Heinrich J.G., Aldinger F. (Eds.) Ceramic Materials and Components for Engines
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DESIGN STANDARD
FOR
ADVANCED CERAMIC MATERIALS
AND COMPONENTS
Hiroshi Kawamoto*, Minoru Matsui**, and Hideo Kobayashi***
*,
**
Japan Fine Ceramics Center, Nagoya,
456-8587
Japan
***
Tokyo Institute
of
Technology, Tokyo,
152-8552
Japan
*
Sending Researcher from Toyota Motor Corporation
**
Sending Researcher from NGK Insulators, LTD.
ABSTRACT
Design technologies for advanced ceramics could be
required to confirm sufficient strength and reliability of
the components under the service conditions.
Consequently, developing and applying these design
methods, widespread uses of the materials and
components should be accelerated
in
various structural
systems. Whereas design methods suitable for advanced
ceramics have been steadily making progress for the
structural applications, it is extremely meaningful to
incorporate the cumulative experiences
in
the individual
field into design standards. However, there is no
standard for these design methods
in
Japan and
advanced countries until now. Therefore, constructing
the design standards for ensuring reliability of ceramic
materials and components has
so
far been conducted
as
an important subject for the Technology Infrastructure
Project
in
the Ministry of International Trade and
Industry, Japan. This paper introduces the present
activities.
BACKGROUND AND OBJECTIVES
Development efforts have been made
in
automobile
technologies for heat engines to improve the power
performance and the fuel economy, and
so
on. It is well
recognized that ceramic applications are key to succeed
in
such advanced heat engines, owing to the mechanical
and the thermal properties
[l,
21.
Fig.
1
shows ceramic
components developed or under research for
reciprocating engines. Many ceramic components have
been studied for a long time. However, there are not
so
many mass-produced components now. Few attempts
have
so
far been made at practical applications of
ceramic materials and components to present-day
automobiles.
An
example is the amount of Si3N4 for
mass-produced components. From
FY
90
to
FY
93,
the
amount was increased considerably because of the
practical uses of turbocharger rotors. After that, the
amount decreased according to a reduction
in
the
numbers of turbo-charged cars. The use of tribological
components such
as
rollers and roller-bushings
in
fuel
injection pumps for diesel engines has recently
increased.
In
Fig.
2,
R&D
directions for structural ceramics are
recommended for the widespread uses. Those are
materials technologies for the development of high
performance materials and the uses of these materials.
In
systems, it is essential that those components
maintain high reliability with an excellent balance
between cost and performance. Consequently, design
methods are important for making sure of the reliability
in
components.
1)
Accumulating and passing on experiences
in
structural design methods suitable for advanced
ceramics systems.
2)
Constructing Design Standards for the reliability of
ceramics systems
as
an important subject
Objectives ofthis work are
as
follows
:
;
Mass-produced
component
Rockerarm seat
Rocker-arm
Pad
Shim
,\
\
/Cam roller
Swirl chamber
Fig.
I
Ceramic components developed or under
research for automobile reciprocating engines
Material technologies balanced between
performance, reliability and cost
Low
cost manufacturing
High damage tolerance materials
Technologies in uses
of
materials
for fracture resistance
Processing, Machining, Inspection
Analyses
a
simulations
Fig.
2
R&D
items expected from now on
273
for the Technology Infrastructure Projects in the
Japanese government.
3) Accelerating the wide-spread practical uses of
ceramic materials and components in various structural
systems. Applying the standards to these systems.
FY1997
I
FYl998
I
FY1999
I
FY2000
1
FY2001
Gener8l Design Standard
Stationary components
ACTIVITIES ON TECHNOLOGY
INFRASTRUCTURE PROJECTS
FY2002
Technology infrastructure projects in the Japan Fine
Ceramic Center are basically composed of development
or construction of design standards, testing methods and
data bases. These are supported
as
important subjects
for the Technology Infrastructure Projects
in
the
Japanese government.
Members
in
the committee of design standards are
as
follows. Most of these members have experience with
R&D
on advanced ceramic components
in
private
companies
or
enterprises.
Design Standard under
Particular
Service
Conditions
Contact components
Pileup structure components
Chairperson
Members
H.
Kobayashi (Tokyo Institute of Technology)
N. Okabe (Ehime University)
T. It0 (Tokai University)
Y.
Kimura (Kogakuin University)
N.
