Computational Methods For Creep Modeling
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Open Access Theses & Dissertations
2020-01-01
Computational Methods For Creep Modeling Computational Methods For Creep Modeling
Ricardo Vega University of Texas at El Paso
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COMPUTATIONAL METHODS FOR CREEP MODELING
RICARDO VEGA JR
Master’s Program in Mechanical Engineering
APPROVED:
Calvin M. Stewart, Ph.D., Chair
Yirong Lin, Ph.D.
Soheil Nazarian, Ph.D.
Stephen L. Crites, Jr., Ph.D.
Dean of the Graduate School
Copyright ©
by
Ricardo Vega Jr
2020
Dedication
Dedicado a mis amigos y mi familia, especialmente a mi mamá grande, quien me acompaña y ve
graduar desde su lugar en el cielo.
COMPUTATIONAL METHODS FOR CREEP MODELING
by
RICARDO VEGA JR, B.Sc.
THESIS
Presented to the Faculty of the Graduate School of
The University of Texas at El Paso
in Partial Fulfillment
of the Requirements
for the Degree of
MASTER OF SCIENCE
Department of Mechanical Engineering
THE UNIVERSITY OF TEXAS AT EL PASO
May 2020
v
Acknowledgements
I would like to acknowledge and thank Dr. Calvin M. Stewart for all these years of being
my advisor and my mentor through which he offered me help and guidance in my professional and
personal life. I would like to acknowledge and thank my family who are always there for me with
their unconditional love and support, no matter which path I traversed. I would like to thank my
close friends for always being there for me, pushing me to achieve more, and being my best
supporters. Lastly, I would like to thank the Department of Energy (DOE) National Energy
Technology Laboratory for funding my research under Award Number(s) DE-FE0027581.
vi
Abstract
Creep is an important phenomenon present in the power generation and aerospace industry.
Creep is a mechanism that degrades the quality of a part, such as the turbines used in different
industries, when elevated temperatures and loads are applied. Creep damage can cause critical
failures that can cost millions of dollars and as such, prevention is important. A multitude of creep
models exist to help with design and failure prevention. These models are applied using
computational tools which allows the engineer to properly predict the life of a part or other material
properties. In this work various creep numerical optimization methods are explored. In the first
study, minimum-creep-strain rate (MCSR) models are collected to create a “metamodel”. A
metamodel is a model of models which in this study, is composes of 9 MCSR models. The MCSR
is an important value for creep design as it tends to be the base of more complex models. Finding
the best data fitting MCSR model is important as it will help build complex models properly by
finding the corresponding MCSR based on needed stresses and temperatures. The metamodel can
be regressed to the original MCSR models to rapidly and consistently fit any of the base component
models; this is called the constrained metamodeling approach. The metamodel is also used with a
pseudo-constrained approach, a metamodeling approach that aims to automatically find the best
model autonomously while exploring new possible model forms. In the second study, the collected
MCSR models are used to create “material specific” creep continuum damage mechanics-based
constitutive models. Herein, material specific is defined as a constitutive model based on the
mechanism-informed MCSR equations found in deformation mechanism maps and calibrated to
available material data. The material specific models are created by finding the best MCSR model
for a dataset. Once the best MCSR model is found, the Monkman Grant inverse relationship
between the MCSR and rupture time is employed to derive a rupture equation. The equations are
vii
substituted into continuum damage mechanics-based creep strain rate and damage evolution
equations to furnish predictions of creep deformation and damage. Material specific modeling
allows for the derivation of creep constitutive models that are better tuned to the specific material
of interest and available material data. The material specific framework is also advantageous since
it has a systematic framework that moves from finding the best MCSR model, to rupture time, to
damage evolution and, creep strain rate. In the final study, objective functions for creep
constitutive models are evaluated and the best objective function for different categories of creep
data are determined. A plethora of creep constitutive models have been developed to predict the
stress-rupture, minimum-creep-strain-rate, creep deformation, and stress relaxation of materials.
Sometimes these models can be calibrated analytically but oftentimes numerical optimization is
required to generate the best predictions. In numerical optimization, the model, calibration data,
optimization algorithm, objective function, and error tolerance influence the accuracy of
predictions. The objective function is the function that compares the calibration data to model
predictions and is either minimizing or maximizing to a desired error tolerance. In the creep
modeling community, the objective function and error tolerance are rarely reported. Without this
information, it is extremely difficult to reproduce research published by the community. In this
study, twelve objective functions for creep are compiled. Four categories of creep data are
collected. A detailed analysis is performed to determine the best objective function and error
tolerance for each category of creep data.
viii
Table of Contents
Dedication ...................................................................................................................................... iii
Acknowledgements ..........................................................................................................................v
Abstract .......................................................................................................................................... vi
Table of Contents ......................................................................................................................... viii
List of Tables ................................................................................................................................ xii
List of Figures .............................................................................................................................. xiii
Chapter 1: Introduction ....................................................................................................................1
1.1 Motivation .........................................................................................................................1
1.2 Research Objectives ..........................................................................................................3
Development and Application of Minimum Creep Strain Rate
Metamodeling ..............................................................................................3
Development of “Material Specific” Creep Continuum Damage Mechanics-
Based Constitutive Equations ......................................................................3
Selection Process of Objective Functions for Creep Models................................4
ix
1.3 Organization ......................................................................................................................4
Chapter 2: Background ....................................................................................................................5
2.1 Creep and Creep Stages ....................................................................................................5
2.2 Secondary Creep ...............................................................................................................7
2.3 Creep Models ..................................................................................................................14
2.4 Computational Requirements..........................................................................................15
Chapter 3: MCSR Metamodeling ..................................................................................................18
3.1 Metamodeling .................................................................................................................18
3.2 Metamodel ......................................................................................................................19
3.3 Calibration.......................................................................................................................21
3.4 Material Database ...........................................................................................................23
3.5 Results and Discussion ...................................................................................................25
Isothermal ....................................................................................................................25
Arrhenius......................................................................................................................32
Chapter 4: “Material Specific” Constitutive Equations .................................................................39
4.1 Methodology and Data ....................................................................................................39
Material Deformation Mechanisms .............................................................................40
x
MCSR and Creep Rupture Models ..............................................................................40
Creep Strain and Damage Modeling ............................................................................41
Material Data ...............................................................................................................42
4.2 Results and Discussion ...................................................................................................44
Material Deformation Map ..........................................................................................44
MCSR and Creep Rupture Predictions ........................................................................45
Creep Strain and Damage ............................................................................................48
Chapter 5: Selection Process for Objective Functions with Creep Modeling ...............................52
5.1 Methodology ...................................................................................................................52
Model Selection ...........................................................................................................53
Solver Method and Settings .........................................................................................53
Generated Results and Evaluation Criteria ..................................................................55
5.2 Data .................................................................................................................................57
5.3 Results .............................................................................................................................58
Chapter 6: Conclusions and Future Work ......................................................................................65
6.1 Conclusions and Future Work ........................................................................................65
Development and Application of Minimum Creep Strain Rate
Metamodeling ............................................................................................65
xi
Development of “Material Specific” Creep Continuum Damage Mechanics-
Based Constitutive Equations ....................................................................66
Selection Process of Objective Functions for Creep Models..............................67
References ......................................................................................................................................68
Vita 73
xii
List of Tables
Table 2.1: MCSR Models ............................................................................................................. 11
Table 2.2: Deformation mechanisms and relationship to mathematical function [11] ................. 12
Table 2.3: Creep Rupture Models ................................................................................................. 13
Table 2.4: Objective Functions ..................................................................................................... 17
Table 3.1: Metamodel Regression Conditions .............................................................................. 20
Table 3.2: Material constant boundaries ....................................................................................... 22
Table 3.3: Material Properties for P91 [76] .................................................................................. 24
Table 3.4: NMSE Values for Isothermally Constrained and Pseudo-Constrained Models .......... 31
Table 3.5: NMSE Values for Arrhenius Constrained and Pseudo-Constrained Models .............. 38
Table 4.1: Nominal chemical composition (mass percentage) of Heat MGC for alloy P91[79] .. 43
Table 4.2: Creep data for P91 [79] ................................................................................................ 44
Table 4.3: MCSR models and their error values ........................................................................... 45
Table 4.4: Material constants for the JHK pair ............................................................................. 47
Table 4.5: Simulated MCSR with error differences ..................................................................... 47
Table 4.6: Simulated creep rupture with error differences ........................................................... 48
Table 4.7: Sinh Constants ............................................................................................................. 49
Table 4.8: Simulated final strains and actual final strains differences ......................................... 51
Table 5.2: Percent errors for objective functions used for the Wilshire MCSR model ................ 59
Table 5.3: Percent errors for objective functions used for the Wilshire creep rupture model ...... 60
Table 5.4: Material constants for the MCSR Wilshire model ...................................................... 62
Table 5.5: Material constants for the creep rupture Wilshire model ............................................ 63
xiii
List of Figures
Figure 1.1 – Example of an Industrial Gas Turbine (GE Heavy Duty Gas Turbine 7HA.03) [8] .. 2
Figure 1.2 – Creep Failure on an industrial gas turbine (Berkeley Research Company, Berkeley
California) [9] ................................................................................................................................. 2
Figure 2.1 – Creep strain curves varied with stress and temperature ............................................. 5
Figure 2.2 – Creep strain curve separated by regimes .................................................................... 6
Figure 2.3 – MCSR Map ................................................................................................................. 8
Figure 2.4 – Example of Deformation Mechanism Map ................................................................ 9
Figure 3.1 – MCSR Metamodel Flowchart................................................................................... 19
Figure 3.2 – MCSR Metamodel Flowchart ASTM P91 Data [76] ............................................... 23
Figure 3.3 – Simplified Norton MCSR Isothermally Constrained Fit .......................................... 25
Figure 3.4 – Norton MCSR Isothermally Constrained Fit ............................................................ 26
Figure 3.5 – JHK MCSR Isothermally Constrained Fit ................................................................ 26
Figure 3.6 – Nadai MCSR Isothermally Constrained Fit ............................................................. 27
Figure 3.7 – Soderberg MCSR Isothermally Constrained Fit....................................................... 27
Figure 3.8 – Dorn MCSR Isothermally Constrained Fit ............................................................... 28
Figure 3.9 – McVetty MCSR Isothermally Constrained Fit ......................................................... 28
Figure 3.10 – Garofalo MCSR Isothermally Constrained Fit ....................................................... 29
Figure 3.11 – Wilshire MCSR Isothermally Constrained Fit ....................................................... 29
Figure 3.12 – Pseudo-Constrained Isothermal Metamodel ........................................................... 30
Figure 3.13 – Simplified Norton MCSR Arrhenius Constrained Fit ............................................ 32
Figure 3.14 – Norton MCSR Arrhenius Constrained Fit .............................................................. 32
Figure 3.15 – JHK MCSR Arrhenius Constrained Fit .................................................................. 33
xiv
Figure 3.16 – Nadai MCSR Arrhenius Constrained Fit ................................................................ 33
Figure 3.17 – Soderberg MCSR Arrhenius Constrained Fit ......................................................... 34
Figure 3.18 – Dorn MCSR Arrhenius Constrained Fit ................................................................. 34
Figure 3.19 – McVetty MCSR Arrhenius Constrained Fit ........................................................... 35
Figure 3.20 – Garofalo MCSR Arrhenius Constrained Fit ........................................................... 35
Figure 3.21 – Wilshire MCSR Arrhenius Constrained Fit ........................................................... 36
Figure 3.22 – Pseudo-Constrained Isothermal Metamodel ........................................................... 37
Figure 4.1 – “Material specific” modeling framework ................................................................. 39
Figure 4.2 – Creep deformation curves for P91 at 100, 110, 120, 140, 160, and 200 MPa and
600°C [79] ..................................................................................................................................... 43
Figure 4.3 – Deformation Mechanism Map for P91 with experimental data plotted [11,80-81] . 44
Figure 4.4 – JHK MCSR model prediction with extra simulated temperatures 550, 650, and
700°C ............................................................................................................................................ 46
Figure 4.5 – JHK creep rupture prediction model prediction with extra simulated temperatures
550, 650, and 700°C ..................................................................................................................... 46
Figure 4.6 – Creep strain curves for P91 data for 100, 110, 120, 140, 160, and 200 MPa at 600°C
....................................................................................................................................................... 50
Figure 4.7 – Damage curves for P91 data for 100, 110, 120, 140, 160, and 200 MPa at 600°C .. 50
Figure 5.1 – Systematic procedure for finding the best objective function .................................. 52
Figure 5.2 – Example percent error data trend graph with positive slope and correlation ........... 56
Figure 5.3 – Raw 316SS creep rupture data from NIMS [76] ...................................................... 57
Figure 5.4 – Raw 304SS MCSR data from NIMS [76] ................................................................ 58
Figure 5.5 – 304SS MCSR Wilshire numerical and analytical fits .............................................. 61
xv
Figure 5.6 – 316SS creep rupture Wilshire numerical and analytical fits .................................... 62
Figure 5.7 – 304SS MCSR percent error trends ........................................................................... 63
Figure 5.8 – 316SS creep rupture percent error trends ................................................................. 64
1
Chapter 1: Introduction
1.1 Motivation
There is an increase in demand for improving the efficiency of energy generation in fossil
energy plants and Advanced Ultrasupercritical power plant. In order to increase the efficiency of
these power plants, high temperatures that can be above 1400°F and pressures above 4000 psi are
necessary. These elevated conditions can potentially increase the performance in power generation
plants by about 10%, however these elevated conditions can cause problems [1]. The industrial
gas turbines that are used such as the one shown in Figure 1.1 can experience various means of
failure because of the conditions and usage, but some of the main concerns, especially due to the
elevated temperatures and consistent loads cycles, are creep and fatigue failure[2-5]. Creep is
mechanical deformation due to a material experiencing a constant load under elevated
temperatures and fatigue is mechanical wear down due to a material experiencing alternating load
cycles. These mechanisms can cause failure in industrial gas turbines, which can sometimes prove
critical, such as that shown in Figure 1.2. The elevated temperatures can drastically cut the life of
the components in the gas turbines, as such constant maintenance is required in order to prevent
critical failure [3]. The maintenance that is performed on these gas turbines, specifically the
sections which are most susceptible to creep, make up 50-70% of the cost to maintaining the gas
turbine [6]. These routine maintenances are important as they happen frequently: they occur about
every 2 years for the components that experience the elevated temperatures directly, and every 4
to 5 years for major inspections [7].
