2006_tesis_powers-panolkaltsis_mechanical behavior of ceramics at high temperatures

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MECHANICAL BEHAVIOR OF CERAMICS AT HIGH TEMPERATURES:

CONSTITUTIVE MODELING AND NUMERICAL IMPLEMENTATION

by

LYNN MARIE POWERS

Submitted in partial fulfillment of the requirements

For the degree of Doctor of Philosophy

Dissertation Advisers: Dr. Vassilis Panoskaltsis and Dr. Dario Gasparini

Department of Civil Engineering

CASE WESTERN RESERVE UNIVERSITY

August, 2006


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CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

______________________________________________________

candidate for the Ph.D. degree *.

(signed)_______________________________________________

(chair of the committee)

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

(date) _______________________

*We also certify that written approval has been obtained for any

proprietary material contained therein.

Lynn Marie Powers

Prof. Vassilis P. Panoskaltsis

Prof. Dario A. Gasparini

Prof. Robert L. Mullen

Prof. John J. Lewandowski

5 May 2006


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Copyright 2006 by Lynn Marie Powers

All rights reserved


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Table of Contents

Table of Contents............................................................................................................... ivList of Tables ..................................................................................................................... vi

List of Figures ................................................................................................................... vii

Acknowledgements.......................................................................................................... xiiiAbstract..............................................................................................................................xv

Chapter 1 - Introduction ...................................................................................................1

1.1 High-Temperature Applications of Ceramic Materials ...........................................11.2 Objectives and Scope of Research...........................................................................7

1.3 References................................................................................................................7

Chapter 2 Observed Material Behavior .......................................................................9

2.1 Introduction..............................................................................................................9

2.2 Viscous Flow .........................................................................................................122.3 Microstructural Features ........................................................................................19

2.4 Asymmetry.............................................................................................................26

2.5 Temperature ...........................................................................................................302.6 Damage ..................................................................................................................36

2.7 Randomness ...........................................................................................................38

2.8 Summary................................................................................................................43

2.9 References..............................................................................................................44

Chapter 3 Modeling Review.........................................................................................47

3.1 Introduction............................................................................................................47

3.2 Mechanical Behavior .............................................................................................473.3 Damage ..................................................................................................................51

3.3.1 One Dimensional Damage ............................................................................523.3.2 Multi-Dimensional Damage..........................................................................54

3.4 Simulation Techniques...........................................................................................55

3.4.1 Quantitative Modeling ..................................................................................56

3.4.2 Review of Microstructural Simulations ........................................................633.5 Summary................................................................................................................74

3.6 References..............................................................................................................75

Chapter 4 Constitutive Model......................................................................................80

4.1 One-Dimensional Linear Viscoelastic Model........................................................804.2 Nonlinear One-Dimensional Viscoelastic Model ..................................................914.3 Multidimensional Viscoelastic Model .................................................................103

4.4 Nonlinear Multidimensional Viscoelastic Model ................................................113

4.4.1 Volumetric Component of the Model.........................................................1134.4.2 Deviatoric Component of the Model ..........................................................117

4.5 Damage ................................................................................................................1264.5.1 One-Dimensional Damage..........................................................................126


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4.5.2 Multi-Dimensional Damage........................................................................143

4.6 Temperature .........................................................................................................1544.7 Summary..............................................................................................................164

4.8 References............................................................................................................165

Chapter 5 Numerical Implementation ......................................................................167 5.1 Material Models...................................................................................................167

5.1.1 One-Dimensional Constitutive Models.......................................................168

5.1.1.1 Linear Viscoelasticity ........................................................................1685.1.1.2 Noninear Viscoelasticity....................................................................172

5.1.2 Multi-Dimensional Constitutive Models ....................................................180

5.1.2.1 Linear Viscoelastic Models................................................................1895.1.2.2 Nonlinear Viscoelastic Model, Volumetric .......................................197

5.1.2.3 Nonlinear Viscoelastic Model, Deviatoric.........................................203

5.2 Parameter Estimation...........................................................................................205

5.3 References............................................................................................................207

Chapter 6 Applications...............................................................................................209

6.1 Predictions for Different Load Conditions ..........................................................2106.1.1 Stress and Strain Response .........................................................................215

6.1.2 Life Prediction ............................................................................................231

6.2 Temperature Dependent Viscoelasticity..............................................................2446.3 Two-Phase Model ................................................................................................270

6.4 References............................................................................................................281

Chapter 7 Conclusions and Future Work ................................................................282 7.1 Conclusions..........................................................................................................2827.2 Future Work.........................................................................................................287

Bibliography ...................................................................................................................290


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List of Tables

1.1 Benefits of ceramics in aerospace systems ....................................................................5

3.1 Statistical values for a 2D Voronoi diagrams ..............................................................58

4.1 Response for the standard linear solid .........................................................................84

4.2 Stress and strain for test configuration.......................................................................107

4.3 Response for the deviatoric standard linear solid model in the xx direction.............108

4.4 Response for the volumetric standard linear solid model..........................................1084.5 Empirical linear viscoelastic material properties.......................................................109

5.1 Calculation of stress and internal variable for a one-dimensional viscoelasticmodel..........................................................................................................................171

5.2 Calculation of stress, internal variable and damage for a one-dimensional

viscoelastic model......................................................................................................1795.3 Calculation of stress and internal variables for a multi-dimensional

viscoelastic model......................................................................................................182

5.4 Calculation of stress, internal variables and damage for a multi-dimensionalviscoelastic model......................................................................................................183

5.5 Calculation of stress, damage and internal variables for a multi-dimensional

viscoelastic model with plane stress conditions.........................................................185

6.1 Test Matrix for SN88.................................................................................................245

6.2 Material Parameters ...................................................................................................252


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List of Figures

1.1 Turbine engine with ceramic composite components....................................................21.2 Engine efficiency as a function of turbine inlet temperature.........................................3

1.3 Stress rupture limits as a function of temperature and year for various materials.........4

1.4 Solid oxide fuel cell .......................................................................................................6

2.1 Comprehensive fracture map for MgO doped HPSN tested in flexure in air..............10

2.2 Transmission electron micrographs showing microstructural features .......................11

2.3 Effect of strain rates on engineering stress/strain curves tested in tension at 1200C.132.4 Strain as a function of time for several uniaxial creep tests at 1371C .......................14

2.5 Isochrones for silicon nitride (NT154) ........................................................................15

2.6 a) Deformation of nano-crystalline Si-B-C-N ceramics as a function of time forcompressive loads at a test temperature of 1400C, b) Isochrone for one day and

one week for the tests presented in a) ..........................................................................16

2.7 Creep test with unload at 1200C and 70 MPa. The left axis represents the truestress and the right axis the true strain.........................................................................17

2.8 Strain as a function of time for a silicon nitride under 200 MPa for 60 hours at

1300C, after 60 hours, the load is removed................................................................182.9 Scanning electron micrograph of (a) an as-sintered specimen and (b) a deformed

tensile specimen illustrating the retention of equiaxed grains and concurrent

cavitation; the tensile axis is horizontal.......................................................................20

