Computational Methods For Creep Modeling

University of Texas at El Paso University of Texas at El Paso

ScholarWorks@UTEP ScholarWorks@UTEP

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

Follow this and additional works at: https://scholarworks.utep.edu/open_etd

Part of the Mechanical Engineering Commons

Recommended Citation Recommended Citation Vega, Ricardo, "Computational Methods For Creep Modeling" (2020). Open Access Theses & Dissertations. 3057. https://scholarworks.utep.edu/open_etd/3057

This is brought to you for free and open access by ScholarWorks@UTEP. It has been accepted for inclusion in Open Access Theses & Dissertations by an authorized administrator of ScholarWorks@UTEP. For more information, please contact [email protected].

https://scholarworks.utep.edu/

https://scholarworks.utep.edu/open_etd

https://scholarworks.utep.edu/open_etd?utm_source=scholarworks.utep.edu%2Fopen_etd%2F3057&utm_medium=PDF&utm_campaign=PDFCoverPages

http://network.bepress.com/hgg/discipline/293?utm_source=scholarworks.utep.edu%2Fopen_etd%2F3057&utm_medium=PDF&utm_campaign=PDFCoverPages

https://scholarworks.utep.edu/open_etd/3057?utm_source=scholarworks.utep.edu%2Fopen_etd%2F3057&utm_medium=PDF&utm_campaign=PDFCoverPages

mailto:[email protected]

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.

73

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:

[email protected]

Computational Methods For Creep Modeling

Documents

Transcript of Computational Methods For Creep Modeling