2
Figure 1.1 – Example of an Industrial Gas Turbine (GE Heavy Duty Gas Turbine 7HA.03) [8]
Figure 1.2 – Creep Failure on an industrial gas turbine (Berkeley Research Company, Berkeley
California) [9]
Based on the damage, the cost of the needed maintenance can vary but if the damage is
controlled and kept to an acceptable minimum, the large costs can be drastically reduced, and
possible critical failures can be avoided. Due to the benefits and safety concerns related to
preventing failure, modeling, and designing for the life of the gas turbine components is an
important endeavor. However, these mechanisms have limited reference data due to their nature
of needing long periods of time [10]. Currently there are several models that have been developed
to aide in the interpretation and prediction of various mechanical phenomena from which later
some will be discussed. However, oftentimes engineers face the problem of trying to define which
model would aid them for their project or situation. The engineer needs a model that fits the
available data properly, generates realistic results, and offers proper insight into the properties that
3
the engineer is interested such as creep rupture. There is a need for developing tools and processes
that engineers can consistently and easily use to determine the best models for predicting the
properties they are interested in.
1.2 Research Objectives
The objective of the research is to explore different computational methods and apply them
to model creep and its various parameters. Three different studies are conducted and they each
hold their own objectives.
Development and Application of Minimum Creep Strain Rate Metamodeling
The objective of this study is to create a “metamodel” out of 9 MCSR models. The
metamodel will be used under a constrained modeling approach to regress the model into its base
forms and find the best model for a given data set manually. The metamodel will also be used
under a pseudo-constrained approach to attempt to find the best model for a given dataset
autonomously. The pseudo-constrained approach will also be used to try and find new form models
generated from the metamodel.
Development of “Material Specific” Creep Continuum Damage Mechanics-Based Constitutive
Equations
The objective of this study is to demonstrate a framework by which one can develop
“material specific” creep continuum damage mechanics-based constitutive equations. This is done
by evaluating the available MCSR models and determining their material mechanism affinity. The
best fit and proper material mechanism MCSR model is found and used to determine the
corresponding creep rupture mode. The found MCSR and creep rupture constants are used to
generate creep deformation and damage plots using the Sinh model.
4
Selection Process of Objective Functions for Creep Models
The objective of this study is to find the best objective function for four different categories
of creep data that are collected. Twelve different objective functions will be evaluated and the ones
that generate the best numerical fit will be recommended. The best numerical fits will be compared
to analytical fits when possible to back up generated fits.
1.3 Organization
The present work is organized as follows. Chapter 2 offers background information on
creep and creep mechanics. The different stages of creep are lightly covered with a focus on
secondary creep. Secondary creep is covered in depth and various MCSR models are presented.
Deformation mechanism maps, along with their relationship with the MCSR are discussed. Creep
damage mechanics models are also discussed. Chapter 3 follows the “metamodeling” study. The
creation of the metamodel and its properties are covered in depth. The results of the constrained
and pseudo-constrained approaches using the metamodel are presented and discussed. Chapter 4
covers the development of the “material specific” creep continuum damage mechanics based
constitutive equations. The chapter covers the process of creating the necessary modeling
equations and the models that are used. The process covers material data, material deformation
map, and the generated creep constants. The generated results are presented and evaluated. Chapter
5 covers the study performed on objective functions. Different objective functions are collected
and evaluated when applied to creep data. The results are evaluated and compared to analytical
results when possible. The final chapter, chapter 6 covers the conclusion and future work for all
the studies that were performed.
5
Chapter 2: Background
2.1 Creep and Creep Stages
Creep is a mechanism that happens when a material is subjected to an elevated temperature
(usually referred to as the activation temperature which can be between 0.3 0.6m mT T T where
mT is the melting temperature,) and a constant applied stress that causes the material to plastically
deform under strain [11]. Creep is a thermally activated mechanism and as such it is heavily
dependent on temperature. Other creep factors include time, applied stress, material composition,
and material shape. Creep failure works in the manner that at high temperatures, under a constant
applied stress, the material will fail at a quicker rate due to degradation of the material
microstructure, whereas at a low temperature, the material will fail at a at a later time, sometimes
by a difference of years.
Figure 2.1 – Creep strain curves varied with stress and temperature
Cre
ep
Str
ain
, ε c
r
Time, t
min,1 3 2 3
3 2 1
min,3 min,2 min,1
T T T
1 1,T
2 2,T
3 3,T
min,2
min,3
6
Figure 2.2 – Creep strain curve separated by regimes
Creep deformation can be split into three principal regimes: primary, secondary, and
tertiary [12]. The regimes each represent the conditions and processes a material experiences under
the different temperature, stress, and strain rate conditions. A typical creep deformation plot varied
with temperature and stress is depicted in Figure 2.1 and the region splits between all 3 sections is
depicted in Figure 2.2.
The first stage is the primary creep stage which is also called the transient creep stage. In
this stage, there is hardening behavior which is shown by a high scaling strain rate which slows
down as the regime moves on [11]. This stage can be short and instantaneous which has made it
such that in some modeling instances, the primary regime is ignored. The next stage is the
secondary creep regime, or steady-state creep. This regime tends to be one of the largest
components of creep damage, except when dealing with high stresses and temperatures where the
region can be collapsed into a single point. This region is defined by the constant strain rate because
of a balance in hardening and recovery mechanisms [11]. The minimum-creep-strain-rate (MCSR)
is derived from this material behavior as it is the slope of the secondary regime. The MCSR is an
7
important value and will be covered in more depth in a later section as it is a foundation block for
the formation of more complex creep models. In the final region, the tertiary regime, or the
accelerating creep, the creep rate speeds up to the point of failure. This region holds the point of
failure or rupture, and as such holds a large portion of the expansion or formation of failure driving
mechanisms such as crack propagations or void formation and expansion. Based on material
properties, this region can be large or small, but it is always defined by the failure of the material
[13-14].
2.2 Secondary Creep
The minimum-creep-strain-rate (MCSR) is one of the earliest creep parameters measured
from materials. Since the 1929 Norton-power law, researchers have developed MCSR models to
predict the MCSR creep behavior [15]. The MCSR is the slowest strain rate observed during an
isostress-isotherm creep deformation test and is invariant at a set isostress and isotherm. In creep
constitutive models, the MCR is exploited as a basis from which strain hardening, recovery, and
microstructural damage mechanisms may be evolved to predict the full creep deformation curve.
For example, the 2015 Sinh Model is a continuum-damage-mechanics based model capable of
predicting the secondary and tertiary creep regimes. The Sinh creep-strain-rate takes the form
( )3 2sinh expcr
s
A
=
(1)
where A and s are material constants, is a unitless material constant, and is damage that
ranges from 0 1 where 1 = indicates rupture [16-19]. The 1943 McVetty MCSR law exists
inside of Sinh [Eq.(2)] as
min sinhs
A
=
(2)
8
where temperature-dependence is not explicitly defined. In this example, the accuracy of the
MCSR model directly affects the accuracy of creep deformation predictions. This situation is true
for most creep deformation constitutive models [18,20-23].
Figure 2.3 – MCSR Map
The MSCR arises from three creep mechanisms including diffusional-flow, power-law,
and breakdown with distinct slopes, in as illustrated in Figure 2.3. Diffusional flow occurs at low
stress and a wide temperature range. Diffusional flow, also called Harper-Dorn creep, is often
segregated into the boundary and lattice diffusion corresponding to Nabarro-Herring and Coble
creep respectively [24-26]. Diffusional flow is controversial as some parties support diffusional
flow only in pure metals, some argue that it is present within all alloys, while others doubt its
existence entirely [27]. The slope, n tends to unity [25]. The power-law regime, also called five-
power-law, contains the largest amount of experimental data and is defined by moderate stress and
UTS
minM
inim
um
Cre
ep
Str
ain
Ra
teNote: log-log scale
BD
3n
1n
Diffusional
Flow
Stress
2n
1T2T
3T4T
Power-Law Breakdown
9
temperature. The slope, n , is typically 5 but can exist between 2 and 12 depending on the creep
resistance of a material [26-33]. Breakdown is observed at elevated temperature and stress, where
there is a shift from climb-controlled to glide-controlled flow. The slope, n is above 12, even up
to 40, depending on the alloy [34].
Figure 2.4 – Example of Deformation Mechanism Map
Deformation mechanism maps are maps that were developed by Frost and Ashby and are
related to the MCSR maps as mechanisms are shared [24]. The maps depict the mechanisms that
a material of interest can undergo along with required conditions for said mechanism such as stress,
temperature, and strain-rate as depicted in Figure 2.4.
Deformation mechanism maps plot strain-rate as a function of stress and temperature.
Strain-rate is plotted as isolines. When there is observed a shift in the slope of the isolines, a
mechanism transition has occurred. Although these maps can be conditional in reliance and are
not exact, they offer useful insight for knowing what kind of mechanisms a material of interest is
undergoing, which in most cases can be more than one. With the assistance of a deformation map,
Dislocation Glide
Dislocation Creep
Nabarro-Herring Creep
Coble Creep
Theoretical StrengthStress Ratio
Temperature Ratio
/ G
/ mT T
10
experimental data can be categorized into the dominant mechanism which allows for proper model
selection, specifically when determining the best MCSR model with optimal active mechanism.
Numerous constitutive laws have been developed to predict MCSR as a function of stress
and temperature. The functional form of phenomenological and mechanistic MCSR models are
similar; diverging only in how the material constants are determined [35]. Nine phenomenological
MCSR models are listed in Table 2.1. The functional form of these MCSR models differ; responses
ranging from linear to nonlinear on a log-log scale. Many MCSR models are not capable of
modeling all three creep mechanisms illustrated in the MCSR map and the deformation mechanism
map as some models have an affinity to certain mechanisms [11,36-40]. Ultimately, the fit ability
of a MCSR model is dependent on its functional form, the expected material behavior, and data
available for fitting. For example, if data is only available for a single mechanism, a linear model
will produce superior predictions to a nonlinear model. By matching the experimental data of
interest to the correct mechanism-based model, a more realistic and material informed result can
be obtained if desired. The model form and their corresponding mechanism are demonstrated in
Table 2.1 and Table 2.2.
11
Table 2.1: MCSR Models
Name Model Eq.