2.10 Grain boundary sliding mechanisms illustrating out-of-plane, separation,rotation and crack growth in a-d, respectively...........................................................21

2.11 Histograms of film thickness distribution of grain boundaries of the

experimental materials crept at 1430C with a stress of 40 MPa for 690 h:

(a) uncrept grip end; (b) crept gauge section ............................................................222.12 High-resolution lattice fringe in the grip end showing a film thickness of

0.74 nm ......................................................................................................................232.13 High-resolution lattice fringe in the grip end showing a film thickness of

a) 0.5 nm and b) 1.25 nm at different grain boundaries.............................................24

2.14 Histograms of intergranular film thickness distributions in materials (a) as-hot-

pressed, and (b) after compressive deformation ........................................................242.15 TEM micrographs ......................................................................................................25

2.16 Strain as a function of time for a tensile and compressive creep test at

temperatures and stresses shown in each graph.........................................................272.17 Microstructure of specimens tested at 1371C after exposure to a stress

of 300 MPa is a) tension and b) compression............................................................282.18 Strain as a function of distance from the tensile surface of flexure specimens .........302.19 (a) Stress strain curves of fine-grained Ti3SiC2samples

(b) The effect of temperature on the ultimate tensile strength and strains to failure .32

2.20 Isochrones of strain as a function of stress and temperature for NT154 siliconnitride after 10 hours under load................................................................................33

2.21 Time to failure as a function of stress and temperature for NT154 silicon nitride....332.22 Strain as a function of time and temperature with log scale for stress of 150 MPa...34


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2.23 Relaxation and recovery at different temperatures....................................................35

2.24 Interstitial cavities in different silicon nitrides ..........................................................372.25 Low-magnification TEM micrograph of the microstructural damage.......................38

2.26 Strain as a function of time for a tensile creep test at 1371C and a stress of

150 MPa in air............................................................................................................39

2.27 Fifteen sets of creep curves for this study. Each subfigure lists the times to failureand minimum creep rates for the specimens..............................................................41

2.28 Time to failure for the 14 laboratories .......................................................................42

2.29 Potential contributions to random behavior for a typical strain time curve...............422.30 Cavity development in flexure specimens tested at 1300C......................................43

3.1 Schematic of strain as a function of time illustrating the three creep regimes ............493.2 Sample of a random network.......................................................................................59

3.3 Generation of a random network with Mathematica ...................................................60

3.4 An example of the Johnson-Mehl model.....................................................................63

3.5 A heterogeneous structure with various levels ............................................................64

3.6 Schematic of modeling hierarchy ................................................................................673.7 Distribution of the principal material directions..........................................................68

3.8 An open cell Voronoi foam..........................................................................................693.9 Mesh for a Voronoi network with grain boundaries....................................................70

3.10 Ferritic-pearlitic simulation .......................................................................................71

3.11 Two phase simulations...............................................................................................723.12 Voronoi network superimposed onto a square meshed grid......................................73

3.13 Two meshes of the same microstructure....................................................................74

4.1 One dimensional standard linear solid model..............................................................81

4.2 Strain as a function of time for several creep tests ......................................................854.3 Isochrones for the creep curves shown in Figure4.2....................................................86

4.4 Stress as a function of time for several relaxation tests ...............................................88

4.5 Stress strain curves as a function of the strain rate for a constant strain rate test........89

4.6 Strain response as a function of the dashpot parameter for a creep test ......................904.7 Strain response as a function of the spring constant for a creep test ...........................90

4.8 Isochrones of strain at time equal to infinity as a function of time and

asymmetry constant .....................................................................................................944.9 Strain as a function of time for several creep tests ......................................................95

4.10 Isochrones for the creep curves shown in Figure4.9..................................................97

4.11 Strain response as a function of the dashpot parameter for a 50 MPa creep test.......984.12 Strain response as a function of the inelastic spring constant ENLfor a

50 MPa creep test.......................................................................................................99

4.13 Strain response as a function of the parameter for a 50 MPa tensile creep test. ..1004.14 Strain response as a function of the constant C0for a 50 MPa creep test and a

-50 MPa creep test ...................................................................................................1014.15 Standard linear solid for a multidimensional model................................................103

4.16 Stress strain curves for constant strain rate tests for a) deviatoric, b) volumetric

and c) total as a function of total strain rate for titanium silicocarbonate................111


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4.40 The creep compliance on a log scale as a function time on a log scale for

various temperatures ................................................................................................1574.41 The shift factor as a function of temperature. ..........................................................1574.42 Strain as a function of time for a creep test of 100 MPa at different temperatures

for silicon nitride with no damage ...........................................................................160

4.43 Strain as a function of time for a creep test of 100 MPa at different temperaturesfor silicon nitride with damage ................................................................................162

6.1 Experimental Configurations.....................................................................................2126.2 Isochrones of strain as a function of stress after one week and one month under

load.............................................................................................................................213

6.3 Strain as a function of time for uniaxial specimens...................................................2146.4 Finite element mesh for the flexure beam with load and boundary conditions.........216

6.5 Stress distribution at midspan through the thickness of a 4-point bend specimen

after 1000 hours for the A) linear viscoelastic, B) nonlinear viscoelastic and C)

asymmetric nonlinear viscoelastic material models ..................................................216

6.6 Deflection as a function of time for 4 point bend specimens ....................................2176.7 Deviatoric stress in the xx-direction at four times: a) 1 hour, b) 10 hours,

c) 100 hours and d) 300 hours....................................................................................2186.8 Volumetric stress at four times: a) 1 hour, b) 10 hours, c) 100 hours and

d) 300 hours ..............................................................................................................219

6.9 Total stress in the xx-direction at four times: a) 1 hour, b) 10 hours,c) 100 hours and d) 300 hours....................................................................................220

6.10 Deviatoric stress at the midspan in the xx-direction as a function of position

and time....................................................................................................................2226.11 Normal stress at the midspan in the xx-direction as a function of position

and time....................................................................................................................2226.12 Volumetric stress at the midspan as a function of position and time.......................223

6.13 The square root of the second invariant of the deviatoric stress at the midspan

as a function of position and time............................................................................223

6.14 Internal variable, deviatoric inelastic strain, at the midspan in the xx-directionas a function of position and time............................................................................223

6.15 Internal variable, volumetric inelastic strain, at the midspan as a function of

position and time......................................................................................................2236.16 Deviatoric strain in the xx-direction at the midspan as a function of position

and time....................................................................................................................226

6.17 Volumetric strain at the midspan as a function of position and time.......................2266.18 Total strain at the midspan in the xx-direction as a function of position and time..227

6.19 Axisymmetric finite element mesh for the ball-on-ring specimen with load and

boundary conditions.................................................................................................2286.20 Volumetric stress in the ball-on-ring specimens at a) 0 hrs and b) 500 hrs.............229

6.21 Deflection as a function of time for ball-on-ring specimens, solid lines are

experimental data and dashed lines are analytical predictions ................................230

6.22 Mesh without symmetry boundary conditions for the flexure beam.......................2336.23 Deflection as a function of time for 4-point bend beam specimen with element

