FINITE ELEMENT ANALYSIS AND LIFE ESTIMATION OF A PERMANENT
CYLINDRICAL MOLD WITH ANSYS AND COFFIN-MANSON APPROACH
Except where reference is made to the work of others, the work described in this thesis
is my own or was done in collaboration with my advisory committee. This thesis does
not include proprietary or classified information.
_________________________
Ajay K Roy
Certificate of Approval:
_____________________________
Dr. J T. Black
Professor Emeritus
Industrial and Systems Engineering
_____________________________
Dr. Lewis N. Payton, Chair
Assistant Research Professor
Industrial and Systems Engineering
_____________________________
Dr. Ruel A. Overfelt
Associate Professor
Materials Engineering
______________________________
Dr. Winfred A. Foster Jr.
Professor
Aerospace Engineering
______________________________
Dr. Jerry Davis
Assistant Research Professor
Industrial and Systems Engineering
______________________________
Steven L. McFarland
Acting Dean
Graduate School
FINITE ELEMENT ANALYSIS AND LIFE ESTIMATION OF A PERMANENT
CYLIDRICAL MOLD WITH ANSYS AND COFFIN-MANSON APPROACH
Ajay K Roy
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
May, 2005
iii
FINITE ELEMENT ANALYSIS AND LIFE ESTIMATION OF A PERMANENT
CYLIDRICAL MOLD WITH ANSYS AND COFFIN-MANSON APPROACH
Ajay K Roy
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon request of individuals or institutions and at their expense. The author
reserves all publication rights.
______________________________
Signature of Author
______________________________
Date
Copy sent to:
Name Date
iv
VITA
Ajay K. Roy, son of Baikunth Roy and Kalawati Devi, was born October 1977, in
Patna, Bihar, India. He graduated from St. Xavier?s High School in 1993. He attended
Science College in Patna, Bihar, India for two years. After working as a sales supervisor
for two years, he entered Pune University, College of Engineering (Sangamner)?s
Department of Production Engineering in July 1997 and graduated with distinction with
a Bachelor of Engineering degree in Production Engineering, 2001. After working for
one year as a Service and Manufacturing Engineer in Alfa Laval India Ltd., he entered
the Graduate School at Auburn University, Department of Industrial and Systems
Engineering, in August 2002.
v
THESIS ABSTRACT
FINITE ELEMENT ANALYSIS AND LIFE ESTIMATION OF A PERMANENT
CYLIDRICAL MOLD WITH ANSYS AND COFFIN-MANSON APPROACH
Ajay K Roy
Master of Science, Industrial and Systems Engineering
Auburn University, 2005
(B.E. Pune University, 2001)
Directed by: Dr. Lewis Payton
Thermal fatigue is the most severe problem encountered by a permanent mold,
leading to heat checking and cracking which affects the dimensional stability of the
mold. Developing a methodology to determine the optimal diameter to thickness ratio to
ensure the dimensional stability of a permanent cylindrical mold exposed to cyclic
thermal loading is the focus of this work. In this research, thermal stress analysis was
performed for multilayered cylindrical molds made up of 2 ? % Cr 1% Mo steel and
99% pure copper and cylindrical molds made up of 2 ? % Cr 1% Mo steel. Heating and
cooling cycles of 10 and 25 seconds were applied to the inside surface, while the
outside surface was water cooled. A 2-D (Plane strain) coupled-field analysis was
performed using a thermal-elastic-plastic model accounting for the elastic, as wel-l as,
the plastic deformation with ANSYS. The Coffin-Manson equation was then used to
vi
calculate fatigue life utilizing the strain amplitude was obtained from the finite element
analysis. The results of the finite element analysis and the calculated fatigue life were
validated against a widely accepted mathematical model?s result and empirical
industrial data. The method estimated the actual fatigue life observed in industry
conservatively (within 5%).
vii
ACKNOWLEDGMENTS
I wish to express my sincere appreciation to my committee members Dr. J T.
Black, Dr. Lewis N. Payton, Dr. Ruel A. Overfelt, Dr. Winfred A. Foster and Dr. Jerry
Davis.
Special recognition, however, goes to Dr. J T Black for encouraging and
supporting me to start my career at Auburn University. I also thank him for his valuable
guidance throughout my studies and for giving me the opportunity to gain some
teaching experience. My gratitude extends as well to Dr. Lewis N. Payton for his
constant interest and guidance.
I am most grateful to my parents for their support throughout all my endeavors.
They have always been my source of inspiration. I am thankful to my friends and family
members for always supporting me and boosting my confidence when my spirits were
low.
viii
Style manual or journal used: SME (Society of Manufacturing Engineers)
Computer software used: Microsoft Word, ANSYS, Microsoft Excel, Maple 9.5
ix
TABLE OF CONTENTS
LIST OF FIGURES ??????????????????????..
x
LIST OF TABLES ???????????????????????
xiv
NOTATIONS ???????????????????????......
xv
CHAPTER 1: INTRODUCTION ????????????????...
1
CHAPTER 2: LITERATURE REVIEW ?????...????????.
5
CHAPTER 3: ANSYS 9.0 FEATURES AND SYNTAX
16
CHAPTER 4: METHODOLOGY, MODELS AND PARAMETERS???.
35
CHAPTER 5: RESULTS AND DISCUSSION?????????.??...
45
CHAPTER 6: CONCLUSIONS AND FUTURE WORK ?..??????..
57
REFERENCES ????????????????????????
59
APPENDIX A (MATERIAL PROPERTIES)?..???????????.
63
APPENDIX B (LOG FILE, ANSYS INPUT FILE)..?????????..
71
APPENDIX C (RESULTS FOR MONOLAYER CYLINDER, T=0.52? ?.. 91
APPENDIX D (RESULTS FOR MONOLAYER CYLINDER, T=0.95??... 97
APPENDIX E (RESULTS FOR VALIDATION, OLIVER 1988) ???.....
103
x
LIST OF FIGURES
Figure 1.1 Schematic of centrifugal casting
Figure 2.1 Interaction between material, function, shape and process
Figure 3.1 Geometry of Plane13 element
Figure 3.2 Types of hardening rules
Figure 3.3 Stress-strain behavior of multilinear isotropic plastic material
Figure 3.4 Uniaxial behavior
Figure 4.1 30.00 foot long mold
Figure 4.2 2-D drawing of multilayer cylinder
Figure 4.3 Cross section of cylinder showing dimensions
Figure 4.4 Load curve
Figure 4.5 Load and boundary conditions
Figure 4.6 Stress-strain curve of steel (2?% Cr 1% Mo) at different temperature
Figure 4.7 Stress-strain curve of oxygen free high conductivity copper at different
temperature
Figure 5.1 Radial temperature (C) in multilayer cylinder at time 1,10 and 35 second
Figure 5.2 Radial stress (Pa) in multilayer cylinder at time 1,10 and 35second
Figure 5.3 Tangential stress (Pa) in multilayer cylinder at time 1,10 and 35 second
Figure 5.4 Axial stress (Pa) in multilayer cylinder at time 1,10 and 35 second
Figure 5.5 Equivalent stress (Pa) in multilayer cylinder at time 1,10 and 35 second
Figure 5.6 Net displacement (m) in multilayer cylinder at time 1,10 and 35 second
Figure 5.7 Equivalent plastic strain in multilayer cylinder at time 1,10 and 35 second
Figure 5.8 Equivalent plastic strain in multilayer cylinder over time
Figure 5.9 Total strain in multilayer cylinder over time
Figure 5.10 Total strain difference in multilayer cylinder over time
Figure 5.11 Fatigue life of mold vs thickness of mold
xi
Figure A.1 Density of steel (2 ? % Cr 1% Mo) vs temperature
Figure A.1 Specific heat of steel (2 ? % Cr 1% Mo) vs temperature
Figure A.3 Elastic modulus of steel (2 ? % Cr 1% Mo) vs temperature
Figure A.4 Poisson?s ratio of steel (2 ? % Cr 1% Mo) vs temperature
Figure A.5 Coefficient of expansion of steel (2 ? % Cr 1% Mo) vs temperature
Figure A.6 Thermal conductivity of steel (2 ? % Cr 1% Mo) vs temperature
Figure A.7 Density of copper vs temperature
Figure A.8 Specific heat of copper vs temperature
Figure A.9 Elastic modulus of copper vs temperature
Figure A.10 Poission ratio of copper vs temperature
Figure A.11 Coefficient of expansion of copper vs temperature
Figure A.12 Thermal conductivity of copper vs temperature
Figure C.1 Radial temperature (C) in monolayer cylinder at time 1,10 and 35 second
Figure C.2 Radial stress (Pa) in monolayer cylinder at time 1,10 and 35 second
Figure C.3 Tangential stress (Pa) in monolayer cylinder at time 1,10 and 35 second
Figure C.4 Axial stress (Pa) in monolayer cylinder at time 1,10 and 35 second
Figure C.5 Equivalent stress (Pa) in monolayer cylinder at time 1,10 and 35 second
Figure C.6 Net displacement in monolayer cylinder at time 1,10 and 35 second
Figure C.7 Equivalent plastic strain in monolayer cylinder at time 1, 10 and 35 second
Figure C.8 Equivalent plastic strain in monolayer cylinder over time
Figure C.9 Total strain in monolayer cylinder over time
Figure C.10 Total strain difference in monolayer cylinder over time
Figure D.1 Radial temperature (C) in monolayer cylinder at time 1,10 and 35 second
Figure D.2 Radial stress (Pa) in monolayer cylinder at time 1,10 and 35 second
Figure D.3 Tangential stress (Pa) in monolayer cylinder at time 1,10 and 35 second
Figure D.4 Axial stress (Pa) in monolayer cylinder at time 1,10 and 35 second
Figure D.5 Equivalent stress (Pa) in monolayer cylinder at time 1,10 and 35 second
Figure D.6 Net displacement in monolayer cylinder at time 1,10 and 35 second
Figure D.7 Equivalent plastic strain in monolayer cylinder at time 1, 10 and 35 second
xii
Figure D.8 Equivalent plastic strain in monolayer cylinder over time
Figure D.9 Total strain in monolayer cylinder over time
Figure D.10 Total strain difference in monolayer cylinder over time
Figure E.1 Radial temperature profile in a composite tube with a 6 mm Steel OD layer
? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.2 Radial stress in a composite tube with a 6 mm Steel OD layer ? 18 mm Cu
middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.3 Tangential stress in a composite tube with a 6 mm Steel OD layer ? 18 mm
Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.4 Axial stress in a composite tube with a 6 mm Steel OD layer ? 18 mm Cu
middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.5 Equivalent stress in a composite tube with a 6 mm Steel OD layer ? 18 mm
Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.6 Equivalent plastic strain in a composite tube with a 6 mm Steel OD layer ?
18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.7 Radial temperature profile in a composite tube with a 6 mm Steel OD layer
? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.8 Radial stress in a composite tube with a 6 mm Steel OD layer ? 18 mm Cu
middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.9 Tangential stress in a composite tube with a 6 mm Steel OD layer ? 18 mm
Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.10 Axial stress in a composite tube with a 6 mm Steel OD layer ? 18 mm Cu
middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.11 Equivalent stress in a composite tube with a 6 mm Steel OD layer ? 18 mm
Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.12 Equivalent plastic strain in a composite tube with a 6 mm Steel OD layer ?
18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.13 Radial temperature in a composite tube with a 6 mm Steel OD layer ? 18
mm Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.14 Radial stress in a composite tube with a 6 mm Steel OD layer ? 18 mm Cu
middle layer ? 6 mm Steel ID Layer at 10 second
xiii
Figure E.15 Tangential stress in a composite tube with a 6 mm Steel OD layer ? 18 mm
Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.16 Axial stress in a composite tube with a 6 mm Steel OD layer ? 18 mm Cu
middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.17 Equivalent stress in a composite tube with a 6 mm Steel OD layer ? 18 mm
Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.18 Equivalent plastic strain in a composite tube with a 6 mm Steel OD layer ?
18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.19 Total strain in a composite tube with a 6 mm Steel OD layer ? 18 mm Cu
middle layer ? 6 mm Steel ID Layer at 10 second
Figure E.20 Total strain difference in a composite tube with a 6 mm Steel OD layer ?
18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second
xiv
LIST OF TABLES
Table 3.1 Summary of multilinear isotropic plastic option ??????? 25
Table 5.1 Results for multilayer cylinder analysis....?????????.. 46
Table 5.2 Results available in Appendix C ????????????.. 46
Table 5.3 Results available in Appendix D????????????... 47
Table 5.4 Calculation and validity of fatigue life??????????? 55
xv
NOTATIONS
Variable Definition
{?el} elastic strains
?
density (DENS)
C specific heat (C)
EX elastic modulus (E)
PRXY
poisson ratio (?)
ALPX
coefficient of expansion (?)
T temperature (Temp)
t Time
{?pl} plastic strains
{?tr} trial strain
{?th} thermal strain
{?cr} creep strain
{?sw} swelling strain
{?} total strain
pl
^
?
equivalent plastic strain
{?} Stresses
?e equivalent stress
?y material yield parameter
?m mean stress
^
pl
e
?
equivalent stress parameter
?
plastic multiplier
xvi
Variable Definition
{?} yield surface translation
?
plastic work
C translation multiplier
[D] stress-strain matrix
ET tangent modulus
F yield criterion
N stress ratio
Q plastic potential
{S} deviatoric stress
q heat flux (heat flow)
heat generation rate per unit volume
{L} vector operator
v Velocity
K thermal conductivity
h
f
film coefficient
1
CHAPTER 1: INTRODUCTION
Casting is the process wherein molten material is poured into a mold and
allowed to solidify. Centrifugal casting (Figure 1.1) is one of the casting methods
commonly used to make parts, in particular axi-symmetrical parts, hollow parts and
structures with large diameters such as pipes, pressure vessels and cylindrical liners.
Centrifugal force is exerted on liquid molten metal by spinning the mold, whereupon
liquid metal solidifies on the inner wall of mold producing a sound casting. The mold is
generally a long, hollow tube lined with a centrifugally applied refractory material in a
slurry form, which is subsequently dried and baked.
This study focuses on the permanent cylindrical mold, its thermal management,
thermal fatigue life and dimensional stability. Thermal fatigue is encountered by molds
leading to heat checking and cracking. The time taken to develop the crack determines
the life of the mold. The size and thickness of the mold determines the thermal and
mechanical load it can withstand and cooling rate or cycle time it requires for a given
material. Thus selecting the optimal diameter to thickness ratio or inside diameter to
thickness ratio is a vital step in the customization of a cylindrical mold. A methodology
is developed to determine the optimal diameter to thickness ratio to ensure the
dimensional stability of a permanent cylindrical mold that will be exposed to a cyclic
thermal loading on the inner surface with cooling on the outer surface.
