.11 Snapshots of the second-order sensitivity coefficients of the temperature
with respect to k3k3, cp2c
p2 and cp2k3 for the welded joint shown in
Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.1 Strip drawing configuration: Unit[mm] . . . . . . . . . . . . . . . . . . . 75
xiii
8.2 x-directional velocity from FEE for Example 1 (Elastic mat. and fric-
tionless BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.3 y-directional velocity from FEE for Example 1 (Elastic mat. and fric-
tionless BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.4 Mises’ stress from FEE for Example 1 (Elastic mat. and frictionless BC):
Unit[MPa] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.5 x-directional velocity from FEE for Example 2 (Elasto-plastic mat. and
frictionless BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.6 y-directional velocity from FEE for Example 2 (Elasto-plastic mat. and
frictionless BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.7 Mises’ stress from FEE for Example 2 (Elasto-plastic mat. and friction-
less BC): Unit[MPa] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.8 σyy
stress from FEE for Example 2 (Elasto-plastic mat. and frictionless
BC): Unit[MPa] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.9 Equivalent plastic strain from FEE for Example 2 (Elasto-plastic mat.
and frictionless BC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.10 x-directional velocity from FEE for Example 3 (Elastic mat. and velocity
BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.11 x-directional velocity from FRE for Example 3 (Elastic mat. and velocity
BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.12 y-directional velocity from FEE for Example 3 (Elastic mat. and velocity
BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
xiv
8.13 y-directional velocity from FRE for Example 3 (Elastic mat. and velocity
BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
8.14 Mises’ stress from FEE for Example 3 (Elastic mat. and velocity BC):
Unit[MPa] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.15 Mises’ stress from FRE for Example 3 (Elastic mat. and velocity BC):
Unit[MPa] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.16 Eulerian configuration for FSW analysis . . . . . . . . . . . . . . . . . . 87
8.17 x-directional velocity for FSW from FRE (Elasto-plastic mat. and ve-
locity BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.18 z-directional velocity for FSW from FRE (Elasto-plastic mat. and veloc-
ity BC): Unit[mm/s] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.19 Mises’ stress for FSW (Elasto-plastic mat. and velocity BC): Unit[MPa] 91
8.20 Equivalent plastic strain for FSW (Elasto-plastic mat. and velocity BC) 91
xv
Acknowledgments
I am most grateful and indebted to my thesis advisor, Panagiotis Michaleris, for
the large doses of guidance, patience, and encouragement he has shown me during my
time here at Penn State. I am also grateful and indebted to all of my labmates, for
inspiration and enlightening discussions on a wide variety of topics. I am especially
indebted for the financial support which they have provided to me over the years. I
thank my other committee members, Richard C. Benson, Ashok D. Belegundu, and
Tarasankar Debroy, for their insightful commentary on my work.
1
Chapter 1
Introduction
Residual stress, which affects the strength and deformation of the processed prod-
ucts, occurs in various material processes: fusion welding, laser forming, extrusion,
rolling, and friction stir welding . In order to determine appropriate process parameters,
numerical methodologies to predict residual stress are necessary. From the numerical
analysis formulations, sensitivity evaluation algorithm can also be developed. Design
parameter sensitivity supplies information for appropriate parameter modifications and
enables gradient optimization of the parameters.
This thesis mainly focuses on residual stress in fusion welding and FSW. Fusion
welding processes have been successfully analyzed using weakly-coupled thermo-elasto-
plastic Finite Element (FE) formulations in Lagrangian frames [71, 74, 4, 39, 12, 57,
67, 70, 9]. However, a validated numerical residual stress prediction algorithm for FSW,
which requires fully-coupled thermal-mechanical analysis because of considerable heat
generation through large plastic dissipation, is not available.
In this thesis, a systematic numerical algorithm is develop to optimize side heaters
in a fusion welding process for minimum residual stress sensitivity and optimization
procedures are developed from the weakly-coupled thermal-mechanical FE analysis for-
mulations to minimize residual stress in a fusion welding and velocity-base Eulerian
thermo-elasto-plastic formulations are developed for residual stress prediction in FSW.
2
An automatic differentiation facility, ADIFOR [15, 11, 10], is also explored and demon-
strated for the evaluation of material property sensitivity to welding residual stress in a
fusion welding.
The Objectives of this thesis are as follows:
1. Develop a systematic algorithm to optimize side heaters in a fusion welding process
for minimum residual stress.
2. Present and explore a computational procedure for evaluating sensitivity by using
ADIFOR, an automatic differentiation facility.
3. Develop an efficient computational algorithm to predict residual stress in friction
stir welding process.
1.1 Optimization of Side Heaters in a Fusion Welding Process
As a mechanical joining process, welding has many advantages in design flexibility,
cost savings, reduced overall weight, and enhanced structural performance. However,
welding results in residual stresses which have undesirable effects on the performance of
the welded structure [35, 75, 70, 19]. For example, strength degradation, various types
of distortion, and even buckling can be caused by residual stress.
It is, of course, possible to control the welding residual stress by reducing the
welding heat input and/or modifying the structural dimensions. However, design con-
sideration may impose limits on such modifications. In that case, transient thermal
tensioning technique can be used for the same purpose without modifying design specifi-
cations [13, 14]. This technique can be implemented by applying side heaters that move
3
along with welding torches, which is experimentally realized by M. V. Deo [26]. The side
heaters can be applied as shown in Figure 1.1. Under the same welding conditions, the
welded structure without side heaters has buckling distortion (Figure 1.2) whereas the
other with side heaters has only some angular distortion (Figure 1.3).
Fig. 1.1. Welding setup with side heaters
Application of side heaters to the welding process requires determining the di-
mensions of side heaters, such as the heat input, the size, and the relative positions of
side heaters to the welding torches for minimum residual stress. Empirical approaches
for this optimization problem are generally time consuming and costly. Therefore, a
systematic computational optimization methodology is required.
The welding process has been widely analyzed as a weakly coupled thermo-elasto-
plastic problem using nonlinear finite element technique in Lagrangian frames [71, 74, 4,
4
Fig. 1.2. Welded structure without side heaters (12” X 12” X 1/8”)
Fig. 1.3. Welded structure with side heaters (12” X 12” X 1/8”)
5
39, 12, 57, 67, 70, 9]. Considering the heavy computational load of welding analysis, non-
gradient methods such as genetic algorithm are computationally too expensive for this
side heater optimization problem. Gradient optimization methods are computationally
more efficient since they require far less objective function evaluation. However, they
require the calculation of sensitivities of the objective function and constraint functions
with respect to each design variable.
Sensitivity analysis has been widely used in many design optimization problems
[81, 38, 25, 86, 54, 5, 85, 7]. Sensitivity analysis can be performed by analytical or by
finite difference techniques [37]. Finite difference methods have round-off or truncation
errors and require additional objective function evaluation for all design variables. Thus,
analytical methods are more accurate and computationally more efficient than finite dif-
ference method. Analytical sensitivities can be computed either by direct differentiation
or by adjoint method [38]. Direct differentiation method is computationally more effi-
cient than adjoint method if the optimization problem has more constraints than design
variables.
Sensitivity analysis for coupled systems is presented in reference [56]. Sensitivity
analysis of thermo-elasto-plastic processes in two dimensional frames with the assump-
tion of generalized plane strain has been implemented in minimizing welding residual
stress and distortion in reference [55]. Sensitivity analysis for thermo-elasto-plastic pro-
cesses in Eulerian reference frames has been developed to optimize the laser forming
process in reference [61]. Critical assumptions are inherent in the formulation of these
two approaches. The two dimensional approach is limited in accounting for three dimen-
sional effects and the Eulerian approach is applicable only for steady-state processes.
6
Michaleris et al. have demonstrated a sensitivity analysis for thermo-elaso-plastic pro-
cesses in Lagrangian reference frames [57]. However, the solution algorithm employed
(iteration-subiteration) is computationally less efficient than radial return algorithm [71]
and ,thus, the sensitivity algorithm is implemented only for a two dimensional exam-
ple. Therefore, it is necessary to develop sensitivity equations for thermo-elasto-plastic
processes with the radial return algorithm in three dimensional Lagrangian reference
frames.
1.2 Evaluation of Material Property Sensitivity to Welding Residual
Stress using ADIFOR
Numerical simulation techniques to study the various phenomena associated with
welding have been developed. For example, weld-pool physics, heat and fluid flow, heat
source-metal interactions, weld solidification microstructures, phase transformations, and
residual stresses and distortions have been studied. Recent studies of residual stresses
and distortions in welded structures are reported in [31, 74, 4, 39, 12, 2, 9, 83, 66, 67, 16].
In these numerical studies of welding, the accuracy of temperature-dependent material
properties plays an important role in the accuracy of predicted residual stresses.
Since current measurement technology does not allow the accurate determination
of the material parameters that are used in the analytical models, it is useful to assess
the sensitivity of the thermomechanical responses of welded joints to variations in the
various material parameters. The present study focuses on this topic. Specifically, the
objective of this research is to present a computational procedure for evaluating the
sensitivity coefficients of the quasi-static response of welded joints by using the direct
7
differentiation method in conjunction with the automatic differentiation software facility
ADIFOR [15, 11, 10].
1.3 Residual Stress Prediction in Friction Stir Welding
Friction stir welding (FSW) is a new joining technology developed at The Welding
Institute (TWI) in England [76]. A schematic of a FSW operation for joining two flat
Fig. 1.4. A schematic of friction stir welding operation
plates is shown in Figure 1.4. The bottom of the plates is usually supported by a die and
both outer sides of the plates are clamped rigidly. Rotating a speed of several hundred
rpms, the FSW tool, consisting of a shoulder and a pin, advances longitudinally with a
velocity of several millimeters per second during welding, a little slower than the torch
travel speed in fusion welding processes. The tool pin is usually threaded to supply a
8
larger friction heating area and tilted by several degrees from the vertical to facilitate
consolidation of the weld. The tool shoulder, with diameter up to three times of that of
the pin, prevents the material from being expelled.
Although some applications use additional heat [44] and some report local tran-
sient melting on contact surfaces [58], the weld is usually performed mechanically in
the solid state without melting [50]. These welded structures have improved strength
and fatigue properties compared with conventional fusion welding in which solidification
cracking, porosity, oxidation, and other defects typically occur due to melting. High
restraint in FSW limits the formation of angular distortion. However, an experimental
investigation [63] has revealed that the residual stress of FSW is comparable to that of
fusion welding. Since high residual stress for large parts may lead to buckling distortion,
there is a need for numerical capability to predict residual stress in FSW.
FSW has been primarily utilized to join aluminum alloys. Along with the de-
velopment of the FSW tool, this technique has been shown to be applicable to joining
copper, magnesium, lead, titanium, zinc, plastics, mild steel, and even mid- and high-
strength steels. Although considerable experimental work has been reported on FSW
[53, 41, 18, 60, 52], few analytical modeling works can be found in the published litera-
ture for residual stress formation in FSW. A validated analytical model is necessary to
efficiently optimize FSW processes for various types of materials and environments.
The temperature field has been analyzed independently of the mechanical field,
assuming heat is generated only from friction in the contact surface between the tool and
material in References [53, 41, 18, 60]. However, this independency assumption cannot
9
be adopted for a thorough analysis of FSW because of the considerable flow rate and
plastic dissipation heat generation.
Heurtier et al. [62] suggested a simplified thermo-mechanical model to under-
stand material movement during FSW. However, their closed-form sloution is obtained
from many simplifying assumptions such that the solution is rather a guess than an
analysis result. Chao and Qi [17] have analyzed residual stress in FSW process using a
thermo-elasto-plastic anlysis scheme. However, they did not consider pin tool geometry,
plastic dissipatation heat generation, and convective heat flow due to material movement.
Ulysse [80] has performed fully-coupled thermal-mechanical analysis with pin geometry.
However, his model cannot evaluate residual stress, since the viscoplastic constitutive
model does not consider elastic effects.
A fully coupled thermal-mechanical model is more appropriate in modeling the
FSW process because the plastic dissipation heat generation may be too large to neglect
in the thermal analysis. Heat transfer Finite Element (FE) formulations in Eulerian
frames have been well developed for laminar flow of materials with a known velocity
field [88]. Therefore, this research is mainly focused on mechanical analysis of the FSW
process.
Various models have been developed in Lagrangian frames to estimate residual
stress during history-dependent material processes, such as conventional fusion welding
with rate-independent plasticity model [47] or rate-dependent viscoplastic model [4, 59,
84], or combined model (rate-independent at lower temperature and rate-dependent at
higher temperature) [30]. Small deformation and weak thermal-mechanical coupling
are assumed in these models and incremental analysis is performed. However, these
10
Lagrangian models may not be directly applied for FSW analysis since the workpiece
deforms severely in the FSW process and Lagrangian elements may be similarly distorted.
Arbitrary Lagrangian Eulerian (ALE) formulations have been developed by Haber[36]
and Koh [43] for history-independent problems and by Gosh [29] for history-dependent
problems. Although ALE formulations can resolve element distortion, they still require
incremental analysis for history-dependent problems.
If the coordinate system is selected to move with the FSW tool, an Eulerian
steady-state formulation can be applied so that the solution field can be obtained in a
steady state analysis. Eulerian formulations have been used mainly for forming, extru-
sion, and rolling processes. Visco-elastic models [20, 21, 27] are appropriate for polymer
melts and viscoplastic models [1, 22, 42, 65, 82] for metals. However, these formulations
cannot predict residual stress since the elastic strain is neglected. Lee and Dawson [45]
have evaluated residual stress where elasticity is neglected on loading and recovered after
loading is removed. However, this method sacrifices accuracy since the plastic evolution
is ignored during unloading. Elastic-viscoplastic models in Eulerian frames have also
been developed assuming incompressible elasticity [3, 23, 77]. Multiplicative elastic and
plastic strain decomposition has also been incorporated into this elastic-viscoplastic ma-
terial model in Reference [51]. However, an elastic-viscoplastic model is reported to be
numerically unstable when the elastic response becomes large [78].
Although extensive research has been published for Eulerian elastic-viscoplastic
(rate-dependent plasticity) material models, limited publications are available for Eu-
lerian elasto-plastic (rate-independent plasticity) models. A displacement-based mixed
11
formulation in undeformed moving reference frames was first introduced by Balagangad-
har et al. [6] for a laser surfacing problem with small deformation and rate-independent
elasto-plasticity and has been further developed by Shanghvi et al. [69] for the analysis of
laser forming. Large deformation formulations with multiplicative decomposition of the
deformation gradient have also been developed by Balagangadhar et al. [24]. However,
the velocity prescribed boundary condition essential for FSW cannot be incorporated
into this displacement-based formulation. Thus, a velocity-based Eulerian formulation
with elasto-plastic material model is more appropriate for modeling FSW. Publications
for velocity-based elasto-plastic material model in Eulerian frames are rare except the
one by Thompson et al. [78] in which a Flow Rate equilibrium Equation, or FRE method,
is proposed.
In this thesis, the FRE is investigated and a novel thermo-elasto-plastic Eule-
rian formulation based on the standard Flow Equilibrium Equation (FEE) is developed.
The performance of the two formulations is explored by simulating strip drawing exam-
ples. An application approach of these algorithms to modeling the FSW process is also
discussed.
1.4 Thesis Layout
The overall thesis structure is presented in Chapter 1.
The side heater optimization procedure is presented through Chapter 2, Chapter
3, and Chapter 4 and published as [72]. Finite element equations for the weakly coupled
thermo-elasto-plastic process in three dimensional Lagrangian frames are reviewed in
Chapter 2. Sensitivity equations are developed in Chapter 3 based on the finite element
12
equations in Chapter 2. In Chapter 4, the developed gradient optimization algorithms
are implemented to optimize side heaters in a fusion welding process to minimize welding
residual stress.