Removal ...................................................................................................................233


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6.24 Flexural stress distribution on the 4-point bend beam as a function of time...........234

6.25 Volumetric damage for the 4-point bend beam as a function of time when thedamage model constants are deterministic ..............................................................236

6.26 Flexural stress distribution on the 4-point bend beam as a function of time with

an assumed deterministic damage model.................................................................237

6.27 Flexural stress distribution on the 4-point bend beam as a function of time...........2406.28 Volumetric damage for the 4-point bend beam as a function of time when the

constants are uniform...............................................................................................241

6.29 Deviatoric damage for the 4-point bend beam as a function of time when theconstants are uniform...............................................................................................242

6.30 Predicted deflection as a function of time for a set of ten 4-point bend

specimens with element removal.............................................................................2436.31 SR76 tensile specimen used for creep tests in the first round robin ........................245

6.32 Strain as a function of time and stress at 1400C ....................................................246



6.35 Strain as a function of time and stress at 1250C ....................................................2486.36 Strain as a function of time and stress at 1200C ....................................................248

6.37 Strain as a function of time and stress at 1150C ....................................................2496.38 Strain as a function of time and stress at 1400C ....................................................249

6.39 Strain as a function of time and temperature with log scale for stress of 150 MPa.250

6.40 Strain as a function of time and temperature with log scale for stress of 200 MPa.2516.41 Time to reach a strain of 0.01 as a function of temperature for creep tests with a

stress of 150 MPa.....................................................................................................251

6.42 Time to reach a strain of 0.01 as a function of temperature for creep tests with astress of 200 MPa.....................................................................................................253

6.43 Failure time as a function of stress and temperature. Solid lines show failure timefor damage equal to one. Symbols represent data....................................................256

6.44 Strain as a function of time and temperature for tensile specimens at 150 MPa.....257

6.45 Strain as a function of time for tensile specimens at 1300C ..................................257

6.46 Finite element mesh for a Voronoi tessellation of the uniaxial tensile specimen

with =1000.............................................................................................................260

6.47 Boundary conditions for the Voronoi tessellation ...................................................261

6.48 Deviatoric stress, sxx, for the =1000 tessellation at a) 0 hours, b) 0.96 hours

and c) 1.32 hours......................................................................................................262

6.49 Volumetric stress, , for the =1000 tessellation at a) 0 hours, b) 0.96 hours

and c) 1.32 hours......................................................................................................263

6.50 Stress, xx, for the =1000 tessellation at a) 0 hours, b) 0.96 hours and

c) 1.32 hours.............................................................................................................2646.51 Stress, zz, for the =1000 tessellation at a) 0 hours, b) 0.96 hours and

c) 1.32 hours.............................................................................................................265

6.52 Volumetric damage, DK, for the =1000 tessellation at a) 0.96 hours and

b) 1.32 hours ............................................................................................................267

6.53 Volumetric damage, DK, for the =1000 tessellation at a) 0.96 hours and

b) 1.32 hours. The scale is setup to show elements whose damage is less than 0.1268


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6.54 Deviatoric damage, DG, for the =1000 tessellation at a) 0.96 hours andb) 1.32 hours ............................................................................................................269

6.55 Mesh for the two-phase model.................................................................................271

6.56 Boundary conditions on the two-phase model.........................................................273

6.57 Strain, xx, for the two-phase network at a) 0 hours and b) 50 hours.......................274

6.58 Strain, xx, for the two-phase network at a) 100 hours and b) 120 hours.................2756.59 Strain, xx, for the two-phase network at a) 125 hours and b) at failure,

129 hours..................................................................................................................276

6.60 Stress, xx, for the two-phase network at a) 0 hours and b) 50 hours......................278

6.61 Stress, xx, for the two-phase network at a) 100 hours and b) 120 hours................279

6.62 Stress, xx, for the two-phase network at a) 125 hours and b) at failure,129 hours..................................................................................................................280


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Acknowledgements

First, I would like to thank my two advisors for their continued support and assistance

with the completion of this project. Dr. Vassilis Panoskatsis was particularly helpful in

partaking in long technical discussions with me while Dr. Dario Gasparini was my

invaluable resource when it came to problem solving. Both were also helpful in engaging

with me in the cultural debate over Greek versus Roman influence and discussing which

society contributed more technological advances to the work at hand. Without them, this

project could not have been completed.

Thanks must also be given to Dr. Robert Mullen and Dr. John Lewandowski for

serving on my review committee. Their time and advice was a valuable asset during the

final stages of this project and contributed greatly toward its completion.

It would be a dire mistake if I did not also thank all those at the NASA Glenn

Research Center for their constant support and encouragement. If it was not for Dr.

Bernard Gross and his continual insistence that I take on this project, none of this work

would have started, let alone come to fruition. Dr. David Thomas, Dr. Louis Ghosn, Jane

Manderscheid and Fred Holland also offered their assistance with technical help, well

thought out advice, and of course the occasional ride to the train station. Without all of

you, this project would still only be an aspiration.

Finally, I must thank my adoring family for their emotional support throughout this

journey. While each member has been supportive, I must especially thank my parents,


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Dave and Carole, for all they have done for me. Whether it was picking me up from

NASA or running countless copies of my project down to Case Western, they always

offered a willing hand. I must also thank my brother, Scott, for continually bugging me

to finish my thesis. He never let me give up and if it hadnt been for his pestering, I

might not be here today. I must also thank my niece, Elizabeth, for her assistance in

completing these acknowledgements. Thank goodness we now have a writer in the

family.


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Mechanical Behavior of Ceramics at High Temperatures:

Constitutive Modeling and Numerical Implementation

Abstract

by

LYNN MARIE POWERS

High-temperature creep behavior of ceramics is characterized by nonlinear time-

dependent responses, asymmetric behavior in tension and compression, temperature

dependent, and nucleation and coalescence of voids leading to creep rupture. Moreover,

creep rupture experiments show considerable scatter or randomness in fatigue lives of

nominally equal specimens. Failure is caused by the nucleation and growth of voids at the

grain boundaries.

To capture the nonlinear, asymmetric, time-dependent behavior, the standard linear

viscoelastic solid model is modified. Nonlinearity and asymmetry are introduced in the

volumetric components by using a nonlinear function similar to a hyperbolic sine

function but modified to model asymmetry. Temperature is accounted for in the model

through temperature-dependent parameters. The nonlinear viscoelastic model is

implemented in an ABAQUS user material subroutine.

Damage is modeled using two scalar internal variables, one for the deviatoric

component and the other for the volumetric component. Each damage internal variable is

assumed to be governed by a nonlinear, first order ODE that is a function of stress and

two parameters. Each element is assigned damage parameters sampled from a lognormal

distribution. An element is deleted when damage is equal to one. Temporal increases in


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strains produce a sequential loss of elements (a model for void nucleation and growth),

which in turn leads to failure.