Nakazawa (Tsukuba College of Technology)
T. Teramae (Tokyo Electric Power Company)
A. Suzuki (Ishikawajima-Harima Heavy Industry)
T. Machida (Hitachi)
T. Tatsumi (Kawasaki Heavy Industry)
M. Hattori (NGK Insulators)
S.
Nakagawa (Kyocera Corporation)
S.
Suzuki (Isuzu Ceramics Research Institute)
M. Matsui (Japan Fine Ceramics Center)
H.
Kawamoto (Japan Fine Ceramics Center)
Secretariat
I
I
One of the activities is to construct the design
standard according to the program shown in Fig. 3. The
design standard has been almost constructed for
fundamental design methods and components under
steady state service conditions.
Fig. 3 Program for design standard development
CONTENTS
OF
DESIGN STANDARD
The design standard consists of following technology
elements.
1.
scope
2.
Design methods
Development and design
Design by rules
Design by tests
3.
Materials
4.
Analyses of service loads and stresses
5.
Design standards
Temperature restriction
Stress restriction for static fracture
Stress restriction for time-dependent fracture
Proof tests
Corrosion and oxidation
Acceleration effects depending on service conditions
and the evaluation processes
Residual stresses
Machining and surface roughness
Design processes
Evaluation items
Static strength
Time-dependent strength
Resistance to contact damage
Resistance to foreign objects
Degradation and corrosion
6.
Evaluation for environmental factors
7.
Evaluation for machining factors of materials
8. Component designs
9. Appendix
In the appendix, there are the following explanations
for these technology elements.
A2. Design methods
A3. Materials
Comparison between metals and ceramics
Kinds of structural ceramics
A4.
Analyses of service loads and stresses
AS. Design standards
A6. Evaluation for environmental factors
A7. Evaluation for machining factors of materials
A8. Component designs
Stationary components
Rotational components
Automobile components
Blades for ceramic gas-turbines
Ceramic components for a gas-turbine CGT302
A9. Design examples
DESIGN METHODS
Design methods are applied to structural designs for
advanced monolithic ceramics. Design processes for
ceramic components are generally expressed in the flow
chart shown in Fig.
4.
This flow
is
divided into design
by rules and design by tests. The design by rules has the
processes in Fig.
5.
The design by
tests
is classified into
component evaluation and assembly evaluation. After
these, evaluations of equipment and system follow.
274
.
I:
Equipment
&?
system
Durability evaluation
Design
by
rests
Total evaluation
Fig.
4
Design processes for ceramic components
Objective
Portion
b
Service
conditions
I
t
I
specifications
Material, Structure, Component
I
I
t
Load analysis
Stress
analysis
Stress
restriction
Environment evaluation
Design
for manufacturing
-
I
I
Trial
manufacturing
Fig.
5
Processes for design by rules
Design by tests is mainly composed of confirmation
tests and proof tests,
as
shown
in
Table
1.
Testing
samples should be selected for simple components or
models, and for those components fixed
in
assemblies.
As
the testing apparatuses, simplified systems and
actual systems are employed for strength and durability
tests. Loading methods are for stationary and non-
stationary conditions. Non-stationary loading is
important for the reliability of ceramic components due
to the low fracture toughness and the large scattering of
strength.
Proof tests should be done for assuring the minimum
strength or the life-tiqe of components, because of the
limitations for detecting the defects. Fig.
6
shows the
mechanical meaning of proof tests
[I,
21.
The proof
stress should be higher than the applied maximum stress
to assure the design objective. Proof testing damage
must be considered, when the unloading rate is slow.
There is a critical unloading rate not to fracture
in
the
unloading process
[3].
Ceramics have the property that
strength degrades according to applied stress time.
Consequently, the strength must be estimated after the
proof testing
to
establish whether that is lower than the
proof
stress.
The critical rate depends on fracture
toughness and crack propagation properties of the
material, and the proof stress level
[2,
31.
Kinds
of
tests
Sonfirmation
tests
Proof tests
v)
Table
1
Design by tests
Testing methods
Samples
Simple components
Models
Assemblies
Number of samples
Testing apparatuses
Simplified systems
Actual systems
Loading methods
Stationary
Non-stationary
Analyses
of
results
Objectives
Design
qualities
Production
qualities
Initial
static strength
Mroof
stress
0,
S-
t
curve
S
N
curve
2t,
vice
Time
t,
Number
of
cycles
N
a) Stress
-
strength model under proof test
Time
t
b)
Proof test pattern and strength after the test
Fig.