The material property sensitivities to welding phenomena are evaluated using the
automatic differentiation facility in Chapter 5 and also published in a journal [73].
The computational algorithms to predict residual stress in a friction stir welding
process are developed through Chapter 6 and Chapter 8. In Chapter 6, weak formula-
tions of two Eulerian thermo-elasto-plastic for a thermo-elastic rate independent plastic
material model are proposed to predict residual stress in a friction stir welding process.
In Chapter 7, finite element implementation procedure for the weak formulations is pre-
sented. In Chapter 8, the developed two formulations are verified and compared and an
application approach to the friction stir welding analysis is discussed.
Conclusions of this thesis is presented in Chapter 9.
13
Chapter 2
Lagrangian FE Equations
for Weakly-Coupled thermal-mechanical processes
Finite element formulations for quasi-static thermo-elasto-plastic processes in La-
grangian reference frames have been widely used in analyzing fusion welding processes
[71, 74, 4, 39, 12, 57, 67, 70, 9]. Thermal analysis is assumed to be transient while
mechanical analysis remains quasi-static. Thermo-elasto-plastic processes are typically
assumed to be weakly coupled, that is, the temperature profile is assumed to be inde-
pendent of stresses and strains. Thus, a heat transfer analysis is performed initially
and the temperature history is imported as loading in the mechanical analysis. Both
thermal and mechanical problems are nonlinear due to temperature-dependent material
properties and plasticity, respectively.
2.1 Transient Thermal Analysis
For the Lagrangian coordinate X fixed to the body, and time t, the governing
equation for transient heat conduction analysis is given as,
ρCp
∂T
∂t= ∇ · [k∇T ] + Q in the entire volume V of the material (2.1)
14
where ρ is the density of the body, Cp
is the specific heat capacity, T is the temperature,
k is the temperature-dependent thermal conductivity matrix, Q is the internal heat
generation rate, and ∇ is the Lagrangian gradient operator.
The initial temperature field is given by
T = T0 in the entire volume V (2.2)
where T0 is the prescribed initial temperature. The following boundary conditions are
applied on the surface:
T = T on the surface AT , with prescribed temperatures T (2.3)
q = q on the surface Aq, with prescribed heat fluxes q (2.4)
Multiplication of Equations (2.1) and (2.4) by any kinematically admissible func-
tion T , integration over the volume and surface, integration by parts and application of
divergence theorem yields the following weak statement:
∫V
{−∇T
T k∇T + T
[Q − ρC
p
∂T
∂t
]}dV −
∫Aq
T qdA = 0 (2.5)
By applying finite element discretization to Equation (2.5), the global residual
vector R can be assembled from the element residual vector R as follows:
R( nT ) =∑e
BR( nT) = 0 (2.6)
15
where T is the global node temperature vector, T is the element node temperature
vector, left superscript n represents quantities evaluated at the time increment nt, B is
the wapping operator from each element components to the global components, and e is
the element number. The element residual vector R can be evaluated as follows:
R( nT) =∑GV
e
{BT kB nT − NT
Q + NT NρCp
nT − n−1Tnt − n−1t
}WJ
+∑GAq
e
NTqwj (2.7)
where GVe
and GAq
edenote Gauss points in the element volume V
eand on the element
surface Aq
erespectively, left superscript n−1 represents quantities evaluated at the time
increment of n−1t, N and B are the usual matrices which interpolate the temperature
T and temperature gradient ∇T in an element; J and j are the volume and the area
Jacobian components; and the Gaussian weighting is represented as W for the volume
and w for the surface integration. Equation (2.6) is solved in an incremental, iterative
fashion. For each time increment from time n−1t to n
t, nT is updated iteratively with
the known temperature n−1T until R becomes small enough:
δT = −[
dRd nT
∣∣∣∣ nT I
]−1R( nT I ) (2.8)
nT I+1 = nT I + δT (2.9)
Similar to Equation (2.6), the global stiffness dRd nT is assembled from element stiffnesses
dRd nT which can be obtained from Equation (2.7):
16
dRd nT =
∑e
B dRd nT
BT (2.10)
where
dRd nT
=∑GV
e
[BT kB + BT ∂k
∂TB nTN − NT ∂Q
∂TN + NT NρC
p
1nt − n−1t
+NT Nρ∂C
p
∂TN
nT − n−1
Tnt − n−1t
]WJ +
∑GAq
e
NT ∂q
∂TNwj (2.11)
2.2 Mechanical Analysis
The equilibrium equation in a volume of material V with boundary A can be
written as,
∇ · S + b = 0 in V (2.12)
where S is the second-order stress tensor and b the body force vector. The boundary
conditions are given as,
u = u on surface Au (2.13)
S · n = t on surface At (2.14)
where u is the prescribed displacement on surface Au, t is the prescribed traction on
surface At, and n is the unit outward normal to the surface A
t. Using small deformation
theory, the total strain tensor E can be related with the displacement vector u as follows:
17
E =12
{∇u + [∇u]T
}(2.15)
Because of the symmetry, the stress tensor S and strain tensor E are commonly
expressed in the vector form σ and ε for computational efficiency.
Due to the small deformation assumption and the thermo-elasto-plasticity, the
total strain ε can be decomposed into the elastic strain εe, the plastic strain ε
p, and the
thermal strain εt:
ε = εe
+ εp
+ εt
(2.16)
The initial conditions can be described as follows:
u = u0 (2.17)
εp
= εp0 (2.18)
εq
= εq0 (2.19)
where εq
is the equivalent plastic strain.
The stress strain relationship is given by
σ = Cεe
= C[ε − ε
p− ε
t
](2.20)
where C is the temperature-dependent elasticity tensor.
18
Similar to the thermal formulation, a weak formulation and finite element dis-
cretization yields the element residual R:
R( nU) =∑GV
e
[BT n
σ − NT b]WJ −
∑GAt
e
NT twj (2.21)
where U is the element displacement vector. The stress at time t can be decomposed as,
nσ = n−1
σ + ∆σ (2.22)
where ∆ represents an increment during the time increment from n−1t to n
t. Evaluating
Equation (2.20) at the times n−1t and n
t and taking the difference yields the following
equation:
∆σ = nC[∆ε − ∆ε
p− ∆ε
t
]+ ∆C n−1
εe
(2.23)
where
∆ε = B[ nU − n−1U] (2.24)
∆εp
= ∆εqa (2.25)
∆εt
= [ nεt− n−1
εt]h (2.26)
={
nα[
nT − T
ref]− n−1
α[
n−1T − T
ref]}
h (2.27)
h = [ 1 1 1 0 0 0 ]T (2.28)
19
a is the flow vector, α is the temperature-dependent thermal expansion coefficient, and
Tref is the reference temperature.
The elastic predictor σB
and corresponding elastic strain εBe
are defined as
follows:
σB
= n−1σ + nC
[∆ε − ∆ε
t
]+ ∆C n−1
εe
(2.29)
εBe
= n−1εe
+ ∆ε − ∆εt
(2.30)
Using associative J2 plasticity [49], the yield function f is given as follows:
f = σm
− σY
(2.31)
where σm
and σY
are the Mises stress and yield stress. Active yielding occurs when
f ≥ 0. In the case of non-active yielding, σB
and εBe
simply become nσ and n
εe. In
the case of active yielding, the evolution of ∆εq
can be evaluated by the radial return
algorithm [71] (see appendix A) .
The element stiffness matrix is given by the following expression:
dRd nU
=∑GV
e
[BT d
nσ
d nε− NT db
d nU
]WJ −
∑GAt
e
NT dtd nU
wj (2.32)
In case of non-active yielding, dnσ
d nεis equal to nC. In case of active yielding:
dnσ
d nε= λ
effhhT + 2G
effL−1 +
[3GH
3G + H− 3G
eff
]mmT (2.33)
20
where
m =23L−1a (2.34)
Geff
= GσY
σBm
(2.35)
λeff
= [3k − 2Geff
]/3 (2.36)
L = diag( 1 1 1 2 2 2 ) (2.37)
G is the shear modulus, k is the bulk modulus, H is the isotropic hardening coefficient,
σBm
is the Mises stress of the elastic predictor σB
, and m is another form of the flow
vector defined in this study for simplicity.
21
Chapter 3
Sensitivity Formulations
of the Thermo-Elasto-Plastic Process
An optimization problem is generally formulated as follows:
minimize F (φ1, φ2, ..., φr) (3.1)
subject to gj(φ1, φ2, ..., φ
r) ≤ 0, j = 1, 2, ..., s (3.2)
and hk(φ1, φ2, ..., φ
r) = 0, k = 1, 2, ..., p (3.3)
where F is an objective function; φi
is the ith design variable; g
jis the j
th inequality
constraint function; gk
is the kth equality constraint function; and r, s, and p are the
numbers of design variables, inequality constraints, and equality constraints respectively.
In thermo-mechanical processes, for example, design variables can be chosen from any
process parameter and material property and contribute to an objective function directly
and/or indirectly through solution fields such as temperature, displacement, strain, and
stress.
In this section, sensitivity formulations for thermo-elasto-plastic processes are
developed with the direct differentiation method so that gradient optimization methods
can be utilized as discussed in Section 1.1. In the sensitivity formulations, all finite
element variables (not design variables) are divided into two classes, that is, the primary
solution variable ( nT for thermal and nU for the mechanical systems) and others. D
22
is defined to denote derivatives with respect to all variables except the primary solution
at time nt where d represents the usual total derivative, that is,
dYdφ
i
=∂Y∂X
dXdφ
i
+DYDφ
i
(3.4)
where X is the primary solution variable and Y is an arbitrary function.
3.1 Thermal Sensitivity
Differentiating Equation (2.6) with respect to each design variable φi
yields:
dnT
dφi
= −
[
dRd nT
]−1
constant φi
DR
Dφi
(3.5)
where D stands for the differentiation with respect to all variables except nT . Note
that the global stiffness has already been assembled to evaluate the global temperature
in Equation (2.10) and DRDφ
ican be assembled from DR
Dφi. Differentiation of Equation
(2.7) yields:
DRDφ
i
=∑GV
e
{[dBdφ
i
TkB nT + BT k
dBdφ
i
nT + BT dkdφ
i
B nT − NT dQ
dφi
+NT NdρC
p
dφi
nT − n−1Tnt − n−1t
+ NT NρCp
−1nt − n−1t
dn−1Tdφ
i
]WJ
+
[BT kB nT − NT
Q + NT NρCp
nT − n−1Tnt − n−1t
]W
dJ
dφi
}
+∑GAq
e
[NT dq
dφi
wj + NTqw
dj
dφi
](3.6)
23
3.2 Mechanical Sensitivity
In this section, it is assumed that temperatures and their sensitivities are available
(Equations (2.6) and (3.5)). Furthermore, mechanical forces are not considered, that is,
the temperature change is the only loading in this sensitivity formulation. The left
superscript n is dropped for simplicity. Only the main procedure and final equations
of the mechanical sensitivity formulations are presented. More detailed derivations are
presented in Appendix B.
Differentiating the global residual equation with respect to each design variable
φi
yields:
dUdφ
i
= −
[dRdU
]−1
constant φi
DR
Dφi
(3.7)
Similar to the thermal sensitivity formulation, the only term that needs to be evaluated
for displacement sensitivity is DRDφ
i. If the body force b and traction t are zero, Equation
(2.21) becomes:
R =∑GV
e
BTσWJ (3.8)
Recalling the definition of D and noting that B and J are independent of U, the following
equation is derived from Equation (3.8):
DRDφ
i
=∑GV
e
[dBT
dφi
σWJ + BT Dσ
Dφi
WJ + BTσW
dJ
dφi
](3.9)
Now every other term except DσDφ
iin Equation (3.9) can be easily evaluated.
24
Differentiation of Equations (2.29) and (2.30) using the relations of Equations
(2.24) and (2.27) yields:
DσB
Dφi
=d
n−1σ
dφi
+ ∆Cd
n−1εe
dφi
+
{∂C∂T
[n−1
εe
+ ∆ε − ∆εt
]− C
∂∆εt
∂T
}dT
dφi
−[C
∂∆εt
∂ n−1T+
∂n−1C
∂ n−1T
n−1εe
]d
n−1T
dφi
+ CdBdφ
i
[U − n−1U
]− CB
dn−1Udφ
i
(3.10)
DεBe
Dφi
=d
n−1εe
dφi
−∂∆ε
t∂T
dT
dφi
−∂∆ε
t
∂ n−1T
dn−1
T
dφi
+dBdφ
i
[U − n−1U
]− B
dn−1Udφ
i
(3.11)
where
∂∆εt
∂T={
α +∂α
∂T
[T − T
ref]}
j (3.12)
∂∆εt
∂ n−1T= −
{n−1
α +∂
n−1α
∂ n−1T
[n−1
T − Tref]}
j (3.13)
In the case of the non-active yielding:
Dσ
Dφi
=Dσ
BDφ
i
(3.14)
In the case of the active yielding, σ can be expressed as follows from the radial return
algorithm (see Appendix A):
σ = σh
+ σY
m (3.15)
where σh
is the hydrostatic stress of σ . Then DσDφ
iis evaluated as follows:
Dσ
Dφi
=Dσ
hDφ
i
+ σY
DmDφ
i
+ mDσ
YDφ
i
(3.16)
25
where (see Appendix B)
Dσh
Dφi
=13jjT
DσB
Dφi
(3.17)
DmDφ
i
=1
σBm
[I − 1
3jjT − maT
] DσB
Dφi
(3.18)
DσY
Dφi
=
aT Dσ
BDφ
i
+3G
H
H
dn−1
εq
dφi
+∂σ
Y∂T
dT
dφi
− 3∆ε
q
∂G
∂T
dT
dφi
/
[1 +
3G
H
](3.19)
where I is the identity matrix andDσ
BDφ
iis evaluated in Equation (3.10).
Now, DσDφ
iis available for both cases of non-active and active yielding so that dU
dφi
can be evaluated. However, n−1 dσdφ
i, n−1 dε
edφ
i, and n−1 dε
qdφ
iare necessary to evaluate
DσDφ
iin Equations (3.10), (3.11), and (3.19). Therefore, once the displacement sensitivity
dUdφ
i(called primary sensitivity) is evaluated, the stress sensitivity dσ
dφi, elastic strain sen-
sitivitydε
edφ
i, and equivalent plastic strain sensitivity
dεq
dφi
(called secondary sensitivities)
should also be evaluated in the current increment for use in the next increment sensitiv-
ity evaluation. Using the definition of D in Equation (3.4), the secondary sensitivities
can be expressed as follows:
dσ
dφi
=∂σ
∂UdUdφ
i
+Dσ
Dφi
(3.20)
dεe
dφi
=∂ε
e∂U
dUdφ
i
+Dε
eDφ
i
(3.21)
dεq
dφi
=∂ε
q
∂UdUdφ
i
+Dε
q
Dφi
(3.22)
26
In Equations (3.20), (3.21), and (3.22), dUdφ
iand Dσ
Dφi
are already available. Thus, it
is necessary to evaluate the other five terms in the righthand side of Equations (3.20),
(3.21), and (3.22).