Nonlinear viscoelastic model parameters are determined from uniaxial tensile and

compressive creep experiments on silicon nitride. The model is then used to predict the

deformation of four-point bending and ball-on-ring specimens. Simulation is used to

predict statistical moments of creep rupture lives. Numerical simulation results compare

well with results of experiments of four-point bending specimens. A Voronoi simulation

of a tensile creep test is used to study the effects of temperature, stress and damage and to

evaluate model predictions. A preliminary simulation of a two-phase material is

presented.


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Chapter 1

Introduction

Time-dependent deformation characteristics of ceramic materials are important for

design. Applications at high temperatures and those utilizing porous ceramics need to

consider long term exposure to load, the resulting deformation and potential failure

processes. The overall focus of this work is to improve constitutive modeling and to

advance understanding of the behavior of ceramic materials in high temperature

environments.

1.1 High-Temperature Applications of Ceramic Materials

The gas turbine engine environment presents challenges to material technology.

Critical components include the rotors, nozzle guide vanes and the combustor liner. Load

and operating conditions include high temperature, thermal stress, centrifugal stress,

contact stress, high and low frequency cyclic fatigue, creep, stress rupture, oxidation and

corrosion (Anson and Richerson 2002). An application of a ceramic and a ceramic matrix

composite with an operation environment of 3000F (1650C) is shown in Figure 1.1.

Higher operating temperatures in gas turbine engines lead to increased efficiencies

and decreased harmful emissions (Takehara et al. 2002). Benefits include lowering the

NOxemission to 40.3% below IACO (International Civil Aviation Organization) rule and

a 5.4 billion lbs decrease in CO2in the atmosphere (Brewer 2006). Figure 1.2 shows the


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per cent efficiency as a function of the turbine inlet temperature for two different pressure

ratios (Anson and Richerson 2002). The overall efficiency increases as the temperature

increases for both pressure ratios.

CMC Combustor Liner

CMC Vane3000oF

CMC

System

NOx Reduction

CO2 Reduct ion

Compressor/

Turbine Disk

Turbine Airfoil Alloy

and Thermal Barrier

Coating (TBC) SystemCMC Combustor Liner

CMC Vane3000oF

CMC

System

NOx Reduction

CO2 Reduct ion

Compressor/

Turbine Disk

Turbine Airfoil Alloy

and Thermal Barrier

Coating (TBC) System

Figure 1.1: Turbine engine with ceramic composite components. Courtesy of David

Brewer, NASA Glenn Research Center.


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Figure 1.3 gives a history of materials development for high temperature applications

(Gray 2000). For each material system, its appearance on the chart indicates that it is

capable of sustaining a load of 150 MPa for 1000 hours. The plot shows the application

temperature as a function of time of development starting with 1950. Ceramics are a

candidate material for high temperature applications. The advantages of ceramics at high

temperatures are given in Table 1.1 (Gauthier 2002).

Figure 1.2: Engine efficiency as a function of turbine inlet temperature (Anson and

Richerson 2002).


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Figure 1.3: Stress rupture limits as a function of temperature and year for various

materials. Courtesy of Hugh Gray, NASA Glenn Research Center.


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Table 1.1 Benefits of ceramics in aerospace systems (Gauthier 2002)Ceramic Property System Benefit

Low density Reduce system weightHigh specific stiffness and strength High thrust-to-weight ratioProperty retention at high temperatures Thermal efficiency

Low coefficient of thermal expansion Dimensional stability

Environmental durability Long lifeThermal conductivity and electrical properties Applications and material specific

Fuel cells represent another significant high temperature application of ceramics. Fuel

cells are devices that can continuously generate electricity by a reaction between a fuel

and an oxidant (Nagamoto 2003). A fuel cell consists of two electrodes sandwiched

around an electrolyte as shown in Figure 1.4. Solid oxide fuel cells (SOFC) use a porous

ceramic solid-phase electrolyte that reduces corrosion and eliminates electrolyte

management problems found with other materials (Nagamoto 2003). Operating

temperatures for SOFCs are high compared with other fuel cell systems. The porous

material at this operating temperature is characterized by viscoelastic behavior (Dotelli

and Mari 2002; Routbort et al. 2000).

Applications for gas turbine engines and fuel cells place demands on ceramic

materials that generate nonlinear responses to a variety of load conditions. Constitutive

models are necessary to capture this nonlinear, temperature-dependent behavior.


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Figure 1.4: Solid oxide fuel cell (Nagamoto 2003).


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1.2 Objectives and Scope of Research

The primary objective of this research is to develop constitutive models that capture

the observed mechanical behavior of ceramics at high temperatures. The stress and strain

response for these materials is nonlinear, asymmetric in tension and compression, and

temperature-dependent. Ultimately, failure is a function of damage, usually evidenced by

the coalescence and growth of voids. The goal is to implement constitutive models into a

commercial finite element package and verify the models. Applications chosen highlight

the predictive capabilities of the viscoelastic model and life prediction as a function of

damage.

This thesis is divided into seven chapters. The second and third chapters are reviews

of the experimental behavior and model development, respectively. The fourth chapter

provides detail on the proposed constitutive models and illustrates their characteristics.

The following chapter describes the implementation of the constitutive model into a finite

element code. Applications are presented in the sixth chapter. The final chapter contains

the conclusions of this research as well as suggested future work.

1.3 References

Anson, D., and Richerson, D. W. (2002). "The Results and Challenges of the Use of

Ceramics in Gas Turbines." Ceramic Gas Turbine Design and Test Expierence,

M. van Roode, M. K. Ferber, and D. W. Richerson, eds., ASME Press, New York.Brewer, D. (2006). "Private Communication." NASA Glenn Research Center, Cleveland.

OH.

Dotelli, G., and Mari, C. M. (2002). "Modelling and simulation of the mechanical

properties of YSZ/Al2O3 composites: a preliminary study." Solid State Ionics,148(3-4), 527-531.

Gauthier, M. M. (2002). "Structural Applications for Advanced Ceramics." EngineeredMaterials Handbook, ASM International.


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Gray, H. (2000). "Private Communication." NASA Glenn Research Center, Cleveland.

OH.Nagamoto, H. (2003). "Fuel Cells: Electrochemical Reactions." Encyclopedia of

Materials: Science and Technology, Elsevier Science Ltd, Oxford, 3359-3367.

Routbort, J. L., Goretta, K. C., Cook, R. E., and Wolfenstine, J. (2000). "Deformation of

perovskite electronic ceramics - a review." Solid State Ionics, 129(1-4), 53-62.Takehara, I., Tatsumi, T., and Ichikawa, Y. (2002). "Summary of CGT302 ceramic gas

turbine research and development program."Journal of Engineering for Gas

Turbines and Power-Transactions of the Asme, 124(3), 627-635.


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Chapter 2

Observed Material Behavior

2.1 Introduction

Brittle materials, including ceramics such as silicon nitride and silicon carbide,

exhibit unique deformation and failure characteristics at high temperatures. These

materials are characterized by nonlinear time-dependent responses, asymmetric behavior

in tension and compression, and nucleation and coalescence of voids leading to rupture.