Simplified Norton
(SN), 1929
[15] min
nA = (3)
Norton (N), 1929
[15] min0
n
A
= (4)
Nadai (Na), 1931
[41] min0
1expA c
= + (5)
Soderberg (S), 1936
[42] min0
exp 1A
= − (6)
McVetty (M), 1943
[43] min0
sinhA
= (7)
Dorn (D), 1955
[44] min0
expA
= (8)
Johnson-Henderson-
Kahn (JHK), 1936
[45]
1 2
1 20
n0
mi[ ]
n n
A A
= + (9)
Garofalo (G), 1965
[37] min0
sinh
n
A
= (10)
Wilshire (W), 2007
[46]
1
2minln( )
v
TS
k
= − (11)
12
Table 2.2: Deformation mechanisms and relationship to mathematical function [11]
Deformation
Mechanism
Functional Form Ref
Power-Law creep exp n
cr
Q
kT
−
[15,24,47]
Diffusional flow expcr
Q
kT
−
[25,48-52]
Linear+ Power-Law ( )exp sinhcr
QA
kT
−
[53-54]
Power-Law
Breakdown ( )exp expcr
QC
kT
−
[35]
Power-law +
Breakdown ( )exp lncr
QB
kT
− −
[46,55-57]
Temperature dependence is taken into consideration by using the Arrhenius function. The
Arrhenius function is derived by plotting the natural logarithm ( ln ) of min , the MCSR, against
the reciprocal of the absolute temperature (1/ )T at constant stress, [28]:
min exp( )cQ
CRT
−
= (12)
In the Arrhenius function, cQ is the creep activation energy with units of J/mol, R is the universal
gas constant with a value of 8.31 J/mol*K, T is the temperature of interest in Kelvin. The variable
C is replaced with the stress dependent function, or the MCSR model of interest. The activation
energy, cQ is the amount of energy needed for the creep process to start which varies from one
material to another. Although cQ is a material constant, its value is dependent on the active
mechanisms. If you move regions in the deformation mechanism map, the cQ can change.
[36,40,58].
The MCSR models can be used to find creep rupture models using the Monkman-Grant
equation [59].
13
min
MGr
kt
= (13)
where r
t is the time to rupture, min is the MCSR and
MGk is the Monkman-Grant constant. The
inverse relationship allows for the determination of a corresponding creep rupture equation by
rearranging the equation. Since many MCSR equations exist, a collection of rupture models for a
range of creep mechanisms can be acquired through the inverse method. The modified creep
rupture equations are presented in Table 2.3.
Table 2.3: Creep Rupture Models
Name Model Eq.
Simplified Norton
(SN), 1929
[15] ( )
1
r
mt B−
= (14)
Norton (N), 1929
[15]
1
r
t
m
Bt
−
= (15)
Nadai (Na), 1931
[41]
1
1exp
r
t
t B d
−
= + (16)
Soderberg (S), 1936
[42]
1
exp 1r
t
t B
−
= − (17)
McVetty (M), 1943
[43]
1
sinhr
t
t B
−
= (18)
Dorn (D), 1955
[44] exp
r
t
t B
= (19)
Johnson-Henderson-
Kahn (JHK), 1936
[45]
1
1 2
1 2[ ]
r
m m
t t
t B B
−
= + (20)
Garofalo (G), 1965
[37]
1
sinhr
m
t
t B
−
= (21)
Wilshire (W), 2007
[46]
11
1ln( )
r
u
TS
t k
−
= − (22)
14
2.3 Creep Models
Models which can incorporate multiple creep parameters or phenomena such as those that
are multistage can be used to have a better insight of creep behavior. Models which can describe
multiple sections of the characteristic creep deformation curve such as the MPC Omega model and
the Theta projection model [20-21]. The MPC Omega model focuses on modeling the secondary
and tertiary creep regime. The model is reliable in predicting creep as recognized by the American
Petroleum Institute (API 579, Fitness-for-service, 2000), however it does have its limitations such
as not being able to properly capture the primary creep regime in materials for which various
adaptations have been made [60]. The Theta projection model can predict the complete creep
curve: the primary, secondary, and tertiary creep regimes.
Models have also been created by using continuum damage mechanics (CDM), which is a
concept that the previously discussed models have not incorporated. Continuum damage
mechanics focuses on the meso-scale of a material, in which a representative volume undergoes
average deformities that happen at a smaller scale [61]. The discrete defects such as cracks and
voids are averaged over a volume such that damage can be taken into consideration when using
CDM. Kachanov used CDM in this approach, in which damage is taken as an average
representation on the representative volume [62]. Rabotnov applied it to creep phenomena,
creating the Kachanov-Rabotnov model [63]. The Kachanov-Rabotnov model has been used and
studied in multiple instances to produce acceptable results; however, limitations have also been
found. A model that uses CDM and tackles the Kachanov-Rabotnov model’s problems is the Sinh
model [22]. The Sinh model’s reliability and ability to model creep has been demonstrated and
compared to other creep models in previous studies [16].
15
2.4 Computational Requirements
When using any creep model, the constants can be found trough analytical means or
through numerical and computational means. When finding the constants analytically, there is
usually an established procedure that can be followed step-by-step to find the material constants
of a model.
Analytical methods are effective. Cano’s paper uses analytical methods to find the material
constants for the Wilshire model to find creep strain fits of high fidelity [64]. Although analytical
methods are effective due to their consistent procedures and results, this approach is not always
possible since some models do not lend themselves to analytical solutions. The numerical (called
computational in some instances) approach is used in instances where the analytical approach is
not possible, when proper constant ranges are known, or when an algorithm for solving the models
can be created and used effectively. Computational methods rely on an optimization algorithm that
uses initial conditions, constant boundaries, and an objective function to find the optimized results.
Many algorithms exist and each of them can differ in form, application, and requirements. Some
examples of optimization algorithms are the branch and bound, ant colony, and the particle swarm
algorithm [65-67]. Initial conditions are the starting initial guesses for material constants that are
used by the algorithm. The initial conditions in an algorithm are needed for the purpose of having
an initial set of values that the algorithm can use to solve a given problem. The initial conditions
should reflect similar values of what is to be expected of the solutions found for the given problem
as improper initial conditions can generate inaccurate or even at times, unrealistic results. The
constant boundaries are a set of boundaries created such that the algorithm does not go outside of
the realistically possible values for the solution of the problem. These boundaries can be found
trough different methods such as using known max and mins for material constants that are known,
16
or through mathematical analysis of the equations used where one should even use infinite
conditions. Finally, the objective function is the criteria of convergence for the optimization
algorithm. The objective function will be optimized, meaning that it will be minimized,
maximized, or set to another given value, using the chosen optimization algorithm. Once the value
of the objective function comes as close as possible to the given convergence point, the solution
to the problem is found.
The objective functions can take many forms, but one of the most common and basic
equations for it is
exp, ,
1
1 N
i sim i
i
Obj X XN =
= − (23)
which can be called the mean difference error. The N represent the number of points, the exp,iX
refers to the experimental data or actual data and, ,sim iX is the simulated data point. Usually, when
using this type of objective function, an optimization algorithm aims to reduce the value of the
objective or make it as close as possible to 0. This process brings the experimental and simulated
values closer together. It is important that the number of points used is taken into consideration
since multiple data sets with varying data point quantitates can be studied using the same objective
function. The objective function can potentially pose as an obstacle, and as such multiple objective
functions should be considered. A collection of possible objective functions is given in Table 2.4.
It is important to note that the table does not contain all the possible objective function that can be
used. The table is offered as a starter set of objective functions and other functions can be used or
developed, however it is always important to take into consideration which objective function
might suit a specific problem as they can all possibly yield different results, some of which are
better than others.
17
Table 2.4: Objective Functions
Name Objective Function
Mean Difference
Error exp, ,
1
1 N
i sim i
i
X XN =
− (24)
Absolute
Difference Error exp, ,
1
1 N
i sim i
i
X XN =
− (25)
Squared Mean
Difference Error ( )
2
exp, ,
1
1 N
i sim i
i
X XN =
− (26)
Logged Difference
Error exp, ,
1
1log( ) log( )
N
i sim i
i
X XN =
− (27)
Normalized Mean
Squared Error
2
exp, ,
exp1
exp exp,
1
,
1
( )1
1( )
1( )
Ni sim i
simi
N
i
i
N
sim sim i
i
X X
N X X
X XN
X XN
=
=
=
−
=
=
(28)
Logged
Normalized Mean
Squared Error
2
exp, ,
exp1
exp exp,
1
,
1
(log( ) log( ))1
1log( )
1log( )
Ni sim i
simi
N
i
i
N
sim sim i
i
X X
N X X
X XN
X XN
=
=
=
−
=
=
(29)
Root Logged Mean
Squared Difference
Error
2
exp, ,
1
1(log( ) log( ))
N
i sim i
i
X XN =
− (30)
Normalized by the
Max Squared Error
2
exp, ,
1 exp,
( )1 Ni sim i
i Max
X X
N X=
−
(31)
Normalized by the
Min Squared Error
2
exp, ,
1 exp,
( )1 Ni sim i
i Min
X X
N X=
−
(32)
Normalized by the
Range Squared
Error
2
exp, ,
1 exp, exp,
( )1 Ni sim i
i Max Min
X X
N X X=
− −
(33)
Normalized by the
Max Logged
Squared Error
2
exp, ,
1 exp,
(log( ) log( ))1
log( )
Ni sim i
i Max
X X
N X=
−
(34)
Normalized by the
Min Logged
Squared Error
2
exp, ,
1 exp,
(log( ) log( ))1
log( )
Ni sim i
i Min
X X
N X=
−
(35)
Normalized by the
Range Logged
Squared Error
2
exp, ,
1 exp, exp,
(log( ) log( ))1
log( ) log( )
Ni sim i
i Max Min
X X
N X X=
− −
(36)
18
Chapter 3: MCSR Metamodeling
3.1 Metamodeling
Metamodeling is the process of applying mathematical rules and constraints to generate
models-of-models. These models-of-models, or “metamodels”, exist as a mathematical
combination of known models that can regress back into each known model under prescribed
constraints. When using a metamodel, the calibration process for each known model becomes
singular and thus simplified. Additionally, metamodels can be employed in an unconstrained or
pseudo-constrained manner to identify unique MCSR models that exist between the known
models.
Metamodeling has been applied to other creep problems in the past. The successful
metamodel of creep constitutive models can only be achieved by first metamodeling the MCSR
problem. Gorash et al have shown the development of models akin to the metamodel that are
formed through the base MCSR models [11,68-69]. The metamodel was designed to reflect the
three MCSR regimes. Early work focused on metamodeling Time-Temperature-Parameters (TTP)
for creep rupture-prediction [70-73]. Recently, a TTP metamodel capable of combining and
regressing into twelve known TTP models was developed and exploited to determine the optimal
TTP model for several alloys [74]. Metamodeling has been attempted on other creep constitutive
models with a framework for metamodeling the MPC Omega, Theta, and Sin-Hyperbolic
constitutive models producing limited results [75].
19
3.2 Metamodel
Figure 3.1 – MCSR Metamodel Flowchart
A unified MCSR metamodel is derived from the seven MCSR models listed in Table 2.1
following the flowchart illustrated in Figure 3.1. The individual MCSR models are collected into
level 1 metamodels associated with functional form: power, exponential, and sin-hyperbolic.
The level 1 metamodels are linearly summed to create the unified metamodel (level 2)
below
1 2 3
1
3.
1min 1 2 3 4 2
2
ln
sinh exp
v
n n n
TS
o o o o s
aa
A A A A ac k
= + + + + − +
(37)
where 1 2 3 4 0 1 2 3, , , , , , , , , ,sA A A A n n n c v and 2k are material constants along 1 2, , and 3 which are
Heaviside functions that step between zero and one. The unified metamodel can be regressed back
into one of the seven MCSR models by setting the constraints listed in Table 3.1 where the unlisted
variables are set to zero. The metamodel holds 15 material constants. To add temperature
20
dependence, the metamodel is just simply multiplied by the Arrhenius function, creating the
temperature dependent metamodel.
1 2 3
1
3 *.1
min 1 2 3 4 2
2
ln
sinh exp exp
v
n n n
TS c
o o o o s
aQa
A A A A ac k RT
= + + + + − + −
(38)
To regress the metamodels into the appropriate constitutive base model, different
conditions need to be applied to the constants. These conditions are highlighted in Table 3.1. It is
important to note that unlisted variables are set to 0 except for 2, ,sc k and v which stay as 1
otherwise stated. These constants are set to 1 since setting them to 0 would mathematically break
the model as these constants are denominators.