Differentiating Equations (2.29) and (2.30) with respect to U after applying the
relations of Equation (2.24) yields the following equations:
∂σB
∂U= CB (3.23)
∂εBe
∂U= B (3.24)
In the case of the non-active yielding, σB
and εBe
are equal to σ and εe. Furthermore,
there is no plastic evolution. Therefore, the secondary sensitivity equations become:
dσ
dφi
=dσ
Bdφ
i
(3.25)
dεe
dφi
=dε
Bedφ
i
(3.26)
dεq
dφi
=d
n−1εq
dφi
(3.27)
In the case of the active yielding, at first, ∂σ∂U in Equation (3.20) can be obtained
by differentiating Equation (3.15):
∂σ
∂U=
∂σh
∂U+ m
∂σY
∂U+ σ
Y
∂m∂U
(3.28)
27
where (see Appendix B)
∂σh
∂U=
13jjT
∂σB
∂U(3.29)
∂m∂U
=1
σBm
[I − 1
3jjT − maT
] ∂σB
∂U(3.30)
∂σY
∂U=
[aT ∂σ
B∂U
]/
[1 +
3G
H
](3.31)
where∂σ
B∂U is already evaluated in Equation (3.23). Next, consider the following relation
which can be obtained from Equations (2.30), (2.25), and (2.34):
εe
= εBe
− ∆εp
= εBe
− ∆εq
32Lm (3.32)
Then, the two unknown terms in Equation (3.21) are obtained by differentiating Equation
(3.32) respectively:
∂εe
∂U=
∂εBe
∂U− 3
2L
[m
∂∆εq
∂U+ ∆ε
q
∂m∂U
](3.33)
Dεe
Dφi
=Dε
BeDφ
i
− 32L
[m
D∆εq
Dφi
+ ∆εq
DmDφ
i
](3.34)
where (see Appendix B)
∂∆εq
∂U=
1H
∂σY
∂U(3.35)
D∆εq
Dφi
=1
3G
[aT Dσ
BDφ
i
− 3∆εq
∂G
∂T
dT
dφi
−Dσ
YDφ
i
](3.36)
28
and∂ε
Be∂U ,
∂σY
∂U , ∂m∂U , DεBe
Dφi
,Dσ
BDφ
i,Dσ
YDφ
i, and Dm
Dφi
are evaluated in Equations (3.24),
(3.31), (3.30), (3.11), (3.10), (3.19), and (3.18). Finally, the two unknown terms in (3.22)
are obtained as follows:
∂εq
∂U=
∂∆εq
∂U(3.37)
Dεq
Dφi
=d
n−1εq
dφi
+D∆ε
q
Dφi
(3.38)
where∂∆ε
q∂U and
D∆εq
Dφi
are evaluated in Equations (3.35) and (3.36).
29
Chapter 4
Numerical Implementations
for Side Heater Optimization
Welding is a complex process in which various electromagnetic and thermo-mechanical
phenomena take place. Modeling and simulation of the mechanical effects of welding has
now been an on-going research effort for three decades [32, 31, 33, 79, 87, 64, 46, 47, 48].
Most commonly, in welding process modeling, the physics that accounts for heat gen-
eration is not analyzed. Empirical heat generation models are used instead [32]. Fur-
thermore, the majority of welding simulations neglect the molten metal flow in the weld
pool. To consider the convective heat flow in the molten metal, artificially high thermal
conductivity values are assigned to regions having temperatures that exceed the melting
point. A rate-independent, deviatoric plasticity model with von Mises yield condition
and associated flow rule has been used with success in most welding simulations [47].
Some works have also used visco-plastic models [4, 59, 84] or combined rate-independent
plasticity at lower temperatures with visco-plastic models at higher temperatures [30].
In this section, the heat source and the positions of side heaters are optimized
with other variables fixed for minimum residual stress using the sensitivity equations
developed in the previous section. No constraints except the explicit region of design
variables are considered in this example.
30
4.1 Welding Conditions
A schematic of the welding configuration in this simulation is shown in Fig 4.1.
Side heaters travel along the plate and are followed by two welding torches.
Convection boundary conditions are assigned to all free surfaces. Radiation heat
transfer, melting, and solidification are not considered in this simulation because of their
negligible influence upon the residual stress field [74, 4, 67, 70]. The rate of internal heat
generation by the welding torch, modeled as the “double ellipsoid” heat source model
[34], is given by,
Q =6√
3Qw
ηw
f
abcπ√
πe−[3x
2
a2 +3y2
b2+3z
2
c2]
[W/mm3] (4.1)
where Qw
(2680.35 W/mm3) is the welding heat input; η
w(1.0) is the welding efficiency,
x, y, and z are the local coordinates of the double ellipsoid model aligned with the weld
fillet; a (5√
2 mm) is the weld width; b (5√
2 mm) is the weld penetration; c is the weld
ellipsoid length; f is the weld heat input density distribution factor; and v (6.35 mm/s)
is the torch travel speed. The numbers in parentheses are the values which are used
for this implementation. Goldak et al. [34] used c = a and f = 0.6 before the torch
passes the analysis region, and c = 4a and f = 1.4 after the torch pases the analysis
region. However, in this paper, a more distributed heat source with c = 4a and f = 1.0
is used instead to improve the convergence in the simulation. In fact, these factors have
a measurable effect on the temperature field but have negligible effect upon the residual
stress.
31
Torch1
φ2
3.5"
φ3
Ls
Weldingdirection
Side heaters :* Moving along with the torches
* Heat power = φ1
P2
P1
P3
Transverse Center Line
L
Bs
B
x
y
z
2"
Objective Region
Torch2
Fig. 4.1. Configuration of welding and side heating setup.
32
Cp
h
k
Fig. 4.2. Conductivity (k), specific heat (Cp), and air convection (h) for A36.
Fig. 4.3. Elastic modulus (E), Poission’s ratio (ν), thermal expansion coefficient (α),and yield strength (σ
Y) for A36.
33
The side heat source is applied on the top surface of the plate as shown in Fig 4.1
and is defined as follows:
q(x, z) =Q
sηs
2BsL
s
MxM
z(4.2)
Mx
={
tanh(S
x2[x + φ2 + Bs/2])− tanh
(S
x1[x + φ2 − Bs/2])
+ tanh(S
x1[x − φ2 + Bs/2])− tanh
(S
x2[x − φ2 − Bs/2])}
/2 (4.3)
Mz
={
tanh(S
z1[z − Ls/2])− tanh
(S
z2[z + Ls/2])}
/2 (4.4)
where x and z are the local coordinates from the center of the side heating; Qs(W/mm
2)
is the side heating input, ηs
(1.0) is the side heating efficiency; Bs
(6”) and Ls(1”) are
the band width and length of the side heating; and Sx1 (0.2), S
x2 (0.2), Sz1 (0.2), and
Sz2 (0.2) are used to control the gradient of heat flux in the side heater edges. This side
heater shape is shown in Figure 4.4. The numbers in parentheses are the values which
are used in this simulation.
The material properties of A36 steel used in this simulation are shown in Figure
4.2 for the thermal analysis and in Figure 4.3 for the mechanical analysis. The isotropic
hardening coefficient is assumed to be 8000 [MPa] at any temperature.
A finite element model is developed as shown in Figure 4.5 based on reference
[32]. The dimensions are 12” × 12” × 1/8” for the base plate and 12” × 2” × 1/8” for
the stiffener. This model has 13864 nodes and 2352 20-noded brick elements. Since high
temperature gradients are prevalent at the welding region, the mesh is finer along the
welding torch path and coarser away from it. Boundary conditions for the mechanical
analysis are shown in Table 4.1.
34
−150 −100 −50 0 50 100 1500
0.2
0.4
0.6
0.8
1
z [mm]
Mz
Sz1=Sz2=1.0Sz1=Sz2=0.2
−150 −100 −50 0 50 100 1500
0.2
0.4
0.6
0.8
1
x [mm]M
x Sx1=Sx2=1.0Sx1=Sx2=0.2
Fig. 4.4. Side heater shape parameters Mx
and Mz
( see Equation (4.3) and Equation(4.4))
X
Y
ZX
Y
Z
Fig. 4.5. 3D Lagrangian analysis model : 12” × 12” × 1/8”
35
Constrained point Displacement constrained directionP1 X Y ZP2 XP3 X Y
Table 4.1. Boundary conditions for mechanical analysis ( see Figure 4.1 for P1, P2, andP3 ).
4.2 Optimization
Since the residual longitudinal compressive stress away from the weld zone can
be used as a criterion for welding induced buckling [26], the objective region (see Figure
4.1) is selected over the side plate and the magnitude of the residual longitudinal stress
component in the region needs to be minimized. Since each component of stress can be
positive or negative, the objective function is defined as square means of the residual
longitudinal stress component in the objective region. The element stress value is multi-
plied by element length to consider the element size difference in the finite element model.
Therefore, the mathematical expression for the residual longitudinal stress minimization
problem can be defined as follows:
F =∑e
(leσe
zz)2 (4.5)
where σe
zzis the longitudinal residual stress at the centroid of element e in the objective
region shown in Figure 4.1, and le is the x-direction length of the element. The gradient
36
of the objective function F is obtained as follows:
∂F
∂φi
= 2∑e
(le)2σe
zz
∂σe
zz∂φ
i
(4.6)
The design variables are the side heat source Qs(= φ1), transverse position of the side
heater φ2 and the distance between the side heater and the first welding torch φ3 as
shown in Equation (4.2) and Figure 4.1.
The optimization loop is implemented using the BFGS line search method pro-
vided in the DOT package [28].
Thermal and mechanical analyses and their sensitivity analyses are performed in
an in-house SMP FORTRAN 90 code.
4.3 Results
The results of numerical optimization are summarized in Table 4.2. The total
analysis time for each side heating configuration is set up to 3000 seconds for both the
thermal and mechanical analyses. Considering that the welding guns and side heaters
pass through the model within 25 seconds, this analysis time is long enough to ensure
sufficient cooling. Furthermore, to ensure consistency during the optimization problem,
an additional increment performed in the mechanical analysis sets the temperature back
to room temperature. The result plots for temperature analysis are chosen when all of
the heat sources appear in the analysis model. The result plots for mechanical analysis
are chosen at the final increment. Each sensitivity is normalized by multiplying it by the
corresponding design variable interval, (φi)max
− (φi)min
, (see Table 4.2).
37
The temperature profile at the initial design point is shown in Figure 4.6. Sensi-
tivity of the temperature field with respect to the side heat source at the initial design
point is shown in Figure 4.7. Increase in side heat source will result in temperature
increase in the side panel. The sensitivities of the temperature with respect to the trans-
verse and the longitudinal position of the side heater, shown in Figures 4.8 and 4.9,
indicate that temperature is significantly sensitive at the edge of the side heaters. The
longitudinal residual stress profile at the initial design point is shown in Figure 4.10. The
stress is tensile near the welding torch path and compressive away from it. Figures 4.11-
4.13 show the longitudinal residual sensitivities with respect to the side heat source, the
transverse position, and the longitudinal position. The stress in the objective region (see
Figure 4.1) is compressive, as shown in Figure 4.10, so that positive sensitivity is the
desirable direction for minimum residual stress in that region.
Figures 4.14-4.16 show the difference between direct sensitivity analysis and for-
ward finite difference sensitivity analysis, using the error function Error defined as fol-
lows:
Error =S
f− S
d
Sf
(4.7)
where Sf
is normalized sensitivity by forward finite difference method and Sd
is nor-
malized sensitivity by direct differentiation method. All the sensitivity difference plots
show less than 2 percent error over most of the model. The actual numbers that indicate
each sensitivity at the initial design point in the objective region are listed in Table 4.3
and show enough similarity to validate the development of direct differentiation sensi-
tivity formulations. The finite difference sensitivity was simulated for both thermal and
38
mechanical analyses, with perturbations ranging from 10−6 to 10−2. No considerable
deviation was observed in the sensitivity results with the variation of the perturbation.
Sensitivity with perturbation 10−3 is used in the calculations of Figures 4.14-4.16, and
Table 4.3.
The longitudinal residual stress field with the optimum design variables is shown
in Figure 4.17. Compared with Figure 4.10, the residual stress reduction is observed
not only in the objective region but also over the outside panel. Figure 4.18 shows
the longitudinal residual stresses along the “Transverse Center Line” (see Figure 4.1)
for three cases. Residual stress in the objective region is successfully reduced for the
optimum side heater. The vertical dotted-line in the left side from the axis line indicates
where the objective region starts.
The variation of the objective function defined in Equation (4.5) during this opti-
mization is shown in Figure 4.19, where the objective function is normalized by dividing
by its initial value F0. A total of 28 function calls and 6 gradient calls were made dur-
ing the entire optimization. Table 4.4 shows the runtime of the thermal and mechanical
problems at the initial design point on an IBM RS/6000 44P Model 270 system with four
375 MHz POWER3-II 64-bit processors and 8 GB RAM. Sensitivity analysis takes about
28% of the total computation time for the mechanical problem and 33% for the thermal
problem. In this simulation, to perform the sensitivity analysis, the global stiffness in
Equations (3.5) and (3.7) are assembled and decomposed once more at the final solution
of each increment. This computation time of the sensitivity analysis can be decreased if
the global stiffness decomposed at the last iteration of the increment is used, as discussed
in Section 3.
39
Design Variable Initial Val. Minimum Maximum Optimumφi
(φi)0 (φ
i)min
(φi)max
(φi)opt
φ1 : Heat input (Qs) [W ] 5000.00 0.00 10000.00 9288.01
φ2 : Side offset [mm] 50.80 0.00 150.00 54.24φ3 : Long. offset [mm] 50.80 -100.00 100.00 47.00
Table 4.2. Design variables (see Figure 4.1)
4.4 Conclusions and Future Work
Direct sensitivity formulations for thermo-elasto-plastic processes in three dimen-
sional Lagrangian reference frames have been developed. The sensitivity results from
these formulations show good agreement with the results from the finite difference sen-
sitivity analysis method. The sensitivity formulations are successfully implemented in
an optimization procedure to determine the optimal side heater heat input power and
positions for minimum welding residual stress in the transient thermal tensioning pro-
cess. In this simulation, only three design variables (side heat source, positions of side
heater in transverse and longitudinal directions) are considered. However, if necessary,
more design variables such as side heat shape can be considered without much effort.
The addition of constraints such as peak temperature in an objective region can also be
easily incorporated in the optimization.
Since welding residual displacement usually exceeds the small strain range, large
deformation theory needs to be implemented in this analysis procedure as future work.
Adaptive meshing may also reduce the analysis time for large structural problems.
40
X Y
Z
1.75+03
1.64+03
1.52+03
1.41+03
1.29+03
1.18+03
1.06+03
9.45+02
8.30+02
7.15+02
6.00+02
4.85+02
3.70+02
2.55+02
1.40+02
2.50+01
X Y
Z
Fig. 4.6. Temperature profile of the initial design [ ◦C]
X Y
Z
4.50+02
4.20+02
3.90+02
3.60+02
3.30+02
3.00+02
2.70+02
2.40+02
2.10+02
1.80+02
1.50+02
1.20+02
9.00+01
6.00+01
3.00+01
0.
X Y
Z
Fig. 4.7. Normalized sensitivity of temperature with respect to side heat source (φ1) of
the initial design [ ◦C]
41
X Y
Z
0.
-9.00+01
-1.80+02
-2.70+02
-3.60+02
-4.50+02
-5.40+02
-6.30+02
-7.20+02
-8.10+02
-9.00+02
-9.90+02
-1.08+03
-1.17+03
-1.26+03
-1.35+03
X Y
Z
Fig. 4.8. Normalized sensitivity of temperature with respect to transverse position (φ2)
of the initial design [ ◦C]
X Y
Z
1.60+03
1.48+03
1.36+03
1.24+03
1.12+03
1.00+03
8.80+02
7.60+02
6.40+02
5.20+02
4.00+02
2.80+02
1.60+02
4.00+01
-8.00+01
-2.00+02
X Y
Z
Fig. 4.9. Normalized sensitivity of temperature with respect to longitudinal distance(φ3) of the initial design [ ◦C]
42
X Y
Z
4.40+02
4.00+02
3.60+02
3.20+02
2.80+02
2.40+02
2.00+02
1.60+02
1.20+02
8.00+01
4.00+01
0.
-4.00+01
-8.00+01
-1.20+02
-1.60+02
X Y
Z
Fig. 4.10. Longitudinal residual stress of the initial design [MPa]
X Y
Z
2.00+02
1.60+02
1.20+02
8.00+01
4.00+01
0.