6
Mechanical meaning of proof tests
[
1,
21
275
DESIGN STANDARD
BY
STRESS
RESTRICTION
‘lass
’
Design methods for ceramic components are
restricted by the service conditions of temperature and
stress for fracture modes
[4].
Considering the
temperature, the restriction is for non-creep behavior or
creep behavior.
Stress restriction is defined
as
the following equation
for static and fatigue fracture. The design stress has to
be lower than the maximum equivalent stress
am.
Classification for fracture of components
The fracture leads to
a
failure of the whole
system.
The fracture leads to
a
local failure of the
-
UmaX
;
Maximum equivalent stress,
S
;
Minimum design
strength, Koand
KT
;
Basic and classifying safety factors,
/3
and
7
;
Design factors for stress gradient and the
region,
u
and
u
N
;
Distribution and nominal stresses,
SU
;
Specifled
minlmum statlc stength
SW;
Fatigue iimR
Sr
;
Specified minimum tlme-dependent
strength
tr
;
Deslgn life
The slopes
of
static and cyclic fatigue
curves are obtained by fatigue tests
using specimens
with
knoop cracks.
m
W
tl tf
Static fatigue strength
Time
t
R
;
Stress
ratio
R
<o
;sw
=
0.5
su
tl
tt
Cyclic fatigue strength
Time
t
Fig.
7
Minimum design strength
[4]
m
;
Weibull shape parameter,
Vmt
and
VE
;
Reference
and effective volumes,
V;
Volume,
5
;
Equivalent
stress,
u
T,
u
2
and
4
3
;
Principal stresses,
t
IIIU
;
Maximum shear stress,
u
R;
Vertical stress
;
Shear parameter,
As
shown in the diagram between strength and time
in
Fig.
7,
the minimum design strength
s
is defined in the
range from static strength to time-dependent strength
under the non-creep behavior of static fatigue and cyclic
fatigue
[4].
For the example of
Sf,
this strength is a
time-dependent one, corresponding to the time
tr.
sw
expresses the fatigue limit for static and cyclic fatigue.
The slopes of the static and cyclic fatigue curves are
obtained by fatigue tests using specimens with hoop
cracks. These values are based on many experimental
data for Si3N4, and
so
on.
Su
is the specified minimum
design tensile strength. The non-fracture probability is
more than
99%
under the condition that the stress
in
components is lower than the stress value of
Su.
Su
is
calculated from the mean flexural strength and the
Weibull probability technique.
Table
2
shows a classification for important
components
[4].
From class
1
to
class
4,
safety factors
should be defined according to the importance
of
fracture. For example, for class
1,
the safety factor
K1
has
the highest value, because the fracture leads to a
failure of the whole system.
Table
2
Classification for important components
[4]
system.
‘lass
21The operation must be suspended for the1
lremoval of the damaged subsystem.
lThe fracture leads to
a
local failure of the
system.
‘lass
lThe operation of system can be resumedl
lafter the replacement of the component.
lThe fracture leads to decreasing of the
I
function.
The reoair of the comoonent can be made
Class
4
1
lwithoui the system stop.
A DESIGN EXAMPLE
OF
AUTOMOBILE
COMPONENTS
A
strength design example is introduced
in
Fig.
8
for
automobile components and the design methods.
Various service loads are generated
for
every
component
in
automobile systems.
So,
strength design
methods are applied to confirm the reliability under the
service conditions, such
as
design by rules, simulations
and tests. On these occasions, safety factors, endurance
limits for fatigue and empirical rules are employed. The
design by tests includes the fracture behavior analysis
under more severe conditions than those for vehicles
that are usually used. In the tests, there are methods
276
using components only, those put
in
assemblies and
in
vehicles.
Fig. 9 shows the representative mass-produced
ceramic components, such
as
turbocharger rotor, swirl
chamber and exhaust gas control valve
[5,
6, 7,
8,
91.
The Si3N4 turbocharger rotor is one of the most mass-
produced ceramic components. However, the amount
is less than we would have expected it to be. This is due
to a lack of balance between cost and performance, Fig.
10
shows a fuel injection
pump
of
the distributor type
for diesel engines.
in
this
pump,
Si3N4 roller bushings
are installed for the purpose of increasing the durability
performance
[
101.