Table 3.1: Metamodel Regression Conditions
MCSR Model Constraints Material
Constants
Simplified Norton
(SN) 1 1 00, 1, 1A n = 3
Norton (N) 1 1 00, 1, 1A n 3
Johnson,
Henderson, Kahn
(JHK) 1 1 0 2 20, 1, 0, 0, 1A n A n 5
Nadai (Na) 4 0 1
10, 1, 1, , 1s
s
A c cc
= = 4
Soderberg (S) 4 0 00, 1, 0A = 3
Dorn (D) 4 00, 0A 2
McVetty (M) 3 3 00, 1, 0A n = 3
Garofalo (G) 3 3 00, 1, 0A n 3
Wilshire (W) 2 0 31, 0, 1, 1k v = = 4
In the pseudo-constrained mode, the unified metamodel [Eq.(37)] is modified with the
addition of seven Heaviside functions as follows
21
31 2
min 1 1 2 2 3 3
0 0 0
1
7 3
5 14 4 6 2
2
( ) ( ) ( ) ( ) ( ) sinh( )
( ) ln( )
( ) exp ( ) ,
0 0
( ) 1 0
1 0
2
nn n
v
TS
o s
i
i i
i
H x A H x A H x A
H x aH x a
H x A H x ac k
x
H x x
x
= + + +
+ − +
= =
(39)
where the discrete variables 1,2,3,4,5,6,7ix = must be optimized. The Heaviside functions act as
switches that enable/disable terms within the metamodel. The unified metamodel [Eq.(39)] can be
used to identify novel MCSR models that exist at the interface between the seven base models.
Just as the previous metamodel from [Eq.(38)] without the Heaviside functions, [Eq.(39)] can take
into account temperature dependence by simply multiplying the metamodel with the Arrhenius
function.
31 2
1 1 2 2 3 3
0 0 0
1*
min7 3
5 14 4 6 2
2
( ) ( ) ( ) ( ) ( ) sinh( )
exp( ) ln( )
( ) exp ( )
nn n
vc
TS
o s
H x A H x A H x A
QH x a
RTH x aH x A H x a
c k
+ + +
= − + − +
(40)
3.3 Calibration
The calibration process for the metamodel and subsequent seven MCR models is written into
the MATLAB programming language. Calibration is performed by minimizing an objective
function per isotherm. The vector-valued objective function, ( )f x follows
22
( )
( ) ( ) ( ) ( )
exp
1 2
log( ) log( ) ,
;
sim
n
f x x x
f x f x f x f x
= −
=
(41)
where expx and simx are vectors of the experimental and simulated min respectively. The objective
function uses a logarithm of base ten such that small and large min are given a more equal weight.
The objective function is minimized using the built-in nonlinear least-squares solver (lsqnonlin)
2 2 2 2
1 22min ( ) min( ( ) ( ) ... ( ) )n
x xf x f x f x f x= + + + (42)
where ( )f x is the vector-valued objective function, 2
2( )f x is the scalar-valued sum of the
squares, and n denotes the length of the vector. The upper and lower bounds for the material
constants are defined in Table 3.2 where [ , ] denotes a closed interval and [ .. ] denotes an integer
interval.
Table 3.2: Material constant boundaries
iA in 1 0 ix
( 1hr− ) unitless unitless (MPa) unitless
[0., 1] [1, 40] [0., 1] [1, ] [0., 1]
Post-optimization, the error is reported using the Normalized-Mean Square Error, NMSE .
The Normalized-Mean Square Error, which is modified to use logarithmic values is as follows
2/ exp /exp, exp,
/ exp /expexp
2, exp,
exp1
log( )( )1
log( )
( )1
lsim l simsim i i
lsim l simsim
Nsim i i
simi
x xx xOBJ
N x x x x
x xNMSE
N x x=
=−=
=
−=
(43)
where expx and simx are vectors of the experimental and simulated min respectively which are
logged and where expx and simx are the mean of the vectors which are also logged. The NMSE is
23
used for comparative analysis due to being normalized by the number of data points and average
product of the simulate and experimental vectors.
3.4 Material Database
Figure 3.2 – MCSR Metamodel Flowchart ASTM P91 Data [76]
Alloy 9Cr-1Mo-V-Nb (ASTM P91) was selected to demonstrate the utility of the unified
metamodel. The MCSR was gathered from the National Institute of Material Science (NIMS) [76].
The material data held a total of 135 points with temperatures ranging from 450°C to 700°C. The
data points and the range of the temperatures is presented in Figure 3.2. The material properties
are also given and found in Table 3.3. Values with an asterisk next to them were found trough
interpolation.
24
Table 3.3: Material Properties for P91 [76]
Temp (°C)
Tube Plate Pipe
Yield
(MPa)
Tensile
(MPa)
Yield
(MPa)
Tensile
(MPa)
Yield
(MPa)
Tensile
(MPa)
Room 53 707.2 523.8 685 503 667
100 505.2 658.3 494.5 635.8 478 621
200 485.7 620.8 470.8 589.3 453 583
300 478.3 592.5 455.3 561 444 554
400 453.5 568.7 436.5 542.3 424 533
450 425.4* 527.7* 414.5 516.8 410 512
500 397.3 486.7 391.3 469.8 379 464
550 345.8 418 345.8 414 337 402
575 309.4* 382.1* 305.8* 375.5* 305* 367.5*
600 273 346.2 265.8 337 273 333
625 227.4* 309.1* 223* 306.1* 231.5* 299.5*
650 181.8 272 180.3 275.3 190 266
700 124.2 139.3 115.5 205.3 120 200
* Indicates Interpolated Values
An important material constant was also found before the algorithm was applied. An
analytical method was used for finding the activation energy needed for the Arrhenius function.
The value that was used was that of 286.65 kJmol-1 which was found in Cano’s study that used the
same database [77].
25
3.5 Results and Discussion
The results are presented by first doing an isothermal analysis with the constrained and
pseudo-constrained approach. The isothermal analysis uses [Eq.(37)] with the constrained
modeling approach and [Eq.(39)] for the isothermal pseudo-constrained approach. The Arrhenius
temperature dependent modeling is then applied using the constrained approach with [Eq.(38)] and
the pseudo-constrained approach is used on [Eq.(40)].
Isothermal
Figure 3.3 – Simplified Norton MCSR Isothermally Constrained Fit
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
600°C Raw
650°C Raw
450°C Sim
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
26
Figure 3.4 – Norton MCSR Isothermally Constrained Fit
Figure 3.5 – JHK MCSR Isothermally Constrained Fit
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
600°C Raw
650°C Raw
450°C Sim
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
600°C Raw
650°C Raw
450°C Sim
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
27
Figure 3.6 – Nadai MCSR Isothermally Constrained Fit
Figure 3.7 – Soderberg MCSR Isothermally Constrained Fit
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
600°C Raw
650°C Raw
450°C Sim
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
600°C Raw
650°C Raw
450°C Sim
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
28
Figure 3.8 – Dorn MCSR Isothermally Constrained Fit
Figure 3.9 – McVetty MCSR Isothermally Constrained Fit
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
600°C Raw
650°C Raw
450°C Sim
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
600°C Raw
650°C Raw
450°C Sim
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
29
Figure 3.10 – Garofalo MCSR Isothermally Constrained Fit
Stress, (MPa)
100
MC
R,
min
(%h
r-1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
600°C Raw
650°C Raw
450°C Sim
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
Figure 3.11 – Wilshire MCSR Isothermally Constrained Fit
The generated fits for all 9 model after they have been regressed are presented in Figure
3.3-Figure 3.11. It is important to note that some isotherms from the raw data had to be culled due
to the amount of material constants that [Eq.(37)] holds. Isotherms that did not have enough points
for each material constants had to be culled. There were also problems fitting the simplified Norton
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
600°C Raw
650°C Raw
450°C Sim
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
30
model with this modeling approach. To compensate, the model was fit using other means and
should not be considered since its calibration approach was different than that of the other models.
It is also important to note than on many of the model fits the some of the isothermal lines intersect
with each other, which is an unrealistic creep behavior.
Figure 3.12 – Pseudo-Constrained Isothermal Metamodel
The metamodel from [Eq.(39)] was solved using the pseudo-constrained approach and the
generated fits are given in Figure 3.12. The metamodel ended up autonomously regressing to the
form of
3
1
3
1min 3 4
0 2
1( ) ln2
sinh( ) exp
v
n o
o s
aa
A Ac k
= + + +
(44)
where the Heaviside functions were able to successfully turn off some sections of the model while
the final Heaviside function failed to calibrate, leaving behind a 1/2. The pseudo-constrained
approach for the isothermal approach didn’t completely work as only isotherms 650°C and 500°C
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
500°C Raw
550°C Raw
600°C Raw
650°C Raw
500°C Sim
550°C Sim
600°C Sim
650°C Sim
.
31
were successfully fit while the other isotherms failed. Since the whole data set was unable to be
fit, the pseudo-constrained approach is considered to have failed, even though a new intermediate
model was found. An interesting find was that the initial condition that was given to the pseudo-
constrained modeling approach altered the fits drastically, indicating that the initial conditions
have high sensitivity when dealing with the metamodel.
The errors per isotherm for each of the models available are presented in Table 3.4. The
best models were found to be the Wilshire and Garofalo model as they had the lowest NMSE
values. The pseudo-constrained metamodel has the worst NMSE since it was unable to fit most of
the isotherms.
Table 3.4: NMSE Values for Isothermally Constrained and Pseudo-Constrained Models
Model NMSE
500°C
NMSE
550°C
NMSE
600°C
NMSE
650°C
Overall
NMSE
Simplified
Norton 0.0052 0.0121 0.0139 0.0179 0.0491
Norton 0.0046 0.0016 0.0029 0.0060 0.0151
Johnson-
Henderson-
Kahn
0.0033 0.0079 0.0038 0.0108 0.0258
Nadai 0.0015 0.0028 0.0025 0.0050 0.0118
Soderberg 0.0025 0.0026 0.0021 0.0050 0.0122
Dorn 0.0018 0.0029 0.0021 0.0050 0.0118
McVetty 0.0016 0.0027 0.0021 0.0046 0.0110
Garofalo 0.0016 0.0022 0.0021 0.0046 0.0105
Wilshire 0.0015 0.0021 0.0022 0.0048 0.0106
Pseudo-
Constrained
Metamodel
0.0021 0.8061 0.7567 0.0051 1.57
32
Arrhenius
Figure 3.13 – Simplified Norton MCSR Arrhenius Constrained Fit
Figure 3.14 – Norton MCSR Arrhenius Constrained Fit
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
33
Figure 3.15 – JHK MCSR Arrhenius Constrained Fit
Figure 3.16 – Nadai MCSR Arrhenius Constrained Fit
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
34
Figure 3.17 – Soderberg MCSR Arrhenius Constrained Fit
Figure 3.18 – Dorn MCSR Arrhenius Constrained Fit
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
35
Figure 3.19 – McVetty MCSR Arrhenius Constrained Fit
Figure 3.20 – Garofalo MCSR Arrhenius Constrained Fit
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
36
Figure 3.21 – Wilshire MCSR Arrhenius Constrained Fit
The generated fits for all 9 model after they have been regressed are presented in Figure
3.13-Figure 3.21. There were also problems fitting the simplified Norton, Norton, and JHK model
with this modeling approach due to some modeling constraints with the algorithm. To compensate,
the models were fit using other means and should not be considered since its calibration approach
was different than that of the other models. This approach holds the benefit of having all isotherm
fits be separated from each other and not intersect. The data fits also intersect through the middle
of each data isotherm.
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
37
Figure 3.22 – Pseudo-Constrained Isothermal Metamodel
The metamodel from [Eq.(40)] was solved using the pseudo-constrained approach and the
generated fits are given in Figure 3.22. The metamodel ended up autonomously regressing to the
form of
31 2
min 1 2 3
0 0 0
1
31
4 2
2
1 1 1( ) ( ) ( ) ( ) ( ) sinh( )2 2 2
11 ( ) ln( ) 21 12( ) exp ( )2 2
nn n
v
TS
o s
A A A
aaA a
c k
= + +
+ + − +
(45)
where the Heaviside functions were completely unable to turn on or off any sections of the
metamodel, leaving behind a 1/2 in all sections where the Heaviside functions were located. The
pseudo-constrained approach for the Arrhenius metamodel completely failed as no model was
found and the data fits were not even close to the actual data. Since the whole data set was unable
to be fit, the pseudo-constrained approach is considered to have failed for this approach as well,
although this approach failed to a greater degree than the previous metamodel since no isotherms
Stress, (MPa)
100
MC
R,
min
(%hr-
1)
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
1e+2
1e+3
450°C Raw
500°C Raw
550°C Raw
575°C Raw
600°C Raw
625°C Raw
650°C Raw
700°C Raw
450°C Sim
500°C Sim
550°C Sim
575°C Sim
600°C Sim
625°C Sim
650°C Sim
700°C Sim.