2
Figure 1.1 Schematic of centrifugal casting ( Janco, 1988)
To improve the method used to select the most appropriate mold material and
estimate the location of failure and the life of the mold, a two step approach was used in
this study. The steps are listed below and discussed briefly in the following paragraphs:
1). Finite element analysis
2). Prediction of fatigue life
In the finite element method, a structure is broken down into many small, simple
blocks or elements. The behavior of the individual elements can be described with a
relatively simple set of equations. Just as the set of elements would be joined together to
build the whole structure, the equations describing the behavior of individual elements
are joined into an extremely large set of equations that describes the behavior of the
whole structure. A computer can then be used to solve this large set of equations.
There are many software packages available for finite element analysis. In this
study, the ANSYS FEA package was used. ANSYS is a general purpose finite element
modeling package that can be used to numerically solve a wide variety of mechanical
problems. The centrifugal casting process involves rapid changes of the mold
temperature with time and because of this temperature variation, steep thermal gradients
Mold
3
and high stresses are generated. When stresses pass the elastic limit or yield stress, the
body experiences an irreversible strain (plastic strain). If this load is cyclic it
accumulates enough plastic strain over time, to initiate failure or cracking. A coupled-
field analysis using ANSYS was performed, which takes into account the interaction
(coupling) between two or more fields of engineering. Thermal-stress analysis and
fluid-structure analysis are both examples of coupled field analysis. This analysis
provides detailed results, including information on the deformation, radial stress,
tangential stress, equivalent stress, elastic strain, plastic strain, thermal strain, heat flow,
temperature distribution, thermal gradient and stress ratio. These results were validated
against results reported in literature. Using the results obtained from the finite element
analysis, the life of the mold was estimated using the Coffin-Manson equation. Coffin
and Manson worked independently on the thermal fatigue problem and proposed a
characterization of fatigue life based on plastic strain amplitude (low cycle fatigue)
(Suresh 1991).
Chapter 2 reviews the literature on stress analysis, fatigue life determination,
finite element analysis and methods to improve the life of structures. Chapter 3 gives a
brief overview of the finite element method and the ANSYS features used for modeling
and describe the formulation of the system equation for the features. Chapter 3 also
highlights the method followed by ANSYS to obtain the solution (implementation).
Chapter 4 describes the methodology for this study and gives details of the model and
parameters used for the finite element analysis. Chapter 5 contains the results and
discussion of the multilayer (steel-copper-steel) cylinder as well as results of the
monolayer (steel) cylinder of thickness 0.52? and 0.95?.
4
CHAPTER 2: LITERATURE REVIEW
Thermal fatigue is a severe problem encountered in permanent molds. It results
from the cyclic, rapid and asymmetric heating and cooling experienced by molds and is
one of the major causes of poor quality casting and mold failures. Cyclic temperatures
induce stress and strain conditions due to the thermal expansion or contraction of a
section that is restrained by the surrounding material. Thermal stress develops as the
result of the thermal gradient across a section. This thermal gradient arises because of
the heating and cooling of the surface during the pouring, cooling, ejecting, lubricant
coating and spraying stages of casting cycle. When the molten material is poured into
the mold, the mold surface heats up considerably more than the inner section of the
mold, setting up a steep thermal gradient. As a result, the outer surface expands more
than the inner section, but as the interior is more massive, it prevents the outer surface
from expanding. Plastic deformation will occur when these stresses exceed the yield
strength of the mold material, and with sufficient cycling, micro-cracks will nucleate.
Cracking starts after a number of cycles, but once initiated the cracks will propagate at a
very fast rate. Heat checking also affects the surface, leading to defective castings and
mold failure. This is a thermal fatigue phenomenon resulting from the rapid cyclic
expansion of the mold surface layer as it comes into contact with the molten metal and
5
the constraint of the surface by the much cooler inner portion of the mold (Tanka, et al.
1993).
In order to understand the physics behind plastic deformation, heat checking and
the life of the mold, it is necessary to study the subject in detail. This study applied the
finite element method to analyze a cylindrical mold subjected to a thermal and
mechanical load. This chapter reviews the literature on the stresses generated in
cylindrical shapes subjected to thermal and mechanical loads. The critical factors
affecting the life of cylindrical structures and the methods and techniques used to
analyze the phenomenon are discussed along with the measures to prevent or delay the
crack initiation, so that the life of the mold can be extended.
Duhamel conducted the first study on a long circular cylinder with a
symmetrical temperature distribution about the axis in 1838 and proposed a method to
calculate the elastic stress. Timoshenko and Goodier (1970) also reported the elastic
solutions for several cases of thermal loading of cylindrical shapes. Both these studies
calculated the values of the displacement (U), stresses (?), and elastic strain, but these
solutions are valid only in the elastic region, where the applied stresses do not exceed
the yield strength of the material. Unfortunately in many real-world cases, the applied
stress exceeds the yield strength of the material, at this point the plastic deformation
occurs, and any value of stress calculated based on elasticity theory will be in error.
Wohler conducted systematic investigations of fatigue failure during the period
1852-1869 in Berlin (Suresh 1991). His work led to the characterization of fatigue
behavior in terms of stress amplitude life (S-N) curves, which many fatigue life
predictions used today are based on.
6
To better approximate the stresses and strains when yielding occurs, Mendelson
and Manson (1956) developed a technique which accounts for plastic deformation due
to thermal loading. Their technique consists of deriving strain equations in terms of
temperature and plastic yielding from the equilibrium, compatibility, and stress-strain
relationships for the geometric shape and materials under investigation. The strains are
then calculated by an iterative technique, taking into account the plastic flow of the
material being analyzed. Stresses are then calculated from the general stress- strain
equations.
Coffin (1954) studied cyclic strain and fatigue failure arising from cyclic
thermal stresses. A cyclic temperature was experimentally imposed on a thin, tubular
347 stainless steel (annealed) test specimen subjected to longitudinal constraint. The
studies examined the effect of thermal stress cycling on strain hardening and life to
failure for a fixed mean temperature, effect of degree and kind of previous cold work on
strain hardening and cycles to failure, effect of mean temperature on thermal stress
cycling, effect of period of cycles on cycles to failure and effect of prior strain cycling
on stress-strain characteristics. Coffin concluded that strain hardening is not an
important factor in the problem and discussed the concept of total plastic strain. Coffin
and Manson established a notion that plastic strains are responsible for cyclic damage
(Suresh 1991). They noted that when the logarithm of the plastic strain amplitude,
??
p
/2 was plotted against the logarithm of the number of load reversals to failure, 2N
f
,
a linear relationship resulted for metallic materials, i.e.
cNffp )^2('2/ =???
2-1
7
Where ?
f
? is the fatigue ductility coefficient and c is the fatigue ductility exponent. In
general, ?
f
? is approximately equal to the true fracture ductility ?
f
in monotonic
tension, and c is in the range of -0.5 to -0.7 for most metals.
Hanson (1958) used the Mendelson and Manson technique to compare the
deformation and incremental theories of plasticity in the solution of two boundary value
problems. The deformation theory assumes that the state of stress and strain existing in
the body depends only on the current load. Thus, this theory does not account for the
prior plastic strain due to prior loading and is therefore load-path independent. Using
the incremental theory, the loading and unloading cycle is divided into several small
load increments. Stresses and strains are then calculated based on these small load
increments, and any plastic strains that occur during a load increment or during prior
load increments has a accumulative effect on the stress-strain state. The incremental
theory?s ability to account for plasticity due to small changes in load makes stress strain
calculations load path dependent.
The first problem Hanson addressed concerned the stress analysis of a solid
cylindrical rod of 18-8 stainless steel quenched from 538
o
C. A stress analysis was
performed using both the deformation and incremental theories of plasticity, and the
results of each analysis were compared. The second problem in the study was the stress
analysis, again applying both the deformation and incremental theories of plasticity, of a
thin circular disc when heated on the outside diameter. Hanson found that the results of
the analyses from both the deformation and incremental theories were in agreement
until unloading occurred. Upon unloading however, the values of stress calculated from
8
each theory differed. Hanson?s experimental evidence indicated that the incremental
theory was more predictive of stress-strain state than was the deformation theory.
Manson and Robert (1981) extended the Mendelson and Manson (1956)
technique in order to determine the thermal fatigue life of a rotating solid disc when
subjected to thermal loading. The load conditions were similar to those experienced by
discs in jet engines. Strain range values were calculated based on the mechanical and
thermal loading cycles and a thermal fatigue life was predicted.
Gene Oliver (1988) analyzed and optimized a multilayer tube?s thermal fatigue
life when subjected to cyclic thermal loading on the inside surface. In this study the
thermal load was taken as a constraint and held constant. The thermal gradient, on the
other hand, could be reduced, either by the selection of a high thermal conductivity
material or by thinner shell design. To achieve a lower thermal gradient, enhance the
dimensional stability and increase the abrasion resistance, a three layered sandwich
design was proposed. Total strain equations expressed in terms of temperature and
plastic strain for a cylindrical tube were derived. An equation to determine the radial
temperature profile as a function of time, heat input, interface heat transfer coefficients,
alloy thermal properties and thickness of the layers was also derived. The finite
difference method and incremental theory was used to solve the derived equation for a
multilayered cylinder. The temperature profile obtained for the inside surface of the
hollow cylinder by Oliver (1988) was used in this work to describe the load curve
applied on the inside surface of the cylinder as discussed and plotted in Chapter 3.
Givens (1996) developed a method for a detailed stress/strain analysis of a
coker burner pressurized vessels and the laminated cylindrical shells with multi-
9
directional lay-up angles. To accomplish this method he first developed and tested finite
element models for use with ANSYS program, then obtained the stress/strain response
of burner vessels subjected to internal pressure and laminated shells subjected to an
internal pressure and axial load. The analysis reported here used the cut boundary
displacement method to develop a finite element model and the appropriate submodels
to analyze the entire burner vessel including all major openings. The peak stress was
compared with the far-field stresses determined using the finite element results for
internal pressure simulation and found to be 50% lower than those found from physical
experiments.
Wang (2000) used the finite element method to estimate the life of the dies used
in casting, namely the number of thermal cycles before the die surface reached a failure
level. He performed a thermal-visco-elastic stress analysis using FEM for a 1-D simple
die casting model and a more complex 3-D dumbbell die casting structure. Wang used
MMO software for his analysis and evaluated the effective plastic strain at the die
surface. The point where the maximum effective plastic strain increment occurred was
assumed to be the point of failure in the die. The Coffin-Manson equation was used to
estimate the life of the die, which was composed of H13 steel and Aluminum 380.
Elements with 8 nodes and 20 nodes were developed and compared for accuracy of
result obtained, when the developed elements are used for mesh generation. The
interactions between the thermal and mechanical processes are included by coupling the
thermal and stress analysis.
Sirinterlikci (2000) focused on the thermal and structural issues to develop an
analytical approach, diagnosing the areas susceptible to heat checking in order to
10
estimate the number of shots before the onset of heat checking. He also attempted to
identify the critical factors associated with the structural state (stress/strain) within the
die tool itself. Spray tests were conducted to determine the heat transfer coefficient
during the lubricating spray phase. He examined factors such as the tempering, cyclic
loading, elevated temperature material properties, temperature cycling and influence of
mechanical loading on the structural state within the tooling. His thermal and structural
analysis yielded acceptable results, although the fatigue calculations overestimated the
number of cycles to produce heat checking. However, the structural and fatigue
analyses were successful in the diagnosis of critical tooling areas which are prone to
heat checking. Structural analysis was identified as the most effective approach to
reduce the structural abuse.
To explore the differences between the Tresca and Von Mises yield criteria for
loading-unloading-reloading plastic analyses, Ortega (1993) conducted a parametric
study using finite element method. A program was written for the one dimensional,
elastic plastic analysis of cylinders under positive internal and negative external
pressure. An isotropic hardening condition was assumed. The results obtained from the
analysis were then compared with the commercially available finite element analysis
package - ANSYS 44A. The effect of the order of polynomials used to approximate the
solution was studied, but the results were valid only for the one-dimensional finite
element analysis of thick walled cylinders. Ortega found that the Von Mises yield
criteria is very simple to apply and that the Tresca yield surface grows faster than the
Von Mises surface. Thus, if a cylinder is subjected to pressure loading that causes
yielding, the difference between the two criteria is decreased for the pure shear
11
condition. For uniaxial tension, the Tresca and VonMises criteria is equivalent, but after
loading, these criteria diverge.
Alimi (1989) developed a numerical method to analyze thermal stresses and
displacement in cylindrical shells. The thermal fatigue of a section of a heavy duty
brake drum was studied to illustrate the application of the method developed. This study
also included a custom program written in Fortran IV. Alimi proposed the use of a three
layered (cast iron- copper- cast iron) composite tube rather than brake drums made of
pure cast iron. This study demonstrated that the multilayer cylinder may reduce the
thermal fatigue problem in brake drums and concluded that stainless steel would be a
better material to use for outer layer, although the cost of stainless steel is much higher
than that of cast iron.
Sirkis (1988) presented a two-dimensional hybrid experimental-numerical
technique for elastic-plastic stress analysis, combining two techniques: the boundary
element method and image processing. ?Displacement Pattern Matching?, which is a
pattern recognition scheme, determines the boundary conditions to be used in an elastic-
plastic boundary element code. This is a very interesting study because boundary
conditions are acquired from the actual specimen by comparing digitized images with a
double exposure format. The results obtained using this method compared well with an
ANSYS analysis and many experimental solutions like a perforated strip tensile test, V-
notched specimen tensile test. However, this method works well only with static
systems, i.e. when motion of the body is negligible. The specimen size used in this
study was small, making it easier to process the image but if the sample size is large as
12
in a mold, which may be up to 30 feet long with a radius of between 8 inches to 60
inches, it is often very difficult to process the image and obtain the boundary conditions.
Hah (1997) studied the most proficient modeling techniques for moment
resisting frames by applying nonlinear finite element techniques and the concept of
parallel elasto-plastic material modeling into a nonlinear finite element analysis. The
goal of this dissertation was to make the commercial nonlinear finite element
application packages more practical, user friendly and economical. A simple cantilever
beam with a concentrated tip load that is capable of producing a plastic hinge at the
cantilever support was used as an example in the study. A special set of moment
resisting frames, used to make up the skeleton of the structure to support a two floor
facility was also examined. Hah concluded that to account for dynamic effects in the
structure behavior, the number of elements should be sufficient to account for mass
distribution. Parallel processing is advantageous for such problems because it reduces
the effort needed to add the elements that account for material nonlinearity. Plates, shell
buckling and snap-through problems may also benefit from parallel element techniques.
The auto-stepping algorithm, used by most FEA packages proved to be very useful in
the analysis.
Ruan (1990) investigated the solidification stage of casting, to produce results
that may be useful in the design of solidification processes with a specific freezing front
motion. Ruan presented a two-dimensional finite element model of the heat transfer and
thermo mechanical behaviors associated with the solidification process, that is very
useful in tracking the solid/liquid interface, allowing the and thermal and mechanical
conditions to be modeled with ease. This work may be very useful in obtaining the
13
thermal load curve on the mold in the current study. However, while modeling the
solidification process, the effect of air-gap formation was neglected, thus affecting the
heat transfer and thermal stress.