Fig.
11
shows a strength design example for ceramic
components for automobile engines. First, the design
in
common with materials is done for strength and life-
time predictions. Secondly, the design suitable for
ceramics should be conducted to over-loading and
damage-tolerance. The criteria are mostly that the
strength and the service life should exceed those of
metal components. Thirdly, the assurance by proof tests
must be performed by mechanical and thermal loadings.
Fig. 12 shows a design example by tests for wear
durability of Si3N4 roller-bushings [lo]. Taking the
maximum load where no seizure takes place for the
SUJZ bushing to be
1,
it can be seen that the ceramic
bushing does not seize even under a load which is three
times
as
large
as
that for the metal one. The amount of
wear for metal roller-pin, ceramic roller-bushing and
metal roller is expressed
as
a ratio ofthe prescribed
j
Components
1
Power
plants
Englne,TIM,
TIC,
W,
Shafts
chassis
-
3ervlce loads
i
Engine torque
'
Engine vibration
Thermal loads
etc.
(Loadsunder
1
servlce
conditionsl
_I
I
II
[Design Methods
Design
by
rules
FEM
simulations
Testing8
Components
Assemblles
vehicles
,
Static strength
i
Fatigue strength
I
Deformation
Fig.
8
Automobile components and the design methods
A
strength design example
;
Fig. 9
automobile engines,
as
turbocharger rotor, swirl
chamber and exhaust gas control valve [5,6,7,8,9]
Mass-produced Si3N4 components for
value for the conventional SUJZ bushing, which is
defined to be
1.
Based on the design criteria of seizure
load
and wear rate for metal bushings, ceramic bushings
have three times durability performances
of
metal ones.
/Roller
bushing
Fuel Injection-pump assembly
Roller
Bushing
Fig.
10
diesel engines and Si3N4 roller-bushings
[
101
A
fuel injection pump of distributor type for
Components
Roller-bushing
for
fuel
pump
Swirl
chamber
Criteria
Service
loads
and
under
fluctuations
Thermal stress
Thermal shock
Contact
load
and
Impact
load
Pressure
f
luctuatlons
Foreign
object
damage
Strength design items
a)
Design
in
common
with
materials
Strength and
life
predictions
b)
Design suitable
for
ceramics
Over loading, Damage-tolerance
c)
Assurance
by
proof
tests
Fracture tests (Mechanical, Thermal)
Non-destructive inspections
Fig.
11
of ceramic components for automobile engines
A
strength design example
;
Design methods
277
1
Test
Fuel
:
Kerosene
(90%)
I
-
Design criteria
0
Non-sebute
x
seizure
expected to become a
JIS
(Japanese Industrial
Standard).
5)
A Design Handbook is also going
to
be put
intc
circulation after several years, expanding and fulfilling
the examples
in
the design standard for structural
ceramic materials and components.
SU J2
s3N4
Matelials
of
roller-bushing
Seizure load for ceramic and metal roller
bushings
Ro
Basic
wear
rate
0
Roller-pin
0
Roller-bushlng (Inner)
0
Roller-bushlng
(outer)
A
Roller
0
1.0
K
K
\
Test
ti
me,
(h)
Wear rate
vs.
test time for ceramic and
metal roller bushings
Fig. 12
Si3N4 roller-bushings
[
101
A
design example by tests
;
Wear durability of
In
general, the design criteria for ceramic components
are established, compared with the durability of metal
components.
SUMMARIES
1)
A
Design Standard for advanced ceramic materials
and components for the reliability of ceramic systems
has been almost constructed for fundamental design
methods and components under service conditions
in
steady state.
2) Widespread uses of the Design Standard are expected
for the research and development of ceramics materials
and components.
3)
Consequently, many applications of practical
components would be accelerated
in
various energy and
environment systems.
4)
A
Design Standard for particular service conditions
has
been investigated to complete the standard. This
Design Standard includes contact components, pileup
structure components, and
so
on. Finally, the standard is
Well, this standard is written down
in
Japanese at the
present time.
REFERENCES
[I]
H.
Kawamoto, “Development of Structural Ceramic
Components for Automobile Applications”,
Ceramic
Transactions
Vol.
49,
Manufacture
of
Ceramic
Components,
The American Ceramic Society,
(l995),
173.
[2]
H.