-4.00+01
-8.00+01
-1.20+02
-1.60+02
-2.00+02
-2.40+02
-2.80+02
-3.20+02
-3.60+02
-4.00+02
X Y
Z
Fig. 4.11. Normalized sensitivity of longitudinal residual stress with respect to side heatsource (φ1) of the initial design [MPa]
43
X Y
Z
3.00+02
2.60+02
2.20+02
1.80+02
1.40+02
1.00+02
6.00+01
2.00+01
-2.00+01
-6.00+01
-1.00+02
-1.40+02
-1.80+02
-2.20+02
-2.60+02
-3.00+02
X Y
Z
Fig. 4.12. Normalized sensitivity of longitudinal residual stress with respect to trans-verse position (φ2) of the initial design [MPa]
X Y
Z
6.00+01
5.00+01
4.00+01
3.00+01
2.00+01
1.00+01
0.
-1.00+01
-2.00+01
-3.00+01
-4.00+01
-5.00+01
-6.00+01
-7.00+01
-8.00+01
-9.00+01
X Y
Z
Fig. 4.13. Normalized sensitivity of longitudinal residual stress with respect to longitu-dinal distance (φ3) of the initial design [MPa]
44
X Y
Z
1.50-02
1.30-02
1.10-02
9.00-03
7.00-03
5.00-03
3.00-03
1.00-03
-1.00-03
-3.00-03
-5.00-03
-7.00-03
-9.00-03
-1.10-02
-1.30-02
-1.50-02
X Y
Z
Fig. 4.14. Error of sensitivity in longitudinal stress w.r.t. φ1
X Y
Z
1.50-02
1.30-02
1.10-02
9.00-03
7.00-03
5.00-03
3.00-03
1.00-03
-1.00-03
-3.00-03
-5.00-03
-7.00-03
-9.00-03
-1.10-02
-1.30-02
-1.50-02
X Y
Z
Fig. 4.15. Error of sensitivity in longitudinal stress w.r.t. φ2
45
X Y
Z
1.50-02
1.30-02
1.10-02
9.00-03
7.00-03
5.00-03
3.00-03
1.00-03
-1.00-03
-3.00-03
-5.00-03
-7.00-03
-9.00-03
-1.10-02
-1.30-02
-1.50-02
X Y
Z
Fig. 4.16. Error of sensitivity in longitudinal stress w.r.t. φ3
Elem. Long. Res. Normalized Long.Res. Stress Sensitivity [MPa]Num. Str. [MPa] w.r.t. φ
1w.r.t. φ
2w.r.t. φ
3
(e) σe
zz· 10
−2S
d· 10
−2S
f· 10
−2S
d· 10
−2S
f· 10
−2S
d· 10
−2S
f· 10
−2
1170 -1.429378 1.534790 1.534793 -1.545589 -1.545593 -.4419802 -.4420532
1174 -1.136784 1.226007 1.226012 -.8681098 -.8681133 -.2690298 -.2690666
1175 -.7276272 0.8724073 0.8724127 0.1806813 0.1806781 -.04463780 -.04462288
1176 -.2540701 0.5337367 0.5337401 1.791179 1.791177 0.2623870 0.2624793
1461 -1.420768 1.488860 1.488867 1.648537 -1.648543 -.3766052 -.3766837
1465 -1.130927 1.221829 1.221834 -.9176395 -.9176443 -.2352484 -.2352935
1466 -.7268130 0.8810058 0.8810122 0.1871546 0.1871505 -.04261226 -.04260130
1467 -.2606105 0.5485422 0.5485510 1.865630 1.865627 0.2400113 0.2401171
Table 4.3. Normalized longitudinal residual stress sensitivities of initial design bydirect(S
d) and finite difference(S
f) methods in the objective region
46
X Y
Z
4.40+02
4.00+02
3.60+02
3.20+02
2.80+02
2.40+02
2.00+02
1.60+02
1.20+02
8.00+01
4.00+01
0.
-4.00+01
-8.00+01
-1.20+02
-1.60+02
X Y
Z
Fig. 4.17. Longitudinal residual stress of the optimum design [MPa]
Fig. 4.18. Longitudinal residual stress comparison along the ”Transverse Center Line”(see Figure 4.1)
47
0 1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
ITERATION
No
rma
ilze
d O
bje
cti
ve
, F
/F0
Fig. 4.19. Variation of the normalized objective function F/F0 during optimization
Computation time (seconds)Analysis type Analysis only Analysis with Sensitivity
Real time∗ CPU time Real time∗ CPU timeThermal analysis 3072.78 9288.64 4611.21 13450.21
Mechanical analysis 4147.30 14339.43 5732.49 18980.30
Table 4.4. Computation times for the initial design (Real time∗ reflects the efficiencyof the parallelization)
48
Chapter 5
Evaluation of Material Property Sensitivity
using ADIFOR
A computational procedure for evaluating the sensitivity coefficients of the quasi-
static response of welded joints with ADIFOR is presented. Two-dimensional weakly-
coupled thermal-mechanical FE analysis is performed. A rate independent, small de-
formation thermo-elasto-plastic material model with temperature-dependent material
properties is adopted.
5.1 FE Analysis
The thermal and mechanical FE equations used in this study are basically equiv-
alent to those in Chapter 2 except for two-dimensional characteristics and generalized
plane-strain assumption. The generalized plane-strain condition is assumed to account
for the out-of-plane expansion in two-dimensional model. The out-of-plane strain εz
is
assumed to have a linear distribution over the analysis plane:
εz
= e − xφy
+ yφx
(5.1)
where e is the out-of-plane strain at the origin of the coordinate system and φx
and φy
are the strain variations in the y and x directions, respectively.
49
5.2 Sensitivity Evaluation
The sensitivity coefficients, which are the derivatives of the various thermal and
mechanical response quantities with respect to the material parameters, are evaluated by
using the direct differentiation method in conjunction with the automatic differentiation
software facility ADIFOR [15, 11, 10]. The sensitivity coefficients obtained by ADIFOR
were validated by comparing them with those obtained by finite difference approxima-
tions. The sensitivity information can be used to (see, for example [68]): (a) assess the
importance of the parameters used in describing the thermal and mechanical properties
of the material on the time histories of the temperature and residual stresses. This, in
turn, can help both in refining the material models and in the design of improved materi-
als. (b) assess the effects of uncertainties in the material parameters on the time-history
response of welded structures; and (c) predict the changes in the time-history response
of welded structures due to changes in the material parameters.
5.3 Numerical Studies
The computational procedure described in the preceding sections is applied to
study the temperature and residual stress-time histories and their sensitivity coefficients
for a double fillet conventional welding of a stiffener and a base plate made of stainless
steel AL-6XN (see Figure 5.1). The variations of the thermal and mechanical proper-
ties of AL-6XN with temperature are shown in Figure 5.2. Each of the thermal and
mechanical properties is approximated by the piecewise linear variation shown in Figure
5.2.
50
Numerical results are presented for the temperature - and residual stress - time
histories and their sensitivity coefficients for a double fillet conventional welding of a
stiffener and a base plate made of stainless steel AL-6XN. A two-dimensional generalized
plane strain model is used, which is adequate for predicting the residual stresses.
Torch1
3.5"
Weldingdirection
L
12"
x
y
z
2"
Torch2
L
12”
2”
Fig. 5.1. Welding conditions for double fillet welding
5.3.1 Welding Process and Welding Conditions
The schematic welding configuration used in the present study is shown in the
Figure 5.1. The width(B) of the base plate is 12”, the height of the stiffener is 2” and
the thickness of each of the base plate and stiffener is 1/8”.
Double fillet welding is used with one welding gun on either side of the stiffener.
The guns are 3.5” offset from each other with one gun following the other as shown in
51
0 1000 2000 3000 4000 500010
20
30
40k
Cp
h
500
600
700
0
2.5
5
7.5
10
k(W
/m/o
C)
Cp
(J/K
g/o
C)
h(W
/m2
/oC
)
Temperature (oC)
a) thermal properties
0 500 1000 15000
50
100
150
200 E ν
α
σy0
0
0.1
0.2
0.3
0
100
200
300
400
15
18
21
E(G
Pa)
ν
σ y0
(MPa
)
α*1
0-6
(1/o
C)
b) mechanical properties
Temperature (oC)
12
33
2
1
2
2 3
3
4
4
4
4
5
5
x10
6
1
1
2
2
3
3 4 5
4 54
3
Fig. 5.2. Temperature-dependent thermal and mechanical properties for AL-6XN; (a)Conductivity k, specific heat C
p, and convection coefficient h (b) Elastic modulus E,
yield strength σy, Poisson’s ratio ν, and thermal expansion coefficient α
52
Figure 5.1. The details of the welding process and welding conditions are described in
[26].
5.3.2 Finite Element Model
3.175 mm
5 mm
3.175 mm
Fig. 5.3. 2D Lagrangian analysis model
The finite element model used in each of the thermal and mechanical analyses
is shown in Figure 5.3. The model has 388 8-node quadratic elements and 1343 nodes.
In the thermal analysis all the free surfaces are taken as convective surfaces. In the
mechanical analysis the constraints shown in Figure 5.3 are applied. Convergence studies
were performed by using successively refined grids. The results obtained by the model
shown in Figure 5.3 were found to be in close agreement with those obtained by finer
53
grids. Typical results are shown in Figure 5.4 and Figure 5.5 for the response studies,
and in Figures 6 through 11 for the sensitivity studies, and are described subsequently.
5.3.3 Response studies
The time histories of the temperature and von-Mises stress at 6 points are shown
in Figure 5.4. The maximum temperatures occur at point 1 at time t=14.55 seconds, and
at point 4 at time t=0.55 seconds. The maximum values of the von-Mises stress occur
near points 1 and 4 at t=3000 seconds. Contour plots for the normalized temperature
at t=0.4, 0.55 and 14.55 seconds, and for the von-Mises stress at t=0.56, 14.73, and
3000 seconds, are shown in Figure 5.5. Each contour plot is normalized with respect to
the maximum absolute value of the function represented, and consequently the contour
intervals are bounded by 0 and 1. An examination of Figure 5.4 and Figure 5.5 reveals
that the maximum values of the temperature and von-Mises stress occur in the weld
zones.
5.3.4 Sensitivity Studies
The time histories of the first-order sensitivity coefficients of the temperature with
respect to the three sets of parameters ki, c
pi, and h
iat points 1 and 4 are shown in
Figure 5.6. Corresponding time histories of the first-order sensitivity coefficients of von-
Mises stress with respect to the three sets of parameters Ei, σ
y0iand α
iat the same
points are shown in Figure 5.7. Each sensitivity coefficient is normalized by multiplying
by the same parameter, with respect to which the sensitivity is evaluated. Contour plots
of the largest normalized sensitivity coefficients ∂T/∂k4 and ∂T/∂cp4 at t=0.55, 2.51
54
0
700
1400
2100
2800
3500Point 1Point 2Point 3
Time, sec.
T
.1 1 10 100 3000
(oC)
a) Temperature - time history
b) von-Mises - time history
Point 4Point 5Point 6
Time, sec..1 1 10 100 3000
Point 4Point 5Point 6
Time, sec..1 1 10 100 3000
0
300
600
900Point 1Point 2Point 3
σm
Time, sec..1 1 10 100 3000
(MPa)
12
3
45
6
Fig. 5.4. Time histories of the temperature and von-Mises stress for the welded jointshown in Figure 5.1
TTmax
σmσm, max
t (sec) = .56 14.73 3 000
t (sec) = .4 . 55 14.55
1.0
.5
0
0.1 0.1 0.2
0.1
0.2
0.20.2
Fig. 5.5. Snapshots of the normalized temperature and von-Mises stress for the weldedjoint shown in Figure 5.1
55
-1500
-1000
-500
0
500
Time, sec..1 1 10 100 3000
-1200
-900
-600
-300
0
300
Time, sec..1 1 10 100 3000
-60
-40
-20
0
20
Time, sec..1 1 10 100 3000
-1500
-1000
-500
0
500
4321
Time, sec..1 1 10 100 3000
i
-1200
-900
-600
-300
0
300
4213
i
Time, sec..1 1 10 100 3000
-60
-40
-20
0
20
23145
i
Time, sec..1 1 10 100 3000
∂T∂hi
hi
∂T∂ki
∂T∂cpi
cpi
ki
1 4
Fig. 5.6. Time histories of the first-order sensitivity coefficients of the temperature withrespect to thermal properties k
i, c
pi, and h
ifor the welded joint shown in Figure 5.1
56
-250
0
250
500
32514
.1 1 10 100 3000Time, sec.
i
-800
-400
0
400
800
.1 1 10 100 3000Time, sec.
-80
0
80
160
240
320 12345
.1 1 10 100 3000Time, sec.
i
-160
0
160
320
.1 1 10 100 3000Time, sec.
-500
-250
0
250
500
4321
TimeTime, sec..1 1 10 100 3000
i
-600
-300
0
300
600
900
.1 1 10 100 3000Time, sec.
∂σm∂Εi
Εi
∂σm∂σy
σy
αi∂σm∂αi
0i
0i
1 4
Fig. 5.7. Time histories of the first-order sensitivity coefficients of von-Mises stress withrespect to mechanical properties E
i, σ
y0i, and α
ifor the welded joint shown in Figure
5.1
57
∂T∂c
∂T∂c max
∂T∂k4
∂T∂k4 max-
-
t (sec) = .55 2.51 14.55
p4 p4
-0.1
-0.10.0
0.0
-0.10.0
-0.1
-0.1
0.00.0
-0.1
1.0
0
-1.00.1 0.0 0.0
Fig. 5.8. Snapshots of the first-order sensitivity coefficients of the temperature withrespect to thermal properties k4 and c
p4 for the welded joint shown in Figure 5.1
1.0
0
-1.0
0.00.00.0
0.0
0.1
0.10.0
-0.1
0.0-0.1
-0.2
0.0
∂σm∂σy
∂σm∂σy max
∂σm∂E5
∂σm∂E5 max
t (sec) = .56 22.43 3 000
05 05
Fig. 5.9. Snapshots of the first-order sensitivity coefficients of von-Mises stress withrespect to mechanical properties E5 and σ
y05 for the welded joint shown in Figure 5.1
58
-1
0
1
2
3
4
5 x 103
Time, sec..1 1 10 100 3000 -8
-6
-4
-2
0
2
.1 1 10 100 3000Time, sec.
3x 10
-8
-4
0
4
8
12
16
.1 1 10 100 3000Time, sec.
3x 10
-50
-40
-30
-20
-10
0
10
.1 1 10 100 3000Time, sec.
3x 10
-4
0
4
8
.1 1 10 100 3000Time, sec.
3x 10
-20
-15
-10
-5
0
5
.1 1 10 100 3000Time, sec.
3x 10
∂2T∂c ∂k3
c k3
∂2T∂k3
2k32
∂2T∂c2
c2p2
p2
p2
p2
12
3
45
6
Point 1Point 2Point 3
Point 4Point 5Point 6
Fig. 5.10. Time histories of the second-order sensitivity coefficients of the temperaturewith respect to k3k3, c
p2cp2 and c
p2k3 for the welded joint shown in Figure 5.1
59
1.0
0
-1.0
0.0 0.0 0.0
0.0 0.0 0.0
∂2T∂c
∂2T∂c max
∂2T∂k3
2
∂2T∂k3
2max
t (sec) = .4 . 42 .48
p22 p2
2
∂2T∂c ∂k3 maxp2
∂2T∂c ∂k3p2
Fig. 5.11. Snapshots of the second-order sensitivity coefficients of the temperature withrespect to k3k3, c
p2cp2 and c
p2k3 for the welded joint shown in Figure 5.1
and 14.55 seconds are shown in Figure 5.8. Contour plots for the largest sensitivity
coefficients ∂σm
/∂E5 and ∂σm
/∂σy05
at t=0.56, 22.43 and 3000 seconds are shown in
Figure 5.9. Time histories of the largest normalized second-order sensitivity coefficients of
the temperature ∂2T/∂k
2
3, ∂
2T/∂c
2
p2, and ∂
2T/∂c
p2∂k3 at 6 points are shown in Figure
5.10. Contour plots of the maximum second-order sensitivity coefficients ∂2T/∂k
2
3, and
∂2T/∂c
2
p2, and of the mixed second-order sensitivity coefficients ∂
2T/∂c
p2∂k3, at t=0.4,
0.42, 0.48 seconds are shown in Figure 5.11.