White (1997) modeled multilayer thin film structures using analytical solution
and hybrid finite element methods. Sharp edges and multimaterial wedges were
included in this treatment. The hybrid finite element method formulates a special
element containing a singularity to be employed in the region of the free edges, while
regions beyond the free edges are represented by conventional elements. Combining
classical techniques and finite element solutions helps to obtain the stress field with the
least number of elements. Multilayer Structures (MLS) were used to provide solutions
for both steady state and transient problems. MLS also assisted in predicting the
location of crack initiation and adhesion failures between layers. A structure consisting
of a thin film of AIN (substrate) of 0.64 mm (25mil) thickness, BCB (dielectric) of
10?m thickness, and copper (metallization) of thickness 5?m was modeled using MLS.
The program developed for this study could be used to study the thin oxide layer
formed on the inside surface of a cylindrical mold.
Okono (1978) used the finite element method to conduct a heat transfer analysis
in solid propellent rocket motors modeled as long, hollow, circular cylinders subjected
to a randomly varying temperature. For this analysis, the nonhomogeneous cylindrical
structure was subdivided into homogeneous concentric regions and treated as
thermorheologically simple viscoelastic rings. A probabilistic approach was used for the
description of strength and induced thermal stress, since a random thermal response and
a random material strength were produced by the random environmental temperature.
14
Using the assumption of a plane strain condition, the end effects were neglected. The
probability of failure was calculated based on probability distributions of material
strength and induced thermal stress using Maximum Stress Theory and the Maximum
Strain Theory. A normal distribution was hypothesized and a chi-square test was
performed to verify the normality assumption. The estimation of the service life was
based on the assumption that the annual thermal history would be repetitive. Poisson?s
ratio was taken to be 0.49 which implies that material is slightly compressible. In this
investigation, tangential stress was found to have a profound effect on the failure of the
motor, although the method took a long time to conduct the stress analysis and to obtain
the data needed for a meaningful failure prediction. Mechanical and thermal loads could
not be combined by this method as it is only good for thermal loads and hence was
limited in application.
Numerous studies have examined different aspects of the casting process.
Numerical solutions have been obtained from custom programs, written in different
computer languages, to analyze a cylinder subjected to thermal and mechanical loads. In
order to incorporate new parameters the programs developed are generally very difficult
to modify and moreover the codes are not widely available. However the thermal
behavior of a cylindrical mold can be conveniently analyzed using the finite element
method. In the past, the finite element method has been used to study pressure vessels
(Given 1996), dies (Wang 2000), solid propellants in rocket motors (Okono 1978) and
brake drums (Alimi 1989), all of which involved commercially available finite element
modeling software. There are many commercial finite element modeling software
packages available which can be used to analyze the permanent cylindrical mold, such
15
as ANSYS, ABACUS, NASTRAN and ALGOR to name a few. Finite element analysis
of the cylindrical mold using any one of the commercially available packages will
greatly simplify the otherwise complex computer programming need for this project.
This study will make an effort to verify that the available finite element modeling
packages have the capability to model complex material behavior with reasonable
accuracy.
16
CHAPTER 3: ANSYS 9.0 FEATURES AND SYNTAX
ANSYS is a finite element modeling and analysis software package that can be
used to analyze complex problems in mechanical structures, thermal processes,
computational fluid dynamics, magnetics and electrical fields, to mention just a few of
its applications. The capability to automate common tasks or even build a model in
terms of the parameters using ?ANSYS Parametric Design Language (APDL)?, makes
it more flexible. APDL also encompasses a wide range of features such as repeating a
command, macros, choice of parallel processing, if-then-else branching, do-loops, and
scalar, vector and matrix operations. ANSYS provides a rich graphics capability that
can be used to display the results of the analysis on a high-resolution graphics
workstation. This chapter, which is mainly based on the material provided in ANSYS
9.0 documentation (2004), summarizes the equations used by ANSYS features (used in
this study like rate independent plasticity, multilinear isotopic hardening (MISO),
implementation and thermal boundary conditions) to formulate the system equation.
This chapter also gives a brief overview of finite element method.
3.1 Finite Element Method
The finite element method is a numerical procedure for solving physics
problems governed by a differential equation as an energy theorem. It has two
characteristics that distinguish it from other numerical procedures (Huebner et al. 2001):
17
1). The method utilizes an integral formulation to generate a system of algebraic
equations.
2). The method uses continuous piecewise smooth functions (interpolation) for
approximating the unknown quantity or quantities.
The finite element method can be subdivided into four basic steps. These steps
are listed here (Huebner et al. 2001).
1). Discretization of the region: This includes the location and numbering of the
nodes, as well as specifying their co-ordinate values.
2). Specification of the approximation equation (interpolation function): The
order of the approximation, whether linear or quadratic, must be written in terms of the
unknown nodal values. An equation is written for each element. It is important to
choose a proper interpolation function, which satisfies certain convergence
requirements. Polynomials are the preferred approximating functions for the following
reasons
? A polynomial of infinite order corresponds to the exact solution, but it is
sufficient to obtain approximate solution using finite order polynomials.
? It is easy to perform differentiation and integration with polynomials, hence it is
easier to formulate a model by using a polynomial function, which is highly
compatible with the computer.
? It is possible to increase the accuracy of the solution by simply increasing the
order of the polynomial.
3). Development of the system of equations: The weighting function for each of
the unknown nodal values is defined and the weighted residual integral is evaluated.
18
This generates one equation for each unknown nodal value. In the potential formulation,
the potential energy of the system is written in terms of the nodal displacement and then
is minimized, which in turn gives one equation for each of the unknown displacements.
4). Calculation of the quantities of interest: These quantities are usually related
to the derivative of the parameter and include the stress components, heat flow and fluid
velocities.
3.2 ANSYS
3.2.1 Basic flow of ANSYS
? Pre-Processor
1) Defines material properties and their behavior.
2) Defines the geometry and the important features.
3) Handles discretization (meshing).
4) Defines load and boundary conditions.
? Processor (Solution): This component handles the analysis and
preprocessing of data, which involves the computation of element properties,
assemblage of elements and solution of equations of equilibrium.
? Post-Processor: Reviewing the result is probably the most important step in
the analysis, because it helps the user to understand the affect of an applied load on the
design, the quality of the finite element mesh, and so on. The post-processor accepts the
results of the analysis, computes stress and handles post-processing of the results
through the generation of graphs, tables and pictures.Two post-processors may be used
to review the results: the general post-processor (POST 1), and the time-history post-
processor (POST 26). The general post-processor allows the user to review the results
19
over the entire model at specific load steps and substeps or at specific time-points or
frequencies. The time-history post-processor allows the user to review the results over
time.
3.2.2 Analysis capabilities and range of applications
ANSYS is capable of performing structural, thermal, fluid, electromagnetic and
coupled field (CF) analyses. The thermal and structural analysis, which is used in this
study is described briefly.
? Structural Analysis
This type of analysis is the most common application of FEA and is used
primarily for mechanical and civil engineering applications. Structural analysis is
possible in the many areas, two of them are described briefly in this chapter.
- Static Analysis: This ignores the effect of time varying loads, although it can
include a time-varying load by approximating it as a static equivalent load. Static
analysis is used to determine displacements, stresses, strains and forces induced by
loads that do not cause significant inertia and damping effects.
- Transient Dynamic Analysis: This analysis can be used to determine the time-
varying displacements, strains, stresses, and forces in a structure as it responds to any
combination of static, transient, and harmonic loads. The time scale of the loading is
such that the inertia or damping effects are considered to be important. The basic
equation of motion solved by a transient dynamic analysis is
(M){u&& } + (C){u& } + (K){u} = {F(t)} (3-1)
where:
(M) = mass matrix
20
(C) = damping matrix
(K) = stiffness matrix
{u&& } = nodal acceleration vector
{u& } = nodal velocity vector
{u} = nodal displacement vector
{F(t)} = load vector
? Thermal Analysis
This type of analysis is used to calculate the thermal gradients, temperature
distribution, heat transfer and thermal flux of an object. The analysis can be performed
using conduction, convection and radiation heat transfer modes. Here a thermal analysis
is followed by stress analysis in order to calculate the thermal stresses caused by
thermal expansions or contractions. The analysis can be performed in the following two
areas.
- Steady-State Thermal Analysis: A steady-state thermal analysis calculates the
effects of steady thermal loads on a system or component. Steady-state thermal analysis
can be used to determine temperatures, thermal gradients, heat flow rates, and heat
fluxes in an object that are caused by thermal loads that do not vary over time. A steady
state condition allows varying heat storage effects over time to be ignored.
- Transient Thermal Analysis: Transient thermal analysis is used to determine
temperatures and other thermal quantities that vary over time. Loads in a transient
analysis are functions of time. Transient thermal analysis can be used to determine
temperatures, thermal gradients, heat flow rates, and heat fluxes in an object that are
caused by thermal loads that vary over time. To specify time-dependent loads, the
function tool can be used to define an equation or function describing the curve and
21
then the function is either applied as a boundary condition, or the load-versus-time
curve can be divided into load steps. In this study a time dependent load curve
(described in chapter 4) is specified.
? Coupled-field analysis
A coupled-field analysis is an analysis that takes into account the interaction
(coupling) between two or more disciplines (fields) of engineering. A thermal-stress
analysis, for example, handles the interaction between the structural and thermal fields:
it can be used to solve for the stress distribution due to applied temperature, or vice
versa. Other examples of coupled-field analyses are piezoelectric analysis, thermal-
electric analysis, and fluid-structure analysis.
3.2.3 Element library
The ANSYS element library consists of more than 100 different element
formulations or types. An element type is identified by a name (8 characters maximum).
An element is selected from the library for use in the analysis by selecting from a drop
down menu or inputting its name in the element type command. Elements are connected
to the nodes in a specific sequence and orientation. This connectivity can be defined by
automatic meshing, or may be input directly by the user. Each element type has a
degree of freedom set, which constitute the primary nodal unknowns to be determined
by the analysis. These may be displacements, rotations, temperatures, pressures,
voltages, etc. Derived results, such as stresses, heat flows, etc., are computed from these
degree of freedom results.
The element used for this analysis was PLANE13. This element has a two-
dimensional structural and thermal field capability with coupling between the fields. It
22
also has magnetic, electrical and piezoelectric capability. This element is defined by
four nodes with four degrees of freedom per node. Figure 3.1 shows the geometry of the
element. U
x
, U
y
, Temp, A
z
and volts can be used as degree of freedom for the nodes, but
as discussed earlier only four of these degrees of freedom can be used per node. It can
have convection or heat flow as surface load, but not both. If both are applied on the
same surface, convection supersedes heat flow. The element will accept temperature
and heat generation as the body load. A special feature of this element is that it takes
care of large deflection, large strain, stress stiffening and birth and death. The
PLANE13 element can behave as plane strain, plane stress or axisymmetric element.
3.2.4 Structures with Material Nonlinearities
Material nonlinearities arises due to the nonlinear relationship between stress
and strain, that is, the stress is a nonlinear function of the strain. The relationship is also
path dependent (except for the case of nonlinear elasticity and hyperelasticity), so that
the stress depends on the strain history as well as the strain itself. The program can
account for the material nonlinearities like rate-independent plasticity, creep,
I
K
L
J
Y
X
1
2
3
4
Figure 3.1: Geometry of PLANE13 element
23
hyperelasticity and viscoealsticity. In this study, rate independent plasticity, which is
characterized by the irreversible straining that occurs in a material once a certain level
of stress is reached, is used to account for material nonlinearities. The input for this
feature is discussed in Chapter 4 and the formulation is described further in this chapter.
For the case of nonlinear materials, the elastic strain is given by:
{?
el
}= ?- {?
th
}- {?
pl
}- {?
cr
}- {?
sw
}
(3?2)
where:
?
el
= elastic strain vector
? = total strain vector
?
th
= thermal strain vector
?
pl
= plastic strain vector
?
cr
= creep strain vector
?
sw
= swelling strain vector
And the total strain is :
{?
tot
}= {?
pl
}+ {?
th
}+ {?
cr
} (3?3)
where:
?
tot
= component total strain
3.2.5 Rate-independent plasticity
Rate-independent plasticity is characterized by the irreversible straining that
occurs in a material once a certain level of stress is reached. Plastic strains are assumed
to develop instantaneously, independent of time. Plasticity theory provides a
mathematical relationship that characterizes the elastoplastic response of materials.
24
There are three components involved in the rate-independent plasticity theory: the yield
criterion, flow rule and the hardening rule. These will be discussed in detail below:
? Yield Criterion
The yield criterion determines the stress level at which yielding is initiated.
For multi-component stresses, this is represented as a function of the individual
components, f({?}), which can be interpreted as an equivalent stress ?
e
:
?
e
=f({?}) (3?4)
where:
{?} = stress vector
When the equivalent stress is equal to a material yield parameter ?
y
,
F({?})=?
y
(3?5)
the material will develop plastic strains. If ?
e
is less than ?
y
, the material is elastic and
the stresses will develop according to the elastic stress-strain relationship.
? Flow Rule
The flow rule determines the direction of plastic strain and is given by:
?
?
?
?
?
?
?
?
=
?
Q
?}{d?
pl
(3?6)
where:
? = plastic multiplier (which determines the amount of plastic strain)
Q = a function of stress termed the plastic potential (which determines the
direction of plastic strain)
25
? Hardening Rule
The hardening rule describes the change in the yield surface with progressive
yielding, so that the conditions (i.e. stress states) for subsequent yielding can be
established. Two hardening rules are possible in ANSYS: work (or isotropic)
hardening and kinematic hardening. In work hardening, the yield surface remains
centered about its initial centerline and expands in size as the plastic strains develop.
For materials with isotropic plastic behavior, this is termed isotropic hardening and
is shown in Figure 3.2 (a). Kinematic hardening assumes that the yield surface
remains constant in size and the surface translates in stress space with progressive
yielding, as shown in Figure 3.2 (b). The ANSYS program provides seven options
with which to characterize different types of material behaviors. In this study,
multilinear isotropic hardening rule ?MISO? (Figure 3.3) was used to characterize
the material behavior (refer to appendix A and appendix B for the input files for
ANSYS). Table 3.1 summarizes the yield criterion, flow rule and hardening rule for
multilinear isotropic plasticity option.
Name Multilinear isotropic hardening
Yield Criterion Von Mises/ Hill
Flow Rule Associative
Hardening Rule Work
Material Multilinear
Table 3.1: Summary of multilinear isotropic plastic option
26
3.2.6 Plastic strain increment
If the equivalent stress computed using elastic properties exceeds the material
yield, then plastic straining must occur. Plastic strains reduce the stress state so that it
satisfies the yield criterion given in, Equation 3-5. Based on the theory discussed earlier,
the plastic strain increment is calculated. The hardening rule states that the yield
criterion changes with work hardening and/or with kinematic hardening.