Kawamoto, “Strength and Fracture of Ceramic
Components for Automobile Engines”,
Science
of
Machine,
V01.40,
No.1, Yokendo, LTD, Tokyo,
(l988),
13
1
(in Japanese).
[3]
M. Ichikawa, “Extremely Sensitive Dependence of
Strength after Proof Testing of Ceramics on Initial
Strength”,
Material Science Research International,
Vol.1,
No.1,
(1995), 59.
[4]
A.
Suzuki, “Design Standard for Preventing Static
Fracture of Fine Ceramic Components”,
IHI
Technical
Review,
28-2,
(1
988),
82.
[5]
H.
Kawamoto, T. Shimizu, and
H.
Miyazaki,
“Strength and Reliability of Silicon Nitride Ceramic
Turbocharger Rotor for High Performance Automotive
Engines”,
Proceedings
of
International Symposium on
Reliability and Maintainability
1990
-
Tokyo,
(1
990),
199.
[6] K. Takama,
S.
Sasaki, T. Shimizu, and N. Kamiya,
“Design and Evaluation of Silicon Nitride Turbocharger
[7]
H.
Kawamoto, T. Shimizu,
M.
Suzuki, and
H.
Miyazaki, “Strength Analysis of Silicon Nitride Swirl
Chamber for High-Power Turbocharged Diesel
Engines”,
Ceramic Materials
and
Components for
Engines,
Deutsche Keramische Gesellschafi,
(
1986),
1035.
[8] S. Kamiya, M. Murachi,
H.
Kawamoto,
S.
Kato,
S.
Kawakami, and
Y.
Suzuki, “Silicon Nitride Swirl Lower
-Chamber for High Power Turbocharged Diesel
Engines”, I985
SAE Transactions,
No.
850523,3.894.
[9]
T. Shimizu,
Y.
Hirata, and
K.
Tanaka,
“Development of Ceramic Exhaust
Gas
Control Valve
for the Two-way Twin Turbo Engines”,
Proceedings
of
Society OfAutomotive Engineers ofJapan,
No.
9433533,
(1
994).
[lo] K. Noda,
S.
Kamiya, T. Fujimura, and K.
Taniguchi, “Development of Ceramic Roller-bushings
for Diesel Distributor-type Fuel injection Pump”,
JSAE
Review,
Society of Automotive Engineers of Japan, 20
(1
999), 197.
Rotom”,
91-GT-258,
ASME,
(1991).
278
STATIC
AND
CYCLIC STRESS-LIFETIME CURVES
OF
CERAMICS
Hideo Awaji,
Jin
Gang, Atsushi Honjoh
Nagoya Institute
of
Technology,
466-8555,
Gokiso-cho, Showa-ku, Nagoya, Japan
ABSTRACT
A
statistical technique for estimating static and cyclic
fatigue limits of stress-lifetime curves was proposed.
Fatigue tests were performed on alumina smooth-
surface specimens to estimate stress-lifetime curves by
means of a three-point flexure test in air at room
temperature. These tests were carried out under several
stress levels with constant maximum stresses, and the
data of fatigue lifetimes under each constant stress level
including imperfect data were analyzed statistically to
estimate the Weibull’s original distributions of the
lifetime. The statistically defined fatigue limits were
estimated from the relationship between the residual
crack length in the specimens surviving after fatigue
tests and the stress level applied. The results revealed
that the ratio of the fatigue limit to the flexure strength
in air was
0.53
for the cyclic fatigue and 0.71 for the
static fatigue.
INTRODUCTION
Recent investigations on the cyclic fatigue behavior
of structural ceramics clarified the following
mechanisms of crack growth[
1-51.
(A) Time-dependent
crack extension due to an environmentally activated
process under tensile stress cycles,
(B)
crack extension
caused by degradation of stress shielding in the process
zone wake under cyclic loads, and
(C)
mechanical crack
growth caused by debris or asperity of the crack surface
under an unloading cycle. Mechanism
(A)
operates
when a material is sensitive to the testing environment,
while mechanism
(B)
operates when a material exhibits
a rising crack-resistance curve (R-curve). Additional
effects
on
the crack extension might be produced by
temperature increase at the crack tip caused by a
hysteresis loop of a cyclic stress-strain curve[6].