38
were fit. The initial conditions also had a high sensitivity and changed the fits but to no benefit or
improvement of the fits.
The average errors for all the models under the Arrhenius condition are presented in Table
3.5. The best model for this approach was the Wilshire model since it had the lowest NMSE. The
pseudo-constrained metamodel has the worst NMSE since it was unable to fit any of the isotherms.
It should also be pointed that the models with asterisk next to them should not be considered due
to them being optimized trough different computational methods.
Table 3.5: NMSE Values for Arrhenius Constrained and Pseudo-Constrained Models
Model Overall NMSE
Simplified Norton* 0.05159
Norton* 0.05159
Johnson-Henderson-Kahn* 0.04267
Nadai 0.04124
Soderberg 0.04125
Dorn 0.04125
McVetty 0.04221
Garofalo 0.04087
Wilshire 0.0288
Pseudo-Constrained Metamodel 5.49E6
39
Chapter 4: “Material Specific” Constitutive Equations
4.1 Methodology and Data
Figure 4.1 – “Material specific” modeling framework
MCSR Model
min
nA =
min
0
sinhA
=
min
0
expA
=
1st
Power-law
2
Rupture Model 3
( )1
minrt −
=
CDM-based Model 4
3
2min expcr
=
( )( )
1 exp 1exp
rt
− − =
1
23
t
5
G
cr
/ mT TRaw
1Mechanism Map
Time,
Creep
Strain
40
Through a systematic evaluation and application process, a MCSR model and
corresponding creep rupture equation can be applied to make material specific CDM-based
models. In order to generate the creep strain and damage predictions, the systematic process shown
in the flow chart depicted in Figure 4.1.
Material Deformation Mechanisms
Step 1 requires the assessment of the strain-rate data that will be used. The corresponding
stress and temperature values from the strain-rate data will be used and plotted on top of a
mechanism map that corresponds to the material of interest. It is important to note that maps can
vary even for the same material and they can sometimes be hard to obtain due to them being created
with experimental data. Due to the limitations of the mechanism maps and their inexactitudes, they
will be used more as an estimate and a qualitative tool rather than an exact quantitative one. When
a map is unavailable for a material, 2 options are possible. The first option is to generate the desired
map by collecting the necessary material data. This option will generate the best map for the
material that is being used, however it is also the one that requires the most resources. The
alternative option is to use the map for a material that is similar in application and composition,
which is the easier option but can yield improper results.
MCSR and Creep Rupture Models
Step 2 compares various MCSR models. A total of 9 MCSR models will be taken into
consideration. The MCSR models are included in Table 2.1. The models are each associated with
different material mechanism which are shown in Table 2.2. Each of the models will account for
temperature dependence by using the Arrhenius expression.
The objective function for optimization follows
41
( ) ( )2
, ,
1
1log log
N
raw i sim i
i
RMSLE X XN =
= − (46)
where the ,raw i
X is the raw data value and ,sim i
X is the simulated value acquired from the model.
Constant N is the number of points used in the study. The error is then minimized using Excel’s
solver application. The solver uses a GRG Nonlinear method where the material constants are
changed to reduce the value of the error as close as possible to 0. The error values are then
compared and ultimately, the model which has the lowest error and represents the correct
deformation mechanism is selected.
In step 3, the Monkman-Grant inverse relationship is exploited to produce a corresponding
creep rupture equation. The creep rupture models are shown in Table 2.3. The creep rupture
equation is calibrated using the same objective function and solver method to find the
corresponding material constants.
Creep Strain and Damage Modeling
Step 4 uses the values that were acquired from step 2 and 3. Various creep strain and
damage models exist such as the already discussed Kachanov-Rabotnov and Sinh model. This
study focuses on using the Sinh model as a tool for modeling creep strain and damage as it has
shown to have proven reliability in other studies [23,78]. This does not mean that other models
cannot be used. If the researcher wishes to do so, other models that use MCSR and creep rupture
can be replaced with the Sinh model. The system allows for substitution, especially if a model is
found that is better suited to the material in question. An example can be a model that focuses on
the primary creep regime which is replaced with the Sinh model since the material or data in
question exhibit’s a large primary creep regime.
42
The Sinh model is represented by the set of equations
3
2min expcr
=
(47)
( )( )
1 exp 1exp
rt
− − = (48)
( ) ( )
1ln 1 1 exp
r
tt
t
−= − − −
(49)
where cr
is creep strain rate, is damage rate, is current damage, and is a model constant,
min is the MCSR which was found in step 2, and
rt is creep rupture time which was found in step
3. The model is calibrated by using strain, time, rupture data, and applying the MCSR and creep
rupture results that were found. Step 5 is finally achieved after the model is calibrated, the strain
curves are generated.
Material Data
The material that is studied is 9Cr-1Mo-V-Nb (P91). Experimental data is obtained from
literature [79]. The composition of the alloy HEAT MGC is shown in Table 4.1. The material form
is tube. The data was collected at 600°C for which the material has a yield strength of 289 MPa
and tensile strength 357 MPa. Creep deformation data consists of 6 stress level curves for 100,
110, 120, 140, 160, and 200 MPa which are depicted in Figure 4.2. A general summary of the
creep data is provided in Table 4.2.
43
Table 4.1: Nominal chemical composition (mass percentage) of Heat MGC for alloy P91[79]
Element Mass percent (mass%)
Fe Bal.
C 0.09
Si 0.29
Mn 0.35
P 0.009
S 0.002
Ni 0.28
Cr 8.70
Mo 0.90
Cu 0.032
V 0.22
Nb* 0.072
N 0.044
Al* 0.001
Figure 4.2 – Creep deformation curves for P91 at 100, 110, 120, 140, 160, and 200 MPa and
600°C [79]
Time, t (hr)
10-2 10-1 100 101 102 103 104 105
Cre
ep
Str
ain
, (m
m/m
m)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
100 MPa
110 MPa
120 MPa
140 MPa
160 MPa
200 MPa
600°C
44
Table 4.2: Creep data for P91 [79]
Temperature T
Stress
MCSR
min
Final-creep-
strain-rate
final
Rupture time
rt
°C MPa %/hr %/hr hr
600 100 5.12E-07 1.26E-04 34121.4
600 110 8.42E-07 1.60E-04 21147.2
600 120 1.78E-06 2.64E-04 12796.5
600 140 7.57E-06 5.77E-04 3460.8
600 160 5.64E-05 2.55E-03 943.5
600 200 1.81E-03 6.48E-02 40.1
4.2 Results and Discussion
Material Deformation Map
Figure 4.3 – Deformation Mechanism Map for P91 with experimental data plotted [11,80-81]
The initial acquired data is plotted over a material deformation map. The deformation map
that is used was acquired from another study that also looked at P91 [11]. Although it is limited in
the mechanism and area that it shows, it is enough to find the mechanisms that are present. The
data is plotted over the map as depicted in Figure 4.3. The solid line indicates the point at which
the mechanism transitions from linear creep to power-law creep. There are 5 points on the linear
Temperature, T (°C)
560 580 600 620 640 660 680 700
Str
es
s,
(M
Pa
)
50
100
150
200
Mechanism Transition Line
Exp Data
Power-law Creep
Linear (Viscous) Creep
45
side and 1 point on the power-law side. Since there are 2 mechanisms present, a model with power-
law and linear modeling capabilities should be chosen.
MCSR and Creep Rupture Predictions
The experimental data is used to find the best MCSR model and consequently, the best
creep rupture model. It is important to note that only 1 isotherm is present, which is that of 600°C.
Initially, all the 9 MCSR models are fit to the experimental MCSR data. The error value for all
MCSR models is shown in Table 4.3.
Table 4.3: MCSR models and their error values
Model RMSLE
Norton 2.04E-1
Simplified Norton 5.34E-1
Johnson-Henderson-Kahn 2.27E-2
Soderberg 1.228
Dorn 1.228
Nadai 2.04E-1
McVetty 1.508
Garofalo 2.04E-1
Wilshire 5.14E-1
The model with the lowest error value is the JHK model depicted in [Eq.(9)]. The JHK
model uses 2 power laws which allows it to easily model power-law creep with one section of the
model. The other model section can be used to predict another mechanism, which in this case the
material constants will be adapted to model the linear (viscous) creep regime. The double power-
law model is best used due to the limited amount of data available and the presence of 2
mechanisms, one for each power law section. Another factor to take into consideration is the
number of constants that the JHK model has, since it has the most at 5, the data fit can be better
than other models that struggle to fit “bends” such as the ne present after the first 3 points in the
data. The JHK model’s prediction is shown in Figure 4.4. Once the best MCSR model was found,
its creep rupture inverse pair was used and fit to generate Figure 4.5. The material constants for
46
both the MCSR and creep rupture model are shown in Table 4.4. The fits appear to go over the
data properly for both models.
Figure 4.4 – JHK MCSR model prediction with extra simulated temperatures 550, 650, and
700°C
Figure 4.5 – JHK creep rupture prediction model prediction with extra simulated temperatures
550, 650, and 700°C
Stress, (MPa)10 100 1000
MC
SR
, m
in (
%/h
)
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Exp 600°C
Sim 600°C
Sim 550°C
Sim 650°C
Sim 700°C.
Time to Rupture, tr (hr)
100 101 102 103 104 105 106
Str
es
s,
(M
Pa
)
1
10
100
1000
Exp 600°C
Sim 600°C
Sim 550°C
Sim 650°C
Sim 700°C
47
Table 4.4: Material constants for the JHK pair
General Q
(kJ/mol*K) 230
MCSR
1A
(1/hr) 0.0791
1n 16.23
0
(MPa) 35.81
2A
(1/hr) 1718E2
2n 4.959
Creep Rupture
1B
(hr) 4.688E-9
1m 15.22
t
(MPa) 9.206
2B
(hr) 4377
2m 5.256
The acquired MCSR and time to rupture values are presented in Table 4.5 and Table 4.6.
The percent error difference is used to gauge the quality of the simulated values along with an
evaluation for it the values are overpredicted or underpredicted.
Table 4.5: Simulated MCSR with error differences
Temp T
Stress
MCSR
min
Sim MCSR
min,sim
Percent
Error
Diff
Prediction
°C MPa %/hr %/hr %
600 100 5.12E-7 5.07E-7 9.77E-1 Under
600 110 8.42E-7 8.87E-7 5.34 Over
600 120 1.78E-6 1.65E-6 7.30 Under
600 140 7.57E-6 8.13E-6 7.40 Over
600 160 5.64E-5 5.36E-5 4.96 Under
600 200 1.81E-3 1.83E-3 1.10 Over
48
Table 4.6: Simulated creep rupture with error differences
Temp T
Stress
Rupt time
rt
Sim
Rupt time
,r simt
Percent
Error
Diff
Prediction
°C MPa hr hr %
600 100 34121.4 35166.0 3.06 Over
600 110 21147.2 20662.6 2.29 Under
600 120 12796.5 12187.2 4.76 Under
600 140 3460.8 3762.3 8.71 Over
600 160 943.5 892.9 5.36 Under
600 200 40.1 40.7 1.50 Over
The predictions for both models yielded small percent errors with no large differences from
the simulated values to the actual values as all errors were below 10% difference. The MCSR and
creep rupture values are half over predicted and underpredicted with no true pattern. The smallest
percent error is found at 100 MPa while the largest is found at 140 MPA. The largest error can be
attributed to the first point that is spread from the first 3 stress levels and having to be used to
model the fit the curve that the JHK model generates. The creep rupture values had larger percent
errors than that of the MCSR. The smallest error was found at 200 MPa while the largest was also
found at 140 MPa. Since the highest error is at 140 MPa as well, it indicates that when using the
JHK model, the points that hold the highest errors are the points in the middle of the range since
the JHK model has a curve bend. The MCSR and creep rupture predictions both behaved properly
since the MCSR did increase with stress increases and the creep rupture values did decrease with
stress increases.
Creep Strain and Damage
Once the MCSR and creep rupture constants were found they were applied to obtain the
simulated MCSR and creep rupture values for the applied stresses of 100, 110, 120, 140, 160, and
200 MPa at 600°C. The simulated results will be used to find the creep strain and damage fits that
49
can be generated using the Sinh model. The Sinh model will have the constant for fixed as it
can be found by
min
lnfinal
=
(50)
The values that are used are those presented in Table 4.2. The constants are found through
numerical optimization using the RMSLE objective function and applying it to the strain data and
a simulated strain data using the Sinh model, the raw time values, and a backwards solution
integration method. The material constants and the errors used for the Sinh model are shown in
Table 4.7. The generated creep strain and damage predictions are presented in Figure 4.6 and
Figure 4.7 respectively.