Moreover, rupture experiments show considerable scatter or randomness in fatigue lives

of nominally equal specimens. This chapter reviews the literature on high temperature

mechanical behavior of ceramics, globally and at the microstructural level.

Ceramic materials have changing mechanical properties and failure mechanisms over

the temperature range at which these materials are used in design. A diagram showing the

failure mechanisms for silicon nitride as a function of temperature is shown in Figure 2.1

(Quinn 1990). At room or low temperature, these materials are brittle and linear elastic

showing no time-dependent response under load. Their failure is generally due to a single

flaw or crack which is a part of a distribution of flaws. This material is modeled with

Weibull statistics and the confidence intervals are shown in Figure 2.1. A gradual

weakening of the material occurs above 900C. As temperature increases, their failure

mechanism, though still based on a single flaw, changes to include slow crack growth. At


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higher temperatures and low loads, these materials exhibit time dependent deformation

behavior and are referred to on Figure 2.1 as creep fracture.

Creep behavior depends on the presence or absence of a grain boundary phase. A

typical microstructure for each of these materials is shown in Figure 2.2 (Lewis and

Dobedoe 2003). Figure 2.2a shows a fine grained hot-pressed alumina with no grain

boundary phase. Figure 2.2b shows a microstructure of a silicon nitride with a grain

boundary phase. Where an amorphous intergranular phase is present, the composition and

quantity of this phase become critical in determining the creep performance. The more

refractory the intergranular phase, the more resistant the ceramic will be to creep

Figure 2.1: Comprehensive fracture map for MgO doped HPSN tested in flexure in

air (Quinn 1990).


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Temperature effects are presented. The lifetime predictions including microstructural

damage are presented. Both the deformation behavior and the lifetime are statistical in

nature. Studies conducted to quantify this are presented.

2.2 Viscous Flow

In the mechanics of deformable media, the response behavior of an elastic solid is

captured by the classical theory of elasticity. In a simple uniaxial test, the load

deformation curve follows the same path for both increasing and decreasing load. Under

a constant level of load, the deformation is constant, i.e. time independent. Viscous flow

is often assumed to be Newtonian with the stress proportional to the rate of strain and

independent of the strain itself. This behavior is time dependent.

The theories of elasticity and Newtonian fluids do not adequately describe the

response behavior and flow of most real materials. Between these two responses, a real

material may exhibit combined response characteristics of solids and fluids. Attempts to

characterize the behavior of real materials under the action of external loads gave rise to

the science of rheology. The phenomenon labeled viscoelasticity is defined as

mechanical behavior combining response characteristics of both an elastic solid and a

viscous fluid. A viscoelastic material is characterized by a level of rigidity of an elastic

solid body and at the same time it flows and dissipates energy as a viscous fluid (Haddad

1995).

Ceramic materials at high temperatures are viscous. Evidence of this behavior is

found by changing load rates and by observing responses during unloading. The stress

strain curves for several uniaxial tensile specimens of titanium silicocarbide, Ti3SiC2at


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1200C are shown in Fig. 2.3 (Radovic et al. 2000). The strain rates range from

1.3710-5

/s to 6.8510-4

/s. At the fastest strain rate the material behaves as an elastic

body. As the strain rate decreases, the stress/strain behavior deviates from elastic.

In addition to constant load rate tests to measure strain response, another common test

is the creep experiment. In this experiment, a constant load is applied for some duration.

The load up is generally ramped quickly enough so that no viscoelastic behavior occurs.

Figure 2.4 shows strain as a function of time for uniaxial creep tests conducted on silicon

Figure 2.3: Effect of strain rates on engineering stress/strain curves tested in tension at

1200C. (Radovic et al. 2000).


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nitride (NT154) specimens at 1371C (Menon et al. 1994a; Menon et al. 1994b; Menon

et al. 1994c). The strain response at any one time is not proportional to the applied stress;

that is, the strain is a nonlinear function of the applied stress. Also the response is

different in tension and compression. To further study these effects, strains are plotted at

fixed times of 1, 10 and 100 hours for various stress levels; these isochrones are shown in

Figure 2.5 (Menon 1994). As shown in this figure, for a prescribed stress amplitude, the

tensile strain is higher than the compressive strain. Replicate tests conducted on tensile

specimens at 150 MPa also show scatter in the strain response.

Figure 2.4: Strain as a function of time for several uniaxial creep tests at 1371CMenon 1994 .

-0.008

-0.004

0.000

0.004

0.008

0.012

0.016

0 50 100 150 200 250 300

Time (hr)

Stra

in

180

150

140

130

125

-30

-100

-200

-300

-400

-500


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Figure 2.5: Isochrones for silicon nitride (NT154) after a) 1 hour, b) 10 hours and

c) 100 hours at 1371C (Menon 1994).


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Compressive creep tests also demonstrate a nonlinear behavior versus stress for a

nano-crystalline Si-B-C-N ceramic (Kumar et al. 2004). The strain response as a function

of time and load is shown in Figure 2.6a. Isochrones for these tests are shown in Figure

2.6b. A positive magnitude has been used for compression in Figure 2.6.

Figure 2.6: a) Deformation of nano-crystalline Si-B-C-N ceramics as a function of

time for compressive loads at a test temperature of 1400C. b) Isochrone for one day

and one week for the tests presented in a) (Kumar et al. 2004).

0

0.005

0.01

0.015

0 20 40 60 80 100

Stress, MPa

Strain

day

week

a)

b)


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Another important aspect of the mechanical behavior of viscoelastic materials is their

response to the removal of load. The strain response of titanium silicocarbide, Ti3SiC2, to

a load/unload test at 1200C is shown in Figure 2.7 (Radovic et al. 2000). The load

history is shown on the left side of the graph. A creep test was conducted for 80 hours at

70 MPa. At that time the load was removed and the strain was measured for some time.

The strain is shown on the right side of Figure 2.7. The total strain resulting from the

creep test has both a permanent and a reversible part. This is shown by the partial

recovery of the strain.

Figure 2.7: Creep test with unload at 1200C and 70 MPa. The left axis represents thetrue stress and the ri ht axis the true strain Radovic et al. 2000 .


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Silicon nitride shows similar behavior. One conventional creep test was run at

1300C and 200 MPa with creep recovery measured for about the same time as the

forward creep as shown in Figure 2.8 (Woodford 1998). At least 40% of the creep strain

was fully recoverable in 70 h.

These examples of silicon nitride, titanium silicocarbide, and nano-crystalline Si-B-

C-N ceramic demonstrate that classes of ceramic materials are viscoelastic at high

temperatures. Their deformation response is a function of time and these materials show

partial recovery on load removal. The following section describes these phenomena at the

microstructural level.

Figure 2.8: Strain as a function of time for a silicon nitride under 200 MPa for 60

hours at 1300C, after 60 hours, the load is removed (Woodford 1998).