Incorporating these dependencies into Equation 3-5, and rearranging, we get:
(a) Isotropic work hardening (b) Kinematic hardening
?
1
?
2
?
1
?
2
Initial yield surface
Subsequent yield
surface
Subsequent yield
surface
Initial yield surface
Figure 3.2: Types of hardening rules
?
max
?
1
?
2
2 ?
max
?
?
Figure 3.3 : Stress ?strain curve for MISO behavior
27
F({?},?, {?})=0 (3?7)
where:
? = plastic work
{?} = translation of yield surface
The plastic work is the sum of the plastic work done over the history of loading
and expressed as:
{ } [ ]{ }
?
=
plT
d?M??
(3?8)
where:
[]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
200000
020000
002000
000100
000010
000001
M
The translation (or shift) of the yield surface is also history dependent and is
given as:
{ } { }
?
=
pl
d?C?
(3?9)
where:
C = material parameter
Equation 3-7 can be differentiated so that the consistency condition is:
[]{} []{}0d?M
?
F
d?
?
F
d?M
?
F
dF
TT
=
?
?
?
?
?
?
?
?
+
?
?
+
?
?
?
?
?
?
?
?
= (3?10)
from Equation 3-8 differential of plastic work is given as:
{ } [ ]{ }
plT
d?M?d? =
(3?11)
28
and from Equation 3-9 differential of translation is given as:
{ } { }
pl
d?Cd? = (3?12)
Then Equation 3-10 becomes
[]{} {}[]{} []{}0d?M
?
F
Cd?M?
?
F
d?M
?
F
pl
T
plT
T
=
?
?
?
?
?
?
?
?
+
?
?
+
?
?
?
?
?
?
?
?
(3?13)
The stress increment can be computed via the elastic stress-strain relations
{ } [ ]{ }
pl
d?Dd? = (3?14)
Where D is the stress-strain matrix
with
{ } { } { }
plel
d?d?d? ?= (3?15)
since the total strain increment can be divided into an elastic and plastic part.
Substituting Equation 3-6 into Equation 3-13 and Equation 3-15 and combining
Equation 3-13, Equation 3-14, and Equation 3-15 yields
[][]{}
{}[] [] [][]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
Q
DM
?
F
?
Q
M
?
F
?
Q
M?
?
F
d?DM
?
F
?
TT
T
T
(3?16)
The size of the plastic strain increment is therefore related to the total increment
in strain, the current stress state, and the specific forms of the yield and potential
surfaces. The plastic strain increment is then computed using Equation 3-6:
{ }
?
?
?
?
?
?
?
?
=
?
Q
?d?
pl
(3?17)
29
3.2.7 Implementation
A Euler backward scheme is used to enforce the consistency condition in Equation
3-10. This ensures that the updated stress, strains and internal variables are all on the
yield surface. The algorithm proceeds as follows:
1. The material parameter ?
y
is determined for this time step (i.e. the yield stress at
the current temperature).
2. The stresses are computed based on the trial strain {?
tr
}, which is the total strain
minus the plastic strain from the previous time point.
{ } { } { }
pl
1tt
pl
t
???
?
?= (3?18)
where the superscripts refer to the time point. The trial stress is
{?
tr
}=[D]{?
tr
} (3?19)
3. The equivalent stress ?
e
is evaluated at this stress level by Equation 3-4. If ?
e
is
less than ?
y
the material is elastic and no plastic strain increment is computed.
4. If the stress exceeds the material yield, the plastic multiplier (?) is determined as
explained earlier.
5. Plastic strain increment (??
pl
) is computed via Equation 3-17.
6. The current plastic strain is updated
{ } { } { }
plpl
1n
pl
t
??? ??=
?
(3?20)
where:
{ }
pl
t
? = current plastic strains
and the elastic strain computed
{ } { } { }
pltrel
??? ??=
(3?21)
where:
30
?
el
= elastic strains
The stress vector is:
{?}=[D]{?
el
} (3?22)
where:
{?} = stresses
7. The increments in the plastic work ?? and the center of the yield surface {??}
are computed via Equation 3-11 and Equation 3-12 and the current values are
updated
????
1tt
?=
?
(3?23)
and
????
1tt
?=
?
(3?24)
where the subscript t-1 refers to the values at the previous time point. For output
purposes, an equivalent plastic strain (
pl
^
? ), equivalent plastic strain increment
(
pl
^
? ), equivalent stress parameter
pl
e
^
? are computed. The equivalent plastic
strain increment is given by:
{}[]{}
2
1
pl
T
pl
pl
^
??M??
3
2
??
?
?
?
?
?
?
=
(3?25)
Accumulated plastic strain is given by:
pl
^
pl
^
?? ??=
(3?26)
31
3.2.8 Multilinear Isotropic Hardening
As explained earlier the multilinear isotropic hardening option uses the von
Mises yield criterion with the associated flow rule and isotropic (work) hardening.
The equivalent stress Equation 3?4 is:
{}[]{}
?
?
?
?
?
?
= SMS
3
2
?
T
e
(3?27)
where {s} is the deviatoric stress represented as:
{} {}
??
T
m
000111??S ?= (3?28)
When ?
e
is equal to the current yield stress ?
k
the material is assumed to yield.
The yield criterion is:
{}[]{} 0SMS
3
2
F
k
T
=?
?
?
?
?
?
?
= ?
(3?29)
For work hardening, ?
k
is a function of the amount of plastic work done. For the
case of isotropic plasticity assumed here, ?
k
can be determined directly from the
equivalent plastic strain of Equation 3?29 and the uniaxial stress-strain curve in Figure
3.4. Here, ?
k
is output as the equivalent stress parameter. For temperature-dependent
curves with the MISO option, ?
k
is determined by temperature interpolation of the input
curves after they have been converted to stress-plastic strain format.
3.1.9 Thermal Analysis
The first law of thermodynamics states that thermal energy is conserved.
Applying this to a differential control volume:
{} {} {} {} qqT^LTLT^v
t
T
?c &&&=+
?
?
?
?
?
?
+
?
?
(3-30)
where:
32
? = density (input as DENS in ANSYS)
C = specific heat (input as C in ANSYS)
T = temperature (T(x,y,z,t))
T = time
{q} = heat flux vector (output as TFX, TFY, and TFZ)
q&&& = heat generation rate per unit volume
{} =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
z
y
x
L
{v}= =
?
?
?
?
?
?
?
?
?
?
Vz
Vy
Vx
Velocity vector for mass transport of heat.
Next, Fourier's law is used to relate the heat flux vector to the thermal gradients:
{ } [ ]{ }TLDq ?=
(3?31)
?
1
?
2
?
3
?
4
?
2
?
3
?
4
?
5
?
1
?
5
?
?
Figure 3.4: Uniaxial Behavior
Strain
Stress
Vector operator
33
where:
?
?
?
?
?
?
?
?
?
?
=
zz
yy
xx
K00
0K0
00K
D
K
xx
, K
yy
, K
zz
= conductivity in the element x, y, and z directions, respectively
Combining Equation 3-30 and Equation 3-31,
{}{} {} []{} qT)LD(LTLv
t
T
?c
TT
&&&+=
?
?
?
?
?
?
+
?
?
(3?32)
Expanding Equation 3-32 to its more familiar form:
?
?
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
?
?
?
+=?
?
?
?
?
?
?
?
+
?
?
+
?
?
+
?
?
X
T
K
XX
T
K
XX
T
K
X
q
Z
T
V
Y
T
V
X
T
V
t
T
c
XXXZYX
&&&?
(3?33)
It will be assumed that all effects are in the global Cartesian system.
Three types of boundary conditions are considered.
1. Specified temperatures acting over surface S
1
:
T=T* (3?34)
where T* is the specified temperature.
2. Specified heat flows acting over surface S
2
:
{q}
T
{?}= -q*
(3?35)
where:
{?} = unit outward normal vector
Q* = specified heat flow
3. Specified convection surfaces acting over surface S
3
(Newton's law of cooling):
{q}
T
{?}= - h
f
(T
S
-T
B
) (3?36)
where:
34
h
f
= convective film coefficient
T
B
= bulk temperature of the adjacent fluid
T
S
= temperature at the surface of the model
Combining Equation 3-31 with Equation 3-32 and Equation 3-36 yields
{ } [ ]{ } *qTLD?
T
= (3?37)
{ } [ ]{ } ( )TThTLD?
Bf
T
?= (3?38)
Multiplying Equation4-32 by a virtual change in temperature, integrating
over the volume of the element, and combining with Equation 3-37, and
Equation 3-38, yields:
{}{} {} []{}()()
() ()
?? ?
++
=
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
+
?
?
?
?
d(vol)q?TST)d(T?ThSd*?Tq
voldTLD?TLTLV
t
T
?c?T
vol3Bf
3
S2
2
S
TT
vol
&&&
(3?39)
where:
Vol = volume of the element
?T = an allowable virtual temperature =?T(x,y,z,t))
35
CHAPTER 4: METHODOLOGY, MODELS AND PARAMETERS
This chapter describes the overall methodology and then details the model
including its geometry, load, boundary conditions, material properties, assumptions and
solution options.
4.1 Methodology
The finite element method is used to deal with complex structures that are
difficult to solve analytically. The finite difference method has been used in the past to
study permanent cylindrical molds but the achievements and developments in the finite
element technique have made it preferable for large-scale computations. The finite
difference method is more suitable for moving boundary conditions, whereas the finite
element method is suitable for solid structures such as permanent cylindrical mold, dies,
pressure vessels and heat exchangers. The finite difference method uses an orthogonal
mesh to represent the geometry. It is very difficult to model a thin-walled cylinder and
obtain an acceptable solution, with a reasonable number of elements. Even the use of
thousands of elements may not improve the analysis results. In contrast, the mesh in the
finite element method can be non-orthogonal, generated by linear, triangular or
quadrilateral elements. A thermal stress analysis of the mold can be performed with
finite element method with no modification of the mesh. Thus, the finite element
method is best suited for studying the mold. The finite element analysis used in this
36
study will provide information on the temperature distribution, stresses and strains in
the mold. Only a single cycle of the thermal load is analyzed. By itself, the use of a
single thermal cycle would not give adequate information to assess the life of the mold
but the finite element analysis produces a result that is sufficient to estimate the life of
mold by applying the Coffin-Manson equation. The Coffin-Manson equation uses
plastic strain amplitude to estimate the life of a mold as discussed in Chapter 2. The
plastic strain amplitude was obtained from the simulation results and substituted in the
Coffin-Manson equation in order to calculate the life of the mold. The results obtained
were validated against empirical industry data (i.e. the number of parts produced before
the mold is discarded) and the mathematical model?s results reported by Oliver (1988).
It is always economically desirable to improve the life of the mold. A sandwich
(multilayer) model of steel-copper-steel has been suggested in the literature as a way to
reduce the thermal gradient and stresses in mold or to improve the life of the mold
(Oliver 1988, Alimi 1989). This multilayer cylindrical model was analyzed using the
above mentioned finite element method and the results are discussed in Chapter 5.
4.2 Geometry and Boundary Conditions
The solid geometries of an actual mold and a multilayer cylinder are shown in
Figures 4.1 and 4.2. Two models have been developed using an actual industrial
geometry and a multi-layered cylinder of steel-copper-steel, which is the main focus of
this study. This combination can lower the thermal gradient, provide dimensional
stability and increase abrasion resistance. This design may be thought of as the thermal
equivalent of the mechanical ?I beam? (Oliver 1988). The outer and inner layer of steel
37
provides strength and stiffness, while the middle layer of copper provides a high
conductivity path for heat flow.
For this analysis, the end effects are neglected. As shown in Figure 4.1 there are
protruding and curved shapes on both ends of the mold, although most of the mold is
cylindrical in shape. The effect of the ends (which are curved and protrude) will be
different than that of the long symmetrical cylindrical shape. Even for a long
symmetrical cylinder, the stresses at the ends are 25% higher than at points away from
the end (Ugura 1999). However the mold must be discarded if the crack develops in the
center, i.e. away from the ends. Figure 4.2 represents the 2-D geometry which was built
in ANSYS for the analysis, which is a small part of cross section A-A shown in Figure
4.1. The inner gray layer is steel (thickness 6 mm), the middle orange layer is copper
(thickness 18 mm) and the outer gray layer is steel (thickness 6 mm) in Figure 4.2. This
is a two dimensional representation of a multilayered cylinder. The values for the
cylinder dimensions noted in Figure 4.3, are:
R1 = 9.84 in, cylinder inside radius.
R2 = 10.07 in, outside radius of steel inner layer and inside radius of middle
copper layer.
R3 = 10.69 in, outside radius of middle copper layer and inside radius of outer
steel layer.
R4 = 11.02 in, cylinder outside radius.
t = 1.18 in, thickness of cylinder.
38
Figure 4.2: 2-D drawing of multilayer cylinder
Steel Steel Copper
AA
A
a
Section A-A
Figure 4.1: 30.00 foot long mold
A
A
39
The thermal constraints imposed are:
1. The initial temperature of cylinder is 25
o
C.
2. The load is applied on the inner surface of the cylinder. The applied load curve
is obtained by heating the inside surface of the cylinder by a flame for 10
seconds, after which the flame is turned off (Oliver 1988). The body is cooled
by water at 25
o
C on the outer surface for next 25 seconds. The load curve
obtained with this method is shown in Figure 4.4. The applied load curve
replicates the phenomenon of pouring of the molten material (when the
temperature of the mold inner surface rises considerably) and solidification of
the material (when the temperature of the mold inner surface decreases).
3. The heat transfer coefficient on the outside surface is a function of the surface
temperature. If the outer surface temperature is below 100
o
C, the heat transfer
coefficient is 1575 watts/m
2
, but if the surface temperature exceeds 100
o
C then
the heat transfer coefficient is 3150 watts/m
2
.
R1 R2
R3
R4
t
Figure 4.3: Cross section of cylinder showing dimensions
Steel
Copper
Steel
40
4. The total cycle time is 35 sec. First 10 sec is the heating phase and last 25 sec is
the cooling phase.
The structural constraints are:
1. Plane strain.
2. Symmetrical boundary condition.
Figure 4.5 shows the load and boundary conditions applied to a multilayer
cylinder. The inner layer is subjected to the thermal load curve shown in Figure 4.4,
while the outer layer is subjected to convection cooling. The upper and lower layer are
subjected to symmetrical boundary conditions. The thermal conductivity for steel and
copper is defined in terms of the material properties and the mode of heat transfer is
assumed to be mainly conduction and convection.