Table 4.7: Sinh Constants
Temperature T
Stress
Lambda
Phi RMSLE
°C MPa
600 100 5.35 3.540 5.18E-1
600 110 5.09 2.625 4.91E-1
600 120 4.84 1.931 5.40E-1
600 140 4.39 2.176 5.30E-1
600 160 3.98 1.925 4.33E-1
600 200 3.26 3.576 1.53E-1
50
Figure 4.6 – Creep strain curves for P91 data for 100, 110, 120, 140, 160, and 200 MPa at 600°C
Figure 4.7 – Damage curves for P91 data for 100, 110, 120, 140, 160, and 200 MPa at 600°C
The plots that were generated were of proper quality as they were able to mostly capture
the secondary regime and the starting curvature of the tertiary regime. The predictions do fit the
general shape of creep deformation curves that the Sinh models can generate. The predictions were
visually conservative regarding the ductility as most predictions were not able to reach the final
point of data and finished in between the final 2 points of raw data. The fits final point for stress
Time, t (hr)
10-2 10-1 100 101 102 103 104 105
Cre
ep
Str
ain
, (
mm
/mm
)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Sim 100 MPa
Sim 110 MPa
Sim 120 MPa
Sim 140 MPa
Sim 160 MPa
Sim 200 MPa
600°C
Time, t (hr)10-1 100 101 102 103 104 105
Da
ma
ge
,
0.0
0.2
0.4
0.6
0.8
1.0
1.2
100 MPa
110 MPa
120 MPa
140 MPa
160 MPa
200 MPa
600°C
51
levels 100, 140, and 200 MPa end up with the final prediction point to the right as life was over
predicted by small degrees. The stress levels for 110, 120, and 160 have their final prediction
points to the left of the final raw data point since they under predicted rupture. The final simulated
strains are presented to compare with the actual final strains in Table 4.8. All predictions were
underpredicted, however the best fits for ductility were for stresses 140 and 160 MPa.
Table 4.8: Simulated final strains and actual final strains differences
Stress
MPa
Actual Final
Strain
final
%
Simulated Final
Strain
sim
%
Percent
Error
%
Prediction
100 2.13E-1 1.34E-1 37.1 Under
110 2.56E-1 1.64E-1 35.9 Under
120 2.74E-1 2.00E-1 27.0 Under
140 2.57E-1 2.42E-1 5.84 Under
160 4.03E-1 3.88E-1 3.72 Under
200 2.96E-1 2.36E-1 20.3 Under
52
Chapter 5: Selection Process for Objective Functions With Creep Modeling
5.1 Methodology
Figure 5.1 – Systematic procedure for finding the best objective function
A systematic approach can be used to determine the most optimal objective function for
creep modeling. The approach is depicted in Figure 5.1 where the necessary steps to find the best
53
possible objective function for a certain model are shown in order from 1-4. It is important to note
that this procedure can be applied to other mechanisms or models unrelated to creep, or even
engineering as the aim of this process is to find the best mathematical objective for a given model
of interest.
Model Selection
Step 1 involves the selection of the model for which the objective function is to be
determined. The model can be of any mechanism, however for this study the models that will be
used will be those for MCSR and creep rupture. The models selected for both the MCSR and creep
rupture is the set of the Wilshire equations which are depicted in [Eq.(11)] and [Eq.(22)]. These
equations were chosen as they had previously been determined in other studies to be effective in
computational and analytical modeling procedures. Any other model from Table 2.1 or Table 2.3
could have been chosen for the respective mechanism they cover as the study is not tied to the
Wilshire model.
The methods employed in this study are primarily computational. It is important to take
note, that the best objective function for a model of interest can better be determined and gauged
in accuracy if the chosen model has a method of solution other than computational. The alternate
solution will offer another mean of data comparison to determine if the numerical result is realistic.
Models that hold various methods of solution end up being optimal for this study process.
Solver Method and Settings
Step 2 requires for the user to select a solver algorithm that is to be used for finding the
material constants of the selected models. The solver algorithm should be kept consistent when
testing all the objective functions as the focus is finding the most optimal objective function and
not solver algorithm. The metamodel study discusses some possible benefits of using different
solver algorithms and a similar process such as the one in this study could possibly be employed,
however that is a subject for another study.
54
The solver algorithm chosen is one of the basic ones included in Excel, which is the main
tool used for this study. The GRG nonlinear solver is used to find the material constants of the
models using the 12 objective functions. The objective functions are set to be optimized to
converge as close as possible to the value of 0. The convergence criteria can be different based on
the study. The value of 0 is chosen since the objective function is directly related to the relationship
between raw data and computational data. A value of 0 indicates that there is no difference between
the actual raw data and the generated simulation data, which would be the best possible solution
scenario.
The boundary conditions and initial conditions are also set in this step since they are
required before any optimizations are performed. The boundary conditions are to be based on
material constant knowledge and mathematical solutions for the models. The lower and upper
boundaries for the model’s material constants along with the initial conditions are presented in
Table 5.1. The initial conditions were set by taking into consideration the Wilshire model’s general
constant solutions found in studies such as Cano’s [77]. The only constant which is not solved for
with the rest of the material constants is the activation energy. The activation energy uses the raw
data and the approach used by Cano since the activation energy is a constant that should be always
found first since it is important in creep.
Table 5.1: Solver conditions for Wilshire model optimization
Mechanism Constant Lower
Boundary
Upper
Boundary
Initial
Condition
MCSR
2k
(1/hr) 1 20 10
v -1 -0.0001 -0.1
Creep Rupture
1k
(hr) 1 1000 30
u 0.0001 1 0.1
55
Generated Results and Evaluation Criteria
Step 3 and 4 can be grouped together as they are the steps pertaining to the generated
results. Step 3 is used to evaluate the generated numerical results. The objective functions are all
compared against each other by using the percent error formulas
( )
( )
exp
exp
exp, ,
1 exp,
% 100
1% 100
sim
Ni sim i
i i
X XError
X
X XIsotherm Error
N X=
−=
−=
(51)
where the result is presented as a percent which is how much the simulated result, simX , deviates
from the actual raw data, expX . The average percent error per isotherm of data is found by adding
all the percent error of a given isotherm and dividing by the total amount of points. This
comparison allows for isotherms to be evaluated individually against each other. The final percent
error used is an average of the isotherm percent error which is used to collectible gauge the
accuracy of the objective functions and their effect on the data set as a whole.
The percent errors for each objective function for the complete data set are plotted against
affecting variables such as stress. A linear fit uses the basic linear model
y mx b= + (52)
which results in finding the corresponding data slope, m , and R-squared value. The slope is used
to determine if the objective function predictions have a preferential solution approach. A
preferential solution means that an objective function might have a batter affinity for certain values
and will calibrate those values to a higher degree than others.
56
Figure 5.2 – Example percent error data trend graph with positive slope and correlation
In Figure 5.2, the data has a positive slope and a R-Squared value that comes close to 1.
The positive slope indicates that the objective function did not reduce the error at high stress values
properly which is a consistent and continuous trend, indicated by the R-squared correlation value.
The opposite can be true, where a model has higher error at lower stresses with a negative slope
and a negative R-squared value. Ideally, the plot should generate a fit that is as close as possible
to a horizontal line (slope that is close to 0) with an R-squared value close to 0 which would be
indicative of a model that reduces the percent error equally at all stress stages without any data
trends. Comparing the percent errors, the slopes, and R-squared values yields the best possible
objective functions by finding the lowest values. Along with the quantitative results, the model
constants found can then be used to find the generated data fits and be plotted to qualitatively asses
the results.
The material constants, model fits, and percent errors should also be compared to analytical
results when possible. Analytical results allow for determining if the best objective function found
generates realistic results. The analytical results found in this study were found using the process
that was used by Cano in his Wilshire model study [77].
57
5.2 Data
Alloys 316SS and 304SS were selected to demonstrate the objective function procedure.
Alloy 316SS data was collected to be used with the creep rupture model while 304SS for the
MCSR. Both data sets were extracted from the National Institute of Material Science (NIMS)
database [76]. The material data held a total of 313 data points for 316SS pertaining to creep
rupture and 20 points for 304SS pertaining to the MCSR. The activation energy is found using the
raw data, which is found to be 230kJ/molK for 316SS and 171kJ/molK for 304SS. The
corresponding raw data for both alloys is shown in Figure 5.3 and Figure 5.4.
Time, t (hr)
101 102 103 104 105 106
Str
ess,
(M
Pa)
0
50
100
150
200
250
300
Raw 600°C
Raw 650°C
Raw 700°C
Raw 750°C
Figure 5.3 – Raw 316SS creep rupture data from NIMS [76]
58
Stress, (MPa) 1e+1 1e+2 1e+3
MC
SR
, (
%/h
r)
1e-7
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
Raw 600°C
Raw 650°C
Raw 700°C
.
Figure 5.4 – Raw 304SS MCSR data from NIMS [76]
5.3 Results
The percent errors for each of the objective functions pertaining to the Wilshire MCSR and
creep rupture model prediction fits are presented in Table 5.2 and Table 5.3 correspondingly.
59
Table 5.2: Percent errors for objective functions used for the Wilshire MCSR model
Objective
Function
600°C
Percent Error
(%)
650°C
Percent Error
(%)
700°C
Percent Error
(%)
Average
Percent Error
(%)
Error Mean
Difference 84.70 94.00 96.91 91.76
Error Absolute
Difference 562.9 438.9 349.7 451.1
Error Squared
Mean Difference 3967 3724 3473 3722
Error Logged
Difference 98.63 25.54 37.84 55.43
NMSE 4016 3769 3514 3766
Logged
NMSE 45.29E11 12.33E8 10.10E8 15.86E11
Root Error Log
Mean Squared
Difference
100.1 23.98 37.63 55.39
NMaxSE 3967 3724 3474 3722
NMinsSe 3967 3724 3474 3722
NRangeSE 3967 3724 3474 3722
Logged
NMaxSE 100.1 23.98 37.63 55.39
Logged NMinSE 100.1 23.98 37.63 55.39
Logged
NRangeSE 100.1 23.98 37.63 55.39
60
Table 5.3: Percent errors for objective functions used for the Wilshire creep rupture model
Objective
Function
600°C
Percent
Error
(%)
650°C
Percent
Error
(%)
700°C
Percent
Error
(%)
750°C
Percent
Error
(%)
Average
Percent
Error
(%)
Error Mean
Difference 25.11 41.78 57.46 54.70 43.43
Error Absolute
Difference 47.82 75.25 68.55 41.99 60.49
Error Squared
Mean
Difference
87.49 115.0 88.57 46.16 89.23
Error Logged
Difference 36.36 55.94 93.58 95.18 66.99
NMSE 123.6 147.6 108.5 54.21 115.5
Logged
NMSE 24.39 46.35 60.98 53.33 45.29
Root Error
Log Mean
Squared
Difference
24.18 43.77 57.89 51.32 43.33
NMaxSE 87.49 115.0 88.57 46.16 89.23
NMinsSe 87.49 115.0 88.57 46.16 89.23
NRangeSE 87.49 115.0 88.57 46.16 89.22
Logged
NMaxSE 24.18 43.77 57.89 51.32 43.33
Logged
NMinSE 24.18 43.77 57.89 51.32 43.33
Logged
NRangeSE 24.18 43.77 57.89 51.32 43.33
The value of interest that will be used for evaluation purposes will be the average percent
errors. Based on the results presented from the tables, several objective functions had very similar
or equal percent errors. For the MCSR, the best models were the error logged difference, root error
logged mean squared difference, logged NMaxSE, logged NMinSE, and logged NRangeSE. All
these objective functions held a percent error that was the same and the online difference that could
be discerned had to be to a decimal place of at least the ten thousandths. Based on that degree of
accuracy, the “best” models are the root error logged mean squared difference, which is [Eq(30)],
61
for the MCSR and the logged NMinSE for the creep rupture data, [Eq(35)] , however any of the
other models already mentioned to be of similar error can prove to be as effective. The models that
performed the worse are notably less effective regarding the percent error and can easily be
distinguished from the rest of the objective functions due to their considerably larger errors
compared to the other objective functions. The worst models were the NMSE, which is [Eq.(28)],
for creep rupture and the logged NMSE, [Eq.(29)],for the MCSR.