Due to the inherent brittleness of ceramics, the static
and cyclic fatigue lifetime data of structural ceramic
specimens with smooth surfaces exhibit quite a wide
scatter, which sometimes obscures the fatigue tendency
of the materials[7,8].
Thus
statistical treatment is
advantageous for the analysis of the fatigue data of
ceramics.
Determination of a fatigue limit
in
stress-lifetime
(S-
t) curves is essential for evaluating the macroscopic
fatigue phenomena occurring in the industrial structural
ceramics. Several researchers have estimated the fatigue
limits
of
ceramic specimens with smooth surfaces, but
the techniques used are not convenient because they
require a large number of specimens and are time-
consuming[6,9,
lo].
In this study, static and cyclic fatigue tests were
performed in air at room temperature, using a three-
point flexure specimen with smooth surface of alumina
ceramics which is expected to show both time-
dependent and cycle-dependent crack extension
behavior. The tests were conducted under several
constant stress levels[7], and the fatigue lifetimes for
each stress level including imperfect data were analyzed
statistically to estimate the original statistical
distribution function of the fatigue lifetimes. Residual
crack lengths in the specimens surviving after the
running time is also estimated from the retained strength
distribution measured by a three-point flexure test in an
inert environment. Then the statistically defined fatigue
limits of static and cyclic fatigue lifetimes are estimated
fiom the relationship between the residual crack lengths
in specimens surviving after the fatigue tests and the
stress levels applied.
EXPERIMENTAL PROCEDURES
The material used in this study was a commercial
polycrystalline alumina exhibiting
99.5%
purity (Mitsui
Mining
Co.,
Ltd.) and mean grain size of about
2
1.1
m.
The mean values and standard deviations of several
mechanical properties are shown in Table
1.
Specimens
3
X
4
X
40
mm
in size were machined out, chamfered
and finished by polishing with diamond paste. The
intrinsic or initial fracture toughness of the alumina with
rising R-curve behavior,
K,c,
measured by the
SEVNB
(single-edge V-notched-beam) method[ 11,121 in an
inert environment (in dry
N,
gas) was
3.19
MPam”*.
Static and cyclic fatigue tests were performed in
three-point flexure (outer span
30
mm) using a
piezoelectric bimorph-type instrument[
131.
The cyclic
stress was a sinusoidal wave with a frequency of 127
279
Table 1 Mechanical properties of alumina
Three-point
flexure strength
in air
/MPa
Lifetimesh
Three-point Intrinsic fracture
inert flexure toughness in inert
strength environment
/MPa /MParn"'
51 0-1-58.0
51
4k49.0
3.1 9f0.10
Hz,
a constant amplitude and a stress ratio
R
of 0.2.
The
fatigue tests were conducted at room temperature in air
for a running time of
lo5
s.
The retained strength of the
specimens surviving after the running time was also
estimated to obtain the residual crack length in the
surviving specimens using a three-point flexure test in a
dry nitrogen atmosphere.
RESULTS
S-t
CURVES
During fatigue tests on smooth-surface specimens of
ceramics, some specimens will fail before the maximum
stress amplitude is reached, and some specimens will
survive even after the running time. Data for the
specimens which failed before the maximum testing
stress was reached and the specimens surviving after the
running time are termed imperfect data in this study. All
the data including the imperfect
data
should be analyzed
together to obtain the original statistical distribution
function of the lifetimes, fiom the viewpoint of the
scattered distributions of flaw population in the
specimens.
Figure
1
shows the Weibull plot for the lifetimes of
alumina in the cyclic fatigue test carried out under the
constant maximum stress level of 412 MPa, assuming
that the lifetimes can be described by a two-parameter
Weibull distribution. The solid circle
(0)
with arrow
pointing to the left
(+)
indicates the imperfect data for
the specimen which failed before the maximum stress
level was reached, the solid circles with arrows pointing
to the right
(+)
denote the imperfect data for the
specimens surviving after the running time, and the
other circles with no arrow indicate the perfect data for
the lifetimes of the cyclic fatigue tests. Only the
perfect data are used to estimate the Weibull's
distribution function using a least-square approximation,
and the imperfect data are taken up only for counting
the order. Cumulative failure probability was calculated
using the following symmetric rank method,
i
-
0.5
n
F,
=-
where
i
is the order of the lifetimes,
n
the total number
of the specimens including the imperfect
data,
and
F',
the
cumulative failure probability. The shape parameter,
m,
and the scale parameter,
,
for the two-parameter
Weibull distribution of the cyclic fatigue lifetimes are
shown in the figure, and the value of in the figure
indicates the median of the distribution function. All of
loo
10'
lo2
lo3
104
99.9-j
0.1
CP
0
5
10
lntl
s
Fig.