An examination of Figures 5 to 11 reveals:
1. The first-order sensitivity coefficients of the temperature with respect to the
parameters k4, h2 and cp4 are larger than those with respect to the other parameters in
each category.
60
2. The maximum absolute value of the first-order sensitivity coefficients of T with
respect to k4 and cp4 occur at the same point and nearly at the same time as those for
T .
3. The first-order sensitivity coefficients of von-Mises stress σm
with respect to the
parameters E5, σy05
, α3 and ν1 are larger than those with respect to the corresponding
parameters in each category.
4. The maximum absolute values of the first-order sensitivity coefficients of σm
occur at different points, and at different times from those of T .
5. The second-order sensitivity coefficients of the temperature with respect to
k3 and cp2 are larger than the second-order sensitivity coefficients with respect to the
corresponding parameters in each category.
5.4 Concluding Remarks
A computational procedure is presented for evaluating the sensitivity coefficients
of the quasi-static response of welded structures. Uncoupled thermo-mechanical analysis
is performed. The temperature field is assumed to be independent of stresses and strains.
The heat transfer equations emanating from a finite element semi-discretization are inte-
grated using an implicict backward difference scheme to generate the time-history of the
temperature. The mechanical response during welding is then calculated by solving a
generalized plane strain problem. A rate independent, small deformation thermo-elasto-
plastic material model with temperature-dependent material properties is adopted in the
study.
61
First- and second-order sensitivity coefficients of the thermal and mechanical re-
sponse quantities are evaluated by using a direct differentiation approach in conjuction
with an automatic differentiation software facility.
Numerical results are presented for a double-fillet conventional welding of a stiff-
ener and a base plate made of stainless steel AL-6XN material. Time histories of the
response and sensitivity coefficients, and their spatial distributions at selected times are
presented. The first- and second-order sensitivity coefficients can be used to generate
taylor series approximations for the quasi-static response for welded joints with slightly
different material parameters.
62
Chapter 6
Eulerian fully-Coupled thermal-Mechanical Analysis
for FSW Process
The state of a thermo-elastic rate-independent plastic material of FSW can be
described by four field variables: the temperature T , velocity v, stress σ , and equivalent
plastic strain εq fields. The temperature field can be determined iteratively from a
thermal analysis with given mechanical fields, which can be determined from mechanical
analysis with given temperature field. Since heat transfer analysis for a given velocity
field is well established [88] this research focuses on the mechanical analysis.
6.1 Heat Transfer Model
The governing equation and boundary conditions for steady-state convective heat
transfer in Eulerian frames are given as,
ρcpvi
∂T
∂xi
= −∂q
i∂x
i
+ Q (6.1)
where
qi
= −kij
∂T
∂xj
(6.2)
x is the spatial coordinate, q is the heat flux, k is the thermal conductivity, cp
is the
specific heat, and Q is the heat generation rate per unit volume. The boundary conditions
63
can be written as,
T = T on ST (6.3)
qini
= q on Sq (6.4)
where T is the prescribed temperature on surface ST , n is the normal vector, and q is a
prescribed heat flux on surface Sq. As can be seen in Equation (6.1), the velocity field
is required for thermal analysis.
6.2 Mechanical Formulation
In mechanical analysis, the temperature field is considered as known, and the
velocity, stress, and equivalent plastic strain fields are determined from the equilibrium
or rate equilibrium, stress evolution, and plastic strain evolution equations. The weak
form of each equation is presented using kinematically admissible functions, v, D, σ ,
and εq for velocity, rate of deformation, stress, and equivalent plastic strain, respectively.
6.2.1 Flow Equilibrium Equation (FEE)
Neglecting inertia and assuming steady state conditions, the linear momentum
balance equation becomes the equilibrium equation. The equilibrium equation with
neglecting body force can be written as,
∂σij
∂xi
= 0 (6.5)
64
where σ is the Cauchy stress. The weak form of Equation (6.5) is
∫V
Dij
σij
dV −∫S
vj
niσij
dS = 0 (6.6)
where V is the control volume and S is the boundary surface.
6.2.2 Flow Rate Equilibrium Equation (FRE)
Thomson and Yu [78] derived a rate equilibrium equation from Equation (6.5).
d
dt
[∂σ
ij
∂xi
]=
∂Pij
∂xi
= 0 (6.7)
where
Pij
≡ vk
∂σij
∂xk
−∂v
i∂x
k
σkj
+∂v
k∂x
k
σij
(6.8)
and v is the particle velocity. The weak form of Equation (6.7) can be written as,
∫V
∂vi
∂xj
Pij
dV −∫S
vj
niP
ijdS = 0 (6.9)
6.2.3 Constitutive Equation
A hypo-elastic, rate-independent associative J2 plastic material with isotropic
strain hardening is considered in this research. This material model is expressed with
material time derivatives for Lagrangian formulations in most available literatures [8].
All the material time derivatives are transformed into steady-state spatial expressions
65
for Eulerian formulations. The Jaumann rate of Cauchy stress σ∇J can be determined
from the material behavior tensor C, and the elastic rate of deformation De as follows:
σ∇J
ij= C
ijklD
e
kl(6.10)
where
σ∇J
ij= v
k
∂σij
∂xk
− σik
Wkj
− σjk
Wki
(6.11)
De
ij= D
ij− D
p
ij− D
th
ij(6.12)
Dij
=12
[∂v
j
∂xi
+∂v
i∂x
j
](6.13)
Wij
=12
[∂v
j
∂xi
−∂v
i∂x
j
](6.14)
Cijkl
= λδij
δkl
+ µ[δik
δjl
+ δilδjk
](6.15)
λ =νE
(1 + ν)(1 − 2ν)(6.16)
µ =E
2(1 + ν)(6.17)
λ and µ are the Lame’ constants; E and ν are the elastic moduli; Dp and Dth are the
plastic and thermal rate of deformation, which can be evaluated as follows:
Dp
ij= v
k
∂εq
∂xk
aij
(6.18)
Dth
ij= β v
k
∂T
∂xk
δij
(6.19)
66
where εq is the equivalent plastic strain, T is the temperature, and the plastic flow tensor
a and thermal strain strain coefficient β are given as,
aij
=32σ
σd
ij(6.20)
β =∂α
∂T
[T − T
ref]
+ α (6.21)
where α is the thermal expansion coefficient, Tref is the Reference temperature, and
Mises’ stress σ and deviatoric stress σd are
σ =
√32σd
ijσd
ij(6.22)
σd
ij= σ
ij− 1
3σkk
δij
(6.23)
Since Dp has only deviatoric components, the following relationship can be obtained:
Cijkl
Dp
kl= 2 µ v
k
∂εq
∂xk
aij
(6.24)
The yield function f for isotropic linear hardening materials can be described as,
f = σ − σY = σ − σ
Y 0 − Hεq (6.25)
where, σY is the yield stress, σ
Y 0 is the initial yield stress, and H is the linear hardening
coefficient. In case of active yielding, the yield function should remain on the yield
67
surface, that is,
df
dt= 0 (6.26)
From Equations (6.25) and (6.26), the plastic evolution equation can be written as,
vk
∂εq
∂xk
=γ
H
a
ijvk
∂σd
ij
∂xk
− vk
∂σY 0
∂xk
− vk
∂H
∂xk
εq
(6.27)
where
γ =
1 if f ≥ 0 and dfdt ≥ 0
0 otherwise(6.28)
The characteristics of stress and equivalent plastic strain evolution equations are hy-
perbolic and this class of equations is susceptible to numerical oscillation. Therefore,
the SUPG stabilizing technique [40, 8] is used for the weak formulations of stress and
equivalent plastic strain evolution equations:
∫V
[σij
+ τck
∂σij
∂xk
]{vk
∂σij
∂xk
− Cijkl
De
kl− σ
ikW
kj− σ
jkW
ki
}dV = 0 (6.29)
∫V
[εq + τc
k
∂εq
∂xk
]v
k
∂εq
∂xk
− γ
H
a
ijvk
∂σd
ij
∂xk
− vk
∂σY 0
∂xk
− vk
∂H
∂xk
εq
dV = 0 (6.30)
68
where the stabilization factor τ and the convective velocity ci
are evaluated as follows:
τ =h
2(6.31)
ci
=vi√
vjvj
(6.32)
where h is the characteristic element length.
6.2.4 Boundary Conditions (BC) for FEE and FRE
Equations (6.29) and (6.30), as commonly used for both FEE and FRE, only
require that the stress and equivalent plastic strain should be known on the inlet surface
where the material enters control volume. The FEE is characterized by Equation (6.6),
the FRE by Equation (6.9), and BC for Equations (6.6) and (6.9) are applied through
their second terms, which can be rewritten as follows:
∫S
vjniσij
dS =∫S
vj
tj
dS (6.33)
∫S
vjniP
ijdS =
∫S
vj
[vk
∂tj
∂xk
− ni
∂vi
∂xk
σkj
+∂v
k∂x
k
tj
]dS (6.34)
where t is the traction. Equations (6.33) and (6.34) vanish for the velocity described
boundary, thus, no additional consideration is required. For the traction prescribed
boundary, Equation (6.34) is still dependent on the field variables, velocity and stress,
while Equation (6.33) simply becomes constant.
69
Chapter 7
Numerical Implementations
of the Eulerian Thermo-Elasto-Plastic FE Formulations
7.1 Voigt Transformations
Considering the balance of angular momentum, the stress tensor and the rate of
deformation tensor can be transformed into Voigt form. The details of Voigt transforma-
tion are shown in Appendix C. The Voigt-transformed weak forms of Equations (6.6),
(6.9), (6.29), and (6.30) can be written as follows:
Rve
=∫V
DiσidV −
∫S
vpnqM
pqkσkdS (7.1)
Rvr
=∫V
{D
i
[Yi+ D
khkσi
]−
∂vi
∂xj
∂vi
∂xk
Mkjl
σl
}dV −
∫S
vpnqP
qpdS (7.2)
Rσ
=∫V
[σi+ τc
k
∂σi
∂xk
]{vp
∂σi
∂xp
− Yi
}dV (7.3)
Rq
=∫V
[εq + τc
k
∂εq
∂xk
]{vp
∂εq
∂xp
− Gpvp
}dV (7.4)
where
Yi
≡ Cik
[D
k− D
th
k
]− 2µG
kvkm
i+ A
ikσk
(7.5)
Gi
≡ γ
HF
i(7.6)
Fi
≡ ak
∂σd
k∂x
i
− ∂σY 0
∂xi
− εq ∂H
∂xi
(7.7)
70
7.2 Mixed Formulation and Smoothing Function
Finite element discretization is applied to the Voigt form of residual equations
((7.1) - (7.4)) to obtain the element residual and stiffness equations using the relation-
ships shown at Appendix D. The assembly of the element residual vector R and the
element stiffness matrix ∂R∂U can be obtained as follows:
R =
Rv
Rσ
Rq
; U =
V
S
Q
;∂R∂U
=
∂Rv
∂V∂Rv
∂S∂Rv
∂Q
∂Rσ
∂V∂Rσ
∂S∂Rσ
∂Q
∂Rq
∂V∂Rq
∂S∂Rq
∂Q
(7.8)
where Rv is either Rve for FEE, or Rvr for FRE. ∂Rve
∂V becomes zero, whereas ∂Rvr
∂V
does not. Thus, FRE equations can be solved by iterative method as in Thompson
and Yu [78], but FEE equations cannot. However, FEE also can be solved if mixed
formulation technique is applied. In this research, both equations are solved by using
the mixed formulation technique, as shown in Equation (7.8).
The residual equations need to be differentiable to obtain stiffness equations like
Equation (7.8). However, γ in Equation (7.4) takes discrete values, as can be seen in
Equation (6.28). In order to allow the discontinuous residual equation to be differen-
tiable, γ is replaced by a smoothing function originally introduced by Shanghvi et al.
[69]:
γ =14γaγb (7.9)
γa = tanh
(S
a1[
σ
σY− 1]
+ Sa2)
+ 1 (7.10)
71
γb = tanh
(S
b1[
∂f
∂xp
vp
]+ S
b2)
+ 1 (7.11)
where Sa1, S
a2, Sb1, and S
b2 are parameters determining the shape of the smoothing
function.
7.3 Finite Element Equations
Each component of the element residual and stiffness is presented in this sec-
tion. The surface integral terms of Equations (7.1) and (7.2) are not considered for
implementation simplicity. In fact, the boundary conditions for the example problems
presented in Section 8 are selected so that the surface integral terms vanish like velocity
prescribed boundary conditions. The summation symbol over Gauss Points is dropped
for simplicity.
The element residual vector for Equation (7.1) is
Rve
i= B
ilσlWJ (7.12)
and the corresponding stiffness components are:
∂Rv
i∂V
j
= 0 (7.13)
∂Rve
i∂S
j
= BliN
s
ljWJ (7.14)
∂Rve
i∂Q
j
= 0 (7.15)
72
where J is determinant of the volume Jacobian and W is weighting for Gaussian inte-
gration. These symbols are also dropped to simplify the following residual and stiffness
expression. The element residual vector for Equation (7.2) is
Rvr
i= B
li
[Yl+ D
phpσl
]− B
v
pqi
∂vp
∂xr
Mrql
σl
(7.16)
Each stiffness for this residual equation is
∂Rvr
i∂V
j
= Bli
∂Yl
∂Vj
+ BliB
pjhpσl− B
v
pqiB
v
prjM
rqlσl
(7.17)
∂Rvr
i∂S
j
= Bli
∂Yl
∂Sj
+ BliD
phpN
σ
lj− B
v
pqi
∂vp
∂xr
Mrql
Nσ
lj(7.18)
∂Rvr
i∂Q
j
= Bli
∂Yl
∂Qj
(7.19)
where
∂Yi
∂Vj
= Cik
[B
kj− βh
p
∂T
∂xq
Nqj
]− 2µ
∂[G
kvk
]∂V
j
mi+ B
a
ikjσk
(7.20)
∂Yi
∂Sj
= −2µ∂[G
kvk
]∂S
j
mi− 2µG
kvk
∂mi
∂Sj
+ Aik
Nσ
kj(7.21)
∂Yi
∂Qj
= −2µ∂[G
kvk
]∂Q
j
mi
(7.22)
∂[G
kvk
]∂V
j
=
[γdb
H
]∂f
∂xp
Npj
+ GpN
pj(7.23)
73
∂[G
kvk
]∂S
j
=[ γ
H
]Lqr
∂mr
∂Sj
∂σd
q
∂xp
vp
+ aqB
σ
qpjvp
+
[γda
HσY
]apN
σ
pj+
[γdb
H
]∂F
p
∂Sj
vp(7.24)
∂[G
kvk
]∂Q
j
= −[ γ
H
] ∂H
∂xp
vpN
q
j−[
σγda
σY 2
]N
q
j−[
γdb
H
][∂H
∂xp
vpN
q
j+ HB
q
pjvp
](7.25)
∂f
∂xi
= Fi− H
∂εq
∂xi
(7.26)
∂mk
∂Sj
=[− 3
2σ2σd
kar
+32σ
Zkr
]N
σ
rj(7.27)
and
γda ≡
[F
pvp
] [γbS
a1
4
] [1 −[γa − 1
]2](7.28)
γdb ≡
[F
pvp
] [γaS
b1
4
] [1 −[γb − 1
]2](7.29)
For Equation (7.3), the element residual becomes
Rσ
i=[N
σ
ki+ τB
σ
klicl
]{∂σk
∂xp
vp− Y
k
}(7.30)
The stiffness components for this residual equation are
∂Rσ
i∂V
j
=[N
σ
ki+ τB
σ
klicl
]{∂σk
∂xp
Npj
−∂Y
k∂V
j
}(7.31)
∂Rσ
i∂S
j
=[N
σ
ki+ τB
σ
klicl
]{B
σ
kpjvp−
∂Yk
∂Sj
}(7.32)
∂Rσ
i∂Q
j
=[N
σ
ki+ τB
σ
klicl
]{−
∂Yk
∂Qj
}(7.33)
74
The element residual vector from Equation (7.4) is:
Rq
i=[N
q
i+ τB
q
licl
]{∂ε
q
∂xp
vp− G
pvp
}(7.34)
The stiffness components for Rq are
∂Rq
i∂V
j
=[N
q
i+ τB
q
licl
]
∂εq
∂xp
Npj
−∂[G
pvp
]∂V
j
(7.35)
∂Rq
i∂S
j
=[N
q
i+ τB
q
licl
]−
∂[G
pvp
]∂S
j
(7.36)
∂Rq
i∂Q
j
=[N
q
i+ τB
q
licl
]B
q
pjvp−
∂[G
pvp
]∂Q
j
(7.37)
75
Chapter 8
Numerical Examples
for the Eulerian Thermo-Elasto-Plastic FE Formulations
OMP FORTRAN 90 code based computer programs for both FEE and FRE are
developed for 8-noded brick elements. Strip drawing examples are shown to verify the
validity of FEE formulation and to compare the performance of the two formulations.