Stress, (MPa)
10 100
MC
SR
, (
%/h
r)
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
Raw 600°C
Raw 650°C
Raw 700°C
Analytical 600°C
Analytical 650°C
Analytical 700°C
Numerical 600°C
Numerical 650°C
Numerical 700°C
Figure 5.5 – 304SS MCSR Wilshire numerical and analytical fits
62
Time, t (hr)
101 102 103 104 105 106
Str
ess,
(M
Pa)
0
100
200
300
400
500
Raw 600°C
Raw 650°C
Raw 700°C
Raw 750°C
Analytical 600°C
Analytical 650°C
Analytical 700°C
Analytical 750°C
Numerical 600°C
Numerical 650°C
Numerical 700°C
Numerical 700°C
Figure 5.6 – 316SS creep rupture Wilshire numerical and analytical fits
The root error logged mean squared difference Wilshire fit for the MCSR is shown in
Figure 5.5 while the NMinSE creep rupture fits in Figure 5.6. The numerical fit is compared to the
analytical fit as they are both plotted on top of the raw data. Qualitatively, the generated fits run
through the data and are practically on top of each other, demonstrating no large discernible
difference. The material constants for both approaches and mechanisms are shown in Table 5.4
and Table 5.5. The material constants for the analytical and numerical approach for both
mechanisms are also like each other, being different by less than 15%.
Table 5.4: Material constants for the MCSR Wilshire model
Constants Analytical Numerical
2k (1/hr) 4.4 4.684
v -0.0967 -0.1019
63
Table 5.5: Material constants for the creep rupture Wilshire model
Constants Analytical Numerical
1k (hr) 29.49 34.31
u 0.1557 0.1629
Stress, (MPa)
0 50 100 150 200 250
Perc
ent
Err
or
(%)
0
50
100
150
200
250
Analytical Error
Analytical Regression
Numerical Error
Numerical Regression
Numerical
m=0.2581777224
r ²=0.0447475218
Analytical
m=0.5909360584
r ²=0.1857779255
Figure 5.7 – 304SS MCSR percent error trends
64
Stress, (MPa)
0 50 100 150 200 250 300
Perc
ent
Err
or
(%)
0
100
200
300
400
500
600
Analytical Error
Analytical Regression
Numerical Error
Numerical Regression
Analytical
m=-0.296255885
r ²=0.0747933758
Numerical
m=-0.2223992025
r ²=0.0626788891
Figure 5.8 – 316SS creep rupture percent error trends
The percent error plots for both mechanisms are shown in Figure 5.7 and Figure 5.8. Both
plots have the percent errors for the analytical and numerical comparisons included. The MCSR
errors hold low positive slopes and low R-Squared values, indicating that the prediction methods
do not really hold any relation to the different stress levels. As stress increases or decreases, the
percent errors stay varied and spread. It should be noted that the R-squared is lower for the
numerical approach than the analytical approach by a slight amount, however it is not enough for
it to be called a better approach. All the conclusions are also true for the creep rupture errors,
except that for this mechanism, the trend is slightly negative. The average percent errors for the
MCSR analytical and numerical approaches are 58.62% and 55.50% respectively. The numerical
approach is slightly better than the analytical approach. The average percent errors for the creep
rupture analytical and numerical approaches are 45.50% and 43.13% respectively. The numerical
approach is slightly better than the analytical approach.
65
Chapter 6: Conclusions and Future Work
Various studies were conducted and reported in this collection of works. The conclusions
and future work for each of the studies will be reported separately from each other to keep the
coherency.
6.1 Conclusions and Future Work
Development and Application of Minimum Creep Strain Rate Metamodeling
The objectives of the study were partially met:
• The developed “metamodel” was able to successfully regress into any of the 9 base MCSR
models using the software tool to find the best fit using the constrained-modeling approach
• The pseudo-constrained modeling approach was able to generate new intermediate models
that could fit some of the material data, but in general the approach failed to completely
and effectively fit the data
The study found that the best MCSR model for the constrained metamodeling isothermal
approach and available data were the Wilshire and Garofalo model. The Wilshire model was the
best model for the constrained metamodeling Arrhenius approach. The pseudo-constrained
approach required modifications in order to work.
In the future, the study should use other solver mechanisms that are able to find global
solutions. It was found that the initial conditions used for finding the solutions to the pseudo-
constrained modeling approach had drastic changes if the values were altered. This shows that the
pseudo-constrained modeling approach might need a different algorithm such as simulated
annealing, which uses random initial values per iteration to find a solution to an optimization
problem. By testing the algorithm and making it functional, the pseudo-constrained approach could
66
possibly work. If the algorithm is effective, then the same metamodeling process can be applied to
other material models such as creep-fatigue and creep strain.
Development of “Material Specific” Creep Continuum Damage Mechanics-Based Constitutive
Equations
The objectives of the study were met:
• A framework for developing the “material specific” CDM constitutive equations was
developed
• The material constants for the mode were found and used to generate creep strain fits
• The goodness of fit was of high quality as the generated results were able to go over the
creep strain data properly
The study generated results that were close to the actual creep strain data. The generated
final strains were underpredicted, with the largest deviation being of 37.1% to the actual data.
Although the final strains were underpredicted by a small degree, the process can be used since
underpredicting life benefits an engineer in some instances. By underpredicting the final creep
strain values by a small degree, a window of prevention can be given before a part fails.
In the future, the study should perform parametric simulations on the generated material
constants. Parametric simulations are helpful as they can indicate if a model lack the ability to
properly interpolate between data and extrapolate outside of the available data. If the
parametric simulations are proven to work, then the approach and generated model can be said
to properly work for various conditions. The approach should also be expanded to be used with
other CDM models. Ideally, the study can be expanded, and better results can be found if more
creep data is acquired. It can potentially be worthwhile to redo the study with a larger dataset.
67
Selection Process of Objective Functions for Creep Models
The objectives of the study were met:
• A framework for finding the best objective functions for creep rupture and the MCSR was
developed and used
• The 12 objective functions were compared against each other and the best ones were found
for each mechanism
• The objective functions were proven to be effective and able to generate good predictions
since they were compared to analytical approach results which were similar in fits
The study found the best objective functions for the MCSR and creep rupture Wilshire
models. Multiple objective functions were found to be effective where the common trend
amongst the best models was that all of them used logged values. This makes sense as the
nature of MCSR and creep rupture data is that of having values that are different in magnitude.
By logging the data, the difference in magnitude is drastically reduced, being able to give better
balanced weight and significance to large and small values that would otherwise not be
optimized.
In the future, the study should be expanded to include other creep data such as creep
strain and creep relaxation data. Creep strain and creep relaxation are prevalent in creep
modeling and finding the best possible objective function for them can help beginner modelers.
Another approach can be taken where a custom objective function is created and tested along
with the other 12 objective functions.
68
References
[1] Schilke, P. W., Foster, A.D., Pepe, J. J., and Beltran, A. M., 1992, “Advanced Materials Propel
Progress in LAND-BASED GAS TURBINES,” Advanced Materials and Processes, 141(4), pp.
22-30.
[2] Kassner, M. E., & Smith, K. (2014). Low temperature creep plasticity. Journal of materials
research and technology, 3(3), 280-288.
[3] Meher-Homji, C. B., & Gabriles, G. (1998). Gas Turbine Blade Failures-Causes, Avoidance,
And Troubleshooting. In Proceedings of the 27th turbomachinery symposium. Texas A&M
University. Turbomachinery Laboratories.
[4] Manu, C. C. 2008. Finite Element Analysis of Stress Rupture in
Pressure Vessels Exposed to Accidental Fire, MS Thesis, Queen’s University, Kingston, Ontari
o, Canada.
[5] Stewart, C. M. 2013. A Hybrid Constitutive Model For Creep, Fatigue, And Creep-
Fatuigue Damage. PhD Dissertation. University of Central Florida. Florida. USA.
[6] Bernstein, H. L. 1998. Materials issues for users of gas turbines.In
Proceedings of the Twenty-
Seventh Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University,
College Station, Texas (pp. 129-179).
[7] Janawitz, J., Masso, J., & Childs, C. (2015). Heavy-duty gas turbine operating and maintenance
considerations. GER: Atlanta, GA, USA.
[8] General Electric. Heavy-Duty Gas Turbine 7HA.03. https://www.ge.com/power/gas/gas-
turbines/7ha/7ha-03
[9] Cuzzillo, B. R., & Fulton, L. K. (2014). Failure Analysis Case Studies. Retrieved October 27,
2019, from http://www.berkeleyrc.com/FAcasestudies.html#top.
[10] Haque, M.S, and Stewart, C. M., 2018, “The Disparate Data Problem: The Calibration of
Creep Laws Across Test Type and Stress, Temperature, and Time Scales,” Theoretical and
Applied Fracture Mechanics, (under review). TAFMEC_2018_166.
[11] Gorash, Yevgen. "Development of a creep-damage model for non-isothermal long-term
strength analysis of high-temperature components operating in a wide stress range." Martin Luther
University of Halle-Winttenberg, Halle Germany(2008).
[12] Lemaitre, J., Chaboche. 1985. J.L. Mechanics of Solid Materials. Paris: Cambridge University
Press.
[13] Evans, H. E. (1984). Mechanisms of creep fracture. Elsevier Applied Science Publishers Ltd.,
1984,, 319.
[14] Meyers, M. A., & Chawla, K. K. (2008). Mechanical behavior of materials. Cambridge
university press.
[15] Norton, Frederick Harwood. The Creep of Steel at High Temperature. McGraw-Hill Book
Company Incorporated. New York .(1929)
[16] Haque, M. S. “An improved Sin-hyperbolic constitutive model for creep deformation and
damage” AAI1591958. Master’s Thesis, University of Texas at El Paso, El Paso, Texas.(2015)
[17] Haque, M. S., and Stewart, C. M., “Finite Element Analysis of Waspaloy Using Sinh Creep-
Damage Constitutive Model under Triaxial Stress State,” ASME Journal of Pressure Vessel
Technology, 138(3), 2016. doi: 10.1115/1.4032704
69
[18] Haque, M. S., and Stewart, C. M., 2016, “Modeling the Creep Deformation, Damage, and
Rupture of Hastelloy X using MPC Omega, Theta, and Sin-Hyperbolic Models,” ASME PVP
2016, PVP2016-63029, Vancouver, BC, Canada, July 17-21, 2016.
[19] Haque, M. S., and Stewart, C. M., 2017, “The Stress-Sensitivity, Mesh-Dependence, and
Convergence of Continuum Damage Mechanics Models for Creep,” ASME Journal of Pressure
Vessel Technology, 139(4). doi:10.1115/1.4036142.
[20] Prager, M., 1995, “Development of the MPC Omega Method for Life Assessment in the Creep
Range,” Journal of Pressure Vessel Technology, 117, pp. 95-95.
[21] Evans, R. W., Parker, J. D., and Wilshire, B., 1992, “The Θ Projection Concept—
A Model-Based Approach to Design and Life Extension Of Engineering Plant,” International
Journal of Pressure Vessels and Piping, 50(1), pp. 147-160.
[22] Haque, M. S., and Stewart, C. M., 2015, “Comparison Of A New Sin-Hyperbolic Creep
Damage Constitutive Model With The Classic Kachanov-Rabotnov Model Using Theoretical And
Numerical Analysis,” 2015 TMS Annual Meeting & Exhibition, Orlando, FL, March 15-19, 2015.
[23] Haque, M. S., and Stewart, C. M., 2015, “A Novel Sin-Hyperbolic Creep Damage Model to
Overcome the Mesh Dependency of Classic Local approach Kachanov-Rabotnov Model” ASME
IMECE 2015, IMECE2015-50427, Houston, TX, November 13-19, 2015.
[24] Frost, Harold J., and Michael F. Ashby. Deformation mechanism maps: the plasticity and
creep of metals and ceramics. Pergamon press, Oxford, 1982.
[25] Harper, J., and John Emil Dorn. "Viscous creep of aluminum near its melting
temperature." Acta Metallurgica 5, no. 11 (1957): pp.654-665
[26] Mohammed, F. A., Murty, K.L. and Morris, J. W., Jr. In: The John E. Dorn Memorial
Symposium. Cleveland, Ohio, ASM(1973).
[27] Ruano, O. A., J. Wadsworth, and O. D. Sherby. "Harper-Dorn creep in pure metals." Acta
Metallurgica 36, no. 4 (1988): pp.1117-1128.
