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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
6.33 Strain as a function of time and stress at 1350C ....................................................247
6.34 Strain as a function of time and stress at 1300C ....................................................247
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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9
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
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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.
Chuang, T. J., Wang, Z. D., and Wu, D. D. (1992). "Analysis of Creep in a Si-Sic C-Ring
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Crampon, J., Duclos, R., Peni, F., Guicciardi, S., and DePortu, G. (1997). "Compressivecreep and creep failure of 8Y(2)O(3)/3Al(2)O(3)- doped hot-pressed siliconnitride."Journal of the American Ceramic Society, 80(1), 85-91.
Fields, B. A., and Wiederhorn, S. M. (1996). "Creep cavitation in a siliconized silicon
carbide tested in tension and flexure."Journal of the American Ceramic Society,79(4), 977-986.
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Jin, Q., Wilkinson, D. S., Weatherly, G. C., Luecke, W. E., and Wiederhorn, S. M.
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Kottada, R. S., and Chokshi, A. H. (2000). "The high temperature tensile and
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Kraus, H. (1980). Creep Analysis, John Wiley & Sons, Inc., New York.
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Lofaj, F., Okada, A., and Kawamoto, H. (1997). "Cavitational strain contribution to
tensile creep in vitreous bonded ceramics."Journal of the American Ceramic
Society, 80(6), 1619-1623.Lofaj, F., Okada, A., Usami, H., and Kawamoto, H. (1999). "Creep damage in an
advanced self-reinforced silicon nitride: Part I, cavitation in the amorphousboundary phase."Journal of the American Ceramic Society, 82(4), 1009-1019.
Luecke, W. E. (2002). "Results of an international round-robin for tensile creep rupture
of silicon nitride."Journal of the American Ceramic Society, 85(2), 408-414.Luecke, W. E., and Wiederhorn, S. M. (1997). "Interlaboratory verification of silicon
nitride tensile creep properties."Journal of the American Ceramic Society, 80(4),
831-838.Luecke, W. E., and Wiederhorn, S. M. (1999). "A new model for tensile creep of silicon
nitride."Journal of the American Ceramic Society, 82(10), 2769-2778.Luecke, W. E., Wiederhorn, S. M., Hockey, B. J., Krause, R. F., and Long, G. G. (1995).
"Cavitation Contributes Substantially to Tensile Creep in Silicon-Nitride."
Journal of the American Ceramic Society, 78(8), 2085-2096.
Menon, M. N. (1994). "Private Communication, Honeywell." Phoenix, AZ.Menon, M. N., Fang, H. T., Wu, D. C., Jenkins, M. G., and Ferber, M. K. (1994a).
"Creep and Stress Rupture Behavior of an Advanced Silicon- Nitride .2. Creep
Rate Behavior."Journal of the American Ceramic Society, 77(5), 1228-1234.Menon, M. N., Fang, H. T., Wu, D. C., Jenkins, M. G., and Ferber, M. K. (1994b).
"Creep and Stress Rupture Behavior of an Advanced Silicon- Nitride .3. Stress
Rupture and the Monkman-Grant Relationship."Journal of the AmericanCeramic Society, 77(5), 1235-1241.
Menon, M. N., Fang, H. T., Wu, D. C., Jenkins, M. G., Ferber, M. K., More, K. L.,
Hubbard, C. R., and Nolan, T. A. (1994c). "Creep and Stress Rupture Behavior ofan Advanced Silicon- Nitride .1. Experimental-Observations."Journal of the
American Ceramic Society, 77(5), 1217-1227.
Quinn, G. D. (1990). "Fracture Mechanism Maps for Advanced Structural Ceramics .1.
Methodology and Hot-Pressed Silicon-Nitride Results."Journal of MaterialsScience, 25(10), 4361-4376.
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Radovic, M., Barsoum, M. W., El-Raghy, T., Seidensticker, J., and Wiederhorn, S.
(2000). "Tensile properties of Ti3SiC2 i