1
Weibull plot for the lifetimes of alumina in cyclic
fatigue tests carried out under a
o
max
of
4
12 MPa
the Weibull plots in this study were constructed
in
the
same way.
Figure
2
shows the S-t curve of alumina obtained
in
the cyclic fatigue tests conducted under the four
constant stress levels of 450,412,
354
and 290 MPa, in
which the solid circles
(0)
indicate the cyclic fatigue
lifetimes of each specimen, and the empty circles
(0)
the median of the distribution of the cyclic fatigue
lifetimes for each stress level, estimated by means of the
technique described above. The empty circle with an
error-bar' denotes the median and the standard
deviations of the strength distribution of alumina
measured using a three-point flexure test in air, where
the median value was 514 MPa. It is noticeable that the
fatigue lifetimes dispersed over a wide range, hence
statistical treatments are a requisite for analyzing the
6001
I
I
I
I I
I
I
b
m
m
2
4-l
m
500
-
400
-
5
300-
E
.d
Lifetime
Is
Fig. 2 Stress-lifetime curve for alumina
in the cyclic fatigue test
280
fatigue properties. The cyclic fatigue parameter,
n,,
was
calculated to be 14, using Munz and Fett's method[
141.
S/MPa
FATIGUE LIMITS
A
new statistical technique for estimating fatigue
limits defined upon a statistical aspect is proposed here.
The retained strengths in the specimens surviving after
the running time in the cyclic fatigue tests were
measured using a three-point flexure test in an inert
environment. Figure
3
shows the Weibull plot for the
original distribution of the retained strength in the
specimens surviving after the running time in the cyclic
fatigue tests carried out under the maximum stress level
of 412 MPa. Of the ten specimens in the figure, one
specimen failed before the stress level intended was
reached, five specimens fractured during the cyclic
fatigue tests, and four specimens survived after the
running time,
as
shown in Fig.
2.
Then the Weibull's
distribution function was analyzed using the four
retained strengths. The shape and scale parameters,
shown in Fig.
3,
indicate the Weibull parameters of the
original distribution of the retained strength
in the
surviving specimens, and
So
denotes the median
of
the
distribution function.
S(MPa)
540
99.91-
s
\
I
I
0.11'
'1
I I
''4
6 6.1 6.2 6.3 6.4
1nS
Fig.
3
Weibull plot for the inert retained strengths in the
surviving specimens after cyclic fatigue tests
carried out under a
u
IMx
of 4
12
MPa
Figure 4 shows the original distributions of the
retained strength in the surviving specimens exposed to
cyclic fatigue loads with the maximum stress levels
of
412, 352 and
290
MPa, together with the inert strength
distribution shown by the broken line and the strength
distribution in air shown by the dot-dashed line. The
three solid lines indicate the original distributions of the
retained strength
in
the specimens surviving after the
I
-1
0.1
6 6.1 6.2 6.3 6.4
1nS
/MPa
Fig. 4 Weibull plot for the original retained strength
distributions in the surviving specimens after the
running time in the cyclic fatigue tests
running time in the cyclic fatigue tests. By comparing
the median of these distributions, it is clear that the
medians of each strength distribution hnction increase
with a decrease in the stress level loaded.
DISCUSSION
Under the assumption of small scale yielding,
subcritical crack growth rates of ceramics
in
region
I
of
crack extension rate vs. stress intensity factor relation
fblfill a power-law rule,
and the stress intensity factor
is
given as
K,
=
YOU'
'
(3
1
where
a
is
a crack length,
c
fatigue time,
K,(.
intrinsic
or
initial fracture toughness of the materials with
R-
curve[8],
K,
stress intensity factor,
A
and
n
fatigue
parameters,
IJ
applied stress and
Y
shape factor of the
specimen used. Assuming that the fatigue lifetime and
the strength of ceramics are caused by the same flaw,
the initial pre-existing crack length,
ai,
can be conducted
directly from the inert strength,
S,,
and the intrinsic
fracture toughness,
K,.
u,
=[$)
(4)
If the inert strength can be described by the two-
parameter Weibull distributions,
28
1