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2.3 Microstructural Features

Ceramic microstructures are characterized based on the presence or absence of a grain

boundary phase. Their viscoelastic and failure behavior is dictated by this characteristic

(Lewis and Dobedoe 2003). A fine grained hot-pressed alumina is an example of a

ceramic with no grain boundary phase. Direct grain to grain contact of the sub micron

-Al2O3grains is observed as shown in Figure 2.2a. Ceramics with a grain boundary

phase include silicon nitride which has a thin film of glass between the-Si3N4grains as

shown in Figure 2.2b (Lewis and Dobedoe 2003). The glass layer dominates high-

temperature performance.

An example of a ceramic microstructure with a grain boundary phase is a magnesium

doped alumina (Kottada and Chokshi 2000). The relative densities of the as-sintered

specimens were estimated to be > 99%. Fig. 2.9a shows a scanning electron micrograph

of the as-sintered specimen. Measurements revealed an average grain size of 2.0 0.1

m, and an aspect ratio of 1.06 0.05. The grains were clearly equiaxed, and

measurements made by the Kottada and Chokshi (2000) indicated that the grain size

distribution was log-normal.

Fig. 2.9b is a scanning electron micrograph of a specimen with a grain size of 3.6 m

tested to fracture at 1550C and 3.510-5

/s. It is clear that the grains remain essentially

equiaxed after considerable deformation. There is also evidence for the nucleation,

growth and interlinkage of cavities to form large macroscopic cracks perpendicular to the

tensile axis. Kottada and Chokshi (2000) also observed some cavitation in tests

conducted in compression.


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Grain boundary sliding mechanisms have also been examined in alumina by

Blanchard and coworkers (1998). These are shown in Figure 2.10. The tensile specimen

was under 35 MPa load for 8 hours at 1500C. The applied load is in the horizontal

direction in the figure. Figure 2.10a shows out of plane grain boundary sliding. Grain

boundary separation is shown in Figure 2.10b. Figure 2.10c shows rotation of a grain. In-

plane grain rotation observed is indicated by the arrows. Arrow 1 points out the left side

of the boundary, which is closed, while arrow 2 indicates a gradual opening of the

boundary moving toward the right. Finally, microcracks, such as those shown in Fig.

2.10d, were generally observed to develop perpendicular to the tensile stress axis, as

Figure 2.9: Scanning electron micrograph of (a) an as-sintered specimen and (b) adeformed tensile specimen illustrating the retention of equiaxed grains and concurrent

cavitation; the tensile axis is horizontal (Kottada and Chokshi 2000).


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expected. These microcracks were observed to nucleate in the gauge section at the edges

of a specimen, most likely due to the presence of machining flaws and the lack of a

chamfer at the specimen corners.

Figure 2.10: Grain boundary sliding mechanisms illustrating out-of-plane, separation,

rotation and crack growth in a-d, respectively (Blanchard et al. 1998).

a) c)

b)d)


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Another study by Jin and coworkers (2001) examined the statistical distribution of the

grain boundary thicknesses before and after a sample is exposed to a creep test. Figure

2.11a shows the results measured as the sample crept at 40 MPa for 690 h. At the uncrept

grip end, the data show a Gaussian distribution with a mean value of 0.720.13 nm. This

suggests that there exists a characteristic value of the grain-boundary film widths in the

undeformed material, independent of grain orientation. In the crept gauge section,

however, the film widths exhibit a bimodal distribution, with the first peak around 0.52

nm and a second peak around 1.33 nm (Fig. 2.11b), i.e., some grain boundaries become

thinner while others become thicker after creep. For modeling purposes, it is important to

note that boundary-phase thicknesses are about three orders of magnitude smaller than

grain size.

Figure 2.11: Histograms of film thickness distribution of grain boundaries of the

experimental materials crept at 1430C with a stress of 40 MPa for 690 h: (a) uncrept

grip end; (b) crept gauge section (Jin et al. 2001).


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A typical film for unloaded material is shown in Figure 2.12. Its thickness is 0.74 nm.

Images of a thinner film (0.5 nm) and a thicker film (1.2 nm) in the gauge section after

loading are shown in Fig. 2.14 (Jin et al. 2001).

Figure 2.13: High-resolution lattice fringe in the grip end showing a film thickness of

a 0.5 nm and b 1.25 nm at different rain boundaries Jin et al. 2001 .

Figure 2.12: High-resolution lattice fringe in the grip end showing a film thickness of

0.74 nm (Jin et al. 2001).


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Similar studies have been conducted for silicon nitride under compressive creep

conditions (Wang et al. 1997). A histogram of the film thickness distribution before and

after exposure to load is shown in Figure 2.14. When compared with the histogram for

the tensile specimen, less change is seen. The material had a greater number of

boundaries free of film and an increase in thick boundaries.

The microstructure after compressive creep tests has also been studied by Crampon

and coworkers (1997). At 1300C under 175 MPa, TEM of a crept sample revealed a

partial crystallization of the glassy phase and a cavity nucleation and growth in the triple

grain junctions as shown in Figure 2.15a. Cavities were wedge-shaped as shown in Fig.

Figure 2.14: Histograms of intergranular film thickness distributions in materials (a) as-

hot-pressed, and (b) after compressive deformation (Wang et al. 1997).


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2.15b. Bubble-like cavities also were evidenced in some cases as shown in Figure 2.15c.

These cavities were not present on the undeformed samples and were therefore related to

the deformation. The growth of cavities was sometimes observed through the pockets and

along two contiguous grain boundaries where the glassy film was rather thick (Fig.

2.15d). In such a case, where cavities form on contiguous boundaries, coalescence will

probably occur.

Figure 2.15: TEM micrographs (a) in a thick portion of the foil, illustrating a largenumber of cavities produced during compressive creep, (b) of typical wedge-shaped

cavity produced during compressive creep, (c) of bubblelike cavities produced during

compressive creep, and (d) showing the growth of cavities through the glass pockets. In

each figure, the scale bar is 0.5 mm (Crampon et al. 1997).


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2.4 Asymmetry

Differences in the microstructural response to load in tension and compression will

affect the mechanical behavior at the macrostructural level. The strain response is

expected to be greater in tension than it is in compression. These differences are

examined for a uniform load as well as a flexural beam where the stress state is not

uniform and contains both tensile and compressive regions.

Tensile and compressive response for silicon nitride was studied by Wereszczak and

coworkers (1999b). The strain response was always greater in tension than in

compression for an equal magnitude of stress. Examples of this creep asymmetry are

illustrated in Figs 2.16(a-e) for 1316C:125 MPa, 1371C:30 MPa, 1371C:200 MPa,

1399C:25 MPa, and 1399C:100 MPa, respectively. The applied stress given is both in

tension and compression. The creep histories in Figure 2.16 show that the tensile and

compressive curves start to diverge at the beginning of the test. The ratio of the strain in

tension to that in compression for the same magnitude of stress increased with the

magnitude of stress. Temperature also increases the ratio of tensile to compressive strain.

Post-testing microstructural analysis revealed that differences in the amounts of

tensile- and compressive-stress-induced cavitation accounted for the creep strain

asymmetry. Multigrain junction cavities also formed in compressively crept specimens

tested at 1371C:200MPa and 1399C:-100MPa as shown in Figure 2.17 (Wereszczak et

al. 1999b). In addition to cavity-concentration differences, trends in cavity-type, size, and

location also provided insights into the tensile and compressive creep deformation

behavior.