Load Curve
0
100
200
300
400
500
600
700
800
10203040
Time(s)
Te
m
p
(
F
)
Temp (C)
Figure 4.4: Load curve
41
Figure 4.5: Load and boundary conditions
4.3 Material Properties
The material properties are a function of temperature except for Poisson?s Ratio,
which is assumed to be constant at all temperatures. Material number 1 is steel (2? %
Cr 1% Mo) and material number 2 is oxygen free high conductivity (OFHC) copper.
The material properties used are in SI units (MCS) and are listed below:
1. Density
2. Specific heat
3. Linear isotropic property
? Elastic modulus
? Poisson ratio
4. Thermal expansion
5. Thermal conductivity
Steel Copper Steel
HTC
Symmetry BC
Load Applied
42
6. Multilinear isotropic property (Stress- strain curve)
Appendix A contains a complete list of the material properties used in the
analysis for steel and copper. Figures 4.7 and 5.8 are representative plots for material
properties (refer to Appendix A, Figure A.1 ? A.12 for other material properties).
Figures 4.6 and 4.7 show the stress-strain curves (input for multilinear isotropic
hardening) for steel and copper, respectively.
Fatigue life, defined as number of times molten material can be poured before
crack initiates, was calculated for the inner surface, which is subjected to the highest
load. As discussed earlier, this calculation is based on the Coffin-Manson equation
(discussed in Chapter 2). In order to use this equation, fatigue ductility coefficients (the
failure strain for a single load reversal and it is an empirical constant) and fatigue
ductility exponent (slope of SN curve and it is also an empirical constant) are required,
which are listed as below (Suresh 1991):
?
f
(fatigue ductility coefficient) =0.73
C (fatigue ductility exponent) = -0.62
4.4 Assumptions
1. Plane strain
2. Heat is transferred by conduction and convection. Radiation effects are
neglected, because the outer surface is surrounded by a water jacket.
3. One thermal cycle lasts 35 sec, consisting of 10 seconds of heating followed by
25 second of cooling.
4. There is no heat abstraction by the air within the hollow cylinder.
5. No oxide layer is present on the inner surface of the mold.
43
Figure 4.6: Stress-strain curve of steel (2? % Cr 1% Mo) at different temperature
Figure 4.7: Stress-strain curve of oxygen free high conductivity copper at
different temperature
44
6. There is no gaseous gap between the inner surface of the mold and the molten
material.
7. Material properties are a function of temperature.
8. Contact surface (i.e. the interface between steel and copper) is always bonded.
9. Fatigue life is deterministic
4.5 Solution options
In order to obtain the solution, the following settings were invoked in ANSYS.
The method used to introduce the settings is detailed earlier and also in Chapter 3. For
example, to incorporate plastic material properties the stress strain curve is defined as
input for multilinear isotropic hardening (MISO).
1. Problem Dimensionality ??????...2-D
2. Degree of Freedom?????????U
x
U
y
Temp
3. Analysis Type???????????Transient
4. Nonlinear Geometry Effects????.? On
5. Units ?????????????.? SI (MCS)
6. Plastic Material Properties Included ??Yes (MISO)
7. Time at End of Load Step ?????? 35 sec
8. Time Step Size ??????????. 0.25 sec
45
CHAPTER 5: RESULTS AND DISCUSSION
Finite element models of a multilayer cylindrical mold (steel?copper?steel) and
two monolayer cylindrical molds (steel) were subjected to the load and boundary
condition detailed in Chapter 4. The finite element analysis was performed with
ANSYS Release 9.0. Analysis data was obtained for every time step but plotted only at
critical time points i.e. time = 1 second (start of cycle), time = 10 seconds (end of
heating phase and start of cooling phase) and time = 35 seconds (end of cycle). The
time 1,10 and 35 seconds are also considered because the validation results are only
available for the above mentioned time points.
The plots for the multilayer cylindrical mold ? steel 2 ?% Cr 1% Mo (thickness
= 0.236 in), oxygen free high conductivity copper (thickness = 0.708 in), steel 2 ?% Cr
1% Mo (thickness = 0.236 in) ? are produced and discussed in this chapter. The results
shown in this Chapter and the plots in Appendices are summarized in Tables 5.1, 5.2
and 5.3. The results produced and the validation status of each result is shown in the
above mentioned tables. For example, X (Y) means that the result for the corresponding
entity is mentioned in the respective appendix and it has also been validated against the
result in Appendix E. Complete results from ?An analytical method to optimize the
thermal fatigue life of multilayered cylindrical shells? by Gene Oliver (1988) are given
in Appendix E. As mentioned earlier, Oliver?s study was used for validation.
46
Time
Sr. No. Analysis Result
1 second 10 second 35 second
1). Radial temperature profile X (Y) X (Y) X (Y)
2). Radial stress X (Y) X (Y) X (Y)
3). Tangential stress X (Y) X (Y) X (Y)
4). Axial stress X (Y) X (Y) X (Y)
5). Equivalent stress X (Y) X (Y) X (Y)
6). Equivalent plastic strain X (Y) X (Y) X (Y)
7). Net displacement X
8). Equivalent plastic strain over time X (Y)
9). Total strain over time X (Y)
10). Total strain difference over time X (Y)
Table 5.1: Results of multilayer cylinder analysis*
? X ? Plot generated from data obtained by analysis performed in ANSYS, Y ? Plot validated against (Oliver, 1988)
Time
Sr. No. Analysis Result
1 second 10 second 35 second
1). Radial temperature profile X X X
2). Radial stress X X X
3). Tangential stress X X X
4). Axial stress X X X
5). Equivalent stress X X X
6). Equivalent plastic strain X X X
7). Net displacement X
8). Equivalent plastic strain over time X
9). Total strain over time X
10). Total strain difference over time X
Table 5.2: Results available in Appendix C*
* X ? Plot generated from data obtained by analysis performed in ANSYS, Y ? Plot validated against (Oliver, 1988)
47
Time
Sr. No. Analysis Result
1 second 10 second 35 second
1). Radial temperature profile X X X
2). Radial stress X X X
3). Tangential stress X X X
4). Axial stress X X X
5). Equivalent stress X X X
6). Equivalent plastic strain X X X
7). Net displacement X
8). Equivalent plastic strain over time X
9). Total strain over time X
10). Total strain difference over time X
Table 5.3: Results available in Appendix D *
* X ? Plot generated from data obtained by analysis performed in ANSYS, Y ? Plot validated against (Oliver, 1988)
Figure 5.1: Radial temperature (C) distribution at time 1,10 and
35 second in multilayer cylinder (Steel-Cu-Steel)
0
100
200
300
400
500
600
700
800
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Distance from inner surface (m)
T
e
m
per
at
ur
e (
C
)
Time = 10second
Time = 35second
Time = 1second
Radial temperature distribution plotted in Figure 5.1 verifies that in First 10
seconds the temperature of the body is rising and in next 25 seconds the temperature of
48
the body decreases. At the end of the cycle i.e. at time 35 second the highest
temperature is in the copper layer (middle section).
Figure 5.2: Radial stress (Pa) at time 1, 10 and 35 second in multilayer
cylinder (steel-Cu-Steel)
-30.00E+06
-20.00E+06
-10.00E+06
00.00E+00
10.00E+06
20.00E+06
30.00E+06
0 0.005 0.01 0.015 0.02 0.025 0.03
Distance from inner surface (m)
R
a
d
i
a
l
st
r
e
ss
(
P
a
)
Time =35second
Time =1second
Time =10second
The results from the finite element analysis were analyzed to determine how
well they conformed to the criteria that must be satisfied. The conditions that must be
met in order to give the model credibility are listed below:
1. The radial stress should be zero on inside and outside surface.
2. The radial stress distribution should be continuous.
The above mentioned conditions can be verified by examining the plots in
Figure 5.2. The radial stress is zero on inside and outside surface and is significantly
smaller than either the tangential stress (Figure 5.3) or the axial stress (Figure 5.4) . The
radial stress also appears to be continuous.
49
Figure 5.3: Tangetial Stress (Pa) at time 1, 10 and 35 second in
multilayer cylinder (Steel -Cu- Steel)
-800.00E+06
-600.00E+06
-400.00E+06
-200.00E+06
000.00E+00
200.00E+06
400.00E+06
600.00E+06
0 0.005 0.01 0.015 0.02 0.025 0.03
Distance from inner surface(m)
T
a
n
g
en
tial S
t
res
s
(
P
a
)
Time = 1second
Time = 35second
Time = 10second
Figure 5.4: Axial Stress(Pa) at time 1, 10 and 35 second in multilayer
cylinder (Steel-Cu-Steel)
-800.00E+06
-600.00E+06
-400.00E+06
-200.00E+06
000.00E+00
200.00E+06
400.00E+06
600.00E+06
0 0.005 0.01 0.015 0.02 0.025 0.03
Distance from inner surface (m)
Ax
ial St
r
e
ss
(Pa)
Time = 1second
Time = 35second
Time = 10second
The results from this study are generally in good agreement with the results in
Appendix E but the tangential stress and axial stress are not equal. The model
developed for this study assumes a plane strain condition, i.e. the displacement
50
component in the axial direction is zero and the other displacement components are
independent of any displacement in the axial direction. For this condition the tangential
stress is given by (Johns 1965):
?
z
= ? (?
r
+ ?
?
) ? E ? ?T
5-1
This shows that axial stress (?
z
) and tangential stress (?
?
) will not be equal if no
additional constraints have been imposed. From above equation the axial stress and
tangential stress will be equal only if Poisson ratio (?) is one and radial stress (?
r
),
modulus of elasticity, coefficient of expansion and temperature difference is zero.
Figure 5.5: Equivalent stress (Pa) at time 1 ,10 and 35 second
in multilayer cylinder (Steel-Cu-Steel)
000.00E+00
100.00E+06
200.00E+06
300.00E+06
400.00E+06
500.00E+06
600.00E+06
700.00E+06
0 0.005 0.01 0.015 0.02 0.025 0.03
Distance from inside surface(m)
E
qui
val
e
n
t
stress
(P
a)
Time 1
Time 10
Time 35
Unlike the results in Appendix E, the location of the highest equivalent stress is
on the inside surface rather than being slightly away from the inside surface (towards
the outer surface). The method of defining the load cycle in this study may have
contributed to this phenomenon. It is interesting to note that at the beginning of the
cycle (i.e. at time = 1 second), the location of the highest equivalent stress is on the
51
inside surface (Figure 5.5), but as the cycle proceeds (i.e. at time = 10 seconds), the
highest equivalent stress front has moved toward the outside surface, and at the end of
the cycle (i.e. at time 35 seconds) it again regains its original position (Figure 5.5). This
agrees with the results in Appendix E. The equivalent stress in this study is calculated as
(ANSYS 9.0 documetation,2004)
{}
2
1
2
?z
2
?z
2
r?
2
?z
2
z?
2
?re
)??(?*6)?(?)?(?)?(?
2
1
?
?
?
?
?
?
?
+++?+?+?=
(5?2)
It is clear from the Figure 5.5 that equivalent stresses are discontinuous in the
multilayer model. Figure 5.6 (Equivalent plastic strain) show that yielding has occurred
on the inside surface, hence it can be concluded that the equivalent stress at that location
is above the yield stress or yield strength.
Figure 5.6: Equivalent plastic strain at time 1, 10 and 35
second in multilayer cylinder (Steel-Cu-Steel)
0.00E+00
1.00E-03
2.00E-03
3.00E-03
4.00E-03
5.00E-03
6.00E-03
7.00E-03
8.00E-03
9.00E-03
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Distance from inner surface(m)
Equivalent plast
ic
s
t
rain
Time = 1second
Time = 35second
Time = 10second
52
The magnitude of the equivalent stress in the outer layer is lower than the yield
stress. The plot for plastic strain (Figures 5.6) indicates that there is no plastic strain in
the outer layer of steel, hence the yield stress has not been exceeded.
Figure 5.7: Net Displacement(m) at time 1, 10 and 35 second in
multilayer cylinder (Steel-Cu-Steel)
0.00E+00
5.00E-05
1.00E-04
1.50E-04
2.00E-04
2.50E-04
3.00E-04
3.50E-04
4.00E-04
4.50E-04
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Distance from inside surface (m)
Ne
t
d
i
sp
la
cem
e
n
t
(m
)
Time = 1Second
Time = 10Second
Time = 35Second
Figure 5.7 shows the expansion and contraction of the multilayer cylinder. The
body expands radially during heating phase i.e. from time 1 second to time 35 second
and in cooling phase body tries to regain its original position but at the end of the cycle
i.e. time 35 second as shown in plot the body does not regains its original position.
Figure 5.9 shows a graphical representation of radial strain, tangential strain and
axial strain. One of the assumptions for this model is plane strain condition, which
indicates that axial strain must be zero. It can be concluded from the figure that the axial
strain is zero, which must be satisfied in order to give the model credibility. The total
strain differential ((?
r
- ?
?
), (?
?
- ?
z
) and (?
z
- ?
r
)) is plotted in Figure 5.10 and the data
53
obtained (Table 5.2) from the plot is used for fatigue life calculations for the cylindrical
mold.
?
r
?
?
?
z
Figure 5.8: Equivalent plastic strain in multilayer cylinder over time
Figure 5.9: Total strain in multilayer cylinder over time
54
5.1 Fatigue Life
As discussed earlier in Chapter 2 the Coffin and Manson (Suresh 1991) studied
low cycle fatigue, demonstrating that plastic strain is the major cause of low cycle
fatigue failure. They gave an empirical formula to estimate the fatigue life (N
f
) based on
plastic strain approach, which is as below:
1/2*?
p
= ?
f
*(2N
f
)
c
5-3
Where ?
p
(plastic strain amplitude) is calculated using following expression (Wang,
2000):
[][][]
222
*
3
2
rzzrp
???????
??
?+?+?= 5-4
(?
r
- ?
?
)
(?
z
-?
r
)
(?
?
-?
z
)
Figure 5.10: Total Strain differential in multilayer cylinder over time
55
Using Equations 5-2 and 5-3 the fatigue life (i.e. the number of pours before the
first crack will develop) can be calculated. The calculated results and results obtained
from the finite element analysis are summarized in Table 5.4.
Fatigue life is calculated using the Coffin-Manson equation, as discussed in
Chapter 2. Table 5.4 shows the fatigue life obtained for different permanent cylindrical
mold geometries.
Sr. No. 1 2 3
Material Combination
Steel-
Cu-Steel
Steel Steel
Inner Radius 9.84 3.42 6.69
Outer Radius 11.02 4.15 7.63
Thickness 1.12 0.52 0.94
?
p (plastic strain amplitude)
from ANSYS run
0.00588 0.00463 0.00544
# of Cycles (calculated) 3646 5361 4134
Actual industrial value 3800 5800 4300
Conservative Error (%) 4.0 7.5 3.8
Table 5.4: Calculation and validity of fatigue life
Figure 5.11: Fatigue life of the mold vs thickness of the mold
3000
3500
4000
4500
5000
5500
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Thickness (in)
Fatigue life (# of
cyc
les)
56
Increasing the thickness increases the distance involved in heat transfer, and if
the cooling condition of the outer surface is kept constant, the body acts as a heat sink
and the temperature is trapped between the inside surface and the outside surface. The
temperature of the inside and outside surface is less than temperature between them
(Figures 5.1, C.1 and D.1) at the end of the cycle i.e. at time 35 second. The use of the
multilayer cylinder does reduce the stress level in the outer layer but the primary
concern is to reduce the stress level on the inner surface, where the crack initiates.