An FSW example is simulated to show the potential of the programs for FSW analysis.
8.1 Strip Drawing Examples
Although the computer programs are developed for 3-dimensional problems, im-
plicit plane strain examples with zero z-directional velocity are simulated to show the
characteristics of the formulations more clearly. Figure 8.1 shows the geometric feature
y
A
C
D
B
EE
20
6
40
5
Fig. 8.1. Strip drawing configuration: Unit[mm]
76
of the example. A fourth order polynomial function is used to describe the curvature
such that the slope is zero at both ends and midpoint of surface D. Surface A is the
inlet where zero stress and equivalent plastic strain are assumed. Surface B is the outlet
where uniform normal velocity 4 mm/sec and zero tangential velocity are imposed, and
surface C is for symmetric BC where zero normal velocity and zero tangential traction
are prescribed. Temperature is assumed to be 20 ◦C everywhere. Material properties
for Young’s modulus E, Poisson’s ratio, thermal expansion coefficient α, yield stress
without hardening σY 0, and hardening coefficient H are assigned with 6.06 GPa, 0.442,
1.17 × 10−5 / ◦C, 64 MPa, and 1 GPa.
Examples with frictionless surface D and E are adequate for the verification of
the formulations since the resultant stress field is quite predictable. However, the second
term of Equation (6.9) for FRE does not vanish for the frictionless boundary condition
if the surface has curvature [78], whereas the surface integral has not been implemented
in this research. Therefore, the following two frictionless examples (Example 1 and 2)
are simulated only by FEE. Example 1 is presented for verification of the elastic part of
the FEE formulation and Example 2 for demonstration of the elasto-plastic capability.
However, the results from FEE in the examples are comparable with the results from
FRE in Thompson and Yu’s [78], since both have similar geometry and boundary condi-
tions and the strip drawing simulations in this research are implicitly two-dimensional.
In Example 3, all velocity components are prescribed on surface D and E (Example 3)
such that both second terms of Equations (6.6) and (6.9) become zero. Thus, Exam-
ple 3 can be simulated by both FEE and FRE formulations developed in this research
and is presented in order to compare both formulations directly. The material model,
77
boundary conditions, and simulation algorithms for the three strip drawing examples are
summarized in Table 8.1.
Material BC on surface D and E Sim. AlgorithmExample 1 Purely Elastic Frictionless FEEExample 2 Elasto-Plastic Frictionless FEEExample 3 Purely Elastic All Velocity Components FEE, FRE
Table 8.1. Specifications of the strip drawing examples
8.1.1 Example 1: Purely Elastic Example with Frictionless Surface
The material is assumed to be purely elastic in this example problem. Zero normal
velocity and tangential traction on surfaces D and E are imposed for the frictionless
condition. This example is analyzed using the FEE formulation. Figures 8.2 and 8.3
show the velocity field and Figure 8.4 displays the Mises’ stress field. Velocity and stress
fields are conserved between inlet and outlet since no energy is dissipated during the
frictionless process with purely elastic material.
8.1.2 Example 2: Elasto-Plastic Example with Frictionless Surface
In this example, the boundary conditions are the same as those of Example 1
and the FEE formulation is used for the simulation. The material is changed to elasto-
plastic. Figures 8.5-8.9 present the velocity, stress, and equivalent plastic strain field
for this process. The overall stress level in Figure 8.7 is lower than that in Figure 8.4
78
X
Y
Z
4.75
4.70
4.65
4.60
4.55
4.50
4.45
4.40
4.35
4.30
4.25
4.20
4.15
4.10
4.05
4.00
X
Y
Z
Fig. 8.2. x-directional velocity from FEE for Example 1 (Elastic mat. and frictionlessBC): Unit[mm/s]
X
Y
Z
.8
.7
.6
.5
.4
.3
.2
.1
-.0
-.1
-.2
-.3
-.4
-.5
-.6
-.7
X
Y
Z
Fig. 8.3. y-directional velocity from FEE for Example 1 (Elastic mat. and frictionlessBC): Unit[mm/s]
79
X
Y
Z
1500.
1400.
1300.
1200.
1100.
1000.
900.
800.
700.
600.
500.
400.
300.
200.
100.
0.
X
Y
Z
MSC/PATRAN Version 9.0 09-Sep-04 21:03:34
Fig. 8.4. Mises’ stress from FEE for Example 1 (Elastic mat. and frictionless BC):Unit[MPa]
X
Y
Z
4.75
4.70
4.65
4.60
4.55
4.50
4.45
4.40
4.35
4.30
4.25
4.20
4.15
4.10
4.05
4.00
X
Y
Z
Fig. 8.5. x-directional velocity from FEE for Example 2 (Elasto-plastic mat. andfrictionless BC): Unit[mm/s]
80
X
Y
Z
.8
.7
.6
.5
.4
.3
.2
.1
-.0
-.1
-.2
-.3
-.4
-.5
-.6
-.7
X
Y
Z
Fig. 8.6. y-directional velocity from FEE for Example 2 (Elasto-plastic mat. andfrictionless BC): Unit[mm/s]
X
Y
Z
450.
420.
390.
360.
330.
300.
270.
240.
210.
180.
150.
120.
90.
60.
30.
0.
X
Y
Z
Fig. 8.7. Mises’ stress from FEE for Example 2 (Elasto-plastic mat. and frictionlessBC): Unit[MPa]
81
X
Y
Z
450.
400.
350.
300.
250.
200.
150.
100.
50.
0.
-50.
-100.
-150.
-200.
-250.
-300.
X
Y
Z
Fig. 8.8. σyy
stress from FEE for Example 2 (Elasto-plastic mat. and frictionless BC):
Unit[MPa]
X
Y
Z
.30
.28
.26
.24
.22
.20
.18
.16
.14
.12
.10
.08
.06
.04
.02
-.00
X
Y
Z
Fig. 8.9. Equivalent plastic strain from FEE for Example 2 (Elasto-plastic mat. andfrictionless BC)
82
because of stress relaxation through plastic deformation. The y-directional normal stress
after the narrow region becomes tensional, as shown in Figure 8.8, since the height of
the material flow is imposed to be the same for inlet and outlet, although the plastic
strain development is as shown in Figure 8.9.
8.1.3 Example 3: Purely Elastic Example with Velocity Prescribed BC
In this example, the material is assumed to be purely elastic and all components
of velocity are constrained on surfaces D and E such that the velocity component of the
normal direction to the surfaces becomes zero,
vx
= vo
x× 6
y(8.1)
vy
=dy
dx× v
x(8.2)
where x and y are the coordinate components of a point on the boundary D and E, vx
and vy
are the velocity components on D and E, and vo
xis the given outlet velocity.
The resulting plots for Example 3 are presented in Figures 8.11-8.14. Since the material
is purely elastic, the stress on the outlet should be the same as the stress on the inlet.
From this point of view, the results from FEE are more reasonable than those from FRE.
This error is inherent from the characteristics of FRE formulation and FEA modeling.
Some influx along the curved boundary is unavoidable because normal direction of the
curved boundary cannot be continuous in the FE modeling. The result of FRE is much
more sensitive to this virtual influx since FRE imposes the rate equilibrium, instead of
83
X
Y
Z
4.75
4.70
4.65
4.60
4.55
4.50
4.45
4.40
4.35
4.30
4.25
4.20
4.15
4.10
4.05
4.00
X
Y
Z
Fig. 8.10. x-directional velocity from FEE for Example 3 (Elastic mat. and velocityBC): Unit[mm/s]
X
Y
Z
4.75
4.70
4.65
4.60
4.55
4.50
4.45
4.40
4.35
4.30
4.25
4.20
4.15
4.10
4.05
4.00
X
Y
Z
Fig. 8.11. x-directional velocity from FRE for Example 3 (Elastic mat. and velocityBC): Unit[mm/s]
84
X
Y
Z
.8
.7
.6
.5
.4
.3
.2
.1
-.0
-.1
-.2
-.3
-.4
-.5
-.6
-.7
X
Y
Z
Fig. 8.12. y-directional velocity from FEE for Example 3 (Elastic mat. and velocityBC): Unit[mm/s]
X
Y
Z
.8
.7
.6
.5
.4
.3
.2
.1
-.0
-.1
-.2
-.3
-.4
-.5
-.6
-.7
X
Y
Z
Fig. 8.13. y-directional velocity from FRE for Example 3 (Elastic mat. and velocityBC): Unit[mm/s]
85
X
Y
Z
1500.
1400.
1300.
1200.
1100.
1000.
900.
800.
700.
600.
500.
400.
300.
200.
100.
0.
X
Y
Z
Fig. 8.14. Mises’ stress from FEE for Example 3 (Elastic mat. and velocity BC):Unit[MPa]
X
Y
Z
1500.
1400.
1300.
1200.
1100.
1000.
900.
800.
700.
600.
500.
400.
300.
200.
100.
0.
X
Y
Z
Fig. 8.15. Mises’ stress from FRE for Example 3 (Elastic mat. and velocity BC):Unit[MPa]
86
equilibrium itself. The FEE and FRE converged after 5 and 6 Newton-Raphson iterations
for this example, respectively.
8.2 FSW Analysis
Assuming no slip on the spinning tool contacting surface, considerable plastic
strain is expected. However, elasto-plastic analysis algorithms hardly converge for prob-
lems with such large plastic strains. Moreover, high temperature is expected in the re-
gion around the spinning tool where rate-dependent plasticity is more appropriate [30].
Fully coupled thermal-mechanical analyses using viscoplasticity are relatively well devel-
oped and easily incorporate large plastic strain evolution although they yield no residual
stress. Therefore, we suggest a combined thermal-viscoplastic and thermo-elasto-plastic
procedure to analyze FSW process. In this analysis procedure, a fully-coupled thermal-
viscoplastic analysis is performed first , as in Reference [80]. Then, in order to analyze
residual stress formation, the thermo-elasto-plastic algorithms, either by FRE or FEE,
can be performed after high plastic strain evolution region is excluded from the control
volume where the boundary conditions for separation surface can be obtained from the
viscoplastic analysis.
In this section, proper boundary conditions for FSW analysis are discussed. Al-
though the mechanical analysis of FSW is focused in this research, thermal boundary
conditions are also considered for future research. A FSW simulation example is pre-
sented to show the potential of the thermo-elasto-plastic formulations for the combined
FSW analysis procedure.
87
Spining tool
Spining tool shoulder
X
Y
Z
Eulerian CoordinateFixed to the tool pin center
Material Moving Direction
Inlet Surface
Outlet Surface
Side Surface
Fig. 8.16. Eulerian configuration for FSW analysis
8.2.1 Boundary Conditions for FSW analysis
• Thermal Boundary Conditions
Heat generation in FSW has been modeled as frictional heat in the contact surface
or as plastic dissipation. The frictional heat flux per area can be modeled as,
q = κPvr
i(8.3)
where q is the heat flux per unit area per unit time, κ is the friction coefficient, P is the
contact pressure which is the magnitude of σij
nj, and vr is the relative velocity between
a particle and the tool. Thus, the heat flux is also dependent on the mechanical field.
In order to eliminate this dependency, P is assumed to be constant and tool velocity is
approximated for relative velocity [53, 41, 18, 60]. Then, Equation (8.3) can be rewritten
88
as,
q = κPξω (8.4)
where ω is the angular velocity of the tool and ξ is the distance from the FSW tool
center. The plastically dissipated heat is given as [62, 80],
Q = Dp
ijσij
= σvi
∂εq
∂xi
(8.5)
where Q is the heat source per unit volume.
• Mechanical Boundary Conditions
The control volume moves with uniform velocity and the FSW tool rotates with
center fixed in Eulerian frames, as shown in Figure 8.16. Boundary conditions for the
inlet and outlet surfaces should be prescribed as discussed in Section 6. The outlet BC
can be applied for both side surfaces since the material is rigidly constrained during
FSW process. The bottom plate is supported so as to remain vertically stationary. No
slip condition for the surface contacting the tool pin can be applied as,
v = [zω, 0, −xω]T (8.6)
where x, y, and z are local coordinates with the origin at the tool center. The tool shoul-
der surface is also vertically motionless and traction may be applied from the frictional
89
load as follows:
vini
= 0
ti
= −κPvr
i√vr
jvr
j
(8.7)
8.2.2 FSW Example
The FE model (200× 1× 700 mm) is developed with removed large plastic strain
region (radius of 50 mm) around the spinning tool. The weld speed vweld of −4 mm/s is
applied for the outlet and the side surfaces, and zero stress and equivalent plastic strain
for the inlet surface. The velocity on the separation surface vs is assumed as,
vs = [zsω, 0, −x
sω + v
weld] (8.8)
where xs, y
s, and zs are local coordinates for the separation surface with the origin at
the tool center, and 10−3 rad/s is used for ω. The y-directional velocity is constrained
to be zero everywhere, so that the problem becomes one of implicit plane strain. The
material properties of the previous strip drawing examples are also used in this simu-
lation. Figures 8.17-8.20 show the velocity, stress, and equivalent plastic strain fields
for this FSW example from FRE. It should be noted that this FSW example is pre-
sented only to show the potential of FRE and FEE algorithms for FSW analysis with
arbitrary boundary conditions on the separation boundary, and thus, the results do not
reflect real FSW phenomena. Since the material flows across the separation boundary,
stress and equivalent boundary conditions should be prescribed on the material entering
90
X
YZ
.040
.035
.030
.025
.020
.015
.010
.005
.000
-.005
-.010
-.015
-.020
-.025
-.030
-.035X
YZ
Fig. 8.17. x-directional velocity for FSW from FRE (Elasto-plastic mat. and velocityBC): Unit[mm/s]
X
YZ
-3.960
-3.965
-3.970
-3.975
-3.980
-3.985
-3.990
-3.995
-4.000
-4.005
-4.010
-4.015
-4.020
-4.025
-4.030
-4.035X
YZ
Fig. 8.18. z-directional velocity for FSW from FRE (Elasto-plastic mat. and velocityBC): Unit[mm/s]
91
X
YZ
150.