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Figure 2.16: Strain as a function of time for a tensile and compressive creep test at

temperatures and stresses shown in each graph (Wereszczak et al. 1999b).

a) c)

b) d)

e)


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Wereszczak and coworkers (1999b) concluded that multigrain junction cavities

formed in all tensile crept specimens, with larger concentrations found in tensile

specimens which accumulated greater amounts of tensile strain. Multigrain cavities also

formed in compressively crept specimens tested at relatively high temperatures and

stresses, but their concentrations were far less than their tensile specimen counterparts

tested at the same magnitude of stress.

Figure 2.17: Microstructure of specimens tested at 1371C after exposure to a stress

of 300 MPa is a) tension and b) compression (Wereszczak et al. 1999b).

a)

b)


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Lofaj and coworkers (1997) studied the contribution of cavitation to asymmetry in

tension and compression. They concluded that cavities contribute to the deformation in

uniaxial tension regardless of their shape and orientation. Cavities do not contribute to

compressive deformation. Creep asymmetry follows from such preferential contribution.

In compression, any cavitation is due to the stress perpendicular to the load. Lofaj (1997)

also surmised that the contribution of cavitation to tensile deformation was found to be

proportional to the volume fraction of cavities.

With the asymmetry characteristic of these materials, it is important to investigate

specimens with nonuniform states of stress similar to those found in actual applications.

In particular, experimentalists sought specimens that were easy to fabricate, laboratory

tests that were easy to conduct, and with an elastic solution containing both tension and

compression stress states. Two of these tests are the flexure beam (Fields and Wiederhorn

1996) and the C-ring (Chuang et al. 1992). Both specimens have a stress state with a

dominant stress similar to a uniaxial stress state. For the elastic solution, the magnitudes

of the maximum tensile and compressive stresses are equal.

For viscoelastic materials the stresses and strains are a function of time. The strains in

a flexure beam have been recorded as a function of time and position and are shown in

Figure 2.18 (Fields and Wiederhorn 1996). The elastic solution (time=0) for the midspan

(which is not shown in the figure) is symmetric in tension and compression with a neutral

axis at the center. As time passes, asymmetry in tension and compression causes a neutral

axis shift (Choi and Salem 1994). The neutral axis is shifting away from the tensile

surface as shown in Figure 2.18.


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2.5 Temperature

The response of a material to load is a function of temperature (Kraus 1980). Radovic

and coworkers (2000) have studied the changing stress-strain response of titanium

silicocarbide in tension over a wide temperature range. The stress strain response and the

failure strains were investigated. Figure 2.19a shows stress-strain curves over the 25-

1300C temperature range. The strain rate for these tests was 1.3710-4

/s. The authors of

this study also presented the stress strain behavior as a function of load rate (Figure 2.3).

Figure 2-19a illustrates the linear elastic behavior at temperatures up to 1100C. Above

this temperature the material is viscoelastic at this load rate.

Figure 2.18: Strain as a function of distance from the tensile surface of flexure

specimens. The specimens shown here were tested for 2, 5 and 10 hours at 1300C and a

load of 205 N (max stress of 120 MPa) (Fields and Wiederhorn 1996).


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The ultimate tensile strength and the failure strain are shown in Figure 2.19b for this

material (Radovic et al. 2000). The ultimate tensile strength gradually decreases in the

linear elastic region below 1000C. Above 1100C, the ultimate tensile strength

decreases rapidly. The failure strain remains relatively constant in the linear elastic

temperature range. At temperatures where the material is viscoelastic, the failure strain

increases with temperature.

The mechanical behavior of silicon nitride has been investigated over the temperature

range 1200-1400C (Menon et al. 1994a; Menon et al. 1994b; Menon et al. 1994c). Creep

tests were conducted on silicon nitride NT154. This data set consisted of approximately

100 specimens; eighty were tested to failure in tension and 20 in compression. This

database also included 3 sets of replicate tests. The duration of the compressive tests was

approximately one week. Isochrones for strain as a function of stress and temperature

after 10 hours under load are shown in Figure 2.20. The tension/compression asymmetry

is apparent in this figure. The strain increases as a function of temperature. The reported

strain for the replicate tests is the average value. Figure 2.21 shows the time to failure for

the creep tests as a function of stress and temperature. The time to failure decreases as the

temperature increases. Failure times for the replicate tests are plotted individually.

The effect of temperature on individual creep tests is shown in Figure 2.22. The log

of strain is plotted as a function of the log of time for two creep tests at 150 MPa. The

two tests were conducted at 1371C and 1400C. The strain response is shifted for the

creep test at a higher temperature. This shift is uniform throughout the duration of the

test.


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Figure 2.19: (a) Stress strain curves of fine-grained Ti3SiC2samples using a strain rate

of 1.3710-4

/s. (b) The effect of temperature on the ultimate tensile strength and strains

to failure (Radovic et al. 2000).


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Figure 2.20: Isochrones of strain as a function of stress and temperature for NT154

silicon nitride after 10 hours under load.

Figure 2.21: Time to failure as a function of stress and temperature for NT154 siliconnitride.


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Another study on silicon nitride, SN88, was conducted by Woodford (1998). To

determine whether there is a well-defined temperature above which viscoelastic behavior

occurs, a series of loading and unloading experiments were conducted on a single

specimen at increasing temperature. The specimen was loaded at 10 MPa/s to 300 MPa

starting at room temperature and increasing to 1300C. The stress was allowed to relax

for 1 day at each temperature, unloaded, and then held at zero stress

Figure 2.22: Strain as a function of time for a 150 MPa creep test at two temperatures,

1371C and 1400C. The strain and the time are shown on a log scale.

0.001

0.01

0.1

1 10 100 1000

Time, hr

Strai 1371

1400

Temperature

C


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for 1 day to measure strain recovery, before loading to the next higher stress. The results

are shown in Figure 2.23 (Woodford 1998). The first significant relaxation was observed

at 800C and progressively increased to 1300C. The creep recovery during the 1 day

hold was approximately 40% over this temperature range. It appears that linear elastic

behavior may be assumed up to about 800C and that viscoelastic behavior becomes

increasingly important at higher temperatures.

Figure 2.23: Relaxation and recovery at different temperatures (Woodford 1998).


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2.6 Damage

In Section 2.3, the initial effects of load on the microstructure were examined. As the

microstructure deforms and approaches its failure limit, microstructural changes are more

dramatic. One of the most common features in ceramics is the growth of voids at grain

junctions.

Figure 2.24 shows the nucleation and growth of cavities primarily in the interstitial

pockets of silicate located at multigrain junctions (Luecke and Wiederhorn 1999). For

silicon nitride, Luecke and coworkers postulated that once the silicate phase has

completely left the pocket, the cavity stops growing (Luecke et al. 1995). Their study has

shown that the continuous formation of new cavities, rather than the growth of existing

cavities, dominates the volume growth.