However, an increase in thickness may be accompanied by an increase in convection
coefficient for more heat removal. For example spray cooling is commonly used to
increase the convection coefficient, particularly in continuous casting and even for
larger cylindrical molds for fast heat removal.
57
CHAPTER 6: CONCLUSIONS AND FUTURE WORK
As a result of this study, it was found that the permanent cylindrical mold could
accurately be studied using a commercial FEA package, ANSYS. This reduces the
dependency on expert programming skills in developing the numerical solutions. The
model can be modified very easily to incorporate new parameters. The reduction in the
time needed to perform two-dimensional analyses is an added advantage. It takes no
longer than 15 minutes to obtain the solution for a significantly low time step (0.001
second) on a Celeron processor.
When these results were compared with the numerical results reported by earlier
researchers, it was found that the stresses in the outer layer of steel were significantly
different. This indicates the necessity of verifying the results experimentally. During
this study it became clear that obtaining suitable material properties at high
temperatures is a daunting task. Very few expressions relating material properties and
temperature were found in the literature. An extensive study of material properties at
elevated temperatures (in the range of 1000
o
C - 2000
o
C) should be conducted. This
should further improve the accuracy of the Coffin-Manson approach
In this work, the interface between the steel and copper was assumed to be
always bonded. However, due to differences in the coefficients of thermal expansion
and the temperature distribution, this assumption may not be valid. An initial stress
58
condition or incorporating the glue (bonding material to hold the steel and copper
together) property, would help to better understand the model. A detailed investigation
of contact surface between layers will be a step forward.
In real molds an oxide layer is formed on the inside surface of cylinder, acting as
an insulating medium between molten material and mold. A study of this layer and its
properties would help to define the thermal load that is applied at the inner surface
when subjected to different pouring conditions.
As discussed earlier spray cooling is a viable option for larger cylindrical molds
to increase the heat transfer coefficient. Spray cooling is widely used in continuous
casting. There have been few studies to understand the effect of nozzle design and fluid
velocity on the distribution of heat transfer coefficient in continuous casting. The parts
produced by continuous casting are mainly flat slabs. Because of cylindrical shape of
the mold, problems such as ?out of roundness? and ?problems while pulling the
solidified pipe? are often encountered in industry while introducing the spray cooling.
The effect of spray cooling on permanent cylindrical mold should thus be studied.
59
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th
edition, Prentice Hall.
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France.
6. Gao J. W. and C. Y. Wang 1999. Modeling the solidification of functionally graded
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incremental theories, Ph.D. diss., Iowa state university.
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10. Hattel J. H. and P. N. Hansen 1994. A 1-D Analytical model for the thermally
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11. Huebner K. H., Dewhirst D. L. and Smith D. E., Ted G. Byrom 2001. The Finite
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th
edition, John Wiley & Sons, Inc.
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st
edition.
14. Lin J. Y. and H.T. Chen, 1994. Numerical solution of hyperbolic heat conduction in
cylindrical and spherical systems, Appl. Math Modelling, Vol. 18, p384-390.
15. Logan D. L. 1986. A First Course in the Finite Element Method, PWS Engineering.
16. Manson S.S. and Robert E. 1981. Thermal Stress and Low-cycle Fatigue, Krieger
Publishing Company, Florida pp. 224-240.
17. Mendelson A. and Manson S.S. October 1956. Practical solution of plastic
deformation problems in elastic range ASME, Paper No. 56-A-202.
18. Okono A. J. 1978. Thermal stress analysis and life prediction of a time-temperature-
dependent viscoelastic hollow cylinder subjected to a time-dependent temperature,
Ph.D. diss., Virginia Polytechnique and State University.
19. Oliver G. L. 1988. An analytical method to optimize the thermal fatigue life of
multilayered cylindrical shells, Ph.D. diss. The University of Alabama at
Birmingham.
20. Ortega R. 1993. Finite element analysis of a thick-walled cylinder elastic-plastic
behavior, MSE, The University of Alabama in Huntsville.
61
21. Park J. K., B. G. Thomas, I. V. Samarasekera and U. S. Yoon 2002. Thermal and
Mechanical behavior of copper molds during thin-slab casting: mold crack
formation, Metallurgical and Materials Transaction, Vol 33B, p426-436.
22. Ruan Y. 1990. Analysis of temperature fields and thermal stresses in solidifying
bodies using the finite element method, Ph.D. diss., The University of Minnesota.
23. S. Suresh 1991. Fatigue of Material 2
nd
edition, Cambridge University Press.
24. Serajzadeh S. 2004. Modeling of temperature history and phase transformation
during cooling of steel, Journal of Materials Processing Technology 146, p311-317.
25. Simo, J. C. and Taylor, R. L. 1985. Consistent Tangent Operators for Rate-
Independent Elastoplasticity, Computer Methods in Applied Mechanics and
Engineering, Vol. 48, pp. 101-118.
26. Sirikis J. S. 1988. A two-dimensional hybrid experimental-numerical technique for
elasto-plastic stress analysis, Ph.D. diss., University of Florida.
27. Sirinterlikci A., 2000. Thermal management and prediction of heat checking in die
casting, Ph.D. diss., The Ohio State University.
28. Tanaka M., T. Matusmoto and Q. F. Y. Shinshu 1994. Time-stepping boundary
element method applied to2-D transient heat conduction problems, Appl. Math
Modelling, Vol 18, p569-576.
29. Timoshenko S.P. and Goodier J.N. 1970. Theory of Elasticity 3
rd
edition, McGraw-
Hill Publications.
30. Trovent M. and S. Argyropoulos 1998. The implementation of a mathematical
model to characterize mold metal interface effects in metal casting, Canadian
Mettalurgical Quarterly, Vol 37, p185-196.
62
31. Ugura A. C. 1999. Stresses in Plates and Cylinders, 2
nd
edition, WCB/McGraw-Hill.
32. Wang B. December 2000. The prediction of low cycle fatigue for die casting with
FEM, MEng, Carleton University, Ottawa, Canada.
33. White D. Y. 1997. Stress analysis in multilayer structures, Ph.D. diss., The Florida
State University.
63
APPENDIX A
(MATERIAL PROPERTIES)
64
APPENDIX A
This appendix contains plots of material properties for steel (Figures A.1 ? A.6) and
copper (Figures A.7 ? A.12)
Figure A.1: Density of steel (2? % Cr 1% Mo) vs temperature
65
Figure A.2: Specific heat of steel (2? % Cr 1% Mo) vs temperature
Figure A.3: Elastic modulus of steel (2? % Cr 1% Mo) vs temperature
66
Figure A.4: Poisson ratio of steel (2? % Cr 1% Mo) vs temperature
Figure A.5: Coefficient of expansion of steel (2? % Cr 1% Mo) vs temperature
67
Figure A.6: Thermal conductivity of steel (2? % Cr 1% Mo) vs temperature
Figure A.7: Density of copper vs temperature
68
Figure A.8: Specific heat of copper vs temperature
Figure A.9: Elastic Modulus of copper vs temperature
69
Figure A.10: Poisson ratio of copper vs temperature
Figure A.11: Coefficient of expansion of copper vs temperature
70
Figure A.12: Thermal conductivity of copper vs temperature
71
APPENDIX B
( INPUT FILE (ANSYS) FOR FINITE ELEMENT ANALYSIS)
72
APPENDIX B
This appendix contains the log file generated by ANSYS to define elements,
create the geometry, define the material properties, assign attributes to the geometry,
mesh the geometry, define loads and boundary conditions and define solution options
such as time steps, data output and convergence control. This input file is for pre
processing and defining solution options. Once the model has been created and solution
option has been defined, the user is expected to check the solution options and start to
solve the current load set. After obtaining the solution, the user can postprocess any
results of interest.
/BATCH
/COM,ANSYS RELEASE 9.0 UP20021010 10:12:51 03/16/2005
/input,menust,tmp,'',,,,,,,,,,,,,,,,1
/GRA,POWER
/GST,ON
/PLO,INFO,3
/GRO,CURL,ON
/REPLOT,RESIZE
/PREP7
!*
/ BUILD GEOMMETRY (CHAPTER 4.2)
CSYS,1
ET,1,PLANE13
K,1,0.25,-2.5,0,
K,2,0.256,-2.5,0,
K,3,0.256,2.5,0,
K,4,0.25,2.5,0,
K,5,0.256,-2.5,0,
73
K,6,0.274,-2.5,0,
K,7,0.274,2.5,0,
K,8,0.256,2.5,0,
K,9,0.274,-2.5,0,
K,10,0.28,-2.5,0,
K,11,0.28,2.5,0,
K,12,0.274,2.5,0,
L, 1, 2
L, 2, 3
L, 3, 4
L, 4, 1
L, 5, 6
L, 6, 7
L, 7, 8
L, 8, 5
L, 9, 10
L, 10, 11
L, 11, 12
L, 12, 9
FLST,2,4,4
FITEM,2,1
FITEM,2,2
FITEM,2,3
FITEM,2,4
AL,P51X
FLST,2,4,4
FITEM,2,5
FITEM,2,6
FITEM,2,7
FITEM,2,8
AL,P51X
FLST,2,4,4
FITEM,2,9
FITEM,2,10
FITEM,2,11
FITEM,2,12
AL,P51X
FLST,5,4,4,ORDE,4
FITEM,5,1
FITEM,5,3
FITEM,5,9
FITEM,5,11
FLST,5,2,5,ORDE,2
FITEM,5,1
FITEM,5,3
74
/ DEFINE MATERIAL PROPERTIES ( CHAPTER 4.5)
TOFFST,273
MPTEMP
MPTEMP,1,25,204,427,538,593
MPTEMP,6,649,704,760,
MPDATA,EX,1,1,193e9,172e9,160.34e9,146.49e9,133.14e9
MPDATA,EX,1,6,101.8e9,79.8e9,66.2e9,
MPTEMP
MPTEMP,1,25,204,427,538,593
MPTEMP,6,649,704,760,
MPDATA,NUXY,1,1,0.285,0.285,0.285,0.285,0.285
MPDATA,NUXY,1,6,0.285,0.285,0.285,
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,ALPX,1,1,0.47e-5,0.48e-5,0.49e-5,0.52e-5,0.54e-5
MPDATA,ALPX,1,6,0.55e-05,0.56e-5,
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,DENS,1,1,8940,8936,8933,8923,8917
MPDATA,DENS,1,6,8914,8910,
MPTEMP
MPTEMP,1,25,
MPDATA,MU,1,1,1,
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,KXX,1,1,40.08,39.76,39.33,37.44,35.68
MPDATA,KXX,1,6,34.65,33.51,
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,C,1,1,460,484,507,578,625