140.
130.
120.
110.
100.
90.
80.
70.
60.
50.
40.
30.
20.
10.
0.X
YZ
Fig. 8.19. Mises’ stress for FSW (Elasto-plastic mat. and velocity BC): Unit[MPa]
X
YZ
.045
.042
.039
.036
.033
.030
.027
.024
.021
.018
.015
.012
.009
.006
.003
.000X
YZ
Fig. 8.20. Equivalent plastic strain for FSW (Elasto-plastic mat. and velocity BC)
92
boundary in the combined analysis scheme, as has been stated in Section 6.2.4. The
FEE simulation for this example shows convergence difficulty and no result has been
obtained. The convergence difficulty is attributed to the fact that the FEE formulation
requires equilibrium boundary conditions more strongly than the FRE formulation. The
FEE formulation may still be applicable to FSW analysis if boundary conditions satisfy
equilibrium.
8.3 Conclusions and Future Works
Two Eulerian elasto-plastic FE algorithms with SUPG-mixed formulation tech-
nique are developed and investigated to predict residual stress in FSW process. From
strip drawing simulations, it is concluded that FEE more easily incorporates boundary
conditions and yields more reasonable results than FRE. However, the FEE requires
more strict equilibrium BC. A combined elasto-plastic and viscoplastic analysis scheme
is suggested to analyze FSW process. A FSW example shows the potential usefulness of
the FEE and FRE for FSW analysis in the combined analysis procedure.
For the complete FSW analysis by the combined elasto-plastic and viscoplastic
analysis scheme, a fully-coupled thermal-mechanical (viscoplastic) algorithm which can
incorporate the FSW boundary conditions described in Section 8.2.1 is needed. Surface
integral capability for FEE and FRE is necessary to apply boundary conditions on the
separation surface using vicoplastic analysis results.
93
Chapter 9
Conclusions
Numerical modeling and optimization of welding residual stress is studied in this
thesis.
Fusion welding process is analyzed using nonlinear finite element analysis in
weakly-coupled thermal-mechanical scheme where thermal filed is analyzed as transient
heat conduction and mechanical filed as thermo-elasto-plastic. Sensitivity equations
of both thermal and mechanical analysis for the gradient optimization of thermo-elasto-
plastic process. The direct sensitivity algorithms are verified by comparing with the finite
difference sensitivity. The side heaters for transient thermal tensioning are successfully
optimized for minimum welding residual stress. The proposed numerical sensitivity and
optimization scheme can be applied for other thermo-elasto-plastic process such as laser
forming.
Material property sensitivity (derivative of temperature and Mises’ stress with re-
spect to each base material property from which the material properties are interpolated
linearly for entire temperature range) in fusion welding process is evaluated using the au-
tomatic differentiation facility, ADIFOR. The first and second derivatives are evaluated
and the importance of each material property is assessed. Although automatic differen-
tiation produces sensitivity as accurately as the formulated sensitivity and convenient to
94
implement, the automatically generated sensitivity programs are computationally ineffi-
cient especially for nonlinear problems.
Numerical modeling for residual stress prediction in friction stir welding process
is studied. Two Eulerian thermo-elasto-plastic formulations are developed: Equilibrium
based FEE and rate-equilibrium based FRE. SUPG stabilization technique is used for
the weak formulations of stress and equivalent plastic evolution equation which are char-
acterized as hyperbolic partial differentiation equations to relieve numerical instability.
Mixed formulation technique is applied to overcome the limits of iterative solution proce-
dure for FEE. Strip drawing examples are simulated to verify and compare the Eulerian
thermo-elasto-plastic formulations: FEE predicts the mechanical fields (velocity, stress,
equivalent plastic strain) reasonably, the results from FRE can be distorted by the vir-
tual influx of materials on the velocity prescribed curved surfaces, and both FEE and
FRE do not converge if the difference in velocity on the boundary relative to overall flow
velocity exceeds certain limits as in FSW. In order to overcome the convergence problem,
a combined thermal-viscoplastic and thermo-elasto-plastic analysis scheme is suggested
for the complete analysis of residual stress in FSW. An FSW example is simulated using
FRE to demonstrate the combined analysis scheme.
95
Appendix A
Basic Plasticity
A.1 Isotropic hardening plasticity in deviatoric space
The stress vector σ can be decomposed into its volumetric and deviatoric com-
ponents.
σ = σh
+ s (A.1)
where
σh
=13jjT σ (A.2)
The yield function in Equation (2.31) can be rewritten as follows:
f =
√32
[sT Ls
]1/2 − σY
(εq, T ) (A.3)
= σm
(s) − σY
(εq, T ) (A.4)
The flow vector a can be expressed by definition and above equations as follows:
a =∂f
∂σ=
∂σm
∂s=
32σ
m
Ls (A.5)
96
The following relation holds for an isotropic material:
Ca =3G
σm
s (A.6)
A.2 Summary of the radial return algorithm
In radial return algorithm theory, the flow vector a does not change during the
constitutive iterations and in conjunction with Equation (A.5):
a = aB
=3
2σBm
LsB
(A.7)
Note that every variable which has subscript B comes from σB
. From Equations (2.22),
(2.23) and (2.29):
σ = σB
− C∆εp
(A.8)
There is no plastic evolution for the volumetric stress.
σh
= σBh
=13jjT σ
B(A.9)
Application of Equations (2.25), (A.6), (A.7), and (A.9) into Equation (A.8) yields
s =
[σBm
− 3G∆εq
σBm
]sB
(A.10)
97
Substitution of Equation (A.10) into Equation (A.3) yields
f = σBm
− 3G∆εq− σ
Y= 0 (A.11)
∆εq
can be evaluated by solving Equation (A.11) iteratively. Finally, the following
relation can be obtained:
σY
= σBm
− 3G∆εq
(A.12)
Therefore, Equation (3.15) can be obtained by applying Equations (A.12), (2.34), (A.5),
and (A.10) to Equation (A.1).
98
Appendix B
Detailed Derivation of Plastic Sensitivity Equations
B.1Dσ
hDφ
iand
∂σh
∂U
Equations (3.17) and (3.29) are straightforwardly obtained from Equation (A.9).
B.2 DmDφ
iand ∂m
∂U
In order to obtain Equation (3.18), Equation (2.34) can be rewritten from Equa-
tions (2.34), (A.1), and (A.7) as follows:
m =1
σBm
[σ
B− σ
Bh
](B.1)
Thus,
DmDφ
i
=1
σBm
[Dσ
BDφ
i
−Dσ
BhDφ
i
]−
DσBm
Dφi
sB
[σBm
]2(B.2)
where
DσBm
Dφi
=∂σ
Bm∂σ
B
DσB
Dφi
= aT DσB
Dφi
(B.3)
Equation (3.30) can be obtained by the same procedure.
99
B.3Dσ
YDφ
iand
∂σY
∂U
Note that σY
is a function of the temperature and the equivalent plastic strain.
DσY
Dφi
=∂σ
Y∂T
dT
dφi
+∂σ
Y∂ε
q
Dεq
Dφi
=∂σ
Y∂T
dT
dφi
+ H
d
n−1εq
dφi
+D∆ε
q
Dφi
(B.4)
From Equation (A.12):
D∆εq
Dφi
=1
3G
[Dσ
BmDφ
i
− 3∆εq
DG
Dφi
−Dσ
YDφ
i
](B.5)
where
DG
Dφi
=∂G
∂T
dT
dφi
(B.6)
Thus, Equation (3.19) can be obtained by substituting Equation (B.5) into Equation
(B.4). OnceDσ
YDφ
iis evaluated,
D∆εq
Dφi
can be obtained from Equation (B.5), which
becomes Equation (3.36).
The weakly coupled thermo-mechanical analysis imposes the condition that the
change of displacement does not affect the temperature field.
∂T
∂U= 0 (B.7)
Furthermore, the current displacement field does not affect any of the previous fields.
∂σY
∂U=
∂σY
∂T
∂T
∂U+
∂σY
∂εq
∂εq
∂U= H
∂∆εq
∂U(B.8)
100
From Equation (A.12) :∂∆ε
q
∂U=
13G
[∂σ
Bm∂U
−∂σ
Y∂U
](B.9)
Equations (3.31) and (3.35) can be obtained from Equations (B.8) and (B.9).
101
Appendix C
Voigt Transformation
Using the Voigt notation, σ and D are transformed to column vectors as follows:
σ =[σ1, σ2, σ3, σ4, σ5, σ6
]T
=[σ11, σ22, σ33, σ12, σ23, σ31
]T
(C.1)
D =[D1, D2, D3, D4, D5, D6
]T
=[D11, D22, D33, D12, D23, D31
]T
(C.2)
and the other terms should also be transformed consistently:
σd
i= Z
ijσj
(C.3)
∂σd
i∂x
j
= Zik
∂σk
∂xj
(C.4)
σ = σd
iL
ijσd
j(C.5)
ai
=32σ
Lij
σd
j(C.6)
mi
=32σ
σd
i(C.7)
Cij
= λhihj
+ 2µL−1
ij(C.8)
Dth
i= β
[∂T
∂xj
vj
]hi
(C.9)
102
where h and L are constant tensors as defined in Chapter 2, and another constant tensor
Z is defined as,
Zij
= δiδj− 1
3hihj
(C.10)
The terms, σik
Wkj
− σjk
Wki
, in Equation (6.11) also can be transformed into Voigt
form by defining a new tensor A such that
Aij
σj
= [η11, η22, η33, η12, η23, η31]T (C.11)
where
ηij
= Wik
σkj
+ Wjk
σik
, i, j, k = 1, 2, 3 (C.12)
Then the resultant A is obtained as follows:
A =
0 0 0 2W12 0 2W13
0 0 0 2W21 2W23 0
0 0 0 0 2W32 2W31
W21 W12 0 0 W13 W23
0 W32 W23 W31 0 W21
W31 0 W13 W32 W12 0
(C.13)
103
The tensor form of stress can be restored from Voigt form of stress using the following
constant matrix.
Mijk
=∂σ
ij
∂σk
(C.14)
where Mijk
is zero except
M111 = M222 = M333 = M124 = M214 = M235 = M325 = M136 = M316 = 1 (C.15)
104
Appendix D
FE Discretization
D.1 Mapping to The Master Element
The spatial coordinate x is interpolated from the shape function N and the ele-
ment nodal coordinate vector X.
xi
= Nij
Xj
(D.1)
The Jacobian J of the mapping from the element nodal vector X to the master element
coordinate r is evaluated as follows:
Jij
=∂x
i∂r
j
=∂N
ik∂r
j
Xk
(D.2)
The determinant and inverse of J are calculated as follows:
J = det(J) (D.3)
J−1
ij=(J−1)
ij(D.4)
105
D.2 Field Variable Interpolators
vi
= Nij
Vj
(D.5)
σi
= Nσ
ijS
j(D.6)
εq = N
q
iQ
i(D.7)
T = Nth
iTi
(D.8)
where V, S, and Q are the element nodal velocity, stress, and equivalent plastic strain
vectors; N, Nσ, and Nq are corresponding shape functions.
D.3 Gradient Interpolators
The spatial derivatives can be evaluated as follows:
∂
∂xj
= J−1
ij
∂
∂rj
(D.9)
In this manner, the gradient interpolators B, Ba, Bv, Bσ, Bq, and Bth can be evaluated
as follows:
Di= B
ijVj
(D.10)
106
where
Bij
=
J−1
il
∂Nij
∂rl
for i = 1, 2, 3
[J−1
1l
∂N2j
∂rl
+ J−1
2l
∂N1j
∂rl
]for i = 4
[J−1
2l
∂N3j
∂rl
+ J−1
3l
∂N2j
∂rl
]for i = 5
[J−1
3l
∂N1j
∂rl
+ J−1
1l
∂N3j
∂rl
]for i = 6
Aij
= Ba
ijkVk
(D.11)
where Ba
ijk= 0 except
Ba
14k= J
−1
2l
∂N1k∂r
l
− J−1
1l
∂N2k∂r
l
Ba
24k= −B
a
14k; B
a
65k= B
a
42k=
12B
a
14k; B
a
56k= B
a
41k= −1
2B
a
14k
Ba
16k= J
−1
3l
∂N1k∂r
l
− J−1
1l
∂N3k∂r
l
Ba
36k= −B
a
16k; B
a
63k= B
a
45k=
12B
a
16k; B
a
54k= B
a
61k= −1
2B
a
16k
Ba
25k= J
−1
3l
∂N2k∂r
l
− J−1
2l
∂N3k∂r
l
Ba
35k= −B
a
25k; B
a
53k= B
a
46k=
12B
a
25k; B
a
52k= B
a
64k= −1
2B
a
25k
107
∂vi
∂xj
= Bv
ijkVk
(D.12)
∂σi
∂xj
= Bσ
ijkS
k(D.13)
∂εq
∂xi
= Bq
ijQ
j(D.14)
∂T
∂xi
= Bth
ij(D.15)
where
Bv
ijk= J
−1
jl
∂Nik
∂rl
Bσ
ijk= J
−1
jl
∂Nσ
ik∂r
l
Bq
ij= J
−1
il
∂Nq
j
∂rl
Bth
ij= J
−1
il
∂Nth
j
∂rl
108
References
[1] P. R. Dawson A. Agrawal. A comparison of galerkin and streamline techniques for inte-
grating strains from an eulerian flow field. International Journal of numerical methods in
Engineering, 21:853–881, 1985.
[2] J. M. J. McDill A. S. Oddy and J. A.Goldak. Consistent strain fields in 3d finite element
analysis of welds. ASME Journal of Pressure Vessel Technology, 112(3):309–311, 1990.
[3] M. Abeo-Elkhier, G. A. Oravas, and M. A. Dokainish. A consisitent eulerian formulation for
large deformation analysis with reference to metal-extrusion process. International Journal
of Non-Linear Mechanics, 23:37–52, 1988.
[4] J. H. Argyris, J. Szimmat, and K. J. Willam. Computational Aspects of Welding Stress
Analysis. Computer Methods in Applied Mechanics and Engineering, 33:635–666, 1982.
[5] S. Badrinarayanan and N. Zabaras. Prefrom design in metal forming. In NUMIFORM,
Simulation of Materials Processing: Theory, Methods, and Applications, pages 533–538.
Balkema, 1995.
[6] D. Balagangadhar, G.A. Dorai, and D.A. Tortorelli. A displacement-based eulerian steady-
state formulation suitable for thermo-elasto-plastic material models. International Journal
of Solids and Structures, 36(16):2397–2416, 1999.
[7] D. Balagangadhar and D.A. Tortorelli. Design of steady processes via a displacement based
reference frame formulation. In NUMIFORM, Simulation of Materials Processing: Theory,
Methods, and Applications, pages 609–613. Balkema, 1998.
[8] Ted Belytschko, Wing Kam Liu, and Brian Moran. Nonlinear Finite Elements for Continua
and Structures. John Wiley and Sons, 2000.
[9] L. A. Bertram and A. R Ortega. Automated Thermomechanical Modeling of Welds Using
Interface Elements for 3D Metal Deposition. In Manuscript for Proceedings of ABAQUS
User’s Conference, Oxford: Hibbit Karlsson and Sorensen Inc., 1991.
[10] C. Bischof, A. Carle, G. Corliss, and P Hovland. Generating derivative codes from Fortran
programs. Scientific Programing, 3:1–29, 1992.
[11] C. Bischof, A. Carle, P. Khademi, and A. Mauer. Automatic differentiation of FORTRAN.
Computational Science and Engineering, 3(3):18–32, 1996.
109
[12] H. J. Braudel, M. Abouaf, and J. L. Chenot. An Implicit and Incremental Formulation
for the Solution of Elastoplastic Problems by the Finite Element Method. Computers and
Structures, 22(5):801–814, 1986.
[13] Ya. I. Burak, L. P. Besedina, Ya. P. Romanchuk, A. A. Kazimirov, and V. P. Morgun.
Controlling the longitudinal plastic shrinkage of metal during welding. Avt. Svarka, 3:27–
29, 1977.
[14] Ya. I. Burak, Ya. P. Romanchuk, A.A. Kazimirov, and V. P. Morgun. Selection of the
optimum fields for preheating plates before welding. Avt. Svarka, 5:5–9, 1979.
[15] A. Carle, M. Fagan, and L. Green. Preliminary results from the application of automatic
adjoint code generation to CFL3D. AIAA Paper, pages 98–4807, 1869.
[16] A. P. Chakravati, L. M. Malik, and J. A. Goldak. Prediction of Distortion and Residual
Stresses in Panel Welds. In Computer modelling of fabrication processess and constitutive
behaviour of metals, pages 547–561, Ottawa, Ontario, 1986.
[17] Y.J. Chao and X. Qi. Thermal and thermo-mechanical modeling of friction stir welding of
aluminum alloy 6061-t6. Journal of materials processing & manufacturing science, 7:215–
233, 1998.
[18] Yuh J. Chao, X. Qi, and W. Tang. Heat transfer in friction stir welding-experimental and
numerical studies. Journal of manufacturing science and engineering, 125:138–145, 2003.
[19] L. P. Connor, editor. Welding Handbook. American Welding Society, Miami, FL, eighth
edition, 1987.
[20] M. J. Crochet and M. Bezy. Numerical solutions for the flow of viscoelastic fluids. J.
Non-Newtonoan Fluid Mech., 5:281–288, 1979.
[21] P. R. Dawson. Viscoplastic finite element analysis of steady-state processes including strain
history and stress flux dependence. Applications of Numerical Methods to Forming Processes,
28:55–67, 1978.
[22] P. R. Dawson. On modeling of mechanical property changes during flat rolling of aluminum.
International Journal of Solids and Structures, 23:947–968, 1987.
[23] P. R. Dawson and E. G. Thompson. Finite element analysis in steady-state elasto-visco-
plastic flow by the initial stress-rate method. International Journal of Numerical Methods
in Engineering, 12:47–57, 1978.
[24] D.Balagangadhar and D.A.Tortorelli. Design of large-deformation steady elastoplastic manu-
facturing processes. part i: A displacement-based reference frame formulation. International
Journal for Numerical Methods in Engineering, 49:899–932, 2000.
110
[25] K. Dems and Z. Mroz. Variational Approach to Sensitivity Analysis in Thermoelasticity.
Journal of Thermal Stress, 10:283–306, 1987.
[26] M. V. Deo and P. Michaleris. Mitigation of welding induced buckling distortion using
transient thermal tensioning. Science and Technology in Welding, 8(1):49–53, 2003.
[27] O. C. Zienkiewicz E. G. Thompson, J. F. T. Pittman. Some integration techniques for
the analysis of viscoelastic flows. International Journal for numerical methods in Fluids,
3:165–177, 1983.
[28] VMA Engineering. DOT User’s Manual, Version 3.00. Vanderplaats, Miura and Associates,
Inc., Goleta, CA, 1992.
[29] S. Ghosh and N. Kikuchi. An arbitrary eulerian-lagrangian finite element for large defor-
mation analysis of elastic-viscoplastic solids. Computer Methods in Applied Mechanics and
Engineering, 86:127–188, 1991.
[30] J. Goldak. Thermal stress analysis in solids near the liquid region in welds. In Mathematical
Modelling of Weld Phenomena 3, Graz, Austria, 1997.
[31] J. Goldak and M. Bibby. Computational Thermal Analysis of Welds: Current Status and
Future Directions. In A. F. Giamei and G. J. Abbaschian, editors, Modeling of Casting
and Weldin Processes IV, pages 153–166, Palm Coast, FL, 1988. The Minerals & Materials
Society.
[32] J. Goldak, M. Bibby, J. Moore, R. House, and B. Patel. Computer Modeling of Heat Flows
in Welds. Metallurgical Transactions B, 17B:587–600, 1986.
[33] J. Goldak, V. Breiguine, and N. Dai. Computational weld mechanics: A progress report on
ten grand challenges. In 4th Int Conf on Trends in Welding Research, Gatlinburg, USA,
1995.
[34] J. Goldak, A. Chakravarti, and M. Bibby. A New Finite Element Model for Welding Heat
Sources. Metallurgical Transactions B, 15B:299–305, 1984.
[35] R. Gunnert. Residual Welding Stressed. Almqvist & Wiksell, Stockholm, 1955.
[36] R. B. Haber. A mixed eulerian-lagrangian displacement model for large derformation anal-
ysis in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 43:277–
292, 1984.
[37] R. T. Haftka and Z. Gurdal. Elements of Structural Optimization. Kluwer Academic Pub-
lishers, The Netherlands, 1992.
111
[38] E. J. Haug, K. K. Choi, and V. Komkov. Design Sensitivity Analysis of Structural Systems.
Academic Press, New York, 1986.
[39] H. Hibbitt and P. V. Marcal. A Numerical, Thermo-Mechanical Model for the Welding and
Subsequent Loading of a Fabricated Structure. Computers & Structures, 3(1145-1174):1145–
1174, 1973.
[40] T. J. R. Hughes and A. Brooks. A theoretical framework for petrov-galerkin methods with
discontinuous weighting functions: Application to the streamline-upwind procedure. In
Finite Elements in Fluids, pages 47–65. John Wiley and Sons, 1982.
[41] J.Gould and Z.Feng. Heat flow model for fricion stir welding of aluminum alloys. Journal
of materials processing & manufacturing science, 7:185–194, 1998.
[42] M. Kawahara and N. Takenchi. Mixed finite element method for analysis of viscoplastic
fluid flow. Comp. Fluids, 5:33–45, 1977.
[43] H. M. Koh and R. B. Haber. Elastodynamic formulation of the eulerian-lagrangian kinematic
description. ASME Journal of Applied Mechanics, 53:839–845, 1986.
[44] P. J. Konkol, J. A. Mathers, R. Johnson, and J. R. Pickens. Friction stir welding of of
hsla-65 steel for shipbuilding. Journal of Ship Production, 19:159–164, 2003.
[45] Young-Shin Lee and Paul R. Dawson. Obtaining residual stresses in metal forming after
neglecting elasticity on loading. Journal of Applied Mechanics, Transactions ASME, 56:318–
327, 1989.
[46] L.-E. Lindgren. Finite element modelling and simulation of welding, Part 1 Increased com-
plexity. Journal of Thermal Stresses, 24:141–192, 2001.
[47] L.-E. Lindgren. Finite element modelling and simulation of welding, Part 2 Improved ma-
terial modelling. Journal of Thermal Stresses, 24:195–231, 2001.
[48] L.-E. Lindgren. Finite element modelling and simulation of welding, Part 3 Efficiency and
integration. Journal of Thermal Stresses, 24:305–334, 2001.
[49] J. Lubliner. Plasticity Theory. Macmillan Publishing Company, New York, 1st edition,
1990.
[50] M. W. Mahoney. Properties of friction-stir-welded 7075 t651 aluminum. Metallurgical and
Materials Transactions A: Physical Metallurgy and Materials Science, 29A:1955–1964, 1998.
[51] A. M. Maniatty, P.R.Dawson, and G. G. Weber. An eulerian elasto-viscoplastic formulation
for steady-state forming processes. International Journal of Mechanical Sciences, 33:361–
377, 1991.
112
[52] M.A.Sutton, A.P.Reynolds, D.-Q.Wang, and C.R.Hubbard. A study of residual stresses
and microstructure in 2024-t3 alumunum friction stir butt welds. Journal of Engineering
Materials and Technology, 124:215–221, 2002.
[53] J.C. McClure, W. Tang, L.E. Murr, X. Guo, Z. Fang, and J.E. Gould. A thermal model of
friction stir welding. In Proceedings of the 5th International Conference : Trends in Welding
Research, pages 590–595, Pine Mountain, GA, 1998.
[54] R. A. Meric. Sensitivity Analysis of Functionals With Respect To Shape For Dynami-
cally Loaded Nonlocal Thermoelastic Solids. International Journal of Engineering Science,
26(7):703–711, 1988.
[55] P. Michaleris, J. A. Dantzig, and D. A. Tortorelli. Minimization of Welding Residual Stress
and Distortion in Large Structures. Welding Journal, 78(11):361–366s, 1999.
[56] P. Michaleris, D. A. Tortorelli, and C. A. Vidal. Tangent Operators and Design Sensi-
tivity Formulations for Transient Nonlinear Coupled Problems with Applications to Elasto-
Plasticity. International Journal for Numerical Methods in Engineering, 37:2471–2499, 1994.
[57] P. Michaleris, D. A. Tortorelli, and C. A. Vidal. Analysis and Optimization of Weakly Cou-
pled Thermo-Elasto-Plastic Systems with Applications to Weldment Design. International
Journal for Numerical Methods in Engineering, 38(8):1259–1285, 1995.
[58] O. T. Midling and O. Grong. A process model for friction stir welding. Acta Metallurgica
et Materialia, 42:1595–1609, 1994.
[59] O. Myhr. Modeling of microstructure evolution and residual stresses in processing and
welding of 6082 and 7108 aluminium alloys. In H. D. Brody and D. Apelian, editors, 5th
International Conference on Trends in Welding Research, Pine Mountain, Georgia, USA,
1998.
[60] O.Frigaard, O.Grong, and O.T.Milding. A process model for friction stir welding of age hard-
ening aluminum alloys. Metallurgical and Materials Transactions A : Physical metallurgy
and Materials Science, 32:1189–1200, 2001.
[61] S. Paul, P. Michaleris, and J.Shanghvi. Optimization of Thermo-Elasto-Plastic Finite Ele-
ment Analysis using an Eulerian Formulation. International Journal for Numerical Methods
in Engineering, 56:1125–1150, 2003.
[62] P.Heurtier, C.Desrayaud, and F.Montheillet. A thermomechanical analysis of the friction
stir welding process. Materials Science Forum, 396-402:1537–1542, 2002.
113
[63] M. Posada, J. DeLoach, A.P. Reynolds, S.R. Bhide, and P. Michaleris. Evaluation of Friction
Stir Welded HSLA-65. In Advanced Marine MaterialMaterials: Technology and Applications,
London, UK, October 2003. Royal Institution of Naval Architects (RINA).
[64] D. Radaj. Heat effects of welding. Springer-Verlag, Berlin, 1992.
[65] N. Rebelo and S. Kobayashi. A coupled analysis of viscoplastic deformation and heat transfer
- theoretical considerations. International Journal of Mechanical Sciences, 22:699–705, 1980.
[66] J. B. Roelens and F. Maltrud. Determination of residual stresses in submerged arc multi-pass
welds by means of numerical simulation and comparison with experimental measurements.
In IIW Annual Assembly, Glasgow, 1993. Doc. X-1279.
[67] E. F. Rybicki and R. B. Stonesifer. Computation of Residual Stresses due to Multipass
Welds in Piping Systems. Journal of Pressure Vessel Technology, 101:149–154, 1979.
[68] A. Saltelli, K. Chan, and E. M. Scott. Sensitivity Analysis. John Wiley and Sons Inc.,
Hoboken, NJ, 2000.
[69] J. Shanghvi and P. Michaleris. Thermo-Elasto-Plastic Finite Element Analysis of Quasi-
State Processes in Eulerian Reference Frames. International Journal for Numerical Methods
in Engineering, 53:1533–1556, 2002.
[70] Y. Shim, Z. Feng, S. Lee, D.S. Kim, J. Jaeger, J. C. Paparitan, and C. L. Tsai. Determination
of Residual Stress in Thick-Section Weldments. Welding Journal, 71:305s–312s, 1992.
[71] J. C. Simo and R. L. Taylor. Consistent Tangent Operators for Rate-Independent Elasto-
Plasticity. Computer Methods in Applied Mechanics and Engineering, 48:101–118, 1985.
[72] J. Song, P. Michaleris, and J. Shanghvi. Optimization of of Thermo-Elasto-Plastic Menu-
facturing Processes. to appear, 2004.
[73] J. Song, J. Peters, A. Noor, and P. Michaleris. Sensitivity analysis fo the thermomechanical
response of welded joints. International Journal of Solids and Structures, 40:4167–4180,
2003.
[74] P. Tekriwal and J. Mazumder. Transient and Residual Thermal Strain-Stress Analysis of
GMAW. Journal of Engineering Materials and Technology, 113:336–343, 1991.
[75] K. Terai. Study on Prevention of Welding Deformation in Thin-Skin Plate Structures.
Technical Report 61, Kawasaki, 1978.
[76] M. W. Thomas, J. Nicholas, J. C. Needham, M. G. Murch, P. Templesmith, and C. J.
Dawes. Friction stir butt welding. GB Patent Application 9125978.8 December 1991; US
Patent 5460317, October 1995.
114
[77] E. G. Thomson and H. M. Berman. Steady-state analysis of elasto-viscoplastic flow during
rolling. In J. F. T. Pittman, O. C. Zienkiewicz, R. D. Wood, and J. M. Alexander, editors,
Numerical Analysis of Forming Processes, New York, 1984. John Wiley.
[78] Erik G. Thomson and Szu-Wei Yu. A flow formulation for rate equilibrium equations.
International Journal for Numerical Methods in Engineering, 30:1619–1632, 1990.
[79] Y. Ueda. Establishment of computational weld mechanics. Transactions of JWRI, 24:73–86,
1995.
[80] P. Ulysse. Three-dimensional modeling of the friction stir-welding process. International
Journal of Machine Tools and Manufacture, 48:1549–1557, 2002.
[81] V. B. Venkayya. Structural Optimization: A Review and Some Recomendations. Interna-
tional Journal for Numerical Methods in Engineering, 13:203–228, 1978.
[82] M. Viriyauthakorn and B. Caswell. Finite element simulation of viscoelastic flow. Journal
of Non-Newtonian Fluid Mechanics, 6:245–267, 1980.
[83] J. Wang and H. Murakawa. A 3-d fem analysis of buckling distortion during welding in thin
plate. In 5th International Conference in Trends in Welding Research, Pine Mountain, GA,
1998.
[84] Z. Wang and T. Inoue. Viscoplastic constitutive relation incorporating phase transformation
- application to welding. Material Science and Technology, 1:899–903, 1985.
[85] K. Wiechmann and F.-J. Barthold. Remarks on variational design sensitivity analysis of
structures with large elasto-plastic deformations. In 7th AIAA/USAF/NASA/ISSMO Sym-
posium on Multidisciplinary Analysis and Optimization, pages 349–358. AIAA, 1998.
[86] R. J. Yang. Shape Design Sensitivity Analysis of Thermoelasticity Problems. Computer
Methods in Applied Mechanics and Engineering, 102:41–60, 1993.
[87] N. Yurioka and T. Koseki. Modeling activities in japan. In Mathematical Modelling of Weld
Phenomena 3, Graz, Austria, 1997. The Institute of Materials.
[88] O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method. McGraw-Hill, UK, fourth
edition, 1991.
Vita
Jinseop Song received his B.S. degree in naval architecture and ocean engineering
at Seoul National University in 1995. He worked as a ship-hull designer in Dawoo
Shipbuilding and Marine Engineering until 1998. He received his master degree in Naval
Architecture and Ocean Engineering at Seoul National University in 2000. In Aug 2000,
he enrolled in the graduate program of Penn State University and began to pursue
his Ph. D. degree. His research interests include nonlinear finite element analysis,
sensitivity analysis, and optimization of solid mechanics, rate independent/dependent
plastic process, heat transfer, and structural dynamics.