The relationship between the microstructural properties and the cavity characteristics

was studied by Lofaj (1999). Interstitial cavities in the pockets between Si3N4grains were

very easily observed in each creep-tested specimen, because of their high density and

size. Figures 2.25 illustrate a group of cavities that were formed at the junctions of the

matrix grains. Figure 2.25a shows a microstructure of the damaged silicon nitride after a

200 MPa stress was applied for 11,114 hours at 1250C and figure 2.25b shows a

microstructure for the same material after 1682 hours under a load of 155 MPa at

1300C. Their shapes are varied, depending on the geometry of the grains and the

possibility for grain sliding. Relatively large, irregularly shaped cavities are observed

simultaneously with smaller triangular cavities. The local density and the size of the

cavities in the specimens after extremely long creep tests at 1250C (>10000 h) were

very similar to those after considerably shorter tests at higher temperatures. Note that the


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dimensions of the individual multigrain-junction cavities are comparable to the size of the

matrix grains. The coalescence of the cavities may lead to the formation of microcracks

that are larger than the size of the matrix grain.

Figure 2.24: Interstitial cavities in (a) NT154, a HIPed Si3N4crept for 689 h at 1430C

under 75 MPa to a failure strain of 0.020, and (b) SN-88, a gas-pressure-sinteredSi3N4,18 crept for 477 h at 1400C under 100 MPa to a failure strain of 0.042 (Luecke

and Wiederhorn 1999).


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2.7 Randomness

To quantify the scatter in time and strain to failure for ceramics at high temperatures,

round robin replicate tests on identical specimens have been performed at various

laboratories (Menon et al. 1994b; Wereszczak et al. 1999a). Two round robins were

organized by Luecke at the National Institute of Standards and Technology (NIST)

involving several laboratories from around the world (Luecke 2002; Luecke and

Figure 2.25: (A) Low-magnification TEM micrograph of the microstructural damage in

the studied silicon nitride after creep testing at 1250C under a stress of 200 MPa,interrupted after 11114 h. (B) Similar cavities are the most-often-observed type of

cavities also in shorter tests (1300C, 155 MPa, 1692 h) (Lofaj et al. 1999).


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Wiederhorn 1997). Both exercises used a commercially available silicon nitride (SN88)

with highly repeatable high temperature mechanical properties. For the first round robin,

participating laboratories were given tensile specimens of the same size and the

instructions given prescribed as little of the test procedure as possible. The second round

robin would have similar rules; however, different size specimens were tested.

Five laboratories participated in the first round robin which tested circular cross

sectioned tensile specimens in creep with a stress of 150 MPa at 1400C (Luecke and

Wiederhorn 1997). Each specimen was placed under creep conditions until failure.

Advance tests at NIST demonstrated that the failure time should be less than 100 hours.

The strain as a function of time for the combined set of tests is shown in Figure 2.26. The

mean time to failure is 75.8 hours with a coefficient of variation of 0.095. The mean

failure strain is 0.0286 with a coefficient of variation of 0.136. Both ranges are within

expected limits for this material.

Figure 2.26: Strain as a function of time for a tensile creep test at 1371C and a stress of

150 MPa in air (Luecke and Wiederhorn 1997).

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0 10 20 30 40 50 60 70 80 90

Time, hr


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The second round robin produced different results (Luecke 2002). It involved a larger

number of laboratories and the samples tested were not identical in size. All specimens

were subjected to a creep test at 200 MPa and 1375C. Figure 2.27 shows the strain time

curves by laboratory on axes with the same scale. A much larger variation is present in

this study. The time to failure for all laboratories is shown in Figure 2.28. Data from each

laboratory are plotted vertically. The coefficient of variation is printed at the top. Solid

symbols represent tests on the large-diameter, buttonhead specimens and open symbols

represent tests on small cross-section specimens. The variation in the time to failure for

the SN88 material is similar to that found for NT154 as shown in Figure 2.21.

Luecke gives several potential sources for the randomness in the time or strain to

failure (Luecke 2002). These were summarized in the schematic shown in Figure 2.29.

These sources are divided into 3 groups: the experiment, the material, and size effect.

Experimental sources include variability in temperature and load, specimen alignment

and strain measurement. It was concluded that although experimental sources may have

contributed to the scatter, they were probably not the primary cause. Material features

such as subtle chemical and physical differences can alter mechanical behavior.

For this round robin, evidence points to the size effect as significant. As a group, the

small cross-section specimens lasted about 5 times longer and the deformation rate was

about 3 times slower than the large cross-section buttonhead specimens. Luecke (2002)

cites a possible reason for this is the oxidizing layer of the specimen. The oxidizing

process effects the second phase at the grain boundaries, making them resistant to

deformation. For the large specimens, the bulk of the specimen is not affected. The bulk

of the small specimen is oxidized.


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Figure 2.27: Fifteen sets of creep curves for this study. Each subfigure lists the times to

failure and minimum creep rates for the specimens (Luecke 2002).


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Figure 2.28: Time to failure for the 14 laboratories. The dashed lines represent the

mean value for all laboratories. Solid symbols represent tests on the large-diameter,buttonhead specimens and open symbols represent tests on small cross-section

specimens (Luecke 2002).

Figure 2.29: Potential contributions to random behavior for a typical strain time curve

(Luecke 2002).


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2.8 Summary

The mechanical behavior of ceramics at high temperatures has shown several

important characteristics. These materials are nonlinear viscoelastic, asymmetric in

tension and compression and their deformation is a function of temperature.

Microstructural features show the presence of voids and the damage introduced by those

voids. This void nucleation mechanism contributes to the randomness in both the time to

failure and the deformation. Replicate tests have shown considerable scatter.

Figure 2.30: Cavity development in flexure specimens tested at 1300C and an initial

maximum tensile stress of 250 MPa. Three different specimens at a) 1.2%, 1.75% and c)

2% max strain Fields and Wiederhorn 1996 . The tensile surface is at the to .

100 m


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The void nucleation and growth in a flexure beam illustrates most of these

characteristics. Figure 2.30 shows the cavity development in flexure specimens (Fields

and Wiederhorn 1996). The tensile surfaces for three different specimens at different

strains are shown. The thickness of the beams is 3 mm, therefore the area shown in each

micrograph is one tenth the thickness (300 m). As the strain increases, the density of

cavities increases. The voids form near the tensile surface and their location is random.

As time evolves and the strain increases, cavities begin to appear farther away from the

tensile surface at the top in the figure.

The goal of this research is to model the mechanical behavior of ceramics at high

temperatures. A review of the modeling efforts for these materials is presented in the next

chapter.

2.9 References

Blanchard, C. R., Lin, H. T., and Becher, P. F. (1998). "Grain boundary sliding

measurements during tensile creep of a single-phase alumina."Journal of theAmerican Ceramic Society, 81(6), 1429-1436.

Choi, S. R., and Salem, J. A. "Creep Behaviour of Silicon Nitride Evalated by

Deformation Curvature and Neutral Axis Shift Determinations." Silicon-Based

Structural Ceramics. Proc.Symp. Honolulu, 7-10 November 1993, p.285-293.Ceram.Trans.Vol.42.

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