MPDATA,C,1,6,648,672,
MPTEMP
MPTEMP,1,25,204,427,538,593
MPTEMP,6,649,704,760,
MPDATA,PRXY,1,1,0.285,0.285,0.285,0.285,0.285
MPDATA,PRXY,1,6,0.285,0.285,0.285,
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,REFT,1,1,25,25,25,25,25
MPDATA,REFT,1,6,25,25,
MPTEMP
75
MPTEMP,1,25,93,150,204,260
MPTEMP,6,315,371,427,605,730,
MPDATA,EX,2,1,95.3e9,92.4e9,92.1e9,87.79e9,79.13e9
MPDATA,EX,2,6,80e9,80e9,80e9,80e9,80e9
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,ALPX,2,1,0.37e-5,0.37e-5,0.37e-5,0.39e-5,0.397e-5
MPDATA,ALPX,2,6,0.40e-5,0.40e-5,
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,DENS,2,1,7860,7858,7856,7851,7848
MPDATA,DENS,2,6,7846,7844,
MPTEMP
MPTEMP,1,25,
MPDATA,MU,2,1,1,
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,KXX,2,1,376.85,374,370,356,343
MPDATA,KXX,2,6,336,328,
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,C,2,1,386,392,397,413,423
MPDATA,C,2,6,429,434,
MPTEMP
MPTEMP,1,25,93,150,204,260
MPTEMP,6,315,371,427,605,730,
MPDATA,PRXY,2,1,0.337,0.337,0.337,0.337,0.337
MPDATA,PRXY,2,6,0.337,0.337,0.337,0.337,0.337,
MPTEMP
MPTEMP,1,0,100,200,500,700
MPTEMP,6,800,900,
MPDATA,REFT,2,1,25,25,25,25,25
MPDATA,REFT,2,6,25,25,
/ DEFINE STRESS-STRAIN CURVE FOR MULTILINEAR ISOTROPIC
HARDENING BEHAVIOR (CHAPTER 3.1.4 AND CHAPTER 4.5)
TB,MISO,1,8,7,
TBTEMP,25
TBPT,,0.0033,637.35e6
TBPT,,0.0042,670.97e6
TBPT,,0.0067,711.22e6
76
TBPT,,0.01,737.63e6
TBPT,,0.014,755.03e6
TBPT,,0.018,764.93e6
TBPT,,0.02,766.18e6
TBTEMP,204
TBPT,,0.0035,602.43e6
TBPT,,0.0042,638.28e6
TBPT,,0.0067,679.84e6
TBPT,,0.01,708.49e6
TBPT,,0.014,727.16e6
TBPT,,0.018,737.42e6
TBPT,,0.02,739.5e6
TBTEMP,427
TBPT,,0.0035,561.18e6
TBPT,,0.0042,591e6
TBPT,,0.0067,629.93e6
TBPT,,0.01,656.39e6
TBPT,,0.014,673.84e6
TBPT,,0.018,682.79e6
TBPT,,0.020,684.94e6
TBTEMP,538
TBPT,,0.0035,512.74e6
TBPT,,0.0042,541.3e6
TBPT,,0.0067,581.47e6
TBPT,,0.01,606.68e6
TBPT,,0.014,625.35e6
TBPT,,0.018,634.29e6
TBPT,,0.02,636.45e6
TBTEMP,593
TBPT,,0.003,399.44e6
TBPT,,0.0035,426.65e6
TBPT,,0.0067,479.63e6
TBPT,,0.01,506.06e6
TBPT,,0.014,524.73e6
TBPT,,0.018,534e6
TBPT,,0.020,537.04e6
TBTEMP,649
TBPT,,0.0022,224.13e6
TBPT,,0.0035,264.21e6
TBPT,,0.0067,303e6
TBPT,,0.01,323.05e6
TBPT,,0.014,338.25e6
TBPT,,0.018,345e6
TBPT,,0.020,347.96e6
TBTEMP,704
TBPT,,0.0016,127.72e6
77
TBPT,,0.0022,143e6
TBPT,,0.0042,165e6
TBPT,,0.0067,182e6
TBPT,,0.01,195.79e6
TBPT,,0.018,210.5e6
TBPT,,0.02,212.18e6
TBTEMP,760
TBPT,,0.00093,61.57e6
TBPT,,0.0022,72.59e6
TBPT,,0.0042,78.5e6
TBPT,,0.0067,85.5e6
TBPT,,0.01,90.32e6
TBPT,,0.018,96.123e6
TBPT,,0.020,98.19e6
TB,MISO,2,8,7,
TBTEMP,25
TBPT,,0.0027,257.28e6
TBPT,,0.0032,269.56e6
TBPT,,0.0067,294.96e6
TBPT,,0.01,306.97e6
TBPT,,0.014,315.58e6
TBPT,,0.017,319.02e6
TBPT,,0.02,320.03e6
TBTEMP,93
TBPT,,0.0026,240.26e6
TBPT,,0.0032,251.98e6
TBPT,,0.0067,273.75e6
TBPT,,0.01,283.34e6
TBPT,,0.014,291e6
TBPT,,0.017,294.25e6
TBPT,,0.02,295.25e6
TBTEMP,149
TBPT,,0.0024,222.02e6
TBPT,,0.0032,234e6
TBPT,,0.0067,256.1e6
TBPT,,0.01,266.37e6
TBPT,,0.014,275e6
TBPT,,0.017,278.4e6
TBPT,,0.02,279.4e6
TBTEMP,204
TBPT,,0.0021,184.36e6
TBPT,,0.0032,204.71e6
TBPT,,0.0067,229.5e6
TBPT,,0.01,241.52e6
TBPT,,0.014,251.34e6
TBPT,,0.017,254.76e6
78
TBPT,,0.02,256.37e6
TBTEMP,260
TBPT,,0.0022,174.1e6
TBPT,,0.0032,190e6
TBPT,,0.0067,209.5e6
TBPT,,0.01,219.1e6
TBPT,,0.014,226e6
TBPT,,0.017,230e6
TBPT,,0.02,230.9e6
TBTEMP,315
TBPT,,0.0001,8e6
TBPT,,0.0022,32.05e6
TBPT,,0.0042,42.7e6
TBPT,,0.01,58.485e6
TBPT,,0.014,64.667e6
TBPT,,0.017,67.0e6
TBPT,,0.02,67.883e6
TBTEMP,371
TBPT,,0.0001,8e6
TBPT,,0.0022,28.01e6
TBPT,,0.0042,37.153e6
TBPT,,0.01,50.607e6
TBPT,,0.014,56e6
TBPT,,0.017,58.4e6
TBPT,,0.02,59.4e6
TBTEMP,427
TBPT,,0.0001,8e6
TBPT,,0.0022,23.76e6
TBPT,,0.0042,30.49e6
TBPT,,0.01,41.517e6
TBPT,,0.014,45.88e6
TBPT,,0.017,47.68e6
TBPT,,0.02,48.5e6
/ DEFINE ATTRIBUTE TO THE LAYERS (CHAPTER 4.5)
CM,_Y,AREA
ASEL, , , ,P51X
CM,_Y1,AREA
CMSEL,S,_Y
!*
CMSEL,S,_Y1
AATT, 1, , 1, 0,
CMSEL,S,_Y
CMDELE,_Y
CMDELE,_Y1
79
!*
CM,_Y,AREA
ASEL, , , , 2
CM,_Y1,AREA
CMSEL,S,_Y
!*
CMSEL,S,_Y1
AATT, 2, , 1, 0,
CMSEL,S,_Y
CMDELE,_Y
CMDELE,_Y1
/ DESCRITIZE THE CONTINUUM (CHAPTER 4.3)
FLST,5,4,4,ORDE,4
FITEM,5,1
FITEM,5,3
FITEM,5,9
FITEM,5,11
CM,_Y,LINE
LSEL, , , ,P51X
CM,_Y1,LINE
CMSEL,,_Y
!*
LESIZE,_Y1, , ,12,1, , , ,1
!*
FLST,5,2,4,ORDE,2
FITEM,5,5
FITEM,5,7
CM,_Y,LINE
LSEL, , , ,P51X
CM,_Y1,LINE
CMSEL,,_Y
!*
LESIZE,_Y1, , ,36,1, , , ,1
!*
FLST,5,6,4,ORDE,6
FITEM,5,2
FITEM,5,4
FITEM,5,6
FITEM,5,8
FITEM,5,10
FITEM,5,12
CM,_Y,LINE
LSEL, , , ,P51X
CM,_Y1,LINE
80
CMSEL,,_Y
!*
LESIZE,_Y1, ,0.5, ,1, , , ,1
!*
MSHAPE,0,2D
MSHKEY,0
!*
FLST,5,3,5,ORDE,2
FITEM,5,1
FITEM,5,-3
CM,_Y,AREA
ASEL, , , ,P51X
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
!*
AMESH,_Y1
!*
CMDELE,_Y
CMDELE,_Y1
CMDELE,_Y2
!*
/UI,MESH,OFF
SAVE
!*
!*
/ DEFINE THAT CONTACT SURFACE IS ALWAYS BONDED(CHAPTER 4.6)
/COM, CONTACT PAIR CREATION - START
CM,_NODECM,NODE
CM,_ELEMCM,ELEM
CM,_KPCM,KP
CM,_LINECM,LINE
CM,_AREACM,AREA
CM,_VOLUCM,VOLU
/GSAV,cwz,gsav,,temp
MP,MU,2,1.0
MAT,2
MP,EMIS,2,
R,3
REAL,3
ET,2,169
ET,3,172
R,3,,,0.5,0.1,0,
RMORE,0.1,0.1,1.0E20,0.0,1.0,0.5
81
RMORE,0.0,340,1.0,,1.0,0.5
RMORE,0,1.0,1.0,0.0,,1.0
KEYOPT,3,3,0
KEYOPT,3,4,0
KEYOPT,3,5,1
KEYOPT,3,7,2
KEYOPT,3,8,0
KEYOPT,3,9,0
KEYOPT,3,10,1
KEYOPT,3,11,0
KEYOPT,3,12,5
KEYOPT,3,2,0
KEYOPT,3,1,1
! Generate the target surface
LSEL,S,,,2
CM,_TARGET,LINE
TYPE,2
NSLL,S,1
ESLN,S,0
ESURF,ALL
CMSEL,S,_ELEMCM
! Generate the contact surface
LSEL,S,,,8
CM,_CONTACT,LINE
TYPE,3
NSLL,S,1
ESLN,S,0
ESURF,ALL
ALLSEL
ESEL,ALL
ESEL,S,TYPE,,2
ESEL,A,TYPE,,3
ESEL,R,REAL,,3
/PSYMB,ESYS,1
/PNUM,TYPE,1
/NUM,1
EPLOT
ESEL,ALL
ESEL,S,TYPE,,2
ESEL,A,TYPE,,3
ESEL,R,REAL,,3
CMSEL,A,_NODECM
CMDEL,_NODECM
CMSEL,A,_ELEMCM
CMDEL,_ELEMCM
CMSEL,S,_KPCM
82
CMDEL,_KPCM
CMSEL,S,_LINECM
CMDEL,_LINECM
CMSEL,S,_AREACM
CMDEL,_AREACM
CMSEL,S,_VOLUCM
CMDEL,_VOLUCM
/GRES,cwz,gsav
CMDEL,_TARGET
CMDEL,_CONTACT
/COM, CONTACT PAIR CREATION - END
CWZDELE,3,1,''
!*
!*
/COM, CONTACT PAIR CREATION - START
CM,_NODECM,NODE
CM,_ELEMCM,ELEM
CM,_KPCM,KP
CM,_LINECM,LINE
CM,_AREACM,AREA
CM,_VOLUCM,VOLU
/GSAV,cwz,gsav,,temp
MP,MU,2,1
MAT,2
MP,EMIS,2,0
R,4
REAL,4
ET,4,169
ET,5,175
R,4,,,0.5,0.1,0,
RMORE,0.1,0.1,1.0E20,0.0,1.0,0.5
RMORE,0.0,340,1.0,,1.0,0.5
RMORE,0,1.0,1.0,0.0,,1.0
RMORE,10.0
KEYOPT,5,3,0
KEYOPT,5,4,0
KEYOPT,5,5,1
KEYOPT,5,7,2
KEYOPT,5,8,0
KEYOPT,5,9,0
KEYOPT,5,10,1
KEYOPT,5,11,0
KEYOPT,5,12,5
KEYOPT,5,2,0
KEYOPT,5,1,1
! Generate the target surface
83
LSEL,S,,,2
CM,_TARGET,LINE
TYPE,4
NSLL,S,1
ESLN,S,0
ESURF,ALL
CMSEL,S,_ELEMCM
! Generate the contact surface
LSEL,S,,,8
CM,_CONTACT,LINE
TYPE,5
NSLL,S,1
ESLN,S,0
ESURF,ALL
ALLSEL
ESEL,ALL
ESEL,S,TYPE,,4
ESEL,A,TYPE,,5
ESEL,R,REAL,,4
/PSYMB,ESYS,1
/PNUM,TYPE,1
/NUM,1
EPLOT
ESEL,ALL
ESEL,S,TYPE,,4
ESEL,A,TYPE,,5
ESEL,R,REAL,,4
CMSEL,A,_NODECM
CMDEL,_NODECM
CMSEL,A,_ELEMCM
CMDEL,_ELEMCM
CMSEL,S,_KPCM
CMDEL,_KPCM
CMSEL,S,_LINECM
CMDEL,_LINECM
CMSEL,S,_AREACM
CMDEL,_AREACM
CMSEL,S,_VOLUCM
CMDEL,_VOLUCM
/GRES,cwz,gsav
CMDEL,_TARGET
CMDEL,_CONTACT
/COM, CONTACT PAIR CREATION - END
!*
!*
/COM, CONTACT PAIR CREATION - START
84
CM,_NODECM,NODE
CM,_ELEMCM,ELEM
CM,_KPCM,KP
CM,_LINECM,LINE
CM,_AREACM,AREA
CM,_VOLUCM,VOLU
/GSAV,cwz,gsav,,temp
MP,MU,1,1.0
MAT,1
MP,EMIS,1,
R,5
REAL,5
ET,6,169
ET,7,175
R,5,,,0.5,0.1,0,
RMORE,0.1,0.1,1.0E20,0.0,1.0,0.5
RMORE,0.0,40,1.0,,1.0,0.5
RMORE,0,1.0,1.0,0.0,,1.0
RMORE,10.0
KEYOPT,7,3,0
KEYOPT,7,4,0
KEYOPT,7,5,1
KEYOPT,7,7,2
KEYOPT,7,8,0
KEYOPT,7,9,0
KEYOPT,7,10,1
KEYOPT,7,11,0
KEYOPT,7,12,5
KEYOPT,7,2,0
KEYOPT,7,1,1
! Generate the target surface
LSEL,S,,,6
CM,_TARGET,LINE
TYPE,6
NSLL,S,1
ESLN,S,0
ESURF,ALL
CMSEL,S,_ELEMCM
! Generate the contact surface
LSEL,S,,,12
CM,_CONTACT,LINE
TYPE,7
NSLL,S,1
ESLN,S,0
ESURF,ALL
ALLSEL
85
ESEL,ALL
ESEL,S,TYPE,,6
ESEL,A,TYPE,,7
ESEL,R,REAL,,5
/PSYMB,ESYS,1
/PNUM,TYPE,1
/NUM,1
EPLOT
ESEL,ALL
ESEL,S,TYPE,,6
ESEL,A,TYPE,,7
ESEL,R,REAL,,5
CMSEL,A,_NODECM
CMDEL,_NODECM
CMSEL,A,_ELEMCM
CMDEL,_ELEMCM
CMSEL,S,_KPCM
CMDEL,_KPCM
CMSEL,S,_LINECM
CMDEL,_LINECM
CMSEL,S,_AREACM
CMDEL,_AREACM
CMSEL,S,_VOLUCM
CMDEL,_VOLUCM
/GRES,cwz,gsav
CMDEL,_TARGET
CMDEL,_CONTACT
/COM, CONTACT PAIR CREATION - END
/MREP,EPLOT
SAVE
!*
/ DEFINE LOAD AND BOUNDARY CONDITIONS (CHAPTER 4.2)
ANTYPE,4
!*
TRNOPT,FULL
LUMPM,0
!*
FLST,2,385,1,ORDE,2
FITEM,2,1
FITEM,2,-385
IC,P51X,TEMP,25, ,
FLST,2,6,4,ORDE,6
FITEM,2,1
FITEM,2,3
86
FITEM,2,5
FITEM,2,7
FITEM,2,9
FITEM,2,11
DL,P51X, ,SYMM
FLST,2,1,4,ORDE,1
FITEM,2,4
SAVE
*DIM,Load,TABLE,58,1,1,time, ,
!*
*DIM,conv,TABLE,3,1,1,time, ,
!*
*SET,LOAD(1,0,1) , 0
*SET,LOAD(1,1,1) , 25
*SET,LOAD(2,0,1) , 0.7
*SET,LOAD(2,1,1) , 357.28
*SET,LOAD(3,0,1) , 0.76
*SET,LOAD(3,1,1) , 425
*SET,LOAD(4,0,1) , 1
*SET,LOAD(4,1,1) , 462.5
*SET,LOAD(5,0,1) , 1.52
*SET,LOAD(5,1,1) , 506.45
*SET,LOAD(6,0,1) , 2.16
*SET,LOAD(6,1,1) , 544.27
*SET,LOAD(7,0,1) , 2.8
*SET,LOAD(7,1,1) , 573.3
*SET,LOAD(8,0,1) , 3.2
*SET,LOAD(8,1,1) , 598.5
*SET,LOAD(9,0,1) , 3.5
*SET,LOAD(9,1,1) , 615.32
*SET,LOAD(10,0,1) , 4.2
*SET,LOAD(10,1,1) , 632.12
*SET,LOAD(11,0,1) , 4.37
*SET,LOAD(11,1,1) , 646.64
*SET,LOAD(12,0,1) , 4.9
*SET,LOAD(12,1,1) , 658.09
*SET,LOAD(13,0,1) , 5.5
*SET,LOAD(13,1,1) , 671.83
*SET,LOAD(14,0,1) , 5.8
*SET,LOAD(14,1,1) , 680.23
*SET,LOAD(15,0,1) , 6.5
*SET,LOAD(15,1,1) , 688.61
*SET,LOAD(16,0,1) , 7
*SET,LOAD(16,1,1) , 697
*SET,LOAD(17,0,1) , 7.6
*SET,LOAD(17,1,1) , 707
87
*SET,LOAD(18,0,1) , 8.5
*SET,LOAD(18,1,1) , 715.3
*SET,LOAD(19,0,1) , 8.9
*SET,LOAD(19,1,1) , 723.7
*SET,LOAD(20,0,1) , 10
*SET,LOAD(20,1,1) , 735.5
*SET,LOAD(21,0,1) , 10.5
*SET,LOAD(21,1,1) , 602
*SET,LOAD(22,0,1) , 10.75
*SET,LOAD(22,1,1) , 552
*SET,LOAD(23,0,1) , 11.5
*SET,LOAD(23,1,1) , 518.7
*SET,LOAD(24,0,1) , 12.2
*SET,LOAD(24,1,1) , 489.63
*SET,LOAD(25,0,1) , 12.95
*SET,LOAD(25,1,1) , 463.62
*SET,LOAD(26,0,1) , 13.07
*SET,LOAD(26,1,1) , 446.79
*SET,LOAD(27,0,1) , 13.67
*SET,LOAD(27,1,1) , 429.96
*SET,LOAD(28,0,1) , 14.4
*SET,LOAD(28,1,1) , 412.34
*SET,LOAD(29,0,1) , 14.55
*SET,LOAD(29,1,1) , 398.6
*SET,LOAD(30,0,1) , 15.25
*SET,LOAD(30,1,1) , 384.79
*SET,LOAD(31,0,1) , 15.85
*SET,LOAD(31,1,1) , 378.65
*SET,LOAD(32,0,1) , 16
*SET,LOAD(32,1,1) , 369.5
*SET,LOAD(33,0,1) , 16.5
*SET,LOAD(33,1,1) , 357.2
*SET,LOAD(34,0,1) , 17.5
*SET,LOAD(34,1,1) , 349.5
*SET,LOAD(35,0,1) , 18
*SET,LOAD(35,1,1) , 338.8
*SET,LOAD(36,0,1) , 18.75
*SET,LOAD(36,1,1) , 328.09
*SET,LOAD(37,0,1) , 19.4
*SET,LOAD(37,1,1) , 319.66
*SET,LOAD(38,0,1) , 20.1
*SET,LOAD(38,1,1) , 307.4
*SET,LOAD(39,0,1) , 21
*SET,LOAD(39,1,1) , 294.3
*SET,LOAD(40,0,1) , 22
*SET,LOAD(40,1,1) , 285.17
88
*SET,LOAD(41,0,1) , 22.7
*SET,LOAD(41,1,1) , 274
*SET,LOAD(42,0,1) , 23.2
*SET,LOAD(42,1,1) , 269
*SET,LOAD(43,0,1) , 23.8
*SET,LOAD(43,1,1) , 261.4
*SET,LOAD(44,0,1) , 24.4
*SET,LOAD(44,1,1) , 255
*SET,LOAD(45,0,1) , 25
*SET,LOAD(45,1,1) , 248.36
*SET,LOAD(46,0,1) , 26
*SET,LOAD(46,1,1) , 238.39
*SET,LOAD(47,0,1) , 26.5
*SET,LOAD(47,1,1) , 233.02
*SET,LOAD(48,0,1) , 27.2
*SET,LOAD(48,1,1) , 226.1
*SET,LOAD(49,0,1) , 28
*SET,LOAD(49,1,1) , 218.4
*SET,LOAD(50,0,1) , 29
*SET,LOAD(50,1,1) , 209.2
*SET,LOAD(51,0,1) , 29.75
*SET,LOAD(51,1,1) , 203.8
*SET,LOAD(52,0,1) , 30.3
*SET,LOAD(52,1,1) , 196.1
*SET,LOAD(53,0,1) , 31.5
*SET,LOAD(53,1,1) , 186
*SET,LOAD(54,0,1) , 32.6
*SET,LOAD(54,1,1) , 181.6
*SET,LOAD(55,0,1) , 33
*SET,LOAD(55,1,1) , 176
*SET,LOAD(56,0,1) , 33.6
*SET,LOAD(56,1,1) , 172.38
*SET,LOAD(57,0,1) , 34.1
*SET,LOAD(57,1,1) , 167.7
*SET,LOAD(58,0,1) , 35
*SET,LOAD(58,1,1) , 161
*SET,CONV(1,0,1) , 0
*SET,CONV(1,1,1) , 1575
*SET,CONV(2,0,1) , 6.5
*SET,CONV(2,1,1) , 3150
*SET,CONV(3,0,1) , 35
*SET,CONV(3,1,1) , 3150
SAVE
!*
!*
/GO
89
DL,P51X, ,TEMP, %LOAD%
FLST,2,1,4,ORDE,1
FITEM,2,10
/GO
!*
!*
SFL,P51X,CONV, %CONV% , ,100,
SAVE
FINISH
/DEFINE SOLUTION OPTIONS (CHAPTER 5)
/SOL
!*
ANTYPE,4
!*
TRNOPT,FULL
LUMPM,0
!*
ANTYPE,4
NLGEOM,1
DELTIM,0.25,0,0.25
OUTRES,ERASE
OUTRES,ALL,1
KBC,0
NCNV,0,0,0,0,0
TIME,35
/GST,1,0
!*
OUTRES,ALL,ALL,
!*
SOLCONTROL,ON,1,
!*
!*
TIME,35
AUTOTS,-1
DELTIM,0.25,0,0.25,1
KBC,0
!*
/PNUM,KP,0
/PNUM,LINE,1
/PNUM,AREA,1
/PNUM,VOLU,0
/PNUM,NODE,0
/PNUM,TABN,0
/PNUM,SVAL,0
90
/NUMBER,1
!*
/PNUM,MAT,1
/REPLOT
!*
TSRES,ERASE
/STATUS,SOLU
91
APPENDIX C
RESULTS FOR MONOLAYER CYLIDRICAL MOLD ( IR = 3.42??, OR = 4.15??,
THICKNESS = 0.52??)
92
APPENDIX C
Figure C.1: Radial temperature (C) in monolayer cylinder
at time 1,10 and 35 second
0
100
200
300
400
500
600
700
800
900
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Distance from inside surface (m)
T
e
mper
a
t
ur
e (C)
Time = 1second
Time = 10second
Time = 35second
Figure C.2: Radial stress (Pa) in monolayer cylinder at
time 1,10 and 35 second
-30E+06
-20E+06
-10E+06
00E+00
10E+06
20E+06
30E+06
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Distance from inside surface (m)
Rad
i
al
st
re
ss
(
P
a)
Time=1second
Time=35second
Time=10second
93
Figure C.3: Tangential stress (Pa) in monolayer cylinder
at time 1,10 and 35 second
-800E+06
-600E+06
-400E+06
-200E+06
000E+00
200E+06
400E+06
600E+06
800E+06
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Distance from inside surface (m)
Tange
n
tial stres
s
(P
a)
Time = 1second
Time = 10second
Time = 35second
Figure C.4: Axial stress (Pa) in monolayer cylinder at
time 1,10 and 35 second
-800E+06
-600E+06
-400E+06
-200E+06
000E+00
200E+06
400E+06
600E+06
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Distance from inside surface (m)
Axial str
ess (P
a)
Time = 1second
Time = 10second
Time = 1second
94
Figure C.5: Equivalent stress in monolayer cylinder at
time 1,10 and 35 second
000E+00
100E+06
200E+06
300E+06
400E+06
500E+06
600E+06
700E+06
800E+06
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Distance from inside surface (m)
Equ
i
va
len
t
st
re
ss
(
P
a
)
Time = 35second
Time = 10second
Time = 1second
Figure C.6: Net displacement (m) in monolayer cylinder
at time 1,10 and 35 second
0.00E+00
5.00E-05
1.00E-04
1.50E-04
2.00E-04
2.50E-04
3.00E-04
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Distance from inside surface (m)
Net
di
spl
a
ceme
n
t (
m
)
Time = 10second
Time = 35second
Time = 1second
95
Figure C.6: Equivalent plastic strain in monolayer
cylinder at time 1,10 and 35 second
-0.005
0
0.005
0.01
0.015
0.02
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Distance from inside surface (m)
E
q
u
i
valent plastic
st
rain
Time = 1second
Time = 10second
Time = 35second
Figure C.8: Equivalent plastic strain in monolayer cylinder over time
96
Figure C.9: Total strain in monolayer cylinder over time
Figure C.10: Total strain difference in monolayer cylinder over time
?
r
?
z
?
?
?
r
- ?
?
?
z
- ?
r
?
?
- ?
z
97
APPENDIX D
RESULTS FOR MONOLAYER CYLIDRICAL MOLD (IR = 6.69??, OR = 7.63??,
THICKNESS = 0.95??)
98
APPENDIX D
Figure D.1: Radial temperature (C) profile in monolayer
cylinder at time 1,10 and 35 second
0
100
200
300
400
500
600
700
800
900
0 0.005 0.01 0.015 0.02 0.025
Distance from inside surface (m)
Temperature (C
)
Time = 1second
Time = 35second
Time = 10second
Figure D.2: Radial stress (Pa) in monolayer cylinder at
time 1,10 and 35 second
-30E+06
-20E+06
-10E+06
00E+00
10E+06
20E+06
30E+06
0 0.005 0.01 0.015 0.02 0.025
Distance from inside surface (m)
Ra
di
al
st
res
s
(Pa
)
Time = 1second
Time = 1second
Time = 1second
99
Figure D.3: Tangential stress (Pa) in monolayer cylinder
at time 1,10 and 35 second
-800E+06
-600E+06
-400E+06
-200E+06
000E+00
200E+06
400E+06
600E+06
800E+06
0 0.005 0.01 0.015 0.02 0.025
Distance from inside surface (m)
Tangentia
l
stre
ss (P
a)
Time = 1second
Time = 35second
Time = 10second
Figure D.4: Axial stress (Pa) in monolayer cylinder at
time 1,10 and 35 second
-800E+06
-600E+06
-400E+06
-200E+06
000E+00
200E+06
400E+06
600E+06
800E+06
0 0.005 0.01 0.015 0.02 0.025
Distance from inside surface (m)
Ax
ial st
ress
(Pa)
Time= 1second
Time= 35second
Time= 10second
100
Figure D.5: Equivalent stress (Pa) in monolayer cylinder
at time 1,10 and 35 second
000E+0
100E+6
200E+6
300E+6
400E+6
500E+6
600E+6
700E+6
800E+6
0 0.005 0.01 0.015 0.02 0.025
Distance from inside surface (m)
Equiva
le
nt
st
re
ss
(P
a
)
Time = 1second
Time = 35second
Time = 10second
Figure D.6: Net displacement(m) in monolayer
cylinder at time 1,10 and 35 second
0.00E+00
5.00E-05
1.00E-04
1.50E-04
2.00E-04
2.50E-04
3.00E-04
3.50E-04
0 0.005 0.01 0.015 0.02 0.025
Distance from inside surface (m)
Net displacment
(
m
)
Time = 1second
Time = 10second
Time = 35second
101
Figure D.7: Equivalent plastic strain in monolayer
cylinder at time 1,10 and 35 second
-5.00E-03
0.00E+00
5.00E-03
1.00E-02
1.50E-02
2.00E-02
0 0.005 0.01 0.015 0.02 0.025
Distance from inside surface (m)
Eq
uiv
a
le
nt pla
s
ti
c
s
t
ra
in
Time = 1second
Time = 10second
Time = 35second
Figure D.8: Equivalent plastic strain in monolayer cylinder over time
102
Figure D.9: Total strain in monolayer cylinder over time
Figure D.10: Total strain difference in monolayer cylinder over time
?
r
?
?
?
z
?
r
- ?
?
?
z
- ?
r
?
?
- ?
z
103
APPENDIX E
RESULT FOR VALIDATION [12]
104
APPENDIX E
?
Figure E.1: Radial temperature profile in a composite tube with a 6 mm Steel OD
layer ? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 1second (Oliver, 1988)
Figure E.2: Radial stress in a composite tube with a 6 mm Steel OD layer ?
18 mm Cu middle layer ? 6 mm Steel ID Layer at 1second (Oliver, 1988)
105
Figure E.4: stress in a composite tube with a 6 mm Steel OD layer ? 18 mm Cu
middle layer ? 6 mm Steel ID Layer at 1 second (Oliver, 1988)
Figure E.3: Tangential stress in a composite tube with a 6 mm Steel OD layer
? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 1second (Oliver, 1988)
106
m Steel OD layer ?
Figure E.5: Equivalent stress in a composite tube with a 6 mm Steel OD layer
? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 1 second (Oliver, 1988)
Figure E.6: Equivalent plastic strain in a composite tube with a 6 mm Steel OD
layer ? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 1 second (Oliver, 1988)
107
Figure E.8: Radial stress in a composite tube with a 6 mm Steel OD layer ? 18
mm Cu middle layer ? 6 mm Steel ID Layer at 10 second (Oliver, 1988)
Figure E.7: Radial temperature profile in a composite tube with a 6 mm Steel OD
layer ? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second (Oliver, 1988)
108
Figure E.9: Tangential stress in a composite tube with a 6 mm Steel OD layer ? 18
mm Cu middle layer ? 6 mm Steel ID Layer at 10 second (Oliver, 1988)
Figure E.10: Axial stress in a composite tube with a 6 mm Steel OD layer ? 18 mm
Cu middle layer ? 6 mm Steel ID Layer at 10 second (Oliver, 1988)
109
Figure E.11: Equivalent stress in a composite tube with a 6 mm Steel OD layer ?
18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second (Oliver, 1988)
Figure E.12: Equivalent plastic strain in a composite tube with a 6 mm Steel OD
layer ? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second (Oliver, 1988)
110
Figure E.13: Radial temperature profile in a composite tube with a 6 mm Steel OD
layer ? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 35 second (Oliver, 1988)
Figure E.14: Radial stress in a composite tube with a 6 mm Steel OD layer ? 18
mm Cu middle layer ? 6 mm Steel ID Layer at 35 second (Oliver, 1988)
111
Figure E.15: Tangential stress in a composite tube with a 6 mm Steel OD layer ? 18
mm Cu middle layer ? 6 mm Steel ID Layer at 35 second (Oliver, 1988)
Figure E.16: Axial stress in a composite tube with a 6 mm Steel OD layer ? 18 mm
Cu middle layer ? 6 mm Steel ID Layer at 35 second (Oliver, 1988)
112
Figure E.17: Equivalent stress in a composite tube with a 6 mm Steel OD layer ?
18 mm Cu middle layer ? 6 mm Steel ID Layer at 35 second (Oliver, 1988)
Figure E.18: Equivalent plastic strain in a composite tube with a 6 mm Steel OD
layer ? 18 mm Cu middle layer ? 6 mm Steel ID Layer at 10 second (Oliver, 1988)
113
Figure E.19: Total strain in a composite tube with a 6 mm Steel OD layer ? 18 mm
Cu middle layer ? 6 mm Steel ID Layer over time (Oliver, 1988)
Figure E.20: Total strain difference in a composite tube with a 6 mm Steel OD
layer ? 18 mm Cu middle layer ? 6 mm Steel ID Layer over time (Oliver, 1988)