[28] Ashby, M., Shercliff, H., and Cebon, D., Materials: Engineering, Science, Processing and
Design. Butterworth-Heinemann, Oxford. 2013.
[29] Ashby, M. F. and Jones, D. R. H., “Engineering Materials 1: An Introduction to Their
Properties and Applications”. Oxford: Butterworth Heinemann, 2nd ed., 1996.
[30] Dimmler, G., P. Weinert, and H. Cerjak. "Investigations and analysis on the stationary creep
behaviour of 9–12% chromium ferritic martensitic steels." Materials for Advanced Power
Engineering 2002 (2002): pp.1539-1550.
[31] Dimmler, G., P. Weinert, and H. Cerjak. "Extrapolation of short-term creep rupture data—
The potential risk of over-estimation." International journal of pressure vessels and piping 85, no.
1 (2008): pp.55-62.
[32] Langdon, Terence G. "Creep at low stresses: An evaluation of diffusion creep and Harper-
Dorn creep as viable creep mechanisms." Metallurgical and Materials Transactions A 33, no. 2
(2002): pp.249-259.
[33] Yavari, Parviz, and Terence G. Langdon. "An examination of the breakdown in creep by
viscous glide in solid solution alloys at high stress levels." Acta Metallurgica 30, no. 12 (1982):
pp.2181-2196.
[34] Chen, Y. X., W. Yan, W. Wang, Y. Y. Shan, and K. Yang. "Constitutive equations of the
minimum creep rate for 9% Cr heat resistant steels." Materials Science and Engineering: A534
(2012): 649-653.
[35] Sherby, Oleg D., and Peter M. Burke. "Mechanical behavior of crystalline solids at elevated
temperature." Progress in Materials Science 13 (1968): pp.323-390.
70
[36] Earthman, J. C., Gibeling, J. C., Hayes, R. W., and et. al., “Creep and stress relaxation testing,”
in Mechanical Testing and Evaluation (Kuhn, H. and Medlin, D., eds.), no. 8 in ASM Handbook,
chapter 5, pp. 783 – 938, Materials Park, OH, USA: ASM International, 2000.
[37] Garofalo, F., Fundamentals of Creep and Creep-Rupture in Metals. New York: The
Macmillan Co., 1965.
[38] Naumenko, K. and Altenbach, H., Modelling of Creep for Structural Analysis. Berlin et al.:
Springer-Verlag, 2007.
[39] Penny, R. K. and Mariott, D. L., Design for Creep. London: Chapman & Hall, 1995.
[40] Vishwanathan, R., Damage Mechanisms and Life Assessment of High-Temperature
Components. Metals Park, Ohio: ASM International, 1989.
[41] Nádai, Arpád, and Arthur M. Wahl. Plasticity. McGraw-Hill Book Company, inc., 1931.
[42] Soderberg, C. R. (1936). The Interpretation of Creep Tests for Machine Design. Trans.
ASME, 58(8): 733-743.
[43] McVetty, P. G. (1943). Creep of Metals at Elevated Temperatures-The Hyperbolic Sine
Relation between Stress and Creep Rate. Trans. ASME, 65: 761.
[44] Dorn, J. E. (1962). Progress in Understanding High-Temperature Creep, H. W. Gillet Mem.
Lecture. Philadelphia: ASTM.
[45] Johnson, A.E., Henderson, J. and Kahn, B. (1963) “Multiaxial creep strain/complex
stress/time relations for metallic alloys with some applications to structures”. ASME/ASTM/I
Mech E, Proceedings Conference on Creep. Inst. Mech. E., New York/London.
[46] Wilshire, B., & Scharning, P. J. (2008). A new methodology for analysis of creep and creep
fracture data for 9–12% chromium steels. International materials reviews, 53(2), 91-104.
[47] Bailey, R. W., “Creep of steel under simple and compound stress,” Engineering, vol. 121, pp.
129 – 265, 1930.
[48] Coble, R. I., “A model for boundary diffusion controlled creep in polycrystalline materials,”
J. Appl. Phys., vol. 34, pp. 1679 – 1682, 1963.
[49] Harper, J. G., Shepard, L. A., and Dprn, J. E., “Creep of aluminum under extremely small
stresses,” Acta Metallurgica, vol. 6, pp. 509 – 518, 1958.
[50] Herring, C., “Diffusional viscosity of a polycrystalline solid,” J. Appl. Phys., vol. 21, no. 5,
pp. 437 – 445, 1950.
[51] Nabarro, F. R. N., “Deformation of crystals by the motion of single ions,” in Report of a
Conference on the Strength of Solids (Bristol), no. 38, (London, U.K.), pp. 75 – 87, The Physical
Society, 1948.
[52] Lifshitz, I. M., “On the theory of diffusion-viscous flow of polycrystalline bodies,” Soviet
Physics. Journal of Experimental and Theoretical Physics, vol. 17, pp. 909 – 920,
1963.
[53] Dyson, B. F. and McLean, M., “Microstructural evolution and its effects on the creep
performance of high temperature alloys,” in Microstructural Stability of Creep Resistant
Alloys for High Temperature Plant Applications (Strang, A., Cawley, J., and Greenwood, G. W.,
eds.), pp. 371 – 393, Cambridge: Cambridge University Press, 1998.
[54] Dyson, B. F. and McLean, M., “Micromechanism-quantification for creep constitutive
equations,” in IUTAM Symposium on Creep in Structures (Murakami, S. and
Ohno, N., eds.), pp. 3–16, Dordrecht: Kluwer, 2001.
[55] Wilshire, B., & Battenbough, A. J. (2007). Creep and creep fracture of polycrystalline
copper. Materials science and engineering: a, 443(1-2), 156-166.
71
[56] Wilshire, B., & Scharning, P. J. (2007). Long-term creep life prediction for a high chromium
steel. Scripta Materialia, 56(8), 701-704.
[57] Wilshire, B., & Scharning, P. J. (2008). Prediction of long-term creep data for forged 1Cr–
1Mo–0· 25V steel. Materials science and technology, 24(1), 1-9.
[58] Viswanathan, R., “Effect of stress and temperature on the creep and rupture behavior
of a 1.25%chromium – 0.5%molybdenum steel,” Metall. Trans. A, vol. 8, no. A, pp. 877
– 884, 1977.
[59] Monkman, F. and Grant, N., 1956, “An Empirical Relationship Between Rupture Life and
Minimum Creep Rate in Creep-Rupture Tests,” Proceedings of ASTM, 56, p. 593-596.
[60] T.L. Anderson, D.A. Osage, API 579: A comprehensive fitness-for-service guide, Int. J. Press.
Vessel. Pip. 77 (2001) 953–963. doi:10.1016/S0308-0161(01)00018-7.
[61] A. Cauvin, R. B. Testa. 1999. Damage mechanics: basic variables in continuum theories,”
International Journal of Solids and Structures, 36, 747-761.
[62] L. M. Kachanov, The Theory of Creep, National Lending Library for Science and
Technology, (Boston Spa, England, Chaps. IX, X, 1967).
[63] Y. N. Rabotnov, Creep Problems in Structural Members, (Amsterdam, Weinheim,North
Holland, WILEY-VCH Verlag GmbH & Co. KGaA, 1969).
[64] Cano, J. A., "A Modified Wilshire Model For Creep Deformation And Damage Prediction"
(2019). Open Access Theses & Dissertations. 2836. https://scholarworks.utep.edu/open_etd/2836
65 Clausen, Jens (1999). Branch and Bound Algorithms—Principles and
Examples (PDF) (Technical report). University of Copenhagen. Archived from the
original (PDF) on 2015-09-23. Retrieved 2014-08-13.
[66] Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. ISBN 0-
262-04219-3
[67] Kennedy, J.; Eberhart, R. (1995). "Particle Swarm Optimization". Proceedings of IEEE
International Conference on Neural Networks. IV. pp. 1942–
1948. doi:10.1109/ICNN.1995.488968.
[68] Gorash, Yevgen, and Donald MacKenzie. "On cyclic yield strength in definition of limits for
characterisation of fatigue and creep behaviour." Open Engineering 7, no. 1 (2017): 126-140.
[69] Altenbach, Holm, Y. Gorash, and K. Naumenko. "Steady-state creep of a pressurized thick
cylinder in both the linear and the power law ranges." Acta Mechanica 195, no. 1-4 (2008): 263-
274.
[70] S.S. Manson, C.R. Ensign, A Quarter-Century of Progress in the Development of Correlation
and Extrapolation Methods for Creep Rupture Data, Journal of Engineering Materials and
Technology, 101 (1979) 1–9.
[71] S.R. Holdsworth, R.B. Davies, Recent advance in the assessment of creep rupture data,
Nuclear Engineering and Design ,190 (1999) 287–296. doi:10.1016/S0029-5493(99)00038-2.
[72] D.R. Eno, G.A. Young, T.-L. Sham, A Unified View of Engineering Creep Parameters, in:
Am. Soc. Mech. Eng, Pressure Vessel Piping Division PVP, (2008). doi:10.1115/PVP2008-61129.
[73] D. Šeruga, M. Nagode, Unification of the most commonly used time-temperature creep
parameters, Material Science and Engineering A. 528 (2011) 2804–2811.
doi:10.1016/j.msea.2010.12.034.
[74] Haque, M.S., and Stewart C.M., 2018, “Metamodeling Time-Temperature Parameters for
Creep,” Nuclear Engineering and Design.
72
[75] Haque, M. S., and Stewart, C. M., 2016, “Exploiting Functional Relationships between MPC
Omega, Theta, and Sinh-Hyperbolic Models” ASME PVP 2016, PVP2016-63089, Vancouver,
BC, Canada, July 17-21, 2016.
[76] National Institute of Material Science – Database for Creep materials,
https://smds.nims.go.jp/MSDS/en/sheet/Creep.html
[77] Cano, J., and Stewart, C.M., 2019, “Application of the Wilshire Stress-Rupture and
Minimum-Creep-Strain-Rate Prediction Models for Alloy P91 in Tube, Plate And Pipe Form,”
ASME TurboExpo 2019, Phoenix, Arizona, June 17-21, 2019.
[78] M. S. Haque, C. M. Stewart. Theoretical and Numerical Analysis Of A New Sin-Hyperbolic
Creep Damage Constitutive Model. Journal of Mechanics Physics of Solids.
[79] Kimura, K., Kushima, H., & Sawada, K. (2009). Long-term creep deformation property of
modified 9Cr–1Mo steel. Materials Science and Engineering: A, 510, 58-63.
[80] Kloc, L. and Skleniˇcka , V., “Transition from power-law to viscous creep behaviour of P-91
type heat-resistant steel,” Mater. Sci. & Eng., vol. A234-A236, pp. 962 – 965,1997.
[81] Kloc, L., Skleniˇcka , V., Dlouh´y, A. A., and KuchaˇRov´A, K., “Power-law and viscous
creep in advanced 9%Cr steel,” in Microstructural Stability of Creep Resistant Alloys for High
Temperature Plant Applications (Strang, A., Cawley, J., and Greenwood, G. W., eds.), no. 2 in
Microstructure of High Temperature Materials, pp. 445 – 455, Cambridge: Cambridge University
Press, 1998.
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Vita
I am Ricardo Vega Jr. I received my Bachelor of Science degree in Mechanical Engineering
from the University of Texas at El Paso in 2018 with magna cum laude honors. This document is
submitted as a requirement for the completion of a Master of Science in Mechanical Engineering
at the University of Texas at El Paso. My studies were performed under the instruction and
supervision of my advisor Dr. Calvin M. Stewart. The document holds a collection of the work
done as a graduate and undereducated research assistant with the Materials at Extremes Research
Group. In the group, I conducted studies on creep modeling and computational methods.
The publications related to this work are:
1. Vega, Ricardo and Stewart, Calvin. “Validation of the Development and Application of
Minimum Creep Strain Rate Metamodeling” ASME 2019 Turbomachinery Technical
Conference and Exposition. Phoenix, AZ, June 17-21, 2019.
2. Vega, Ricardo, Cano, A., Jaime, and Stewart, Calvin. “Development of ‘Material Specific’
Creep Continuum Damage Mechanics-Based Constitutive Equations” PVP 2020 Pressure
Vessels and Piping Conference. Minneapolis, Minnesota, July 19-24, 2020.
3. Vega, Ricardo, Cano, A., Jaime, and Stewart, Calvin. “Selection Process of Objective
Functions for Creep Modeling”. (Manuscript in preparation).
Contact Information: Ricardo Vega Jr, institute email: [email protected], personal email: